Is Mathematics Invented or Discovered?

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I have been fascinated by mathematics my entire life, and it is a pleasure to come to you to discuss two of its fundamental aspects. its incredible capacity to describe reality, and then what mathematics really is. So let's start with the first. How accurately does math describe the physical world?

Well, it is extraordinarily precise. In different areas, more precise. In some areas, we know less about it. But I think people often find it puzzling that something abstract like mathematics could really... describe reality as we understand it. I mean, reality, you think of something like a chair or something, something made of solid stuff. And then you say, well, what's our best scientific understanding of what that is? Well, you say it's made of fibers.

cells and so on. And these are made of molecules, and those molecules are made of atoms. Those atoms are made out of nuclei and electrons going around. And then you say, well, what's a nucleus? And you say, well, it's a... protons and neutrons and they're held together by things called gluons and they're neutrons and protons and they do things called quarks and so on. And then you say, well, what is an electron and what's a quark? And at that stage...

The best you can do is to describe some mathematical structure. You say there are things that satisfy the Dirac equation or something like that, which you can't understand what that means without mathematics. I mean, the mathematical description... of reality is where we're always led. And these equations are fantastically accurate. The Dirac equation, which describes the electron or quarks, is a very precise equation. And for example...

there's a calculation which describes the magnetic moment. That is, electrons behave like little magnets. And the magnetic moment, the strength of that magnet, can be described in terms of other parameters. And there's a calculation which gets the accuracy of that. Well, Feynman had a very good description. He said it describes the distance between New York and Los Angeles. to an accuracy of less than the thickness of a human hair so that's pretty precise that's unbelievable

And that's describing the microstructure of atoms and... Yeah, but these are the particles. The electron and the gluons. The electrons. Quarks. This is specifically electrons and quarks, the things which are called... spin-half particles, but don't worry about that. Gluons are slightly different. Now, mathematics can also describe things in the ordinary physical world, the gravitational attraction, electromagnetic attraction.

with the same kind of descriptive accuracy? Well, there's another, yeah, even more in a certain sense, because gravity, according to Einstein's theory, I mean, Newton's theory already, had a precision of something like one part in... 10 to the 7, so that's 10 million. One part in that. And then there were discrepancies seen in the behavior of mercury and so on.

And that's where you start to see differences with Newton's scheme. And then Einstein comes along and produces a theory which is now known to have a precision something like 10 to the power 14. And that precision is a measure of how accurate there's a particular system of two stars going around, special kinds of stars called neutron stars, very dense objects.

And these stars, one of them is what's called a pulsar. It emits pulses of signals, which can be timed extremely precisely. And over a period of, well, I suppose it's... maybe more than 30 years now, I can't remember, they've been observing this thing. And in that period of time, the accuracy over that length of time is known to something like one part in 10 to the 14. Wow. the agreement between Einstein's theory.

and the observations. So it's telling you these are very, very precise theories. So whether we're talking about the structure of very large entities, neutron stars, over great distances in the universe with gravity, or... the structure of an electron. That's right. Mathematics, in both cases, is able to describe it with that kind of incredible precision. Exactly. Yep. And these are small equations. I mean, they're not...

That's right. They're relatively small equations. I mean, they're a little difficult to understand. Einstein's theory is certainly subtle. It's not complicated in the sense that the ideas... Okay, you have to understand about curved space and that sort of thing, which is not an easy thing to get your mind around. But once you get over that... Once you get over that, it's about the simplest thing you could write down. Wow.

in that kind of term. So we have this extraordinary precision between mathematics on the macroscopic level with neutron stars and at the microscopic level with the nature of the electron.

And mathematics is incredibly precise in both cases. So what does that now begin to tell us about what mathematics really is? Yes, well, in a sense, this is telling us that... Our picture of physical reality depends on something in the sense which is more precise, at least in our understanding of it, than how we think about the world. And this precision... really dates back to the ancient Greeks, the time of Pythagoras and later, where they developed the mathematical ideas as a field of study.

stimulated to some degree by physical reality, because the geometry of Euclid, which was very much part of the mathematics that was... being studied then, which we know nano isn't extraordinarily precise. I mean, it is extraordinarily precise, but it's not as precise as Einstein's theory. So one has to go a little bit beyond the...

geometry that they had. I don't think they quite appreciated that they were doing physics because they didn't realize that the geometry of the world could have been anything else. But they developed this mathematical scheme purely as a study on its own. And so mathematics was studied as a pure intellectual activity and without necessarily it being related to the structure of the physical world, although geometry clearly was a big input.

But then the properties of numbers and how you add and multiply and the notions of prime numbers, the fact there are infinitely many prime numbers, that goes back to Euclid and earlier. These things just about numbers were developed very much from the time of the Greeks. And ever since then, mathematics has been the subject which you can study for its own sake. It has its own life, in a sense.

And certainly mathematicians view it this way. It's something out there which seems to have a reality independent of the ordinary kind of reality, like things like chairs and so on, which are what we normally think of as real. Okay, the mathematical reality is something different. It's sometimes referred to as a platonic world, a platonic reality. And sometimes people have a lot of trouble thinking of that as real. I mean, philosophers...

worry about that and so on. What would that mean, a platonic reality? Well, I think it's a different kind of reality from the reality of the physical world. I tend to think of there being different. ways of looking at reality. There's the reality of our mental experience, which, okay, interrelates with the physical reality, but so then does the mathematical reality of this platonic world, which...

gives reality to these notions. So if you like mathematical facts, like there is no largest prime number, it's something independent of ourselves. It's always been true. It doesn't... It didn't somehow become true as soon as somebody saw how to prove it. It's always been true. They're wonderful examples like the... And it would have been true if nobody ever proved it. Exactly, yes. And in a sense, that had to be so.

Because if the physical world depended so precisely on these mathematical laws, I couldn't have known what to do in a certain sense if the mathematics hadn't already been there. It's not us that imposes this on the world. It's out there. Sometimes people think that maybe the reason we have good mathematical laws of physics is that's the best way we can come to understand the world.

Something more than that. It really is out there in the world. Well, that's the argument. Whether mathematics is invented by us, by human beings, trying to impose our way of thinking on the physical world. or whether it is discovered because it's already out there and we're finding it because it's already there. Those are the two polar views. Sometimes people do argue. They say, well, you know, it's just our way of organizing.

what we see about us. But I really don't think that's good enough because Newton, for example, the observations probably had about three figures and three decimal places. And he produced this theory which kept on working. until about seven figures, you see. Then there were discrepancies seen. Einstein produced his theory mostly out of his head, appealing to things that were known to Galileo and so on. But apart from that...

It was not much more empirical evidence. But he produced this theory, which extended far beyond anything that the observations at that time told us about. And they keep on agreeing with the observations. So that theory... which is, if you like, a Platonic absolute thing, a mathematical thing, seems to be... inbuilt into the way the world operates. It's not as though you see a new effect and say, okay, we never think of a better theory to accommodate that one. Sometimes science is like that.

But these really good physical theories are not like that. You're revealing something in the way the world operates, which is there all the time. And I don't think there's any way of understanding that just in terms of our trying to understand. what we see around us. A critical fact really seems to be what you said, that when these mathematical theories were discovered, the accuracy that the observation that they had at the time was...

small compared to the accuracy that those theories then produced. That's right. I mean, Einstein's case, okay, seven figures of decimal were known perhaps in the planetary motions, but there's another seven out there. The precision is over and above that. Ten million. Yes. Ten million, ten millions. It's ten million, ten millions. Yes, that's right. Yes, yes. I mean, that's... Unbelievable. Yes, it's incredible. So push it further. What does that mean? Because mathematics is...

is almost infinite in terms of all the different relationships and expressions and things that we already know. Yes. What does that mean in terms of how much mathematics is sitting out there? Well, that's a good point, because there's an awful lot of mathematics which doesn't seem to have any clear relation to the physical world.

The way I like to picture it is there is this world of mathematics, and only a small part of that, and it's a very fruitful part, it's an extraordinarily fruitful part, has relevance to the physical world. There's an awful lot out there which, as far as we know...

has no relation to physical behavior. Well, of course, people said some of that in the past, and then we've been surprised to find some other things that later have. But still, there's so much math out there, and so much... bizarre I don't know how else to put it structures that it would seem impossible that that could relate to the physical world but what does that mean about if it is out there in some platonic world

What is out there? In other words, all these infinite ideas and structures and possibilities? Yes. Well, sometimes people think of these as mental creations, you see. But it doesn't really explain... And there's this wonderful example of the Mandelbrot set. Extraordinarily complicated. The fractional. Yes, and you can magnify little bits of it and you see all this incredible detail. And that's all there in a very simple...

mathematical idea, and it's encompassed by this very simple piece of mathematics. How does that give your own sense of what mathematics really is? Well, I think there are two aspects to mathematics, at least how I look at it. Some people are just exploring the mathematics, and that's their real interest. And it's the beauty in the subject often. And that's why they're doing it, because they find it exhilarating.

something they find really wonderful to do. But there's the other side of it, which is how it relates to the physical world. And there is this extraordinary precision that we find when you get the mathematics right.

It really mirrors the behavior of the physical world to an unbelievable degree. And so there's these two sides to mathematics. It has this reality which you can study quite independently of its role in physics. And the other side, which is... how it really does seem to reveal how the real world operates, in a certain sense what the world is, as far as we can understand it.

Stephen, everybody knows what mathematics is, but what most people don't realize is that when mathematicians or philosophers begin to talk about mathematics, they do it with a sense of awe. And the wonder is whether mathematics is something that is so fundamental to reality that it has an eternal, necessary existence. as some people say, in some platonic heaven.

And it's always there. Or is it something that is a regularity that humans have sort of invented that sort of describes the world and works very well but has no deeper reality? How do we begin to take those two dramatically different positions and evaluate them? I think I've been interested for a long time in questions about what is the essence of mathematics.

I make my living building this thing called Mathematica which attempts to cover, in the broadest possible sense, the kinds of things that mathematics might encompass. But so a question that I've been interested in also from the point of view of basic science is, is the mathematics that we sort of practice today the only possible mathematics, or is it a mathematics that is sort of a great artifact of our civilization?

but sort of a historical accident artifact. The conclusion that I've sort of resoundingly come to is that the mathematics that we have today is in fact really a historical artifact. Now, that's not historically... in the tradition of mathematics itself that's not what people have tended to conclude they've tended to think that mathematics is sort of the most general possible formal abstract system

If you look at the history of mathematics, that's certainly not how it originally started out. I mean, in ancient Babylon, you know, there was arithmetic for commerce and other things, and there was geometry for land surveying. And what I think... has really been the history of mathematics as the progressive generalization of arithmetic and geometry, plus one key methodological idea. And the key methodological idea is the idea that one can make theorems and abstract proofs of those theorems.

And I think that methodology is kind of what's driven the development of mathematics. Mathematics has been the set of things about which one can reasonably make theorems. Now, inevitably... that has sort of limited the kinds of generalizations that are possible in mathematics because, in a sense, when one generalizes from the integers to the real numbers to the complex numbers to the quaternions, et cetera, et cetera, et cetera, one is doing that typically.

at every step, saying, let's figure out some theorem that we really like, and let's then find the most general kind of thing that satisfies that type of theorem. So in a sense, one is always keeping very close to this thing where certain theorems will be true and will be provable. Now, one of the things that comes about, one can ask the question, if one just sort of arbitrarily looks at formal systems,

will they tend to have the character of mathematics as we know it today? Will they tend to have the feature that most of the things one asks about, one can successfully prove theorems about? I think in both cases, the answer is no, not really. So, for example, one thing one can do is to kind of ultimately deconstruct mathematics. If one looks at, you know, there are maybe three million papers that have been published about mathematics, okay? And these are all based on a certain set of axioms.

The axioms are what you grow mathematics from. The axioms are quite simple. They're a couple of pages. You can give all of the axioms that are commonly in use in mathematics. And so the question is then, Is that, are those the only possible axioms, or can one sort of look at other possible axioms? So one thing one can do, each of these axioms is represented by some sequence of symbols you can write down on the page. You can just enumerate.

all possible such sequences of symbols. And one can ask, what's true about these other kinds of axioms? These are sort of a universe of possible mathematics. Our particular mathematics is the particular set of axioms you can write down on these couple of pages, but there's a whole universe of possible mathematics is out there. What are they like?

First question might be, where does our particular mathematics lie in this universe of possible mathematics? Is it possible mathematics number one? Is it possible mathematics number 10? Is it possible mathematics number a quintillion? Where does it lie? I was curious about this question for logic, for example. We always think of logic as being this very absolute thing, but in fact, it's just a particular axiom system.

that lives in the space of all possible axiom systems, space of all possible formal systems. And so I ask this question, where is it in that space? The answer is... Depends on exactly how you enumerate the space. But roughly, it's about the 50,000th possible axiom system. So right there, in the universe of possible axiom systems, the universe of possible mathematics, there's logic. But wouldn't it happen that...

all of the others are self-contradictory in some way? No. No, many of the others, all of the others are perfectly valid axiom systems on which you could start building a mathematics. Now, some of them will have features that some of them are less rich than our mathematics. logic itself is much less rich than set theory or the formal theories of arithmetic. But logic has had quite a good run for its little axiom system.

So these other possible mathematics, as you look at them, it's an interesting question. Can you recognize which one could be? Is there something special about our particular mathematics? I don't think so. I think if the aliens delivered a different possible mathematics...

I don't think we would be able to immediately say that's not a reasonable, valid mathematics. Because it would be self-consistent, even though it would be radically different. Yes. So we can talk also about things like Gödel's theorem. And we can ask the question, see, one of the things that happens in these axiom systems is we can ask, from an axiom system, can we figure out what's true based on that axiom system?

So, in a sense, the most fundamental fact about mathematics is that it's hard. It might be the case that given an axiom system, we could immediately deduce what all the consequences of that axiom system would be. But there's this sort of very fundamental fact.

that I call computational irreducibility, that kind of makes it the case that from a particular axiom system or from a particular set of rules, there's a certain irreducible difficulty in working out what the consequences of those rules should be. Same thing with axiom systems. That's sort of the most fundamental fact about mathematics, is there's sort of an irreducible difficulty in figuring out from the axiom systems what are the true facts of that area of mathematics.

This is something and so the... This has the consequence that there are facts. Given an axiom system, there will be things that are undecidable from that axiom system that are, in a sense, infinitely far away from the axiom system, infinitely difficult to determine. There's a certain... raft of technicalities associated with this, okay? But one of the fundamental questions is, when we see that there are, if we look at the history of mathematics, mathematics has had all these unsolved problems.

Things like Fermat's last theorem, the Riemann hypothesis, things like this. There's a question. Are these things not solved because we just haven't developed the mathematical technology to be able to solve them, or are they fundamentally unsolvable?

Now, what's happened in the history of mathematics is that most things people have been interested in have ended up being eventually solvable, although sometimes with effort and centuries of work and so on. But one of the things that I suspect is that that's actually... not really the way that is the true reality to mathematics. That really, if we were to just sort of ask mathematical questions arbitrarily, that the vast majority of them would end up turning out to be unsolvable.

In fact, that unsolvability is actually very close at hand in mathematics. We just don't see it because the particular way that mathematics has progressed historically has tended to avoid it.

but mathematics is a good model of the natural world, and mathematics has been sort of driven by modeling the natural world. I think there's kind of a circular argument, because what's happened is that those things which have been successfully addressed in science and studying the natural world are just those things.

that methods like mathematics have successfully allowed us to address. And so this idea that sort of mathematics has been wonderfully successful in modeling the natural world, and so that's all we need to think about in mathematics is what we need to model the natural world.

It's been constrained by the fact that these are the particular methods we've used, they've worked in particular ways, and so on. I think that, in fact, a lot of the things that we see in the natural world that we haven't been successfully able to model with mathematics... They're really the things that are associated with those parts of mathematics that we've never reached historically.

Partly they're the other mathematicss that haven't been the ones that were developed historically. Partly they're questions, even in our existing mathematics, that we haven't chosen to ask because they're not things that have arisen historically. So I think one of the exciting things that one realizes... is that human mathematics, it's one of the great artifacts of our civilization. It's one of the perfect, wonderful things that's been produced by a huge amount of human effort.

But it's an artifact. But it's an artifact. And there is much more out there in the sort of space of all possible mathematics. And I think in the future, we will see an increasing kind of realization and an increasing ability to explore all that other universe of mathematics is. And it will be profoundly important for not only for... mathematics but for our science and for our technology.

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I think I'll wait inside. Max, the question of mathematics, what is it really? Is it something that humans impose on the physical world, much like... the system of taxonomy and biology, which seems to work very well as we classify animals and plants in different kingdoms and all that. Or is it something that is out there, it's always existed, and we're somehow let into little pieces at a time that we can access?

How do these two ideas work together and what's the significance of whatever answer you have? It's very important to not conflate the language of mathematics, which we do invent, with the structures of mathematics. which we discover. For example, when Plato and his contemporaries started getting really interested in how many regular three-dimensional shapes there were with flat sides, they discovered that there were exactly five of them.

the tetrahedron, the cube, the octahedron, the dodecahedron, and the icoshedron. They invented the name. They were free to invent the name dodecahedron, for example, for one of them. They could have called it the schmodecahedron or the zoodecahedron if they wanted, but they were not free to invent the sixth one. There is no sixth one. It just doesn't exist. And it's exactly the same way in physics.

when we discover things out there and then invent names for them. My dad asked me once when I was a little kid, he said, Max, how can we know that the planet Jupiter is actually called Jupiter when no one has ever gone there and checked? I went back and thought. I brought this for a while and then I came back really excited. Daddy, Daddy, I figured it out. We humans invented the name. So what I'm saying in my book is that the things that we have discovered in our external physical reality.

correspond to things that we discover in mathematics, structures like the dodecahedron, except, of course, much more complicated ones. And this is a... sentiment that most of my mathematicians friends also share. David Vogan at MIT, for example, has a beautiful poster he's put up on the wall of his office of E8.

this mathematical structure he spent a decade of his life studying and he would be really pissed at me if i insinuated that he just made this up he feels he's discovering it and of course if you take two people who both moved to New York and lived there for a decade, they're not going to discover exactly the same streets, but they're both going to discover Central Park and Times Square and the main through-fairs.

And it's the same way if you take our Earth-based civilization and some alien civilization, eventually we're all going to discover the basic things like the integers and plus and multiplications and the platonic solids and so on. and then as we gradually map out the map of New York City or the map of the platonic math landscape where we do what kind of obscure faraway places we find will depend on our cultural interests but we are still discovering

rather than inventing. So if we're discovering that implicitly means that there's something there, and you define it platonically, something in a platonic existence is there and we're uncovering, how much stuff is in that? platonic existence. This platonic reality of all mathematical structures is vast. Is it vast or is it infinite? There are infinitely many different mathematical structures. What's so nice is that it's still...

Not some kind of vague anything goes, that anything I can think about exists. It's very hard to prove that a mathematical structure is actually self-consistent. The famous mathematician David Hilbert said that mathematical... existence really is freedom from contradiction. And mathematicians work very hard and probably publish papers sometimes just to prove that something actually exists mathematically and is consistent.

Imagine in the future writing a program for a super advanced computer to generate an atlas which just has all the mathematical structures in there organized by increasing complexity where on page one you'll have some really simple stuff.

like the empty set and then you get eventually to the cube and the dodecahedron and then eventually you get to three plus one dimensional pseudo-Ramonian manifolds and Hilbert spaces and the kind of stuff which we physicists work on now. But it's so much more vast than that. I mean, this is just a...

tiny, minuscule fraction of all the possibilities. I mean, just the number theory alone, how many prime numbers, if that itself is infinite, that's all out there in your platonic space. Even though there are infinitely many... Counting numbers, 1, 2, 3, 4, 5, 6, they together form a single mathematical structure, though, that mathematicians like to call the integers. But there are indeed also vast numbers of them.

This raises a fascinating question. It seems like nature prefers simplicity because it seems like the mathematical structure that we physicists seem to find ourselves in here is actually much simpler. than a lot of things that you could cook up. We live in a structure with enormous mats of symmetry, which strikes us as beautiful and elegant. And this is a deep mystery.

which I feel we still don't fully understand. Why is this exactly? So in your platonic existence of mathematics, what exists there? Is it every... every counting number, every prime number, or is it the concept of, you know, take n and add 1 and do that dot, dot, dot. When it's time to scale your business, it's time for Shopify. Get everything you need to grow the way you want. Like all the way. Stack more sales with the best converting checkout on the planet.

Track your cha-chings from every channel right in one spot and turn real-time reporting into big-time opportunities. Take your business to a whole new level. Switch to Shopify. Start your free trial today. n plus 1. What actually exists there? The mathematical structures that exist in this platonic reality that I call our level 4 multiverse.

consists of abstract elements with relations between them. And this includes, for example... So not all the specific generations. That's right. The number five itself is not the mathematical structure. The name doesn't mean anything. The name doesn't mean anything, but the concept of five things. Some popular examples of mathematical structures are different kinds of numbers. The integers are a mathematical structure, 1, 2, 3, 4, 5, etc. The real numbers, the complex numbers.

Then there are a lot of mathematical structures of mathematicians called spaces of different kinds. We have Euclidean space. two dimensions with three dimensions, four dimensions, and so on. We have Minkowski space, which is Einstein's famous space-time. We have curved spaces. known as pseudo-Rimonian manifolds, which is the space that Einstein said we live in. So there's a vast...

variety of different kinds of structures. But I'm trying to understand, within each one of these structures, is it just the description of the structure and an algorithm for producing elements within that structure, or do each elements exist? in this platonics or just take the counting numbers so what i'm positing is that we but our physical reality is one particular mathematical structure of this sort so for example

Euclidean space is one mathematical structure which has within it infinitely many points, which would correspond to all the different physical points. So you would say this point here corresponds to a mathematical point in this Euclidean space, and this point there corresponds to that other point.

And this length here corresponds to blah. And we know, of course, that this is not really Euclidean space because we realize that space is curved and so on. But there is another kind of mathematical structure. known as a pseudo-Riemannian manifold, which can take care of that. And more broadly, what I'm saying is that for every single physical entity that we think of as something we can touch or measure with a detector, there's a...

there is a corresponding mathematical entity there in a mathematical structure. So, for example, if you take a thermometer, you can measure a little number at each point in the air here, which we call the temperature. and you can measure a pressure with a barometer, etc. And when we make weather forecasts, what we do is we divide space in all these three-dimensional pixels we call voxels, put this in a really big computer and try to calculate whether it's going to rain tomorrow.

These numbers are not fundamental, but the magnetic field, for example, that you can measure by holding a compass there and check which direction the needle lines up to, etc. which is also described by a bunch of numbers throughout space is as far as we can tell something very fundamental similarly the electric field and we have all these other fields which tells us about quarks and electrons and so on and it's very much like a weather forecast again

At each point in space, we think of there as being all these numbers there. And by putting this into the computer, we've successfully managed to, as physicists, calculate all sorts of properties of protons and atoms and... and most other things that we care about. And this is again an example of how everything here can be described by mathematics and therefore correspond to a mathematical structure. A mathematical structure

I think is most easily thought of as something which has no properties at all except mathematical properties. If you specify all these numbers saying how strong is the magnetic field here and there and there and you specify everything there is to say about the world.

David, what accounts for the remarkable capacity of mathematics, abstract symbols, to reflect the real world? I mean, I've noted that, and it's beautiful indeed. I think it's totally unsurprising. I would be astonished if mathematics were not to describe, be very useful in describing the real world. Because after all, what is mathematics? is just a highly developed form of language that we use as human beings.

to describe regularities. And I believe that mathematicians discovering mathematics are not inventing. some new abstract structure, they are discovering some part of reality, of the world, the physical world. So I'm a Platonist in that sense. Mathematics is something that exists, and mathematicians discover it. And they, after all, are human beings who have evolved language. Mathematics is a higher form of language by natural selection. In order to be successful and survive in the real world,

obvious that the tools they would develop would be those that are appropriate for describing the real world. Mathematics, though, can exist even if there was no physical world. plus 2 would always equal 4, even if there had never been anything physical. Maybe. So to some extent, I suppose I must... Admit that I believe that reality includes mathematics in addition to... But, you know, mathematics is not...

I truly believe is not an independent creation of... There are different opinions. Some people... I've asked mathematicians, but, you know... What would happen if we made contact with an extraterrestrial civilization, the other side of the galaxy? And we asked them about their mathematics. Would they have the same mathematics as us?

Most mathematicians believe that they would have the same mathematics. They wouldn't have a whole new kind, a different kind of mathematics. I even believe, although not everyone agrees with that, that By and large, the historical development of their mathematics would have been similar. But just like the laws of physics throughout the universe are the same, I believe the laws of mathematics...

throughout the universe or different civilizations throughout the universe would be the same. But mathematics is something that is in the structure of reality. So I don't find it, therefore, too surprising that it works, nor too surprising that we discovered it, often in league with physics and other sciences, by the way, because we... are trying to understand the reality that we observe around us and the tools we develop, such as language and mathematics.

were developed in order to succeed at that task. So it's not surprising why it works so well and even why we find it beautiful.

George, I've been speaking to mathematicians, and there's really great debate whether mathematics is invented or discovered. And it's more than a philosophical debate, if you will, because it really speaks to... What is reality? Is there some kind of other reality beyond that which we see in the physical world? From the standpoint of the embodied mind which you have elucidated, how does mathematics fit? Mathematics...

is not either invented or discovered. The dichotomy is false. It's like the false dichotomy between being a realist and a relativist. If the mind is embodied... You're connected to the world. You're interacting with it. You don't have a direct understanding of it, but you evolved to fit the world well, to function well in it. And the same is true with mathematics.

We have in our brains the ability to be able to tell one from two, from three, from four. And there's a part of our brain that does that, very simply. We are able... as infants to be able to do what's called baby arithmetic. You have a child that's a few days old. It's sitting there happily sucking on a bottle. You have somebody checking where its eye movements goes. You take a stage. It goes up. There's a puppet on there.

Then the stage goes down, the mother puts another puppet on, it goes up, there are two puppets, the baby's happy. If it goes up and there are three puppets, the baby sucks like mad and stares. The baby... from birth, can tell one and one is two. Very simply. There's stuff in the mind for very, very simple arithmetic, not in terms of numbers or calculation or things like that, but there's basic arithmetic there.

Now, that capacity gets extended via the rest of the mind and brain, and it gets extended very largely via metaphor. For example, When a child is very young, there's a stage in which children are playing with containers. They're putting things in and taking them out and getting into stuff and, you know, constantly putting things, finding places to hide stuff and putting them in and taking them out.

a good month or two when babies are just doing that. And what they're doing as they do that, they can tell how many things are going in and how many things are coming out. They're constructing a metaphor for arithmetic, unconsciously, automatically, their brains are doing this, in which numbers have to do with collections of objects. and that the size of a collection is the size of a number. That's one of our major metaphors for arithmetic.

There's another. When they take steps or they move, there's two steps, three steps, etc. They are what are called subitizing. That is, they are being able to tell one and one is two and so on very simply. And then they have a notion of... as being steps in space, where the place you start is your zero and you continue.

There are four, I won't go through all of them, basic metaphors for arithmetic that work like that, and they all have pretty much the same entailments. Metaphors are things you reason in terms with. And you have the same entailments for basic arithmetic. But then there are other things in arithmetic that don't come out of that. Like, have you ever wondered why minus 1 times minus 1 is plus 1?

Well, there's a cognitive reason for this, and it does not come out of those metaphors. What does come out of the metaphor of being able to see numbers as points on a line in space is that you can go backwards the other way, and that gives you negative numbers. Now, when you think about it and you visualize that,

You can do metal rotations. We know from Roger Shepard's experiments we do metal rotations. And you can rotate 180 degrees and get the positive numbers going times the negative numbers so that... If you do that, rotation by 180 degrees is multiplication by minus 1. You do it twice, and you get back to plus 1. Mental rotation added to those metaphors gives you minus 1 times minus 1 is plus 1.

And then that ultimately leads to the imaginary numbers, because when you do i times the square root of minus 1 times the square root of minus 1, it's 90 degrees and 90 degrees. And that's why in imaginary arithmetic... When you multiply by i, the square root of minus 1, it's a rotation of 90 degrees. It's very simple stuff coming out of the nature of the brain and the mind and the way the mind works. This is something that's built into our brains.

and it's a natural capacity. But it's also, once you set up those metaphors, they have entailments. Metaphors can be entirely precise, and mathematics must be precise. It can be regular, and we think regularly in terms of them. And you can define exactly what's going on in terms of a system that is grounded in the body, where the concepts are clear.

from the way they're grounded and where the metaphors give entailments. And it turns out that once you ground them, you can then discover what the entailments are. So what mathematicians do, once they set up a branch of mathematics, they are discovering that. Now, metaphors are crucial in setting up branches of mathematics. Think, for example, of the number line.

In that, there you have a metaphor. Numbers are points on a line. They don't have to be points on a line, they could just be numbers. In set theory, numbers are sets of a certain kind. Zero is the empty set, one is the set containing the empty set, and so on. That's a metaphor.

It's not out there in the world. But once you do that, you can discover the properties of that metaphor and you get a whole branch of mathematics. And this is true in general for all branches of mathematics. Take the issue of infinity. A crucial idea, how do you get the infinite set of numbers, just of integers, for example? It goes on and on and on. Well, we have two different concepts of infinity.

There's one that doesn't end. No end. That's infinity. It goes on and on and on. But that's not the one that's in traditional mathematics, though there is an unconventional form of mathematics that uses that. In traditional mathematics, there is a complete set of all the numbers even though they don't end. How is that possible? The answer is metaphor. And what's interesting about it is that Rafael Nunez and I discovered that there is one metaphor for what we call actual infinity.

in all branches of mathematics. And it applies differently in different branches, but it gives the answer, for example, to why you can have infinite numbers. It gives answers to, for example, how calculus works. And it's the same metaphor for infinity that's involved in understanding calculus. Then you can say, well, how do you do this? How is it possible to understand this? And the answer is very interesting, and it comes out of cognitive science and linguistics.

We have discovered in linguistics and in study of neural computation that the idea of aspect in linguistics, which is the structure of events, has to do with a certain kind of neural structure that allows you to sequence events as wholes and to have a starting position, a central part, an ending position, and so on.

In every language, you distinguish between concepts that have an end point, like I walked a mile or I jumped when you finish when you come down, versus I breathed, where you keep going on and on. One is completed, the other is incompleted. Now, what happens with infinity that goes on and on and on is it's incomplete.

Actual infinity metaphorically understands that in terms of completed aspect, like an action that has an end point where it goes on and on and on, but it imposes the end point metaphorically. And when you formulate that precisely, and one thing that we do in the theory of metaphor is give precise, mathematically precise, formulations of these metaphors, you can show that what actual infinity is.

And then you can show it's something that can apply not only in arithmetic, but it can apply in geometry. That is, for example, what is a real number? What is pi? Well, you go from 3.1, 3.14, 3.145, etc., until you get the infinite expansion of pi. Well, that's the same metaphor going to infinity. The same thing happens in other branches of mathematics, as we show in our book, that mathematics is metaphorical, but

When you set up a new branch of mathematics, what you're doing is putting together metaphors from other branches of mathematics. Let's say I grant everything that you say. There still was a time. That is 99.999, et cetera, of universal history where there were no minds, where there were no... brains around, that the physical world worked, and that physical world seems to be described by mathematics. So one is still confronted with the question that even...

As we apprehend mathematics through metaphor, assuming I grant everything you say, there is still a mathematics in the world, even if human beings never existed or if brains never existed. It's not in the world. The world is as it is. Let's take a very simple case. Take a spiral nebula, and you can show that there are logarithmic spirals in nature. That's the way they put it, as if the spiral...

the mathematical formula were out there in nature. But it turns out when you understand what's going on in such a nebula, it's pretty straightforward. What you have is a constant rotation.

And you have an explosion that goes outward. And when you put those two together, when you take the metaphors for understanding rotations and the cognitive mechanisms for that, and the cognitive mechanisms for understanding it, going outward and you take the mathematics of those two it's a logarithmic spiral you can show exactly why that is the logarithmic spiral is not in the nebula it's in your understanding of the nebula

The marvelous thing about mathematics is that we can create mathematics with our brains that fit phenomena in the world remarkably. It is not a miracle. that that's the case because we have the capacity to see and understand the world, to categorize it in terms of what our brains do, and then we can create a mathematics out of that.

in a systematic way using what our brains allow us. It is not out there in the world. The flowers may fit certain kinds of series, but the series are not in the flowers. To watch complete conversations with over 100 of the world's leading thinkers on cosmos, consciousness and meaning, Most people would rather assemble a 300-piece cabinet than search for insurance. That's why The Zebra searches for you, comparing over 100 insurance companies to find savings no one else can.