Instant Genius - How the maths of sameness and difference can change the way you view the world

Episode Date: July 6, 2025

Anyone who has studied maths even briefly will no doubt be familiar with the ‘equals’ sign. But there’s much more to this seemingly simple symbol and the concepts that it represents than may first meet the eye. In this episode, we speak to mathematician and author Dr Eugenia Cheng about her latest book Unequal – The Maths of When Things Do (and Don’t) Add Up. She tells us why maths doesn’t have to be intimidating, the importance of recognising different viewpoints in maths, and how a gaining a deeper understanding of maths can help us live our daily lives more effectively. To get the exclusive gift box from Shokz, order via this link: ⁠https://bit.ly/4kFt10l⁠ Learn more about your ad choices. Visit podcastchoices.com/adchoices

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Every Monday and Friday you hear world-leading scientists and experts talking about the most fascinating ideas in science and technology today. I'm Jason Goodyear, commissioning editor of BBC Science Focus. Anyone who has studied maths even briefly will no doubt be familiar with the equal sign. But there's much more to this seemingly simple symbol and the concepts that it represents that may first meet the eye. In this episode, we speak to mathematician and author

Dr. Eugenia Chink about her latest book, Unequal, The Maths of When Things Do and Don't Add Up. She tells us why maths doesn't have to be intimidating, the importance of recognising different viewpoints in maths, and how gaining a deeper understanding of maths, can help us live our daily lives more effectively. So welcome to the podcast. much for joining us. Thanks for having me. So today we're talking all about your book, Unequal,

the maths of when things do and don't add up. So for someone who's thinking of picking up a copy, let's start with, what's the premise of the book? The premise is I wanted to talk about equality and also inequality in maths and in life. And maths is famous for equations, which sound like just things being equal. But what I want to stress is that there's always something equal and not equal at the same time. And we usually only focus on the equal part and not the unequal part. And actually, we make decisions about where the line is in between those. And there are lots of interesting techniques in maths for doing that, which I think can help us think about tricky issues in life as well. So you mentioned that equations. Let's start with this then. So what exactly is an

equation? Good question. Well, some people, think they're just things to torture children with. And unfortunately, that is also true. But really, it's giving us two points of view on the same thing. Right. There's a left-hand side of the equation on the right-hand side. And what they're saying is this is two different points of view that are somehow the same in some way. Like, one plus three equals three plus one. So you might go, oh, yeah, well, that's obvious because they're both four. But what it's saying is that there are two different ways of making four, and that they both make four through a different process. So you also talk about something called bijection. I think we should like define what that is.

What do you mean by that? Oh wow, that's a big leap down the line. So that's the big main concept about sameness that I talk about in the first part of the book, because maths is about when things do and don't count as the same and what that can mean. And by ejection, which is a very technical sounding word, what it means is that you have two sets of things that kind of match up with each other perfectly. And so it's like saying if you invite a whole lot of people to a party, are there enough chairs for them? And so there are two possibilities. Maybe there are more people than chairs, in which case some people would have to, you know, sit on each other's laps or stand. Or maybe there are more chairs than people, in which case there'll be some empty chairs. But maybe

you want there to be exactly the same number of chairs as people. Maybe it's a concert and the ideal situation is that every seat is sold out and nobody has to share a seat, but there's also no empty seats, in which case there's a perfect matching up between the people and the seats. And that is called a byjection in maths. And it's where the concept of numbers comes from, because if two things perfectly match up, it's kind of because there's the same number of them. So sort of if you'll indulge me, sticking with the sort of definitions, you also talk about functions. So what's a function?

Oh, you're diving straight into the technical things here. There's quite a lot of technical stuff in this book that I wanted to show because although many people, unfortunately, find it a little scary and off-putting, I want to show that it can be helpful. And I want to stress that I'm not expecting everyone to understand everything in the book, but just to kind of see it and appreciate how it works, because mathematicians come up with these rigorous techniques in order to make sure that we really know what we're talking about.

And if you're not worried about how you know what you're talking about, then maybe you don't care. You know, if you just go around making things up and you don't need to back your arguments up with anything, then it doesn't really matter where your thoughts came from. But mathematicians try to back everything up very carefully. And a function is a starting point for thinking about when things match up perfectly. So a function, again, can be thought of as something used to torture children

at school. And when it is being used to torture children at school, and I don't mean that everyone is torturing children at school, but when it's used at school, it's often in the form of some kind of formula like x squared or x plus one or two x plus three or something. But a function is really just a way of taking some inputs and producing some outputs. And I like to think it's a bit like a vending machine where there's a load of buttons and you press buttons in different combinations and hopefully out pops the food that you want. And so that's how I like to think of a function. Just coming off the back of that, well, you mentioned there. So a lot of people who studied at school that, you know, they'll know basic arithmetic or etc. But when we start using letters

instead of numbers, some people get really intimidated. So what's your view on that? Yeah, I sympathise with that. And unfortunately, maths is often presented in a way that is quite intimidating. And some people get a lot of self-esteem from being able to do this difficult thing. And unfortunately, that can often make the other people feel worse. And if the people who can do it are very self-satisfied about their ability to do it, they will want to gloat over the people who find it difficult. So that is one of the reasons it can be intimidated. And another reason it can be intimidating, if nobody's ever really explained to you that it's supposed to be there to help you. It's not just there to test you or to put people in a hierarchy by test scores. The reason that we use letters instead of numbers is so that we can, first of all, refer to things before we know what they are. So if we're trying to understand, we're trying to figure out what number something is, but we don't know what it is yet, then we need a way to refer to it. It's a bit like a pronoun when we are talking about a person and we don't know who they are yet.

So we can refer to a number as X and then we can explore some of the things that X satisfies without knowing what it is and then we can unravel it to work out what X actually is. And another reason we use letters is so that we can refer to a lot of numbers at the same time so that we can use our brains more efficiently. A lot of maths is about working out ways to use our poor little things. finite brains more efficiently to understand the very complicated world around us. And so we can make general statements. So instead of saying, 1 plus 3 equals 3 plus 1, and also 1 plus 4 equals 4 plus 1, and also 2 plus 5

equals 5 plus 2, and also 2 plus 2 equals 3 plus 2, which is pretty tedious, we can instead say X plus y equals y plus x for any numbers x and y. and that's a much more efficient way of expressing it. That's a great explanation. So in the book you talk a lot about different points of view and how they're importance in maths. So what do you mean by that? Well, going back to the very basic example I gave at the beginning

of 1 plus 3 equaling 3 plus 1, if we go back and think about how we might explain that to small children, it's quite interesting because we do a lot of physical manipulations when we're doing maths with small children. We get counting blocks or we get objects so that they can see and feel things. And then when they're supposed to grow up and not do that anymore,

which I think is a real shame because those physical objects actually have a lot of insight embedded in them. So, for example, if we're showing that 1 plus 3 equals 3 plus 1, one way we can do it is we can take one thing, like a block or a cube,

and then we could take three things, maybe of a different colour, So perhaps we take one red thing and three blue things, and that's one plus three. And then to turn it into three plus one, we can slide them around each other. Or we can just walk around to the other side, in which case we'll see it the other way around. So we will be literally seeing it from a different point of view. So from one side of the table, it's one thing and three things. But if we walk round to the other side of the table, it's three things and one thing, which is a different point of view.

Now that's a very literal different point of view, but other equations are always about changing the point of view in some way, whether it's about changing the order in which we did something or changing the groupings. If you remember anything about multiplying out things in brackets, what we're doing is that we're grouping things in different ways and showing that the two different ways are just different points of view on the same thing. And that can help us. And I think that having different points of view on the same thing. And I think that having different points of view on the same. thing helps us in life as well. And if we practice it in maths, then it becomes something that comes to us very easily in life too. So you're sort of sticking with this for a while, but expanding it out. You talk about local and global viewpoints. So what's the key point there that you're making? That's another question about different points of view where it's about zooming in and zooming out. And if you zoom in really close up to something, then you see a lot of detail, but you don't see very far. But that can be important, especially if it's something to do with the world around you. So this is true in life as well. It's important to know what's going around us

right here so that we don't, you know, walk into traffic and have traffic come and hit us, or so that we can have good interactions with the people that we see in our daily lives. But then zooming out, there's a question of how all those local parts fit together to make up a hole. And in the field of topology in mathematics, that's the field studying shapes of spaces. And what we see is that we can take the same local shapes. So we can take, for example, some very small flat shapes, like tiny pieces of paper. And we can stick them together in infinitely many different ways to make really, really different overall shapes. It's a bit like doing papier-mache, where you tear up pieces of paper and you soak them in glue. Incidentally, I haven't done this for ages.

but it sounds really fun right now. And then you stick them together and you can make all sorts of shapes. You can make a sphere, a ball, or you could make a donut shape with a hole, or you could make a dinosaur, or you could make an octopus. You know, you can use tiny flat shapes

to fit together to all sorts of things. And so then the point is, if you only looked at the shape very zoomed in, you wouldn't be able to tell what it looked like from far away. And so being able to zoom in and out and understand the relationship between those scales is something that we learn in maths that can also help us in life. So what about this notion of the manifold which you talk about? Yeah, that is one of the more advanced pieces of maths that I talk about in this book. And I really, I didn't want to hold back too much on showing difficult things because I think that these are concepts that everyone can.

see something of. It's true that you might not be able to understand it entirely, but I keep saying throughout the book, you can think of this as like gazing at abstract art, and just like you can go into an art gallery and look at art, even if you don't really understand it or you can't produce it yourself. I think we should do more of that with maths as well, and show maths as something to gaze at, and manifolds are something that we can sort of metaphorically and also literally gaze at because they are a way of sticking together little flat shapes to make very different surfaces. So if we do this with two-dimensional things, perhaps you can imagine a little graph with X and Y axes, like a what we call a Cartesian plane, like a normal graph shape.

If we take a whole load of those and stick them together, then what we need to know in order to understand this situation is how the different X and Y axes, line up with each other along the edges. It's a bit like if you're making clothes, you're making them usually out of flat pieces of material, but the pattern tells you how to join those pieces of material up. And if you joined them up in a different way, you would make a very different overall shape of clothing.

And so what I talk about with manifolds is how mathematicians have sat down and decided what different shapes are possible, because some shapes are very similar to each other enough that we count them as the same. So, for example, if you imagine a donut shape with a hole in it, that's technically the surface of it is technically called a torus. And a torus could be very elongated.

It could be very skinny. It could have a really huge hole. It could have a really small hole. But in all those cases in the field of manifolds, we would count them as kind of the same because they're not really interestingly different. And there is a technical way to define that. but in maths we call it this very fancy-sounding thing, homotopy equivalence.

And I acknowledge that sometimes the words in maths can be very off-putting, but I think it's nice to see them anyway. The point is that if you had a donut shape or a bagel with two holes in it, that would be very different from one with one hole in it, in a way that maths makes precise. And so there is a sense in where, the different one-hold shapes can be different because they can be bigger and smaller and elongated. But there is a sense in which they're not really that different.

And the key in maths is to decide which differences really matter and which ones don't really matter for the current purposes. Ambition comes in all shapes and sizes. At First Citizens Bank, we roll with your goals because we're built for what you're building. fit for your ambition for Citizens Bank. Study and play. Come together on a Windows 11 PC. And for a limited time, college students get the best of both worlds.

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Today, in partnership with French acoustic specialist's focal, name audio creates systems that deliver exceptional sound, and unforgettable listening experiences at home. Try it for yourself at a focal powered by name boutique. Visit focal powered by name.com. for more information. So let's move on to something else you talk about, which is logic and the equivalence of logical statements. I think this is really, really interesting. So can you just give us a sort of

Cliff's Notes version of that? Yeah, maths depends on logic. Logic is the entire way that mathematical arguments proceed. And that's different from other fields. Every academic field has its own framework for deciding what counts as true. Maths uses. logic. And logical equivalence is when two logical statements have the same logical content. And so it means that if you know that one of them is true, then the other one must be true and also the other way round, so that it becomes logically equivalent whether you know one or the other. So for example, if you say that a number is divisible by six, that is logically equivalent to knowing that is divisible by two and by three, because if a number is

divisible by six, it's definitely divisible by two and three. And the other way around, if it's divisible by two and three, it's definitely divisible by six. And the thing is that statements can be logically equivalent, but not emotionally equivalent. So with the case of numbers, it's quite easy to tell if a number is divisible by two, because that's about even numbers. And hopefully we all know that you can just look at the last digit of a number and see if that's even, and then you know that the entire number is even. And it's actually not that difficult to tell if a number is divisible by three, either, because there's a little trick where you add the digits together, and if the sum of the digits is divisible by three, then the number is divisible by three. And so maybe that's

easier than testing to see if something is divisible by six. Now, that's a pretty basic mathematical example, and you might think, well, that's not very interesting. And I admit, it. It's not that interesting in itself, but often mathematical examples are there, not because they're interesting for themselves, but because they help us get started with some big thoughts or some big ideas. And often what happens in life, there's two things. There's when someone is having an argument and they replace the argument with something else that actually isn't logically equivalent, but sounds like it might be. And that's often where a straw person argument, comes in, where they replace your argument with something that they're claiming is equivalent,

that it isn't equivalent. And that is one way in which understanding logical equivalents can help us, because it can help us debunk those kinds of arguments. Say, for example, people who did believe in wearing masks during the pandemic, and other people might have said, oh, so you don't believe in personal freedom. Whereas, in fact, wearing a mask is not logically equivalent to not having any personal freedom. Another case where this can come into play in normal life is where someone uses a logically equivalent statement that is genuinely logically equivalent, but is not emotionally equivalent. And so they're doing it to emotionally manipulate somebody into not believing something. So this happens in the US a lot with the healthcare system called Obama

care because some people hate Obama so much that anything with his name on it, they'll just automatically hate regardless of what it actually is, when in fact, if they just called it something very logical, neutral sounding, like the Affordable Care Act, they wouldn't object to it nearly as much. And so being aware of the difference between logical equivalence and emotional impact can help us defend ourselves against being manipulated emotionally, but it can also help us to convince people of things. If we want to try to re-express something in a way that we'll convince them, but we're still using something that's logically equivalent. So that's what's called a sort of false equivalence argument. So, I mean, are there any pitfalls we can look out for to protect ourselves against

falling for these? Well, I really think that maths is a way of strengthening our core intelligence. A lot of people complain at school that the maths they're doing at school is never going to be useful in their daily lives. And indeed, people grow up and go off into life and go, oh, well, all the maths we studied at school was completely useless because I never used triangles in my daily life. And that's true. But the point of studying triangles, for example, at school, isn't because we need triangles in our daily lives. It's to strengthen our brains. Just like when we do muscles to strengthen our core, say if we go to the gym and do core exercises, it's not because there's any particular activity in life that just uses our core muscles,

but it's that having a strong core is good for us. It helps us access the rest of our strength. It helps us not fall over, for example. And having a good strong core of our brain is similar, even if the specific things we studied aren't going to be directly used in our daily lives, it's the training our brain in logical thinking that can really help us sometimes see past the emotional content if we're very upset by what someone is arguing. We can see past that if we're very good at seeing logic. And then we can understand what's really going on in the argument. And then we can have a more sensible and more productive decision.

discussion with people. Absolutely. So one of the sort of other headline things that you discuss is something called category theory. So I think quite a lot of people won't have heard of that. So can you run us through that? Yeah. And I would like to say that quite a lot of mathematicians might not have heard of it either because it's quite a new field. It's a new field of maths that grew up in the middle of the last century. So people who studied maths last century, which sounds really old, but here we are, we're a quarter of the way through this century. But even people who studied maths last century might not have encountered it. Category theory is a way of studying things via their relationships with each other.

And this is quite radical because up until the 1950s, people weren't really doing that. They were really studying the intrinsic characteristics of things. And then mathematicians realized that what really matters often is not the intrinsic characteristics, but the relationships between them. And so this is like studying humans by looking at how they relate to other humans. And it's also actually a bit like how large language models work. Because those models don't really understand what words are or what they mean. What they do is see how words are related to other words.

And if you understand in one language, say in English, that perhaps the word ice is related to water and it's related to freezing. And then you find those same relationship. relationships in another language. And so in order to translate that to say French, you don't need to understand what ice is. You just need to find the place where there's a relationship between words in the same way as the relationship between ice and water and frozen happened. And so the language becomes a big shape instead of being words with meanings. It's a shape of interconnected things with relationships with each other. And then we can map to another language by mapping the

shape rather than the meanings. And this is a really big deal in maths because it means that we can move between different contexts and understand how things fit together without having to go into the details. And when the details get really complicated, this can be really, really powerful. So it's another example of how maths enables us to use our poor little finite brains in a more powerful way to deal with the really complicated world around us. So we've talked about an awful lot there. So other than, of course, buying the book, is there a headline message you'd like to leave our listeners with? I think there are two messages. There's a specific one and their general one. So the specific one is that even in maths, equality and inequality, that is sameness and

difference are kind of subtle. And we make decisions about what we're going to count as the same and what we're going to count as different. And I show how we do that in a lot of different areas of maths. And since even in maths, that's a decision that we make, I think that shows that in life, it is even more a decision that we make because life is way more complicated than maths. And because it's a decision we're making, we should take responsibility for it and show that the decisions we make are things that we've done and that other people might do it differently and so instead of getting angry with them

we should just talk about why we've made those decisions and how we could make them differently. So the more general remark I'd like to make is you might think that you're not going to understand any of this and I'd like to encourage you to think that that's fine because actually mathematicians don't understand it either but we don't let that put us off and so I wanted to show some things that are

quite difficult in this book because I don't feel like we should keep them away from people. I think it's a bit patronising to say, I'm not going to tell you this thing because it's too difficult. You're not going to understand it. Instead, I want to say, you know what, everyone has a right to see these things. I love these things and I would like to show them to everyone to wet your appetite or so that you can be aware of what's going on or so that you have a way in. And starting to find a way in is much more important than what. worrying about whether you're going to make it to the end. Because there is no end, there's always more maths for us to try to understand,

always more things for us to be confused about. And it's just a chance always for us to become more intelligent. Thank you for listening to this episode of Instant Genius, brought to you from the team behind BBC Science Focus. That was Dr. Eugenia Cheng. To discover more about the topics we've just discussed, check out her latest book, Unequal, the maths of when things do and don't add up. If you liked what you just heard, then please do consider subscribing to Instit Genius

on your preferred podcast platform. If you'd like to see our guests and hosts in person, then please also check out our YouTube channel at ScienceFocus. The current issue of BBC's Science Focus magazine is out now. Pick up a copy wherever you buy your favourite magazines or downloaders on your app store of choice. You can also find us on Apple News or online at sciencefocus.com.

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