UNEQUAL

Also by Eugenia Cheng

How to Bake Pi

Beyond Infinity

The Art of Logic

x + y

Is Maths Real?

UNEQUAL

The Maths of When Things Do (and Don’t) Add Up

EUGENIA CHENG

First published in Great Britain in 2025 by

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eISBN 978 1 80522 308 5

CONTENTS

Introduction

1 When are things the same?

2 Equations

3 The stories hiding behind numbers

4 Equality between functions

5 Why the Earth could be a bagel

6 Interchangeable lids and mathematical glue

7 Logic, emotions, insight

8 Sameness in category theory

9 Sameness in branches of maths

10 From Sheffield to Buffalo

11 More dimensions, more forgiveness

Epilogue

Acknowledgements

Index

INTRODUCTION

What’s the first thing you think of when you think of maths? It might be arithmetic and times tables – things with clear answers like ‘2 + 2 = 4’ and ‘6 × 9 = 54’. It might be equations where you have to ‘find x’, by a series of manipulations that you may or may not remember or understand. It might be equations for a straight line in the form y = mx + c, or quadratic equations, or equations involving trigonometry. Or maybe some famous equations pop into your head, like Pythagoras’s a2 + b2 = c2, or Einstein’s E = mc2, or, for those who have gone a bit further into maths, Euler’s mysterious- looking equation eiπ = −1.

Maths is famous for its equations, but it’s about so much more than that. And even the equations themselves are much more than they seem. At first sight, equations seem very clear-cut: they are just situations where something is the same as something else.

Or are they?

When it comes to the kind of maths that is further along the usual education process – abstract mathematics, for example – things start to look different. Deep down, equations are hiding much more beyond just something being the same as something else. It’s true that 1 does equal 1, and all that ‘1 = 1’ means is that 1 is the same as 1, and likewise 2 = 2 and 3 = 3, and so on. But this is not enlightening. In fact, it’s rather boring. No new understanding emerges if we declare 1 = 1; no lightbulbs go off, no doors open, no new possibilities arise, no connections are made. So equations like 1 = 1, while true, are not the point of maths.

What is the point of maths?

The point of maths is to gain new understanding and illuminate different points of view. When things are very obviously the same, it doesn’t get us anywhere to know that or state it. It’s when things are not obviously the same that the situation gets interesting. And this is the secret behind equations: they are not just about things being the same, they are also about things being different. They’re about the senses in which things are both equal and not equal, at the same time. And things can be equal and unequal in many different ways. Equations are not supposed to be obvious, but they appear obvious to us if we only see the ways in which the two sides are equal and forget about the ways in which they’re not. If we think the equation is obvious, we might proceed more quickly to a straightforward answer, but we will miss some rich opportunities for understanding along the way.

When equations are about numbers, ‘equality’ does have quite a straightforward meaning, but maths is about many more complicated, interesting, fun and illuminating things than numbers. Maths is also about shapes, symmetries, logical ideas, and more. If we start considering those other objects and the different worlds they live in, ‘sameness’ and ‘difference’ open up a whole new world of understanding. As mathematical ideas get more complex, so does the concept of ‘equality’, and the boundary between things being equal and unequal becomes more of a grey area, or perhaps a rainbow of beautiful, vibrant, subtly nuanced colours. A big part of my research field of category theory is about finding more nuanced and insightful ways to say when things can count as the same even though they’re different, and this involves making choices about how we want to view the things around us. It’s a delicate balance, a dance that is too often hidden in our haste to find clear right answers. But the dance is beautiful, it’s fascinating, and it can help us think much more clearly about not just mathematics but our lives and the world.

In this book I am going to introduce increasingly complex and subtle mathematical ideas, and the increasingly complex and subtle approaches to sameness and equality that go with them. Maths is an abstract thinking ground, a place to practise developing our thought processes, a controlled environment for brain work. By contrast, life is far more complex and nuanced than any mathematical ideas, and so our approaches to sameness and equality in life need even more nuance. Understanding these subtleties in mathematics is a powerful starting point for us to understand them in life too, so that instead of oversimplifying into black-and-white arguments, we can look for and embrace the grey areas of equality where different potentially valid points of view can live.

Maths is about much more than an equation on a page, and even an equation on a page is more than it seems. It’s much more interesting, and much more powerful, when we see that, in fact, = does not just mean =. We see that mathematical thinking is not just about rigid answers, but about flexibility, and the discipline of seeing different points of view and holding them in our heads at the same time. We see that almost everything can be considered equal and unequal at the same time, whether it’s numbers, shapes, patterns, transformations, structures, words, meanings or people. And it’s up to us what we do about it.

Mathematics isn’t a series of rules, facts or answers. It’s an invitation into a way of thinking. Welcome in! I’m glad you’re here.

1 WHEN ARE THINGS THE SAME?

In maths it might seem that equations and equalities are very clear-cut and unambiguous. However, in this book I am going to argue that this is an oversimplified view of what maths is, and that, actually, there is plenty of grey area around the concept of equality, even in maths. Much of the power, the efficacy and the fun of maths comes from the very fact that sameness and difference are hazier notions than they might first appear. We are probably more used to having grey areas around what equality means in life, but if the idea seems strange in maths I think that’s a sign that we’re presenting maths as too separate from normal life. Unfortunately, approaches to fixing that problem often involve shoehorning maths problems into very contrived ‘word problems’ about life; it is less common to look at how we intuitively think about things in life and then relate that to maths. But abstract maths is often about investigating our intuition and giving it a rigorous framework, and equality or ‘sameness’ is a key example of that.

Quite often in normal life we say that things are ‘the same’ but we don’t mean it entirely rigidly, and we mean different things in different contexts. We are actually extraordinarily flexible about what ‘the same’ means, while also being remarkably precise in our contextual understanding and interpretation.

Take cats and dogs, for example. Most children learn to recognise cats and dogs when they’re very young. We can point out a cat to a small child, and they will quite reliably be able to point out other cats afterwards, not mistaking them for dogs. So they know which animals are ‘the same’ animal, and which ones are different. But what is it that makes some animals the same animal and some different? It’s quite hard to articulate that. You could say it’s about the sound they make, but children can tell cats and dogs apart even if they’re not meowing or barking. Perhaps it’s the way they move; but children can tell them apart even if the animals are asleep or are just in pictures. In fact, children can distinguish cats and dogs even in cartoon-like drawings, which are not at all realistic but somehow capture the essential cat-ness and dog-ness. That is, after all, one of the great skills of a cartoonist or any kind of visual artist: to capture the essence of something without necessarily depicting all its details.

I think that’s the key point: that we as humans are able to see beyond details, deep into the essence of things. This means we can say things are ‘the same’ when they are the same in essence, even if they differ in some details. Importantly, we’re also able to be flexible about what level of detail we count as important in any given context, so that the boundary between ‘surface details’ and ‘deep essence’ is itself movable. It’s not that we’re being ambiguous, it’s that we’re recognising different things as important in different situations. If you see posters about a lost cat, you’ll be able to be much more specific about whether another cat you see is ‘the same’ as the cat in the picture. But at some level of detail it becomes much more like a learnt skill. I’m not at all a cat expert, so I might think a cat looks like the one in the picture if it has the same general colouring, but someone who is much more attuned to the exact features of cats might be able to tell immediately that those are different cats.

The same general idea causes difficulty in learning new languages, when you’re trying to tune in to sounds that you’re not used to. You might think you’ve pronounced something ‘the same’ as your teacher, but perhaps there is some nuance that you’re not detecting, and so to a native speaker your pronunciation would sound completely different. I once attempted to learn Russian but gave up after a particularly frustrating lesson during which I was attempting to pronounce the difficult letter щ. The session with my teacher went something like this:

Me: sh

Teacher: no

Me: sh

Teacher: no

Me: sh

Teacher: yes!

Me: sh

Teacher: no

Me: sh

Teacher: yes!

Me: sh

Teacher: yes!

Me: sh

Teacher: no

Me: sh

Teacher: no

Me: sh

Teacher: yes!

At least, that’s how it sounded to my ears: as far as I could discern, I was saying it exactly the same each time, but evidently to her I was saying it differently in some critical way that I just didn’t know how to detect.

This is a particularly noticeable issue with tonal languages, for someone who is trying to learn them but whose first language is not tonal. In English, for example, the word ‘soup’ means the same thing whether we say it low, high or with a scooping up tone because we’re indicating a question. However, in Cantonese the word for soup sounds something like ‘tong’, but you have to say it with the right pitch because otherwise you might be saying sugar, or iron, or candy, or fillet. In one language ‘same’ just means the vowels and consonants, but in another it also means the pitch; conversely in Cantonese the consonants ‘l’ and ‘n’ are more or less interchangeable, so ‘lay’ and ‘nay’ both mean ‘you’ (if said with the right pitch) but in English those are different words.*

These subtleties around sameness and difference come to the fore if we think about asking computers or artificial intelligence (AI) to do certain things that we take very much for granted, such as recognising cats and dogs; AI was very bad at that for a long time, needing to be trained on thousands and thousands of images rather than the one or two that children need. Understanding handwriting is also very difficult for AI. It’s a bit of a mystery to me how we’re able to recognise so many different renditions of a letter as that letter. For example, you can probably recognise all the following as a letter ‘f’, but perhaps you’d be hard pushed to describe in words how you do that:

Context definitely helps: we can recognise a letter in a word in a sentence more easily than out of context. This makes shopping lists harder to decipher than a paragraph in a letter – though perhaps nobody writes either of those by hand any more. It does make it harder for children to learn to write, when they haven’t yet understood what the tolerances are for something to count as ‘the same’ as the image they’re trying to reproduce as the letter ‘a’. So a crudely drawn ‘a’ counts, but a perfect reproduction of a typed ‘a’ in mirror image does not. On the other hand, if you look at yourself in a mirror, you can still identify yourself as you even though the image is flipped. You can also probably recognise yourself upsidedown, or from the back, but if you’re trying to recognise someone you know less well that might not work.

Changing literal point of view

When we are looking at things that are less complicated and more predictable than humans, we are adept at extrapolating from what we see, to work out what’s going on where we can’t see. For example, we can recognise a cube from many different angles:

However, we can be tricked by optical illusions such as so-called ‘impossible objects’. These are objects we can draw in two dimensions and which our brain spontaneously tries to interpret in three dimensions, except that the object in three dimensions would be impossible. My two favourites are the impossible triangle and the impossible fork.

Impossible structures were made famous by the artist M. C. Escher, but they were pioneered much earlier in 1934 by Swedish graphic artist Oscar Reutersvärd when he was just eighteen years old and still at school.

If we just think about what an ordinary (possible) triangle could look like from different directions, we might think that the triangle could be rotated but all its lengths will stay the same. However, if we move further and further away from the triangle, it will look smaller and smaller. Our brains may be able to adjust for the distance and understand that the triangle hasn’t actually shrunk, but only if there is some context to help us.

When the triangle looks smaller it will still have the same angles, it’s just that the lengths of the sides will have shrunk.

Triangles with the same angles as each other but different overall sizes are called ‘similar’ in maths, whereas those that have exactly the same size and angles are called ‘congruent’.

If we also think about looking at a triangle obliquely, rather than straight on, we will find that the angles change as well. An equilateral triangle flat on a table will only look equilateral if we look at it from above; if we put our eye close to the table and look across it, the triangle will appear wide and short.

This principle is invoked when warning words are painted on roads. Drivers who see them aren’t actually eye-level with these markings, but if they’re driving at speed, then, effectively, the surface shrinks in the direction of travel according to how fast they’re moving, so the words need to be elongated in order to be legible. The word below looks very elongated, but if it were on the road and you were driving over it, it might just look like normal text.

We might find that all triangles can look like each other if we view them from different angles, so perhaps all triangles are in some sense the same. We’ll come back to this – and we’ll find that in some mathematical contexts all triangles really do count as the same.

Projective geometry is a field of maths that studies what shapes look like when projected in different directions. It grew out of the study of perspective by architect Filippo Brunelleschi in the fifteenth century. Artists had been coming to realise that in order to make their two-dimensional paintings of the three-dimensional world look more realistic, they needed to adjust the angles, so that things with right-angles in the world would not necessarily end up having right-angles in the painting. Brunelleschi and his contemporary Leon Battista Alberti were the first people to undertake a systematic study of this. The techniques can be counterintuitive, but once you get used to it you know that if you’re drawing a cube in perspective, then the back square face needs to be smaller on the page than the front square face, even though they are representing squares that are the same size: things can be the same in one sense but very different in another.

Changing mathematical point of view

Understanding how the same thing looks different from different points of view is an important part of maths. This is very different from the sadly widespread view of maths as something fixed and rigid. Take the idea of the ‘commutativity of addition’, for example, which says that adding numbers together gives the same answer whichever way round we do it; for example:

2 + 3 = 3 + 2

This can be thought of as just a ‘fact’ or a fundamental truth about numbers, but I prefer to understand it as seeing things from a different point of view – quite literally, if you are doing your arithmetic using counting blocks.

The left-hand side of that equation is 2 + 3 and it might look like this in counting blocks:

whereas 3 + 2 looks like this:

At first glance, the two look different, but the second version is just what you would see if you were to walk round to the other side of the first set-up and see it from there:

I like demonstrating this by really walking around, to make vivid the point that we are changing our point of view while also being sure that the configuration isn’t changing.

We can do something similar to think about multiplication too: 2 × 3 and 3 × 2 are different views of this grid:

If you understand the change of point of view that has happened, then you can get a better understanding of the sense in which the two things in question are the same; and this is true in life as well. We could do with more skill at seeing things from other people’s points of view. That’s not usually thought of as something to do with maths, but I think it really is, because of how carefully maths deals with different views of the same concept. However, sometimes we might get so used to the different views counting as ‘the same’ that we forget that the shift in viewpoint is happening, such as if we are so sure that 2 × 3 and 3 × 2 give the same answer that we forget how they’re different. This sameness and difference comes up if we think about the prime factorisations of numbers.

Prime factorisations

Prime numbers are the basic building blocks of numbers, using multiplication. This means they are defined so that ‘every number can be expressed as a product of prime numbers in exactly one way’. However, in order for ‘exactly one way’ to make sense, we have to declare that changing the order in which we write down the product doesn’t count as different. For example, 6 can be expressed as a product of prime numbers as 2 × 3 or 3 × 2. But those aren’t interestingly different, so we declare that they don’t count as different.

It’s up to us what we count as the same and different, and under some circumstances 2 × 3 and 3 × 2 really do count as different. I mean, they are in fact different expressions, for a start: they use symbols in a different order. They also mean different things: if we are counting packets of cookies, then 2 × 3 is two packs of cookies with three cookies each, whereas 3 × 2 is three packs of cookies with two cookies each. Those are different, even though the total number of cookies is the same. (One uses more packaging than the other.)

But when we’re talking about prime factorisation, we’re just trying to understand, abstractly, how numbers break down into smaller parts. Worldly issues such as packaging don’t come into it. Calling 2 × 3 and 3 × 2 different wouldn’t be very illuminating from that point of view. Moreover, it would mean that we can’t make the key statement about ‘unique prime factorisation’, and that statement is interesting and illuminating. So we invoke the more forgiving notion of sameness in order to say something insightful. This idea of choosing a more forgiving and therefore insightful sameness comes increasingly into play with mathematical concepts that are more complicated than numbers, such as patterns.

Sameness of patterns

Here are two 4-by-4 grids filled in with a particular pattern. Can you see the pattern, and do you consider that the two grids have the same pattern?

0 | 1 | 2 | 3 |  |  | a | b | c | d
1 | 2 | 3 | 0 |  |  | b | c | d | a
2 | 3 | 0 | 1 |  |  | c | d | a | b
3 | 0 | 1 | 2 |  |  | d | a | b | c

They are visibly not exactly the same because one has numbers and the other has letters. But the general pattern is the same, because each one has diagonal stripes running from top right to bottom left. One way of expressing this idea more formally is to make a direct translation from the left-hand pattern to the right-hand one by replacing the numbers with letters according to the following scheme:

0 → a

1 → b

2 → c

3 → d

This idea of being able to make some direct superficial replacements to ‘translate’ one thing into another is a key notion of mathematical sameness that we’ll come back to.

That’s all very well, but do you think this next pattern is also the same?

a | b | c | d
d | a | b | c
c | d | a | b
b | c | d | a

This one still has diagonal stripes, but they are now pointing the other way, from top left to bottom right.

There is no correct answer to whether or not this is the same pattern as the previous one because it really depends what you care about in this particular moment. What we count as ‘the same’ involves choices: it’s not something fixed and absolute. Mathematics is about making precise arguments, so instead of just going ‘I think this is the same’ only for someone else to say ‘I think this isn’t’, we try to come up with precise definitions of what we mean by ‘the same’ at any given time. This means it is at least incontrovertible whether two things count as the same according to that particular definition, even though there could well be other definitions according to which we conclude something else. In any given context we choose how much of the sameness to focus on, as well as how much of the difference. This is very explicitly the case when we are dealing with approximation.

Approximation

Approximation is one approach to things being the same in a more nuanced and forgiving way than just equality. The idea of a grey area around equality might seem strange in a mathematical context, but approximation is something we do all the time in our daily lives, possibly without really thinking about how much we’re doing it. It is forced on us by the messy and complicated nature of the world, by our limitations as observers, and by the inaccuracy of our measuring equipment or our brains. This makes it sound like a constraint, but it's also a superpower: the idea of approximation is made possible by our brains’ amazing capacity for accepting difference as part of sameness, even if we don’t realise we’re doing it.

As with many mathematical concepts we can start by thinking about it for numbers, and come up with ways to say that a number is ‘approximately the same as’ another. But we can then apply it to other things as well, to use this way of thinking far beyond numbers. Is a biscuit approximately the same as a cookie? It depends what country you’re in; in the US a biscuit is approximately the same as a British scone, but not exactly the same, and British people are liable to get upset about things masquerading as scones if they’re not exactly how a scone ‘should’ be. Still, some people are more forgiving about scones than others. (I’m not very forgiving at all.)

Here are some pictures of hand-drawn shapes. The first one is my good-faith attempt to draw a free-hand circle. As the shapes progress they become less and less like perfect circles. Which ones would you be prepared to accept as circles?

It probably depends on the context. If I’m cutting out a circle of baking parchment to line a cake tin, I want it to be fairly accurate because it bothers me if there are parts of the circle sticking up getting in the way of the batter, but it doesn’t bother me so much if there are small gaps at the edge of the base. However, if I’m giving a talk about symmetry, then it’s better if the circles I have on my slides are genuine circles, so that they all have the symmetry I’m trying to describe. But even here I am being slightly approximate – there can’t be a genuinely genuine circle on my computer screen as the pictures are made of pixels, and if you zoom in far enough you’ll see that the picture is full of jagged right angles. However, since we can’t see them when zoomed out, that doesn’t bother me, though it might bother other people – we all have different tolerances, and it doesn’t mean some of us are right and some are wrong.

Sometimes we can get fixated on particular features or details of difference for reasons that aren’t entirely rational, but that doesn’t always mean the reasons are invalid. I get upset with American ‘scones’ because they’re often not round, but triangular. I see that cutting scones into triangles is more efficient, because when you cut out circles you end up with bits left over in between the circles. If you don’t want to waste that dough, you then have to re-form and re-roll those leftover parts, and the second and third rollings are never as good as the first. However, I prefer my scones to be delicious rather than efficient, and I personally don’t like the tough, crunchy corners of a wedge-shaped scone. Furthermore, a scone is not just food to me but an emotional connection, and if it’s the wrong shape it doesn’t make that emotional connection in my head. Food is of course a much more expansive and multidimensional concept than numbers, so it’s easier to come up with a rigorous concept of approximation for numbers than for food.

There are various reasons we might want to approximate numbers. Perhaps we don’t know the exact number. Perhaps we can’t measure the exact number beyond a certain accuracy. Perhaps we don’t need to know the exact number beyond a certain accuracy, so it’s not worth the effort of being any more accurate than that. It all depends on context. If I’m making a cake, my measurements don’t need to be that exact, despite the idea that baking has to be very accurate. I try to measure my ingredients to within about 5 grams, but there have been times when I’ve done it entirely by feel because I was somewhere with no scales, and the cake was completely fine. By contrast, if you’re a medical professional measuring a dose of drugs for a patient, then you will need to be a lot more accurate than that, especially if the drug is potent.

What could it mean for two numbers to be approximately the same? There are two main possibilities that we humans use. Each has different benefits and different drawbacks, so when we use them we should be aware of the choices that we’re making and the consequence of those choices. This is a lesson we keep in mind as the maths becomes more complicated and we are making ever more choices about what to count as the same.

Approximation by tolerance interval

One approach to approximation is to say that two numbers will count as roughly the same as each other if they’re within a certain distance of each other. We can then pick what the tolerable distance is, depending on how accurate we want to be. We might say someone is about the same age as us if they’re no more than two or three years different; however, if we’re talking about babies we might need them to be at most two or three months apart, or even two or three days if they’re very young indeed. I am going to refer to this sort of method as approximation by ‘tolerance interval’ because we start by specifying what size of difference we are going to ‘tolerate’ within our concept of ‘approximately the same’. We might pick it to be 10 grams if we’re weighing ingredients for a cake, or 2 grams if we’re weighing ingredients for macarons, which are much more sensitive than cake.

This is fine if you’re just baking something or talking about two people. However, if you try to build longer arguments with this concept you start running into trouble as the distances pile up. You might think you’re roughly the same age as someone who’s three years older than you, and they think someone is roughly the same age who’s three years older than them, and so on, and in a few steps there’s a twenty-year gap.

The question of whether it makes sense to build up long strings of ‘sameness’ is something we will keep in mind throughout this book: it’s called the question of transitivity. If we can make a long string of things that are ‘roughly the same’ and at the end we still have something that’s ‘roughly the same’, the concept of sameness is said to satisfy transitivity, or we say it is transitive.

I think this issue is why it’s so easy to overeat by just eating a tiny bit more, repeatedly. Because if you eat a tiny bit more it feels like it’s not really more, but rather, the total you’ve eaten is more or less the same as before you ate that tiny bit. And then you eat a tiny bit more, which doesn’t really make a different total from the previous amount, and then you keep going until you’ve eaten twice as much as your body actually needed. At least, I do. Some people don’t seem to be prone to this at all.

This is also how people can steal large amounts of money without their victims noticing – they just skim a tiny bit off a large number of accounts, or overcharge by a tiny amount. If the amount is small enough, it won’t be noticeable to the victim, but the criminal can accrue a substantial amount of money because those small parts add up to something large.

While those are negative outcomes, I think we can also turn this concept to our advantage to change our lifestyle for the better, by changing it just a tiny bit repeatedly. Each tiny change can feel like it’s not really that different, but if you keep making tiny changes repeatedly you eventually get to a lifestyle that’s dramatically different from the one you started with. I did this to make my lifestyle healthier, gradually getting used to eating smaller portion sizes and a larger proportion of vegetables, so that now I fill almost my entire plate with vegetables. Twenty-five years ago that would have horrified me.

There are many cases where we implicitly have different tolerance intervals around bigger numbers. A thief may be able to skim more money off a bank account with a large amount in it without someone noticing, rather than from someone who is counting pennies (although some very rich people have become rich in part by taking great care of the pennies). Similarly, eating an extra bite or two of dessert will make much less difference to someone who weighs more or who has a very high metabolism or uses up a lot of energy as an athlete, than to someone who is tiny or has a very slow metabolism.

Perhaps the tolerance interval we allow in an approximation should be a proportion rather than a fixed amount. This method is sometimes used in the rules for election recounts. A recount may occur if the totals for the two leading candidates are close enough together that an inaccuracy in counting could result in a different winner. In the UK there does not seem to be a blanket rule for when there must be a recount, but several US states do have such a rule. In most states with an automatic recount rule, the threshold is 0.5 per cent, that is, there must be a recount if the difference between the two leading candidates is less than 0.5 per cent of votes cast. In some states there’s a fixed number that can trigger a recount as well, such as in Delaware where the threshold is 1,000 votes or 0.5 per cent of votes cast, whichever is smaller.

One of my favourite ever election stories was the Winchester by-election after the UK general election in 1997. At the general election, the Liberal Democrat Mark Oaten was declared the winner by just two votes (26,100 versus 26,098) following repeated recounts and arguments about spoilt ballots. However, the candidate who came second was the Conservative Gerry Malone, the sitting MP. He challenged the result in court and the difference of two votes was deemed not certain enough, given that fifty-four ballots had been declared invalid. So there was a re-run in the form of a by-election. In the re-run, Mark Oaten not only still won, but he won by a rather hilariously large 21,556 vote margin. Perhaps this demonstrates the rationale behind voting systems in which the leading two candidates proceed to a second-round run-off election.

Using proportions instead of a fixed amount to measure differences addresses the need for a sliding scale across values, but it doesn’t fix transitivity: suppose we fix the acceptable difference to be 1 per cent, then we can still find A, B and C where A and B are roughly the same, B and C are roughly the same, but A and C aren’t. We can fix some of the issues of tolerance intervals by using rounding instead.

Approximation by rounding

When we use rounding, instead of saying two numbers are roughly the same if they are within a certain distance, we round every number up or down to fixed reference points, and then everything that rounds to the same reference point counts as the same.

For example, we could round everybody’s age to the nearest multiple of 5. So everyone from 18 to 22 we could call ‘roughly 20’, and everyone from 23 to 27 is ‘roughly 25’, and so on:

18 to 22

roughly 20

23 to 27

roughly 25

28 to 32

roughly 30

33 to 37

roughly 35

38 to 42

roughly 40

⁝

We are actually already doing that when we refer to everyone’s age by a whole number instead of saying ‘They are 33 years 4 months 13 days 5 hours 3 minutes and 41 seconds old. Oh wait, it’s 46 seconds now.’

I always find it endearing when small children give their age more precisely than adults do, because when you’re four-and-a-half, that feels very different from merely being four. We adults can use more accuracy when it matters too. For decimal numbers we can round to any number of decimal places we want. If we are rounding to one decimal place, we are declaring that we really don’t need to know a number more accurately than to the nearest tenth. When I’m converting between a round cake recipe and a square one, I usually take π to be 3.14 because that’s plenty of accuracy for a cake. Actually I could probably take π to be just 3 and it would be fine.

We can round to any chosen number of decimal places, which means we look at the next decimal place after that one to tell us if it should round up or down. If we don’t want this to be centred around whole numbers or decimal places we can set the bands to be anywhere we want, such as when you’re asked for your age on a demographic survey and it gives you bands to choose from like this:

20–24

25–29

30–34

35–39

and so on.

It always feels a bit galling when I’ve just crossed over into a higher one: this is one of the anomalies that we incur by switching from tolerance intervals to rounding.

This anomaly causes great frustration to students when we decide on grade boundaries for exams. When I was last grading with grades, we were allowed to set our boundaries wherever we wanted, so it might be something like this:

A

=

90–100

A–

=

80–89

B+

=

75–79

B

=

70–74

B–

=

65–69

C+

=

60–64

C

=

55–59

The last time I was marking undergraduate exams in the UK the boundaries were:

First

70–100

2:1

60–69

2:2

50–59

Third

40–49

Then, of course, it becomes contentious around the boundary. If someone gets 69 they feel very hard done by and think it’s unfair that they were only 1 away from a first but didn’t get a first. How can we be so sure that they didn’t deserve a first but someone with one more mark did?

And herein lies the issue with this way of attempting to fix the problem we had with tolerance intervals. Yes, we do fix the problem of transitivity, as this form of ‘more or less the same’ is now transitive. But now we have ended up with a different peculiar anomaly, where things that are actually very close together end up counting as far apart where they’re right on a boundary. For example, 2.49 rounds to 2, but 2.5 rounds to 3, even though they are only one hundredth apart. Meanwhile, 1.5 and 2.49 round to the same thing even though they are almost 1 apart.

This leads to all sorts of issues at the boundaries, including students being cross about missing an A by 1 point, or some-one’s disability benefit being cut off when they work 1 more minute per week, or people being ineligible for social support because they earned 1 more pound per year than someone else who was eligible.

One way we fix it for more innocuous situations like age brackets is to be more vague in our terminology. We might use the terms ‘early thirties’, ‘mid thirties’ and ‘late thirties’ without being specific about exactly where the cut-offs are. The Association for Women in Mathematics has produced a lovely pack of mathematical playing cards, called EvenQuads, with each card featuring a picture of a woman mathematician. I’m honoured to be one of them, but am amused that my vague birth date is listed as ‘late twentieth century’ whereas a friend of mine who is only a year older is listed as ‘mid twentieth century’. Funnily enough, I think ‘late twentieth century’ actually makes me sound older than I am, but I can’t really put my finger on why that is, except perhaps my general paranoia about ageing, together with the rational fear of how society judges and dismisses ageing women.

We tried to fix this issue for exam grade boundaries at my job in the UK by holding epic faculty meetings where we discussed each individual case, and if someone was within 2 marks of the boundary we usually bumped them up over it. However, this just created a new anomaly: people who got 68 were bumped up to a first, while people who got 67 weren’t, but they were only 1 away from being bumped up so they felt hard done by. We felt like we were making the boundary more forgiving, but in effect we were just moving it. One might then argue that the people who were within 2 of the new effective boundary also deserved to be bumped up, but then where would it end?

When I was grading exams in the US and had more autonomy over how I did it, I tended to just make sure nobody ended up with a mark that was close to a boundary, so that they wouldn’t feel frustrated and I wouldn’t have to get into arguments with them about it. But then I came up with an entirely different system which was to decide who deserved what grade based on descriptors rather than accruing points. The descriptors were things like ‘Has a strong grasp of logic and can build rigorous formal proofs’, ‘Has a reasonable grasp of logic but writes proofs that have small logical gaps in’ and so on. I would pick descriptors to assign grades, and then assign a numerical mark after assigning the grade (so basically, backwards from the usual way) and made sure that everyone was in the low to mid range so that they didn’t feel there was a near miss with the next grade up.

I now have an even better method than that: I teach at the School of the Art Institute of Chicago, which has no grades at all so I don’t even have to deal with this issue any more.

Another issue that can happen with rounding is if you round too early in the process and then start substituting your answer into further calculations. Substitution is one of the fundamental principles about equality that we will discuss in the next chapter. It is different from the question of transitivity, and is more about whether your notion of sameness is ‘preserved’ by doing various operations on it.

Preservation

When we are talking about approximate sameness of numbers rather than exact sameness, we have to be more careful about whether it will be maintained by various operations.

For example, take two numbers that are within 1 of each other; if we add the same amount to both of them they will still be within 1 of each other. But what if we multiply? For equality this is not a problem, but it is a problem if we take two numbers that are only approximately the same, say 2 and 2.5. If we multiply them both by 10 we get 20 and 25, which are no longer within 1 of each other.

We say that this notion of ‘approximate sameness’ is not preserved by multiplication by 10. That is, we can start with two things that are approximately the same (according to the definition in question), but if we then multiply them both by 10 the results are no longer approximately the same (according to the same definition in question).

This issue still arises if we do rounding instead of tolerance intervals. For example, 2.4 and 1.8 both round to 2, but if we multiply each number by 10 we get 24 and 18, which do not round to the same whole number.

We also run into trouble if we round some numbers before adding them together. For example, if you do 2.4 + 2.4 you get 4.8, which rounds to 5. This means that if you round too soon in your calculation you’ll find yourself doing 2 + 2 and getting 4, whereas if you had rounded later you’ll get the answer 5.

This anomaly is known in some circles as ‘WeightWatchers maths’. The famous (but also vilified) weight-loss programme calculates a number of points for any given food, based on its nutritional content. The idea is to simplify a large amount of information into a simpler single number, so that you don’t get overwhelmed when trying to decide what to eat each day to fit within a certain quota. Regardless of what you think of this system, the rounding errors make for some amusing ‘maths’. Each calculation is rounded to the nearest whole number, so perhaps one slice of bread should have been 2.4 points but gets rounded down to 2. Then in the app if you put in two pieces of bread it logs 5 points, and everyone feels hard done by and says ‘2 + 2 = 5 in WeightWatchers maths’.

I remember being frustrated in physics lessons at school by the whole issue of decimal places, and trying to remember that if you’re aiming for an answer that’s correct to, say, three decimal places, you have to use earlier answers that are correct to perhaps four or even five decimal places. But I could never quite remember how accurate you needed to be and when, and I still can’t; I was much better at maths when it had letters instead of numbers and we didn’t have to deal with these rounding issues. Many people feel horrified when the ‘numbers turn into letters’ in maths, but I am relieved and even delighted.

Choices

There are no right and wrong answers when it comes to approximation. Perhaps I should say this more carefully – there are possible wrong answers, but only if you go against logic, or say you’re doing something you’re not (or measure a dangerous drug too inaccurately). It can turn out ‘wrong’ if you say you’re doing something to a certain accuracy when you’re not, for example. It would be wrong to claim that some notion of sameness is transitive when it isn’t, or to claim that it’s preserved by an operation when it isn’t, or that it works with substitution when it doesn’t. But within those logical constraints there aren’t right and wrong ways to approximate things, nor are there right and wrong answers to what can count as the same as something else or not.

Instead, there are just choices we can make, and consequences to those choices. It is up to us to decide what we want to count as the same and not the same, but we will then have to put up with whatever consequences come from those choices, so it’s better to be aware of what those consequences are, whether it’s non-transitivity or weird boundary behaviour. In different situations different strange behaviours are more tolerable than others. Understanding the differences in the much simpler context of numbers can help us be more ready for them when they come up in life situations that are much more complex.

For example, I wish we had a better way to deal with anomalies about the thresholds for benefits and exam grades. I think if we were all better at dealing with sliding scales we could do that. Even if we think students need to be assessed, it’s not clear why we have to divide them into four distinct categories rather than just putting everyone on a more continuous scale. I’m generally not in favour of the American SAT system, but one thing in its favour is that it does have a much larger range of possible scores (400–1,600) so it’s more like a sliding scale.

Another more complicated scenario involving dividing people into distinct categories comes into play when we think about political parties. Even though this is no longer about numbers, similar phenomena arise from our attempt to group people into separate parties according to them having ‘similar political beliefs’, particularly when there are very few parties to choose from, as in two-party systems.

One way to group people is similar to the tolerance intervals for numbers – maybe we say that people have similar political beliefs if they are within a certain distance of each other, politically. The first problem is that it’s hard to know how to measure that as there are so many different dimensions, and people’s beliefs may be very similar about, say, healthcare, but be very different about university tuition fees or foreign policy.

But the next problem is the non-transitivity: person A might have similar beliefs to person B, who has similar beliefs to person C, who has similar beliefs to person D, and so on, but if you keep going in one direction, then by the time you get to person Z they’re very different from person A. If you then extend that across thousands of people or millions of people, everything can – and does – become extremely incoherent. It can result in different factions of one political party potentially being even more opposed to each other than they are to the other party, especially when the furthest-right portion of a left-wing party is closer to the furthest-left portion of a right-wing party.

An unfortunately much more coherent approach to party politics is to unite around one core common cause or a common enemy and ignore other issues. This is more like rounding to the nearest whole number. If you form a party around the principle of being anti-abortion, or anti-Europe, or anti-immigration, then you automatically achieve transitivity. That is, you might define this:

Person A and Person B have similar political views if they are both anti-immigration.

This relation is then automatically transitive, and so it can unite a party much more easily. This is often something that right-wing parties do more successfully, whereas left-wing parties try to find similarities by something more like a ‘tolerance interval’, and end up fighting each other and unable to unite.

The trouble with the approach of ‘uniting around one core principle’ is then trying to get anything done that isn’t related to that core uniting principle. This might seem a long way from maths, but the principles at work are the same, and maths really is about understanding principles so that we can apply them very broadly.

Perhaps, like with exam grades, it would be more sensible just to stop trying to divide people into clearly delineated parties like that, or at least to have more parties so that the scale can become a bit more sliding and less discrete; this is more the case in some countries other than the ones I know best (the UK and the US). However, this would require a huge change to both the political systems and to the general mentality of all the citizens.

All this discussion of approximation is a key example of how we can look at the senses in which things are both the same and not the same. We might examine a situation and decide that some things are within tolerances of being similar, but then it turns out we’re looking at the wrong criteria. This happens when non-chess-players think a chess configuration is similar to another because it is physically similar, but it’s actually strategically completely unalike.

Thinking about sameness always involves choices about what we’re going to see as the same and what we’re going to see as different. It involves choices about which differences we’re going to see and which we’re going to disregard. This is something we do all the time in our daily lives, to greater and lesser extents in different contexts, depending on our personalities, our preferences and our tolerances; honing our ability to do this in mathematical settings helps us to notice the choices we’re making in life too.

Once we start seeing the nuances of ‘sameness’ and ‘difference’ we realise that there are grey areas everywhere, and there are important lessons we can learn from that about the world around us, especially if we acknowledge that sameness involves choice rather than treating equality and inequality as immutable facts. This can enrich our lives in many ways that aren’t obviously to do with maths. For example, we can listen to the same piece of music but have different experiences from hearing different performances, which is why I love going to live concerts so much more than listening to recordings.

If we all listen to the same piece of music, look at the same painting, eat the same food or swim in the same sea, there is a shared experience, but if we describe it in words we will all come up with different things. There might be some overlap, but some people might have found it thrilling whereas others found it boring. Some people might want to do it again as soon as possible, and some might want to avoid it for the rest of their lives. However, it was still a shared experience. On the other hand, sometimes I have found nothing quite so alienating as being in the same place as a group of people, doing the same thing as them but feeling completely differently about it from everyone else.

I don’t have to describe my feelings the same way as my friends in order to value sharing an experience with them. Perhaps the very fact that we can end up with completely different words to describe the same experience makes our lives rich and rewarding. I think this is what makes maths rich and rewarding too: that there is always an interplay between sameness and difference. That interplay is a whole world to be explored, and that’s what I’m going to do in this book.

We will think about the interplay between sameness and difference in maths, and we will go on to consider sameness and difference in life, and when differences do and don’t matter. When does it matter what someone looks like? If you need to cover for someone at work it only matters that you can fulfil all the roles they do; it doesn’t matter if you look like them or not (unless you’re working as a look-alike or a stunt double). When should people count as having the same qualifications as someone else? When should they count as having the same experience? What should gender pay equity mean? What does it mean to treat people fairly?

We can think about whether it makes a difference if we buy the same thing at a local family-run shop or from a giant chain where everything is cheaper; we can consider what kind of shops we choose to spend our money at, and weigh this up against our desire to spend less of our own money. And we can also choose the ways in which we treat people the same. It is entirely true that no two people are exactly the same – nobody is the same as anybody except themselves. But it’s not very helpful to take either of the extreme views ‘everyone is entirely different’ or ‘we’re all entirely the same’. In maths one extreme would be to declare that no equations hold except ones of the form 5 = 5; the other extreme would be to declare that everything is equal because everything is ‘a thing’. If mathematics did either of those things, it would be rather less interesting, productive or useful than it is. In fact, I hazard that it would be completely boring, unproductive and useless.

Our exploration of a more productive approach to sameness in maths is going to be a journey, starting from concepts that I hope are familiar, but progressing into deep mathematics including open research. I hope that as the journey advances you will rest assured that nobody is expecting you to understand everything. Even mathematicians don’t typically feel that we understand anything fully, but rather than this putting us off, it is what drives us forwards.

I don’t believe we should keep maths hidden because it is ‘too hard’. I hope that even when it gets complicated you will appreciate seeing glimpses of it, or enjoy gazing at it like abstract art. It’s in contemplating things we don’t understand that we expand our minds. Mine has definitely expanded in the process of me writing about these things and pondering all the things I don’t understand either. I will always keep trying to understand more, and I will always keep sharing what I’ve understood as widely as I can.

* However, apparently ‘nay’ is more refined-sounding than ‘lay’, so in that sense they’re not interchangeable.

2 EQUATIONS

The tension or flexibility between sameness and difference is more obvious when we’re thinking about the human experience, or shapes, or patterns. But it is also crucially present when we talk about the most straightforward form of sameness in maths: equations.

Equations might seem like the most clear-cut and inflexible part of maths; in fact, if you think maths is all about equations, and you think equations are rigid black-and-white facts, then you probably think that maths is all rigid and black-and-white. However, throughout this book we’re going to see that there is far more to equations than right and wrong answers.

First of all I want to think about the subtly different roles that the little equals sign can play for us. It can be hiding depths that we might not notice because we’ve become so used to the idea of equations supposedly saying something unambiguous. So, what are the different things that equations do?

Declaring what the answer is

First of all, equations are often used simply to say what the answer is to something. This happens when we’re just calculating the result of an operation, like

2 + 2 = ?

This is perhaps the most basic and obvious use of an equals sign, but it is really just one very small starting point.

Pivoting between different points of view

We also use an equals sign to pivot between different points of view on the same thing. For example, this equation

1 + 5 = 5 + 1

is telling us that if we take 1 thing and then 5 more things, we end up with the same quantity as if we took 5 things and then 1 more thing. That is called the commutativity of addition. Here’s another equation:

(8 + 5) + 5 = 8 + (5 + 5)

which is telling us that we can group the numbers in different ways to add them up, and the answer will still be the same. That’s called the associativity of addition.

In each case, we could also express the situation with two equations. Here’s what we might do for the associativity equation:

(8 + 5) + 5 = 18 8 + (5 + 5) = 18

However, somehow that would be telling us something slightly different, or putting a different emphasis on it. The first version

(8 + 5) + 5 = 8 + (5 + 5)

bypasses the ‘answer’ 18 – it tells us that these two things have the same answer without telling us what the answer is. This comes from a deeper understanding about why the answers are the same, not just that they are the same. I always appreciate things that give us a deeper understanding rather than just telling us a fact, and I find this one satisfyingly sneaky. It’s like bypassing an unnecessary stage. This appeals to my love of laziness, or rather, smart efficiency.

Two things that play different roles are the same

Sometimes we’re not exactly giving two different points of view on the same thing, but we’re saying two things that play different roles turn out to be the same. The remarkable equation known as Euler’s identity

eiπ = −1

falls into this category. We won’t go into detail here about this equation but, for now, suffice to say that it expresses a relationship between several famous quantities that come from different places: −1, the irrational numbers π and e, and the imaginary number i which is defined to be a square root of −1.* It’s not obvious at first sight why these should be related in this way.

Another example is the following expression, known as Leibniz’s formula. It is an infinite sum of fractions that somehow adds up like this:

π4=1−13+15−17+19− . . .

Here, we are equating two things that look different – that is, they are expressed differently. Outside of the mathematical world, we might see this in situations where we observe that two different, unrelated roles are being filled by the same person, for example if your mother is also your piano teacher. Or, more seriously, this can happen if a politician who is in charge of trade agreements also happens to be on the board of a company benefiting from those trade agreements, or someone leading a global conference on climate change turns out to be simultaneously in charge of substantial oil exports for his country. It involves more surprise than just giving two points of view on the same thing, and it involves more illumination than just giving an answer to something.

A starting point for investigation

Equations can also express a starting point of a relationship between things, a relationship that we then investigate further, perhaps to reveal a previously unknown quantity. I have a very clear memory of some number cards we used in primary school, where the first level said things like this:

The idea was for us to fill in the boxes to practise our arithmetic. This one uses an equals sign in the first sense I described: to introduce the answer to something. But then there were some more advanced cards that said things like this:

I remember staring at that and first thinking that the box was just in the wrong place, and that we were still supposed to add up the numbers and write the answer in the box. I was baffled about why the box had moved. But then the teacher explained to us that we were supposed to fill in the box to make the equation make sense. I had the now-familiar sense that my brain was contorting out of my skull; I say it’s ‘now-familiar’ because I still have this feeling every time I encounter a mind-bending mathematical concept that I can’t yet get my head around.

This might sound like a trivial step in kindergarten numeracy, but I am convinced it was a profound moment in the development of my abstract thinking. It was the first time I dipped my toe into the idea that an equals sign doesn’t mean that ‘the answer’ is about to declared. ‘The answer’ in this case was, in some sense, what we were writing in the box – but this came before the equals sign. Later on in education, this is presented as ‘solving an equation’. Colloquially, any string of mathematical symbols might be referred to as an ‘equation’, but really an equation is something with an equals sign in it (and not any other symbol: not < or ≠ or ≅ or anything else). And the point of that equals sign is to express a relationship that is going to help us understand something. If it’s some sort of process on the left and an answer on the right, then the equals sign is indeed announcing the answer to something. But if there’s something complicated on the left and something else complicated on the right, then it’s introducing a relationship rather than an answer.

In a way, ‘solving an equation’ is a strange piece of phrasing, because an equation is just a statement, not a problem or a puzzle. Often what we really mean by ‘solving an equation’ is resolving everything that is unknown about it. There might be an x in there representing a quantity that we’re trying to pin down, and then ‘solving the equation’ consists of pinning it down. The starting point is that we don’t know what something is, but we know how it relates to some other things, so we write down those relationships in an equation or equations, and then we see what we can discover. It’s like a murder mystery, where you get clues about the murderer, and then you gradually pin down who they must be.

Making a definition

So far we’ve thought about using an equals sign to introduce an answer, or to express new points of view, or to express relationships. Other times in maths an equals sign is just being used to make a definition, or to set some notation so that we have a clearly agreed starting point for whatever is going to come next. We might write something like

a2 = a × a

to define a piece of notation, and then

an = a × a × a × … × a [n times]

Or we might write

sin⁡ x⁢=oppositehypotenuse

which could be taken as a definition for sin x (although it’s not the one that is usually used in higher-level mathematics). Or we might write

i⁢=−1

which is a definition of the imaginary number i. (It doesn’t matter if you’ve never seen that before or don’t understand it.) Here we are not so much expressing a relationship or giving an answer, as getting our starting point clear. Sometimes, to emphasise this, mathematicians add a colon before the equals sign, like this:

a2 := a × a

This makes the situation visibly asymmetrical, to remind us that one side is nothing more than a piece of notation.

Getting the starting point clear is a very important part of maths, as we’re trying to build rigorous logical arguments, and you can’t do that unless you have unambiguous starting points. It can make maths seem pedantic, but the precision is there for a reason. Too often in daily life we get into arguments with each other just because we haven’t been clear about the definitions we’re using. Usually it then turns out that we’re all using different definitions. Making those definitions clear is essential in maths, but it’s important in daily life too if we want to avoid misunderstandings; being unclear about definitions is a way to manipulate people in bad faith.

One classic example of this is in arguments about feminism, when people variously self-identify as feminists or not. But more often than not it turns out everyone is using a different definition of ‘feminism’, and talking past one another, sometimes wilfully. Some people use an extremely narrow and somewhat offensive definition in order to denigrate the whole concept of feminism. Perhaps they say that feminists are people who think men are all evil and should be wiped off the planet. At the other extreme some people might use such a broad and vague definition that it almost becomes meaningless as a point of view. Perhaps they say that feminists are people who believe women have the right to decide how to live.

Similar things happen in discussions about the existence of god, or free will, or the perceived dangers of ‘cancel culture’, ‘identity politics’ or ‘wokeness’. By ‘god’ some people mean ‘something more powerful than humans’, which is much easier to believe in than ‘an old man in a cloud with a long white beard who controls everything we do’; however, other people adopt the very restrictive definition just to deny the existence of god without having to think very hard.

Some people just say ‘woke’ to mean ‘being aware of other people’s struggles and the way in which they have suffered through no fault of their own, and believing that we should help people who have suffered in that way’; those who are against it are often not at all clear what they mean by it, as evidenced by some painful videos of politicians railing against wokeness but then going completely tongue-tied when the interviewer asks them what ‘woke’ means. I believe we would have much more sensible discussions about all these things if we began by declaring what we mean when we use these words.

Highlighting a sense in which things are the same

When we use an equation in maths, we’re often really using it to say ‘there’s a sense in which these ostensibly different things are the same – what are we going to do with that insight?’ In maths it’s often that we can use the equivalence to do something more easily, such as if we factorise the expression on the left of the following equation, turning it into the one on the right:

x2 + 2x + 1 = (x + 1)2

This can help us as it shows that the thing on the left is the square of something else. This might help us understand some geometry. Or it might help us understand what the graph of this function looks like: it’s the same as the more familiar graph for x2, just shifted over by 1, because instead of plotting the square of each number, we’re plotting the square of the number that’s 1 to the right of each number.

Recognising a sense in which things are the same even though they’re not entirely the same is at the heart of what we’re doing when we draw an analogy between two situations. The difference between how we do this in maths and how we do it in daily life is largely that in maths we try to express exactly what the ‘essence’ is that we’re talking about, whereas in life we are prone to being more vague about things just being ‘analogous’, and hoping to learn something from that sameness. We are not saying that those situations are exactly the same; we’re just saying that some essence of them is the same and that we can gain insight from that. Unfortunately all too often someone will refute our attempt by saying ‘That’s not the same!’ Indeed, it is not the same, but there is a sameness and a difference, and between them there is something they can teach us.

For example, someone’s opinions might significantly differ from ours, making it hard for us to relate to them, but we still might be able to tap into some part that is analogous at some level, and thereby be able to empathise with them although we disagree. Or we might be able to recognise a pattern of beliefs that we have seen somewhere before, so that we can use what we’ve already understood about that other situation and transfer it to this new one. This is what happened with the equation

x2 + 2x + 1 = (x + 1)2

where we were able to relate the more arbitrary-looking string of symbols on the left to a simpler squared function on the right, and could then use what we already knew about squared functions in general.

I often see patterns such as people holding certain political opinions not because they really believe them, but because they are seeking validation and feel disregarded by all mainstream politicians. Or they hold a position because they have a general distrust of authority, which I can sympathise with: I have a distrust of authority in certain situations, but I draw my lines in different places. Or they’re more worried about false positives than false negatives, or care more about the idea of self-governance than the actual outcomes. I can find a similarity to my own beliefs for that last one, because given the choice between a benevolent, highly efficient and compassionate dictatorship and a vaguely inept democracy, I’d probably still plump for the latter (not because of the actual outcomes, but because of the potential for what might occur down the line if the dictatorship stopped being so benevolent). The point is that at an abstract level I am choosing the process I care about rather than the outcomes.

We’ll come back to the use of sameness as a pivot later on and see that it can be used for both the honourable purpose of trying to understand people better, and also for the more nefarious purpose of trying to manipulate them.

So far we have been thinking about what we are really doing when we declare an equality or write an equals sign in maths. Aside from that question, there are also various different techniques we use when we work with equality in equations. We might not think about these techniques very much; in fact, becoming ‘good at maths’ is to some extent a process of becoming so familiar with certain thought processes that we can do them subconsciously without exhausting our brains (not to be conflated with rote memorisation). However, if we go deeper into maths it’s important to be able to fish those thought processes back out of our subconscious and make ourselves aware that we’re doing them, because in some less familiar situations things might go wrong. It’s a bit like being aware of our unconscious biases so that we can counteract them if we care to.

These techniques we use when working with equality are things we will come back to when we think about more nuanced versions of sameness. We will want to know whether we can still use our familiar techniques even when our concept of sameness has become more subtle; we’ve already seen that this can fail if we’re talking about ‘approximate sameness’.

Different ways we work with equality in equations

We have seen that a basic use of an equals sign in maths is to declare an answer, but in most cases we are not stating an equality as a final conclusion, but as a starting point for developing some thinking. A string of equalities may ensue, and then eventually we might well arrive at a final equality as a conclusion.

Unfortunately, the way the manipulation of equations is taught in mainstream education often leaves the impression that it is just a manipulation of symbols according to some arcane rules, rather than the careful investigation of some statements with meanings, using some logic that we are pursuing to build our understanding.

I do remember being taught how to ‘solve’ equations myself, and being told that ‘everything flips over when it moves to the other side’: plus turns into minus, and vice versa, and times turns into divide, and vice versa. While this did work for me for a while, it’s not an explanation – it’s a rule, or a method, or an algorithm, or an instruction, but not an explanation. And it removes the meaning from the situation, so it doesn’t actively build an understanding of what’s really going on. This sort of algorithm is limiting because while it might result in someone being able to solve problems that look almost exactly the same, it is more likely that they will not be able to apply their understanding to situations that look a bit less the same.

I myself succumbed to this pitfall. I remember being faced with an equation to solve for a secondary school scholarship exam, but because the equation involved letters instead of numbers I couldn’t work out how to do it. My friend and I asked our teacher afterwards, and when she explained it to us I felt a bit silly, because it’s just the same whether you’re doing numbers or letters – you manipulate both sides of the equation in the same way, and as long as you’ve done the same thing to both sides, the equation still holds.

On the one hand, feeling a bit silly when you realise how simple something is is a sign that you’ve become more intelligent; on the other hand, this example shows that I didn’t really understand the full meaning of equations before then. I must have been just doing manipulations and using too much of my sense of numbers. Having such a vivid memory of this means I understand why so many people shudder when they even think about solving an equation, and why they have a complete block when it comes to using letters in maths – they probably never had the real meaning of the situation explained to them enough (or at all).

The meaning of the situation is that we have a statement saying two things are equal, and one principle of equality is that if two things are equal (in some particular sense), then if we do the same thing to them (in an appropriate sense) the results should be the same. This is a form of ‘stability’ in the face of manipulation.

Stability under manipulation

If we know two things are equal, say because they’re on opposite sides of an equals sign, then if we manipulate them both in the same way (in an appropriate sense) they should still be equal. That’s what I mean by ‘stability’ – the equality isn’t disrupted. We’ve already seen that if we’re only doing ‘approximate equality’, then this might go wrong, like when we multiplied two ‘approximately equal’ things by 10 and ended up with things that weren’t close together any more.

Things can still go wrong if we’re considering equality, which is why I said we have to manipulate both things in the same way ‘in an appropriate sense’. This is because unless things are truly equal, as in identical with no difference at all, then there is some sort of difference between them as well as a sameness. If we then do the same thing to both of them but in a way that invokes their difference rather than their sameness, then we could get a different outcome.

For example, we might consider the following equation to be true:

1 + 1 = 2

but we need to remember that the sameness we’re talking about is just the quantities that result from the operation at hand (addition). The way we’ve written the two sides is different. So if we do the same things to the quantities, then we will get the same result on both sides: if we add 3 to both sides, they will still be equal:

1 + 1 + 3 = 2 + 3

If we subtract 1 from both sides (of the original equation), they will still be equal

1 + 1 − 1 = 2 − 1

If we multiply both sides by 5, they will still be equal

(1 + 1) × 5 = 2 × 5

But if we just write a 0 on the right of both sides, this is something to do with orthography and not with quantity, and the two sides will no longer be equal

1 + 10 ≠ 20

We could do something that is not to do with quantity and still luck out and end up with an equality, say perhaps we paint both sides green, or write both sides larger but that was sort of luck: if we do a manipulation invoking different properties that are outside the realm of the ‘sameness’ of the equation, then we are no longer guaranteed to have the same on both sides.

1 + 1 = 2

The principle of doing the same thing to both sides is sometimes depicted as a balance scale, where the two sides balance as long as they weigh the same.

Then if you take the same weight off both sides, or put the same weight on, they will still balance: the equilibrium will not be disturbed, which is another way of thinking about the ‘stability’ we’re talking about.

If we think about it like a scale, then the equation

x + 1 = 3

can then be depicted as an unknown weight x together with 1 kg weight, balancing with 3 kg weight. (It doesn’t matter whether it’s 1 kg or 1 g or 1 of something else, as long as the three things on the right are each the same as the one thing on the left):

We then know that if we take off the 1 kg weight from the left, we need to remove 1 kg from the right as well, to keep the scales balanced, which means the black box will balance with the 2 kg remaining on the right. I personally haven’t set eyes on a balance scale since my childhood, and even then I think I only ever used one for educational purposes. Now that almost everything has gone digital, I wonder if invoking balance scales fails to tap into anyone’s intuition any more.*

We could express this ‘balance’ principle mathematically by saying:

Whenever x = y, we can apply the same function to both x and y and the answers must be equal.

What a ‘function’ is here is rather vague, but I’ll gloss over that for now and say that it’s some sort of operation which produces a new result, and isn’t random, so that whenever you feed it the same input you always get the same output. If we write the function as f, then we might write f (x) for the answer we get when we feed it the input x, and f (y) for the answer when we feed it the input y. Then we can restate the above to express this fundamental principle of equality:

Whenever x = y, we know f (x) = f (y).

This might seem obvious, or it might seem extremely obscure (which is the fate of much of mathematics – seeming obvious to some people and obscure to others). But I’m stressing this point here because it becomes much more critical when we’re thinking about more subtle situations, or higher-dimensional situations. I think it is actually simultaneously obvious and obscure, which is, to me, a wonderful thing about mathematics.

One thing that is certainly true is that there are relations for which this principle doesn’t hold. For example, we could think about the relation ‘less than’ instead of equals, which we often write as <. The question we’re then asking is:

If x < y, do we necessarily know that f (x) < f (y)?

The answer is no: it now depends on what type of function f is. For example, if f is the function ‘add 1’, then it is true:

If x < y, then it follows that x + 1 < y + 1.

It’s also true if f is the function ‘multiply by 2’:

If x < y, then it follows that 2x < 2y.

However, it’s not true for the function ‘take the negative’:

If x < y, then it is not true that −x < −y.

For example, 1 < 2 but −1 > −2. In fact, the direction of the < flips over when we take negatives.

We will investigate this further later on and see that the functions that work well with ‘less than’ in this way are a bit special, and it’s interesting (to mathematicians, at least) to see which ones do and don’t do that. When we start trying to replace ‘equals’ with something more nuanced, we will often be interested in knowing which functions work well with our new version and which ones don’t, so that we can be careful how we apply them.

Stringing together chains of equalities

Another obvious-and-obscure principle of equality is that if one thing is equal to another thing, and if that second thing is also equal to a third thing, then we can conclude that the first and third things are also equal. This is rather a lot of words to follow, and so mathematicians prefer to write it in symbols, perhaps like this:

If a = b and b = c, then a = c.

As before, here we’re using letters because we’re not referring to specific things, but any things, which we’re representing by letters to avoid the tedious process of saying ‘one thing’, ‘another thing’ and ‘a third thing’.

We met this principle briefly when talking about approximation: it’s called transitivity. It’s a property that we can look for in relationships other than equality, which is why it has a general name. Transitivity holds for ‘less than’ as well:

If a < b and b < c, then a < c.

but it doesn’t work for more specific relationships like ‘is 1 less than’:

If a is 1 less than b and b is 1 less than c, then a is 2 less than c, not 1 less than c.

We saw that it doesn’t work for ‘is approximately the same as’ if we define that by tolerance intervals, but it does if we are using rounding.

It also doesn’t work for most family relationships, for example ‘is mother to’:

If a is mother to b and b is mother to c, then a is grandmother to c, not mother.

There are messier real-world situations as well, such as friend-ship. A friend of my friend is not necessarily my friend, for example, although they might be; this is more ambiguous than the mother relation, where the mother of someone’s mother is definitely not their mother, and we can say for sure that they are the grandmother. Transitivity of family relationships can get complicated in blended families. Terminology is already complicated with various types of cousin; it gets more complicated with marriage bringing in-laws into it, even more complicated when there are step-parents or half-siblings, and mind-bending when a man keeps having more children with younger and younger women in his old age.

Transitivity of relations is not just something that is true or not true; it is a concept to investigate, and a lens through which to view relations in order to analyse them. Recently, I was trying to measure the height of a large chalkboard drawing hanging on my wall. It was too high for me to reach with a tape measure, so I sized it up using a broom instead, which was possible because the broom is rigid whereas the tape measure is floppy. I then measured the broom against the tape measure – I was implicitly using transitivity of equality. The property is often used in that type of scenario: we’re trying to show that two things are equal, but it turns out to be easier to show that they’re both equal to a third thing.

We saw a mathematical example of this earlier without really drawing attention to it, when we said

1 + 5 = 5 + 1

‘because’ they both equal 6. We didn’t directly show that the two sides are equal, but instead we showed that both sides are equal to a third thing.

We can state that more abstractly like this:

• We are trying to show that x = y.

• First we show that x = a.

• Then we show that y = a.

• Then we use transitivity to conclude that x = y.

We are also invoking the principle of symmetry there – but we’ll come to that in a moment.

When we start trying to think about more nuanced versions of sameness we’ll see that trying to maintain transitivity is quite tricky, just like when we were trying to think about approximation. Different ways of dealing with this anomaly typically end up producing other anomalies, so we end up having to choose which strange anomalies we want to avoid the most, and which ones we will have to put up with.

Symmetry

The next fundamental principle we might think about is symmetry: when we say a = b, we can simply reverse this equation and say b = a. That is:

Whenever a = b it is also true that b = a.

This says it doesn’t matter which way round we express an equation, which is something we probably take for granted when we write down an equals sign. We might have used both transitivity and symmetry subconsciously when thinking about this:

1 + 4 = 4 + 1 ‘because’ they both equal 5

What we’re really doing there is saying both of these equations are true:

1 + 4 = 5 4 + 1 = 5

We’re then subconsciously invoking symmetry to know that

5 = 4 + 1

and then we’re invoking transitivity to know that

1 + 4 = 5 and 5 = 4 + 1 implies 1 + 4 = 4 + 1

If this seems tedious, pedantic and over the top, I think that’s a wonderful sign of how brilliant our brains are at getting used to these properties and using them without thinking. However, the discipline of mathematics involves being able to do things without thinking when appropriate, but also being able to notice what you’re doing when desirable, so that you can move things between your subconscious and your conscious thinking as needed. The reason we might need to notice those things is in case we start working with concepts that don’t behave so smoothly. Then we need to turn our consciousness back on again and be careful. It’s a bit like going to a different country where they drive on the opposite side of the road. When you’re aware that the rules are different, you make yourself far more aware of which way you’re looking when you cross the road. You might well watch for traffic subconsciously when you’re at home, but it’s important to be able to bring the act into your consciousness again when you’re somewhere else.

The symmetry of ‘equals’ might be something we take even more for granted because it’s built into the design (if we can call it that) of the equals sign, which is symmetrical across the dotted vertical line shown here:

This is unlike the ‘less than’ sign which is not symmetric across that vertical line – if you flip it in the left–right direction it changes.

Indeed the ‘less than’ relation is not symmetric, so if we flip the symbol we flip the meaning as well: a < b means something very different from b < a, so we have to be careful which way round we write those. The thin end of the symbol points to the smaller side, and the bigger end points to the bigger side.

This idea of symmetry is not about visual symmetry, which might be the more obvious concept of symmetry. Rather, it’s about symmetry of ideas. That’s a more abstract form of symmetry, between the roles that things play rather than symmetry of their visible characteristics. We’re asking if two things’ roles are interchangeable in the context we’re thinking about. For example, in the expression

a + b

a and b are playing interchangeable roles. This is more or less just saying that addition is commutative. In the following expression

a2 + 2a + 2b + b2

a and b are also playing interchangeable roles, but a lot more is going on.

It’s similar to the idea of reciprocity in human relationships. If you have a friendship in which you seem to be making all the effort – always being the one to suggest meeting up, or getting in touch after a gap – then it might feel asymmetrical. The symmetry doesn’t have to be exact in order for things to feel balanced though – perhaps one of you makes more effort at getting in touch, but the other makes more effort in arranging things to do once you’re in touch. Good relationships don’t have to involve two people contributing the exact same thing, but perhaps contributing things that are equivalent or appropriate relative to your tastes. If one person loves cooking but hates washing up, and the other person hates cooking but is happy to do the washing up, it would be a bit pointless for the cooking and the washing up to be done in equal amounts by both people, in the name of ‘equality’; it would make much more sense for one person to do all the cooking and the other to do all the washing up. In one sense the two people are not contributing ‘the same’, but in another sense this is an equitable solution. We will explore this idea more later.

One way we can test for the interchangeability of roles – both in maths and in the real world – is simply to interchange them and see what happens. When someone is learning to be a teacher, they first observe classes, then they assist in classes, and then they might actually take the role of the teacher temporarily to see how they get on. In maths the interchangeability is likely to involve switching something in an equation or an expression. For example, in this expression from above

a2 + 2a + 2b + b2

we can try replacing every a with a b, and replacing every b with an a, and we’ll get this

b2 + 2b + 2a + a2

which we can see is the same expression as long as we are allowed to invoke commutativity of addition to change the order back to the first one. In fact, this ‘test’ involved doing some substitution, which is our next fundamental principle of equality.

Substitution

In the example where we tested for interchangeability, we were looking to see if two things played the same role in a particular context. But if two things are actually equal, they should always be interchangeable. You might wonder what I mean by ‘actually equal’ at this point, which is in a way the point of the whole book, but we can start by thinking of equality of numbers, and then go on to consider this principle of substitution as something we desire for any other form of mathematical equality:

If a = b we can substitute b into an equation wherever a was and the equation will still be true.

This should hold no matter what the equation is or what it means.

For example, if we know that this equation is true:

5a3 + sin a = a2

and we know that a = b, then we can switch every a in that equation for a b, and get this:

5b3 + sin b = b2

Or, if we know that a equals something more complicated like b + 3, we can put that in everywhere there was previously an a, and we’ll get this:

5(b + 3)3 + sin (b + 3) = (b + 3)2

We don’t even have to know what the equation means (and the one above is not very meaningful); we can still do the substitution, and the whole thing should still be true.

In fact, this principle of substitution is about the equality in the expression a = b (or whatever we’re substituting), not the equality in the equation that we’re substituting everything into, so we can do the substitution in other situations and not just equations. That is, we can write down anything at all about a, not necessarily an equation, and then substitute something that’s equal to a.

For example, if we know a = b then we can take:

3a < a2

and it will become:

3b < b2

Or we could take:

‘a is a purple elephant’

and it will become:

‘b is a purple elephant’

This is something we can always do when a and b are actually equal. But as with all the other basic principles, things get trickier when we start using more subtle notions of sameness, or have more noticeable differences along with their sameness. We saw this problem when we were thinking about approximation, where we could add two numbers that each round to 2 (to the nearest whole number), but the sum rounds to 5 (to the nearest whole number).

Things get even worse when we start thinking about logical statements that are in some sense equivalent but in some other very crucial senses not at all equivalent. That’s why we avoid this sort of ambiguity in maths; when we build an argument on that sort of ‘equivalence’ in normal life, then the argument is very shaky, and is liable to be a false equivalence argument, where you replace someone’s point with another one that is not really equivalent. In a way, ‘false equivalence’ is itself a shaky term for it, because everything is in some sense equivalent and in some sense not; what’s more the point is whether you’ve invoked too much of the sense in which the things are not equivalent. We will come back to this later on, when we also talk about false accusations of false equivalence. I think ‘false equivalence’ should really be called ‘inappropriately drawn equivalence’ but that’s a lot less punchy than ‘false equivalence’; it’s unfortunate that we need punchy catch-phrases in order to make ideas stick.

All of these basic principles about equality are things we can safely take for granted in maths until we start considering more nuanced notions of sameness. Sometimes we consider the extra nuance because we want to allow more leeway into our situation, but sometimes we do it because we simply can’t pin something down with very great accuracy in the first place. Eventually in maths we acknowledge this by building the leeway into the objects we’re considering, but a method that is more familiar, and in some ways more straightforward, is to throw away all the messy parts of life in our move from life to maths, and this is how we get to numbers. Numbers, and equality between numbers, work extremely precisely and unambiguously, but we have to discard a lot of information to get to that world, as we’ll soon see.

* I say ‘a’ square root as –i would also be a square root of –1.

* There might have been balance scales at the greengrocer’s when I was a child but I haven’t seen a greengrocer’s for years either (although they do still seem to be around in France).

3 THE STORIES HIDING BEHIND NUMBERS

If equations between numbers seem uninteresting to you, I sympathise. In fact, being boring is arguably the whole point of numbers – which you might find comforting, validating or affronting. Their entire raison d’être is to reduce things in the world to objects with no properties at all except quantity, so that we can measure things unambiguously and come up with completely black-and-white answers. The interesting part is in how we reduce the world to these numbers, and that’s also where the ambiguity can come in. For example, when we express pizza sizes by their diameter, that works for the purposes of specifying what pizza you want, but not for the purposes of adding up their areas: two 9-inch pizzas do not have the area of an 18-inch pizza. But that’s not the fault of the numbers; the problem is in how we chose to reduce the pizza to a single number in the first place, and the process we then attempted to apply to that number. Once we’re in the world of numbers, the answers to operations may be unambiguous, but there’s an unlimited quantity of ways of moving between the vast world around us and the more limited world of numbers. There are endless different ways of interpreting thoughts about numbers in relation to the thoughts we’re trying to have about the world around us. Whenever we reduce something in the world to just one number we lose a vast quantity of information, just like when we evaluate students with just one number out of 100, or we rate a book out of five stars, or attempt to sum up a person’s health with their BMI.

Numbers are an extreme simplification of the world. The world is complicated, and when we reduce it to numbers we are trying to focus on one aspect of it in isolation, to understand just that aspect as some sort of starting point. It’s like a controlled experiment, where you eliminate all other variables in order to focus on one thing. Or like drills in music or sport, where you are trying to practise one specific skill at a time, before incorporating it back into the whole.

We suppress all sorts of stories in order to produce a number. When we’re extracting data from the real world there might be real human lives behind those numbers, or more data that was too complex to include, or more dimensions we collapsed into one. When we read a novel we experience the plot, the characters, the pacing, the writing style and more; if we give it a rating out of five we are reducing possibly infinite subtly intertwined criteria to a single number.

When we don’t just reduce ideas to numbers, but then go on to put those numbers into equations, there is even more that might be hidden, as the different numbers might be hiding different things. The answers to operations on numbers are unambiguous, but there are plenty of subtleties that happen on the way to those answers. Take the innocuous- looking equation

1 + 2 = 3

At a basic level it means that if we take one thing and another two things we get three things. But the numbers could be representing different types of thing, for example:

If we start doing something at 1 o’clock and it takes 2 hours, it will take us to 3 o’clock.

or

If we start on the first note of the scale and go up two notes, we will get to the third note of the scale.

In both of those ‘real-life’ cases the ‘one’ and the ‘two’ are playing different roles, but we eliminate that subtlety when we just say 1 + 2 = 3. The numbers have made things simpler, but they have done it by obscuring or removing the original context.

And what about this equation:

1 + 4 = 4 + 1

You might think this equation is true because both sides equal 5. But the above equation is not the same as this equation:

5 = 5

which is even less interesting than the first one. So if both sides of the first equation equal 5, why aren’t those two equations the same? Why is the second equation not telling us anything, while the first one is?

The reason is that the first equation has a bigger grey area in between the difference and the sameness. The two sides are visibly not the same, for a start, and mathematically they’re telling us to do different processes: either start with 1 and add 4, or start with 4 and add 1. So the processes are different, but the end results are the same.

By contrast, with the equation 5 = 5 there is no grey area at all between the difference and the sameness, because there is no difference. The two sides are so exactly the same that we gain absolutely no insight from this equation. Insight comes from having both a sameness and a difference, not just a sameness. The sameness counts towards an equation being true, but the difference is what counts towards it being interesting.

More important than the numbers themselves is the question of where the line is between the sameness and the difference. And usually it’s a grey area, so it’s not a question of where the line between them is, but where it has the potential to be, and where we’re going to consider it to be for now. This becomes more and more of a key idea as the mathematics becomes more nuanced. And the bigger the story that is hiding behind the numbers, the bigger the grey area can be.

Numbers as a simplification

Numbers are sometimes criticised for being too much of a simplification, being too crude a tool or being boring, but in a way that’s the whole point. The power of numbers comes from their simplicity, and it’s not the fault of numbers if we humans use them in too crude a way: it’s our fault.

Whenever we use numbers to describe something in the world, we are necessarily forgetting some details and saying that some situations are the same after forgetting those details. It might be that two students have sat the same exam and have both earned the same mark, as a number out of 100. It is highly unlikely that those two students wrote exactly the same thing throughout their exam (unless they plagiarised very unsubtly). Getting the same mark doesn’t mean they did the same thing, it means that their papers were judged somehow equivalent on a very crude scale.

And the scale really is crude: it’s one-dimensional. Numbers are either equal or not equal, and if two numbers are not equal, we can always say which one is bigger. Or, to be more precise, numbers have a fixed order on a number line which we call ‘less than’ and ‘greater than’ rather than the slightly ambiguous terms ‘smaller’ and ‘bigger’. We can draw the line like this:

and then if we have two different numbers, one of them will always be unambiguously to the right of the other, and we call that one ‘greater than’ the one to the left. We don’t get to say that there’s a sense in which some part of the number is greater than another part, while another part was less, when all we have is individual numbers. In this sense, numbers are one-dimensional – they live on a one-dimensional line.*

Standardised tests in school or for IQ reduce a complex concept (‘intelligence’, whatever that is) into a single number. This isn’t inherently dangerous, but it can become dangerous when humans take that number for something that it isn’t, and then act on it in ways that can cause damage to some people’s lives. Standardised tests are really not a measure of intelligence – they’re a measure of someone’s ability to answer some very specific questions under specific circumstances on a particular day. This might partly reflect intelligence, but it will also reflect a whole host of other things too, including the kind of school they go to, and whether they had private tutors, which in turn reflect – at least to some extent – their parents’ wealth; there are also factors like the language they speak at home, or whether they have any quiet space to study. Besides the questions of home environment, there are also other issues such as contrived time pressure and the resulting stress, and this is aside from whether the test even tests anything worth evaluating. Why are we trying to assign every child a number in this way?

If we use a number to describe something in the real world, it’s almost certain that the story hiding behind the number originally had more dimensions. With a total on an exam it’s quite likely that there were different questions, and perhaps one student got more on questions 1 and 2 but less on question 3. We wipe out all that information, though, when we take the total and list one number as the final result. Again, it’s not the fault of the number that we have lost all that information – it’s our fault for reducing everything to just one number.

Actually, when I sat my undergraduate exams we were given two numbers rather than just one: one number evaluated how many mostly complete solutions we had managed, and the other just added up marks in a conventional way. The idea was that you could go round scraping marks together from a lot of different questions without every really writing a complete proof, and that that is a very different achievement from one where you do fewer questions but write very complete proofs. Judging people using two numbers rather than one is better, but it’s still not very subtle. It’s a bit like the amazing revelation of finding trousers with two measurements rather than one: perhaps the waist and the in-seam. It’s still a ludicrously small amount of information to use to try and identify clothes that fit, but at least it’s better than reducing us all to a one- dimensional scale of single numbers for clothes sizes.

When we use equality (and inequality) between numbers, we often omit details early on for ease, but then we don’t put back the nuance later. We end up acting as though things are the same if they end up the same after eliminating those details, or we rank them in a one-dimensional ranking according to those numbers and nothing else.

In this chapter we’re going to examine what we’re really doing when we reduce everything to a number and compare the results. This is going to involve a deeper dive into what it means when we say that numbers are the same. This might seem obvious once we accept what numbers are, but it turns out to be difficult to say precisely what numbers are. It’s already hard for the basic counting numbers 1, 2, 3, and so on. It gets even harder and more convoluted if we try to go further and think about negative numbers, or fractions, or irrational numbers. And with each further step we take on that journey, we’re hiding more in the concept of equality between numbers, because we’re hiding more in the idea of the numbers themselves.

What are numbers anyway?

People used numbers for a very long time – thousands of years – without having a rigorous definition of what they were. That might sound strange to you, or it might never have occurred to you to need a rigorous definition of numbers. After all, there are plenty of words we use in everyday life without having rigorous definitions of them. We learn to speak when we’re very young and I doubt that we have any definition at all when we start picking up words as children – we just listen to other people using them and we copy them.

But as we get older we start meeting words that describe more complicated or abstract concepts, and it is harder to understand what they mean just by copying, so we need definitions to be sure we are using them correctly. In mathematics, at a certain point mathematicians started wanting a more rigorous definition of numbers, to be sure that the work they were doing was on solid ground. This is a bit different from philosophers wanting to know what numbers are for philosophical reasons. Mathematicians just want to know that their logic is sound.

There are various differently convoluted ways of defining numbers rigorously. You can absolutely get on with using numbers in your daily life without knowing any of these, but I want to introduce one as it sheds some light on this whole question of equality of numbers, in a way that guides us towards interesting further thoughts.

The approach I want to focus on involves taking very seriously the idea of numbers as analogies between collections of things with the same quantity. This is what happens when we teach small children how to count: we don’t tell them what numbers are, we just keep counting things in front of their eyes and hope that they start to get the idea. We probably do it with as many different kinds of object as possible, to reinforce the idea that it doesn’t matter what the objects are, it just matters how many there are, even though we never really explain what ‘how many’ means.

So we show them one banana, two bananas.

We show them one apple, two apples.

We show them one child, two children. (And because children are practically full-time pattern spotters, we also have to convince them that the plural of child isn’t ‘childs’.)

And so on.

Eventually – with luck, persistence and imagination – they get the idea of what this ‘two’ thing is that you keep going on about, without us ever saying what it is (even if we know what it is, which most of us don’t).

One of the definitions of ‘two’ in maths is remarkably similar to this process, but it makes the idea behind it into something describable. It starts from the idea that you can measure the quantity of objects in a collection (known in maths as a ‘set’, even if the objects don’t match in any way), not by defining the concept of ‘quantity’ but by saying when two sets have the same quantity of objects in them.

Here’s the idea. If you show a child two bananas, two apples and two children, they might in their mind start matching them up together so that everything lines up. Or perhaps you actually hand out the bananas and the apples. In fact the two things don’t have to be the same: perhaps there’s one apple and one pear, and the two children get one fruit each.

In maths we do this sort of lining up using ‘perfect pairings’ between sets, where the objects of one are perfectly matched up with the objects of the other. The technical mathematical name for this is a bijection. So if we hand out one fruit to each child, we’ve made a bijection between the set of fruits and the set of children.

Bijections

The concept of a bijection is a key starting point for thinking about more nuanced versions of sameness, and for getting a better understanding of what equality really means in mathematics. A bijection is a way of perfectly pairing up the things in one set with the things in another. Here pairing ‘perfectly’ means that everything in the first set is paired up with exactly one thing in the second set, and also everything in the second set is paired up with exactly one thing in the first set. For the children and the fruit it means that nobody gets two pieces of fruit, nobody ends up with no fruit, nobody has to share fruit, and there’s also no fruit left over.

Here is an example of something that is a bijection. It’s not formal notation, but it’s a picture showing a scheme for pairing things up:

A ‒‒‒‒ 1

B ‒‒‒‒ 2

C ‒‒‒‒ 3

D ‒‒‒‒ 4

E ‒‒‒‒ 5

More specifically, it’s a bijection between the set

{ A, B, C, D, E }

and the set

{ 1, 2, 3, 4, 5 }

We use curly bracket notation to indicate a set: the objects or ‘elements’ of the set are the things inside the curly brackets.

Here is an example of something that is not a bijection:

A ‒‒‒‒ 1

B ‒‒‒‒ 2

C ‒‒‒‒ 3

4

5

This is not giving a bijection between the sets

{ A, B, C } and { 1, 2, 3, 4, 5 }

because the objects 4 and 5 on the right have not been paired with anything on the left.

The concept of a bijection is something that we will keep referring back to later on in the book, so if there’s one technical-sounding word to try to remember while reading this book, I would suggest this one. Terminology can be off-putting in maths, and it can seem that there are simply far too many words with obscure definitions to remember for no particular purpose. However, sometimes it’s just handy to have a word for a concept when you’re going to be referring to it and thinking about it a lot, so that you don’t have to keep using a long phrase to describe what you mean, over and over again.

It might seem obvious that you can make a bijection as long as the two sets have the same number of things in them, but the point here is that we’re trying to come up with a definition of number using things that don’t involve the concept of number. (Another side effect of this approach is that we can also do it with infinite sets, enabling us to define the ‘size’ of different infinities, but that’s a different story.)

However, the intuition about these pairs of sets having the same number of things in them is good intuition, and mathematicians use that to make a rigorous concept of number. We imagine looking at every possible set of things and finding all those that have a bijection with each other. This will classify all the sets into separate ‘piles’, one pile for each size of set.

And then we say ‘OK, this pile of sets that all have bijections with each other count as the same for these purposes’, and that’s a number.

That last part is a substantial leap in imagination, or a leap in abstraction. It would be like teaching a child what the number two is by gathering everything in their house of which they own two, and lining them up, and saying ‘that’s two’. However, that is more or less what we do when we repeatedly count two things in front of a child. At the end of it they don’t exactly know what two is but they have a concept they might be able to work with in practice – they can go round pointing out two of other things all over the place.

Now, this scheme is a bit tedious for various reasons, one of which is that there are infinitely many sets in bijection with each other in each pile. So how do we actually use this in practice? We pick one of those sets as a representative.

Representatives

Usually, in normal life, the most convenient set representing the number two is the set

{ one, two }

which is why when we count two things we point at them and intone ‘one, two’: we are making a bijection with our representing set, even if we don’t usually think of it like that.

When we use our representing set we are forgetting a huge quantity of stuff. We are forgetting everything about every set in that pile, everything other than the fact that they are all in bijection with the set { one, two }. It’s like electing representatives in a democracy, when we choose someone to represent a part of the country in government, rather than sending every individual citizen in person. It’s much more efficient, but also sometimes frustrating and disheartening. It can even be dangerous, if we not only ignore the individuals’ characteristics, but also ignore the fact that we’ve chosen to ignore them.

The key, in both cases, is to try to benefit from the efficiency but also mitigate for the effects of what we’ve ignored. It’s fine to forget details, but we must remember that we forgot them. We must remember that each number is representing infinitely many different sets that happen to be in bijection with each other. Now we can see what ‘equality between numbers’ is hiding.

What numbers are hiding

The concept of ‘equality’ between numbers is hiding bijections between an infinite number of different possible sets. That’s an awful lot of information to be hiding. Equality between numbers usually seems so clear-cut, but that’s because we’ve forgotten all of this interesting information in advance.

Treating sameness with more subtlety is about finding different balances between the amount of information we forget and the clear-cut nature of what we can say at the end. After all, if we can only say something is clear-cut by forgetting a lot of information, was it worth it? The answer is certainly ‘yes’ if you are trying to hide things from people. Perhaps you’re a politician who is trying to win votes by manipulation, or trying to get a policy in place by hiding the negative effects it will have on some people. Perhaps you’re an advertiser trying to persuade someone to buy something by hiding some information about it and focusing on only what is to your benefit. Perhaps you’re a university trying to entice students to study there by hiding the true cost, or the questionable benefit of some degree courses for future life. Perhaps you’re a drug company trying to hide some bad side effects of your drugs. Perhaps you’re a scientist trying to win a Nobel Prize by faking a big breakthrough. Unfortunately there’s an endless array of people trying to hide things, and numbers are a very effective place to do it.

However, that doesn’t mean we should give up on numbers or stop using them; that would be throwing out the baby with the bathwater. Any tool that helps us can also be used for ill, and unfortunately human nature means there will always be people trying to do that, so throwing away the tools is futile. Instead we can try to protect ourselves, and society, by developing a good understanding of how the tools can be used, so that we can detect when they’re being used for ill but still keep using them for good, and even refine them so that they work better for good, and less well for bad. This is what we’ll be doing throughout this book with the notion of equality and sameness.

All of our discussion so far has been about whole numbers, that is, the counting numbers, the ones that represent quantities of things we can count one by one. If we include negative numbers, things become more complicated. This is the next step on a journey towards more and more expressive forms of number, that is, numbers that are capable of expressing more subtle concepts.

Building more expressive forms of number

We defined the counting numbers via bijections between sets of the same size, but we can’t do this with more expressive forms of number because other numbers don’t represent the quantity of objects in a set. What’s a set with a negative number of objects? What’s a set with one-third of an object in it? It gets worse for irrational numbers.

The counting numbers are technically named natural numbers in maths, because they are the numbers that occur to us most naturally. Negative numbers, fractions and irrational numbers involve more abstract ideas. As a result, they’re more difficult for us to get our heads around, both intuitively and technically. They’re defined by building on the idea of natural numbers, rather than by directly extracting ideas from the world. As a result, the concept of equality has to go via the same route.

In fact, negative numbers are defined to be ways of getting back to zero from a positive number, so before we try to think about negative numbers we need to take a look at zero. It’s a slight digression, but it would be unsatisfactory to go on without it.

Zero

In our scheme for defining numbers via bijections between sets, zero is the pile of sets in bijection with the empty set: the set with no objects in it at all. But there is a way to characterise zero according to its relationship with other numbers, and this is the method we use when we define negative numbers. Negative numbers are defined via the idea of getting back to zero from a positive number, and zero is the chosen ‘point of return’ because zero itself has a special property: when you add it to something nothing happens.*

We can write that defining property of zero like this:

For any number a, a + 0 = a.

Incidentally, if we know that we can add numbers any way round and get the same answer (commutativity) we also know this:

For any number a, 0 + a = a.

This property of 0, that when you add it to anything else nothing changes, is called the property of being an additive identity; in maths ‘identity’ usually means ‘leaving something the same in some way’.* This property can be used as a defining or characterising property because we can also prove that in any given context there can only be one additive identity; if there were more than one, then it wouldn’t be a good way to define something because we wouldn’t know which one we were defining. The fact that there’s only one number with the property of being an additive identity is a form of uniqueness – it’s saying that zero is unique in this way. The way mathematicians prove that something is unique is to show that if we had another one, then it would have to be equal to the first one. This sounds a bit odd in normal language because it’s basically saying ‘if we had two of these things they would have to be the same so there’s only one’, but it sounds a bit less contradictory if we use letters:

If x and y are both additive identities, then x = y.

It’s a bit like saying ‘if person A and person B are both the current president of the United States then person A and person B must be the same person’. This sort of thing gets overlooked when, say, someone on an online dating site gets accused of impersonating someone famous, which apparently happened to author and human rights advocate Tara Moss.* It is assumed that the online person must be a fraud, but in fact the online person could be the same person as the famous person.

Incidentally, this way of defining uniqueness shows that uniqueness and equality are intimately related. That might not be entirely obvious when we’re first thinking about uniqueness, but it is something mathematicians think about a lot. It means that as we develop more and more nuanced versions of sameness, we also arrive at more and more nuanced versions of uniqueness.

Here’s how we prove that identities are unique. The reason I’m going to include this argument here is not because it is particularly illuminating or important or that I think you should understand it, but rather, it is to show how careful we try to be when we build up our basic ideas, and how difficult it can turn out to be to pin down something that sounds obvious. So even if you don’t understand the following argument (or don’t feel like you want to understand it) you can at least marvel, perhaps, at how convoluted it looks, when all we’re doing is proving that there is only one number zero. But also, if we understand this proof carefully, then we have automatically understood other uniqueness proofs at the same time, because they all take ‘the same’ form.

First we suppose that we have two numbers satisfying the property of being an additive identity. We’ll call them x and y.

So we know these two things:

Property 1: For all numbers a, a + x = x + a = a

and also

Property 2: For all numbers a, a + y = y + a = a

But both of these things are true for all numbers a, which doesn’t mean ‘for all numbers called a’, but rather, it means ‘for any number, and we will temporarily call it a so that we can refer to it later’. So we can substitute anything for a when we want to use this fact. In particular we know that the first property must hold if we use y in the place of a, that is

y + x = x + y = y (1)

And also we know that the second property must hold if we use x in the place of a, that is

x + y = y + x = x (2)

But look: the first line of equalities (1) above contains the equality

x + y = y

and the second line (2) contains

x + y = x

so by the transitivity of equality we can conclude that x = y. This concludes the proof.

Note that when I said this is ‘not illuminating’ I didn’t mean it’s not illuminating to anyone: evidently there’s a reason mathematicians do this sort of thing. I actually find this proof quite interesting in its ideas, though not so much in its details. The general idea is that we imagine having two objects with some sort of superlative property, like ‘largest number in the set’, and we pit them against each other. If the superlative property is superlative enough, then there can be no draw: the only way that both objects can be superlative is if they’re the same. For example, if you and I both pick the largest number out of this list:

34, 4, 41, 8

then our numbers must be equal.

This is an idea that pervades my field of research, category theory, as we’ll see later. But we’ll also find that when we introduce more nuanced versions of sameness we can have something like a draw, and that in that case two objects can be ‘superlative’ if they are sufficiently equivalent, but not necessarily actually the same object.

Now, let’s return to the original point of thinking about zero carefully, which was to think about negative numbers.

Negative numbers

We can now define negative numbers as a way to get back to zero; this is called the ‘additive inverse’ property. It is stated like this:

Given any number a, an additive inverse for it is a number b such that a + b = b + a = 0

Now, just like we did for the additive identity, we can show that additive inverses are unique, that is, if x and y are both additive inverses for the same number a, then they must be equal. Once again I’ll include the proof so that you can marvel at it if you like.

We start by supposing x and y are both additive inverses for a, which means:

For x: a + x = x + a = 0 (1) For y: a + y = y + a = 0 (2)

Now I’m going to manipulate these equations a bit, being careful to do the same to both sides so that the equation is still true. I’ll start with this equality, which was part of the first line:

x + a = 0

and I’m going to add y to both sides of the equation, on the right:

x + a + y = 0 + y

Now, the right-hand side definitely comes to y because of the defining property of 0, so we have:

x + a + y = y (3)

But let’s look at the left-hand side: we know that a + y = 0 (because of the defining property of y in the second line above) so we can substitute that in for a + y, to get this:

x + (a + y) = x + 0

But now we can see that this comes to x because of the defining property of 0 again. So we’ve got

x + a + y = x (4)

So combining equations (3) and (4) again using transitivity of equality, we conclude that x = y.

I tried to be careful about observing when I used some basic principles of equality there, including substitution and transitivity, but I was admittedly a bit cavalier about symmetry – I didn’t really pay attention to which way round I was writing my equations, which would have been a further level of care that I could have taken if I were being even more precise, or if I were worried about moving to an asymmetrical situation.

Now that we know additive inverses are unique, we can define negative numbers by this property. It might seem odd to define something by a property, but it’s like saying ‘the president of the US is the head of state of the US’ rather than saying ‘the president of the US is Joe Biden’ (which is true at the time of writing, but won’t be true at the time you’re reading this). The former definition means that whenever you encounter someone who is the head of state of the US, you can conclude that they are the president of the US. It’s a bit more stable because it works regardless of which human being is currently holding the position; defining negative numbers as additive inverses is also a bit more stable, not because they change across time, but because they could change across contexts.

This is how we define negative numbers in abstract maths:

For any number a we define −a to be the additive inverse of a.

You might well have lost track of what we were trying to do now, which is often a hazard of thinking hard about maths. In the case of this chapter, we were, deep down, trying to head towards what it means for numbers to be equal. For positive whole numbers we used the concept of bijection, and said that two sets have the same number of things in them if they can be put in bijection with each other. For negative numbers to be equal we can’t use the concept of bijection because negative numbers don’t represent quantities of things. Instead, they are all additive inverses of positive numbers. And then we have to define equality of negative numbers like this:

Two negative numbers are equal if they are both additive inverses of the same positive number.

So the concept of equality between negative numbers still depends on the concept of equality between positive numbers, which itself depends on that principle of bijection between sets, but it is another level more abstract than that.

When we take another step in laying our mathematical foundations, we get to fractions – and things get even more convoluted.

Fractions

Fractions are often introduced via sharing, or slices of pizza, or chopping things up. But abstract mathematicians don’t define fractions via their meaning, just like we didn’t define negative numbers by trying to quantify debt. We define fractions analogously to how we define negative numbers: we defined negative numbers via the concept of additive inverse, and we define fractions via the concept of multiplicative inverse. We use ‘the same’ definition, but we replace the concept of addition by the concept of multiplication everywhere. So, where an additive inverse for a is ‘a number we can add to a to get us back to the additive identity’ (i.e. zero), a multiplicative inverse for a is ‘a number we can multiply a by to get us back to the multiplicative identity’.

The first step is to understand what the multiplicative identity is. If we follow through the analogous definition as for additive identities, we see that a multiplicative identity is a number with the property that ‘when we multiply other numbers by it, nothing happens’. The unique number with this property is 1, so 1 is the multiplicative identity for numbers. We can state this property formally as we did for the additive identity, like this:

For any number a, a × 1 = 1 × a = a.

Then a multiplicative inverse for a is ‘a number we can multiply a by to get us back to 1’, or more formally:

For any number a, a multiplicative inverse for it is a number b such that a × b = b × a = 1.

As with additive inverses, we can then show that multiplicative inverses are unique, so we can define the fraction 1a to be the multiplicative inverse of a, provided a isn’t zero.

We then have to go through further steps to define fractions that don’t have a 1 at the top, like ⅔ or ¾. The technical name mathematicians use for this type of number is ‘rational number’, because they relate to ratios.* I don’t want to go into all those technicalities here; I just want to think about what it then means for two fractions to be ‘equal’.

In the usual orthodoxy about fractions, we consider them to be the same even if they’re expressed differently, perhaps by not being in lowest terms, such as:

12=24=36=…

But this is a choice and a decision, and it’s not always true. There is a sense in which one half of a pie is not the same as two quarters because if someone gives you one quarter and then another quarter there will be an extra cut in it. Perhaps the crust will have crumbled at that point. I sometimes want to buy a whole circular cross-section of Stilton because it’s an aesthetically pleasing thing to serve at a party, but it’s hard in the US because even when I can find Stilton (which is already a challenge) it’s always pre-cut into small wedges. You can attempt to ‘reconstitute’ these into a complete round, but it won’t be the same.

There are other scenarios in which reducing something to lowest terms doesn’t mean it’s the same. For example, if a study says that four out of five people feel better after exercise, one hopes that doesn’t actually mean that they asked five people and four people said ‘yes’, because that would be a very tiny sample size – it would be much better if they had found this to be true of 4,000 people out of 5,000. In that case four out of five people means something quite different from 4,000 out of 5,000 people.

This becomes more noticeable when we’re adding fractions up. The following equation might seem to be true:

12+12=1

But what if we’re talking about marks on a test? If you only get 50 out of 100 on your first test, that’s half; and if you get 50 out of 100 on your second test, that’s also half. But then altogether you have 100 out of 200, which is still half. So in this case:

50100+50100=100200

When students try to add fractions by adding the top numbers and adding the bottom numbers, it is, conventionally speaking, ‘wrong’. But the above example shows that that method is not straightforwardly ‘wrong’ because it is valid in a different scenario. If you’re tallying up what proportion of people in a school need special accommodations for testing, you might ask each teacher to tally up the numbers for their class. You might get these results:

Class 1
6

27

Class 2
8

25

Class 3
4

28

Class 4
7

31

Then the correct way to find the overall proportion in this case is indeed to add up the top numbers and add up the bottom numbers, not to find a common denominator for the fractions.

All this is to say that equality between fractions is hiding more than we might initially think, and in particular we make choices about what it means, depending on what context we’re in. Things get even more complicated if we go as far as irrational numbers.

Irrational numbers

Fractions, or rather, rational numbers are not the end of the road of complication here. Rational numbers are all very well, but they are still not satisfactory. They are ‘better’ than whole numbers in the sense that there’s more we can do with them: if we only have access to whole numbers we can’t do division very well, because we might end up with parts left over. For example, we can do 10 divided by 5, but what about 10 divided by 3 or 4? There isn’t a very satisfactory answer if we can’t divide up the whole numbers to make fractional parts.

There are still things we can’t do even with rational numbers. For example we can’t take square roots of every-thing. We can find square roots of 4, because we know that 2 and −2 will square to 4, that is:

22 = (−2)2 = 4

We can also find the square roots of 9, 16, 25, and so on. We can find the square roots of fractions made from those square numbers, like 49 or 925. But what about the numbers in between? Thousands of years ago, in the ancient Greek empire, mathematicians had already proved that you can’t make a square root of 2 in rational numbers. That is, there is no fraction ab with a and b being whole numbers and (ab)2⁢=2. But those mathematicians of long ago didn’t know how to fix this issue. It took until the invention of calculus and the concept of convergent sequences to get a rigorous definition of that. I don’t really want to attempt all those definitions here (fractions are already hard enough), but at some level the idea is the same as before: we define a new type of number based on the old type, using a defining property. In this case it’s the concept of ‘limit of a convergent sequence’. We start with all the rational numbers and we define the concept of ‘limit of a convergent sequence of rational numbers’. This is a very technical definition, but the idea is that an infinite sequence of numbers might get closer and closer together without ever actually staying in the same place. It might become very stable towards one particular point even though it doesn’t necessarily ever get there, in which case that point is called the ‘limit’. One example is this sequence:

11,12,13,14,15,16,…

These fractions get smaller and smaller forever, getting closer and closer to 0 but without ever actually reaching 0; we might be able to sense this intuitively, but we can also rigorously prove, according to the technical definition, that the limit of this sequence is 0.

We define the real numbers to be ‘limits of convergent sequences of rational numbers’. Some of those limits were already there in the world of rational numbers – as in the example above, where the limit was 0 – so they don’t give us anything new, but some of the limits weren’t there in the world of rational numbers, so they are new creations. They are called irrational numbers, because they’re not rational. This includes numbers like π and the square root of 2.

Just like with additive inverses, we prove that limits are unique (that is, if x and y are both limits of the same sequence, then x = y), and then we have a notion of equality for real numbers: two real numbers are equal if they are both the limit of the same sequence of rational numbers.

You might at this point be dismayed that something as intuitive as equality between numbers has become so convoluted. I hope you will take heart in two things. One is that I am not claiming you should understand any of that (especially because I didn’t give full definitions of anything), and I will not rely on it later in the book. I really just wanted to give a flavour of how intricate these things are if we try to do them really logically.

The second thing to take heart in, I hope, is that it’s really wonderful that we are able to take concepts for granted so intuitively, even though the logic behind them is so difficult to pin down. That doesn’t mean there’s something wrong with the logical approach, and it also doesn’t mean there’s something wrong with the intuitive approach. I think it shows how powerful our intuition is, but it’s also important to remember how much that intuition is hiding. There is simply a lot there to understand. We can depend on our intuition to a certain extent, but it can be limiting, and it can also lead us astray. If we want to develop more subtle and nuanced ways of thinking, one way to do that is to develop our intuition in different ways. And one way to develop our intuition is to understand what’s really going on with it, and start tweaking it a bit.

Going down a dimension

Numbers have taken us down a dimension from sets, and we’ve gone down that dimension to make things simpler. The trade-off is that we lose nuance in the process. In my field of research, category theory, we add dimensions back in, in order to get some nuance back. The trade-off is then the other way round: we gain nuance, but things become more complicated. However, the whole idea of research is to come to an understanding of more and more complicated things. The idea is to get better at handling the complicated concepts so that we can benefit from the nuance, rather than giving up and sticking with the simpler ideas all the time.

I am often tempted to stick with something I know because I can’t face the effort of learning something new. I keep every phone I have until it’s almost non-functional because I so dread the prospect of learning how to use a new one. But then, when I do get a new phone I fairly quickly recognise its benefits, after a few initial days of frustration.

I believe there are sometimes solutions to social issues where some slightly more complicated maths would help, but the barrier is that not enough people understand the maths. For example, the way social support cuts off at a certain level of earning discourages people from trying to get a job, because if the job doesn’t pay above a certain amount they will end up worse off. Some people think the answer is to have no benefits at all, to force people to get jobs if they want to have any income. But a more compassionate solution would be to calibrate the support so that the amount you get tails off instead of abruptly cutting off if you have some income from a job; that way, you will always be better off if you have a job. However, I suspect that if someone tried to introduce such a system, people would feel that it must be hiding something nefarious from them. The opposition can always oppose it by claiming that the numbers are hiding something untoward, and many people who don’t understand it will believe them.

Another example is the ‘alternative vote’ system of voting (also called ranked-choice voting or single transferable vote), which was rejected in a referendum in the UK in 2011. By contrast, it was voted into use in New York City in 2019. In both campaigns those opposing it claimed that it was too complicated for people to understand; but of course those arguing against the proposed new system stood to lose out by it. If something seems too complicated for people to understand, but is arguably fairer, then we should educate people better, not just give up. (If you don’t think it’s fairer that’s a whole different argument.)

More broadly, if something seems too complicated for humans to understand, but arguably will enable us to do something better, then we should try to understand it, not just give up. This is the case with more complicated notions of sameness. Category theory is one approach that humans are taking to try and understand sameness better. We will see more about that later, but for now I want to consider some situations in life where we reduce things to numbers but run risks in doing so.

The risks of going down a dimension

The census involves measuring racial diversity, so people are asked to select a category for their race. But this involves making decisions on what counts as a different race. No matter how we decide to draw those lines, there will always be people who don’t fit into the clear divisions if their biological mother and biological father came from two different classifications. We could include ‘half A and half B’ classifications, but then what about three-quarters and a quarter? An eighth? How fine are we going to make those gradations before we admit that it’s become a bit pointless and we should accept that it’s some kind of continuum? At the same time, there is still discrimination against people who aren’t or don’t look white; in fact, the discrimination is on a continuum as well, with the darkest- skinned non-white people being discriminated against much more than the lighter ones. By choosing to use one way of classifying people, we run the risk of neglecting something that genuinely does affect people’s lives.

The polls in advance of elections are another example. Polls attempt to model the population by taking a ‘representative’ sample of voters, but this involves making choices about which characteristics are key when it comes to the decisions people make in the polling booths. This risks assuming that all the people with that characteristic are ‘the same’. Politicians rightly come under fire for assuming that, say, all Black women are going to vote the same way (and this is usually left-wing politicians assuming they will get the vote of all Black women and so not actually doing anything policy-wise to try to appeal to them).

Another example where numbers are too unsubtle is in counting covid cases. At the beginning of the pandemic it was important to track sheer numbers of cases because there was so little understanding of the disease and how to treat it, and no vaccines. At that time, case numbers were a fairly good indicator of hospitalisations and deaths. But as the scientific understanding of the virus has increased and the treatments and vaccines have improved, the relationship between case numbers, hospitalisations, long covid and deaths has changed. If you choose to focus only on number of cases, you lose the detail on how severe the cases are. Numbers, on their own, can be taken as indicators of other things, but we should not continue using the same indicators blindly without adapting to the context.

If we can use more than one number to represent and understand a situation, that will always give us more information; the trade-off is that it might give us too much information to deal with. However we shouldn’t shy away; instead, we should keep getting better at dealing with more information. One way of dealing with more information than just one number is via functions, and that’s the subject of the next chapter.

* I am only talking about ordinary numbers here, the so-called ‘real numbers’, which include the positive and negative whole numbers, fractions and irrational numbers. It does not include the ‘imaginary numbers’, which do create a two-dimensional system of numbers, but aren’t usually used to quantify things in the real world.

* You might wonder how we define addition of numbers if we’ve defined numbers according to this business of bijections. That is a great question, and we’ll come back to it a bit later. I thought we might need a rest from the whole bijection thing before attempting to do addition with it, so for now feel free to think about addition without thinking about bijections at all.

* 0 is not a multiplicative identity because if you multiply by 0, things do change.

* Berndt Sellheim, ‘The moment I knew: we hadn’t met, I hadn’t seen a picture. But when I read her words a humming started up’, Guardian, 5 August 2023.

* ‘Fraction’ is a broader term as the top and bottom of a fraction can be anything, whereas for a rational number the top and bottom have to be whole numbers.

4 EQUALITY BETWEEN FUNCTIONS

In the previous chapter we built up more and more complicated forms of number, and more and more complicated approaches to ‘equality’ between those numbers. The more complicated types of number enable us to express more concepts and quantify more abstract ideas than the whole numbers, which only describe things that can be counted one by one. However, a number is still just one tiny piece of information, and reducing a situation to one number – no matter what type it is – obliterates a huge amount of the whole story of what is going on.

Numbers are just the start of a vast panoply of mathematical objects that mathematicians have been developing across centuries. We keep building on things that came before so that we have more and more concepts to play with or to help us. In this chapter we’ll look at functions, which are a way of describing the world while maintaining more of the story than just a single number. Looking at a function instead of a single number enables us to preserve more context about what’s going on. Functions give us a way to look at results across time, or across populations, or see how situations respond to different inputs, and compare the whole pattern of behaviour. This opens up many more possibilities for ways to think about sameness.

What is a function?

Functions don’t have to have anything to do with numbers: they are just things that take inputs and produce outputs. The inputs and outputs are often numbers, but they can be other things – in fact they can be anything. If the inputs and outputs are ‘anything at all’ rather than numbers, that makes the concept rather more abstract. As usual with abstraction, that typically makes things harder to think about, but the trade-off is that it enables us to use the idea much more broadly, to much greater effect. We can even apply functions to functions themselves, so that the input is a function and the output is another function, but perhaps that’s getting a bit too far ahead of ourselves.

If we stick to thinking about functions between numbers, a function is something that takes a number as an input, and spews out a number as an output. But it’s not random: it’s entirely predetermined what the output will be for any given input. It’s like having a rule for what happens to each input, but the rule doesn’t have to be something well organised or sensible, nor does it have to be given by a formula – it could just be a list.

If we’re only thinking about a finite set of input numbers, we could make a complete list showing what the output is for any given input, for example this one:

input
output

1

2

2

4

3

6

This is a bit like some hilarious signs that get mocked in memes, showing the price of doughnuts or something:

1 doughnut

£2

2 doughnuts

£4

3 doughnuts

£6

4 doughnuts

£8

5 doughnuts

£10

6 doughnuts

£12

7 doughnuts

£14

8 doughnuts

£16

9 doughnuts

£18

10 doughnuts

£20

11 doughnuts

£22

12 doughnuts

£24

In both tables, you probably noticed that each output is achieved by multiplying the input by 2, but in the case of the doughnuts, it should be rather obvious that if one doughnut costs £2, then two doughnuts cost £4, three doughnuts cost £6, and so on. You might hope that if you get half a dozen or a dozen there’s some kind of discount (I would certainly hope that, especially at £2 per doughnut) but you can scan this whole list and find that there is no such discount – it’s just a multiplication table. It would have been much quicker to say ‘£2 each’. If you were going to be hilariously mathematical about it, you could make a sign saying this:

number of doughnuts

total price

n

£2n

This is a formal mathematical way of saying ‘£2 per doughnut’, expressed as a function. It’s a bit over-the-top for doughnuts, but it’s needed when we have no context for the function. If we just said this:

input

output

1

2

we would have no way of knowing what happens for other inputs because we haven’t been given any indication of any kind of pattern. Instead, we could express it like this:

input

output

n

2n

This is a bit unnecessary if the only possible inputs are 1, 2 and 3, but if there are infinitely many inputs – perhaps every number is a possible input – then it becomes much more important, indeed imperative, to have some way to indicate all the outputs without making a list, because life is literally too short to make an infinite list.

This is why we start using letters and formulae in maths – or at least, it’s one of the reasons. It’s a way of expressing a relationship that holds for infinitely many numbers, without having to write it all out or having to say ‘dot dot dot’ and hoping everyone understands you the same way, like this:

input

output

1

2

2

4

3

6

⁝

⁝

If the function in question is called f, then we could write this function formally by saying:

f (n) = 2n

This is telling us that if we apply the function f to the input n, we get the output 2n. We read the f (n) notation out loud as ‘f of n’.

One of the whole points of dealing with functions is to understand something across an entire range of values, so we need a way of communicating all that to someone without relying on guesswork, psychic powers or prior understanding of the situation. I ran into this issue when I first moved to the US and started teaching undergraduates in the US system for the first time. I had to adjust to a strange new (to me) function: the ‘grade point average’ system, which was completely alien to me coming from the UK. The idea is that grades are converted into numbers, and the teaching guide I was given told us this:

A is 4 A− is 3.7 and so on.

This is so much part of the US education culture that it was not considered to need any further explanation, but to me it just looked like a very peculiar function. As far as I could see from this limited information, the system involved subtracting 0.3 for each increment, which would result in the following function:

A

4.0

A−

3.7

B+

3.4

B

3.1

B−

2.8

C+

2.5

C

2.2

C−

1.9

It seemed very odd to me. But then someone explained to me that the sequence actually goes like this:

A

4.0

A−

3.7

B+

3.3

B

3.0

B−

2.7

C+

2.3

C

2.0

C−

1.7

The pattern for the function isn’t ‘subtract 0.3 for each increment’, but rather, unadorned letter grades correspond to whole numbers, and then the + or − modification takes you up or down by 0.3.

This particular function can’t really be described with a formula because the inputs are letters and not numbers, but it can be described by a ‘rule’ as I did above. Regardless of whether a function is described by a formula or a less formulaic rule, we can still ask what it might mean for two functions to be ‘the same’.

Sameness of functions

The most stringent version of sameness for functions is that two functions f and g only count as the same if they give exactly the same output whenever you feed them the same input. In mathematical notation we’re saying:

Two functions f and g count as the same whenever for all inputs x, f (x) = g (x).

Remember, the notation f (x) means ‘the output you get when you apply the function f to the input x’, and g (x) means ‘the output you get when you apply the function g to the input x’. So the expression is telling us that f and g always give the same output, if you feed them the same input.

It doesn’t matter what the formula or rule for the function is if we think about equality of functions like this – it doesn’t even matter if there is no formula, and only a table of outcomes. Later on we’ll see why it can really help us to have a formula, when we are looking for more subtle forms of sameness and our intuition isn’t subtle or powerful enough to recognise them. But for now I want to draw some pictures of functions and use our visual intuition.

Turning functions into pictures

If the inputs and the outputs of our function are all numbers, we can turn the function into a picture by plotting its graph. This is a wondrous technique for turning something abstract into something visible, giving us a chance to invoke our visual intuition, which, for sighted people, has had rather more practice than our abstract intuition. Our lack of intuition for the abstract world is one of the difficult things about learning maths, but if we can translate the abstract ideas into visual ones, we can gradually build up our abstract intuition with the help of our visual intuition. (However, invoking visual imagery isn’t essential for doing mathematics, and there have been some brilliant blind mathematicians, as well as brilliant mathematicians with aphantasia, the inability to visualise things.)

Turning a function into a picture means we can get some visual intuition for various features of functions, and we can also describe visible features and see what they correspond to in functions. Drawing the graph of a function might not always be the most practical way of building up our abstract intuition, especially once we’re in higher dimensions, but we can always build our intuition for abstract things by spending time with them, just like we can build our intuition for anything as long as we’re open to it and believe it’s possible. The same is true of many new scenarios in life. One of the reasons I find visiting new places daunting is because I don’t feel I have any intuition for the place, the people and the atmosphere, so I don’t have as many ways to sense if something is wrong or dangerous. This makes it much more scary, but after spending some time there I do start to build up some intuition and then I feel progressively less afraid.

The way we usually draw the picture of a function is that we put the inputs along the horizontal axis and plot their associated outputs vertically. We’ll think about specific graphs of specific functions later, but for now I just want to think about these pictures in general, and what it might mean for them to be the same.

When two functions are exactly the same, as we defined just now, they have exactly the same output for every input. If we translate this into graphs, this means that their graphs are exactly the same. This is very different from having just one place where they are the same, in which case the graphs would meet at only one point, like this:

This might seem like a more forgiving form of sameness, but instead of being more nuanced it’s really just more blinkered, because the graphs could be really extremely different everywhere except that one spot, and we’re completely ignoring that.

To be more nuanced we could invoke one of the versions of approximation that we met earlier. So instead of saying the outputs have to be exactly equal everywhere, we could say they are approximately the same everywhere, perhaps to within a certain tolerance interval.

A picture of graphs being close to each other forever might look like the diagram below, where there’s a ‘tolerance interval’ around the function depending on how accurate you want to be, and if a second function stays within that corridor, then you consider the functions to be very similar.

This concept is useful if we’re collecting data from the real world, where there might be some inaccuracies built into our measurements. In that case it would be futile to be too precise about what it means for functions to be the same, and we might just want to look at their general shape rather than the exact numbers. This can help us get an idea of trends, like population trends, or traffic flow, or share prices, or the weather, or virus spread.

In physics we might do an experiment to see how something in the world around us behaves, and get some data. Then we plot a graph of our data and look at its general shape, ignoring slight inaccuracies and deviations.

Looking at general shapes

One of my favourite examples of the general shape of a graph relates to springs (the coiled-up kind, not the fresh-water kind). The idea is to investigate how a spring behaves by hanging various different weights on it and then measuring how much the spring stretches. If you have many small weights that you can add to the spring one at a time, then that works particularly well, but it also doesn’t matter if you add weight in uneven increments because the graph will deal with that for you.

Then you might get something like the following data for your weights and the lengths of your spring:

mass/g

length/cm

0

14

5

18

10

19

20

25

25

27

30

30

40

36

45

38

50

44

55

56

You could now try to come up with a formula for a function that produces those exact numbers, but that would be hard for two reasons. One is you simply have nothing much to go on: it’s just a bunch of numbers. Where would you even start trying to come up with a formula?

The other problem is that it’s possible your measurements aren’t very accurate. It’s harder to spot a pattern in numbers if the numbers are slightly inaccurate, whereas if we plot them on a graph our eyes are remarkably good at evening out the inaccuracies and looking at a general shape rather than individual numbers. It’s one of the ways in which graphs are helpful.

The graph of the above data comes out looking like this:

Now it’s much clearer that it looks like a straight line (as I’ve sketched in), apart from at the top where one point has gone slightly haywire. We might then investigate whether that was an error or a trend that is continuing. So we hang a few more weights on, and discover that the graph bends even more. We hang a few more weights on and then suddenly the spring breaks. Oh no, how are we going to replicate the experiment?

Maybe we have another spring available but it’s not the same kind; never mind, we try it with that one. This time we get this graph:

Now we look at our two graphs and think ‘Aha, they have the same shape’. They are not the same numbers, and they’re not the same graph, but still our brains are able to recognise some similarity between them: each one looks like a straight line with an upward slope, and then curves up at the end before the spring breaks. We are able to see past many details to say that these graphs have ‘the same shape’. Not only do we see past the scale, and the exact steepness of the slope, but we can also see past the slight inaccuracies in the numbers which prevent them lying on an exact straight line – they only lie on an approximate straight line. If one point were particularly far off the line we might shrug and say perhaps we made a mistake in measuring. We might wonder how wonky the points would need to be before we become more doubtful, and suspect that it really wasn’t a straight line and we were being too optimistic about this pattern about the behaviour of springs. We might invoke a tolerance interval as we did with approximations.

The next step is where being able to translate back and forth between functions and pictures really helps: it’s the reverse process of drawing a graph. Given a formula or rules for a function, we can translate that into a picture by drawing a graph, but then, given a picture, can we translate that back into a formula? If so, we can provide a formula for the behaviour of the spring, which helps us predict its behaviour even for weights we didn’t try out, but more importantly, it helps us make a theory about springs in general, and that’s how laws of physics are often developed.

The experiment with springs leads us to the law governing springs called Hooke’s law, and it fascinates me not because of the straight line part but because of the part that doesn’t behave like a straight line. If you hang too much weight on a spring it exits the straight-line part and the slope starts to increase. In effect, the spring has become overstretched and become a different spring. If at that point you took all the weights off, it would not retract to its original length, but it would still act like a spring and stretch with a linear graph as long as you didn’t put any more weight on than you previously did. The straight line will be a different one, because the ‘springiness’ will have changed, but it’s still a spring. However, if you overload it yet more, the spring will overstretch more and more until it reaches its breaking point, and then it breaks – and at that point it’s not a spring any more.

I like this as a metaphor for other things in life, like eras of music, or portions of your life. I always think periods of time in music history are like Hooke’s law, where the music style develops within an era (like Baroque or Romantic) as composers push it further and further. But then at a certain point it enters the curved part of the graph where it’s warping the previous parameters without totally abandoning them. At that point you know it’s about to break, and when it does, the era of music changes completely, and a new era starts.

Although an era of music is not something that is very quantifiable, unlike the length of a spring, it feels to me like it has the same general shape as Hooke’s law. From that shape I can theorise something about the mechanism producing that shape. This is often the point of collecting data and then looking at the shape of the data on a graph.

So far we have been talking quite vaguely about the shapes of graphs, but if we want to be able to say something more precise about the relationship between functions and pictures, it is helpful to understand functions using formulae. Mathematicians try to build up their understanding gradually, starting from the simplest possible cases, so we’ll do that now for functions.

Building up types of function

The simplest form of function is a ‘constant function’, something like:

f (x) = 0

This means that we are calling the function f, and whatever number x we put in as the input, the output is 0. In this situation we’re using the equals sign to make a definition or set some notation.

For this function we’re not exactly using a formula to define the output – the output doesn’t have any x's in it, because it doesn’t depend on x at all. No matter what x you feed into the function as an input, the output is the same: 0. It’s called ‘constant’ because it doesn’t change even when x is different. If you draw it as a graph it’s a flat straight line running along the x axis.

As usual we are taking the input values to be going along the x axis (horizontal) and we plot each output value as a y value (vertical). This is just a convention, and we could have done it the other way round, but then it might have been better to write the function as:

f (y) = 0

so that the letter we’re using for the input matches the standard name given to the vertical axis (although that’s just a convention as well). In that case the graph of the constant function would look like this:

Constant functions don’t have to be always equal to zero. We could have f (x) = 1 or f (x) = 2 or any number at all, which we could write as f (x) = a, for any real number a. We see that all of these functions produce similar graphs: a flat straight line, whose height is whatever the constant number is.

There is some sense in which these are all ‘the same’ picture, just moved up or down a bit. We’ve already talked about situations where two things might look different at first, but then we realise it’s just the same situation viewed from a different point of view. The horizontal line graphs could be depicting the same thing, just on a different scale; perhaps one is depicting a constant temperature of 0 degrees Celsius, but the other is depicting that as 32 degrees Fahrenheit.

We might then wonder what happens if we look at the straight line from a different angle. I repeatedly do a double-take when I remember that the grid of streets in Manhattan does not quite line up with the compass directions north, south, east and west. I live in Chicago, where the grid of streets really does match the compass directions, so when I’m walking around Manhattan I accidentally think of myself as going north when I’m going ‘up’, because it is mostly north-ish. It generally doesn’t matter though, because when I’m walking on the grid it’s just a grid (but sometimes I’m attempting to use the sun to work out which way I’m facing). The same sort of thing is true of straight-line graphs. If we just choose to place the x and y axis facing a slightly different way, it doesn’t change the shape of a straight-line graph as it will still be a straight line. However, it will change the formula for it, so if we’re not careful about how a formula matches up with a picture, we might think something is more different than it is.

The simplest type of straight line that isn’t constant is something of the form:

f (x) = x

You might have thought that the constant function was ‘boring’ because it sent everything to the same number, but there is a sense in which this function is even more boring because it does nothing whatsoever – whatever input you give, it just gives you back the same number. It leaves every number unchanged, so it is called the ‘identity’ function. It is related to the additive identity 0 (which leaves everything the same when you add it) in that the instruction ‘add 0’ is one way to produce the identity function; likewise we could do it by multiplying by the multiplicative identity 1.

The graph of the identity function looks like this:

For each input x, the output is the same number, so we get this diagonal line. Perhaps as a picture it doesn’t look so much like a function that is doing nothing, because we see more how the outputs are different relative to each other, rather than how each output is the same as its corresponding input.

There are now two things we could do to soup this up by one step. We could multiply everything by a fixed number, or we could add a fixed number to everything. If we multiply everything by a fixed number we’re saying things like:

f (x) = 2x or f (x) = 3x or f (x) = bx for some number b

When we look at the graph, we see that this multiplication changes the slope. If we multiply by a number bigger than 1 the slope gets steeper, and if we multiply by something smaller than 1 the slope gets flatter.

By contrast, if we add a number to everything, we’re doing something like this:

f (x) = x + 1

and we get the following graph:

If we draw both the graph of f (x) = x and f (x) = x + 1 on the same axes, we can see that the second one is the same as the first one, just shifted up by one unit.

Perhaps you can see that if we did f (x) = x + 2 it would be the same graph as f (x) = x but shifted up by 2 units. If we did f (x) = x + c for any number c, it would be shifted up by c units, and this means that if c is negative the graph will shift down by c units.

I admit I am still irritated by one question on a maths test when I was at school, where I lost a mark because of some issue about negatives. The question was about the value of a car over time, and there was a graph showing the value going down. We first had to find the slope of the graph, and then say what the slope represented. The slope was negative, and I wrote that it represented the rate of increase of the value of the car. I lost a mark on the grounds that the value was decreasing: I was supposed to write ‘rate of decrease of the value of the car’. I still think this was unfair: the slope did represent the rate of increase of the value of the car. The rate of increase happened to be negative, which meant the value was decreasing. But if the slope had been representing the rate of decrease, then a negative value would have said that the value was increasing.

If we now combine the multiplication and the adding, then we get things like this:

f (x) = 2x + 3

which first makes the slope of the line steeper by a factor of 2, and then shifts the whole line up by 3 units.

More generally we could have things like:

f (x) = ax + b

which is more commonly written:

y = mx + c

or in the US for some reason it is instead typically written as

y = mx + b

which is the famous – and often dreaded – equation for a straight line. The British and American versions are ‘the same’ formula even though they use different letters – the letter c or b at the end isn’t a specific quantity, it’s just us saying ‘and then add a number at the end, which we will refer to using this letter but it could be any number’.

All of these straight lines are ‘the same’ in the sense that they’re just straight lines – we could draw any of them using a pencil and the same ruler. The only difference is our frame of reference, that is, where we happen to have put the axes.

Frames of reference

If I drew a straight line on a circular piece of paper and handed it to a bunch of different people, they could all come up with a different equation for it depending on where they decided to put their axes.

The reason I say I’ve put it on a circular piece of paper is so that there’s no obvious frame of reference for where to put the x axis and the y axis. Actually there’s not a serious reason to make the x and y axes perpendicular to each other – it’s just for convenience. The technicalities of specifying coordinates in two dimensions would still work fine if the two axes were at any angles (as long as they weren’t parallel, in which case we’d have some trouble).

So we have some different contexts in which ‘the same’ could mean different things for a function: does it mean ‘exactly the same formula’ or does it mean ‘the same shape graph, even if you have to change your frame of reference in order to see it’s the same shape’? There is no right answer to this, it just depends what you’re trying to understand. Understanding that context makes a difference is important, and visualising a graph on a piece of paper being turned around can help us develop our intuition about things being ‘the same’ if we’re just looking at them with a different frame of reference, just like looking at counting blocks from a different direction to develop our intuition about commutativity.

Sometimes it really does matter that we all pick the same frame of reference. During the pandemic when we suddenly started having a lot of online events, it became much more important to specify the time zone when publicising an event. For an event in person, you can just say the location and the time, and everyone knows that the time is referring to that location’s time zone. For online events it’s less clear; some organisations decided always to use UTC as their frame of reference. Personally I have to look that up every time because I can never remember what it is.

On the other hand if we’re just having a chat with friends about what time we tend to go to sleep and wake up every day, it doesn’t matter what time zone we’re referring to for each of us. Similarly with functions, sometimes we care about the exact numbers relative to a fixed frame of reference, and sometimes we only care about the shape. We might even pick a different frame of reference to make the formula look simpler, if it’s the shape we’re trying to understand rather than the values themselves. In fact, one of the great insights of Einstein’s Theory of Relativity is the idea that, if we look at an object moving through space under the action of gravity, its path looks curved, but we can think of that as a straight line in a different shape of space. So if we think of the shape of space as being curved by gravity, and use that as our frame of reference, the curve becomes ‘in some sense’ the same as a straight line. If the object is something we’ve thrown in the air, and we use ordinary x and y coordinates, the path looks like a particular kind of curve: a parabola.

Parabolas

A parabola is arguably the next more complicated form of function. It involves not just x's, but perhaps x2 as well. The shape of the graph of f (x) = x2 is called a parabola, and it looks like this:

However, just like with straight lines, we can shift the axes up and down or left and right, and the graph will still have the same shape although its formula will change. Here are some examples of how we shift the graph around by adding constants into the formula:

Note that these parabolas might look like they're getting progressively taller, but that’s only because more of the parabola is depicted in the part of the graph I’ve shown: really these are all infinite.

We could also make the parabola wider or narrower by multiplying by a constant:

It’s easy enough to see that the above two images are different, but you wouldn’t be able to tell the difference between those graphs if I used different scales on the axes:

This is one of the senses in which functions can be the same: we can change the frame of reference by changing the scale.

If we combine all the above transformations of parabolas we get functions of the following form:

f (x) = s(x + h)2 + v

Here s, h and v are three numbers: s tells us how much the parabola is stretched, h tells us how much it’s shifted horizontally, and v tells us how much it’s shifted vertically. (Incidentally if s is negative the parabola will be flipped upside down.)

This general formula is a bit like the general formula for a straight line y = mx + c, except now it’s for a parabola, and has three possible constants, where the straight lines only had two. You might notice that this is a quadratic formula written a bit weirdly: a quadratic formula is one with an x2 in it (because ‘quadratic’ sort of means ‘of or pertaining to squares’), but usually you might think of a quadratic formula as something like this:

ax2 + bx + c

That’s just a different point of view. This last formula can always be rewritten as one in the form

s(x + h)2 + v

which makes it clearer what sort of graph it has. The process of going from the latter to the former is just a case of multiplying out some brackets, but the process of going from the former to the latter is harder because you’re doing the ‘opposite’ of multiplying out brackets – and doing the opposite of multiplying is typically harder than multiplying.

The process of going backwards in this particular case is sometimes called ‘completing the square’, because you’re taking something that isn’t a square (or something squared), filling it in until it is one (to make a convenient-looking formula) and then you say ‘oops, I had to add something to make it a square so I’d better subtract it again to make sure I don’t end up with more than I started with’. It is a perhaps slightly archaic method that used to be drilled into students at a certain level of school as a way of solving quadratic equations. My parents were drilled in it but I wasn’t; I learnt it from them. It can also be depicted as completing a square shape geometrically, but my purpose here is not to explain how to do it. As a technique I don’t find it particularly interesting or useful, but I do think it’s an interesting example of a principle: it’s a way of translating between two different points of view on the same expression.

An expression of the form:

ax2 + bx + c

is called a polynomial. Sometimes it is clearer to see a quadratic function written out like that with the x2 term separate from the x term, separate from the constant. This also makes the relationship with the straight-line formula more obvious, because if we take a = 0 in the quadratic formula, it becomes a straight-line formula.

On the other hand, when you complete the square it makes it clearer how the function is related to the standard parabola shape x2, and it emphasises the fact that it’s the same shape, just with axes that are placed and scaled differently. Those are the two points of view that we are moving between when we ‘complete the square’.

We can also think about rotating the parabola, or rotating the axes that we’re using as a frame of reference. But we can also take this a step further and see a parabola as a slice through a cone shape.

Conic sections

The different shapes we can make by slicing through a cone shape are called conic sections. If you imagine an ice cream cone or a conical water cup or a cone-shaped cake, you might be able to imagine slicing it open in different directions and producing different cross-sections.

If you slice it straight across, you’ll get a circular cross-section. If you make your slice a little bit ‘on the bias’ (slanting), you’ll get an elongated circle, technically called an ellipse in maths. As you get more and more ‘biased’ your ellipse will become more and more elongated, until you are actually parallel with the slope of the cone, at which point your cross-section will no longer meet the ‘other side’ of the cone, and it will instead be a parabola (that is, the edge of it is a parabola). If you now keep sloping more, you get another type of curve called a hyperbola.

A hyperbola might not seem that different from a parabola, but that’s because so far I’ve only told you half of the story of conic sections – in maths the cone shape we’re thinking about here has a positive and a negative part, so it’s really a double cone, like this:

It’s also supposed to be infinite, extending endlessly both up and down.

The hyperbola arises when you slice through this shape at an angle close enough to vertical that you cut through both halves of the cone, so the curve has two parts to it.

The quintessential example of a hyperbola is a graph of the function f(x)⁢=1x, which looks like this:

It has two completely disjoint parts, one for positive values of x and one for negative values. We’ll see more about this graph a bit later. By contrast, the parabola arises when you slice parallel to the slope of the cone’s sides, so you only intersect one half of the cone.

These ellipse, parabola and hyperbola shapes look much more different from each other than the parabola shapes we saw previously with our more familiar-looking graphs, but in a way they are still all different points of view on the same thing: different slices through a cone. And in an even deeper way they can be viewed as straight lines through a curved space.

It would be very hard to ‘see’ any of this sameness if we were relying on our visual intuition alone, and this is why using formulae in maths can help open up a whole new vista of abstract patterns to ‘see’. However, the letters and symbols can be an off-putting barrier to entry, which is unfortunate because if you can make it past that, then there is a whole fascinating world on the other side.

One way in which a formula can help us is if it gives us a way to understand where the shape of the graph is coming from and why. If the graph is coming from some data we’ve extracted from the real world, a formula can help us understand why something is happening in the real world. When we recognise that two situations have a similar formula, we can draw connections between things that might otherwise seem unrelated.

Deducing how things work from looking at the graph

When we were looking at graphs for the behaviour of springs, we saw that the graph was a straight line up until the point at which the spring became overstretched. Once we know how to find the formula for the straight-line part, we can then conjecture that this is a law governing the behaviour that we were studying. It gets harder when the picture is more complicated, but for a straight line it’s not too bad, as long as we remember the principles: we multiply x by a number to change the slope, and we add a number to shift the whole thing up. Now the formula mx + c is telling us something real about springs: the constant c is telling us about the starting length of the spring with nothing hanging on it, and m is telling us how stretchy it is.

The spread of viruses can also be modelled by a formula, but it’s somewhat more complicated than the one for springs. The spread of a virus through a population happens by repeated multiplication – each person infects a certain number of other people, on average. So the rate of spread would seem to be proportional to the number of people so far infected; ‘proportional’ just means that the graph would be a straight line with some constant slope, or, in symbols, one quantity is proportional to the other if one is a multiple of the other. Note that it’s the rate of increase that is proportional to the number of infected people, that is, the gradient of the graph increases as the number of infected people increases. This creates a sort of feedback loop, because as the number of infections increases, so does the rate of increase, which means the number of infections increases even faster, and so on. This is, by definition, how exponential functions work.

However, the spread of a virus also depends on how many uninfected people there are wandering around available to be infected. If everyone is already infected, then clearly the virus numbers can’t go up any more as everyone already has the illness. If there are only ten people left uninfected out of a population of a million, then the virus can still spread, but definitely not as fast as if there are 900,000 people still uninfected.

The upshot is that the rate of spread of a virus is proportional to both the number of infected people and the number of uninfected people. If you work that out as a formula and then turn it into a graph it looks like this:

The graph might be steeper if the virus is more infectious, and it might go higher if there’s a bigger population, and it will change if something intervenes such as a vaccine or other mitigation measures, but otherwise the curve will always look the same and it will always look like this.

This graph has also been used to model the ‘spread’ of viral music videos.* In this case the situation was understood starting from the graph, as we did for the spring. The researchers gathered data about the number of views of viral videos over time, plotted a graph, and then noticed that it had ‘the same’ shape as the graph for viral illnesses. So far this is analogous to my thoughts about eras of music behaving like a spring that stretches and then breaks when overloaded. However, for viral videos the idea has been taken further, and researchers conjecture that the similarity in the graph indicates that the spread of videos is controlled by processes analogous to the spread of viral illnesses: each person who shares the video ‘infects’ a certain number of people on average, but not everyone who is exposed – after all, if a friend of yours reposts a video you won’t necessarily watch it. But the shape of the graph then shows that there’s also a notional upper limit to the population available to be infected, even though it’s spreading online – the upper limit of the available population is not the total online population.

This mathematical model is not a proof that this is how the mechanism of viral videos works, but it is a sort of narrative explanation of why the numbers come out that way. It doesn’t explore why some people are infected and why some aren’t, just like the model for viral illness doesn’t do that either – it just makes a function for the numbers as a basis for further and deeper study. But this is why viral videos are called ‘viral’ – there is this sense in which those situations are ‘the same’, which we can see once we understand something about abstract patterns and the sameness of shapes of functions.

In some situations we can go further, and after analysing the shape of a naturally arising function we can use the analysis to produce the function back again, at least to some level of approximation. This is the case in the synthesis of sounds.

Synthesis

I was very excited when I first encountered a Casio synthesiser keyboard when I was a child. This newfangled technology was like a tiny digital piano, with buttons you could press to make it sound like a violin, flute, trumpet or pipe organ. The sounds were rather approximate back then, but since those days synthesisers have become increasingly realistic.

In fact, pipe organs (real ones) have been around for hundreds of years; they’re essentially early versions of a synthesiser, using the maths of harmonics to build up interesting sound waves. Organs typically have many different sets of pipes, not just one per note. Nowadays they are usually housed in a low-pressure room so that you don’t have to actively blow air through a pipe to make it sound – you just open a gap to the outside, and the pressure difference will make air flow through and hence the pipe will sound.

One pipe by itself will sound quite plain, so organists blend the sound from combinations of pipes to make different timbres of note. The interesting thing is that if you’re playing a C you don’t just blend different types of C together: you might blend a C from the octave above, but then also a G, another C and an E.

These notes are the start of the harmonic series, which comes from a wavelength relationship. The harmonic series in maths is the series of fractions:

1,12,13,14,15,16 . . . .

We can then consider making notes whose wavelengths are in these proportions relative to a starting note. Suppose we take the first note to be a C, then we take the next note to have ½ of the wavelength of that C, and then the next one has ⅓ of the wavelength of that C, and so on. This produces these notes:

These notes are called the harmonics of C, and they are always present in any ‘organically’ produced note in different amounts. That is, a note that is produced by something physical making sound waves has an overall wavelength, but it will always have all the fractions of that wavelength present as well, because those waves also fit in the same physical constraints.* For example, if a string is vibrating with fixed ends, a wave with half the wavelength will still fit between those fixed ends:

So will one with a third of the initial wavelength:

and also a quarter, a fifth, and so on:

Now, when our ear hears all those different wavelengths, our brain blends them together into one sound rather than hearing them all separately. The combined wave might look like this:

which is a graph for:

5 sin(x) + 2 sin(2x) + 2 sin(3x) + 0.5 sin(4x) + 0.2 sin(5x)

Rather than hearing separate notes, we hear ‘timbre’; mathematically, timbre is the shape of the wave in question, whereas pitch is just its frequency or wavelength (the two are inversely related).

Given any particular sounding note, we can analyse it to see what all of its constituent harmonics are, that is, how much of each wave of the harmonic series was present. But then we can also create timbres by putting harmonics together ourselves. This is what organists do when they combine pipes. Each set of pipes is controlled using a little piston that the organist pulls out of a console, called a ‘stop’. If the organist invokes all the pipes at once, then it creates a huge, loud, but also very rich sound. To do that, the organist pulls all the stops out – hence the idiom ‘pull out all the stops’.

Organs have been around for hundreds of years, but this same principle has been used much more recently for digital synthesisers. The idea here is that digital sounds can be produced as actual sine waves, and then we can build them up to make different combinations of harmonics and thereby create different timbres to resemble, say, a piano, or a violin, or a flute.

The key is to pick a sense in which something is sufficiently similar to the sound wave you’re aiming for, to count it as a synthesis. For sound waves, this probably means having a ‘tolerance interval’ around the entire function. The maths of approximating waves using sine waves is a field called Fourier analysis, which provides a way to analyse functions and work out how to model them by piling sine waves on top of each other. It turns out that this works for any function that repeats itself regularly, called ‘periodic’, such as a sound wave. So, no matter what shape a periodic wave is, you can recreate it by adding together different-sized sine waves. Or at least, you can get something more or less the same as the wave you’re trying to model, to within a certain level of accuracy – the more sine waves you pile on, the closer you will get. So you can pick a level of accuracy you want, and get within it. The synthesisers of my childhood were not very accurate, but modern ones are much more so.

Sine waves turn out to be ‘basic building blocks’ for general periodic functions, just like prime numbers are ‘basic building blocks’ for numbers (by multiplication). We will see basic building blocks for other mathematical concepts later. Understanding the basic building blocks for functions can help us find a formula to approximate some data we’ve gathered from the world around us, where we have seen the shape of the data and need to work out how to build it up from functions we already understand. Besides helping us understand what physical phenomena are causing something to happen, having a formula for a function can then help us predict what will happen later. However, we have to be careful not to be led astray by the fact that functions can look very similar in one part, but then look very different in another part.

When functions fool us by looking like something else

Functions can look the same as each other for a really large portion, and then look different in some other part. This is especially true if a function isn’t governed by a neat formula, because then the early part of the function is not in any way a predictor of the later part. This makes it difficult if we’re trying to use a function to predict the future behaviour of something like inflation or unemployment. However, we can try to fit a function to the earlier part of the graph, by conjecturing how that early part was produced, and that will then determine what the later part looks like. The problem is that even when functions have a nice formula, two functions can look very similar in an early portion and very different later on. Take the virus formula, for example. It looks just like an exponential at the start, but looks completely unlike an exponential at the end, when the uninfected population is running low:

Even an exponential looks quite like a constant function at the beginning where it’s very flat. Unfortunately, looking at just that first region of the graph at the start of the covid pandemic led some people to think that the virus wasn’t that dangerous as the graph looked so flat. Even later on, some other people thought the graph was destined to flatten out, because of the general shape of the standard virus graph. What they didn’t realise was that it only flattens out when almost everyone has been infected (or is immune thanks to a vaccine), and given the death rates before we had a better understanding of how to treat the illness (and before we had vaccines), that would have meant an even more tragic number of deaths and probably overwhelming the hospital system.

Sometimes even if you’ve decided on a model for the situation, it won’t reliably predict what’s going to happen in the future just because of the problem of getting accurate data. The function might say ‘given any input a we get this output b’, but sometimes we don’t know exactly what input we have. This is especially true if ‘the input’ is actually a huge quantity of information, such as when we’re trying to predict the weather. This is a case where ‘the input’ might not just be a number, but a vast quantity of numbers, perhaps the entire information of the state of the weather now across the entire world, in every single location, and at every single point in the atmosphere. Of course, it would be impossible for us actually to gather that huge quantity of data. But we can’t even get very close to gathering all that data, so we have to make do with just using some of the data as an input. And the problem is the ‘function’ of how the weather works is extremely unstable relative to small changes in input.

This is unlike simple functions such as f (x) = x + 1. For this function, if we change the input by a tiny bit, then we only change the output by the same tiny bit, so it’s not very sensitive to small inaccuracies in our measurements. But for the weather, a tiny change in input can result in a drastically, catastrophically different output. So if one of your measurements is just a tiny bit off, then your prediction can be wildly different. Or if you’re missing one tiny bit of information, the prediction can be wildly different. It could make the difference of whether or not there’s going to be a huge storm. A mathematical system with this feature is called ‘chaotic’, which, importantly, doesn’t just mean ‘random and all over the place’, but rather, that a tiny change in starting conditions can result in massively different results. Specifically in the case of the weather the system is called a Lorenz system. This has a very remarkable shape that looks a bit like two wings with a small join in the middle:*

The idea is that you start in the middle, and a very slight shift in your trajectory could mean you either go off on one of the wings, or to one of the other, drastically separate wings, and each of those wings has trajectories on it that have nothing to do with the other wing and are very far apart. The trite saying about a ‘butterfly flapping its wings and causing a hurricane on the other side of the world’ isn’t really about the butterfly causing that storm, it’s about the tiny change in starting conditions meaning that we end up on a different part of the Lorenz system. The fact that the short-term weather forecast is often not very accurate is an inevitable symptom of the shape of this system; it doesn’t mean that meteorologists are inept and it doesn’t mean that predictions about long-term climate issues can’t be trusted. (Long-term predictions are very different from short-term ones.)

The moral of this story is that we must be very careful if we try to use partial information to infer anything about some unknown parts of a function. However, there are some situations in which we can do it safely, as long as we start with a lot of information and only try to fill in a very small gap, and only if we’re being very aware of what we’re choosing to ignore or choosing to assume in the process.

Fillable gaps

Sometimes a function might have tiny gaps that we are able to fill in hypothetically. The gaps have to be so tiny that the function is very close to being the same as a function with those gaps filled in, so close that there is really only one way to fill those gaps in ‘sensibly’; we just have to choose what ‘sensibly’ means.

A function might not be defined everywhere because there are some places where it just doesn’t make sense. For example, the function f(x) =1x doesn’t make sense if we try and input x = 0, because 10 isn’t defined in ordinary numbers. If we try drawing the graph of 1x for every value apart from where x = 0, we get this, which we saw earlier when we were talking about the hyperbola shape:

As the graph approaches 0 from the right (through the positive numbers) the outcome goes up and up. But as the graph approaches 0 from the left (through the negative numbers) the graph goes down and down. So there isn’t an obvious sensible value to add in when x = 0 because the two sides are getting so far apart. You could argue that the top and bottom are joined up at ‘infinity’ if we interpret positive infinity and negative infinity to be the same. That’s when we draw a dotted vertical line as a sort of impressionistic depiction of that, and call it an asymptote.

However, if we look at the function x sin (1x) something rather different happens. We still can’t define this function at x = 0, but if we draw the graph everywhere except at x = 0 it looks like this:

Now you can’t even see that there’s a gap at x = 0 because the two sides are so close together – as the graph approaches from the right, the oscillations squash down to 0, and the value of the function heads towards 0. The same thing happens from the left. As a result, the ‘gap’ where x = 0 is imperceptible, and it is pretty sensible to add in the point at 0. That is, we can ‘complete’ this function and declare that:

f (0) = 0

Our brains do this sort of filling in of gaps when all the rest of the information around is pointing at one thing. If one letter is missing in a sign, then we can usually quite easily fill it in without even thinking about it. However, sometimes it can make oddly ambiguous or suggestive possibilities.

You might have seen portions of text with all the vowels removed, showing that it’s nt tht hrd t dcphr wrds wth vwls mssng. Whereas if I left out all the consonants instead, it would be much harder:

o a a o eie o i oe ii

Error correction codes are a mathematical way of setting up information deliberately so that accidental gaps can be filled in. This is important when information is being transmitted electronically and prone to small errors. Systems have been designed so that some of the information being sent is a checking code so that the system can check it has received the information correctly, and then, in some particularly clever cases, correct it if it hasn’t.

Our brains are good at doing this sort of thing spontaneously, but it’s also important to realise when we’re doing it, to be aware that we might have done it incorrectly. When I give talks, people often come and talk to me afterwards. Every so often, someone will approach me who reminds me of someone else who was previously rude and obnoxious. I find that I spontaneously brace myself for an unpleasant encounter – my brain has created a connection based on a sameness that is not necessarily relevant. If the person in front of me turns out to be supportive and kind, I feel a bit silly. It doesn’t matter too much though, as long as I’m not cold to them in advance based on an assumption that they’re going to be rude. We can’t really help ourselves having intuitive reactions to new people we encounter based on people we’ve met before because we are such unstoppable pattern spotters; the important thing is to be aware we’ve done it and make sure we don’t treat someone badly based on that unreliable intuition.

People are much more complicated than any mathematical object we can ever study, but our practice at studying mathematical objects can teach us lessons about how to deal with the much more complicated world around us. It doesn’t enable us to predict the weather with 100 per cent accuracy, and nor can it tell us how a person we’ve just met will behave. But it can help us think in different and more agile ways, and it can help us be ready with more techniques for thinking that might help us when we encounter new scenarios in life.

Sometimes it can seem that people are holding wildly different positions of disagreement, when there might be some way of showing that our views do take some sort of overall similar shape, just with a very different frame of reference, or rotated, or flipped. People of a certain political viewpoint often think that the people on the other side are just stupid, ill-informed, credulous, brainwashed, incapable of seeing through misinformation, and full of hate. But it’s important to remember that the people on the opposite side are likely to think it right back. If we remember to see ourselves from other people’s point of view, we see that it’s a surprisingly symmetric situation and this should give us pause.

* Linda Geddes, ‘Mathematicians discover music really can be infectious – like a virus’, Guardian, 22 September 2021.

* Unless you have constructed a device deliberately to cancel out the harmonics, like a tuning fork.

* Dschwen, CC BY-SA 3.0, via Wikimedia Commons.

5 WHY THE EARTH COULD BE A BAGEL

Our journey so far has taken us through numbers to functions. We observed that when we try to encapsulate some aspect of the world in a single number we hide vast quantities of information. So then, when we think about equality between those numbers, there can be vast differences hiding behind the sameness. When we moved on to functions we saw some ways of keeping a broader view of a situation rather than reducing it to a single number. We saw that sameness of functions can be viewed at individual points or in terms of overall shapes. This is a difference we might call ‘local versus global’, where looking at individual points is very local, whereas looking at overall shapes gives us a global view.

In this chapter we’re going to explore the interaction between local and global viewpoints. We will see that, locally, the Earth looks flat, and so does a bagel, but globally they look very different. ‘Local’ can mean ‘at a particular point’, but it can also mean ‘in a small region around a point’, whereas global means ‘across the entire world you’re thinking about’. In maths these terms don’t necessarily refer to physical localities, but they refer to closeness according to any criterion, which could be physical distance, or it could be time, size or any other measure. In the previous chapter we saw that the graph of an exponential looks very flat at the beginning, and we could say that, locally, the curve looks like a constant function in that region. If the curve is charting the spread of a virus over time, then ‘locally’ is referring to points close together in time, not space.

We are going to see that thinking about the interaction between local and global points of view is crucial, not just mathematically, but in daily life too. Sometimes disagreements arise when local and global effects are in opposition to each other, such as when there is tension between laws or governments operating at different levels. In the UK there is often tension between the Scottish government (and Scots law) and the overarching powers of the UK government; in the US there is tension between the powers of individual states and the federal government. Different people believe in different levels of locality. Some people only want much more local powers and don’t want to be governed in any way at a broader level by something like the EU or the federal government. Other people think that cooperation and coordination at all levels is beneficial, and that some things are better done locally and other things are better done more globally. It took me some time to understand that in the US some surprising things are done state by state, such as education. The local education policies do not patch together in a coherent way.

Other times in life, just as in maths, the local and global thinking isn’t about geographical distance, but about a more abstract form of distance such as impact and knock-on effects. When thinking locally we only think about direct and immediate consequences, whereas when thinking ‘globally’ – in the mathematical sense – we think about longer-term consequences, and more indirect consequences. Perhaps there are direct consequences that cause some other effects that then cause some other consequences, and so on. Local thinking can also consist of thinking about individuals, whereas global thinking involves thinking about how individuals come together to form a community, and therefore how the behaviour of individuals changes a community; it’s also about how the way we treat individuals has an effect on the whole community.

Maths has many ways of building up larger shapes from smaller ones, and is very concerned with then understanding how to transfer knowledge from the smaller areas to the bigger ones. One of the general principles of maths is to understand something complicated by understanding simple versions first, particularly via basic building blocks that can be used to build the more complicated structures. Then we just need to understand the basic building blocks and also the building methods. This can enable us either to build up very intricate structures (like using basic Lego bricks to build things), or to look at a very intricate structure and break it down into simple pieces to understand it.

In maths and in life one of the things we need to be careful about is when different situations can look the same locally but turn out to be very different globally. It is much easier only to look at local effects, just like it’s easier to find your way around your own neighbourhood rather than a whole country, or the world. But that gives us only a limited point of view. Practising our local-to-global thinking in maths is a way to get better at this in a controlled environment. If we get used to thinking about local-to-global issues in a mathematical setting, it helps us to see and understand the issues in the rest of life too.

In the previous chapter we did see some situations in which we could use partial information to extrapolate to global understanding when we filled in gaps in functions. We’ll now look at some more situations in which local information does give us global information, before we move on to see many scenarios in which this doesn’t work so well.

When local does imply global

Sometimes in maths it is possible to ‘fill in the gaps’ without having to know all the information in advance. This means that partial information can be as good as full information, and in that sense might be ‘the same’. However, it usually only happens when a very large amount of information is already available, so the part to be filled in is small. We saw this idea when we talked about ‘fillable gaps’ in functions, when almost all the function is known and there’s just a tiny gap with only one ‘sensible’ way of filling it in. In order for this to be unambiguous, we do have to say what ‘sensible’ means. In the previous chapter we saw that the function x sin (1x) looks like the diagram below, with an imperceptible gap at x = 0 where the output can’t be defined.

In this case there’s only one way to complete the function that will make it continuous, and that’s by defining f (0) = 0. For functions, ‘continuous’ has a rigorous formal definition, but for our current purposes it’s fine to think of it as just meaning that the function doesn’t jump around. If we defined f (0) = 1, for example, the function would have to jump up to get to 1. There are infinite ways to ‘complete’ this function, but there’s a unique way to complete it to be continuous.

This is a bit like seeing a configuration on a chessboard where only one outcome is possible: you don’t need to see the rest of the game to know who is destined to win. And perhaps in that case expert players won’t even bother playing the rest of the game because they know what’s going to happen, and the person who is certain to lose will simply resign at that point. This is because there is a unique way to complete the game according to the rules of chess. If you didn’t know those rules, then you wouldn’t know how to complete the game, or you might think there could still be different ways of producing different possible outcomes. (There might still be different ways to play out the end of the game, but if they all end with the same winner, then it doesn’t matter.)

Sometimes the concept of completing something uniquely can just be about a pattern. For example, a ‘Latin square’ is a pattern involving symbols in a square grid, organised in a particular way. Say it’s a 3-by-3 grid, then you have three symbols and you have to put them into the grid so that there is exactly one of each symbol in each row and column. For example, this is a Latin square:

1

2

3

2

3

1

3

1

2

but this isn’t:

1

2

3

2

3

1

2

1

3

It’s a bit like a very small Sudoku, and we saw some 4-by-4 examples in Chapter 1. Now if I fill in just part of the grid and tell you it’s a Latin square, then there might be only one way to fill it in, provided I’ve given you enough to start with. For example, I feel confident you could fill this one in:

a

b

c

b

c

a

c

a

The following, much emptier one still has only one way to be filled in, it just takes a little more thought:

a

b

c

The whole idea of Sudoku is that the puzzle gives you just a few squares filled in, and the aim is to work out how to fill the rest in according to the rules. A decent Sudoku puzzle is one where enough is given to start with that there is only one way to complete it. The difficulty then depends on how much thought is required to pin down the solution, just like the first one above required very little thought, whereas the second one required a bit more.

In many branches of mathematics, when something has ‘only one way the gaps can be filled in’ to satisfy certain constraints, that is called ‘density’. One example is the rational numbers inside the real numbers. The rational numbers are so close together inside the real numbers that if you’re going to fill in the gaps in between them (in a certain rigorous sense), there’s really only one way to do it, a bit like with the Latin square above. So the rational numbers are called ‘dense’ inside the real numbers. For the Latin squares it did matter what symbols you put in the various positions because the entire definition of a Latin square is about the symbols. But with the rational numbers and the real numbers it’s not about the symbols you’re using to fill in the gaps, it’s about the abstract structure. So, for example, it doesn’t matter if you refer to the square root of 2 as 2, or q, or z, or elephant, as long as you say that its defining feature is its property that when you square it the answer is 2.

Density is about when you can have a smaller amount of information and reconstruct the rest uniquely, so that having less information is ‘the same’ as having all of it. This can make a situation more efficient, or it can make our understanding of it more efficient so that we don’t have to carry around so much information in our heads.

However, as the word suggests, ‘density’ usually only happens when you have a large proportion of the information already. Often we are thinking locally because it’s very hard to see more than a local area at once, and we can only look at a very small area relative to the whole. In those situations we are unlikely to have the benefits of density, so we need to think harder about the relationship between the local structure and the global structure. Sometimes both are important, differently.

Individual pieces of data vs a whole set

When we’re looking at trends, we’re trying to see patterns and overall shapes, like when we looked at the overall shapes of functions. But sometimes the individual pieces of data are important too, and not just the trends, especially when the individual pieces of data represent humans with real lives and experiences.

For example, here’s some hypothetical data on the gender pay gap. It’s hypothetical because it’s hard to access real data – in many countries companies over a certain size are now required to publish data about the gender pay gap, but they’re only required to publish a few calculated metrics, such as means and medians, not individual salaries. The trouble is that a large amount of the story can be hidden behind a mean or indeed a median. I’ve constructed the following example to make a situation in which a company could argue that they have no gender pay gap, because the mean earnings for men and for women are the same:

women

men

500,000

200,000

400,000

200,000

30,000

200,000

30,000

150,000

30,000

100,000

30,000

100,000

30,000

100,000

mean

150,000

150,000

However, we see that the highest earners are women, who are earning much more than the highest-earning men. The mean is severely skewed by those two very high-earning women, and all the other women are doing much worse. If we try to describe this pay structure or analyse it using only one data point our answer will be very incomplete because that’s just not enough information to describe even a slightly complicated situation. It would be better to use two numbers (say, the median as well as the mean) but there could still be plenty of nuance obscured in the process of reducing a large amount of data to two numbers, or indeed to any small collection of numbers, when we’re starting with a large set of data. If we’re interested in describing more complicated situations it becomes important to be able to describe and compare overall shapes of things rather than just individual points. In other words, the global picture is far more useful than the local. However, the local picture is much easier to understand. Sometimes in mathematics and in life we benefit from understanding the local picture first, partly because it’s nearer, and partly because it’s simpler. We then need to understand how those local parts patch together. Crucially, the same local parts can often patch together to make very different wholes. A key example of this comes from the mathematical field of topology, and the study of some shapes of spaces that are built by patching together small flat areas. They are called manifolds.

Manifolds

Topology is the mathematical field studying the general shapes of things, not in terms of distance or geometry, but more in terms of connectedness. Topology treats everything as if it is rubbery and bendy, as well as squashable and expandable, and later we’ll look at the notion of sameness used there.

In topology, a manifold is a type of shape that is built up from the most basic possible building blocks: little flat areas. It’s a bit like building something up from papier-mâché, where you tear up little pieces of paper and stick them all down in layers to form a shape or sculpture. Each piece is a flat little piece of paper, and they all need to overlap with each other so that there are no gaps, but you can make a really very complicated shape with those flat pieces of paper. The important point here is that we start with the ‘same’ building blocks – little flat shapes – but we can still build wildly different global structures. This will drive home the fact that things can look the same locally but very different globally, so if we only think locally we can be led far astray.

Manifolds can be made in any dimension – if we wanted to do something higher-dimensional, then instead of sticking together little two-dimensional pieces, we would need to stick together little three-dimensional pieces. That’s a bit harder to imagine and so it’s better done with the technical formalism of maths; here we’ll stick to thinking about the two-dimensional version which is easier to picture, although we’ll soon see that even in the two-dimensional case there are shapes that are very hard to imagine.

The little portions of flat pieces we’ll stick together are modelled using a coordinate system, so that each one is like a little piece of a graph, like this:

This means that each little piece is no harder to understand than a plane with x and y coordinates, sometimes called a ‘Cartesian plane’ after René Descartes, who published the idea in 1637. The point of the coordinate system is that it’s a simple way to organise and think about points in space, as long as your space is a flat plane – which is a very simple shape in the grand scheme of things.

When we stick these pieces together we need to do it roughly coherently, just like when you make something out of papier-mâché you might try to make sure you lay down each piece of paper smoothly without wrinkles or bubbles. You probably don’t screw a piece up into a ball and smoosh it onto the surface. In the maths of manifolds that means that wherever two small patches overlap, we ask for a sensible way to translate from the coordinate system of one to the coordinate system of the other.

I feel nostalgic for the old UK Ordnance Survey maps that I learnt to use while doing Duke of Edinburgh Award expeditions at school. While I now very much appreciate using GPS on my phone, using a compass and a paper map to find our way around the countryside was a more satisfying mental exercise. Each Ordnance Survey map was a quite detailed map of a portion of the countryside. If you wanted to do a longer expedition across a larger portion of countryside, you would need to piece it together across several Ordnance Survey maps. You’d be fine when you were entirely inside one of them, but when moving from one to another you’d better hope, first of all, that there’s no gap between them, and preferably a bit of overlap so that you can be sure where you’re going when switching over between the maps. Secondly, when switching over you’d hope there would be some sort of key to help you orient yourself on the new map relative to the old one. You might walk off the edge of one map in square A6 and walk into the edge of the other map in square P13 but the scales might be different. This is all fine provided you know how to do the translation.

With Ordnance Survey maps we’re just talking about one country, so if we patch all the maps together we’re still aiming for something more or less flat, which keeps things relatively straightforward. But we could, in theory, patch together maps of the entire world and make a sphere, as long as the maps were made of something slightly stretchy so that they could deal with the curvature. Or else we could consider that the maps are small enough relative to the size of the globe that we’ll be able to make a curved surface out of flat pieces anyway, in the same sort of way that clothes are curved even though they’re made from flat pieces of material sewn together. The skill of draping is an amazing art, where flat pieces of material are joined together to make beautifully shaped garments. This is essentially the same as the mathematical construction of a manifold.

The overall point I’m getting to is that there are many different shapes you can build from small flat pieces of material. All of these shapes look the same locally, because if you zoom into a small area around any point everything just looks flat. But the shapes look very different globally, because of the way they’ve been patched together. Let’s explore the sorts of things we can make with a mathematical patching process.

Mathematical patchwork

We are gradually going to build up an extraordinary and unimaginable range of shapes using the idea of mathematical patchwork, showing that simple flat shapes can make vastly different global spaces depending on how we glue them together. This means we shouldn’t make assumptions about what something looks like in total just by looking at small areas individually, even if we look at the small area around every single place in the space. The same local pieces can build up into very different global shapes.

The most basic examples of things we can make from little flat patches are things we really can make using pieces of paper. Here’s something we can make with just one strip of paper – we can join the ends together to make a cylinder:

But here’s something different we can do – we can join the ends together with a twist, and this makes a Möbius strip:

A Möbius strip is a famous and fascinating shape that has an interesting interaction between its local and global structure. Locally it just looks like a flat piece of paper, but globally it is rather curious. One curious aspect of it is that it only has one side – if you colour it in you can colour in both ‘sides’ without ever taking your pen off the page, which shows that there is in fact only one side. There was originally a front and back to the paper, but when we made the twist the front met up with the back and so they became the same surface.

One difference between the cylinder and the Möbius strip is to do with how we can describe them using coordinates. Both of the shapes are made using a single strip of paper, so we could have put a coordinate system on that piece of paper before we started. Perhaps like this:

Or we could give it compass directions:

Now, when we make the cylinder, the compass directions still make sense, which is how we can have compass directions on the Earth as well. You can put the compass directions at the front, and keep travelling around in, say, the east direction, and when you get back to where you started, there will be no problems with which way up north, south, east and west are.

However, if we make the strip of paper into a Möbius strip, something weird happens to our compass directions: north and south are no longer definable. Supposing you pick north to face ‘up’. Then it is going to flip over as you head into the twist, and when you get back around to where you started, north will now be pointing the other way:

In maths this concept is called ‘non-orientability’ (although admittedly we’ve only seen the vague idea behind the very technical definition, which is more about clockwise and anti-clockwise than compass directions). The cylinder is called ‘orientable’: we can put a globally consistent coordinate system on it. The Möbius strip is ‘non-orientable’ because although we can put a local coordinate system on it, we can’t put a global coordinate system on it: the local structure was twisted when we stuck it together, so it doesn’t transfer to a global structure.

Twisted structures

The Möbius strip can be thought of as a twisted product, which is another way of relating it to a cylinder. I’m using the word ‘product’ here in the mathematical sense: not just something that has been produced, but something that has been produced by multiplying two things together.

A cylinder is a straightforward product: it can be constructed in topology by ‘multiplying’ a circle by a straight line. One way to see this is to think about what a more straightforward product is. The ordinary Cartesian plane with x coordinates and y coordinates is a product of two real lines. The idea is that every point on the real line is just a real number (that is, an ‘ordinary’ number, rational or irrational).* When you take the product, you take the entire set of pairs (x,y), where x and y are each real numbers. These can be realised as Cartesian coordinates, giving us the whole two-dimensional plane.

You might wonder why this counts as a multiplication. One way to think of it informally is that it’s a bit like multiplying, say, 2 and 3 by making a grid that is 2 blocks high and 3 blocks across. It’s just that now we’re doing it for the entire real numbers in both directions. Later we’ll see that if we think about multiplication of numbers via sets and bijections, we construct it by taking pairs like coordinates on a Cartesian plane.

We could do this sort of multiplication for just finite line segments instead of infinite lines, say from 0 to 20 in the x direction, and from 0 to 10 in the y direction, and we would get the rectangle below, where each point has an x coordinate between 0 and 20 giving its distance along the rectangle, and a y coordinate from 0 to 10 giving its height from the bottom.

Now imagine that instead of a line and a line, we have a line and a circle. For the circle, we fix a radius, and then every point on it can be specified by an angle, say 0 to 360, where we understand that 0 and 360 describe the same point:

For the line, perhaps we keep the line segment from 0 to 10. Then the product of the line and the circle still has points described by coordinates (x,y) but this time x is a point on the circle, and y is a point from 0 to 10. So each point has an angle around the circle, and a height above the circle, and in total we have a cylinder.

Now, we can’t do that for a Möbius strip, because of the twist; this is related to what we described before about it being a ‘non-orientable’ surface: we can’t give it a coordinate system. You might want to imagine what happens if you try: the circle part is fine, because every point still has an angle around the circle. But if you try to specify a height above the circle, you get into trouble, because there isn’t a consistent way to decide which way round 0 and 10 are, because of the twist.

There are other situations that can have twisted products as well. One of my favourite examples, which I’ve written and spoken about many times, is the Battenberg cake pattern, and then the iterated Battenberg cake pattern, where you make a Battenberg cake each of whose small cakes was already a Battenberg:

The iterated version is a non-twisted product, in a particular context that we’ll come back to, called group theory. But we could also do a twisted product, where we make an iterated Battenberg but we ‘twist’ the cakes in the iteration. In this case it means that instead of just stacking the two types of small Battenberg the same way up in all positions, you stack them one way up in the top layer, but turn them a different way up in the bottom layer. Here’s a picture of the two versions:

In both cases, the individual Battenbergs we’re using are these two:

In the non-twisted version, the top row of these 2-by-2 Battenbergs and the bottom row are both this way up. But in the twisted version, the cakes in the bottom row are rotated, so they look like this instead:

The fact that the cakes in the bottom row are rotated relative to the ones in the top row is what I’m calling a ‘twist’.

The twist in a Möbius strip means that the up and down directions switch over as we move around the strip. The twist in the different iterated Battenberg cakes means that the cake patterns face in different directions in different parts of the cake. This sort of confusion in directions happens in life as well, metaphorically, and understanding twisted mathematical structures helps me understand twisted life situations.

Twisted structures in life

Structures can get twisted in life where social pressures operate in different directions in different contexts, just like up can be pointing ‘up’ on the Möbius strip, but if you go round the loop you find that ‘up’ is now pointing down. For example, as a woman, I feel pressure from society to dress a certain way when I’m doing something formal like giving a presentation or appearing in public. This usually means wearing a dress, heels, make-up, and having my hair done nicely. However, in the context of the world of maths I have often felt pressure not to dress in a way that could be considered too ‘feminine’ for fear of people not taking me seriously as a woman in mathematics. This means that in maths situations I’ve often felt pressure to absolutely not wear a dress or skirt, or heels, or make-up. It’s as if my life is on a Möbius strip, and I can’t tell which way the pressures are pushing me because it depends where on the Möbius strip I’m considering myself to be.

It is confusing if I want to be a role model for standing against social pressures, because the pressures are pushing me in opposite directions, and so standing against them involves doing two opposite things: it is a (mild) act of resistance to wear trousers and flat shoes as a woman at a formal event in broader society, but it is also a (mild) act of resistance to wear a dress and heels as a woman in mathematics.

Sometimes it is the case that the same social pressures are pushing different types of people in opposite directions, rather than pushing one person in opposite directions in different contexts. For example, there has long been pressure on straight people (especially women) to get married, but gay people have been banned from getting married (that is, they have been banned from marrying the people they want, and still are in some countries). This means that liberation for gay people might well mean finally being allowed to get married, and doing so, whereas liberation for straight people might mean living together as a couple without going through the rigmarole of getting married.

Here it’s not an individual’s life that is like a Möbius strip, but society as a whole, and ‘oppression’ and ‘liberation’ are like the north and south compass directions that we can’t define globally, only locally.

Twists are only one way in which local areas can be pasted together to make different global structures. There are many types of manifold, all made from sticking together the very same flat pieces in different ways. If we don’t take the glueing schemes into account, we’ll erroneously assume that the whole structure looks like the parts. This might happen by accident, by ignorance or by wilful stubbornness, as in the case of flat earthers.

Locally the same but globally very different

I have sympathy for the humans of hundreds of years ago who thought that the Earth was flat. All they could see was the more-or-less flat area around them, and the maths of manifolds had not yet been developed, let alone spread into general consciousness (which still hasn’t happened). So they could not really be expected to know about the huge range of shapes a space can take based on flat local areas.

A sphere is a straightforward example of a manifold. In fact it’s the simplest example of a manifold with no edges, technically called boundaries. This is a more ‘satisfying’ type of surface, because every point can feel like it is in the middle of a flat space, whereas if there were an edge, the points on the edge would feel different from the ones not on the edge.

The fact that a sphere is curved doesn’t matter – we could still imagine making a sphere out of papier-mâché. We can also make ‘holes’ in it without introducing any edges. This might sound strange until you think of the surface of a doughnut – a ring doughnut, not a jam doughnut. Or perhaps a bagel.

This shape of surface is called a torus. A bagel might be thought of as having a hole in it, but its torus-shaped surface doesn’t, exactly. Or does it? Maths uses a different word for this kind of ‘hole’ – to be more precise about it, we call it ‘genus’. You can (perhaps) imagine making the surface of a doughnut shape by taking a bendy tube, like a hosepipe, and joining the ends together. This is how bagels are traditionally formed, but we’re only thinking about the surface here, not the solid dough. This surface starts as a tube with a circular edge at each end, but after we join the ends together, there is no longer an edge – we’ve joined the two circular edges together, so it has no holes in the sense of punctures through the surface. But now it has this other kind of ‘hole’. It’s a hole if you think of the doughnut as a solid piece of dough with a hole punctured through it (which is how ring doughnuts are made), but it’s not a puncture if we’re just thinking about the surface. Mathematicians often describe constructing this surface from a flat square with the edges glued together using a diagram like this:

The A’s and B’s indicate which edges are glued together, and the arrows indicate the direction, to make sure we glue the edges with the correct orientation. If we stick the edges labelled A to each other with the arrows facing the same way, we make a tube or cylinder. The edges marked B become circular boundaries, one at each end of the cylinder, and we then stick them together to make a torus. We could do it by sticking together the edges marked B first instead, and then the edges marked A afterwards; it would still make a torus.

This depiction might be less intuitive than the doughnut-like picture, but it is more precise, especially when we go on to make shapes that are very hard to picture. This often happens in maths: we come up with a way of expressing something that is a bit over the top in the basic examples, but the point is that it can then help us get beyond where our basic intuition can take us.

The torus, or surface of a doughnut, has one of the type of ‘hole’ in question, so mathematicians say its genus is one. This makes it different from a sphere, whose genus is zero. I was once at a restaurant in Manhattan which offered me various bread rolls with my dinner, one of which they called a ‘bagel’ despite the fact that it had no hole and was a sort of square pillow shape. I was there with a mathematician, and we both politely chose a different type of bread, and then whispered to each other ‘the bagel had the wrong homotopy type’.

We can make another version of a torus but with two ‘holes’, or rather, with genus two.

This can also be built from a flat shape with the edges glued together, but now we need more edges, and also more stretchy powers of imagination. The construction can be done with the scheme shown below, and you might find it interesting to try and convince yourself that this works:

One way to see that it works is to imagine cutting it in half along this dotted line:

Then perhaps you can see that each half makes a torus with an extra open edge:

If we then stick the two halves together along their respective open edges, we get a genus two ‘double torus’.

Another way to think about it is that we cut a hole out of each torus – an actual hole, with an edge – and then join those holes together. Whichever way we construct it, the result is still a shape that looks flat in any small local area, but is very definitely not flat globally. If we just observed the mostly flat part of the Earth around us, we could validly imagine that we’re living on a genus two torus.

We can also keep going with the doughnut modification: we can cut a hole in the genus two surface, and stick another torus-with-a-hole onto it, and that will make a genus three ‘triple torus’. We can make it from a flat shape like this:

Perhaps you can now see that we could repeat this process as many times as we want to get a genus n torus for any whole number n. All of these surfaces are orientable. It’s fairly easy to see this with a torus, just by imagining putting some coordinates on its surface. We can make a sort of east–west direction going around the hole, and a sort of north–south direction going through the hole:

This will then work more or less the same way on any torus of any genus. The ‘north’ will end up pointing down when you’ve gone through the hole, but if you go all the way through and come back to where you started, there will be a consistent direction to the ‘north’. As we’ve seen, this is not true in non-orientable situations. The doughnuts with extra holes are all orientable, and they’re all built from the same method of cutting a circle out of the surface and sticking on a torus-with-a-hole-cut-out. Incidentally, the ‘torus with a hole cut out’ is sometimes called a ‘handle’ in topology, because it looks a bit like a handle for a jug or something:

The handle itself isn’t a manifold because it has an edge (technically called a boundary), but the boundary is just a circle, so we can stick it onto any other surface with one circular boundary, and then we’ll have a manifold.

We can use a similar method to build up more and more complicated non-orientable surfaces too, except that now our basic building block is no longer a handle, but – wait for it – a Möbius strip.

Mind-boggling non-orientable surfaces

A Möbius strip is a more mind-boggling sort of building block, but boggling our minds is an important part of doing maths. Abstract maths, in particular, is supposed to be mind-boggling; the idea is that we stretch, confound and contort our brains, and that’s how we reach deeper understanding and make our brains fitter, stronger and more agile.

A Möbius strip is not a manifold because a manifold has to have no edges. But it can be used to build manifolds, in a similar way to how we built up all the genus n toruses by cutting a hole and sticking on a torus-with-a-hole-cut-out.

Now here’s the mind-boggling part, which I find rather fun: a Möbius strip also has one circular boundary. If you find this confusing I encourage you to get yourself a strip of paper, make a Möbius strip, and follow its boundary around to confirm that it’s just a circle, although it does loop back on itself. Perhaps you are used to thinking of a circle geometrically, as something with a fixed radius, in which case the boundary of a Möbius strip will seem more like a folded-over figure 8 than a circle. But in topology, geometry and distances don’t matter. When we say ‘circle’ what we really mean is ‘continuous loop’ or ‘line with the ends joined together’. All of these shapes count as circles:

Later we’ll go into more detail about which things count as ‘the same shape’ in topology, but for now it suffices to imagine a topological circle as being like a hairband. No matter how you loop it up on itself it’s still the same hairband – as long as you don’t cut it open. So even if you make a ponytail by looping the hairband on itself twice, you haven’t fundamentally changed the shape of the hairband – or rather, you haven’t changed its topology, even though you have changed its geometry.

If you loop a hairband over itself in a double loop to make a ponytail, that’s the shape of the edge of a Möbius strip.

Now, here’s the point of thinking of the boundary of a Möbius strip as a circle: we can stick it onto anything else whose boundary is a circle, just as we did with the toruses and the handles. It’s weirder, because the surface will have to ‘go through’ itself, because the Möbius strip boundary loops back on itself. That is to say, we have to invoke a higher number of dimensions for the resulting shape to live in. We’ve moved out of the realm of shapes we can make out of paper.

The idea here is that although we live in three-dimensional physical space, in maths we imagine higher-dimensional space by generalising the concept of a dimension. The third dimension enables us to escape two-dimensional situations, and the fourth dimension enables us to escape three-dimensional situations. For example, if you’re stuck in a field with a fence going round it, you’re only trapped in two dimensions. You can escape as long as you can invoke the third dimension to move vertically and climb over the fence. You only have to invoke the third dimension briefly, to climb over, and then you can come back to the ground again.

Now let’s try doing that up a dimension. If you’re trapped inside a sealed room, then you’re really caught three-dimensionally. But you could invoke a fourth dimension to escape, and just like the fence, you only have to go into the fourth dimension briefly. But what’s the fourth dimension?

Stephen Hawking brought into public consciousness the idea that we can treat time as a fourth dimension, by describing Einstein’s idea of four-dimensional space-time. The fourth dimension doesn’t have to be time, however, if you have a good enough imagination – it can be a fourth spatial dimension in your imagination. But let’s suppose it is time, as that might be the easiest fourth dimension to imagine. Then to escape a room all you have to do is travel in time briefly, perhaps to yesterday when the door to the room was open. Then you can walk out of the room and time travel back into today.

The imagery is a bit fanciful, but the idea is that invoking another dimension enables you to get through barriers in the lower dimension. So if we want to glue a circular piece of paper to the boundary of a Möbius strip, we need to loop the circular piece of paper back on itself as well. It won’t be possible in three dimensions, but we can use a fourth dimension to get the paper to ‘go through’ itself.

This is very difficult to imagine, but we can draw a scheme for building it in a similar manner to the way we drew schemes for building toruses earlier. Whereas for the toruses we were making a scheme for constructing something we could already imagine, now we’re doing something different. We’re going to use this kind of diagram to define spaces, and then we will try and work out what they look like afterwards. The shapes we can define in this way that don’t fit into three-dimensional space are very hard to imagine. In maths, when we’re patching small flat pieces together, we don’t have to do this with physical pieces of paper or fabric, so we can do things that work in theory even though they wouldn’t work in real three-dimensional space.

We might start out having no idea about these shapes, but when we work with them for a while we can start to develop an intuition for them, just like how we develop any kind of intuition. I now do have a certain ability to imagine these shapes in my head, in some imaginary version of higher-dimensional space. Maths turns out to be quite a lot about using our imaginations to create mental imagery for things that don’t exist in the real world, to help us make arguments about them. It’s beautiful and creative, a far cry from repetitive exercises and rule-following.

First of all, here’s a scheme for building a Möbius strip:

The edges marked A are stuck together, and the arrows show the orientation, which is what gives it the twist. Now, the edges marked B and C become the circular boundary. If you look closely you can see that if you travel along B in the direction marked with the arrow, the end of that edge (at the corner) must join up with the beginning of edge C, and then the end of edge C must join up with the beginning of edge B. Our aim was to glue the edge of a Möbius strip to a circular piece of paper, and so far we’ve made a Möbius strip whose circular boundary is labelled with a B followed by a C. So we can glue that to the boundary of this circular piece of paper:

This completes the construction, although it might not get us very far towards understanding what this space looks like.

Here is another, more economical way to construct this shape – more economical, but perhaps even harder to understand (as is often the case with economical descriptions). We just start with a circular shape, and glue the edge to itself according to the following scheme:

I’ve called the edges D here because they’re not the same as the A, B or C from the previous construction. (In fact, the first half of D is the old edge A, and the second half is the old edge B.)

The idea here is to imagine that the Möbius strip gets thinner and thinner – so thin that it disappears. Then the B and C edges will land on top of each other. It’s pretty difficult to see why that should count as the same shape as the previous one we made, but we’ll come back to that when we go into a bit more detail with topology. The space made by this glueing scheme is called a ‘projective plane’. There are plenty of other hard-to-visualise shapes that we can make by just drawing flat pieces of paper and schemes for gluing their edges together. Many examples are understandably difficult to imagine because they are impossible to make in three dimensions. Here’s another one:

Now that we know we can stick a Möbius strip onto any circular boundary, we can build a new manifold from any old one by cutting a circle out and sticking on a Möbius strip. It’s hard to say what these spaces look like overall, but the main point I want to stress for now is that all of these increasingly wild surfaces look the same close up: they are locally the same, although they are globally very different. I think it’s therefore understandable that people used to think the Earth was flat – ‘locally’ it is flat, and back in those days nobody could see any part beyond the local part, although some insightful and radical thinkers found ways to reason about the possibility of a curved surface without being able to see it. This is an important part of how knowledge progresses, with our ability to imagine something we haven’t seen or experienced. If we were only able to imagine things we had experienced ourselves, our lives would stay in the same, local realm all the time. This points to the importance of learning things by education instead of just by experience, so that we can open our minds to be able to imagine and even understand things we haven’t experienced personally.

Likewise, the point of studying manifolds isn’t that we do calculations about manifolds in our daily lives, but that it gives us a way to practise a particular way of thinking, and that way of thinking can then help us understand some issues in our daily lives.

Local-to-global issues in life

The importance of local-to-global understanding extends crucially to our daily lives. Sometimes we think too globally and not locally enough, and sometimes we think too locally and not globally enough. Really, we should think at all scales, and understand how the local parts patch together into the global picture. Once we’ve practised this in the realm of maths, it becomes much easier and more immediate to do it in life too. In life, the local-to-global expansion that we’re thinking of could be about various things. It could be about time, where there are short-term and long-term benefits; in fact I find it helpful to think about medium term, and crucially, ‘micro-term’, which is really about instant gratification. Far too often, instant gratification in a very local time range patches together to make bad medium- to long-term consequences, like when I stuff my face with cake to drown out some painful feelings. It helps me to understand that it does help in the micro-term, but makes things worse in the short term (I feel sick) and also the long term (it’s bad for my health). It probably doesn’t matter that much in the medium term.

Sometimes in life, local-to-global thinking really is about geography. The reversing of the flow of the Chicago River is an example of this. Chicago is situated on the banks of Lake Michigan, a bounteous source of fresh water. The Chicago River flows into the lake, and the location of Chicago is particularly good, not because of the size of its eponymous river, but because of the proximity to the Des Plaines River to the west. There is an old portage that had been used by Native Americans in the area for generations – that is, they knew there was some raised ground between the two rivers that made for a relatively easy overground journey carrying supplies. This meant that the Des Plaines River could effectively connect to the whole Great Lakes system. This was important because the Des Plaines River itself flows into the Mississippi, meaning that the portage connects the great Mississippi river basin with the Great Lakes.

European settlers, not content with benefiting from this natural feature, decided to turn against nature instead of going with its flow – rather literally. As the city grew, they wanted to send their sewage away from the city and not back into Lake Michigan, which was the source of their drinking water. They also wanted a more convenient connection between the two water basins, for increased trade that was becoming too much to carry on foot.

So, in what has been called a ‘great feat of engineering’, they reversed the flow of the Chicago River, essentially by building a canal to join it with the Des Plaines River. The following two maps* show the system of waterways before and after the diversion, with the arrows showing the direction of flow of water. In fact, there are two canals, the Chicago Sanitary and Ship Canal (CSSC), which diverts the Chicago River, and the Calumet-Saganashkee (Cal-Sag) Channel, which diverts the Calumet River, which is to the south of the city of Chicago.

A system of locks now forces the Chicago River to flow away from the lake, down the CSSC canal across the old portage, and into the Des Plaines River. This means that Chicago’s sewage does not pollute its drinking water, which is wonderful for Chicago; however, those engineers did not really stop to think about where the sewage was going instead. Or they did think about it, and they didn’t care that it would thus flow down into the Mississippi.

In further unintended consequences, some invasive carp species have been making their way up the Mississippi River system towards the Great Lakes. They were themselves introduced by humans (trying to interfere with nature again) to feed on and thereby control algae and parasites in Arkansas, for aquaculture and sewage treatment plants. But there is now a danger that they will make it through the canal and into the Great Lakes, where it is possible they will wipe out a large proportion of the local species because they feed and reproduce so aggressively and successfully.

That is an example of some quite literal local thinking that did not extend well to a global scale. Perhaps if people had thought about the broader impact, they wouldn’t have taken local action so boldly.

On the other hand, there are times when thinking about global impact can help us take more action locally. Sometimes I feel daunted trying to make a difference in the world, or working out how to do any good, when there just seems to be so much disaster and tragedy around us. With natural disasters, political disasters, wars, famines, illness, poverty, abuse, and more, there seems to be a disconnect between the local and the global, and I sometimes despair at the futility of whatever I’m doing in my tiny patch of the Earth. But I try to remember that if we all make even a small difference in a small local region around us, then those small regions will patch together and make a difference across the world – we can apply manifold- like thinking to world issues.

We can also think about global effects to help forgive ourselves for some of our local behaviour. Thinking locally, there might be various things we don’t do completely perfectly in our lives. We don’t give to every possible charity. We don’t open our home to everyone who has no roof over their head. We don’t recycle every possible object. We sometimes travel by private car, or fly somewhere although we know it’s contributing to climate change. Some of us increase our carbon footprint by having children, others by eating meat. We can drive ourselves to despair thinking too locally about our individual behaviours, and, worse, some people criticise others for individual localised behaviours, when it would be much fairer and more productive to think about the whole. I might not do every single thing perfectly, but I aim to do good in the world overall, when the sum total of my life is balanced out. It’s not that I think we are judged by a supreme being at the end of our lives, but I would like to look back on my life and think I did some good overall, even if some moments, locally, were less great than others. For this it’s very helpful and nourishing to think more about the global effects than we always do.

Sometimes local-to-global thinking in life isn’t about our impact on a small or a large scale, but rather, it’s about how individuals come together to make a community, and thus how the way we treat individuals has an effect on the community at large. Sometimes, like with non-orientable manifolds, we can patch together small, sensible-looking local behaviours, and end up with something that looks contradictory globally, because the metaphorical compass directions get flippedupside down.

I often think about this when teaching, because I am not just trying to help individuals but to build a community that is then conducive for a certain kind of learning. When I first started teaching classes with more discussion I let the students contribute freely and spontaneously, but I soon found that this resulted in conversations that were always dominated by a few louder voices. I moved to picking students’ names out of a hat at random to ask them to contribute, with the promise that whatever they said would be valued. This made for more equity, and it did enable more students to contribute because it removed the pressure to come up with something ‘good’ before speaking: they no longer had to decide in advance whether they had something good to say, because I would call on them and they would just say it anyway. However, it was very stressful for some students, no matter how much I assured them that I would value whatever they said, so I moved away from that again.

Something similar is true for the system of standardised testing in maths. It may well benefit some individuals, and even some marginalised individuals, who are able to do well in those tests and thereby access better educational opportunities. But I think it is bad for the education system overall. We should give everyone better educational opportunities, not maintain flawed systems on the grounds that they give opportunities to some marginalised people.

Sometimes we end up in bitter disputes in life because people on one side of the argument are thinking locally, and people on the other side are thinking globally. It may look like the global thinkers are ignoring the local issues, when really they’re weighing up the global issues as being more important than the local ones. It’s all part of the same set of issues, but with different scales and perspectives, and it’s the interaction between the different scales that’s important. Mathematically, we could keep arguing about whether the sphere and the torus are different because they’re different overall, or the same because they’re the same locally, but that argument would be completely futile. We should acknowledge that they’re the same locally but different globally, and then think about what matters more to us. This kind of thinking becomes second nature after immersing oneself in the maths of it for a while.

One example in life concerns maternity leave. This is not just about what is fair for individuals, but how it impacts society as a whole. If women aren’t given maternity leave, then it means that women overall are hindered from fully participating in the world, because they might have to choose between having children and having a career. Of course, arguably they still do have to make that choice despite having access to maternity leave, but it’s much worse in the US where many women are not even entitled to maternity leave, or if they take time off to give birth, the company might not be required to hold their job for them. Or, even if the company does hold the job for them, the leave is unpaid.

Some of the most contentious disagreements about abortion rights stem from two wildly different ways of thinking of abortion. On the one side there are those who think abortion is simply murder (or they claim to think that, to back up their argument). On the other side there are those who worry that if women can’t access safe and legal abortions, then they are doomed not to be able to participate fully in society because of the burden that an unwanted pregnancy has on their life and their ability to work.

One of those arguments is local, and the other is global. Those who believe in women’s right to access abortion don’t typically believe that abortion is fine – many of us agree that abortion is horrible and traumatic – but we are taking broader consequences into account and weighing up the various issues. It is valid to weigh the local issues as more important than the global, or vice versa, but we should be aware that this is what we’re doing, rather than just shouting past each other without understanding where the root of our differences really lies.

The issues are complex, but training ourselves to think about them in maths means that we can do it in life without becoming overwhelmed with the complexity and running out of brainpower. Being able to understand more complex situations without having to exhaust our brainpower is one of my favourite aspects of maths. Throughout this book we’re going to see ways we can achieve this for different notions of sameness. Studying mathematical ways of thinking about sameness is like doing a mental workout: it’s not the same as thinking about issues in daily life, but it can get our minds very fit and adept at thought processes that help us in our daily lives. In the next chapter we will dig deeper into how different branches of maths expand the notion of sameness beyond strict equality, to allow and include a little more difference along with the sameness.

* I say ‘ordinary’ numbers here as opposed to ‘complex numbers’, which mix up real numbers with imaginary numbers and produce a two- dimensional space of numbers.

* By United States Geological Survey (USGS) – USGS Fact Sheet 014-99.

6 INTERCHANGEABLE LIDS AND MATHEMATICAL GLUE

I always wash jars out and keep them when I’ve finished the contents. But, despite my best efforts, it can be hard to find a matching lid when I want to use one of these jars. The easiest thing to do would be to keep the jars with the lids screwed on, but I always feel like I need to let the air circulate around them in the drawer.

The most obvious way to find a matching lid is if the jar still has its label on, and the lid is labelled in a matching way. But some lids fit jars that don’t match. I can remember some of the matching pairs (Claussen pickles and Bonne Maman jam come to mind) but to avoid having to remember those I could put new labels on the jars and lids so that Claussen pickles jars and Bonne Maman jam jars now have the same label, and so do their lids. This works fine as long as the compatibility is mutual. It would go wrong if Claussen lids fitted on Bonne Maman jars, but Bonne Maman lids did not fit on Claussen jars. These are the sorts of issues that maths formalises.

The reason mathematicians want to do it formally is so that we can build arguments from it, rather than just vaguely saying ‘I think these things are more or less the same’. When we look more closely at how we make an idea rigorous, we also see some of the things that make it work, and some of the things that might make things messy. We’ve already seen this when we thought about approximation by ‘tolerance interval’ and found that we can build up a long chain of people where any consecutive pair of people are within a couple of years of each other in age, but the people at the ends are generations apart.

Considering things to be ‘the same’ informally can cause strange anomalies, or messy situations, or chaos. It’s harder to do it rigorously, and arguably more tedious, but the pay- off is that we avoid some of the chaos and ambiguity that might otherwise arise later. In this chapter we’ll see one way that maths formalises how we treat things as the same, called quotienting, and one set of conditions that maths has established for when something is a well-behaved form of sameness, called an equivalence relation. This will be an important starting point and model for relaxing our notions of sameness rigorously later.

Sometimes the way we do things intuitively does work well and can point us in the direction of how to do it formally. One example of this is in how we tell the time.

Telling the time

Telling the time is much easier now that digital clocks are the norm. However, there is still some confusion about 24-hour clocks in the US, where the 24-hour clock is not widely used. To convert between the 24-hour clock and the 12-hour clock we need to know that 13 is the same as 1, 14 is the same as 2, and so on, where ‘and so on’ means that if the 24-hour clock time is more than 12, we can subtract 12 from it and get the corresponding 12-hour clock time.

But actually, we do something like this every time it becomes a new day and the time starts again from 0 instead of going up to 25 o’clock, 26 o’clock, and so on. If we want to visualise this, we can imagine we’re taking the infinitely long line of time and wrapping it round the clock over and over itself so that every 12 hours it comes back to the same spot. (It’s possible time doesn’t go on infinitely but that’s a different issue.) At one spot we’ll get 0, 12, 24, 36, 48, and so on. At another spot we’ll get 1, 13, 25, 37, 49, and so on. This almost happens if we wind a tape measure up on itself, except that because a tape measure is a physical object rather than an abstract one the size of the circle keeps increasing with more layers, so the numbers that land back in the same spot will gradually get further and further apart, unlike with the hours on a clock. Even if it might feel like the days get longer and longer as time goes on (especially for teachers, as term progresses), in fact every day is the same length.

There is an exception when some countries of the world change their clocks in the spring and the autumn, which causes weird anomalies. This autumn when it happened it occurred to me that instead of putting the clocks back to get more daylight in work hours, we could just make daytime hours shorter during the winter! I digress.

The way that we tell the time using only 12 or 24 hours instead of infinitely many numbers is something we can then emulate in different mathematical situations. It’s called quotienting, because it’s like dividing, at some very abstract level. For telling the time we are starting with all possible integers, and then we are ‘quotienting’ by the relation that says two different times count as the same time on the clock if they are a multiple of 12 hours apart. We will see that when we quotient by this relation, everything that is related according to that relation ends up counting as ‘the same’.

We can do something similar for constructing spaces by glueing.

Constructing spaces by glueing parts together

Another situation we’ve already seen where we were treating things as the same was in the construction of a Möbius strip, a torus or various other spaces, by glueing the edges of something together (abstractly or physically). If you make a Möbius strip from a piece of paper by twisting it and taping the ends together, it is quite likely that you tape it with a little bit of overlap; the physical world is full of these kinds of quirks, but so is the mathematical world, so it’s a helpful thinking tool. Where the paper overlaps it wouldn’t count as two layers in the final Möbius strip – you’d simply pretend it had melded into one layer.

Even if you don’t make an overlap, then to make a continuous surface you at least need to join the edges together, as we indicated in this diagram:

Joining the edges together means considering that those two edges are now the same, point by point. Similarly, if you’re following a sewing pattern, then when you sew two pieces of fabric together you are connecting those edges so that they become one single seam on the resulting garment.

In this case, we are ‘quotienting’ by a relation that gives us instructions for which points of the space get glued together. There are two equivalent but different ways of doing these quotient constructions. In order to explore them let’s go back to thinking about time.

Quotienting for telling the time

Here’s the idea of what we’re doing, mathematically, when we’re using a 12-hour clock to tell the time. We start with a large number of objects, in this case, all the numbers; we might even include negative numbers, because those could represent times in the past. We then decide which numbers are going to count as ‘the same’. If we’re doing the 12-hour clock, times count as the same if they’re multiples of 12 hours apart. This is a form of relationship between numbers, which we could express like this:

Given any numbers a and b, they represent the same time if their difference is a multiple of 12.

Now we have separated out all the numbers into collections of numbers we want to count as being ‘the same’ time, but what do we do next? This is where we have a choice.

For telling the time, we pick one object from each pile of equivalent numbers, which we are going to use as the label. If we’re doing the 12-hour clock we typically decide we’re going to refer to all hours using the numbers 1 to 12. If we were being a bit hilarious we might decide to refer to them as 25 o’clock all the way to 36 o’clock instead. It wouldn’t make any technical difference, it would just be confusing for humans because it goes against what we’re used to. We could even give them labels that weren’t part of the set, like A, B, C, D, E, F, G, H, I, J, K, L. ‘Let’s meet at half past F’ sounds weird to us because we’re not used to it, but it also wouldn’t make any technical difference. Music notes are often labelled A to G after all, also going round in circles, but in France and many other countries they’re called la, si, do, ré, mi, fa, sol and it makes no material difference unless you’re used to one method and then try to rehearse music with someone used to a different one.

Each collection of numbers (or other labels) that we’re counting as the same is called an equivalence class, because it’s a ‘class’ of things we are considering equivalent. (‘Class’ is just another word for collection, but the idea is to indicate that the things are connected in some way.)* We pick one label for each equivalence class, and it’s called a ‘representative’ of the equivalence class, because each representative is one object chosen to represent a whole load of objects. A similar scenario in daily life is when we set up a representative democracy by dividing a country into constituencies and then pick someone from each constituency to represent all the people in it. In that case, the person really is supposed to be from that constituency, although this is sometimes contentious such as when a celebrity is ‘parachuted’ into a safe seat, or when they fabricate residency by buying a flat but not actually living there.

In any case, when we’re telling the time on the 12-hour clock, the following numbers are all in the same equivalence class: 1, 13, 25, 37, and so on. We usually pick 1 to be the representative of all these times, and so we refer to ‘1 o’clock’.

So much for picking representatives. The other way mathematicians can organise this type of situation, as an alternative to picking a representative, is a bit like putting all the equivalent objects in a bag and carrying the entire bag around. I like to think of this like fun-size bags of Maltesers: there are individual Maltesers, but then if you have a bag of fun-size bags of Maltesers, the small bags can be carried around as individual objects, even though they also contain a collection of objects themselves.

This would be a much stranger thing to do for telling the time, because it would mean that instead of saying ‘It’s 1 o’clock’ we’d say ‘It’s [1, 13, 25, 37, 49, 61, …] o’clock’, as if we needed to make sure everyone understood what number it could be if we hadn’t all agreed to go back to 0 every time we hit 12 o’clock.

Although that would be a very tedious way to tell the time, it’s sometimes more convenient to do things that way in more complicated mathematical situations. The reason is that you might want to remember what all the equivalent objects are, and you might want to keep them at your fingertips so that you can use whichever one is best in a given situation. For telling the time it’s pretty easy to pick representatives that work well under all circumstances, but in other mathematical situations there might be different representatives that work better in different situations. Similarly, we could imagine wanting different people to represent us in government for different purposes: perhaps there’s one person who’s particularly good at listening to the needs of constituents, but there’s another who’s better at arguing in parliament, and another who’s better at negotiating with service providers.

If we work with entire equivalence classes instead of representatives, we need to work with equality a bit differently as well. Instead of saying when objects are equal, we need to know when equivalence classes are equal, which comes down to needing to know when objects are equivalent. So we are working with a chosen notion of equivalence that we care about, in the place of equality. In some mathematical situations this is necessary, as we’ll see later; in others it’s not exactly necessary, but it can give us more insight into what is going on, because we’re maintaining more information about the situation than if we just ‘pretend’ that all the equivalent objects are really the same. One example of this is basic arithmetic, with basic numbers.

Natural numbers via equivalence classes

Let’s think about the basic counting numbers again, that is, the numbers 0, 1, 2, 3, and so on. These are called the ‘natural numbers’. We previously said that they are actually analogies between collections of sets that are in bijection with each other, that is, sets where the objects can be perfectly paired up with each other. The typical way to deal with this is then to pick a representative for each set, and typically in normal basic arithmetic we pick objects ‘0, 1, 2, 3, …’. These aren’t actually sets at all: they’re just labels. So we’re in a situation where we’ve picked a label for the equivalent things we’re talking about, and it’s not actually any one of the things in question.

Another way mathematicians have tried to get at numbers is to pick actual representatives of the equivalence classes in question. This means we need to pick one chosen set with each number of elements to be the representative of all the equivalent sets with that number of objects. It is typically done in the following convoluted manner. I’m including it so that you can marvel at how convoluted it is.

The chosen representative for sets with zero objects is the ‘empty set’ – the set with nothing in it. Mathematicians usually consider that there is only one possible empty set, and it’s written as . But in maths we often write sets using curly brackets and we put the elements of the set in between those brackets. In that case, the empty set is written as a pair of curly brackets with nothing in between them, like this: { }

The chosen representative of all the sets containing one object is the set containing the empty set. That is, it’s a set containing one object, and that object happens to be the empty set. It’s like a fun-size bag of Maltesers containing just one empty bag of Maltesers, which would be a sad situation. We might write it mathematically as this: {  }, or, worse, as { { } }. I’m not sure I should even admit this, because it looks so ridiculous, but also, perhaps I shouldn’t insult it because I do find it quite satisfying. We’re building up all the representatives of the numbers from basically nothing: from the empty set.

Next, the chosen representative for all the sets with two objects is the set containing the chosen representatives for 0 and 1. That is, it’s a set containing two things:

1. The empty set.

2. The set containing the empty set.

We might write this as:

{ ∅, { ∅ } }

or

{ { } , { { } } }

The second way of doing this might be dizzying to look at, but it gets less dizzying when you are familiar with the concepts.

Then, the chosen representative for sets with three objects is the set containing the chosen representatives for 0, 1 and 2. That is, it’s a set containing three things:

1. The empty set.

2. The set containing the empty set.

3. The set containing the above two sets.

We might write this as:

{ ∅, {∅} , { ∅, {∅} } }

or

{ { } , { { } } , { { } , { { } } } }

which is almost completely unreadable. This evidently isn’t a notation designed to be friendly to human readers – it’s trying to be rigorously logical, which is an almost completely different goal. Sometimes I think the goals of ‘intelligibility by humans’ and ‘rigorously logical’ are polar opposites. A rigorous logical proof enables us to be sure that an argument is sound, but even mathematicians often need to convert it into something more intuitive in order to understand it.

By now, you might be able to work out how this continues: once we’ve defined the chosen representative for all the n-element sets, called n, we can define the chosen representative for sets with (n + 1) elements to be the set containing all the representatives from 0 up to n.

This is all very convoluted, but it has several great features to it. One is that it is entirely rigorous, and when it was first proposed in the early twentieth century there had not been any rigorous definition of the natural numbers. It might seem ludicrous that humans had been using numbers for so many millennia before mathematics came up with a rigorous definition of them. This can be seen as a great testimony to humans’ instinctive ability for abstract reasoning: they could do something for thousands of years without really understanding what they were doing. It could also be seen as a sign of the extraordinary disconnect between rigorous logical thinking and basic human intuition. I think it’s both. It’s also a sign of mathematicians’ penchant for critiquing everything that has already been done and wanting to improve on it. The whole thing came about because various mathematicians realised that what had gone before wasn’t completely rigorous. This rigorous definition of natural numbers was part of a drive to put the whole of mathematics on a more rigorous footing, and is part of a framework called Zermelo–Fraenkel set theory, after mathematicians Ernst Zermelo and Abraham Fraenkel.

Another great feature of this strange but rigorous definition of numbers is that it enables us to define infinity very easily. Well, relatively easily. We start by observing that we can always make a new natural number by taking the set of all the preceding natural numbers. We then consider the set of all natural numbers, that is, the set of all the things that can be constructed in this way. This must be a representative of some equivalence class of sets that are in bijection with it, and we declare that to be infinity. Or rather, we declare it to be the first infinity; it turns out there is a hierarchy of bigger and bigger infinities, which is another counterintuitive but rigorous result that came out of this definition of numbers.

We might ask ourselves how we then do arithmetic. I find the answer rather endearing, because it’s basically how we use counting blocks to do arithmetic with small children, just stated rather more abstractly.

Arithmetic via equivalence classes

When we do addition using counting blocks, we take a certain number of blocks, and then we take another number of blocks, gather them together, and see how many there are in this new collection of blocks. The new collection of blocks has been made from the old collections by putting them all together. The formal term for this in maths is the ‘disjoint union’. This means that we make a new set that contains all the elements of the previous sets we were thinking about. It’s a disjoint union rather than a union, because we consider the two sets to be disjoint even if they look like they have some things in common. For example, let’s consider the set {0,1}, the set containing the numbers 0 and 1. (Note that the order doesn’t matter – the set {1,0} counts as the same set.) If we take the disjoint union of the set {0,1} and the set {0,1}, we should get a set with four elements: the 0 and 1 from the first set, and the 0 and 1 from the second set. It’s as if we permanently remember that the 0’s and 1’s came from different starting sets so they’re not actually the same. By contrast, a (non-disjoint) union is like amalgamating two shopping lists where two people in the house might have noticed the same things are needed. Perhaps one person has noticed that we need washing-up liquid, butter and eggs. The other person has noticed that we need butter, eggs and milk. Then the union of those two shopping lists is: washing-up liquid, butter, eggs and milk.

A disjoint union is when each person’s list contains things they need themselves. So the total shopping list in that case would be one bottle of washing-up liquid, two packs of butter (one each), two cartons of eggs (one each) and milk.

The union of the sets {0,1} and {0,1} is just {0,1}, unlike the disjoint union which has four elements. When we take the disjoint union of the sets {0,1} and {0,1}, it’s really like taking two blocks and then another two blocks and then seeing how many we have in total. In a way, the way we do two plus two with blocks is a bit weird: we take two blocks labelled ‘one, two’, and then we take another two labelled ‘one, two’ and then we relabel them ‘one, two, three, four’. No wonder it takes a while for children to get the point. But also, the fact that people can work out how to do this intuitively, by watching others do it, is a marvellous thing.

Here’s how this process works in formal mathematics. It is really the same idea except for this slight technical detail: the representative set called ‘two’ contains the elements called ‘zero’ and ‘one’, not the elements called ‘one’ and ‘two’. We saw earlier that we can define the number 2 as a collection of equivalent sets with a representative {0,1} (the set containing the elements called 0 and 1). It’s a representative, because the number 2 could also be represented by any other set containing two things, like {a,b} or {elephant, banana}. The set {0,1} is the ‘official’ chosen representative, chosen from all the possible representatives. The set {1,2} is also a set with two elements, but we can’t use it to define the number 2, because it uses the number 2, and that would be invalid circular reasoning. So we use the set {0,1} instead.

Next, mathematicians define how to do addition with this convoluted notion of number: it’s done using disjoint unions of representing sets. So to find 2 + 2 we take the disjoint union of the set {0,1} and another set {0,1}. This new set contains the following elements:

0-from-the-first-set

1-from-the-first-set

0-from-the-second-set

1-from-the-second-set

This is evidently not an official representative of any equivalence class of our sets, because all our official representatives are of the form

{ 0, 1, 2, 3, …, n − 1 }

whereas this set looks nothing like that. More specifically, we can see that the new set above has four things in it, and we know that our official representative of sets with four elements is this:

{ 0, 1, 2, 3 }

To show rigorously that this is the official answer to 2 + 2 that we’re looking for, we have to give a bijection between the set we produced from the disjoint union, and this official representative set. And we can do it like this:

0-from-the-ﬁrst-set

‒‒‒‒

0

1-from-the-ﬁrst-set

‒‒‒‒

1

0-from-the-second-set

‒‒‒‒

2

1-from-the-second-set

‒‒‒‒

3

So this disjoint union is in bijection with the set

{ 0, 1, 2, 3 }

which is the official representative of the number 4. This tells us 2 + 2 = 4.

What a palaver! I’m sure that before I learnt about this in an undergraduate set theory class I could not have imagined that proving 2 + 2 = 4 could be so difficult. But again, I think this is a sign of the power of our brains, that we can take for granted things that are very difficult to do rigorously with logic.

Now we see that showing numbers are equal is really about showing sets are in bijection with each other. The nuances that are hidden inside those bijections can be very interesting – at least, much more interesting to me than the straightforward statement that numbers are equal.

Let’s take a look at multiplication. In set theory, as in primary school, this is defined as repeated addition. So to do 2 × 3 we are going to take two copies of the set representing the number 3, according to our list of official chosen representatives. First remember that 2 is represented by the set {0,1} and 3 is represented by the set {0,1,2}. Then we take the disjoint union of several copies of {0,1,2} – one for each element of the set {0,1}. So it’s as if we take one copy of {0,1,2} where we’ve attached a label ‘0’ to each element, and then another where we’ve attached a label ‘1’ to each element.

So we get these elements:

0-labelled-with-0, 1-labelled-with-0, 2-labelled-with-0 0-labelled-with-1, 1-labelled-with-1, 2-labelled-with-1

It’s a bit tedious to keep writing ‘labelled with’, so instead we use a shorthand like this:

(0,0), (1,0), (2,0) (0,1), (1,1), (2,1)

where we understand that the first number of each pair in brackets is the original number, and the second one is the label we’ve stuck onto it.

We could now show that this is in bijection with the representative {0,1,2,3,4,5} of sets of six elements, but what I want to do instead is explore what 3 × 2 is according to this story. This time we take a disjoint union of the set {0,1} (which represents the number 2) where we take one copy for each element of the set {0,1,2} (which represents the number 3). So we get this:

(0,0), (1,0)(0,1), (1,1) (0,2), (1,2)

How does this compare to the six elements we got for 2 × 3? It’s not the same elements, but they are in bijection with each other. Moreover, there is a bijection that is highly canonical, that is, it pops out of the structure without us really having to make any choices. Here’s how.

First note that with sets it doesn’t matter in what order we list the elements, so we can reorder these things in any way we want. For example, we could reorder the second set in ‘alphabetical order’. Perhaps this sounds strange with numbers, but imagine that instead of 0, 1 and 2 we were using a, b and c. Then the pairs of numbers could be written

aa ba ab bb ac bc

and then we would probably have no trouble at all putting them in alphabetical order:

aa ab ac ba bb bc

If we now rewrite the first version with a, b and c, but keep the order given, we get this:

aa ba ca ab bb cb

It is now a bit more evident why these two sets are in bijection with each other: the second one is all the same words as the first one but read backwards. Because they’re backwards, I didn’t list them in alphabetical order – in fact they’re in alphabetical order reading the words backwards.

In maths, alphabetical order is called ‘lexicographic order’, and the version where you’re reading the words backwards is called ‘co-lexicographic order’ because we often stick the prefix ‘co’ onto terminology to indicate we’re doing something backwards.

The move from lexicographic to co-lexicographic ordering is something that fascinates me. It comes up surprisingly often in maths and even in life, and to me it’s a sign of the anomalies that arise when we take something that is trying to be two- dimensional and try to force it to be one-dimensional.

I’m reminded of this when I’m playing Scrabble, which I play for fun online. The app I use has a helpful list of valid two-letter words, which I find particularly useful towards the end when I am trying to get rid of the last few letters by adding them to the board somewhere, often by means of obscure two-letter words I’ve never encountered in normal life, like ob, or za, or ch.

The trouble is that the list on my app is in alphabetical order, which isn’t entirely helpful, because if you’re trying to get rid of, say, an H, you want to know all the two-letter words involving an H whether the H is first or second. It would be more helpful to have two lists, one in lexicographic order and one in co-lexicographic order. Or perhaps a list of every two-letter word containing an A in either position, then all the ones containing a B in either position, and so on.

I ended up making my own list in a grid formation, as if the two letters were coordinates. I can then look along the H row for all the words starting with H, and down the H column for all the words ending with H. (Let’s call this a practical application of abstract maths.)

Lexicographic and co-lexicographic ordering are technically equivalent, but we English speakers are used to reading words from left to right, so it’s much easier for us to think about lexicographic ordering rather than trying to alphabetise from the end of a word. So in the human sense the two ordering schemes are not equivalent. Mathematically, however, they are equivalent ways of turning something that is two-dimensional into something that is one-dimensional. In my Scrabble grid, the alphabetic ordering would be obtained by reading all the way along the first row, and then all the way along the second row, and proceeding down the rows to the bottom. Here’s a ‘schematic’ diagram showing the idea:

By contrast, the co-lexicographic method would involve reading down the first column, and then jumping up to the top of the second column and reading all the way down it, and continuing along all the columns to the right. The schematic diagram would look like this:

which is ‘the same’, just flipped across the diagonal. Aside from for words and humans, it’s an entirely arbitrary decision whether we favour rows or columns first.

My favourite example of this is in how you eat corn on the cob. If you’re holding it horizontally, one way is to start at the left, eat all the way along the kernels to the right, and then come back to the left and munch out another row. The individual kernels are so small that you probably don’t take exactly one row of kernels at a time, but munch out a slightly wider row. I think of this as the lexicographic way of eating corn.

The co-lexicographic way (which I think is less obvious) is to start at the left, and eat all the way around the left, rotating the corn through one full rotation, before moving right a bit, and doing one full rotation there. I always do it the lexicographic way* (and perhaps everyone else does too), and when I was little I enjoyed pretending it was like a (manual) typewriter, where you get to the end of the row and have to do a physical carriage return to return the typing carriage to the left. The typewriter we had at home had some sort of bell so that every time you needed to do a carriage return it went ‘ting!’, so when I was eating corn on the cob I would call out ‘ting!’ every time I got to the end of the row and went back to the left.

In maths when we have an arbitrary choice like lexicographic versus co-lexicographic we call it a ‘non-canonical’ choice; something is canonical if there is some structural reason for doing it, which is somehow predetermined without us having to make a decision. It’s like if there’s a canonical place you go for drinks after work: you don’t have to have a long discussion with everyone about where you’re going, you just always go to the same place.

Picking representatives of equivalence classes is often a non-canonical choice and so mathematicians prefer not to do it. Sometimes it is possible to do it in a somewhat organised manner (such as when we pick the set {0, 1, …, n − 1} to represent sets with n elements), but often it’s just arbitrary, like in our choice of what to call left and right, or clockwise and anticlockwise, or in choosing between lexicographic and co-lexicographic. When we make non-canonical choices in maths things can get messy later on.

This is often why mathematicians want to do maths ‘well’, rather than in a way that just works for the present purposes. Sometimes it’s pragmatic to do something that works for now, but abstract mathematicians are always looking further ahead than just now, and trying to do things in a way that we can build on. This is why we expend so much effort on dealing with sameness in a way that will stand up to some rigorous logic rather than just using our intuition – so that we can build on it. So we shouldn’t just pick any old way of thinking of things as the same, but we should make sure the relationship we’re thinking of is enough like an equivalence, in a sense that we need to make precise.

Equivalence relations

We’ve been exploring ways we can regard things as the same according to some relationship we’re interested in, such as regarding times of day as ‘the same time’ if they’re a multiple of 12 hours apart, or regarding sets as ‘the same size’ if there’s a bijection between them. I also talked about fitting lids onto jars, and mentioned that my method of relabelling the lids and jars that fit together only works if the coherence goes both ways: pickle lids fit jam jars but also jam lids fit pickle jars.

This is all a question of whether or not the relationship we’re thinking about is ‘good enough’ not to make a mess when we quotient by it. What that means in maths is that it needs to be sufficiently like equality even though it isn’t actually equality. We need it to satisfy enough properties that we can use it more or less like how we use equality, and this comes down to three basic principles of equality called ‘reflexivity’, ‘symmetry’ and ‘transitivity’. If a relation satisfies all three, then it is considered to have good enough properties that it can act as a form of ‘sameness’ for us. Relations that satisfy all three of these properties are called equivalence relations.

We discussed symmetry and transitivity in Chapter 2, but we didn’t really mention reflexivity, as it was almost too obvious to mention: it says that everything needs to be the same as itself. This is so fundamental for equality that we didn’t focus on it – in fact we dismissed it as useless. Really, that just means that if equality were not reflexive, things would be very strange indeed. However, there are other perfectly interesting relations that are not reflexive.

For example, the relation ‘is the same age as’ is reflexive, because everyone is the same age as themselves. But the property ‘is mother of’ is not reflexive, because not everyone is mother of themselves – in fact nobody is mother of them-selves. A less extreme non-example is ‘in an orchestra with’. If you are in an orchestra at all, then you are in an orchestra with yourself, but some people (like me at the moment) are not in any orchestra, so in particular they are not in an orchestra with themselves.

The next property we need for an equivalence relation is symmetry. We previously said that when we are talking about equality it doesn’t matter if we say a = b or b = a, because those mean the same thing – equality is symmetric. However, not all relations are symmetric. For example, ‘is mother of’ is not symmetric, but ‘is the same age as’ is symmetric.

The last property that we need, in order for our concept of ‘sameness’ to be something we can work with smoothly, is transitivity. This is the one that says we can build up strings of relations. The relation ‘is the same age as’ is transitive, because if a is the same age as b and b is the same age as c, then the relation transfers across b, so a is the same age as c. However, this does not work for the relation ‘is mother of’, because if a is mother of b and b is mother of c then a is not mother of c. (In fact a would be grandmother of c in that case.)

It shouldn’t surprise you to find that ‘is the same age as’ is a better form of ‘sameness’ than ‘is mother of’. But ‘better’ is a hazy and subjective word, and in maths we are trying to say things that are precisely defined. So we can now say something more precise: ‘is the same age as’ is an equivalence relation, but ‘is mother of’ is not.

Here’s how this all works for hours on the clock. The relationship we’re thinking about here is the one that results in 1 o’clock being equivalent to 13 o’clock, 25 o’clock, and so on, and also 2 o’clock being equivalent to 14 o’clock, 26 o’clock, and so on. We have already seen how to state this generally, without having to keep saying ‘and so on’, like this:

Two numbers a and b count as the same time if their difference is a multiple of 12.

Now we want to check that this is a ‘sensible’ relation, that is, an equivalence relation. So we want to check that it is reflexive, symmetric and transitive.

First we check reflexivity: does every number count as the same time as itself according to this definition? Well, the difference between a number and itself is 0, and that is indeed a multiple of 12. (This might be contentious, because we tend to start our multiplication tables with ‘1 times 12’, but we could start with ‘0 times 12 is 0’ instead.)

Now we check symmetry: this is sort of built into the definition the way I’ve stated it, because I’ve just talked about ‘the difference’ between the two numbers a and b. If we were doing this more formally we would need to say whether we were thinking about the difference b − a or the difference a − b. Or perhaps subconsciously we’re thinking about the difference as ‘the bigger one minus the smaller one’ whichever way round they are, which in maths we write like this:

| a − b |

The vertical bars mean we’re taking the ‘absolute value’ of the answer, that is, we ignore the negative sign if there is one, because we just care about the size of the answer, not whether it’s positive or negative.

Anyway, whichever way we do this makes no difference, because asking for b − a to be a multiple of 12 is the same condition as asking for a − b or |a − b| to be a multiple of 12. We say this formally in maths like this:

Given two whole numbers a and b, the following conditions are equivalent:

1. a − b is a multiple of 12.

2. b − a is a multiple of 12.

3. | a − b | is a multiple of 12.

The upshot is that this relation is symmetric – it doesn’t matter which way round we put a and b in it. If a counts as the same time as b, then b counts as the same time as a.

Finally we can look at transitivity: if a counts as the same time as b, and b counts as the same time as c, then does a count as the same time as c?

The answer is yes. If the difference between a and b is a multiple of 12, and the difference between b and c is too, then the total is a sum of those two multiples of 12, so it is also a multiple of 12.

Here’s how we say that formally. If the difference between a and b is a multiple of 12, we can write it like this:

b − a = 12k for some whole number k.

Now if b and c count as the same time, then their difference is also a multiple of 12, so we can write it like this:

c − b = 12m for some whole number m.

Then we can investigate the difference c − a:

c−a=(c−b)+(b−a)=12m+12k=12(m+k)

So the difference between a and c is also a multiple of 12, so they do count as the same time provided a and b do and b and c also do.

We have shown that this relation is reflexive, symmetric and transitive, so it is an equivalence relation. This means that we can say 13 o’clock is ‘the same as’ 1 o’clock (and also 25 o’clock, 37 o’clock, and so on), and this concept of sameness is still a sensible enough notion that we can work with it, in some ways, as if it were an equality.

Bijection as an equivalence relation

We have seen that bijection is a way to say that two sets count as ‘more or less the same for our purposes’. Here the ‘purpose’ in question is measuring quantity, and a bijection between two sets is a perfect pairing up of the elements of one with the elements of the other, showing that the two sets have the same number of elements in them. We can now show that the relation ‘being in bijection with’ is an equivalence relation on sets. So it is a form of sameness that is well organised in this particular way meaning that we can build on it.

First of all, this relation is reflexive because every set has a bijection with itself – you just pair every object with the same object. Next, it’s symmetric because we’re thinking about pairing up the objects of the two sets perfectly, and so if we flip the sets over the pairing still works. It’s like if we ask everyone to sit on a chair, and there’s exactly the right number of chairs for them; we could instead ask everyone to hold a chair above their heads, and there would still be exactly the right number of chairs.

Transitivity is a bit more involved to prove formally, but the idea is that we have three sets A, B and C, and a way to pair up the objects of A and B perfectly, and also a way to pair up the objects of B and C perfectly. We can then produce a way to pair up the objects of A and C perfectly, because each object of B can sort of bring together the object of A they’re paired with, and the object of C they’re paired with.

One way of visualising what’s going on is to picture the people in set B holding hands with people in set A on one side and people in set C on the other side. Then the people in set B bring those two people from A and C together and step away.

Equivalence relations turn out to be the same as just a partition of your world into compartments, where each compartment contains all the objects you’re considering to be equivalent. Things get messy if a relation isn’t an equivalence relation because then you can’t clearly delineate which things are and aren’t equivalent, or which things are in which compartments. You end up with grey areas and unclear boundaries between things. Most situations in life aren’t as clear-cut as an equivalence relation, especially when it concerns people.

We saw some of this when we were thinking about approximation. We can now be more precise about that and say that the problem with approximation by ‘tolerance intervals’ is that this is not an equivalence relation – specifically, it is not transitive. For example, we might try to group children of roughly the same age together at school, to teach them together in year groups. But if we say two children are ‘roughly the same age’ if they are within a year of each other, then a child who’s just turned six is roughly the same age as one who’s about to turn seven, who is in turn roughly the same age as one who’s almost eight. But the six-year-old and the almost-eight-year-old are not roughly the same age. So instead, we make an age cut-off somewhere for each year group, which results in some children who were born the day before the cut-off being in the year above, and finding themselves almost a whole year younger than the oldest children in the year, which makes a big difference when you’re only six years old.

It gets even messier when we think about politics and our attempts to organise people with similar political views into political parties. Like with siblings-in-law who are thirty years apart, we end up with people supposedly in the same political party who have wildly differing positions on some key policies, and sometimes people in different parties turn out to have positions that are more similar than the people who are being considered to be in the same party.

The mathematical approximations we considered were mostly about numbers, but now I want to go on and discuss something more complicated and subtle – logical statements. All sorts of things can go wrong when we try to think about when two logical statements are ‘equivalent’.

* A class is like a set, except there is a very technical difference that would be a huge digression to explain here. The idea is that we technically can’t allow a ‘set of all sets’ because the self-reference causes a paradox called Russell’s paradox, so if we gather all sets into one collection we have to consider that it’s ‘bigger’ than a mere set, and we call it a class.

* When I was first writing this I couldn’t for the life of me remember whether as an adult I eat corn lexicographically or co-lexicographically or some other way. But I bought some and tried it and can confirm that I do indeed still eat it lexicographically.

7 LOGIC, EMOTIONS, INSIGHT

Mathematics is a framework for reasoning using logic. Mathematicians express ideas in a form appropriate for using logic, and then use logic to build rigorous arguments; the point is to have a way of being sure that the conclusions are reliable.

Pure logic is quite black-and-white: a statement simply does or doesn’t follow logically from a previous one, and a statement just is or isn’t logically equivalent to another one. In order to build rigorous logic arguments, it is crucial to understand logic precisely, and to be able to operate within its rigid constraints.

Pure logic doesn’t leave room for grey areas. However, that doesn’t mean there are no grey areas around logical equivalence: there are grey areas introduced by our human interaction with logic. Humans don’t operate by logic alone. Human life involves logic and emotions, in a complicated interconnected way.

Humans respond to ideas differently depending on how they are expressed, even if the logical content is the same. This isn’t a weakness: it’s deeply interesting and enriching. It means that if we have a strong understanding of what counts as logically equivalent, we can change our point of view on something without changing its logical content, and thereby understand it from different angles. We did this when thinking about looking at shapes and configurations of counting blocks from literally different angles. The mismatch between emotional and logical equivalence means we can present things in different ways depending on what we are trying to achieve. Aside from helping us understand things better for ourselves, this can also help us explain things better to other people, or be more persuasive about what we’re trying to say.

Unfortunately it does mean that people can use this technique disingenuously, and attempt to manipulate or deceive others by presenting an argument that is technically logically equivalent, but emotionally very skewed. The good news is that if we develop a strong understanding of the interaction between logical equivalence and human emotional reaction, then we can protect ourselves against this sort of manipulation. We could call it ‘logical literacy’, as long as we take literacy to be not just about reading text but about understanding its human impact too.

We should not simply try to eliminate emotions and work by logic alone, because emotions are part of what makes us human, what makes us able to empathise with each other and build communities, what makes life worth living. While some people like to think of maths as existing in an emotionless vacuum (and in some sense this is true), as soon as we use maths to understand the world around us, we need to understand our emotional interaction with maths. In fact, we humans are emotional creatures, and so if we are thinking about maths in any way it’s important to understand its emotional impact on us.

Different emotional responses

At a superficial level, we respond to information differently depending on how it’s written, even if, apart from the presentation, it is exactly the same information. For example, if I write ‘one plus one equals two’ you might have one reaction, whereas if I write ‘1 + 1 = 2’ you might have a different one. And what about this?

ONE PLUS ONE EQUALS TWO!!!!

The logical content of each statement is the same, but the emotional response is different. Things connect with human emotions in different ways even when they are logically equivalent, and we shouldn’t dismiss that; this is why people try really hard to pick good names for products, and we try to pick good names for mathematical concepts as well. Sometimes mathematicians try so hard to pick an evocative name that they just end up being confusing. Terms like ‘normal’ or ‘universal’ already have a meaning in daily life but are used slightly differently and rather more precisely in maths. Those words are confusing because they have common usage in life, but other times the problem is that the words are too esoteric, such as ‘prone’ and ‘supine’, which have technical meanings in my research field, category theory. Not only are we supposed to remember which of those means ‘lying on your front’ and which means ‘lying on your back’, but we’re also supposed to remember which of the two abstract concepts is like which; this is the kind of thing where I end up looking up the definition online repeatedly, or I give up and focus my research on something else.

Understanding when statements are and aren’t logically equivalent is a key part of maths. When we get taught (or bullied) about the manipulation of equations, we are really being told something about when statements are and aren’t logically equivalent; unfortunately, this too often doesn’t come across. What we’re really doing is saying ‘OK, we know that these two things are equal’ and then we do some manipulation and go ‘This is logically equivalent to saying that these other things are equal’. However, just like with the two sides of an equation, there must be some way in which the new statement is not equivalent to the old one, along with it being logically equivalent, otherwise there would be no point arriving at the new statement.

For example, we might start with this equation:

x + 3 = 5

and manipulate it by subtracting 3 from both sides; then we know that the above equation is equivalent to this one:

x = 5 − 3

and that in turn is equivalent to this:

x = 2

This is the same logically, but it is different in that it directly tells us what x is. We lean on the logical equivalence to pivot into a new statement that tells us something different about the situation.

This is why we can still make progress even if we keep saying things that are equivalent to each other. Just like equations give us different points of view on the same quantity, the point with logic is to find statements that are equivalent logically, but inequivalent in some other interesting sense; this gives us different points of view on the same concept. The inequivalence could be about expressing the same thing in two different ways to get different insights, or it might be to invoke visual intuition, or it might be about invoking different emotional responses (perhaps even in maths). But first we need to be clear what logical equivalence means.

What is logical equivalence?

Logical arguments are based on the notion of logical implication, and logical equivalence is when logical implication goes both ways. So let’s think about logical implication first. One statement logically implies another if the second one automatically follows from the first one, by sheer logic.

In maths we might call those statements A and B and say ‘A implies B’, which means that whenever A is true, then B has to be true. However, when A isn’t true we get no information out of this implication.

For example, if a number is divisible by 4, then it is definitely divisible by 2. So I could write:

A = ‘x is divisible by 4’ B = ‘x is divisible by 2’ Then A implies B.

However, if x is not divisible by 4 (so A is not true) we can’t tell if x is divisible by 2 or not: it might be (for example, 6 is not divisible by 4 but it is divisible by 2), but it might not be (for example, 5 is not divisible by 4, and it isn’t divisible by 2 either).

Also, if x is divisible by 2 (so B is true) we can’t tell if x is divisible by 4 or not: it might be or it might not be. So in this case we know that A implies B, but B does not imply A. Logic has a direction of flow to it and we need to be careful about that: logical implication is not a symmetric relation. However, it is reflexive and transitive, and we’ll look at those things first.

Logical implication is reflexive, because everything implies itself. It’s not very helpful to know that everything implies itself because, like with equations such as

x = x

we don’t gain any knowledge, understanding or insight. Despite this, people sometimes use somewhat reflexive logical implication in daily life to attempt to justify their beliefs. For example, someone might proclaim, ‘I don’t believe in same-sex marriage because I believe marriage should be between a man and a woman’, but this is really just saying the same thing twice in slightly different ways, and hiding their belief behind a veneer of logic – the logic isn’t wrong, it’s just not deep, and hence not helpful.

Logical implication is also transitive: if A implies B, and B implies C, then we can string those implications together and know that A implies C. This is true intuitively, and in mathematical logic it is taken as a basic assumption. Transitivity is a much more profound and productive way to proceed, as it actually gets us somewhere new, unlike using reflexivity which keeps us in the same place or going round in useless circles. In fact, all logical arguments depend on this transitivity to make progress. We start with some things we know, and we say what they imply, and then we say what those new things imply, and we keep going for as many steps as necessary to reach our conclusion. Sometimes other people find it hard to follow that many steps of logic though, and it’s often because the intuition gets left behind in the process. I think it’s a bit like jetlag, where your body physically crosses time zones but your body clock gets left behind, or as William Gibson said in Pattern Recognition, it’s the soul catching up with the body. In any case, the emotional lag of logical implication does not prevent it from being transitive.

So far we’ve seen that logical implication is reflexive and transitive. However, logical implication is not symmetric: ‘A implies B’ and ‘B implies A’ are completely different statements, and this is key to understanding logical equivalence. Usually logic only flows one way, but if it flows both ways we have a special situation, and that’s logical equivalence.

The statement ‘B implies A’ is called the converse of the statement ‘A implies B’. A statement and its converse are logically independent, which means that knowing whether one of them is true or false does not help us know whether or not the other one is true or false. Another way of saying that is that they could both be true, both be false, or they could be one true and one false.

We already saw an example where one is true and one is false:

A = x is divisible by 4

B = x is divisible by 2

In this case ‘A implies B’ is true, but ‘B implies A’ is false. Here’s an example where they’re both false:

A = person x is a man

B = person x is a harasser

‘A implies B’ is false, and ‘B implies A’ is also false, despite some people’s beliefs to the contrary. It is sometimes true that a man is a harasser, but that doesn’t mean that being a man necessarily implies being a harasser. Likewise, harassers are often men, but this is a statistical or sociological fact, not a logical implication.

Here’s an example where the statement and its converse are both true:

A = x is divisible by 6

B = x is divisible by 2 and 3

If x is divisible by 6, then it is definitely divisible by 2 and 3. Conversely, if x is divisible by 2 and 3, then it is divisible by 6. Here’s another example:

A = person x is a lawful permanent resident of the US

B = person x has a US Green Card

A implies B because if you are a lawful permanent resident you get issued with a Green Card showing that you are. B implies A because if you have a Green Card, it proves that you are a lawful permanent resident. (However, my maths brain is now coming up with loopholes, like perhaps you are a lawful permanent resident but you’ve lost your Green Card; however, I think legally you still abstractly ‘have a Green Card’. What about if you have a fake Green Card? Or you have stolen a Green Card? These things are trickier to pin down in normal language than in maths.)

In these last two cases the statement and its converse were both true because A and B were really saying the same thing just in different ways. This is what logical equivalence is. The technical definition of logical equivalence is this:

Statements A and B are called logically equivalent when A implies B and also B implies A.

The statement ‘A implies B’ involves logic ‘flowing’ from A to B; the statement ‘B implies A’ involves logic ‘flowing’ backwards, from B to A. Logical equivalence is when the logic flows both ways. The meaning of it is that whenever A is true, B has to be true, and whenever B is true, A has to be true. This means that they are true at exactly the same time: either they are both true or both false.

However, there can still be different insights to be gained from the different expression of the statements, and this is one of the points of understanding logical equivalence. Like with other forms of sameness, there may be many other ways in which the statements are different; the way in which they’re the same is very specific.

Different insights

The definition of logical equivalence is one thing; the consequences of it in context are another. It is perfectly possible to know the definition without knowing what the point is. The point is to get different insights from the same logical situation.

Take the example of being divisible by 6. When we unpack that to divisibility by 2 and 3 we’ve drilled down further towards some basic principles. However, the two different characterisations of multiples of 6 might be useful in different situations. In some situations you’re building up knowledge about multiples, so if you already know about multiples of 2 and 3 you can immediately deduce multiples of 6 without having to make much extra effort. In other cases it might be easier just to check for divisibility by 6 directly. This is admittedly a contrived example, but at some level it is like wondering if you have the ingredients in your kitchen to make custard, and knowing that you just have to check if you have eggs, sugar and cream (or milk).

In the case of the Green Card, ‘Lawful Permanent Resident’ is a much more formal, legal-sounding piece of terminology. ‘Green Card’ is more informal and evocative. (And yes, physically descriptive: the card is green.) It makes for a better movie title – a movie entitled ‘Lawful Permanent Resident’ doesn’t sound very catchy, whereas the nineties film Green Card was moderately successful back in the day.

In some mathematical situations logical equivalence gives us different points of view on the same thing. When we were thinking about quadratic equations and parabolas we saw that this equation:

x2 + 2x + 1 = 0

is logically equivalent to this one:

(x + 1)2 = 0

but the first relates our situation to a polynomial, whereas the second gives us more geometric intuition via a parabola.

I remember having a very frustrating tutorial session when I was an undergraduate and taking a course on logic. I found it fascinating, but I was really stuck on this theorem called Completeness, which showed that ‘syntactic entailment is equivalent to semantic entailment’. For the present purposes it doesn’t matter what that means, but essentially those are two ways to establish the truth of something in formal logic. Syntactic entailment is where you establish something is true by making a series of logical deductions, but semantic entailment involves assigning a consistent system of truth values (‘true’ or ‘false’) to all the statements in the system, and seeing what necessarily takes the value ‘true’. A completeness theorem for a particular system of logic says that these two methods of establishing truth are equivalent. One way round is more obvious than the other: in any sensible system of logic it should always be the case that everything you can prove must always take the truth value ‘true’. The other way round is more profound, as it says that anything that necessarily takes the truth value ‘true’ must have a proof. In complicated frameworks, proofs don’t always exist. So if you’re in a system where all true things are formally provable, that’s a big deal, and is called the completeness theorem.

Anyway, really the point is that we were showing ‘fancy concept A’ was equivalent to ‘fancy concept B’. I was really confused by it because I didn’t understand what the difference was between those two concepts in the first place. I asked the tutor and he just gave a knowing look and said, ‘Ah, but that’s the whole point, they’re equivalent’, which didn’t help me at all. The proof was quite long, so I said there must be some way in which they’re different because otherwise we wouldn’t have to prove they’re the same, but the tutor just kept repeating, ‘No, that’s the whole point, they’re equivalent’. I requested a different tutor after that (and to this day I remain confused about which is which between semantic and syntactic entailment).

The thing is that, just like with equations, if there is anything to be gained by knowing that two things are logically equivalent, then there must be something inequivalent about them as well. One way that this comes up in abstract mathematics is when we find different ways of characterising the same concept, such as with the two ways of establishing truth we discussed above, so that we can think flexibly about it and have different ways to use it in different contexts.

Different characterisations of the same thing

Prime numbers are a key example of something that gets characterised in different ways. Most commonly in maths lessons they are characterised as ‘numbers that are only divisible by 1 and themselves, and 1 doesn’t count’. But another way to characterise them is as ‘the numbers that are the basic building blocks by multiplication’. The latter characterisation is stated formally by the so-called Fundamental Theorem of Arithmetic, which says that every whole number can be expressed as a product of prime numbers in exactly one way. The fact that every whole number can be expressed like this is what tells us that prime numbers are good building blocks, and the fact that the expression is unique is what tells us that they are basic building blocks.

This latter characterisation is more of a theoretical characterisation according to the role we want prime numbers to play in the end. The former characterisation (saying what they’re divisible by) is a more concrete one. You might think it’s funny to call something concrete when it’s about numbers, which are inherently abstract, but these things are always relative. I think of the characterisation ‘numbers that are only divisible by 1 and themselves’ as concrete because it’s more hands-on, except there I’ve just said the same thing again using different words. In practice, it means it’s more of a ‘test’ we use to see if something is prime, whereas the ‘role it plays’ characterisation is more about how we can then make use of the fact that something is prime once we know that it is.

There are often two different characterisations along these lines – one that is better for testing to see if something is a certain kind of object, and another that tells us how we can then apply it somewhere. It’s like a driving test as opposed to actual driving skills: the test is how we officially decide whether someone is a competent driver on the road, but the actual driving skills are what we’re supposed to put to use. The trouble is that the skill of passing a driving test is not quite the same as the skill of actually driving safely. They’re not logically equivalent: suppose we write

A = ‘you pass your driving test’

B = ‘you are definitely a safe driver’

To see if these are logically equivalent we can examine whether A implies B, and B implies A. For the first one: if you pass your driving test, does that mean you’re definitely a safe driver? Unfortunately I think no, because it’s such a controlled situation. This is especially the case in the US where the driving test depends heavily on where in the country you are. It’s much more lenient in sparsely populated places (apparently), although even if you pass your test in the middle of nowhere you are then legally allowed to drive on highways and in big cities as well.

What about the converse: if you are a safe driver, does that mean you will definitely pass your driving test? The answer to that is also no, because you could have a mean or pedantic examiner. My instructor taught me that when I looked in the mirrors I must move my whole head to make sure the examiner saw me do it, but I don’t think that’s a necessary condition for being a safe driver; someone who is a safe driver might not have been warned about this aspect of driving tests, and so might not move their head enough to satisfy a pedantic examiner and pass.

I’ve also heard that if you take a driving test in Manhattan, then you not only have to do parallel parking but you have to do it timed – I suppose because traffic is so bad they want people to be able to parallel park quickly and not hold up the traffic. So you could be an excellent driver in general but not be able to parallel park in time. (Admittedly, I heard this back in the nineties and have not been able to verify it recently.)

Mathematical situations often arise where we want an efficient way to test if something is true, but then we will use that thing much more broadly. In maths we need the two characterisations to be really logically equivalent though, in order for the test to be logically valid. That is, we need to know that if the first characterisation is true, then the second one definitely is, and if the second characterisation is true, then the first one definitely is. This can give us two interestingly different points of view on a structure, and is often generally referred to as a coherence theorem.

Coherence theorems

One example of the coherence idea involves associativity. We’ll think about associativity of addition, which says that for any three numbers a, b and c, the following is always true:

(a + b) + c = a + (b + c)

This means that if we are adding three numbers together it doesn’t matter what pairwise grouping we use: the answer will be the same. (It also doesn’t matter what order the a, b and c are in, but that’s a different property: commutativity.)

However, in practice when we use associativity we often want to use it for strings of more than three numbers, say four, or five, or n numbers for any finite number n. So the way we test for associativity is with just three numbers, but the way we eventually want to use it is this:

For any finite number n, and any string of numbers a1, ..., an, there is exactly one way to evaluate the sum of those numbers, regardless of how we parenthesise them.

That is not exactly a rigorous statement, but it’s the general idea. The important thing is that those two characterisations are logically equivalent, and that is called a coherence theorem.

In this case, the fact that the second characterisation implies the first is immediate: if it’s true for any finite number n, then it’s definitely true for n = 3. The reason that the theorem is a theorem and not just an obvious fact is that the implication the other way round is much more profound: it says that if the result is true for n = 3, then it’s true for any finite number n whatsoever. That’s quite brilliant, deep and useful, but it is usually sadly missing from discussions of associativity in maths classes.

It gets even more profound when the structures involved are more complicated and the operation involved is more nuanced than mere addition, and a large part of my research is focused on the question of how to characterise higher-dimensional structures in different ways, where some characterisations are better for testing whether the structure is secure, and others are better for using the structure once you know it’s secure.

One key component of these methods is that we need to be able to manipulate logical equivalences correctly, so that we don’t make mistakes when arguing that two characterisations give us the same result.

How we can manipulate logical equivalence

We can manipulate logical equivalence in similar ways to how we can manipulate ordinary equations, because – unlike logical implication, which has directional flow – logical equivalence is an equivalence relation, flowing in both directions. When we do these kinds of manipulations we do have to be careful that we really have a logical equivalence and not just an implication, because, as we saw earlier, if it’s only an implication we will get reflexivity and transitivity but not symmetry. If we acted like we had symmetry, but we were wrong, we could end up replacing a statement with one that was logically not equivalent, but more powerful, and then we would think we can do more things than we can, like if we act as if we have more money in our bank account than we do.

For example, if we know that x = 2, then this implies x2 = 4. But this is a logical implication and not a logical equivalence, because the converse isn’t true: x2 = 4 does not imply x = 2, because x might be −2. This means that x = 2 is more ‘powerful’ information than x2 = 4. If we replace the latter by the former inequivalent statement, we are claiming more power than we have, and that is not logically valid.

For logical equivalences, things work more like how they do for equality, meaning that we can validly replace one statement by another that is logically equivalent. Using ideas from the previous chapter, we can now say more precisely what we mean by ‘working more like how they do for equality’, which is that logical equivalence is an equivalence relation – it is reflexive, symmetric and transitive. Here’s how we show that this is true.

Logical equivalence is reflexive, because every statement is logically equivalent to itself. It is also symmetric, which is built into the definition of logical equivalence. It takes a bit more effort to show that it’s transitive, but it’s not too hard, as follows:

Suppose:

1. A is logically equivalent to B, and

2. B is logically equivalent to C.

We need to show that A is logically equivalent to C. Now if we unravel the definitions we get this:

1. A implies B and B implies A.

2. B implies C and C implies B.

Now if we combine the first half of each statement we get

A implies B and B implies C,

so by transitivity of implication we conclude that

A implies C.

Similarly, combining the second half of each statement we get

C implies B and B implies A,

so by transitivity of implication we conclude that

C implies A.

So we have A implies C, and C implies A, and therefore A and C are logically equivalent.

When statements are logically equivalent we can manipulate them like equalities in the following ways: we can write them either way round, we can put long strings of them together to deduce things, and we can also substitute one statement for a logically equivalent one, without changing the logic of what we’re doing. This enables us to take a different point of view on situations, while being sure that we are still looking at the same situation. If we can’t be sure it’s really the same situation, we don’t gain anything from changing our point of view.

However, if we aren’t very precise about what is really logically equivalent to what, there are many pitfalls.

Mistakes about logical equivalence

The errors around logical equivalence are sometimes accidental, but sometimes they are wilful, when people are trying to manipulate us. Sometimes people aren’t exactly doing it deliberately, but perhaps they’re just overzealous, and trying so hard to convince us of something that they don’t realise they’ve stretched logical equivalence too far. These mistakes are very noticeable if you are well practised with logical equivalence in a mathematical setting, just like practising anything helps you spot errors more quickly.

I separate these errors out into false positives and false negatives. The false positives are when we claim things are logically equivalent when they aren’t. The false negatives are when we think things aren’t equivalent although they are. Let’s start with the false positives.

Mistaking things for being logically equivalent when they aren’t is the bane of many maths students’ lives, and also a source of huge frustration for their teachers, as they attempt to explain why the student’s work is not in fact logically equivalent to the teacher’s correction. Sometimes two arguments can look very similar if you don’t fully understand the logic, when in fact they are catastrophically different. Catastrophic mathematically, that is, in the sense of the logic not working out at all, not catastrophic in the sense of the wider world.

Here’s an example that you might recognise if you remember anything about the sum of a geometric progression. A geometric progression is a series of numbers produced by multiplying by the same number repeatedly. If the number we’re multiplying by is called a and we start on 1, then the first few terms go like this:

1, a, a2, a3

It turns out you can take the sum of these by means of the following formula:

1+a+a2+a3=a4−1a−1

provided a isn’t 1 (because in that case the right-hand side wouldn’t make sense as it would involve dividing by 0). We might ask our students to prove that. One way we can prove it is like this:

(1+a+a2+a3)(a−1)=a+a2+a3+a4−1−a−a2−a3=a4−1therefore   1+a+a2+a3=a4−1a−1

by dividing both sides by a − 1, which is valid provided a ≠ 1.

The first line involves simply multiplying out the brackets to get the expression on the right, and then the second line involves organising all the things on the right and cancelling out the ones that have a plus and a minus version. This gives an equality that is true, and then we divide both sides by a − 1 (which we can, provided a − 1 isn’t 0).

However, here’s what students are prone to writing:

1+a+a2+a3=a4−1a−1(1+a+a2+a3)(a−1)=a4−1a+a2+a3+a4−1−a−a2−a3=a4−1 a4−1=a4−10=0

The two solutions look superficially similar, and appear to have the same steps in them, including multiplying by a − 1 and cancelling out the plus and minus terms. They even have the same reasoning behind them deep down, but the second solution is not logically equivalent to the first one because it starts with the conclusion and ends with the true but useless equation 0 = 0. If a student writes this second version, it is often hard to convince them it’s not equivalent to the first one because it feels to them that they have gone through the same thought processes as the first one. However, although feelings are valid, they are not a valid way to demonstrate logical equivalence.

Maths exercises are one thing, but erroneous claims of logical equivalence crop up all around us in normal life, when false equivalence arguments happen.

False equivalence arguments

A false equivalence argument is when someone claims that one statement is equivalent to another, when it is inequivalent in some crucial way. In maths this is very easy to debunk, because in maths logical equivalence is very clear-cut. If someone argues mathematically that statement A is equivalent to statement B, then they are arguing that A implies B, and B implies A. If you want to claim they are wrong you just have to show that A doesn’t imply B, or B doesn’t imply A. In a false equivalence argument it is often the case that the logical flow in one direction is true, but the flow does not go back again.

In life things are more complicated because we are not usually dealing in pure logic, so there is more room to perform a sleight of hand and claim that things are equivalent when they aren’t. Also, in life the claimed equivalence is not always a logical equivalence but might be a moral equivalence, such as when people claimed that Donald Trump and Hillary Clinton were ‘the same’ (that is, just as bad as each other) and so declined to vote for either.

But sometimes a false equivalence argument in life does boil down to a mathematical question: does A imply B, and does B imply A? One example that often taxes me is when someone argues that seeing both sides of an argument is ‘both-sidesism’. While this might sound tautological, the fallacy of ‘both- sidesism’ is really about giving two sides of an argument equal weight when they don’t deserve it, not just about observing different sides of an argument. If we point out that both parties in a conflict bear some responsibility, and we try to weigh up the responsibility according to the evidence, that is not equivalent to the sort of ‘both-sidesism’ that consists of giving equal airtime to those who do and don’t believe that the 2020 US presidential election was ‘stolen’ or ‘rigged’. In this case I could express it mathematically like this:

A = weighing up both sides of an argument according to logic and evidence

B = giving both sides of an argument equal weight

In this case A does not imply B, because if you weigh up both sides you might find that they deserve equal weight, but you might find that they don’t. Also, B does not imply A, because you can certainly give both sides equal weight with or without weighing them up.

However, sometimes something more subtle or nefarious is going on: someone uses a broad meaning of ‘both-sidesism’ when accusing someone of doing it, before using a more narrow meaning to conclude that this is bad. This happens when more right-wing people accuse more left-wing people of socialism. Perhaps they declare that universal healthcare is ‘socialist’, and they work under the assumption that socialism is bad, therefore this sounds like an argument that universal healthcare is bad. However, there are many different definitions of socialism, and it’s not logically sound to use a very broad one to accuse someone of being socialist, but then switch to a narrow one to conclude that socialism is ‘bad’. This superficially resembles a coherence theorem in which we use one definition to check that a system satisfies some axioms, and then we use another definition to work with the result; however, the key with coherence theorems is that the two definitions must be logically equivalent, so switching to an inequivalent definition is not logically valid.

Straw person arguments are an example of false equivalence, where someone replaces your actual argument with one that sounds similar but is much easier to knock down (like a straw person). The person then proceeds to knock it down, and then they claim they have defeated your argument, but they haven’t – they’ve defeated an inequivalent argument that nobody was making. Such arguments are often accidentally signposted by someone listening to what you say and retorting ‘That’s like saying …’ or paraphrasing you in a way that is a severe distortion.

When people invoke stereotypes, whether positive or negative, they are making a false equivalence – or perhaps just a false implication – between some identity and some characteristic. For example, if they think that all Chinese people are good at maths, they are making a false implication from ‘being Chinese’ to ‘being good at maths’, even if not a false equivalence. Sometimes – and more hurtfully to me – they make a false equivalence between ‘being a childless woman’ and ‘being a child hater’, without taking into account that some childless women (including me) adore children and have always wanted to be a mother, and are traumatised by having lost all their pregnancies before term.

If you tell someone you believe that women have the right to get an abortion under certain restricted circumstances, and they say ‘So you support murder’, they are making a false equivalence. Even if they believe that abortion is a type of murder, they are making a false equivalence between one supposed type of murder and all types. In maths this would be easy to refute, because you would take statements A and B to be these:

A = abortion under some circumstances

B = murder

Claiming these are equivalent consists of claiming that A implies B, and B implies A. Now, whether or not you think every abortion is a murder, there are definitely many murders that are not abortions. So mathematically it is completely clear that A and B are not equivalent, and therefore supporting A is not equivalent to supporting B. Unfortunately in life the argument is so overwhelmed by heated emotions that many people aren’t able to see the logic behind it any more. If we could all apply mathematical logic to real-life situations it wouldn’t fix everything, but we would have at least clearer arguments.

Recently some white male authors have been criticised for their portrayal of fictional characters who are not white men. Similarly, some actors have been criticised for portraying minority characters when they are not a member of that minority. Other people have pushed back against this, claiming it’s ridiculous to say that authors should only write about people who are the same as them, or to say that actors should only play characters who are the same as them. This is a false equivalence and a straw person argument: nobody is saying that authors should only write about people who are the same as them, or that actors should only act characters who are the same as them. Rather, the problem is when minority authors and actors are overlooked and not given a voice, while other authors and actors are then given the voice to represent or speak for these characters. In the case of authors, all too often the way they represent those characters is offensive, not simply because they are not the same as those characters themselves (which would be an ad hominem argument) but because they reduce the characters to stereotypes, misunderstand or misrepresent their experiences, and worse, just see them through a white patriarchal lens. In the case of actors it’s even worse if make-up and prosthetics are used to make someone look like a minority person when they’re not. It’s partly the past trauma of the use of Blackface and Yellowface to imitate Black and Chinese people (usually not in a complimentary way), but it’s also the upsetting idea that these people can put on that act temporarily and then remove all the make-up and float back into the world as a white person, whereas those of us who really have those identities can never escape them.

This false equivalence argument might seem like a world away from maths, but it can also be expressed in terms of statements A and B, like this:

A = Blackface and Yellowface

B = an actor playing someone who isn’t them

If we treat this mathematically, the claim that these are equivalent means A implies B, and B implies A. Now A does imply B, because if someone needs to don make-up in that way they are evidently playing someone who isn’t them. However, B does not imply A, and we can refute that by providing an example of someone playing someone who isn’t them, but who still isn’t wearing Blackface or Yellowface. Any example of a white person playing a white person other than themselves is an example of B being the case without A being the case, showing that B does not imply A, and that therefore A and B are not equivalent. Therefore one can object to A, without objecting to the entirety of B. Similarly, one can accept some actors playing people who are not themselves, without accepting Blackface and Yellowface. Then the question becomes more interesting: when is it acceptable for someone to act someone else’s identity, and when is it offensive?

I long for a day when everyone is so well represented that we no longer have to worry about these things, but we are not there yet. I remain convinced that if we all had more practice at careful logical reasoning, we would get there faster.

Unfortunately, false equivalence arguments are all around us, but I think it’s important to note that there can also be something I call a ‘meta false equivalence’ argument, when something looks like it’s a false equivalence argument but it actually isn’t.

Meta false equivalence

‘False equivalence’ is a standard term but ‘meta false equivalence’ is a phrase I came up with to describe a phenomenon I think we should discuss more; sometimes it can help to give a concept a name in order to focus our attention on it.

Here’s an example of what I mean by ‘meta false equivalence’: someone might say they don’t believe in universal healthcare because they think everyone should pay for what they use themselves, and nobody should have to pay for what someone else uses. A counterargument might then claim this means that the anti-healthcare person also doesn’t believe in roads, because roads have to be paid for collectively and you can’t really just pay for the roads you use.

This might sound like a false equivalence argument between healthcare and roads, but really it’s pointing out that the first person’s general reasoning for their point of view is not as general as they claim it is.

To express this more abstractly:

Argument pointing out flaw in general reasoning

Person A:I believe X because of general principle P.

Person B:General principle P also implies Y, but you don’t believe Y so you can’t really believe P either.

This is different from a false equivalence argument, which goes like this:

Argument containing false equivalence

Person A:I believe X.

Person B:Then you must also believe Y. [Where they claim Y is equivalent to X, but it isn’t.]

The first version of the argument involves more complicated logic, as we’ll now see by breaking that argument down step by step. It actually invokes the contrapositive, which is a logical statement derived from a basic implication, and is equivalent to it. Person A uses the implication:

P implies Y

The ‘contrapositive’ of this statement is the following:

not Y implies not P

and it is logically equivalent to the first one. For example:

‘Being cheese implies being food’

is logically equivalent to:

‘Not being food implies not being cheese’

Note the change in direction; this is different from saying:

‘Not being cheese implies not being food’

which is not true.

Person B in that first argument uses the contrapositive

not Y implies not P

They argue that person A does not believe Y, and therefore they do not believe P. The trouble with this kind of argument is that the logic is a little more complicated than usual and so it can be hard to get across in normal conversation, as you might have noticed by how convoluted my explanation was. It’s much easier to shout ‘False equivalence!’ at someone and feel clever for knowing about logical fallacies than to understand a more subtle argument.

The false equivalence between ‘weighing up both sides’ and ‘both-sidesism’ is also a form of meta false equivalence: someone calling out ‘both-sidesism’ is probably accusing someone of making a false equivalence between two sides of an argument, when really the person they’re accusing is trying to weigh up both sides in preference to claiming one side as 100 per cent correct and the other 100 per cent incorrect; that doesn’t mean they’re claiming that each side is 50 per cent valid.

Another subtle argument that might falsely be accused of being equivalent to a logical fallacy involves ‘what-aboutery’. It’s this sort of argument:

Argument involving what-aboutery

Person A:You’re late and now we’re going to miss our flight.

Person B: Well, last week you were late and we missed the train.

This kind of argument deflects from the point in question by talking about something else that isn’t relevant but is supposedly equivalent (so perhaps this is also a form of false equivalence argument), perhaps as an accusation of hypocrisy. However, there’s a similar but subtly different type of argument which I think is more about pointing out some potential prejudice or other unfairness. For example:

Argument pointing out potential prejudice

Person A:Susan should not get promoted because she never meets deadlines.

Person B:But James never meets deadlines and he got promoted.

Or, more realistically:

Person A: Hillary! Her emails! Lock her up!

Person B: But what about Trump actually trying to overturn the results of a democratic election?

Both of these might sound like deflection as before, but I think this is more about pointing out the probable prejudice of person A in not applying their principles consistently across different people, perhaps across men and women, or people on opposite sides of a political divide (or both). If you’re invoking a different situation in order to point out injustice or unfairness, then that is not equivalent to a ‘what about’ argument.

I have found myself in this situation several times when I have submitted a research paper for publication in a peer- reviewed journal. The peer referee’s report comes back and I feel that I have been criticised in ways that other (male) mathematicians’ papers are not criticised. In many cases this is hard to quantify but sometimes it is very specific: I am told I should include certain definitions rather than taking them as assumed knowledge, but other (male) mathematicians publish papers without including those definitions. Or I am told there is an error in my work, but actually the error is in a male mathematician’s work, which I am citing in my paper. In both cases the criticism may be valid in isolation; the problem is that the criticism has not been applied to other mathematicians, who happen to be male, so it feels like an uneven use of principles. A particularly egregious case of this involves the maths educator Jo Boaler, who has attracted a truly jaw-dropping quantity of hatred, and even death threats, thanks to her work on the benefits of less rigid teaching methods in maths. She has published research – and cites other research – backing up claims that these less rigid methods improve students’ outcomes, both in tests and also in longer-term life achievements. However, her detractors have gone over her publications with something even finer than a fine-tooth comb, nitpicking her arguments and citations. (I suppose that means they’re using a nit-comb – I remember such things with horror from my childhood.) Some of their objections might have a point, while others are astonishingly pedantic, but the main point I want to make is that if they are not applying such a nit-comb to every other person writing about maths education, then they are not being consistent, and it starts looking awfully like a vendetta rather than good-faith academic discourse. A better refutation would be to produce peer-reviewed research showing different outcomes, but this does not seem to take place, partly because many of the most vehement detractors are mathematicians, not maths educators at all, and they would not know how to do research in maths education. Another possibility is that it doesn’t take place because in fact flexible teaching methods do produce better outcomes overall.

Unfortunately this shows that mathematicians, being humans, can also be overwhelmed by their emotions in a way that prevents them from applying logic in a sound way outside of their research. Really this just emphasises the fact that humans – including mathematicians – are not purely logical creatures, and so our emotions do get involved with, probably, every single thing we do. One of the aims of using logic in maths is to try and cut through that, to be sure that our arguments have good foundations. Since everyone has different emotional responses to things, an argument that invokes emotion is less transferable across people, because someone else is likely to have a different emotional response to it.

However, emotions can also be used to help us. Human emotions are usually needed for us to understand things, although different people have different levels of sheer logic they can deal with (and it’s always possible to get better at it). While sheer logic is good for making unambiguous mathematical arguments that mathematicians can communicate to each other clearly, mathematicians also try to present things in more palatable ways so that the sheer logic can be better absorbed. We can do that in life too. This is where the false negatives of logical equivalence come in – when things might not seem to be logically equivalent because they’re emotionally different, but they really are logically equivalent.

The power of emotions in the face of logical equivalence

At a basic level, human reaction is why it’s so important how mathematical symbols and expressions are laid out on the page. Two expressions can contain the same maths, but one might be much more readable to the human eye than the other, for example this string of symbols:

(a+b)+c=a+(b+c)

as opposed to this version, which is just spaced differently:

(a + b)+ c = a +(b + c)

These expressions are logically equivalent, but the physical spacing changes our human interaction with them.

As with everything in maths, this issue becomes more and more critical as the situations become more complicated. It’s something that we can use to our benefit in maths, specifically because the structures we’re representing are abstract. The abstraction means that we can change the physical or visual way we depict a concept without changing the abstract or logical content, and we can find more visually compelling ways to present the same information to help us grasp it. For example, take a look at this multiplication table. It’s a slightly peculiar one because it only includes the numbers 1, 3, 7 and 9, and I’m only writing down the last digit of the answers. So for 3 × 9 the answer is 27 but I’m only writing down the last digit, which is 7.*

× | 1 | 3 | 7 | 9
1 | 1 | 3 | 7 | 9
3 | 3 | 9 | 1 | 7
7 | 7 | 1 | 9 | 3
9 | 9 | 7 | 3 | 1

It doesn’t have a particularly sensible pattern on it, but I will now switch the order in which I write the numbers. Instead of having the rows and columns listed 1, 3, 7, 9 in size order, I’ll do it 1, 3, 9, 7. Then the table comes out looking like this:

× | 1 | 3 | 9 | 7
1 | 1 | 3 | 9 | 7
3 | 3 | 9 | 7 | 1
9 | 9 | 7 | 1 | 3
7 | 7 | 1 | 3 | 9

In switching the order in which I lay out the table I haven’t changed any of the logical information, but I’ve revealed a visual pattern that is much more compelling, with diagonal stripes of the same number going from top right to bottom left. This also helps us see similarities with other tables, like this one for rotations of a square. The table shows what the result is of doing one rotation followed by another, taking into account the fact that rotating by 360 degrees gives ‘the same’ result as rotating by 0 degrees:

+

0

90

180

270

0

0

90

180

270

90

90

180

270

0

180

180

270

0

90

270

270

0

90

180

Rearranging the 1, 3, 7, 9 diagram physically while maintaining the same logical structure helps us to see a pattern, which helps us to relate it to other situations (such as rotations of a square), which in turn helps us to understand some deeper structure in this situation (in this case, that it is a cyclic group, but it doesn’t matter if you don’t know what that means).

Here’s an example from my research. Again, the meaning is unimportant for the present purposes but I want to demonstrate the visual presentation. This is from a recent research paper of mine (joint with Alexander Corner).* We spent a long time working on the geometry of this diagram to try and get it to make more sense visually. We could have drawn it any old how, as long as the arrows still went to and from the same places, and it would still have contained the same logical information. But this version is much more satisfying and readable; the unreadable version is just an illegible pencil sketch in my notebook:

This idea can be used to help us understand complicated mathematical ideas, but if we’re not careful it can also be used by ill-meaning people to manipulate us. Visually striking presentations of data grab our emotions, but if we look carefully at them we might find they’re misleading. If we don’t want to be manipulated by people it’s helpful to be able to read the logical content of those graphics rather than just responding to the emotional content, so that we can understand what the data is actually saying. This is one of the key reasons I advocate for maths education so hard: not because I think it helps in daily life to be better at mental arithmetic or trigonometry, but because, among other things, it helps people protect themselves against manipulation and trickery. I want everyone to have that form of intellectual self-defence, like martial arts for the brain.

The manipulation or deception can also happen in situations not involving visual representation – words also invoke human emotions. Sometimes information can be stated in a way that is technically logically correct, but the problem is the human interpretation. I have seen people (online) complain that covid vaccines were ‘oversold’, but personally I thought they were sold for exactly what they were – I read the efficacy and effectiveness statistics. What if the problem was that humans misinterpreted those statistics? If you’re told that something is 90 per cent effective but then it doesn’t work for you, that doesn’t mean that the thing was oversold: it could be 99 per cent effective and still not work for you. Perhaps the problem is that you took 90 per cent effective to mean ‘this will work’.

Similarly, people throw a lot of criticism at pollsters for being ‘wrong’ about their predictions of election results. However, if they only give a percentage likelihood, then they’re not necessarily wrong: it’s just that people interpret that incorrectly. When the polls said Hillary Clinton would win the 2016 US presidential election with 60 per cent certainty, and then she lost, the polls weren’t wrong, it was the human interpretation that was wrong: people took 60 per cent certainty to mean ‘Hillary is going to win’.

I’ve also seen this criticism levelled at the IVF industry (because it is now an industry): that it promises you something that might not happen. I’ve seen people complain that the doctors only ever tell you the success rate and not the failure rate. This makes me shudder for the state of statistics education, that people hear a success rate and don’t understand that the failure rate is the rest of the percentage (100 minus the success rate). Apparently hearing the failure rate directly resonates in a different way emotionally. This is despite the fact that the information contained in:

‘success rate’

is logically equivalent to the information:

‘success rate and failure rate’

since they necessarily add up to 100.

I admit that I do exploit these sorts of ideas if I want to pay someone a compliment without lying. If you go to someone’s performance and think it was terrible but don’t want to hurt their feelings, you can say something that’s strictly logically true like ‘That was incredible! Unforgettable!’ I think this is what politeness is about. Some people think it’s dishonest, because you’re expressing something different from what you’re really thinking, but I think there’s a sense in which you are expressing exactly what you’re really thinking, which is that you care about your friend and appreciate their efforts. Logic shouldn’t be used at the expense of our emotional human responses, but neither should emotions be prioritised at the expense of all logic. If we understand both, and the ways in which they interact, we can invoke both without sacrificing either.

* If you know something about modular arithmetic, I’m doing arithmetic modulo 10, and I’m only doing it on numbers that have no non-trivial common factor with 10.

* Eugenia Cheng and Alexander S. Corner. Weak vertical composition II: totalities. Theory and Applications of Categories, 41(31):995–1043, 2024.

8 SAMENESS IN CATEGORY THEORY

In the last chapter we thought about logical equivalence and saw that it is produced from situations in which the logic ‘flows both ways’. We started by thinking about logical implication, which is defined with a direction, like traffic flow. And like with the direction of traffic, there might be a one-way logical implication, or two-way (or none). This means that while logical implication is reflexive and transitive, it is not symmetric, because each logical implication by itself only goes one way. But it is possible for there to be a logical implication in both directions in some situations, and this is what gives us the concept of logical equivalence.

This situation is quite typical in abstract maths. We often have relationships or processes that aren’t symmetric, so they don’t necessarily flow in both directions. But the process might flow in both directions in some places, and when it does, that is a special feature, like a whirlpool in a system of rivers, or like a very specific version of moving from local to (slightly more) global thinking.

Thinking about relationships gives us a way of moving away from things just being either ‘the same’ and ‘not the same’, because in between those extremes there are other types of relationship that we might be interested in. In this chapter I’m going to be a bit daring and describe how abstract mathematics deals with all this formally. I call this ‘daring’ because it’s a subject typically considered very advanced and only suitable for advanced maths students, probably in graduate school. But I’d like to describe it even if you don’t understand all of it; in fact I don’t think anyone really understands mathematics, we just keep understanding it more as we go along, without ever finishing. And sometimes, if we run into situations that call for it, we invent new kinds of maths to cope with it.

In abstract maths we take situations that feel similar to each other, and we find a precise sense in which those situations are similar even though they have different details. We find the ‘essence’ of what is making those situations similar, and that’s the process of abstraction. This means we can build an abstract model of what is going on, and then see where else in the world (or in the world of maths) something similar is happening. The part of abstract maths that does this for relationships, and different kinds of sameness, is my field of research, category theory. It is a relatively modern branch of mathematics, founded in the middle of the twentieth century, which is very modern for mathematics; after all, humans have been developing mathematics for many thousands of years, starting in ancient civilisations.

Category theory is a framework for making very general arguments about mathematical concepts, and it came from noticing similarities in how different branches of maths do things. When mathematicians spot patterns, either visual patterns, logical patterns or patterns in mathematical behaviour, they often want to make a general theory of what’s going on so that they can understand all those scenarios in one go instead of having to do that work repeatedly in different contexts. Fundamentally, we mathematicians are lazy. But it’s not the kind of laziness that results in cutting corners shoddily, it’s the kind that results in finding ways to do things more efficiently. This often takes more front-loading of effort, as we have to develop and implement the more efficient system, before we can reap its rewards. Since mathematicians are keen on making that sort of effort, I suppose we’re not lazy overall, we’re just offended by the idea of unnecessary effort or repetitive actions.

So far in this book we’ve seen several precursors to category theory, and to the concept of sameness in category theory. In each case we were looking at relationships between things, and then examining when the relationships went both ways. We saw that equivalence relations satisfy reflexivity, symmetry and transitivity. By contrast, logical implication only satisfies reflexivity and transitivity, because the relation doesn’t always go both ways; when it does go both ways it is called logical equivalence. We are going to see that the ‘less than or equals to’ relation on numbers is a similar type of relation: it is reflexive and transitive, but the relation doesn’t always go both ways – it is not symmetric. We will see that functions are a type of relation on sets, also satisfying something like reflexivity and transitivity, but not symmetry. When functions do go both ways, in a sense that we’ll make precise in this chapter, we get a new characterisation of bijections, which is one of the reasons the ‘going both ways’ idea is so satisfying.

In order to say rigorously what it means for a relation to ‘go both ways’ we first need to be more precise about what types of relation we’re going to think about in the first place. We want something resembling an equivalence relation, but more forgiving, so that we can include more examples. This brings us to the definition of a ‘category’. This is a formal mathematical structure, although the word ‘category’ might be confusing because it has a different meaning from the meaning in normal life. A category in mathematics is essentially a set of objects with some chosen relations between them, satisfying something like reflexivity and something like transitivity, but leaving symmetry as something we investigate rather than demand. However, the mathematical definition is formal and rigorous so that we can build rigorous arguments from it. In mathematics category theory is not just a theory about categorising things, it’s a field of study that works with these formally defined structures.

The formal definition of a category

The formal definition of a category is an abstraction of many structures that mathematicians study. It arose from mathematicians spotting similarities between the different structures they were studying, and deciding to study those patterns in their own right. Unfortunately many of those structures are themselves rather advanced pieces of mathematics, which means that category theory is often thought of as something you can’t approach until you’ve studied all those other examples first. I say ‘unfortunately’, because this has the unfortunate effect of keeping category theory away from people (or keeping people away from category theory) until they’ve jumped over many of the hurdles of undergraduate mathematics.

This chapter is definitely an intensification of abstraction, complexity and profundity. As the abstraction ramps up, I will keep relating the ideas to daily life to give you something to hold onto in case you feel like you’re floating away. I would urge you to keep going even if you feel like you don’t understand, because there will be things in the following chapters that don’t depend on a full understanding of these details. And you can always come back to this if you want to.

I’m going to include the formal details here although, as usual, it’s not because I necessarily think everyone needs to understand it, but that you might like to marvel at it like a piece of abstract art. And just as you might with a piece of abstract art, you might decide that it’s too abstract for you to appreciate, but I still think it’s worth going to an art gallery and looking at what’s there. With abstract maths we too often withhold it from people unless we think they’re going to be able to do rigorous proofs with it, complete formal exercises and pass an exam. I’d like to change that, so here’s the formal definition of a category for you to gaze at. I’ll then explain what it all means, and show how it relates to some of the concepts we’ve seen already.

A category 𝒞 consists of:

DATA

• Objects: a collection ob 𝒞 of objects

• Arrows: for objects A, B in ob 𝒞, a collection 𝒞(A, B) of arrows A ⟶ B

STRUCTURE

• Identities: for any object A in ob 𝒞 an identity arrow 1AA→A

• Composition: for any arrows A→fB→gC composite arrow A→g⁢ ο⁢ fC

PROPERTIES

• Unit laws: for any arrow A→fB

f ◦ 1A = f = 1B ◦ f

• Associativity: for any arrows A→fB→gC→hD,

(h ◦ g) ◦ f = h ◦ (g ◦ f)

Arrows are also called morphisms, and the composite g ◦ f is also written gf.

Definitions in abstract maths often take this sort of shape, with data, structure and properties. Data is the starting point, the building blocks that we decide we’re going to work with. Structure is the ways we can use the building blocks, sort of like methods for building. Properties are the rules that the whole thing has to follow.

We’ll start here with the data. The idea of category theory is that many situations come down to studying abstract relationships, and so if we can make a theory of relationships, then we can study all of those situations at once. So our data here consists of objects and relationships between them. However, once we’re at this level of abstraction the so-called ‘relationships’ don’t have to be like actual relationships any more. They can be more like processes, or paths, or, well, anything really, as long as they satisfy the terms and conditions above. They can be functions, processes, paths in a space, proofs, transformations, matrices … the possibilities are endless. As the ‘relationships’ don’t have to be like relationships, we often call them arrows instead, and draw them as arrows. That visual aid helps us reason with the relationships, by invoking our visual intuition. It’s also important to note that objects in a category can be related in different ways: these aren’t just ‘yes or no’ relationships, so we don’t just say that two objects are or aren’t related. It’s more like looking for different routes from one place to another, or looking at different possible family relationships. I’ll explain the rest of the definition using family relationships as an example.

Next we come to structure. In our formal definition, we had two points: the first (identities) is like reflexivity, and the second (composition) is like transitivity. The first point says that every object A has to come with an arrow

A→1AA

This is an arrow relating A to itself, which is why I say it’s like reflexivity. However, it’s more subtle because there could be many different arrows from A to itself; this one is a special one called the identity and it is like ‘doing nothing’, which we’ll see is determined by the properties. In terms of family relationships this would be like a specific relationship where we say that everyone is ‘the same person as’ themselves. This might seem a bit redundant when talking about people, but when we revisit functions in a moment we’ll see that the identity function is just one of many possible functions from a set to itself.

The second point says that if A is related to B via an arrow f, and B is related to C via an arrow g, we can derive an arrow from A all the way to C. So this is like transitivity, and enables us to build up long strings of arrows, but it’s more subtle than transitivity and is called composition. We can also think about composition using family relationships as an example. If A is the mother of B, and B is the mother of C, we can compose those relationships into ‘grandmother’, which means ‘the mother of the mother’. By contrast ‘is the mother of’ is not transitive as a relationship, because transitivity would mean that your mother’s mother is your mother. That is how this notion of composition is more general or subtle than transitivity – it includes more examples. Other composite relationships include ‘uncle’, which means ‘brother of parent’, and cousin, which means ‘child of sibling of parent’.

Now we can move on to the properties – unit laws and associativity – that govern the way the identities and composition behave. The law about identities is analogous to the way we defined multiplicative identities back in Chapter 3. We characterised the multiplicative identity 1 as the number that ‘does nothing’ when we multiply it. We can write that formally like this:

For any number a, a × 1 = a = 1 × a.

This is abstractly ‘the same’ as the law for identities in a category:

f ◦ 1A = f = 1B ◦ f

In terms of family relationships this would say that if we know person A has a certain relationship with person B (say, A is the mother of B), we do not change anything by also knowing that A is the same person as A, or that B is the same person as B.

The associativity condition is abstractly ‘the same’ as the associativity of multiplication, and for family relationships it gives us something a bit more interesting than identities did. The question is what happens if we have a string of three relationships, such as this:

A→child⁢ ⁢ ofB→sibling⁢  ofC→parent⁢  ofD

In this case A is the niece/nephew (nibling) of C, who is the parent of D. Or we could say A is the child of B who is the aunt/uncle of D. But each way we think about it gives us the same answer: the nibling of your parent is the child of your aunt/uncle, and so we can use one word for it: cousin.

Associativity for multiplication is about being able to multiply a string of numbers of any length by multiplying them in successive pairs, and knowing that no matter which way you do it you will get the same answer. We have a more efficient way of testing for that, just using strings of three numbers. (We saw this for addition, when we talked about coherence theorems in the previous chapter, but it works just the same for multiplication.) This is the idea behind associativity in a category as well: testing to make sure we can build up long strings of arrows unambiguously.

Before we examine some examples of categories, I want to mention a way in which this is a higher-dimensional form of mathematics.

Relations as a new dimension

When we write symbols on the page in straight lines, our objects are zero-dimensional (they’re just points) and our reasoning is one-dimensional (it goes in lines). Whereas in category theory we begin with objects and arrows, so we have zero-dimensional things and one-dimensional things, on the page or in our heads. Our reasoning is then often in the form of jigsaw-puzzle-like shapes on the page, and is two-dimensional, like the huge diagram we included in the previous chapter. This extra dimension gives us more room for nuance, more expressive power, and the space to ask about senses in which things are true and not true, rather than just saying yes or no.

We have seen that numbers were really sets where we forgot some information. That ‘forgetting’ took us down a dimension. Going back up to sets gives us more nuance and expressive power, and going up to categories gives us even more nuance than that. One of the important and powerful aspects of that nuance is that we get a framework for looking at different possible relationships on the same objects. This means that we can put the same objects in different contexts depending on what we care about for any particular purpose. This is important as it helps us to think flexibly about how things – or people – relate to one another. One example involves paths in a space.

In ordinary life when we’re thinking about the route from A to B we might only care about the distance and want to take the shortest route. However, if we’re driving we might prefer to take the quickest route, which might be longer in terms of distance if it involves taking a detour to avoid traffic. If we’re taking public transport we might prefer the cheapest route, which might be neither the shortest nor the quickest, especially if it’s a flight which is made cheaper by a stopover.

Maths formalises this by the concept of a ‘metric’, which is a generalisation of distance: it’s any way of evaluating the ‘gap’ between two points in a space. It does have to satisfy a few axioms in order to be sensible enough for us to reason with it, but it can basically be measuring anything: distance, cost, time or something else. Some metrics take into account whether you’re going uphill or downhill. This can be important because two distances could be the same but if one is uphill and one is downhill, then they’ll probably take very different times and amounts of effort. You could take this into account by measuring the time it takes instead, but if, say, you have a bad knee you might prefer to walk five minutes uphill rather than five minutes downhill. After I had a knee operation, going down stairs was much worse than going up stairs.

Each different metric gives us a different category structure on those locations, giving us a different context in which to consider those locations. It doesn’t mean any particular context is better than the other, but it does focus our attention on what context we’re using. In fact, part of the discipline of category theory is that we always make it very clear right from the start what category we are working in, and if we change category we make sure to do it in a logically consistent way, as we’ll see in Chapter 10 when we discuss moving between contexts.

This focus on context is important when we measure the distance people have travelled to get to the same place, both literally and figuratively. Two people may have arrived at the same place, but one may have come further than the other. If you meet someone at Land’s End who had just walked over from their house up the street, that wouldn’t be as remarkable as someone who had just walked there from John o’Groats. But also, you might meet someone who got there from John o’Groats but by helicopter. That would also be less impressive than someone who had walked there from the other end of the country – or more impressive if your metric is wealth.

This same way of thinking can be applied to more abstract ‘paths’ that we encounter every day. Indeed it should be applied, and it is easier do so with a certain amount of practice at thinking about abstract paths. When we judge people’s achievements for a job application, for example, there is vociferous disagreement about whether we should help minorities and disadvantaged people overcome the disadvantages they have previously experienced in life. It too often becomes a divisive argument about ‘positive discrimination’, instead of a discussion about what metrics we are considering, and what abstract paths we are taking into account. ‘Positive discrimination’ is a really unfortunate term, because discrimination is indeed still discrimination. Whereas if we think about it as declaring up front what category we’re working in, it becomes clearer that whatever metric we’re using is a choice that we have made, and we should be more intentional in those choices. Maths gives us a way to do this.

In a category, not only does every arrow have a clearly defined starting point, but we have to be clear about what relations or paths we are studying. In life, we should not only think about where people started, but we should also think about their path, including the obstructions they worked against all the way along, just like the difference between someone walking uphill and someone walking downhill. Some people suffer so much more obstruction and difficulty all the way through the formative years of their life, that they have to work much harder to achieve ‘the same’ things; so according to a metric taking that into account, they’ve actually achieved more.

Some people agree that if, say, a white man and a Black woman were applying for the same maths professorship, and if all other things were exactly equal, we should hire the Black woman. But if all other things aren’t exactly equal (and in reality, they never are), those same people object to the idea of giving the Black woman any credit for her struggles.

If we reduce all this to numbers, we could say that if the two candidates both scored exactly 85 according to some metric or other not taking backgrounds into account, then the Black woman would have an ‘effective score’ of more than 85 on our more nuanced metric, because of the extra obstructions she had to fight through to achieve the same things (on paper) as the white man. What is her effective score? Most people agree that it would be definitive enough to tip the decision in her favour at that point. Perhaps her effective score is 90.

But what if the Black woman’s absolute score had been 84, to the white man’s 85. If 85 becomes 90, then wouldn’t 84 become something higher than 85 as well? Perhaps 83 would also end up higher than 85. Perhaps even 82 would. The question then is how much that weighting should be, or what is the lowest absolute score that would result in an effective score of better than 85. It’s a question of how we move from the metric that does not take obstruction into account, to the one that does.

I am not suggesting that everyone be reduced to a single number, I’m just using this as a simple example of the principle. When people admit that ‘if all other things are equal’ we should favour the person who has suffered more obstructions in life, they might be judging people according to many dimensions, rather than just one number, but we can extrapolate the model using one number to a model using many.

I think we should stop thinking of this as ‘positive discrimination’ and certainly stop thinking of it as lowering the standards for minorities, because the reality is that certain minority people have to work to much higher standards just to be accepted at all, let alone be valued. Instead, we should consider more carefully the metrics we use, and the different possible ways in which we count things as the same. If even in the rigorous field of mathematics there are different ways to define ‘sameness’ according to context, then surely there must be even more different ways in life. And rather than decide which way is best, we should be aware of what all the different ways are that we are using, and find ways to compare those approaches as well. We’ll come back to ways of moving between different contexts later.

The definition of a category is all very abstract, and many of the examples we’ve seen already (family relationships, logical implication, equivalence relations) are a bit simplistic in some way. In order to delve into the full power of the definition, I’d like to revisit functions.

Functions as relationships

I’d now like to present functions as a form of relationship between sets. This will give us an important example of a category where the objects are the sets themselves and the arrows are the functions.

As we saw in Chapter 4, a function is something that takes in inputs and produces outputs. So if the set of inputs is called A and the set of outputs is called B, then a function is a relation between A and B. We can also think of it as a process of moving from A to B, just like how logical implication was a flow of logic from a statement A to a statement B. We sometimes write ‘A implies B’ like this to emphasise the directionality:

A ⟹ B

and we can write functions similarly, with an arrow to emphasise the directionality:

A ⟶ B

We can then take functions to be the arrows of a category, and we will now work our way through the definition of a category as presented above, to see how it all works for functions. We start with the data: the objects are going to be all possible sets, and the arrows are all the possible functions.

The next thing to look at is structure. For the identity function on a set A, we can use the function that takes any element of A (as an input) and sends it to itself (as an output). It’s the ‘do nothing’ function, which we saw in Chapter 4 and expressed like this:

f (x) = x

Or, using the notation of category theory, we would write 1A for this function on the set A, and define it like this:

for any element a of A 1A (a) = a

In Chapter 4 the inputs and outputs of our functions were numbers, but we can do this even if the inputs and outputs are anything else, such as letters, people, groceries. This function is more technically called the identity function, and we take it to be the identity arrow in this category we’re making. When we look at the properties, we will see why we needed to take this ‘do nothing’ function as the identity, and not any of the other possible functions from sets to themselves. For example, if we’re thinking about functions from the whole numbers to themselves, there’s a function that sends everything to zero. This is definitely not a ‘do nothing’ function – it does a lot, because it collapses everything to zero.

The next piece of structure to think about is composition. This says that given a function A to B, and a function B to C, we need to produce a function A to C. Using the arrow notation we can indicate the question like this:

The dotted arrow indicates that we are going to build this function out of the solid arrows.

The answer is more precise than just ‘yes, we do get a function from A to C’: specifically, we get a function that is built from the two functions f and g that we started with. We can take an element of A, say a, and we can send it to an output in B using the first function f, and then we can keep going and use that element as an input into the second function g, and send it onwards to an output in C using the second function. Here’s a schematic diagram of that happening:

A→fB→gC

a⁢  ↦f(a)↦g(f(a))

The arrows in the bottom row have little vertical bars at the beginning to indicate that this is an action on individual elements of the set, whereas in the top row the arrows are indicating entire functions, acting on the entire set as a whole.

Formally, we are taking an element a, sending it to f (a), and then using g to send that to g (f (a)). This process is called composition of functions, and the result is a ‘composite’ function which we usually write as g ° f, which goes all the way from A to C.

Formally, the composite is defined like this:

(g ◦ f)(a) = g(f(a))

This is a more interesting statement than transitivity because it tells us a specific way to make a function from A to C based on f and g, rather than just telling us that a relationship from A to C exists. Saying such a relationship exists wouldn’t be interesting at all, because there is always a function from any set to another set (unless the output set is empty).

We can now move on to the properties, the last part of our formal definition of a category: the unit laws and the associativity law. The unit law is what ensures that the identity arrow really has to be the identity function and not something else, because when we compose it with any other function f, the resulting function still has to be just f. In arrow diagrams we’re saying this: for any function

fA→B

these two equations must hold:

or, in symbols:

f ◦ 1A = f = 1B ◦ f

Personally I like the arrow presentation as it gives me a clear picture of what is going on with the directions of the functions, but I like the expression in symbols as it emphasises the analogy with multiplicative identities; it’s an example where the two presentations are logically equivalent but visually help us in different ways.

Incidentally, in this characterisation we are using equality of functions, which we looked at in Chapter 4. This means that, to show these equations are true, we have to check that the outcomes are the same for all elements of A. The argument is as follows; we just have to remember how the identity functions are defined, which is that they ‘do nothing’. So for any element a of A, we have 1A(a) = a. We also have to use the definition of composition of functions.

Then for any element a of A:

(f∘1A)(a)=f(1A(a))=f(a)

Perhaps you can check for yourself that 1B ° f = f .

Finally we can check associativity. This is a bit technical for functions and might make your eyes glaze over, but I’ll write it out anyway so that you can gaze at it. We need to start with functions:

A→fB→gC→hD

and show that these two composite functions are equal:

(h ◦ g) ◦ f = h ◦ (g ◦ f)

We are trying to show that two functions from A to D are equal, so we have to show that, given any input a in A, the two sides produce the same output in D. Here’s how that goes. The left-hand side gives this:

((h∘g)∘f)(a)=(h∘g)(f(a))=h(g(f(a)))

whereas the right-hand side gives this:

(h∘(g∘f))(a)=h((g∘f)(a))=h(g(f(a)))

As the two sides give the same answer, we are done. If that made no sense to you, please don’t worry about it; it was really just a very detailed unravelling of definitions.

We have finished following through the entire definition of a category for this example of sets and functions, and so now the question is: what are we going to do with that? The answer is: we’re going to investigate the one property of equivalence relations that is missing from the definition of a category, which is symmetry. We are going to look for places where the arrows go both ways, and this will give us a notion of sameness. Once we’ve got our framework for relationships in place, one of the whole points of category theory is to deal with more nuanced notions of sameness, and to put it on a secure logical footing rather than just a vaguely intuitive one.

Sameness in a category

The three properties of an equivalence relation are reflexivity, symmetry and transitivity. I think of them in that order because reflexivity involves one object, symmetry involves two, and transitivity involves three; but we might also think of symmetry last as it’s the one that is ‘missing’ from a category, so it’s the one we think about after reflexivity and transitivity. In a category we have identities giving us something like reflexivity, and we have composition giving us something like transitivity. In both cases the category theory version is more nuanced than the version for equivalence relations, because we’re allowing more nuanced types of relations. Now we’ll introduce a more nuanced version of symmetry.

Unlike with reflexivity and transitivity, we don’t require relations to be symmetric in a category, which allows us to express and study a much wider range of situations. However, we do then look for situations where there is symmetry locally, and that is the concept we will now define, called ‘invertibility’. Invertible arrows go back and forth between a pair of objects in a category, exhibiting them as ‘more or less the same’. The objects in question will then be called ‘isomorphic’, and the arrows will be called ‘isomorphisms’.

Just like we ‘copied’ the definition of multiplicative identity to give the definition of identity in a category, we can ‘copy’ the definition of multiplicative inverse, which we saw in Chapter 3:

For any number a, a multiplicative inverse for it is a number b such that a × b = b × a = 1.

We are now doing this for arrows in a category instead of for numbers, and we are thinking about composition instead of multiplication – we have gone up a dimension. As with identities, this makes the situation slightly more complicated because we have to think about the directions of the arrows. It might be tempting to make an analogous definition like this:

For any arrow f, an inverse for it is an arrow g such that

f ° g = g ° f = 1.

but this doesn’t quite ‘type-check’. Type-checking is when you do a basic check of consistency to see whether the symbols you’ve written down actually form a valid statement at all, before even asking yourself whether the statement is true or not. It’s very prevalent when writing code for computers because computers need to know exactly what ‘type’ of variable they are dealing with otherwise they can’t operate. It’s similar to checking syntax and grammar in spoken language. The sentence ‘The moon is made of soft green cheese’ makes sense grammatically, but it just isn’t true. Whereas Stephen Colbert’s satirical book title I Am America (And So Can You!) is hilarious because its syntax and grammar are so wonky. It conveys something, but it doesn’t type-check.

In the case of our attempt at defining an inverse arrow above, a diagram can help us. If we are starting with an arrow f and trying to get an arrow g going backwards, we are thinking about functions pointing in these directions:

Now the composites f ° g and g ° f are like this:

It doesn’t ‘type-check’ to ask them to be equal to the same identity, because they do not start and finish at the same objects. Instead, we ask for f ° g to be equal to the identity on the object B, and we ask for g ° f to be equal to the identity on the object A. This gives the following definition:

For any arrow A→fB in a category, an inverse for it is an arrow B→gA such that f ° g = 1B and g ° f = 1A

We say that f and g are inverses to each other, and we say that f and g are invertible. When two objects have a pair of inverse arrows between them, we consider them to be ‘more or less the same’ in a category, or ‘isomorphic’. The etymology of this is from Greek, where ‘iso’ means ‘equal’ and ‘morphosis’ means ‘to form’ or ‘to shape’, so the idea is that this is something that is simultaneously being shaped and being equal. This is a more nuanced form of sameness than equality because it allows for things to be not exactly the same but still in some sense equivalent. It’s also more nuanced because it gives us scope to look at different contexts and have different forms of sameness between the same pair of objects, depending on the context. That is, we can look at the same objects inside different categories, with different arrows, and get different types of sameness from the different category structures. This is like considering people in different categories according to different metrics. In one category we might measure socioeconomic background, in another, education; two people might be ‘the same’ with respect to their educational qualifications, but very different in terms of socioeconomic background, which might give us some insight.

The concept of isomorphism is possibly the most important point of category theory, and perhaps the most important concept of the latter half of this book. It’s the nuanced version of sameness we’ve been building up to all this time. Where ‘bijection’ was the important technical word to remember earlier on when we were focusing mostly on sets and functions, ‘isomorphism’ is the important technical word to remember now that we have gone into a more abstract world of arrows inside categories.

Before we go any further though, we should check that isomorphism deserves to count as a form of sameness at all. The rest of this chapter will be providing justification for that. First we will look at some examples, and see that this definition gives us something sensible in some example categories; then we’ll think more ideologically, and check that isomorphism has good abstract properties that will allow us to work with it similarly to how we work with equality. We’ll start with a simpler example: inequalities of numbers.

Inequality of numbers

The word ‘inequality’ for numbers is ambiguous, but in maths it has a technical meaning: it isn’t just about numbers not being equal, it is about knowing that one quantity is less than another, for example 1 is less than 2, or in symbols:

1 < 2

In normal language we might also call the following an inequality:

1 ≠ 2

But this doesn’t give us as much information as 1 < 2, as it only tells us that 1 and 2 aren’t equal, whereas the first expression tells us what order they go in, according to size. ‘Less than’ is another example of a relationship in between the black-and-white of ‘the same’ and ‘not the same’.

We can investigate some of the properties of the relationship < just as we have done for other relationships. It is transitive: if a < b, and b < c, then it is definitely true that a < c. The relation is not reflexive because nothing is less than itself, but we can ‘upgrade’ to a reflexive relation by using less-than-or-equal-to instead, which is written with the symbol ≤. This is still transitive, and it is now also reflexive, as for any number a the following is true:

a ≤ a

We can now investigate symmetry. Here symmetry is the same concept as it was for equivalence relations; we don’t need the extra condition on inverses because in this example two numbers simply are or aren’t related. We only need the extra part of the definition when it’s possible to have many different relationships between the same objects.

As with logical implication we see that the relation is not symmetric: if a ≤ b, it does not follow that b ≤ a; for example 1 ≤ 2 but it is not true that 2 ≤ 1. Then, as with logical implication, we can take an interest in when the situation is symmetric: when is it true that a ≤ b and b ≤ a? The answer is: this is true whenever a = b.

In fact, this is a slightly collapsed example of a category. The objects are numbers, and there is an arrow from a to b whenever a ≤ b. Then an isomorphism in this category is just a place where a ≤ b and also b ≤ a, and thus the isomorphisms are exactly the equalities in this case. It’s a collapsed version of a category in the sense that it has a much simpler structure than categories are capable of in general, but it’s still a good practice example.

It might seem convoluted to extract the concept of equality back out of inequality, but there are two main points of doing this. One is that it sometimes turns out to be a useful way of proving that two things are equal when we’re working in tricky parts of higher maths. This often happens when we are trying to prove that something is a bijection, that is, a perfect pairing up of the elements of one set with the elements of another. We need to prove that for every element of the second set, there is exactly one element of the first set that is paired with it. We often start by proving that there is at least one element paired with it, and then we prove that there is at most one element paired with it, thus we conclude that there is exactly one element paired with it. That is, if the number of elements paired with it is n, we have shown that 1 ≤ n, and also n ≤ 1, so we conclude that n = 1.

The second point of looking at inequalities in both directions is that ≤ is a prototype arrow that we can make into a category, and so looking at them in both directions is an example of an isomorphism. The fact that the isomorphisms are the equalities here is a reassuring sign that isomorphism is giving us something like sameness, and this gives us another hint that this idea is widespread in maths and therefore worth abstracting and making into a general theory.

In maths, once we’ve tried something out in a slightly collapsed, redundant or degenerate situation, we move on to trying it somewhere where more is going on. So we’ll try it now with functions, which is a lot more involved.

Invertible functions

In a way, the definition of inverses in a category is a sort of game where we copied a concept we’d already met (multiplicative inverses) with some nuance to make sure it would type-check in the world of arrows. This is satisfying at an abstract level, and is a valid way that abstract mathematicians develop new concepts. But then it’s important to see what the game has actually given us, and in this case it’s satisfying because it ties up with a concept we’ve already seen: bijections.

Before we see that, it’s worth us getting a better grip on why functions aren’t symmetric in the first place; this will help us see why the situation is so special when we are able to identify if and where there is some local symmetry.

The idea is that a function sends one set of elements to another set of elements; here ‘element’ is just a technical maths word for ‘thing in a set’. Here’s a picture depicting the idea of a function. The set of ‘inputs’ is called A and the set of ‘outputs’ is called B, and the arrows show what happens to each input.

Note that the dots all represent different elements although they look the same as each other in this picture; the arrows are depicting the action on individual objects, so they aren’t the arrows in the category we’re thinking about (which act on whole sets) but are just a helpful visualisation.

In a way the outputs should be called ‘potential outputs’, because they don’t necessarily actually get produced by any of the inputs. For example, there is a function from the set of all the whole numbers to the set of all the whole numbers (or, as mathematicians say more succinctly, ‘from the whole numbers to the whole numbers’) that multiplies every number by 2. The only whole numbers that result from doing this are the even numbers, but although none of the odd numbers is actually ‘hit’ by this process, it still counts as a function from the whole numbers to the whole numbers. Here’s a picture of how that function works:

Everything on the left has an arrow starting there, but there are some things on the right with no arrow ending there. This is part of where the asymmetry in functions comes from: every input (on the left) has to get sent to an output (on the right), but it is possible for an output not to be hit by anything.

The other aspect of asymmetry is the fact that multiple inputs can land on the same output, but each input can only go to one output. This is because functions represent fully determined processes, so if you feed an input in you need to know exactly what output it’s going to give – there can’t be two possible choices. This is why ‘square root’ is not a function, as there could be a positive and a negative square root; it’s only a function if we specify that we are definitely taking the positive square root (or indeed only the negative one). However, it’s possible for a function to give the same output for two different inputs; for example the squaring function gives the same answer for 2 and for −2, as they both square to 4. In terms of the picture, this means that it is possible for two arrows to end at the same output (on the right), but it is not allowed for two arrows to start at the same input (on the left). Here’s an example of a valid function with two inputs going to the same output:

If we think of the left as women and the right as men, then it’s like saying that every woman has to be married to exactly one man, but men are allowed to be married to many women or even not be married at all. (So, maybe, like fundamentalist Mormons.) Here is how we formally state the condition for being a function:

For every object a of A there is exactly one object b of B such that f (a) = b.

It may be hard to read that formal definition and understand what it means intuitively. In a way, that’s the whole point of the formality of maths – to pivot us into a realm that is good for applying rigorous logic, which is perhaps the opposite of intuition. When I read a formal definition in maths that is un-familiar to me, I have to ‘translate’ it into intuitive terms slowly. However, once I get more familiar with it, I can internalise its meaning, and then I can look at it and know what it means immediately without having to do a translation. It’s a bit like learning a new language, where at first you have to translate everything back into your own language, but after a while you start to think in the new language rather than having to do a translation all the time.

Note that there are situations in which we do want to allow some of those behaviours that are not allowed for functions; this just means those situations are using a more general concept, not functions. Perhaps you’re a supermarket recording what all your loyalty cardholders bought today – some people bought nothing, and plenty of people bought more than one thing. So that is not a function from ‘cardholders’ to ‘groceries’, but it’s still a valid situation. It could be modelled using a mathematical concept called a ‘relation’ instead (not an equivalence relation). Using functions is much simpler, and so if we can model a situation using functions we do, and we only go to the more complicated concept if we really have to.

Now let’s think about bijections again. We previously said that a bijection between sets A and B is a ‘perfect pairing’ between their elements, so that every element of A is paired with exactly one element of B, and every element of B is paired with exactly one element of A.

In fact here is the technical definition of a bijection. A bijection is a function A→fB satisfying this:

For every object b of B there is exactly one object a of A such that f (a) = b.

This is a sort of ‘reverse’ of the condition for being a function in the first place, which is this:

For every object a of A there is exactly one object b of B such that f(a) = b.

The condition on being a function says that each element on the left has exactly one element on the right that pairs with it, and we saw that this condition is asymmetric because elements on the right could still have no elements on the left pairing with them, or more than one. The condition on being a bijection imposes that symmetry, saying that on top of the condition for being a function, it must also be true that every element on the right has exactly one element on the left pairing with it. If each element on the left is paired with exactly one on the right, and each element on the right is paired with exactly one on the left, this is a ‘perfect pairing’.

So much for thinking about the idea of symmetry for functions; we now want to relate this to the formal definition of an isomorphism, which involves arrows going back and forth. Now, if a function f is a bijection we can define a function g going back from B to A by making use of the definition of bijection as follows:

For each element b of B there is exactly one object a of A such that f(a) = b so we define g (b) to be this element a.

This wouldn’t work if f were not a bijection because:

• there might be no object a of A such that f (a) = b, in which case g wouldn’t send b anywhere, which is not allowed in our definition of function; and

• there might be two objects a and a’ such that f (a) = b and also f (a') = b, and then we would be trying to define g to send b to two different places, which is also not allowed in our definition of function.

You might now want to check that this definition of g really does give an inverse for f, and you might also want to check that if f has an inverse, then it must be a bijection, so that we are sure we really do have two logically equivalent characterisations of the same thing, that is:

Being a bijection implies having an inverse. and Having an inverse implies being a bijection.

And, as with all logically equivalent but different characterisations, we get different insights from the two characterisations, which is the whole point of doing this sort of thing in the first place.

We previously had a way of thinking about sets as being ‘sort of the same’ when they are in bijection with one another. And now we have a different way to describe a bijection, by saying it’s a function with an inverse. Functions with inverses can thus give us a way to say that sets are ‘the same’ as each other in a more nuanced way than equality, just like how logical equivalence – when logic flows both ways – gives us a way to say that statements are ‘the same’ in a more nuanced way than them just being identical.

In the language of category theory what we’ve just done is show that in the category of sets and functions, the isomorphisms are exactly the bijections. What we did before was show that in the category of numbers and inequalities, the isomorphisms are exactly the equalities. We’ve also hinted at the fact that in the category of logical statements and logical implications, the isomorphisms are the logical equivalences. These are all examples showing that isomorphisms give us sensible notions of sameness that might be more nuanced than mere equality. This is a good start, but we also want to know that the concept of isomorphism abstractly shares some of the properties that equality has. This is for two general purposes: ideologically, so that we know we haven’t strayed too far from the original concept of equality; and practically, so that we will still be able to perform calculations and manipulations with isomorphism somewhat as we are used to doing with equalities.

Isomorphism is an equivalence relation

One first thing we could check, before going further, is that isomorphism is an equivalence relation. We have seen that an equivalence relation is a relation that is reflexive, symmetric and transitive. So first of all let’s check that the relation of ‘being isomorphic to’ is reflexive: any object is isomorphic to itself via identity arrows. If we call the object A, then we could draw the situation like this:

Since both of these arrows are identities, both the composites are also identities.

Next we can check that the relation is symmetric, but this is more or less built into the definition. If you look at this pair of objects with a pair of inverses between them:

the objects A and B are playing exactly the same roles as each other in the picture, and so this relation is symmetric.

Now, just like with sets that are in bijection with each other, transitivity is a little more complicated to check. The following proof may seem rather abstract, but that’s the whole point, as it means we can then immediately apply it in a wide range of situations. As usual, you can just gaze at this as a piece of abstract art if you like.

The situation we’re looking at is that we have isomorphic objects A and B, and isomorphic objects B and C, and we want to know if there’s an isomorphism between A and C. We can draw a picture of it like this:

Now, the first thing we have to do is see if there’s a pair of arrows going back and forth between A and C. For this, we can use the composite arrows:

Now we have to check that the composites both ways round give the identity, and they do, because the individual composites both ways round give the identity. In category theory we often do proofs using diagrams where we read the diagram ‘dynamically’, which is like walking around the diagram and recalculating it as we go along. So we might look at this diagram of arrows which we are trying to show composes to the identity:

A→fB→f′C→g′B→gA

and then we see that the middle part sort of ‘collapses’ to the identity, so we fill that in like this:

and then we see that this means the outside part can also ‘collapse’ to the identity, so we fill that in as well, showing that the whole composite is the identity as needed:

We do the composite the other way round similarly. In practice, when we get used to doing this, we just draw the entire diagram at once, so for the composite the other way round we would draw this:

We then understand that each region of the diagram is an equals sign in one stage of the reasoning, and that we can stick all these equals signs together, using transitivity of equality, to get an overall equals sign.

This is how calculations and proofs often work in category theory. I wanted to include the argument here to show how different it is from the kind of maths calculations typically done at school involving numbers and equals signs and answers. The kind of diagrammatic reasoning done in category theory feels a lot more like fitting a jigsaw puzzle together to me, and I find it very satisfying.

I also wanted to include it even if you don’t understand it because the act of doing maths involves a lot of not-understanding things. It might seem that the people who are good at maths are those who understand it all easily, but in the end I think that to make progress in maths one needs to get used to not understanding things but sitting with them anyway. I think a better indicator of your future success at maths is not how quickly you can understand something, but how long you can spend with something you don’t understand.

Incidentally we didn’t really need to draw the second diagram: the situation is entirely symmetric, so the proof for the ‘other way round’ composite follows in an analogous manner. In abstract maths we usually just say ‘similarly’, and understand that this means ‘if we did write it out it would be the same argument, so we don’t actually have to do it because we all agree that it could be done’. It’s a fun thing about abstract maths, that agreeing something can be done is sometimes the same as getting it done. This is not true of, say, building a house, where you first have to agree it can be done and then you actually have to do it.

Anyway, we now know that isomorphism is an equivalence relation (reflexive, symmetric and transitive), but is that enough to mean it can count as a form of sameness?

Isomorphic objects are treated as the same

There is a profound theorem in category theory which says that if two objects in a category are ‘the same’ in this more nuanced sense of isomorphism inside a particular category, then the entire category actually sees them as the same. The category really can’t tell them apart, because they play the same role in the category, and the category only sees things according to the roles they play. Another way of thinking of it is that it’s because they have the same relationships with everything else in the category, and the category only sees relationships, not intrinsic characteristics, so it can’t tell that the objects are different.

I sometimes like to think that this is similar to how we can tell identical twins apart. It is very hard to tell them apart when you’ve just met them, but once you get to know them and develop relationships with them, it becomes quite easy to tell them apart. When I was at school there were two pairs of identical twins I remember. One pair were musicians: one played the flute and the other played the cello. Because I also played music I had no trouble telling them apart, because I played in a string quartet with one sister, but only played in the orchestra with the other sister. However, the other pair of identical twins were not musicians and I didn’t have much to do with them, so I don’t think I ever did feel I could reliably tell them apart, because I had exactly the same relationship with both of them.

I don’t think I’m very good at remembering people’s faces in general, but I am much better at remembering the relationships I have with them. If I meet someone once I might totally forget what they look like and what their name is, but remember our entire conversation. I like to think I’m thinking like a category theorist – remembering the relationship and not the intrinsic characteristics.

Here’s a more formal explanation of why a category thinks of isomorphic objects as the same. Suppose we have a pair of isomorphic objects A and B with the isomorphism given by a pair of inverses f and g:

Now imagine that you are another object in this category, say X. The point is that you don’t ‘see’ any intrinsic characteristics about A or B, you only see your relationships with A and with B. In formal category theory, that means you only see any arrows you have to or from A or B.

Now imagine you have an arrow from X to A, like this:

then you can compose it with f to get a corresponding arrow to B:

But conversely, for any arrow you have from X to B, like this:

you can compose it with g to get a corresponding arrow to A:

Now since f and g are inverses, this makes your arrows to A perfectly match up with your arrows to B. In fact, it’s a bijection, given like this:

This means you have ‘the same’ relationships with A as you do with B. So as far as you’re concerned, A and B are ‘the same’ because they play the same role in this context. You can do the same with the arrows from A and B to X.

This is part of a very profound result in category theory called the Yoneda Lemma, after the Japanese mathematician Nobuo Yoneda. A ‘lemma’ is usually a small technical result in maths which is deemed not to be big enough to be worthy of the status of a ‘theorem’. However, in the case of the Yoneda Lemma I think this is a piece of endearing understatement: this lemma is the basis of many advanced arguments in the field. I suppose it’s true in a sense that it’s ‘just’ a technical result, but it’s a beautiful one that makes many things happen.

In a way, all the Yoneda Lemma is doing is making precise the sense in which a category only cares about the relationships inside it, rather than about what anything actually is. Sometimes I like to describe it like this: there’s a saying ‘If it walks like a duck and talks like a duck, it’s a duck’. But the Yoneda Lemma says it’s actually about whether other ducks think it’s a duck. It reminds me of the fact that when Daniel Craig put on an American accent in the film Knives Out (and subsequently in Glass Onion) many British or even American people derided it as terrible, whereas some Americans told me it was a good rendition of a very specific American accent that I had never heard. I think that convincing people with the same accent is a better test of it than convincing British people, or even other Americans.

The point about who gets to judge others as ‘the same’ can distract us in arguments about equality because no two humans are exactly identical and it shouldn’t be up to any individuals to judge when other people can count as ‘the same’. But really we don’t even need to ask about that because equality of humans should be about when they can play the same role in context, so it’s about their relationship with other people they’re interacting with, not about whether someone else can’t tell them apart.

There’s a deep moral in there for all of us, which is that ‘equality’ of humans isn’t about whether we have the same intrinsic characteristics, because obviously no human has exactly the same intrinsic characteristics as another one; even identical twins have different fingerprints. And crucially, people might be able to play the same role in one context but not another. It turns out sameness isn’t a fixed property, but depends on context. And the context we consider is a choice we make, and so we should make that choice carefully.

9 SAMENESS IN BRANCHES OF MATHS

In this chapter I’d like to do a short tour of some different branches of maths, and look at how the concept of sameness that we defined in the previous chapter comes up in those different fields. These will be topics we’ve touched on earlier, but we will treat them a little more rigorously, using the tools of category theory that we have now met. The key is that once we’ve decided what concepts we’re going to think about, we next decide what type of relationships between them we’re interested in. In the language of the previous chapter, this means we’re deciding what category structure to put them in; more informally, we’re deciding what context to put them in, and what we’re going to count as important for now.

Once we’ve done that, we can look at where the relationships f low in both directions, that is, where there are isomorphisms, which is when category theory tells us that things count as somewhat the same. In the previous chapter we saw that, in the case of sets and functions, the definition of isomorphism gives us back the concept of bijection that we had already defined directly. It is often the case that the first step after defining an abstract mathematical concept is checking that it gives the thing we want in some basic case. This sometimes makes it feel like you’re proving things that are obvious, but I think of it as making sure you didn’t break anything by expanding your world. Or it’s like testing out your new tools on something basic first before trying them on something complicated. Whenever I move house the first thing I bake in the oven is a basic sponge cake, and whenever I sit down at a new piano I play the same piece of Bach that is very familiar to me.

But another aspect of studying isomorphisms is the reverse of the process I just described: we might start with some intuition about what we want to count as ‘the same’ in certain contexts, and then maths gives us a way to make that intuition rigorous. In this chapter we’re going to see some situations in which the isomorphisms of category theory give us back the ‘sameness’ that we felt in our gut, but now we can put that feeling on a much more secure, logical footing. Maths gives us a way to make our intuition precise.

We’ll move on from sets and functions, to sets equipped with some sort of extra structure on them. It’s an additional framework that tells us something about how the elements of the set are related, so that we don’t just have objects in a random pile or floating aimlessly in space. It’s the difference between knowing only the names of everyone at a party and knowing who’s related to whom and how they all know the host. It’s the difference between a crowd of people thronging around a bar and people queuing up to order drinks. It’s the difference between open seating in an auditorium or on a plane, versus people all having an assigned seat so that they know where to go.

In maths, as well as in life, the objects we’re thinking about often have some structure holding them together that is important to how we think about them. So we build up from the world of sets and functions, to worlds of sets with different forms of structure on them. In each of the worlds we think about in this chapter we’ll be interested in a different type of structure on our sets: an ordering, a way of combining things, or a way of measuring how far apart things are. In each case we no longer want to think about any old functions between the sets, but rather, we want to think about functions that ‘preserve’ the structure we’re interested in. This means that the structure doesn’t fall apart when we move to a different set. Another way of thinking about it is that we want to think about pertinent relationships between the sets, and those are ones where we can relate not just the objects of the sets, but also the structure on them too. In each case, this gives us a concept of sameness that is more subtle than bijections between sets: we will be restricting our attention to perfect pairings that don’t just pair up the objects perfectly, but match up the structures between them as well. This is an important general principle of mathematics and therefore of category theory. We’ll start with orderings.

Ordered sets

One piece of structure we might have on a set is an ordering, that is, a way of putting all the objects in an order. Take numbers for example. The set of numbers

{ 1, 2, 3, 4, 5 }

is just a set of five things, but our brains sort of automatically think of them in an order because we’re so used to numbers going in an order. After all, one of the whole points of numbers is to measure quantity so that we can compare different quantities.

When we say that the numbers 1, 2, 3, 4, 5 are ‘in order’ we’re saying that 1 is less than 2, which is less than 3, which is less than 4, which is less than 5. We can write this more succinctly in symbols as follows:

1 < 2 < 3 < 4 < 5

This fact about which numbers are greater and less than the others is still true about those numbers even if we write them down in a different order, say like this:

{ 3, 4, 1, 5, 2 }

Writing it like this makes it harder for our brain to process because the numbers have not been written ‘in order’, that is, the order in which we’ve written them does not match their size order. This idea of whether or not different orderings match up is key, whether it’s size order and written order or any other different orders.

Sets that contain things other than numbers don’t necessarily have an obvious ordering on them. In that sense the set { 1, 2, 3, 4, 5 } is quite different from the set

{ monkey, 99, teapot, *, @ }

even though they’re in bijection with each other. That is, we can match up the objects in one set with the objects in the other, because they have the same number of elements, and according to what we explored in the previous chapter, that means the sets are ‘more or less the same’. However, the first set has an obvious ordering on it, by size, but the latter set doesn’t: it’s a rather random collection of things that are not very obviously related.

We could make it a bit more sensible, perhaps, by writing the objects out as words

{ monkey, ninety-nine, teapot, asterisk, at }

and then putting them in alphabetical order

{ asterisk, at, monkey, ninety-nine, teapot }

It’s still not exactly what I’d call a sensible set, but it does now have an order to it besides the order in which we write it down, just like { 1, 2, 3, 4, 5 }.

What I’ve done here is give the set some extra structure, called an ordering, or rather, I’ve focused on the alphabetical ordering that the set already had if we just thought about it harder. Sets we think about in life often have an ordering on them, whether we consciously notice it or not. Sometimes it’s just because we often put things in alphabetical order, or because we think of numbers in size order. Sometimes it’s because we’re standing in a queue. Sometimes it’s because we have been very explicitly ranked in a competition or a race.

If we do have sets with orderings on them, we often want the functions between the sets to preserve that extra structure as well. One reason is so that the resulting notion of ‘sameness’ takes the ordering into account. In my example with the weird set above, I could then list the objects in alphabetical order like this:

1. asterisk

2. at

3. monkey

4. ninety-nine

5. teapot

I’ve made a bijection between the set of numbers and the set of weird things, but it’s not any old bijection – it’s one that makes the orderings match up. We’re going to see that the concept of isomorphism in category theory gives us these coherent bijections, if we proceed carefully.

We also might want to take the ordering into account even if the function isn’t a bijection. This is one of the issues at the root of why dealing with inequalities (expressions involving a < sign) is harder than dealing with equations. For example, if we’re given this inequality relationship:

2 − 3x < 8

we have to be more careful manipulating it than if we’re given this equation:

2 − 3x = 8

For the equation version, we can subtract 2 from both sides to get this:

−3x = 6

and then we can ‘change the sign’ to get:

3x = −6

and divide both sides by 3 to get:

x=−63=−2

However, with an inequality, that is, < rather than =, we have to be careful about which of those operations actually respects the < relationship. The operation we have to be careful about here is ‘changing the sign’. If we’re starting with this inequality:

2 − 3x < 8

then we can subtract 2 from both sides (just like in the equation version) to get this

−3x < 6

but at the next step we can’t just ‘change the sign’ to produce:

3x < −6

because that’s not a valid logical step: if we know

a < b

it’s not true that:

−a < −b

For example, 1 < 2 but −1 is not less than −2; in fact −2 < −1. So ‘changing the sign’ does not maintain the < relationship; it reverses it. The first step of the above manipulations still works because subtracting 2 respects a less-than sign. That is:

If x < y, then x − 2 < y − 2.

By contrast, when we’re ‘changing the sign’ we’re really multiplying both sides by −1, which reverses a less-than sign.

This idea of respecting relationships can be applied more broadly to any sets with orderings on them, and any function between them – we can ask whether or not the function respects the ordering. This means that if a comes before b as inputs, then f(a) should come before f(b) as outputs: they don’t get out of order when you apply the function. Or, stated formally, a function preserves ordering when the following is true:

If a < b, then f (a) < f (b).

In summary, we have upgraded our framework of sets and functions, to sets-with-an-ordering together with functions-preserving-the-ordering. The technical terms for these concepts are ‘ordered sets’ and ‘order-preserving functions’. In the language of the previous chapter, we are now thinking about a category whose objects are ordered sets and whose arrows are order-preserving functions.

Some examples of functions that are order-preserving are the functions that just add a fixed number (positive or negative) or functions that multiply by a positive number. Functions that don’t work include those that multiply by a fixed negative number. Another function that doesn’t preserve order is the squaring function, because for example −2 < −1, but when we square those we get 4 and 1, so the order is reversed there.

The question of preserving ordering comes up if you’re in a queue at the supermarket and they open up another checkout. You don’t want people who are behind you in the queue to make a dash for the new queue, and end up in front of you – that would be rather annoying, because order would not be preserved. In this scenario, you do want the order to be preserved.

However, in some situations the whole point is to change the order, like if you’re shuffling a deck of cards, or, as one of my students pointed out, in triage. For medical emergencies it’s important to treat people in order of the severity of their needs rather than in order of arrival, so triage changes the ordering, specifically so that it’s not first come first served any more (although it might be, by coincidence, if people actually arrived in order of the severity of their afflictions). I apply this approach to question-and-answer sessions as well – I don’t believe that the person who puts their hand up first should necessarily get to ask the first question. I’d rather sort of ‘triage’ the questions into an ordering according to interestingness.

We can now go back to our original point, which was to think about isomorphisms in this framework of orderings. When we were thinking about ordinary sets and functions, it was quite satisfying that the isomorphisms (the invertible functions) turned out to be exactly the bijections. Now we’re in a more subtle category, the category of ordered sets and order-preserving functions, and it turns out that the isomorphisms here are exactly the bijections that respect the ordering. For example, if we’re looking at our earlier example:

1. asterisk

2. at

3. monkey

4. ninety-nine

5. teapot

this bijection preserves the order between the numbers (in increasing order) and the words (in alphabetical order). However, the following is a bijection, but it does not preserve those orderings:

1. ninety-nine

2. monkey

3. at

4. teapot

5. asterisk

Sometimes when we rank things or people we allow ties, where we say we are unable to decide which comes above the other. Or, more subtly, we admit that some of the things are incomparable. Maybe person A is definitely a better singer than person B in every way, but with person C and person D it’s harder to say, because person C has a better voice but person D is more expressive. This is called a partial order in maths, and it means we don’t have to compare everything with everything else, but we just compare things where we can. We can then think about functions between partially ordered sets and think about whether or not they preserve order. It’s still the same basic idea: if a comes before b on the left, then f(a) should come before f(b) on the right. We just need to remain slightly aware that if a and b were incomparable on the left, we don’t know what will happen to them on the right – they might become comparable, and that won’t violate the idea of being ‘order-preserving’ because they just weren’t ordered before.

Total orderings and partial orderings are similar in that they are both about putting things in order, but they are different in that a partial ordering is a much more flexible and general concept. If we have to put everything in a total order we’re asking for a very strict one-dimensional hierarchy. This is rarely realistic in daily life because usually there are different ways in which different things can be sensibly ordered. Unfortunately we often do try to (or have to) put things in a hierarchy, like when we rank students in exams, or rank people in a competition or for job applications. Sometimes ties are allowed, such as when students can get the same mark in exams, but that doesn’t make the hierarchy a whole lot more nuanced – it’s still one-dimensional.

I think one of the big problems is that, even if we can entirely rank things according to one criterion, when we then combine more than one criterion we don’t get a strict ranking any more. This is like trying to order all the coordinates on an x–y Cartesian plane. The x coordinates have an ordering, because they’re just ordinary numbers in a line. The y coordinates also have an ordering, because they are also just ordinary numbers in a line. But if you take pairs of coordinates, what order should they go in? For example, what should come first, (0,1) or (1,0)?

You could decide to favour the x coordinate, which would be like lexicographic ordering (which we discussed in Chapter 6), or you could decide to favour the y coordinate, which would be like colexicographic ordering. But whichever one you chose would be an arbitrary, non-canonical choice. And if you tried to map from one to the other the order would change, so those two different orderings do not count as ‘the same’.

However, if you just allow them to be a partially ordered set instead of a totally ordered one, you don’t have to decide between (0,1) and (1,0), you just allow them to be incomparable. This means that whenever we’re thinking about two different criteria (or more) we don’t have to put everyone into one strict hierarchy overall – we can just acknowledge that one person might be in front according to one criterion and another according to another. I think in life we have too much tendency to put things into one-dimensional hierarchies rather than becoming better at understanding partially ordered sets and multidimensional orderings. Practising these ways of thinking mathematically gives us more space for nuance even though many people think of maths as being quite rigid; the ways of thinking are still rigorous, it’s just that we can also become rigorous about grey areas instead of keeping everything black and white.

And perhaps we don’t need to put things into so many hierarchies at all, even in many dimensions. Sometimes we can just spot patterns, and compare patterns.

Latin square patterns

Latin square patterns are n-by-n grids filled in with n objects, in such a way that there’s only one of each object in each row and each column. We saw some of these in Chapter 1 (although we just called them patterns at the time) and again in Chapter 5 when we were looking at situations in which a small amount of information would give us a way to fill in the rest uniquely. These grids are in fact a way of representing another type of ‘set-with-structure’, but this time, instead of an ordering, we’re thinking about a way of combining any two elements of the set to produce another one. In abstract algebra these structures are called groups, and they always produce Latin square patterns when we make a grid showing how to combine the elements. (All groups give Latin squares but not all Latin squares come from groups, so groups and Latin squares are not logically equivalent.)

Addition on a 12-hour clock is one example, which we can do by making a 12-by-12 Latin square filled with the numbers 1 to 12. Addition always produces another number from 1 to 12 because we imagine we’re on a clock, and if the answer goes past 12 we start again at 1. The addition square ends up looking like this:

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We saw some examples of Latin square patterns that we said were ‘the same’ in Chapter 1.

0 | 1 | 2 | 3 |  |  |  |  | a | b | c | d
1 | 2 | 3 | 0 |  |  |  |  | b | c | d | a
2 | 3 | 0 | 1 |  |  |  |  | c | d | a | b
3 | 0 | 1 | 2 |  |  |  |  | d | a | b | c

We are now going to find a more rigorous way to say what we mean by these patterns counting as ‘the same’, using the concept of isomorphism in category theory. The process is going to follow some quite standard steps in abstract maths. First we have an idea of the things we want to study, and a gut feeling about which ones we want to count as ‘the same’. Then we organise our objects of study into a category by thinking about a sensible or pertinent concept of relationship between them. Once we’ve done that we can find the isomorphic objects in the category and see if they match our gut instinct about which ones we want to count as ‘the same’, and if they do, then we’ve built ourselves a rigorous framework for our intuition. Then we can do a deeper study of what all the different possible types of object are, which is called ‘classification’.

You might notice that none of this involves real-world applications or problem solving: this part of abstract maths tends to proceed purely from our interest and curiosity about some ideas or patterns we’ve noticed, and a motivation to have a rigorous method for reasoning with them. Once we have that, someone might be able to apply it to something in the real world, but if you’re an abstract mathematician the person applying it is usually somebody else, and the application is not usually what drives you.

In the previous section we built a category from ‘sets equipped with an ordering’ and ‘functions preserving the ordering’. Now instead of an ordering on the sets we’re thinking about a type of structure where we can take any two elements and combine them to produce another element. The Latin squares above are implicitly telling us how to do that; perhaps it’s more obvious if I draw them like this, so that they look like addition tables:

| 0 | 1 | 2 | 3 |  |  |  |  |  | a | b | c | d
0 | 0 | 1 | 2 | 3 |  |  |  |  | a | a | b | c | d
1 | 1 | 2 | 3 | 0 |  |  |  |  | b | b | c | d | a
2 | 2 | 3 | 0 | 1 |  |  |  |  | c | c | d | a | b
3 | 3 | 0 | 1 | 2 |  |  |  |  | d | d | a | b | c

The first one now looks like an addition table for a ‘4-hour clock’, instead of a 12-hour clock. The second one is more abstract as it’s letters rather than numbers, but still gives us a scheme for combining two things to provide a third, if we read the table that way.

These mathematical structures are called groups, and they have to satisfy some other rules to make sure that the way of combining things is ‘coherent’, but the main point I want to talk about now is the idea of a function respecting a group structure. Previously we looked at functions respecting orderings, and now, if we care about group structures at all, then we care about functions respecting those group structures.

For orderings the issue was:

We start with two elements in the first set, where one is less than the other, and then we use the function to send them to the second set. Are the results in the same order after using the function as the elements were before using the function?

For groups the issue is:

We start with two elements in the first set and combine them, and then we use the function to send them to the second set. If we combine them after using the function, is it the same as combining them before using the function?

That means if you take two elements a and b in the first set, you could either combine them and then apply the function, or you could apply the function and then combine them, and those two processes should produce the same answer. I’ve depicted this in the diagram below, with an asterisk denoting the operation of combining elements: it could be +, ×, or something else.

A function that preserves the group structure in this way is called a ‘group homomorphism’, and it is the type of structure-preserving function we want for studying groups. That is to say, when we make a category with groups as the objects, we usually take the arrows to be all the group homomorphisms (that is, this is the kind of relationship we choose to look at), as they preserve the structure we’re interested in.

Here’s a different way of thinking about it: as you’ve probably gathered, I love baking, and it’s not unusual for me to bake an elaborate cake for a special occasion. This leaves me with a choice: should I construct the multi-tiered cake at home and then try to transport it that way, or should I transport the tiers separately and then construct it when I get there?

A good driver would drive so smoothly that my cake doesn’t fall apart – that’s the notion of ‘respecting structure’ here. A bad driver would make too bumpy a ride and the cake might fall apart on the way; a bad driver, therefore, does not respect the structure, and the results are not the same.

Here’s an example of group homomorphisms using the 2-hour clock and the 4-hour clock. On a 2-hour clock, 2 is the same as 0, whereas on the 4-hour clock 4 is the same as 0 (and 2 isn’t). We’ll take the two numbers on the 2-hour clock to be 0 and 1, and the four numbers on the 4-hour clock to be 0, 1, 2, 3. Then we could make a function from one to the other like this:

f (0) = 0 f (1) = 2

To check that this respects addition we need to check that

f (1+1) = f (1) + f (1)

Now on the 2-hour clock 1 + 1 = 0 so:

f (1+1) = f (0) = 0

Meanwhile on the 4-hour clock:

f (1) + f (1) = 2 + 2 = 0

so

f (1+1) = 0 = f (1) + f (1)

Really we should also check the condition holds for f (0+1) and f (0+0) as well, so that we’ve checked all possibilities, but checking those is more straightforward.

However, if we did the following perhaps more obvious function it would not respect addition:

f (0) = 0 f (1) = 1

Now f (1) + f (1) = 2, but f (1+1) = 0, and those are not the same on the 4-hour clock.

Once we have decided on our category structure, we look inside it to see what the isomorphisms are. Here the answer turns out to be: the isomorphisms are those bijections that respect the group structure. This means that we need not only a perfect pairing between the elements of one set and the elements of the other, but also we need it to respect all the ways of combining elements. Crucially, this turns out to mean that the Latin square patterns are the same in exactly the way that we originally thought of, so the formalism has made our intuition precise.

Here’s an example where we’ll think about, on the one hand, addition on a 4-hour clock, and on the other hand, rotations of a square. For the 4-hour clock, that means that 4 goes back to being the same as 0, and 5 is the same as 1, 6 is the same as 2, and so on. For the comparison here, and as we did above, I want to use the numbers 0 to 3 rather than 1 to 4. The two choices are logically equivalent (we’re just using 0 instead of 4), but visually different. The addition table for the numbers 0, 1, 2, 3 looks like this:

+ | 0 | 1 | 2 | 3
0 | 0 | 1 | 2 | 3
1 | 1 | 2 | 3 | 0
2 | 2 | 3 | 0 | 1
3 | 3 | 0 | 1 | 2

This might remind us of the pattern we got for rotations of a square in Chapter 7:

+

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90

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270

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270

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We could describe this pattern in words, by saying there are diagonal stripes of the same thing going down from top right to bottom left, but a more rigorous way of saying it mathematically is to make a bijection between the elements of the 4-hour clock and the rotations of the square that respects the structure. We could do the bijection like this:

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➝

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270

It’s like painting by numbers, except that instead of our paint having different colours, the paint is also numbers, so we ‘paint over’ every 0 in the first table with a 0 in the second table, we ‘paint over’ every 1 in the first table with a 90, we ‘paint over’ every 2 with a 180, and we ‘paint over’ every 3 with a 270. Once we’ve done it, the result is the second table, showing us how to combine rotations of a square.

By contrast, here’s a function that is also a bijection, but it does not respect the group structure:

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➝

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90

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270

If we used this function to ‘paint over’ the numbers 0 to 3 in the table for addition on a 4-hour clock, we would turn the pattern into this one:

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This might look like ‘the same’ pattern in some sense, as it still has diagonal stripes going down from the top right to the bottom left. But if we look at what it’s actually telling us about how to combine rotations, we see that some things have gone wrong. This new table is telling us, for example, that if we do a 90-degree rotation and then another 90-degree rotation we get 0, which is not what happens to a square. It tells us that if we do 180 degrees and then another 180 degrees we get 90 degrees, which is also not what happens to a square.

If we rewrite the information in this new table with the rows and columns in the same order as before this becomes clearer; note that we’re not changing the information here, we’re just reorganising it visually to help us see what’s happened to the pattern:

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Now we can say ‘we have destroyed the diagonal pattern’, but that is still not a very precise way of saying what has happened. What does it mean to say that this new function has not preserved the pattern in the table?

We can describe what has happened quite precisely by saying: ‘the function does not respect the group structure’. If we call the function f, then respecting the group structure means this:

For all numbers a and b on the 4-hour clock: f (a + b) = f (a) + f (b)

To show that this has failed, we just have to find one example of a failure. We can pick a and b to be 2 and 2, and then we will show that

f (2+2) ≠ f (2) + f (2)

Now, on the left-hand side we have: 2 + 2 is the same as 0 on the 4-hour clock, so:

f (2+2) = f (0) = 0

Whereas on the right-hand side we have: f (2) = 90, so:

f (2) + f (2) = 90 + 90 = 180

The point is that while this new function is a valid bijection between the two sets, it doesn’t respect the particular pattern we’re thinking about, so it’s not a structure-preserving function. We’ve found that the isomorphisms in this category of groups and structure-preserving functions are based on the isomorphisms between sets (the bijections) but with this extra condition about respecting patterns.

The next thing mathematicians do is try to understand how many patterns are possible that are genuinely different, that is, actually not isomorphic in this category. This is called classification.

Classification of groups

We have found a group with four elements arising from two different situations: the 4-hour clock and the rotations of a square. We found that these groups are actually isomorphic in the category of groups and their homomorphisms (structure-preserving functions), so they’re not really different. Remember, in category theory, isomorphic objects count as ‘the same’. We might then ask if there are any groups with four elements that are genuinely different, that is, not isomorphic to the one above. The answer turns out to be yes: a different one arises from the symmetries of a rectangle.

A rectangle has four possible symmetries: rotation by 0 degrees, rotation by 180 degrees, reflection across a horizontal line, and reflection across a vertical line. If you equip yourself with a piece of paper and flip it around a bit, you can investigate how these symmetries combine with each other. Note that in maths, reflection is performed as an action, so that reflection across a horizontal line means we flip the page over to the back. If you start with your paper in portrait, then a reflection across a horizontal line is what a printer might describe as ‘flip on short edge’, and a reflection across the vertical is what a printer might call ‘flip on long edge’.

Here is the pattern we get if we combine those moves in all possible combinations, where I’m writing V for ‘reflection across vertical line’ and H for ‘reflection across horizontal line’:

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This pattern doesn’t look like the one for rotations of a square, but can we show it’s genuinely different? How can we be sure there isn’t some way of reordering the rows and columns to produce that same diagonal pattern we had before? Technically what we are trying to prove is that there is no possible isomorphism between these groups – rotations of a square and symmetries of a rectangle – in the category of groups and group homomorphisms. We would have to show that there is no possible bijection between the two that preserves structure, so we could go through every bijection and show that none of them works. However, there are quite a few bijections (4 factorial in fact, which is 24) so it would be rather tedious to write them all out, though you might enjoy it if that’s your kind of thing.

Instead we can do something a bit cleverer or perhaps sneakier. We can put the grids for the two scenarios side by side, and look down the diagonal shown below, which gives the result of combining each element with itself.

In the left-hand diagram we see that there are two possible answers: 0 and 180. In the right-hand one there is only one possible answer: 0. This means that there’s no way to make a bijection between the elements on the left and the ones on the right that respects the pattern, because no matter how you relabel the ones on the left you are doomed to get two different elements on the diagonal marked, unlike on the right. (That is not quite a rigorous argument but is the idea behind the rigorous argument.)

In summary, we have found two groups with the same number of elements but which are genuinely ‘different’ in the sense that they are not isomorphic in the category we’re thinking about, which in this case is the category of groups and group homomorphisms. In fact, these two examples are all the possible groups with four elements – any group with four elements has to be isomorphic to either the group of rotations of a square, or to the group of symmetries of a rectangle. Amazingly, among all possible four-element groups in existence, those are the only two possibilities that are different in this sense.

Mathematicians then go further and ask how many different possible groups of each size there are. There are infinitely many possible groups of each size, just like there are infinitely many possible sets of each size, so we have to find some theor-etical way of working out how many possibilities there are that aren’t isomorphic, without having to check them all one by one. This is called ‘classification’, and it’s one of the general projects of mathematicians when we are thinking about a certain kind of structure. That is, we try to work out how many different possibilities there are that are not ‘the same’ according to the type of sameness we have chosen.

For groups, the classification is quite easy when the sizes are small, and then gets spectacularly difficult when the sizes are large.

If there are two elements there is only one possibility:

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This is actually the only possible 2 × 2 Latin square at all – any other 2 × 2 Latin square pattern you make will be ‘the same’ in the sense above. (This is different from the 4 × 4 case where there are some other Latin square patterns that don’t come from groups.) This Latin square comes from various group structures that pop up all over the place, including addition on a 2-hour clock, rotations of a (non-square) rectangle, multiplication of 1 and −1, and many more.

For three elements here is the only possibility:

a | b | c
b | c | a
c | a | b

This pattern arises, for example, if you think about rotations of an equilateral triangle, or addition on a 3-hour clock. Two rotations can always be combined to produce a third one, but if you go round by 360 degrees that counts as the same as 0.

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On a 3-hour clock, we go back to 0 when we get to 3 rather than waiting until 12, so 4 counts as the same as 1, 5 counts as the same as 2, and so on; you might have met this before as ‘arithmetic modulo 3’. It produces an addition table like this:

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This table is ‘the same’ as the one for rotations of an equilateral triangle. It’s always possible to make an n-by-n Latin square that corresponds to rotations of a shape with n sides, where all the sides are the same length and all the angles are the same. That shape is called a ‘regular n-sided polygon’. It turns out that if n is prime the pattern you get via these rotations is the only possible pattern for a group with n objects, but if n isn’t prime there are many more possibilities. When there are four elements that’s the first time there are two possibilities that count as different, and they are the two we discussed above: the one with diagonal stripes, as exemplified by the rotations of a square, and the one we found from the symmetries of a rectangle.

If n isn’t prime we can express it as p × q for some smaller numbers p and q. It turns out we can always make a group structure for p × q elements by starting with a p-element group and a q-element group. The Latin square comes out like an iterated Latin square, where you make the Latin square for one of the groups, but each entry in it was already a Latin square for the other group. This idea of building up larger groups from smaller ‘basic building blocks’ is analogous to how we build up numbers from prime numbers. The prime numbers are the ‘basic building blocks’ for numbers by multiplication – any whole number can be expressed (uniquely) as a product of prime numbers. The groups with a prime number of elements and diagonal stripe patterns in their Latin squares turn out to be ‘basic building blocks’ for groups.*

That’s somewhat mind-boggling, but one example we can work through is similar to how we used Cartesian coordinates when we were thinking about ordered sets. I’ll just do it for 2 × 2 = 4 for now. The idea is to start with two 2-by-2 Latin squares with the elements 0 and 1, and then make a new Latin square where the elements are like coordinates, except we only have two possibilities for each coordinate: 0 and 1. That means there are four possible pairs, and if we plot them on a plane we get this:

Now, the Latin square for each coordinate individually looks like this:

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You can think of this as addition on a 2-hour clock, so 2 counts as the same as 0. You can also think of it in terms of even and odd numbers: 0 is like ‘even’ and 1 is like ‘odd’. If we add two even numbers or two odd numbers we get an even number, but if we add one of each, then it’s odd.

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Now we can try adding the pairs of coordinates together a bit like vectors. Maybe you don’t remember what that means, but we just add the x coordinates and the y coordinates separately. After all, if you travel some direction east and some direction north, and then some more east and some more north, you can add up the east parts and say that’s your total distance east, and you can add up the north parts and say that’s your total distance north. The only other thing to remember here is that we’re doing this on a 2-hour clock, so 1 + 1 = 0. We get this:

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(0,1)

(1,0)

(1,1)

(0,0)

(0,0)

(0,1)

(1,0)

(1,1)

(0,1)

(0,1)

(0,0)

(1,1)

(1,0)

(1,0)

(1,0)

(1,1)

(0,0)

(0,1)

(1,1)

(1,1)

(1,0)

(0,1)

(0,0)

If you stare at this table a bit perhaps you can see that it is just like the pattern we got from the symmetries of a rectangle, and it is a 2-by-2 Latin square where each individual entry is itself a 2-by-2 Latin square.

When we have bigger grids, coming from groups with more elements in them, there are more possibilities that are not iterated Latin square patterns. Here’s the table for doing all the symmetries of an equilateral triangle. There are six symmetries altogether: three reflections (across the lines shown) and three rotations (by 0, 120 or 240 degrees).

When we combine them, we get this table:

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If you look closely you’ll see that the two bottom quadrants are not actually the same as their diagonally opposite ones on the top – they’re twisted. This is a slight twist that we can get into the situation a bit like the twist in the situation with the Möbius strip. (We saw some twisted iterated Battenberg structures in Chapter 5, but I can now be more precise and say that those were Latin square patterns that did not come from groups.)

By contrast, here is an example of a 6-by-6 table that is an iterated Latin square:

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If we divide this pattern into four 3-by-3 quadrants we see that the top left and bottom right are exactly the same (not twisted), as are the bottom left and top right. Now the question is: is that genuinely different from the pattern in the table for symmetries of a triangle? That is, is a structure-preserving bijection between them possible?

You can try to make a structure-preserving bijection between the sets:

{ 0, 120, 240, a, b, c } and { 1, 2, 3, 4, 5, 6 }

However, there are a lot of possible bijections to try and I will warn you now that mathematicians have proved it’s not possible, that is, that the two groups in question here really aren’t isomorphic.

One way we can see that it’s not possible is by looking at the leading diagonal (from top left to bottom right) again, which shows the result of combining each element with itself. In the table for symmetries of the triangle, there is one element that appears four times: 0. However, in the iterated, non-twisted version there is no such element, which means we will never be able to map those patterns to each other.

This shows that those are two ‘genuinely different’ group structures on a six-element set, as opposed to two patterns that are isomorphic but just appear differently because of how we’ve laid them out on the page.

The classification of groups with a finite number of elements has been a big project in mathematics and some of this classification has been organised into a huge book called the ATLAS of Finite Groups.* It’s huge not in terms of number of pages, but in terms of the size of the pages. As the name suggests, it is like a map of the world, made not with the expectation that everyone who picks it up needs to visit every location, or even that anyone needs to visit every location, but rather, that it is a fundamental human endeavour to understand things around us. After all, if we decided we only needed to understand some but not all of it, where would we stop? In daily life it’s important to stop at some point, just in order to get on with the logistics of life like going to school or going to work or eating food. But in the abstract world, mathematicians – like the explorers who made the first maps of the world – don’t stop. And they keep going into other worlds to explore and understand and classify those too.

Another world in which mathematicians are interested in classifying different types of object is the world of topology.

Topology

In Chapter 5 we started to think about a more nuanced version of sameness for the shapes of spaces. We mentioned the fact that spaces can count as ‘the same’ in topology even if they’re not exactly the same shape, as with these shapes that all count as circles:

Shapes can be bent around differently, as if they’re made of rubber, and they still count as the same. This means we also don’t care about lengths or corners, so all triangles are ‘the same’; indeed they’re all ‘the same’ as squares, pentagons or shapes with any number of sides.

But the rubber can also be thought of as stretchy and compressible, not just bendy. Remember, it doesn’t count as ‘the same’ if you make a new hole in it, or stick separate parts together. You can only morph things continuously. This is an idea that comes from our intuitive ability to think of shapes as ‘the same’ with some allowances built in. If we see a pile of ring doughnuts in a shop, we can probably consider that they’re all the same shape even though they’re not exactly the same shape. However, if one of them had no hole we would probably think that was different – indeed, we probably wouldn’t call it a ring doughnut any more. Ring doughnuts cook differently because of the hole, which enables the heat to get to all parts of the dough more evenly. In abstract maths the considerations are more about understanding than about practicality.

In topology, mathematicians are interested in the different paths you can take around a space, and the ways in which the spaces are connected, or have holes and obstructions. This is where the idea about this rubbery sameness comes from. Evidently that is not at all rigorous a definition, but maths often begins with some intuitive idea that is not at all rigorous, and then we work out a way to express it rigorously. In this case this rubbery sameness is called ‘homotopy equivalence’, and it starts out looking a lot like the definition of an inverse that we saw in the previous chapter, but ends up being much more subtle and technical.

The first task is to say exactly what these ‘spaces’ are that we are thinking about. At root, each one is a set of points. But it’s a set of points with some information about how the points are connected to each other. That information gets formalised into a structure called a ‘topology’ on the set. (It’s not topography, which is about physical features in a landscape.) The formal name for a set equipped with this structure is ‘topological space’. I think it’s beyond the scope of this discussion to explain this definition – it’s quite technical and I find it hard to provide an intuition for it. This is because the intuition comes from working with specific basic examples like the real line or the Cartesian plane, and the definition generalises from the way we work with those technically, rather than from the way we think about them intuitively. For the purposes of this book it doesn’t really matter what the definition is; in fact, moving away from the technical definition is an approach taken by abstract mathematicians too, who try to define the category of these spaces according to how they are related to each other, rather than defining the spaces themselves. That takes very seriously the idea that we understand things by the relationships between them rather than by their intrinsic characteristics. It’s like understanding numbers by their positions on a number line, rather than trying to say directly what they are. After all, the definition of numbers that mathematicians came up with in set theory (which we saw earlier) is arguably very unintuitive and not how we actually end up working with numbers in daily life, or even in daily mathematical life.

Here is the formal definition of topological space for the sake of completeness, but I had to look it up myself as I can never quite remember what it is:

A topological space is a set equipped with a set of subsets called open, satisfying

• Any finite intersection of open sets is also open.

• Any union of open sets is also open.

The idea is that a topological space is a set equipped with a concept of ‘closeness’, so that we have a way of saying which points are close to which. If there is a notion of distance (or metric), then we can use that, but the idea of topology is to move to more flexible situations which don’t rely on metrics; after all, if we allow stretchiness, then we’d better not be too concerned with fixed distances. Instead, we have a concept of ‘open set’, which is a bit like a neighbourhood, but is also quite different for technical reasons.

In order to continue with the framework of category theory we next decide what sort of relationships between these things we want to use, that is, we decide what we want the arrows in the category of topological spaces to be. As we are dealing with some ‘sets with structure’, we gravitate towards ‘structure-preserving functions’ as we did for ordered sets and for groups. These are called continuous functions because they don’t break things apart. That is, if two points were close together before we apply the function, they should still be close together afterwards. Unfortunately for the purposes of exposition, we have to think about this backwards. I’ll include the definition for completeness, but we won’t need it in what follows, so please don’t worry about understanding it:

Given topological spaces X and Y, a function f from X to Y is called continuous if, given any open subset U of Y, the set of points sent to U by f is an open set of X.

If you have ever met continuous functions before, they might have been defined in a more restricted context, like the real numbers; the above definition is a generalisation of that. The real line is a particular example of a topological space, but we are about to see many examples that are not like the real line at all.

Once we have a category, we proceed as we did for ordered sets and for groups, and ask ourselves where some symmetry occurs, that is, we ask ourselves what the isomorphisms in the category are. So we look at a pair of topological spaces A and B and a pair of continuous functions between them:

that compose to the identity both ways round. And unfortunately, unlike for ordered sets and for groups, we find that this definition is too rigid for what we’re looking for – the functions f and g will still have to be bijections, with the extra condition that they preserve structure according to our definition. That is, an isomorphism in this category is a continuous function that is a bijection, whose inverse is also continuous.

The problem with it being a bijection is that we won’t be able to really seriously collapse or stretch things. We wouldn’t be able to squash a line all the way down to a point, for example, because a line has infinitely many points but a point only has one. We also wouldn’t be able to stretch a point out to a line. So we would get the bendiness of rubber, but not the stretchiness of playdough.

Actually the playdough metaphor is breaking down here, because even if you stretch or squash playdough you still have the same playdough molecules, so you could argue that this was still a bijection, molecule to molecule. To put it another way, you still have the same volume of playdough, whereas we want the ability to stretch something so that there is actually more of it.

Anyway, we end up making a yet more relaxed notion of sameness. We’ll go into this in more detail in the final chapter on higher dimensions, but the idea is that instead of saying that the composites both ways round have to be the identity, we introduce another level of forgiveness and say that the composites only have to be ‘more or less the same as’ the identity. The notion of ‘more or less the same as’ here is called homotopy, and it replaces the equalities that we would otherwise have in the definition of inverse. Where the definition of inverse says the composites both ways round have to be equal to the identity, we now say they only have to be homotopic to the identity. A homotopy is like a continuous morphing. Instead of invertible arrows, we get something more forgiving, like saying you might not be able to completely undo a process, but you can get back to something very similar to what you started with.

This gives us a more forgiving notion of sameness between topological spaces called homotopy equivalence. It enables us to say things like: a doughnut shape (with a hole) is homotopy equivalent to a coffee mug (with a handle) because they each have one ‘hole’. A pair of trousers is homotopy equivalent to a figure 8 – if you make the legs shorter and shorter the whole thing will eventually become like this:

Then if you look at it from above, through the waist band, it will look something like this:

and then that shape can be shrunk down to the figure 8 inside:

This gives us a way of thinking about the structure of trousers. If we were concerned with length (as we usually are in daily life), it wouldn’t be helpful. But by broadening our definition of ‘trousers’ we get to home in on a specific interesting aspect of the form of trousers which, for example, makes them definitely different from a skirt. (A skirt is homotopy equivalent to a circle, not a figure 8.)

With this new definition of sameness for topological spaces, we can then move on to think about classification, that is, we start wondering what all the different possible types of topological space are, if you only count them as the same when they’re homotopy equivalent. This is called the ‘classification of homotopy types’.

Here’s how the story goes for some restricted types of topo-logical space, namely, the manifolds we introduced in Chapter 5. A manifold is a type of space where if you zoom in it looks flat everywhere. A two-dimensional manifold looks like a flat plane everywhere if you zoom in, but if you zoom out the little portions of flat space can turn out to be glued together in very complicated and twisted ways. Mathematicians try to classify manifolds, that is, decide what all the genuinely different possible types are, relative to our chosen, more forgiving notion of sameness.

When mathematicians try to classify things into different types, it’s a bit like when biologists classify all living creatures into species. But mathematicians can go a bit further because they’re working with abstract objects rather than concrete ones found in the world – this means they can just use logic to dream up every type of the object that is possible. If you study birds, there isn’t really a way of just dreaming up every type of bird that exists based on logical principles. That’s just not how birds work: a bird exists if you find it in the world, not if you create it in your imagination. (We could invent fictional or mythological birds, but they probably wouldn’t count as birds to an ornithologist.) By contrast, manifolds are abstract, so they exist just if you can define them abstractly. So mathematicians can try to concoct or dream up every different type of manifold that is possible, and this is called the classification of manifolds.

In order to classify manifolds – or anything at all – we have to decide two things. One is what counts as a manifold, and the other is when two manifolds count as ‘the same’. In Deborah Stone’s book Counting she points out that when we count things there’s a sort of pun going on, where we don’t just enumerate things but we also decide what ‘counts’ in the sense of mattering. That’s a question of deciding what will get tallied as 1 rather than 0. In abstract maths, when we count things the question in the other direction is just as important: what will get tallied as 1 rather than 2? This is where defining notions of ‘sameness’ comes into the act of counting things.

When we think about what counts as a manifold, we say that a manifold is supposed to look like flat pieces of paper everywhere, but without an edge. (At least, this is for two-dimensional manifolds.) We sort of say that every point on the space has to feel like it’s in the middle of a flat piece of paper. If the shape had an edge, then the points on the edge itself wouldn’t feel like they were in the middle of a flat piece of paper, because they’d be on an edge. They would feel different. So much for what counts as a manifold. Then, we think about which manifolds we consider to be ‘the same’, and here we use our more nuanced version of sameness for topological spaces – homotopy equivalence, or rubbery bendy sameness. That is, we declare manifolds to be ‘the same’ if they are homotopy equivalent.*

The classification of two-dimensional manifolds can be described using the two constructions we introduced in Chapter 5 for building up manifolds by gluing things together. The first was to take any manifold, cut a circular hole in it, and glue on a ‘handle’, that is, torus with a hole cut out of it:

The other construction was to take any manifold, cut a circular hole in it and glue on a Möbius strip.

It turns out that repeating either of these constructions produces all possible different types of two-dimensional manifold, starting from a sphere. If you keep doing the first construction, where you add on a handle repeatedly, it’s not too hard to picture: you get toruses with more and more holes:†

The version where you keep sticking on more and more Möbius strips is extremely hard to picture, because you get non-orientable spaces that need more dimensions – the surface itself is still two-dimensional, but it no longer fits into three-dimensional ambient space because it has to ‘go through’ itself. The first one is the projective plane that we described before and the second one is the Klein bottle, which can be pictured as below,* as long as you accept that the tube ‘going through itself’ means it has briefly gone into a fourth dimension:

The Klein bottle can be represented by the method we used before of a flat square with the edges glued together:

We’re counting homotopy equivalent spaces as ‘the same’ as each other, and as a result that gluing scheme gives us ‘the same thing’ as if you stick a Möbius strip onto itself. We can make an argument for this by manipulating similar diagrams of flat squares:

It’s even more mind-boggling what happens if you stick on a handle and a Möbius strip. You might wonder why that doesn’t count as one of the manifolds we can construct, and the answer is that they sort of cancel each other out, but in a more complicated way than two negatives cancelling out – if you stick on a Möbius strip and a handle, it’s ‘the same’ as sticking on three Möbius strips. The twist in the Möbius strip sort of converts the handle into a Klein bottle, which is itself two Möbius strips stuck together.

One way to think about it (albeit not very rigorously) is to imagine it in pictures. A handle stuck onto a Möbius strip is a bit like this:

But then as the Möbius strip has ‘only one side’ we can slide one end of that handle around past the twist to get this, which is now basically a Klein bottle stuck onto a Möbius strip:

In any case, this is a form of arithmetic for manifolds, and this is a form of equation, but it only works because we have this more forgiving notion of sameness.

You might wonder what any of this is useful for. This question always makes me a bit sad, because I don’t hear it being asked of music, or a painting in an art gallery, or sport. I think the question stems from two things. The first is that maths is oversold as being useful, and so as a result people want to know how it’s useful. The second, relatedly, is that all too often people don’t find maths interesting. If something isn’t interesting, it had better be useful, otherwise there really is no point to its existence.

The understanding of topological spaces and the classification of homotopy types is in fact directly useful because there are so many shapes of space that we need to understand in the world, both in terms of physical space, and abstract space formed by different criteria producing different dimensions. Topology is used to study air flow around jet engines, to work out how to control robotic arms, to understand gravity. But it’s not those direct applications that interest me about topology or any maths. I’m fascinated by the concepts, and I enjoy exercising my brain to think about them. Aside from that, I value the intellectual strength I develop by thinking about these things. Even if I don’t directly apply any of this in my daily life, I am sure that having studied it and thought about it deeply helps me to better understand many aspects of change and sameness and classification in life.

* At least, for commutative groups, that is, those where it doesn't matter in what order you combine elements.

* J. H. Conway et al., ATLAS of Finite Groups (Oxford University Press, 1985).

* Actually, this is a subtle detail, but the more rigid notion of isomorphism for topological spaces would do just as well here, because we don’t need to do any collapsing of dimensions.

† But only a finite number of holes, for technical reasons that I have glossed over. Basically, we’re adding holes one by one, and we can’t get to infinity by going one by one.

* Credit: Tttrung, CC BY-SA 3.0, via Wikimedia Commons.

10 FROM SHEFFIELD TO BUFFALO

The film The Full Monty depicts a motley group of out-of-work men who turn to stripping to try and make some much-needed money. It’s set in Sheffield, in a time when the traditional steel industry has all but shut down and many steelworkers have lost their jobs. The film was also made into a Broadway musical, and I was amused to learn that the Broadway version was set in Buffalo, New York. As I grew up in the UK but have lived a significant part of my adult life in the US, I am interested in transatlantic ‘translation’, but I had not considered the idea of translating an actual town.

Some critics were scathing about this particular translation, but I was never sure if they were just British people who were offended by the whole idea of making the plot American, or who didn’t understand the role that Buffalo plays within the American psyche (I still don’t). You might wonder what The Full Monty has to do with maths, but this is all about trying to move between two different contexts or two different points of view.

Translating from one language to another – even two versions of ‘English’ – is always tricky because of the roles that words play in different cultures, beyond just their meaning as a word in a language. Sometimes the point of a word is the cultural reference, and then a translation will need to find an analogous cultural reference rather than just translate the word.

Puns are particularly difficult to translate because a pun is a word or sound that plays two different roles in one language. Here is how I think of it mathematically: a pun is when two words are equivalent in one sense (in how they sound) but different in another sense (in what they mean). They are interesting because we can use the sense in which they are the same to pivot to a sense in which they’re different, whether it’s to make a surprising punchline for a joke, or to make a profound point in poetry. The trouble is that when you move into a different language that combination of equivalence and inequivalence is unlikely to be preserved.

There’s a song in French by Francis Poulenc that I have attempted to translate for concert programmes, with great difficulty. The poetry is by Louise de Vilmorin, and one of the poems is called ‘Il vole’ which in French means both ‘He flies’ and ‘He steals’. Both meanings are used in the song, simultaneously. I couldn’t think of a word that would do that in English and so I resorted to the unsatisfactory solution of writing ‘He flies/steals’ which gets the point across but loses the poetry and the subtlety.

By contrast there was a rather brilliant solution for the French translation of my first book. The title in English, How to Bake Pi, involved a pun on pi, the Greek letter and mathematical concept, and pie the food. This pun doesn’t happen in other languages, so when the book was translated into other languages typically the title just entirely lost its pun. Rather spectacularly, the French translation did manage to have a pun in the title: it’s called Comment cuire un 9. This literally means ‘How to cook a nine’, but the number nine in French is ‘neuf’, and ‘un neuf’ is a pun on ‘un œuf’ (an egg).

I love this translation because it demonstrates this point about contextual equivalence: it’s not a literal translation, but it’s a translation at a more abstract level, which is ‘How to cook [something that is a pun between a mathematical concept and a food item]’. In any case, my book was not literally about how to bake the number pi.

I understand that placing things in context is essentially how large language models work. Large language models don’t actually understand what words mean (I’m not sure that the concept of AI ‘understanding’ something has any meaning anyway), but they make a model where every word is put into a very detailed context relative to other words. So for example ‘ice’ would be related to ‘cold’ and ‘water’ and ‘frozen’. This can be done for every language individually, and then each language has a shape, or an abstract structure, which is just how all the words fit together relationally. At that point, like with category theory, it doesn’t matter what the words actually are, it only matters how they fit into the shape relative to all the other words. The amazing thing is if you want to translate something you then don’t have to know what it means, you just have to know where it is in the shape, and then find the same spot in the shape of the language you’re translating into. So it’s about ‘understanding’ words by the role they play relative to each other, rather than by what their actual meanings are. The reason it’s a ‘large’ language model is that this is only going to work decently if a really huge, vast, unimaginable amount of data is used to make the shape. Otherwise the structure will be too vague, or there will be too many similar relationships, and it will be too hard to pin down where you are in the shape. But once we do have a huge amount of data with which to build the models, moving between contexts becomes very productive.

In more abstract mathematical situations we don’t need ‘data’ to build good models, but we do need good understanding.

Moving between different points of view

Being able to move between different points of view is a valuable but underrated skill. This can mean being able to see someone else’s point of view (that you disagree with) or just seeing that you yourself have different points of view on the same thing. We have already seen some ways in which moving between points of view is a key part of mathematics, where notions of sameness come from taking different points of view on the same thing.

In this chapter we’re going to see a way in which maths gives us a framework for examining different points of view by carefully moving between contexts. We will see how we can look at something in a different context to see it in a different light. Or we can find concepts that play analogous roles in different contexts in order to find ways in which things are the same even if they look very different.

This skill gives us a way of empathising with other people: we can move from looking at their experience in their context to looking at something we have experienced in our own context. That can help, although it can also make things worse if we try to claim that the experiences are equivalent because they’re analogous in our context. For example, you might try to empathise with someone else’s trauma by thinking about the worst thing that’s happened in your own life. However, it might not sound comparable to the other person at all, who then might be offended by you claiming to ‘understand’ what they’re going through. I think in general it’s safer to do that inside your head, and then keep it to yourself and not say out loud what it is you’re drawing on to help you empathise.

Still, that is better than looking at someone else’s experience in your context. Too often one person experiences something that’s very bad relative to their own life, but which someone else finds to be not very serious compared with their own experiences. While it’s important for us all to remember that other people suffer worse things than us, I think it’s rare for that to help anyone feel better about what they’re currently going through. The answer in that case might be simply only to seek support from people who have not experienced far worse things. So, for example, if you sprain your ankle it’s probably best not to seek sympathy from someone who had to have both legs amputated.

Similar shifts can also help us understand different cultures in the context of their culture rather than from our point of view. It can be hard to understand other cultures’ customs if ours are the opposite. This might be regarding fairly innocuous things like punctuality, personal space, loudness or taking one’s shoes off in the house. But it might be regarding much more serious things like birth, partnership and death rituals, or looking after one’s family. However, there are also questions about how far we should take this, say if another culture commonly arranges forced marriages for children, or endorses men having multiple wives.

These are all questions of moving our thinking between the different contexts of different people’s life experiences. We can get better at these moves by practising in the framework of mathematics.

Moving between contexts formally

We have seen that category theory provides us a way to focus on relationships to formalise the idea of context. We decide what objects we want to study – numbers, shapes, spaces or more complicated things. We then choose particular relationships between objects to study, and we build a mathematical structure called a category from those relationships. One key example is sets together with functions between them, where functions are viewed as a type of relationship between sets. We have also seen sets with certain mathematical structures on them – such as an ordering, a group structure, or a topology – and the relationships we then think about are functions that respect that structure.

However, this is all just the start of category theory – there is much more to it than just organising things into categories. One powerful aspect of category theory is that it gives us a rigorous framework not just for acknowledging what context we’re thinking about, but for moving between contexts and comparing the features inside the different contexts. This involves a form of relationship between categories themselves, called functors. This is a type of zooming out, where we are now thinking of a sort of ‘meta-category’, where the objects are themselves categories and the arrows are functors (and I’ll explain what functors are in a moment). It’s definitely a bit mind-bending to keep all the different levels straight, but it’s one of the very powerful aspects of category theory: that we can zoom in and out and apply the same way of thinking at different levels. It would be like studying interpersonal relationships within a family, and then using the same ideas to model relationships between entire family units.

So there are two levels of structure here: relationships between objects inside individual categories, and relationships between entire categories, where the categories are now treated as objects themselves. The relationships between categories are defined in the same spirit as for all the sets-with-structure we looked at in the previous chapter – we think of each category as a sort of ‘set with structure’ and then build relationships between categories via functions that preserve that structure.

The idea of zooming in and out happens in life too, for example if we rank students within a school, and then zoom out and rank schools against each other. That’s quite a crude approach; a more subtle comparison would take into account some relationships between students in the individual schools. For example, we could look at the socioeconomic status of the students’ families, and then compare different schools’ outcomes taking those backgrounds into account. Similarly, when studying the efficacy of covid vaccines, comparisons were made of hospitalisation and death rates among vaccinated and unvaccinated populations. But covid was known to affect older people more, so the comparisons had to respect the age distributions in each population, which is a more subtle comparison to do. Unfortunately the subtlety was lost on some vaccine deniers who claimed the maths was just ‘fiddling the numbers’ to make it look like vaccines work.

When we’re working with sets with more structure, it’s always going to be more delicate to make a coherent comparison than if you don’t bother respecting the structure. To compare and relate different categories we are going to define a relationship called a functor, which sounds a bit like function, and indeed it is a bit like a function but more complicated. It has to be more complicated because a category has objects and also arrows between them, so when we map to another category we have to map the objects and also the arrows – it’s a sort of two-level function.

Just like the functions preserving structure that we saw in the previous chapter, our functors, or two-level functions, have to preserve the category structure. In this case, the structure in question is the identities and the composition, so this is what we need to ‘preserve’. I will explain the formal definition here, but just like for groups there’s something more intuitive we can say about functors. For groups we said that ‘patterns should be preserved’, and for categories it’s that ‘diagrams of arrows should be preserved’. But first here’s the formal version.

The idea is that we’re trying to define a functor called F from one category C to another category D. First of all we know that F is a function that maps the objects of C to the objects of D:

If we start with an object a in C then we get an object F(a) in D.

But now we also need to know that it maps the arrows of C to the arrows of D, in the following way:

If we start with an arrow fa→b in C then we get an arrow F(f)F(a)→F(b) in D.

Now we get to the part where F has to respect structure, that is, the identities and composition. We’ll do identities first. Remember that for every object a in a category there has to be an identity arrow on it, 1aa→a. Now, ‘respecting the identities’ means that we can either take the identity in C and then use F to transfer over to D, or we can transfer the object a over to D first (using F) and then take the identity on it – and both ways round should give the same answer. Here’s a diagram of that process:

This is summed up by the following rule, otherwise known as an axiom:

1F(a) = F (1a)

For ‘respecting composition’ we’re saying that if we start with composable arrows f and g, we can either compose them in C and then use F to transfer across to D, or we can transfer f and g across to D first (using F) and then compose them – and both methods should give the same answer. Here’s the scenario in a diagram:

This is summed up in the following axiom:

F(g) ◦ F(f) = F(g ◦ f)

Functors enable us to compare categories. They also enable us to move between categories and see what happens to the ideas we carry with us. This means that we can compare entire worlds, or we can compare features within worlds, or we can compare what ideas look like in different worlds. At a more theoretical level, as functors are a good type of relationship between categories, they enable us to build a category of categories, where the objects are categories themselves, and the arrows are functors between them. We can then do what we did in the previous chapter once we had built a category of things we were interested in: we look to see what the isomorphisms in that category are.

Isomorphisms of categories

When we looked at isomorphisms of groups (that is, isomorphisms in the category of groups and group homomorphisms), we found that the abstract definition boiled down to something more intuitive if we thought about patterns of Latin squares. Two groups turned out to be isomorphic if they had the same Latin square pattern, just perhaps with different names for the objects. This came from making a bijection between the sets of elements that turned one of the Latin squares into the other.

We now have a category of categories and functors, and we can look at isomorphisms inside this category. It turns out that these also have an intuitive characterisation – two categories are isomorphic if they have the same shape, just with perhaps different names for the objects and arrows. This is because there will have to be a bijection between the objects of the two categories, and a bijection between the arrows, in such a way that the category structure is preserved. This is very similar to isomorphic groups having the same Latin square pattern, just perhaps relabelled.

Here is an example of two categories with ‘the same shape’. In both cases the objects are numbers. In the first case the objects are all the factors of 6, and the arrows indicate which factors are also factors of other factors. We don’t need to draw the identities or the results of composition, as they can be deduced. The other category has as objects all the factors of 10, and again the arrows indicate which factors are also factors of other factors.

There are different objects at the corners (in this case the objects are numbers), but the shape of the arrows and how they interact with each other is the same in both cases – it’s this basic shape of category:

I’m now just drawing the objects as dots as it doesn’t matter what they are, it only matters that there are four objects with arrow relationships as shown. All the categories with this diamond shape are isomorphic in the category of categories. This is because if we unravel the definition of isomorphism in the case of functors, we find that a functor that is an isomorphism has to be a bijection on objects and also a bijection on morphisms, as well as respecting identity and composition structures as all functors have to. This means all that the functor can really do is relabel the objects and arrows according to the bijections in question, but without moving any part of the diagram.

We also get this same diamond (or square) shape by thinking about a partial ordering in two dimensions. If we start with one partial ordering with just two elements, 1, 0 (and 1 > 0), we could draw that as this very small category:

1 → 0

Perhaps 1 could represent owning a home and 0 could represent not owning a home.

Now, maybe we have another partial order that looks the same, perhaps with 1 for having a job and 0 for not having a job. We can then combine these total orders into a partial order using both dimensions like this:

Each arrow represents direct loss, and there is no arrow across the middle because going from left to right would represent losing a job but gaining a home, and going from right to left would represent losing a home but gaining a job. Each of those is simultaneously a gain and a loss, rather than a direct loss, so there is no arrow in this category. This is not to say it is impossible to lose a job and gain a home, it’s just that the arrows in this category are the direct losses, and ‘losing a job but gaining a home’ isn’t a direct loss, so it’s not represented as an arrow in this diagram.

This construction of a diamond shape using the potential loss of two separate advantages is just like when we made a partial order using coordinates:

In fact the diamond-shaped structure is the product of two of those one-arrow categories, just like when we take the Cartesian plane with x and y coordinates it’s the product of the real line with itself. (This is ‘product’ in the specific sense of being a result of some sort of multiplication, not just the result of any old process.)

Those diamond-shaped categories are all isomorphic categories, that is, they count as the same in the category of categories. So is this one:

We can say, abstractly, that this shape arises any time we have two properties, A and B, and we look at all the possible combinations of having or not having A and B:

This level of abstraction can help us spot more situations where the same feature is coming up – any time there are two ‘competing’ properties, advantages, privileges or criteria. Importantly, we can also spot this feature inside a bigger category – there can be more things going on in the category, but we can still look for small features inside it.

Features we look for

My favourite part of geography lessons at school was glaciation. I loved learning about the kinds of structure that glaciers made in the landscape, learning to recognise their features, and then looking for them in landscapes. I particularly like inter-locking spurs and truncated spurs. It’s not that you would ever see interlocking spurs in isolation, but we can look at a landscape and spot some interlocking spurs inside it.

In retrospect I like to think I was thinking ‘categorically’, that is, in the spirit of category theory – I enjoyed the idea of these structures as abstract concepts, before looking for them in the world around us. After living in Chicago for a while I was surprised to discover that Chicago was the site of glaciation. I was surprised because it’s so very flat here, and it bears absolutely no resemblance to, say, the Alps, which is what we studied as a classic glaciation landscape, with pointy mountain peaks and fishbone ridges descending from them. (The scale of glaciation is rather larger in the US.)

Finding these features in common doesn’t mean that the landscapes are the same, but it can give us a starting point or a foothold for thinking about similarities between parts of the landscape. This can help us with notable features of ‘abstract landscapes’ too, such as in people’s systems of beliefs. Distilling a feature like a spur or a corrie to its abstract essence is a simplification, but it’s a simplification that can help us recognise it more easily in different situations.

For example, if you understand the abstract concept of a false positive versus a false negative, you might notice that this tension is at the root of a wide variety of arguments in life, including about whether or not to get screened for cancer, whether to let refugees enter a country to claim asylum, whether voting in elections should be compulsory, and whether or not vaccines work.

I’ve found it helpful to notice ‘sports-fan-attitude’, where you pick a team and you support it forever, no matter what the team does. People too often take this approach with politics as well, and with conflicts in general, although it would be more reasonable to support neither side entirely but point out which aspects of what they’re doing you do and don’t support. This often gets conflated with ‘both-sidesism’, which is when different points of view are presented as being equivalent even though evidence or logic strongly points more one way than the other. However, taking one side as entirely good and the other as entirely bad is an unnecessarily opposite extreme.

I’ve also noticed similarities and differences between the way exploitation can happen in the world. Sometimes it can happen because people are in such poverty and so desperate that they will suffer terrible conditions in order to earn any money at all. The people in power then make sure that those in poverty remain in poverty, so that the powerful can still exert control. Another type of exploitation can happen when people are desperate, not because of poverty, but because they so desperately want to get into a certain line of work. It can happen with people who want to be academics, or musicians, or film stars. In this case the people in power can exploit them with the implicit or explicit threat of preventing them from having this desired career. The two situations are in some senses analogous, because they both involve manipulation by means of the desperation of the people who don’t hold power. However, in one case it’s a desperation coming from poverty and in the other it’s a desperation coming from desire. In both cases it is unlikely to be effective to appeal to the morals of the people in power to stop exploiting others. Since we can’t rely on their goodwill, we can try legislating against the exploitation, but another option is to address the desperation instead. In the case of poverty, this means providing a better social safety net so that nobody ever is that desperate. Capitalist power-holders vehemently object to such safety nets, specifically because their power depends on the desperation. They even go on record saying as much.

In the case of ambition it’s more subtle. I wish we could persuade people to be less desperate about wanting to go into those fields, to lessen the power held by the gatekeepers. But it sounds a bit like victim-blaming if we ask ambitious young people to rein in their ambition to stop themselves being exploited. We could instead push for more transparency in those fields as another way to lessen the power of the gate-keepers. And perhaps another approach is to educate everyone better and more broadly so that if they can’t have their dream career, there’s still something else they can do.

These are all examples of how the discipline of abstract mathematics has helped me more clearly see patterns of behaviour and argument in the world around me. Category theory provides a framework for making comparisons between different contexts. It gives us practice at isolating particular features in one context, expressing them succinctly, and then finding them in other contexts.

Features we look for in category theory

So far, the main feature we have looked for inside a category is isomorphisms. These are situations with two objects and two arrows going back and forth like this:

where the composites both ways round are the identity. This is a type of symmetry in the relationships, one that provides a more nuanced version of ‘sameness’ than just equality. But there are many other types of feature we might look for inside a category as well, for various reasons.

The diamond or square shapes we just saw are one feature we can look for inside other categories. For example, here is a cube-shaped category coming from examining three possible types of advantage in life. The objects are the possible combinations of advantages, and the arrows represent the loss of one of the advantages:

In this abstract cube, just like in a physical cube, each ‘face’ is a square shape, and this tells us that each square inside the cube can be understood analogously to the square-shaped categories we looked at before. In fact just as those squares were produced as the product of two individual one-arrow categories like this:

1 → 0

the cube is produced as a product of three individual one-arrow categories, and we’ll come back to that in the next chapter.

Another situation we might look for is loops, with arrows going back and forth like for isomorphisms, but where the composites are not the identity. This can highlight situations like vicious cycles or virtuous cycles, or circular arguments, where we keep going round in circles and get more and more looped up without ever coming back to where we started.

In category theory we often also look for parallel arrows like this:

⋅ ⇉ ⋅

highlighting places where there are two different ways of getting from A to B. We might then want to compare those two different paths, but this is a higher-dimensional situation that we will get to in the next chapter.

Another feature we might look for is diagrams shaped like this:

where there is one object (labelled X in this diagram) simultaneously relating to two others (A and B here), giving an implicit relationship between the two bottom objects although there is no direct relation between them. I often think this way when I am thinking about analogies, to focus my attention on the general principle giving rise to the two analogous things as examples. The general principle is the object at the top of the diagram, which too often gets ignored in normal life when we are just arguing about whether the bottom two objects are analogous.

The features I’ve mentioned so far are just shapes of arrow structures inside a category, like corries and interlocking spurs in a glacial landscape. Category theory then takes a step up in complexity, and looks for concepts that aren’t just a particular shape, but moreover play a special role among all objects of the same shape. It’s like not just looking for a tall person (say, to get something down from a high shelf) but instead looking for the tallest person in the room. This is the idea of universal properties in category theory.

Universal properties in sets

I’d like to look at a couple of universal properties for sets, to show how category theory characterises ‘extremes’. We are going to look at a special property of any set with one element. We have already seen that all one-element sets are in bijection with each other, and that if we go down one dimension from sets to numbers, this gives us the number 1 (either as an equivalence class of all those sets, or as a chosen representative of them). The number 1 can be characterised among other numbers as the ‘multiplicative identity’, that is, the number such that when we multiply things by it, nothing happens. However, in category theory one-element sets are characterised more directly, without having to use a concept of multiplication.

We can do something similar for the empty set. One dimension down, in numbers, the empty set gives us the number 0, which is the additive identity. But in category theory we are going to characterise the empty set without any reference to addition.

The method that category theory uses to characterise the empty set and the one-element set may be less intuitive at first sight, but in some ways it is more direct as it doesn’t rely on the operations of addition or multiplication. It only relies on the arrows in the category of sets, which are the functions. This makes it more widely applicable, as we can then make an analogous characterisation in any category. By contrast, for additive and multiplicative identities, we can only do that in contexts where we have a concept of addition or multiplication. While those are quite widespread, there certainly are categories without those things, such as categories of family relationships or paths through a space. Moreover, category theory tries to construct things using minimal special equipment, so even if we think we will mostly be in places with addition and multiplication, we see if we can work without it just in case.

This new, category theoretic approach characterises the one-element sets by thinking about functions to them. The idea is that, given any set A and any one-element set B, there is only one possible function from A to B, no matter what the element of the set B is. Suppose that the one element of the set is the number 1. Then what functions can we make that act on a set A of inputs, but where the outputs are just the set {1}? To define a function we have to say where every element of A lands. But there’s only one place where each element can land, which is the number 1, so there is only one possible function. If the single element of B is the number 2, there is still only one function because it has to send everything to the number 2. If the single element of B is an elephant, the function has to send everything to the elephant.

You might think there could be several different ways of sending everything to the number 1, for example by dividing everything by itself, or by multiplying by 0 and adding 1. However, in abstract maths, functions count as the same if they produce the same outcome – it doesn’t matter how they do it.

What we’re saying, more formally, is this: let T be any one-element set. Then given any set A there is a unique function:

A → T

It turns out that this slightly more obscure-sounding characterisation really does pin down the one-element sets just as well as the idea of a multiplicative identity.*

Now, what about the empty set? It’s pretty difficult to make a function from a set A to the empty set, because there is nowhere for anything to land. In fact, unless A is also empty there can’t be any function at all from A to the empty set because there are no possible outputs.

However, if we instead start from the empty set, then there actually is a function from the empty set to A. This is a bit counterintuitive, but if we think about it strictly logically maybe it becomes a bit clearer. A function gives, for every input, an output. Now, if there are no inputs, then the ‘for every input’ part becomes redundant. When a condition is satisfied in that manner it is called ‘vacuously satisfied’ – it’s been fulfilled because there is nothing to do. It’s like telling someone to make sure they feed all their pet elephants every day. Then every morning they can wake up and say ‘I’ve already fed all my pet elephants today’ because they don’t have any.

So to define a function you can tell someone ‘Make sure you take every input and send it to an output’ and if there are no inputs they can say ‘I’ve done it!’ straight away, because there was nothing to do.

In conclusion, there is always exactly one function from the empty set to any set A, because it’s the vacuous one. (There is one, but only one, because there’s only one way to do this vacuous thing.)

We can write this in symbols, using the symbol ∅ for the empty set: given any set A there is a unique function:

∅ → A

Just like for the one-element sets, this entirely characterises the empty set.

These definitions only refer to the arrows in the category of sets and functions, and not to the elements of the sets, so we can use these same definitions in any context that has morphisms, that is, in any category. This is one of the big aims of category theory: to express familiar concepts using only arrows in a category, so that we can explore those same concepts in other categories. The technical terms for these particular concepts characterising the one-element set and the empty set are as follows: ‘terminal object’ for the one with a unique morphism going to it from any object, and ‘initial object’ for the one with a unique morphism going from it to any object. These definitions give a way of characterising objects by a special role they play inside a category – not just any old relationship with the objects around them, but a very special one, a sort of extreme one. It’s not a competition, but the definition of terminal object is a bit like saying they’re ‘the best’ object in the category, with respect to the property of being the endpoint of morphisms.

Moreover, we can show that this characterisation pins down objects uniquely in a precise sense, as follows. The question is: if two people use this characterisation to identify an object, can we be sure that their objects are the same? For a start, we should use the right notion of sameness for category theory, which is isomorphism. In fact, we can prove that not only must the two people’s objects be isomorphic, they must be uniquely isomorphic: there is only one possible isomorphism in between them. In maths we say that these objects are ‘unique up to unique isomorphism’.

When we say ‘up to’ in abstract maths we don’t mean that we’re filling something up to a line or something; rather, we mean that the statement in question is true with a slight allowance for nuance. It’s more like when we say something is ‘up to three decimal places’, which is specifying what level of accuracy we’ve got. When we say something is true ‘up to isomorphism’ it means there might be some isomorphisms floating around in there instead of equalities, but we’re fine with that because we’re happy with this more relaxed notion of sameness.

For the terminal objects this means that if T and S are both terminal, then not only is there an isomorphism between them, there is a unique isomorphism between them. It’s a sort of meta-uniqueness. It means that they are isomorphic in a canonical way, where we don’t have to make any choices in constructing the isomorphism between them. This is good because it means the structure is more organic, whereas if we humans have to intervene in the structure things tend to get messier.

I’m going to include the proof here, again because you might be interested in seeing it or just ogling it, not because I think you need to understand it precisely. The proof really follows the same ideas as the other uniqueness proofs we did, where we pit the two objects against each other using the properties we’re thinking about.

Suppose S and T are both terminal objects.

• Since T is terminal there is a unique morphism fS→T.

• since S is terminal there is a unique morphism gT→S.

We now show that these are mutually inverse. First consider the composite

fS→Tg→S.

Now S is terminal, so there is a unique morphism

S → S

which is the identity. Thus the composite equals the identity. The proof for the composite

gT→Sf→T

follows similarly. Thus we have an isomorphism between S and T. We know it is unique because we already said there was a unique morphism S → T and T → S.

We can try this out in the category of sets and functions, where terminal objects are the one-element sets. So in that example this result is telling us there is a unique isomorphism between any pair of one-element sets. This might not sound very earth-shattering, but the result also works in other categories. And crucially, sometimes we can use an amazing technique to study more complicated concepts: instead of making a definition more complicated, we can make the background category more complicated, but keep our definition abstractly the same. It turns out that huge quantities of mathematical structures can be expressed as terminal or initial objects by making the background category more complicated. This includes multiplication and addition of numbers, products and disjoint unions of sets, or indeed non-disjoint unions. Also all the topological spaces we’ve built up by gluing things together.

This is all about studying terminal and initial objects in other categories, and one important aspect of understanding that is understanding what happens if we try to transport or translate terminal or initial objects to other categories.

Universal properties in other contexts

If we move from one category to another via a functor, we might wonder what happens to the terminal and initial objects in the first category. Are they still terminal and initial in the new category, or not?

This is a bit like being a superstar in one country, and then moving to a bigger country and seeing if you can make it. Some British film stars or pop stars have not managed to have big careers in the US, but some have, and have done very well, such as Kate Winslet and Daniel Craig. Yet others have perhaps done even better after moving, as Hollywood may have been (or been perceived to be) faster at moving towards inclusivity of non-white actors, especially Black people.*

You might not be trying to be a ‘star’ but just trying to have what is known, I believe, as a ‘unique selling point’ in order to be in some way indispensable and therefore in demand. You might have a USP with respect to a very small environment like your family or your friend group, but that might not transfer to a larger environment like your neighbourhood, your town, your country or the world. And that’s fine – many people are not trying to be indispensable to anyone except their immediate family, so that is still a universal property.

Universal properties characterise objects by the role they play in a particular context (that is, category). When we move to a different category we can see whether the object becomes universal in the new category, but we can also just look for whatever is playing the analogous role in that new category.

Analogous roles in different contexts

Sometimes we’re not exactly looking for the same structure in different contexts, but for something playing an analogous role in a different context. For example, we have looked at prime numbers as the basic building blocks of numbers by multiplication. We have looked at sine waves as basic building blocks for wave-shaped functions. We have looked at groups with diagonal pattern Latin squares playing an analogous ‘building block’ role for groups. We have also found some basic building blocks for two-dimensional surfaces – the Möbius strip and the ‘handle’. These basic building blocks can be called generators in maths because they generate the rest of the structure if you use them in an appropriate way.

We can also look at the same structure from different points of view by putting it in different contexts. For example, a Möbius strip is, in one context, very different from a cylinder because it has a twist in it. This means it has only one side and only one edge, whereas a cylinder has two sides and two circular edges. If you cut a hole in any surface and glue a cylinder on, there will still be a hole coming from the other end of the cylinder, with a circular edge. Whereas we’ve seen that if you cut a hole out of a closed surface with no edge, and glue a Möbius strip on, then you get a closed surface with no hole and no edge, because the Möbius strip has only one circular boundary. In that sense the cylinder and the Möbius strip are very different.

However, in another sense, a Möbius strip is the same as a circle, and so is a cylinder. That sense is homotopy equivalence, the rubbery bendy sameness we discussed in the previous chapter. It’s not too mind-boggling to imagine squashing a cylinder down to a circle, but this might be less intuitive for a Möbius strip. Perhaps we can see this if we imagine making a really thin Möbius strip, out of a very thin strip of paper.

If we make it even thinner, we will hardly be able to see the twist any more, and then if we make it so thin that it’s just a line, it will turn into a circle. The key is that this is not the same circle as the boundary circle – it’s the central circle of the original Möbius strip. This idea spurred me to knit a Möbius strip on circular needles. I read very complicated instructions about how to cast on with a particular kind of stitch to start, but I successfully did it by just winding the wool round and round the circular needles, representing the central circle. Then you knit into one side of the spiral, and when you get back to the beginning, you knit into the other side, creating the circular edge of the Möbius strip which loops over the central circle twice. It’s very satisfying because you end up knitting a continuous edge that builds up the width of the Möbius strip, looping back and forth around what looks like two ‘edges’ of it. All the time you’re doing it, the wire of the needles is looped back on itself like a hairband. It’s quite hard to picture without doing it; it took me ages to work out that I could just start with a spiral like that, and I tried all sorts of more complicated methods first.

Usually when you use circular needles you go round and round and just build up a cylinder: the original edge where you cast on stays as one circular boundary, and you get another edge where you cast off which is the other circular boundary. Indeed, this more or less shows why a cylinder is also homotopy equivalent to a plain circle. You might wonder if that means a Möbius strip is homotopy equivalent to a cylinder, which sounds a bit strange, but it is true: homotopy equivalence is an equivalence relation, although that’s harder to prove than for equality or isomorphism. But it follows the same general ideas: that you have arrows going back and forth, and you can compose them. We’ll come back to that when we talk about higher dimensions. In any case it means that if we know a Möbius strip is homotopy equivalent to a circle, and a circle is homotopy equivalent to a cylinder, then a Möbius strip is homotopy equivalent to a cylinder. It’s much easier to see this via the circle than to go directly from the Möbius strip to the cylinder, which is why knowing we can coherently manipulate these new versions of ‘sameness’ is important.

Moving structure between contexts

We have been talking about moving to a different context and seeing what plays the analogous role in the new context, such as building blocks. But another thing we can do when moving to a different context is carry something with us and see if it changes role when it gets there, or if it can play the same role. It’s a bit like going to a different country and wanting to know if your appliances will work there: most laptop computers can be plugged into different voltages using an adaptor, but other appliances might need a transformer if you’re dealing with a different voltage.

We have already talked about the idea of functions respecting or preserving structure, but now we’re zooming out – we’re thinking about moving between categories and seeing whether structure inside a category is preserved when you move to a different category.

One reason that isomorphism is a good notion of sameness is that it is always preserved when you move between categories using a functor; or perhaps this is a reason why a functor is a good framework for moving between categories.

We saw in Chapter 2 that equality is always preserved when you apply a function. That is, if you have:

a = b

and you apply a function f, then you will always have:

f (a) = f (b)

This is not true of <; that is, you can have:

a < b

but you might not have f (a) < f (b), as we saw in the previous chapter. If we always have:

a < b implies f (a) < f (b)

then this is an order-preserving function, which is a special type.

Sometimes the thing we’re trying to preserve isn’t exactly a relationship between two objects, but a property or a feature of an object, so that we can study the feature by transforming the object into something more intelligible, safe in the knowledge that the feature we’re trying to study won’t change. A feature that doesn’t change under certain transformations is called an invariant.

Invariance and invariants

Invariance is an important concept in maths. It is about things that stay the same even when something else changes. It can be a helpful way to detect if two things are really the same at some level despite having changed their outward appearance in a way that might be superficial. One basic example is trying to tell if a number is odd or even: we only have to look at its last digit – if the last digit is odd, then the number is odd, and if the last digit is even, then the number is even. This way of checking is much simpler than looking at the entire number. So parity (odd-or-evenness) is invariant under the process of forgetting everything except the last digit of a number.

By contrast, if we want to tell if a number is divisible by 7 we can’t just look at the last digit, as multiples of 7 can have any number as their last digit. So divisibility by 7 is not invariant under taking the last digit of a number.

A complicated (and interesting) example is in the study of mathematical knots. In the study of knots, topologists imagine things that are a bit like tangled up pieces of string, and try to work out which ones are ‘the same’ as others. Unlike for knots in actual pieces of string, topologists imagine the two ends of the string are fused together to make a closed loop after you tie the knot – otherwise you could just undo the knot by pulling the ends through and all knots would be ‘the same’. Here are a few pictures of topological knots:*

Just like in the classification of groups and of manifolds, topologists then set about trying to work out what different possible knots there are. And just like with groups and manifolds, one first step is to decide which knots actually count as different, and which count as ‘the same’. The idea is that knots should count as the same if they’re just being viewed from a different angle. Take a look at these depictions of knots:

Perhaps you can try modelling these with a piece of string (or the nearest USB cable), and see that if they were really made from closed loops of string, you could wave the first one around a bit and turn it into the second or third ones without having to cut it open and re-fuse the ends together. So those count as ‘the same’ knot. It’s called the trefoil knot. It’s quite hard to see that without playing with physical string though.

What about unknotting it? That is, could we turn it into the completely unknotted closed knot, the trivial knot, also called the ‘unknot’:

Unlike with trying to show that the three trefoil knots are the same, it might seem obvious that you can’t unknot the trefoil knot without cutting it, perhaps from your personal experience with knots in real life. However, ‘it’s obvious’ is not a mathematical proof (although it does get uttered by frustrated mathematicians sometimes). It’s often harder to prove that things aren’t equivalent than that they are, and this can be true with knots as well. To show that two knots are equiva-lent, we can demonstrate a series of moves with the string that turn one of the knots into the other. Even if we’re not doing it physically, mathematicians have boiled down all the possible ‘moves’ to three basic ones from which all others can be generated – another example of basic building blocks. To show that two knots are not equivalent, however, we have to show that no sequence of moves will ever turn the first knot into the other one. As there are infinitely many possible sequences of moves to check, that might sound like an impossible task, and this is where invariants can help us.

The idea is to find some quality of a knot that never changes with the presentation of the knot, as long as you don’t change the knot itself. This quality that doesn’t change is called an invariant. It doesn’t necessarily give us a way to tell all knots apart because some non-equivalent knots might have the same invariant. But it does give us a way to prove that some knots are inequivalent, if they don’t have the same invariant. That’s a bit tangled up so let me write out the logic a bit more clearly:

same knot implies same invariant

So now we can take the contrapositive and get:

different invariant implies different knot

What we don’t have is:

same invariant implies same knot

It’s a bit like eye colour for humans. People can change their hair colour and their clothes and their face shape, but not their eye colour.* So if you’re trying to identify a missing person, you know that if they have the wrong eye colour, then they’re the wrong person. You don’t know that if they have the right eye colour, then they’re the right person.

One invariant for knots is ‘tricolorability’. The idea is that you can take a diagram for a knot and see if it is tricolorable, which means that you give each strand one of three colours according to a couple of basic rules. First, at least two colours must be used; and second, at any crossing the three strands must be all different colours or all the same colour. It might sound like a crossing should only involve two strands of string, but in the diagram there are three lines going into the crossing, because the strand going over is an unbroken line but the strand going under is drawn as a line broken in two:

Now, it’s possible to prove that the property of tricolorability is a knot invariant, that is, any time you move a strand of a knot but don’t break it, you don’t change the tricolorability: if it was already tricolorable, it will stay tricolorable, and if it wasn’t tricolorable already, then it won’t miraculously suddenly become tricolorable.

For example, the trefoil knot is tricolorable:

However, the unknot is not tricolorable – it only takes one colour. This means we can use that invariant to show that the trefoil knot and the unknot are definitely not the same, and we didn’t have to check every possible way of waving the knot around.

Evidently this is not a very sensitive invariant because it only divides knots into two types – tricolorable and not tricolorable – a bit like eye colour for humans, which divides us into a few more than two types but still not very many. Mathematicians have developed gradually more and more sensitive invariants that can detect more differences between knots, including invariants that use polynomials and higher-dimensional category theory. As with the study of topological spaces and the classification of manifolds, many mathematicians are interested in knots just out of sheer curiosity and the desire to understand something better that we currently only partially understand, but there also are applications in physics (for the interaction of particles), chemistry (for analysing the structure of molecules) and biology (for studying the tangledness of DNA).

The knot invariant of tricolorability that we saw was not very subtle, and was just a ‘yes’ or ‘no’ answer. In order to count as an invariant, it had to be preserved exactly – ‘yes’ had to stay as ‘yes’, and ‘no’ had to stay as ‘no’. However, now that we have a more nuanced version of sameness, isomorphism, we can ask for more subtle forms of invariance by preserving isomorphisms instead of exact sameness.

Preserving more subtle sameness

We have seen that functions always preserve equality. However, now we have moved into the world of category theory, we have a more nuanced version of sameness than equality: we have isomorphisms. In this world, we still want sameness to be preserved, but now the question is whether isomorphisms are preserved when we move to different categories using functors.

Fortunately everything goes well (not that it has anything to do with luck) because it turns out isomorphisms definitely are preserved by all functors. That is, if C and D are categories and F is a functor from C to D, then given an isomorphism in C:

∼a→b

we get an isomorphism in D:

∼F(a)→F(b)

(The ~ sign is just to indicate that these arrows are isomorphisms.) This is because functors respect the structure of identities and composition in a category. Here’s the complete argument:

Suppose we have an isomorphism given by this pair of morphisms

We can then apply the functor F to the entire diagram:

It then remains to see if F (f) and F(g) are inverses. We need to know if they compose to the identity both ways round. We know that f and g compose to the identity, and we know that F respects composition and identities. That means we can either apply F and then compose, or we can compose and then apply F. If we do that latter we’re saying:

F (f°g) = F(identity)

But F preserves composition, so the left-hand side is the same as F (f)°F(g), and F preserves identities, so the right-hand side is the same as the identity. The argument for F (f)°F(g) is analogous.

This result about isomorphisms being preserved is helpful because it means we can move to a different context that helps us in a different way and know that things that counted as ‘the same’ before will still count as the same afterwards. However, the situation is not so straightforward for more nuanced versions of sameness such as homotopy equivalence, the rubbery version of sameness we looked at between topological spaces. That sameness is not necessarily preserved by functors. Here’s what that means in more detail.

Suppose we have spaces A and B that are homotopy equivalent, that is, they are ‘the same’ according to the playdough-like idea of being able to bend, expand and compress things. This is like the squiggly circles being ‘the same’ as circles, and bagels being ‘the same’ as coffee cups. Now, we might move the spaces to a different context via a functor F and wonder whether the results are ‘the same’ in the new context. The answer is that they might not be: we might get F(A) and F(B) that are not isomorphic in this new context.

There is a ‘joke’* about topology that goes like this:

Mathematicians think a doughnut is the same as a coffee cup, but try eating a coffee cup for breakfast.

Here is my formal interpretation of this joke: a doughnut and a coffee cup are homotopy equivalent (provided that the doughnut has a hole and the coffee cup has a handle with a hole) but if you map them to a category of ‘objects in the kitchen’ or something, then they are very likely not to be isomorphic.

In reality, mathematicians do not actually study doughnuts or coffee cups as kitchen objects, but they do study topological spaces via other structures, such as groups. This is the starting point of the field called ‘algebraic topology’. Groups are a form of algebra, and if you study something via algebra it gets called algebraic; algebraic geometry is another important field. Many modern branches of mathematics have grown from the idea of making a connection between existing fields. This enables us to share knowledge between fields and use the understanding of one field to help the understanding of another. When we do this, one of the starting points is to come up with a good way to move between those different contexts. What counts as a ‘good way’ includes a balance between how easy it is to accomplish and what it enables us to do afterwards. Usually there is a tradeoff between those things. There are many ways to move context that are easy to do but destroy all our structure in the process, or lose all the information about the things we’re interested in studying. It’s like the fact that moving house is a lot easier logistically if you just throw away all your possessions – but then you no longer have any possessions. Or, if you’re not concerned about preserving structure, you don’t bother wrapping your breakables, which certainly makes your move much easier.

At the other extreme, there are ways of moving context that would enable us to do amazing things, except that it’s so impossible to move in that way that we can’t accomplish the move in the first place. This would be like moving house cross-country by carrying one object at a time and putting it directly into its new location.

A sensible trade-off for moving house is to pack things in boxes that more or less respect their positions in the house, to make it easier to unpack them efficiently when you get there. But there are still decisions to be made about packing more tightly to save space in your moving truck, and so on.

This is similar to the ways in which we decide how to move context in maths. One important starting decision is which relationships we are trying to preserve between the things we’re studying, and which we are letting go of. At one extreme we might try to preserve them all, but in that case all we can really do is rename everything or paint it a different colour. It wouldn’t be a very interestingly different context, and we probably wouldn’t learn very much by moving there.

We’ve seen that in category theory functors always preserve the relationship of isomorphism, but there are many other relationships that functors don’t always preserve. When we’re trying to study topology via algebra, we have to work harder to find a move that respects the relationship of homotopy equivalence, and sends it to some form of sameness in the context we’re landing in.

In algebraic topology we find ways to move from the world of topology to the world of groups. But we want to know that when we do so any spaces that are homotopy equivalent will turn into groups that are isomorphic. This is essentially saying that one version of sameness (homotopy equivalence of topological spaces) should map onto another (isomorphism of groups). This is called homotopy invariance.

Homotopy invariance is about preserving a form of sameness that is yet more nuanced than what we have met in our category theory so far in this book. We saw that we can make topological spaces and continuous maps into a category, but that the isomorphisms in that category are stricter than the stretchy, rubbery, compressible kind of sameness that we are interested in. But that more subtle form of sameness is a higher-dimensional notion, just like isomorphism was already a higher-dimensional notion than equality. Category theory is a highly effective framework for putting objects into context by introducing relationships between objects, studying isomorphisms between the objects, and moving between contexts. But it is still limited. We have added one dimension to our way of thinking, but only one. Why stop there? If we want to be more nuanced, and enrich our understanding yet further, we need more dimensions. That’s the subject of the last chapter.

* You might wonder whether those characterisations necessarily coincide. That’s an excellent and quite profound question. It really depends on how ‘good’ the concept of multiplication is in the context we’re studying. When it’s a bit less like ordinary multiplication, then the situation is more complicated. This turns out to be highly relevant to quantum physics applications.

* Tom Seymour, ‘Why black British directors and actors leave the UK for Hollywood’, Guardian, 29 August 2013.

* Credit: jkasd, via Wikimedia Commons.

* At least, this was true until recently. It seems that there are now procedures for surgically changing your eye colour, though it’s not clear how safe they are.

* I don’t think it’s very funny.

11 MORE DIMENSIONS, MORE FORGIVENESS

Throughout this book we’ve been examining what it can mean for things to be ‘the same’ or ‘different’. We’ve seen that it involves making choices about what we care about, and that the different possibilities mean that we can put ideas and concepts in different contexts depending on what we choose to care about. We’ve seen that category theory is a branch of abstract mathematics that provides a rigorous framework for us to make those choices and study the consequences.

Sometimes we start with a context and examine what counts as ‘the same’ in it, according to the existing definition of ‘sameness’. More formally in mathematics this takes the following form, which we have seen several times: we start with a particular category and examine what counts as isomorphic inside it. This is like looking at some part of the world and observing who is being treated equally in it, that is, who is being treated ‘the same’ in some way that we care about.

But sometimes we want to do this the other way round: we decide which things we want to count as the same, and then build a context in which that is the case. In mathematics this takes the following form: we try to build a category in which the things we want to count as the same do turn out to be isomorphic.

This latter way round becomes more a question about our values, and how we can create a world in which our values hold true. For example, when it comes to treating people the same, what do we want that to mean, and how can we create a world in which that is achieved? I once had a job where we were explicitly not allowed to play to our strengths. That is, if one person was good at one thing but someone else was good at something else, we were not allowed to share tasks according to what we were good at, because it was deemed – officially – that that would be unfair. I think that’s futile – a bit like the idea I described in Chapter 2, of having a relationship where one person hates cooking but loves cleaning, and the other hates cleaning but loves cooking, but then somehow deciding that you should both share both tasks.

Once we are thinking about the values we want to choose to hold true, a further question is: can we pick anything we like to count as ‘sameness’? Is it always possible to build a world in which those things are treated as the same? This is a hard question in life, and a hard question in maths too. What we’re going to see in this final chapter is that when we try to do that mathematically, we often find that our framework still isn’t expressive enough, which pushes us into using higher dimensions.

Deciding what can count as sameness

In some sense we can decide to think about anything in any way we feel like; the real question is: what will be the consequences of doing so? We can decide to act as if gravity doesn’t exist, but we’ll very quickly break things or hurt ourselves or just be sorely disappointed when we find we can’t actually fly. By contrast, astronauts in Space have to get used to the lack of gravity, and they’ll have trouble if they act like there is gravity there. But then when they come back to Earth, they are liable to forget and try to leave things floating in front of them, and apparently they find themselves dropping pens or cups of coffee. In daily life we have to work with the physical state of the world around us, but in theoretical physics thought experiments about hypothetical situations can help us, without us having to deal with physical consequences.

In abstract maths, the consequences of what we decide to think are also only abstract, and the worst that happens is some logical contradictions. We have seen that we can treat 1 + 1 as 0 and nothing bad happens; it’s just a different sort of abstract world. But we also can try to treat 1 + 1 as 3, and if we do it in too widespread a way we’ll get caught in all sorts of contradictions, or we might just find that everything equals 0. But when we try to do this in the ‘real world’ and interact with other people, we have to be careful. 1 + 1 is only 0 in certain circumstances in the real world – such as if you turn over a piece of paper and then turn it over again – but not others. If we behave as if 1 + 1 is always 0 we’ll get into trouble: we might steal one car and then steal another car as if that makes a total of zero stolen cars, and then we’ll get arrested.

So what could go ‘wrong’ in maths if we treat something as sameness that didn’t really deserve it?

One thing we’ve seen is that we might want our notion of ‘sameness’ to behave somewhat like equality, so we might want it to satisfy reflexivity, symmetry and transitivity. If we’re looking at something that doesn’t already behave like that, then we might want to add some things in to make it happen. For symmetry, that might be a bit like turning a one-way street into a two-way street. For transitivity, that might be like removing a barrier so that when you can get from A to B and from B to C you can always get all the way from A to C (where previously there was something in the way).

In maths, sometimes if something isn’t already an equivalence relation we can use it to generate an equivalence relation.

Generating an equivalence relation

Generating things in maths is a bit like generating electricity: you start with the means to create something and then you use it to create that thing. The slight difference in maths is that you usually start with part of something and then use it to create the rest. This is different from generating electricity, because a generator in that sense isn’t ‘partial electricity’, it’s a machine to make electricity. The machine and the electricity are different. In maths the machine can be part of the thing it is generating, because everything is abstract.

For example, we use prime numbers to ‘generate’ all the rest of the whole numbers, by multiplication.* There, prime numbers are special kinds of numbers, so they are part of the thing they are generating.

For equivalence relations, we can start with a relation that is only part of an equivalence relation, in the sense that it is not yet enough to be an equivalence relation. We can then generate an equivalence relation from it.

For example, if a is considered to be the same as b, and b is considered to be the same as c, then transitivity says we should also consider a to be the same as c. Perhaps we start with a relation that doesn’t satisfy that, but we can then add in a relationship declaring that a is in fact considered to be the same as c. That is to say, sometimes the first concept of ‘sameness’ we think of doesn’t satisfy transitivity, but we can add transitivity in afterwards.

One example is when we were constructing spaces by glueing squares along their edges. Let’s think again about the construction of the torus, where we glue together the edges that are marked with the same letter, and along the orientations shown by the arrowheads:

That isn’t really a rigorous expression, more of a schematic diagram. If we were to express this rigorously we might start by putting in some coordinates, so that we can refer to every point in the square by some sort of label or name. In maths we often choose coordinates so that the square is a unit square, that is, each side has length 1:

Now the square is defined to be all the points (x, y) such that both x and y are between 0 and 1. Here’s a more formal way of saying that:

{(x, y) where 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1}

Now we want to say we’re glueing the top and bottom edges together in the orientation desired. The points along the bottom edge all have the form (x, 0), because they can have any x value (from 0 to 1) but their y values are all 0. Similarly, the points along the top edge all have the form (x, 1). Now when we do the glueing, we’re glueing every point of the form (x, 0) to the corresponding point (x, 1), that is, the point with the same x coordinate but running along the top edge of the square:

When we’re glueing the left and right edges together we do something very similar, except that now we can see that the left edge consists of points of the form (0, y), and the right edge is points of the form (1, y). Then when we glue those edges together we’re glueing every point of the form (0, y) to the corresponding point (1, y), that is, the point with the same y coordinate, but running up the right-hand edge:

We can express all that mathematically by using the following relation, where the ~ sign means we are declaring the things on the left and right of it to be ‘the same’ or glued together:

for all x, (x, 0) ~ (x, 1) and for all y, (0, y) ~ (1, y)

This gives us a lot of pairs of points that we are considering to be ‘the same’. However, as it stands this is not an equivalence relation: it is not reflexive, it is not symmetric and it is not transitive either.

We can quite easily build reflexivity into it by declaring every point to be equivalent to itself, and we can quite easily build symmetry into it by declaring that the above relations work both ways round. However, we still won’t have transitivity, because of the times when x and y themselves equal 0 or 1.

For example:

1. By putting x = 0 we get (0, 0) ~ (0, 1).

2. By putting y = 1 we get (0, 1) ~ (1, 1).

So by transitivity we should have (0, 0) ~ (1, 1) but that’s not in the form of one of the pairs above: in all the pairs listed, either the x coordinate is the same in both points, or the y coordinate is the same. With (0, 0) and (1, 1), neither coordinate is the same.

We can see this feature geometrically by looking at the square picture again. The two types of relationship are the one involving horizontal pairs and the one involving vertical pairs. But we can use horizontal and vertical pairs successively to get the diagonal corners to be the related:

In fact, if you imagine gluing this square along the edges as shown, perhaps you can see that all four corners end up glued together.

This is perfectly fine, it just means that, in the above expression, we haven’t listed absolutely all the pairs of points that end up glued together. But we have listed enough that the rest can be deduced logically. As we mentioned earlier, when something can be deduced from a smaller amount of information, it’s logically equivalent to give the smaller amount and an indication of how the deduction works. The smaller amount of information is said to generate the rest of the structure.

In this case we are taking a relation and generating an equivalence relation from it, by adding in all the pairs that come from reflexivity, symmetry and transitivity (though transitivity was the most interesting one). The end result is the torus. You might say that it’s only homotopy equivalent to a torus, or you might say that a torus is only defined ‘up to’ homotopy equivalence, so that being homotopy equivalent to a torus is the same as being a torus.

However, there could be some other situations in which you try to generate an equivalence relation and accidentally find that you’ve made everything equivalent to everything else. For example, we might try to declare that people are more or less the same age if they are born within a year of each other. We’ve already seen that this is not an equivalence relation because it fails transitivity: A could be eleven months older than B, and B could be eleven months older than C, and now A and C are more than a year apart.

We could try to use this to generate an equivalence relation. So we will now count A and C as being more or less the same age as each other, because that would need to be true for transitivity to hold. We can do that, but the trouble is if we keep going like that we will end up declaring that everyone is more or less the same age as everyone else, because no matter how far apart two people are in age, we could still find a string of people born within one year of each other to bridge that age gap step by step. Of course, that might be how we want to think of people’s ages in some contexts, such as relative to the age of the universe.

Working out what happens if you start with some relations and consider them to be equivalences is a very hard problem in maths that is the basis of some of the most active current research fields.

Dealing with asymmetry

Things get even harder if we’re starting with relations that aren’t necessarily symmetric. The example using age is inherently symmetric, but we might start with a relationship that isn’t. In that case, generating a relation according to transitivity isn’t enough – we need a sort of backwards transitivity as well. This is now in the realm of quite esoteric mathematics, but here is the essence of it.

If we’re thinking about relations that aren’t necessarily symmetric, then we’re typically now in a category with arrows that aren’t reversible. Recall from Chapter 8 that instead of demanding symmetry everywhere we allow non-reversible arrows, and just look for situations of symmetry inside that category.

In a category, ‘transitivity’ is then expressed as composition, which is when we have consecutive arrows that we can stick together end-to-end to make a ‘composite’ all the way along – we can draw it as this triangle, where here the bottom arrow gf is the composite of f and g:

(Here I’m writing gf as a more succinct notation for the composite, which we previously wrote as g°f.)

For transitivity we might want to say that if f counts as some sort of sameness and g also does, then the composite gf should as well. But we are also trying to treat f and g as if they are symmetric, even if they aren’t, which means we can go backwards along them. Let’s do a little thought experiment. If we could go backwards along them, then:

(a) we could reconstruct g by going backwards along f and then forwards along gf, and

(b) we could reconstruct f by going forwards along gf and then backwards along g.

Now if we apply transitivity to those we get:

(a) If f and gf count as some form of sameness, then g should too.

(b) If gf and g count as some form of sameness, then f should too.

An arrow that counts as some form of sameness in this situation is sometimes called a ‘weak equivalence’. It’s some form of equivalence but weaker than all the ones we’ve seen so far.

So to summarise all the types of ‘transitivity’ we’ve found, we have this:

1. If f and g are weak equivalences then gf should also be a weak equivalence.

2. If f and gf are weak equivalences then g should also be a weak equivalence.

3. If g and gf are weak equivalences then f should also be a weak equivalence.

This can be further summarised as saying ‘If any two arrows in the above triangle are weak equivalences, then the third one should also be a weak equivalence’. This is called the ‘two out of three’ property because it’s about having two out of three arrows and knowing that the third one also has the property.

The idea is that if this condition is satisfied, then you can treat these things as if they have inverses and get a sensible enough structure out at the end. ‘Treating these things as if they have inverses’ usually means making a new category with some extra arrows in it that act as the inverses.

This is a similar idea to what we did very early on in this book to get ourselves the negative numbers. We started with just the natural numbers, that is, the whole numbers 0, 1, 2, 3, and so on. Then we decided we wanted everything to have an additive inverse, so we just invented the inverses and put them in, to make the integers (which are the natural numbers and the negative numbers together). We can make the rational numbers similarly, by throwing in all the multiplicative inverses, but we have to be a bit more careful, because we only throw in inverses for every non-zero number. If we tried to throw in an inverse for zero we would have a problem of collapse, a bit like when we tried to say what it means for people to be more or less the same age as each other and found that including transitivity would mean everyone is more or less the same age as everyone else.

For fractions the collapse would be because of what happens if you throw in a multiplicative inverse for 0. By definition, it would be some number a such that:

a × 0 = 0 × a = 1

However, we also know that for every number a

a × 0 = 0

So if we put those two equalities together (using symmetry and transitivity) we get 1 = 0, and if that’s true then everything equals 0. The upshot is that if we try to treat 0 as invertible we end up collapsing the entire structure to 0.* This is why we have to be a bit careful about what we can choose to count as ‘sameness’ if we don’t want to end up collapsing our entire structure. It doesn’t really matter – we can go ahead and count everything the same as everything else, but it doesn’t get us very far, even though it’s not logically inconsistent and it’s not wrong. Maths turns out to be not just about right and wrong answers – it should also be about what’s helpful.

In fact, all of these machinations are signs that something higher-dimensional is going on. That is, if we’re working in a category in which we want a notion of sameness that is yet more forgiving than the isomorphisms in the category, it’s a sign that really we should be working in a higher-dimensional category.

Ever more nuance

I love reading Agatha Christie, but if I had to name my favourite of her books, I’d be hard-pressed. I don’t like putting things in competition with each other and having to name one ‘winner’. Or, if I do, I’d rather do so via a more layered process. There are various of her books that are favourites of mine for different reasons. I like the ones with amusingly feisty young female characters, such as The Sittaford Mystery, They Came to Baghdad and The Man in the Brown Suit. I have fondness for the ones with unexpected female mathematician characters, such as Cat Among the Pigeons and 4.50 from Paddington. I like certain plot twists such as Endless Night and The Unexpected Guest. But the ones that grew on me slowly are the ones that give wry insight into human nature, like The Hollow. There are so many different dimensions that might carry more or less weight at different times.

Once I’ve established these different ways in which I like those books, I can think about different ways in which I like those different features. If I’m reading to pass time while on a train (which is when I do a lot of my reading) it helps to have something with a lot of engaging action, so that I get engrossed and the journey passes more quickly. But it’s not clear that a book depending on a plot twist has quite the same impact when you already know what’s going to happen, although I still enjoy reliving the twist. For re-reading books, the ones that give wry insight or have beautifully poetic language have more to give upon repetition. I love re-reading novels, as I get much more out of them when I’m not on the edge of my seat wondering what’s going to happen next, and I honestly find suspense a bit stressful. So then the question is whether I prefer books that are good for travelling, or books that are good for re-reading. Probably ultimately I would put ‘good for re-reading’ higher, which is why if forced to do so I would probably name Pride and Prejudice as my all-time favourite book because of its engaging characters, wry insight into human nature, beautifully poetic language and understated wit. It doesn’t have the most exciting or surprising plot, but in the end that doesn’t matter much to me.

To me, that is all about higher-dimensional thinking, where we don’t just compare things, but we have different ways to compare things, and then different ways to compare those different ways, and different ways to compare those, and so on forever.

Higher-dimensional spaces

Another form of higher-dimensional thinking is where we have different ways to compare things and then each one of those is its own dimension. At one basic level we could just assess whether people have various benefits such as a job, access to healthcare, a house. We could put them in a diagram like this according to which things they do have, and it would notionally be a three-dimensional cube, as we saw earlier:

We can perhaps now see a sort of hierarchy where at the top level there are people with all three of the advantages in question, at the next level there are people with two of them – perhaps a different two from each other – at the next level down there are people with just one of those things, and at the bottom there are people with none of those things.

If we were much less nuanced about it we could just count the number of advantages people have (out of the given list of three), and how many groups of people there are with each type of advantage:

no. of things

no. of groups of people

3

1

2

3

1

3

0

1

This is a one-dimensional version, where we have forgotten the nuance of exactly what advantages each group of people has, and we have collapsed all the advantages down to simply counting how many there are instead of counting what they are. This is like the move from sets to numbers where we count all sets of two things as ‘the same’, disregarding the detail of what all those two things are.

We could add a fourth advantage, say access to public transport, and the diagram would become a four-dimensional cube. At this point it becomes hard to fit the labels on each corner, so we could abbreviate with a string of coordinates where each position has a 0 if someone doesn’t have that attribute, and a 1 if they do. We get this:

So starting at the top, there are four possible things to lose, which is why there are four edges coming out of the top point. This is more easily seen to be a four-dimensional cube if we rearrange the diagram physically on the page a little, like this:

This diagram is one of the standard representations of a four-dimensional cube on a two-dimensional page. You might be able to spot four pairs of parallel three-dimensional cubes inside this diagram. Here is one pair:

This representation is intriguing to me in many ways. It is better for ‘seeing’ the cube structure, but it is much harder than the first representation for seeing the hierarchy of how many ‘things’ each person has. If we look back at the first representation to see this, we can see that we have the following table showing the one-dimensional version:

no. of things

no. of groups of people

4

1

3

4

2

6

1

4

0

1

If we add another dimension, that is, a fifth advantage, we get the following table, no matter what the advantages are:

no. of things

no. of groups of people

5

1

4

5

3

10

2

10

1

5

0

1

You might recognise these as the lines of Pascal’s triangle. That’s the triangle of numbers obtained by starting with:

1

1 1

Then we make each new line by starting and ending with 1, and for all the positions in between we add up the two numbers above:

Pascal’s triangle pops up all over the place, as it is related to how many ways there are to choose k objects from a set of n, and in this case we are thinking about having k advantages from a set of n. But what I wanted to say is that as we get further down the triangle, the shape of the resulting graph increasingly resembles a bell curve or normal distribution. Here are a few more rows of the triangle:

And here’s that last row, plotted on top of a bell curve:

A student in my class recently asked me if a normal distribution is really a sign that we’ve taken something infinite-dimensional and examined it one-dimensionally. I thought that was a very interesting possibility.

Ways of reducing dimensions

We have been reducing an n-dimensional situation down to one dimension, turning n-dimensional cubes into things that look more like bell curves. We often simplify higher-dimensional situations down to one dimension for ease, but that means we lose nuance. This happens every time there’s some giant ranking of something complex like universities. Usually the method for ranking them has many different criteria, which are then somehow combined into one by an often rather dubious weighting system. And that’s even before considering that some of the individual criteria are rather strange in the first place: I am sceptical of the criterion that consists of measuring what proportion of each graduating class gives money back to the university. (I know one example of a fairly small university which easily gamed this by having every student give one dollar. The students were highly motivated to take part because it meant their university shot up in the rankings, which in turn meant their degrees were worth more. I suppose they do deserve some credit for figuring this out.)

Even if the individual criteria are sensible, all those nuances get lost when all that information is combined into one dimension to produce one hierarchy. Keeping more dimensions gives us more nuance on how things count as the same or ways in which they might be different. However, this is very hard, so there is a trade-off and a possible case of diminishing returns. Higher-dimensional category theory helps us deal with this.

Higher-dimensional category theory

Higher-dimensional category theory builds on the idea of category theory, where we study all objects via their relationships with each other. The idea is that if we are studying everything via relationships with each other, shouldn’t we also study relationships themselves via their relationships with each other? This gives us an extra layer of nuance that is considered to be an extra ‘dimension’ in category theory:

The zero-dimensional things are just objects.

The one-dimensional things are relationships or arrows between those objects. We draw them like this:

fA→B

The two-dimensional things are relationships or arrows between the one-dimensional arrows. We often draw them like this:

We can think of the zero-dimensional things as points, the one-dimensional ones as lines, and the two-dimensional ones as surfaces. However, they don’t have to be physical lines and surfaces, which is why we think of them as relationships or just ‘arrows’.

We can build up much more interesting and complicated diagrams using these two-dimensional arrows, such as in this diagram taken from a recent research paper of mine* studying ways to organise higher-dimensional categories in terms of lower-dimensional components:

But this is just the start, because we might then want to study the two-dimensional things via relationships between them: three-dimensional arrows. And then we want relationships between those, which are four-dimensional arrows. These things quickly become difficult to draw, so we need ways to reason with them other than by drawing diagrams. This issue is one of the (many) things that make higher-dimensional category theory hard, and it is still a very open research field in the early stages of development. The two-dimensional situation is fairly well understood, but the three-dimensional situation is already very difficult. Then consider that if we really want to be consistent about our philosophy of relationships, we should never stop adding in relationships between our relationships, which means we will have infinite dimensions.

As it’s still open research it’s far beyond our scope here to go into those details, but I want to try and explain how this gives us more and more subtle notions of sameness.

Weaker equivalence

One of the ideas behind adding dimensions to a situation is to give us more ways in which to say things are ‘the same’. In a set, things are just equal or not, but when we move into a category we have the possibility of things being isomorphic via a pair of inverse arrows:

Another principle behind this move is that a category really treats isomorphic objects as the same – it can’t tell them apart, as we saw in Chapter 8. So we don’t really use the concept of equality in a category. This means I chuckle a little when people think maths is all about equations, because one principle of category theory is to get rid of all equations and to go round replacing equalities with isomorphisms instead.

One example of this gives us a weakened version of uniqueness. We have seen that in maths if we want to say there is a unique object with a property P, we often say it like this:

If x and y both have property P then x = y.

But if we replace the concept of ‘equality’ by the concept of ‘isomorphism’, then we have this broader version of uniqueness:

If x and y both have property P then x is isomorphic to y.

We encountered this when we showed that if x and y are both terminal objects in a category, then they must be isomorphic; in fact we showed that there must be a unique isomorphism between them.

Uniqueness in life is sometimes more forgiving than saying there is actually only one person with a certain characteristic. I think this is why it feels like it makes sense to call someone ‘very unique’, as if there are gradations of uniqueness, although pedants declare that unique means ‘only one’ and so you are either unique or you aren’t.

Sometimes when we pit two people against each other there doesn’t have to be only one winner – we’re allowed a tie. I like it when there are joint winners because I don’t think everything needs to be about people beating other people, and because I think we should acknowledge that sometimes there are so many dimensions involved it is futile and beside the point to force things into one dimension. But there was general outrage when the Booker Prize of 2019 was awarded jointly to Margaret Atwood and Bernardine Evaristo; admittedly there may have been objections deeper than simply ‘there should only be one winner’. Elections must have one single winner, although maybe one day we’ll have joint presidents or something. There was an amusing incident in 2014 when the University of Alberta was hiring a new president, and some professors started applying for the job in groups of four to make the point that the salary was ridiculously large compared with normal professors. The idea was that they could split the enormous salary between all four of them, and then the university would get four people’s work for the money instead of just one person’s, and they would probably be much more effective. The first such application was led by Professor Kathy Cawsey, Associate Professor of English at Dalhousie University, and twelve more groups of professors followed. (None of them was offered the job.)*

It turns out it is possible for there to be joint winners of a gold medal at the Olympics. During the postponed Tokyo 2020 Olympics (which actually took place in 2021) high-jumpers Gianmarco Tamberi and Mutaz Barshim requested to share the gold medal rather than compete in a jump-off. There had not been a shared gold medal in an Olympic track and field event since 1912, but apparently a resolution was passed in 2011 officially allowing sharing of medals, and this was what these two great friends requested to do rather than try to beat each other.

In maths we’ve shown that allowing another dimension and more nuance means that we have a new notion of sameness called isomorphism. Now, since uniqueness is linked to the concept of sameness, this means that we also have a more nuanced notion of uniqueness.

However, the definition of isomorphism still involves some equalities; they just happen to be at the next level up. We defined an isomorphism between A and B to be a pair of arrows:

such that the composites both ways round equal the identity, that is:

g ◦ f = 1A and f ◦ g = 1B

That’s because the things on either side of the equals signs are arrows in a one-dimensional category, so in that framework there are no relationships between them other than equality – they’re simply just equal or not. We’ve hit a ceiling in our category and have no more dimensions to work with, so we have to stop and use an equality. However, if we have one more dimension to work with, we can use one more level of isomorphism before quitting at the dimension above.

That is, if we have a notion of a relationship or an arrow between the arrows, we can invoke that now. Instead of asking for the composites both ways round to be equal to the identity, we can ask them to be isomorphic to the identity:

g ◦ f ≅ 1A and f ◦ g ≅ 1B

Here the symbol ≅ means ‘is isomorphic to’.

Let’s now unpack what that means. An isomorphism between one-dimensional arrows is a pair of two-dimensional arrows  and  (we sometimes use Greek letters when we’ve gone up a dimension):

such that both composites equal the identity:

β ◦ α = 1 and α ◦ β = 1

Let me say that again: such that both composites equal the identity. That’s right, we are stuck saying ‘equals’ again as we have run out of dimensions. However, if we have some three-dimensional arrows in between our two-dimensional ones we can put isomorphisms here as well. Note that at the previous dimension we needed one pair of inverse two-dimensional arrows giving the isomorphism g°f ≅ 1 and another pair giving the isomorphism f°g ≅ 1. Each of those pairs has a composite two ways round, and each of those needs a pair of invertible three-dimensional arrows, so we now have twice as many three-dimensional arrows in this situation as we had two-dimensional arrows. And if we have four-dimensional arrows we can then ask for those, and we’ll need twice as many of those as we had three-dimensional arrows.

You might be alarmed that at each dimension we need twice as many arrows to deal with this situation as we did at the level below. This is one sign that the situation gets rapidly complicated as we include more and more dimensions. So with each dimension we include, we gain more nuance, but we also gain more logistical complication. This usually means we should carefully consider the trade-off and think about how many more dimensions it’s really worth including.

I still think it’s worth including some, though. It’s like the fact that movies are much more complicated to make than photographs, including the extra dimension of time. Sculptures are more complicated to make than pictures because they have an extra physical dimension. Modern movies are more complicated to make than silent movies because of the extra dimension of sound. Indeed, stereo recordings are more complicated to make than mono recordings because of the extra dimension of spacing. But evidently humans have decided that the extra effort is worth it. Technology can help us to manage the increasing complexity, but we can also help ourselves by practising and developing this skill. However, despite the advances of technology and of our brains, two-dimensional paintings and non-moving photographs still exist, whereas silent movies have mostly become obsolete. It’s not clear if three-dimensional movies or movies with smells are ever really going to take off, although they do already exist; possibly the higher-dimensional experience has gone more in the direction of virtual or augmented reality.

Humans do have a tendency to try and keep things in lower dimensions – and often we squash them into just one dimension – because it’s easier. Or is it easier? It is simpler on the face of it, but then we end up having to deal with more complicated consequences and antagonisms that come from having made that oversimplification. If we get better at thinking in higher dimensions, then that becomes ‘easier’ too, with the benefit of us gaining nuance at the same time.

We should try to invoke even just a little more nuance in some of our arguments in normal life. We don’t necessarily need to go infinite-dimensional, but at least acknowledging one or two more dimensions would already help. It would be less divisive to acknowledge that there are senses in which different things are true, and then have a discussion about what those different senses are and why we care about them differently, whether that is books, restaurants, medical treatments or politics. The possibilities, in life and in maths, are endless, and the potential for what we can learn from doing this is also endless. More dimensions lead to more nuance, and more nuance leads to us being able to communicate things with more detail, and thereby understand each other more deeply. This might not seem the most obvious aim of mathematics, but I think it’s an important one, which deserves more attention.

* The number 1 counts as a product of zero prime numbers, just like the number 0 is the sum of zero numbers.

* This collapse happens if we maintain all the usual rules of arithmetic for numbers. If we drop some of those rules there are ways to have an inverse for 0 without collapsing everything.

* Eugenia Cheng and Alexander S. Corner. Weak vertical composition II: totalities. Theory and Applications of Categories, 41(31):995–1043, 2024.

* ‘Job sharing: Game of clones’, Nature, 25 June 2014.

EPILOGUE

Somewhere in between working on the third and fourth drafts of this book, I had the extraordinary and surreal experience of chatting with NASA astronaut Dr Jeanette Epps while she was on the International Space Station. I had met and befriended Jeanette at a conference, and in March 2024, as she launched into her first Space mission, I watched the live feed with awe, admiration and not a little trepidation.

One evening a few months later I was astonished to receive a WhatsApp message from her, saying hello and asking if we could do a video call the next day. When I replied, I could hardly believe that I was actually messaging someone in Space. That disbelief was nothing compared with the surreal astonishment I felt the next morning when I clicked on a link she had sent me – using software I had previously used for virtual medical appointments during the pandemic – and there was Jeanette on my laptop screen. I was, as it happened, in a friend’s kitchen in Devon. Jeanette was in Space.

She was, of course, floating around in zero gravity. In the background, various other astronauts periodically drifted past, waving. She turned the camera to point out of the window to show me the view, and there was Earth, like a balloon, floating nonchalantly, perfectly round, shrouded in wispy clouds.

My brain felt like it was exploding, just like it so often does when I’m getting my head around new mathematical concepts. Part of what was so surreal was how calm Jeanette was. She had been there for months, and appeared perfectly comfortable and accustomed to her surroundings, while I was shrieking with excitement. It is perhaps no wonder that some people deny that any of that is real, even when they see pictures, because it is a lot to wrap our heads around when we are so used to our perspective of life on Earth, with its local flatness, its cycles of day and night, and its inexorable, inescapable gravity.

Besides the mind-bending experience of seeing the Earth from Jeanette’s point of view in Space, I was captivated by watching other astronauts float past from one part of the ISS to another, all facing different ways. Some went horizontally, some were vertical, some were diagonal. Except all of those direction descriptions are only relative to my Earth-bound orientation.

Jeanette explained to me that when you are in zero gravity you just have to pick a direction and decide which will be ‘up’ for now. On Earth we are constrained to feel ‘down’ as the direction of the pull of gravity, and that can’t be changed, which is more or less why mirrors switch our left and right but not our up and down. I have only experienced not knowing which way is up very temporarily on a few occasions when swimming in the ocean and being hit by an unadvisedly large wave; I’ve been so buffeted and tossed around that I could no longer tell which way was up, which was particularly disconcerting as I was underwater and needed to find the surface in order to breathe.

Jeanette told me that at first it was disorienting in zero gravity too, that if she tried to turn herself the other way up she would flail around as her brain was attached to the previous way up. But, she said, you just have to convince yourself that the new way up is really up, and in order to do that you first have to convince yourself that you’re going to be able to do it. And then you – or at least, the brilliant, highly accomplished, well-trained astronauts – get better and better at it, and then quite used to it.

For what it’s worth I did get more used to talking to her, across our several video chats. I almost wished I didn’t, because I wanted to feel the excitement of it every time, but also, I am fascinated and perennially awe-struck by our brains’ ability to become accustomed to things that we previously thought impossible. We have that ability, but we do have to believe we have that ability first, just like the astronauts have to believe in their ability to acclimatise themselves to a new direction of up.

Maths is more about changing which way up we are than it may at first seem. It is about zooming in and out, like looking at Earth from Space. It’s about seeing something from someone else’s point of view, or working out how to operate somewhere without your familiar tools (such as gravity), or the amazing idea of deciding to take a new way as up, and turning yourself to face that way.

Most of us don’t get to decide which way is physically up, because we are on Earth operating under gravity, but we do get to decide which way is metaphorically up for some other, more abstract things. We get to decide what is right, what is just, what is fair. We get to decide who we are going to count as equals under what circumstances. We get to decide what differences we are going to take into account and what sameness we are going to accept. But, just like the ‘up’ direction in Space, we first have to convince ourselves that it is a choice. This may be daunting because when something is a choice, we have to choose well, and we have to take responsibility for the choice we make. Taking responsibility for our choice involves taking responsibility for the consequences too, and the consequences of those, and of those, forever, in a long string of transitivity. Maths is a safe, controlled environment for practising this. We can try out different choices and explore their consequences in theory, before putting anything into practice. We can exercise and stretch our intellectual muscles in simpler contexts, so that we can become both strong and flexible for when we need to think about the complicated world outside our brains. And we can create models describing the essence of that complicated world, so that we can recognise when the same thing is going on in different places, and learn from our experiences of one place to help us with another.

When we talk about things being the same and different there is a big danger of getting into black-and-white arguments. Throughout this book I’ve been presenting a mathematical approach that is more fluid, in which we acknowledge the choices we are making when we say that things are the same and different. We are making those choices whether we are doing so consciously or subconsciously, and since we are doing it, it’s beneficial to become conscious of the choices we are making, and the frameworks we’re using for making them.

A great deal of mathematics turns out to be about bringing into consciousness things that we are doing intuitively. Perhaps it’s all a form of intellectual self-awareness. It’s a form of self-awareness that we don’t need if we’re having straightforward thoughts and not having detailed discussions with anyone else. But although excessive self-awareness can sometimes be paralysing, that’s not a good reason to avoid self-awareness. It’s better to gain the self-awareness and also learn to remain pragmatic and to be able to work with that self-awareness, so that we can benefit from the pros and not be too set back by the cons.

The fluidity and flexibility of mathematics is part of its power, so it makes me endlessly sad that maths is thought of as rigid. We have seen that maths is about holding sameness and difference in our heads together, and about being able to move between different points of view giving different types of sameness and difference. I hope we can all learn to do this in life as well. Because ultimately we are all different and we are also all the same. We’re all tiny specks on a small planet orbiting one sun in a giant universe, even if precious few of us get to go into Space and see Earth from that point of view. But we’re also living creatures with hopes, fears, joys, griefs. I hope we can acknowledge our differences, but also put them aside in order to work together to make a better world, especially for those who most need it.

ACKNOWLEDGEMENTS

I would like to thank everyone at Profile for their ongoing support, especially my editor Izzy Everington, my publicist Valentina Zanca, and Andrew Franklin. I am always grateful to my family and friends, especially old friends who are always there for me through copious dramatic ups and downs. Also for someone who doesn’t want official thanks, here’s an apophasis.

INDEX

12-hour clock 168, 277

24-hour clock 168

A

abortion 165, 213

abstract art 32, 232, 259

adaptor 333

additive inverse 76, 355

advantages 319, 358

advertiser 70

AI 309

algebraic topology 342

Alps 319

alternative vote 86

aphantasia 95

apophasis 377

approximation 16, 26, 42, 96, 168, 191

arithmetic modulo 3 290

art gallery 232, 306

associativity 34, 205, 233, 255

astronauts 346

asylum 320

asymptote 126

ATLAS of Finite Groups 295

Atwood, Margaret 367

authority 41

B

baby and bathwater 71

Bach 268

bagel 131, 150, 341

baking 19, 281

baking parchment 17

balance scale 46

Barshim, Mutaz 367

basic building blocks 13, 122, 133, 291, 331, 336

Battenberg cake 146

benefits 28

bijections 66, 145, 174, 190, 231, 251

biologists 301

biology 339

birds 302

biscuit 16

black-and-white 33, 59, 193, 249, 376

Blackface 214

blended families 50

BMI 59

Boaler, Jo 220

Bonne Maman 167

Booker Prize 367

both-sidesism 211, 218, 320

Buffalo 307

C

cake 18, 113, 146, 160, 268, 281

cancel culture 39

canonical 181, 185, 328

carbon footprint 163

Cartesian plane 139, 144, 276, 297, 318

Casio synthesiser 118

category theory 76, 195, 229, census 87

chaotic 124

cheese 217, 247

chemistry 339

chess 30, 134

Chicago 103, 160, 319

Chicago River 160

childless women 213

Christie, Agatha 356

circle 17, 113, 295, 333, 341

classification 279

classification of groups 287

classification of manifolds 301

Claussen pickles 167

coffee cup 341

coherence theorem 205, 212, 236

Colbert, Stephen 247

Comment cuire un 9 308

commutativity 34, 54, 73, 108, 205

completing the square 112

conic sections 113

constant function 101

continuous 134, 298, 343

contrapositive 217, 337

convergent sequence 83

coordinates 109, 139, 142, 182, 276, 318, 349

corn on the cob 184

Corner, Alexander 224, 364

corrie 320

cousin 236

covid 87, 123, 226, 312

Craig, Daniel 265

cube 323, 357

cube, four-dimensional 359

cycles, vicious and virtuous 323

D

Dalhousie University 367

data 233

debt 79

deception 226

democracy 42, 69, 172

density 136

Descartes, René 139

desire 321

desperation 321

diagrammatic reasoning 261

discrimination 87, 238

disjoint union 177

DNA 339

do nothing 241

dot dot dot 92

doughnuts 90, 150, 296, 341

draping 141

driving test 204

drugs 71

duck 265

Duke of Edinburgh Award 140

E

Einstein, Albert 109, 156

Einstein’s Theory of Relativity 109

election 20, 87, 219, 226, 320, 367

elephant 56, 136, 178, 325, 326

ellipse 113

emotions 221

empathise 41, 194, 310

empty set 72, 174, 324

equations 33

equilateral triangle 290

equivalence relation 168, 206, 231, 332, 347

error correction 128

EU 132

Euler’s identity 35

Evaristo, Bernadine 367

EvenQuads 24

exam grades 28

exams 23, 64, 276

exploitation 320

exponential 116, 123, 131

eye colour 337

F

factorial 288

false equivalence 57, 211

false positives and false negatives 42, 209, 320

feminism 39

floating 268, 328, 346, 373

floating away 232

fly 163, 346

four-dimensional cube 359

Fourier analysis 122

fourth dimension 156, 304

fractions 79

Fraenkel, Abraham 176

frames of reference 108

frequency 121

friend 50

Full Monty, The 307

function 47, 89

constant 101

do nothing 241

identity 104

invertible 261

order-preserving 273, 334

functor 312, 330, 339

G

gender pay gap 137

generator 331, 348

genus 150

geometric progression 209

geometry 155

glaciation 319

Glass Onion 265

glueing 149, 159, 169, 348

god 39

goodwill 321

GPS 140

grade point average 92

grammar 247

grandmother 235

graph 95

gravity 109, 306, 346, 373

Green Card 199

Green Card 201

grey area 16, 30, 61, 191, 277

group homomorphism 281, 315

groups 277

H

hair band 155

handle 154, 300, 331, 342

harmonic series 119

harmonics 118

Hawking, Stephen 156

haywire 99

helicopter 238

hierarchy 276, 363

homotopy 151, 296

homotopy equivalence 296, 331, 341, 352

homotopy invariance 343

Hooke’s law 100

How to Bake Pi 308

hurdles 232

hyperbola 113, 126

I

i 38, 144

identical twins 262

identity function 104

identity politics 39

imaginary number 38, 144

imagination 66, 69, 152, 156, 302

incomparable 275

infinities 68, 177

infinity 126, 176, 303

initial object 327

instant gratification 160

intelligence 63

interchangeability 54

intrinsic characteristics 262, 265, 297

invariant 334

inverse, additive 76, 355

inverse, multiplicative 80, 246, 355

invertibility 246

IQ 63

irrational number 35, 63, 71, 83, 144

isomorphism 246

of categories 315

of groups 343

IVF 226

J

jet engine 306

jigsaw puzzle 236, 261

job application 238, 276

John o’Groats 238

joint winners 366

joke 308

‘joke’ 341

K

Klein bottle 304

knee operation 238

Knives Out 265

knot, trefoil 336

knots 335

L

Land’s End 238

large language models 308

Latin square 135, 277, 283, 289, 315, 331

Lawful Permanent Resident 199

laziness 35, 230

Lego 133

Leibniz’s formula 35

lemma 264

lexicographic order 182, 276

limit 84

logic 193, 241, 320, 337

loops 323

Lorenz system 124

luck 46, 339

M

macarons 19

make-up 148, 214

Maltesers 172

Manhattan 103, 151, 204

manifold 138, 154, 301, 335

manipulation 70

maternity leave 165

meat 163

meta-category 312

meteorologists 125

mind-bending 37, 50, 312, 374

mind-boggling 154, 291, 331

Möbius strip 142, 154, 170, 293, 303, 331

monkey 270

Moss, Tara 74

moving house 342

multiplicative inverse 80, 246, 355

murder 37, 165, 213

murder mystery 37

music 30, 60, 101, 117, 171, 262, 306

music videos 117

N

natural numbers 173, 355

negative numbers 71, 76

nibling 236

Nobel Prize 71

non-orientable 143, 146, 154, 163, 303

not-understanding 261

O

obvious and obscure 48

Olympics 367

oops 112

order-preserving 273, 276, 334

ordered set 269

Ordnance Survey 140

orientable 143, 154

ornithologist 302

overeat 19

P

painting by numbers 284

palaver 180

pandemic 87, 109, 123, 373

papier-mâché 139, 150

parabola 109

partial order 275

Pascal’s triangle 361

patchwork 141

pedantic 52, 204

periodic function 122

physics 27, 97, 326, 339, 346

pi 22

pipe organs 118

pitch 121

pixels 17

pizza 59, 79

playdough 299, 341

political parties 28, 192

politician 35, 40, 70, 87

polls 87, 226

polygon 290

power 320

preservation 26

President of the United States 73, 78

Pride and Prejudice 357

prime numbers 122, 203, 290, 331, 348

privileges 319

product 144

of prime numbers 13, 203, 291

of real line 318

of zero prime numbers 348

twisted 144

projective plane 159, 304

prone 195

properties 233

proportional 116

psychic powers 92

pull out all the stops 121

Q

quadratic 111, 201

quotient 168, 186

R

ranked choice voting 86

rational number 83

reciprocity 53

recycle 163

reflexivity 186, 206, 231, 246, 347, 350

representatives 69, 172, 185

robotic arm 306

rote memorisation 42

rounding 21, 49

S

sample size 81

SAT 28

School of the Art Institute of Chicago 25

scone 17

Scottish government 132

Scrabble 182

sea 30

self-governance 42

sewing 170

shopping lists 177

sine wave 121, 331

single transferable vote 86

sliding scales 28

sneaky 35

socialism 212

sport 60, 306, 320

sports-fan-attitude 320

springs 97

spur 319

stability 44

standardised tests 63, 164

stereotypes 213

Stilton 81

Stone, Deborah 302

straw person 213

stretchiness 298

structure 233

stunt double 31

substitution 25, 55, 78

Sudoku 135

superlative 76

superpower 16

supine 195

symmetry 17, 51, 186, 206, 231, 245, 352

syntax 247

synthesis 118

T

Tamberi, Gianmarco 367

tape measure 50, 169

tedious 52, 69, 168, 173, 180, 288

terminal object 327, 366

theoretical physics 346

timbre 119

tolerance 17

tolerance interval 19, 49, 96, 100, 122, 168, 191

topological space 297, 343

topology 138, 154, 295, 341

topology, algebraic 342

torus 150, 170, 303, 348, 352

transformer 333

transitivity 19, 49, 186, 231, 347, 375

tricolorability 338

trousers 64, 148, 301

twisted product 144

two out of three 354

U

uncle 235

unconscious biases 42

unintuitive 297

unique selling point 330

uniquely isomorphic 328

uniqueness 73, 328, 366

unit square 349

universal healthcare 212

universal properties 324

unknot 336

UTC 109

V

vaccines 88, 117, 123, 226, 312, 320

vacuously 326

visual intuition 94, 115, 196, 234

W

wavelength 119

weather 97, 124

WeightWatchers 26

what-aboutery 218

Winchester by-election 21

wokeness 39

Y

Yellowface 214

Yoneda Lemma 264

Yoneda, Nobuo 264

Z

Zermelo, Ernst 176

zero 72, 103, 151

zooming out 312, 333, 375