Is Mathematics Invented or Discovered?

David: Do you think mathematics is something that we discover or just something that we're making up?

RP: Well I certainly belong to the school of thought that it's discovery, but you have to be a little bit careful about this. Mathematical ideas, I think, are things which are in a sense out there, even though ideas are things in our heads, in the sense they're our own thoughts.

But suppose you're trying to prove some mathematical result… There are lots of ways at arriving at the result, and there may be one which appeals to you better than the others, or it may be the one you just first thought of. Now, there's an element of what you might call invention there.

David: But the destination you're getting to already exists?

RP: That's the way I would view it, yes.

David: Right.

RP: And also there's… you see there is a degree. You might say, how can something have a degree of existence or not? Well I think it's true: in a sense you can do this. You could have certain ideas which in mathematics have a deeper existence than others, and you can maybe invent various mathematical schemes, and I use the word ‘invention’ there because they might be interesting to play with.

But then some of these have features which reveal deep truths which you had no conception of before. I think the best example I can think of is the idea of complex numbers. Now, you see, these come about from the crazy idea that the square root of minus one, it exists, if you like.

If you have a negative number and you square it, you get a positive number. If you have a positive number and you square it, you get a positive number. So how can you get a negative number when you square it? Well you have to invent something. So if you want to know what the square root of minus one is, we call it i. Well that looks like a pure invention.

David: Yes, it does.

RP: Now, you see, the thing is, what's deep about it is that if you introduce this notion ‒ which at first sight looks like an invention ‒ it unfolds an entire world that you had no conception of, and you didn't invent that: that came as a gift.

Now, you see, there's an additional piece of mystery or magic here too. Not just does it open a world of mathematics, and give you insights in all sorts of things in mathematics that you hadn't got before, it’s mathematical trickery, if you like, or mathematical magic, and you can do wonderful things. Then along comes quantum mechanics, and this quantum mechanics turns out to be fundamentally based on these complex numbers. And if you didn't have this idea, the mathematical idea of imaginary numbers, you couldn't do quantum mechanics. Quantum mechanics tells us that the world is based on these strange numbers: it's not just the measurements of ordinary distances in one direction.

David: So in some sense they were there woven into the fabric of quantum mechanics, before we even thought of them?

RP: Exactly. They were there all the time. They've been there since the beginning of time. These numbers are embedded in the way the world works at the smallest and, if you like, most basic level.

David: So they're woven into the fabric?

RP: They're very much essentially part of the fabric. The fabric couldn’t exist without them.

David: Do all mathematicians think this way?

Ard: I don't know if all of them do, but if you're a physicist, certainly the way that something like imaginary numbers or complex numbers opens up your understanding of the physical world is really amazing. So this seemed like a very abstract kind of funny little game, and suddenly you apply it and it not only opens up new understanding, but understanding that is so counterintuitive to your day-to-day common sense, there's no way we would've ever come up with it had it not been for the mathematics that guided us. And that kind of experience… it's a very powerful experience, isn't it? It's something really amazing.

David: Roger, what do you say? Because I talked to Marcelo Gleiser and he said, ‘Well, I just don't buy this mathematics is woven in.’ He said you could have a different kind of creature which would develop a completely different kind of mathematics. What do you say to that? I mean, are there certain kinds of mathematics which, no matter what kind of creature you are, you'd have to have prime numbers or...?

RP: Yes, well, it's the question that you can develop all sorts of systems in mathematics. Some of them don't get you very far, some do. Some which do get you very far don't seem to have any connection with the physical world. There is a lot of very sophisticated mathematics which as yet (now I'm not saying that it never will have any connection to the physical world, they might). but, as yet, there is no evident connection with the physical world. And it's only certain, very limited areas of mathematics, important ones, and complex numbers is one of them, linear algebra is another one, group theory is another one… there are certain areas which are important to mathematics, but it's not all of mathematics by any means.

Number theory? Does that have any relevance to the physical world? Maybe, we don't know. There's no particular importance, maybe a few odd things, but nothing of really great importance which seems to have developed in the physical world – that may be a temporary thing.

Even things like infinities, Cantor’s incredible theory of infinities where you have a whole algebra of infinities: big ones, little ones. What's the role of that in physics? Zilch, so far. I mean they may have a role to play, but at the moment this huge area of mathematics doesn't seem to have any real role to play in physics. You have to make a distinction between saying, okay, some mathematics seems to be embedded in the way the physical world works, [but] other mathematics, we have no reason to believe it.

THE JOY OF mathematical DISCOVERY

David: For you, is mathematics something discovered or invented?

GC: You know, there is mathematics that, as Ulam puts it, fills in much-needed gaps. There are pieces of mathematics and when you find them, they seem sort of inevitable afterwards – they weren’t in advance – and when you find a thing like that, then it seems more real, then it seems that you’re discovering it. But let’s face it, from a practical point of view we’re inventing it as we go.

But it’s true that on good days, when you’ve found something that you really love, you say, ‘Oh, my God.’ You have a feeling of inevitability that you’re discovering something, that there’s this beautiful thing out there that it really doesn’t feel like any mortal could have invented it. It seems to be something from the Platonic universe of beautiful ideas or from God’s mind. Who knows?

So that’s if you’re very lucky. The whole thing seems so beautiful and so natural and so fundamental that you say, ‘How come I didn’t see it before? How come nobody saw it before?’

That’s a very wonderful feeling, I have to say. But it took ten years to get there because the mathematics was clumsy and awkward, as new mathematics always is. Then people polish it for 300 years... But discovering mathematics is messy, like making love; it’s messy.

David: But the outcome…

GC: The outcome can be…

David: Extremely pleasurable!

GC: Yeah, and it can also be a baby, another human being, who can be a great artist or something. But there are wonderful moments when, yes, you have this feeling, and then moments like that you say, ‘Well, it wasn’t me. I didn’t discover this. This idea exists out there independently of me and maybe it wanted to use me to express itself.’

But maybe it’s a way of fooling ourselves. But this is the kind of thing that helps you to do good mathematics. You have to be inspired. It’s profoundly emotional. I mean, the best mathematics is an art. It’s totally creative. You have to throw your whole personality at it. And it may be that you discover something because you’re crazy. You were the right crazy person to come up with this crazy idea, but other crazy people don’t find anything because their craziness isn’t in sync with the next discovery that had to be made.

David: I was fascinated when you said, ‘Sometimes you feel like the idea just needed you.’

GC: It needed people to express it. It wanted to incarnate, so to speak.

David: Well, I sometimes think that ideas are like a seed.

GC: Absolutely.

David: And that the mind is like a garden. So when people say, ‘I did this,’ I always think to myself that it would be like a little lump of dirt saying to you, ‘Look at the flower I made.’

GC: I absolutely agree with you, David. I remember Benoit Mandelbrot was in a documentary as he was dying, and you could barely hear his voice – they had to put subtitles – and he was saying, ‘I discovered a beautiful world.’

It feels like it’s out there, and you feel very lucky to have stumbled on it. But I don’t think you can take the credit. It’s like climbing mountains: you climb a mountain to get a better view, to see further, and one always feels that there are other mountain ranges that are higher. In the distance you can see still higher mountains, and the ranges never stop and it always gets higher. The further you see, the more you realise that there are.

So, for example, I’ve worked on Omega, but what is consciousness? What is the mind? How does the brain work? Can you prove that Darwinian evolution works? Where do new species come from? I mean, there’s endless questions, and each question just opens more questions.

IS THERE A MATHEMATICS OF LIFE

Ard: There’s sometimes, I think, a stereotype of scientists trying to unweave the rainbow and trying to, you know… if we once we’ve understood something, then we can reduce it to its components, and then…

GC: That destroys it. That’s reductionist.

Ard: That’s reductionist.

GC: Yeah, we’re destroying the beauty of everything. We’re destroying the sacred and making it secular and prosaic by understanding it. I don’t feel that way. I feel that it’s absolutely inspiring that puny human beings are able to have such beautiful theories.

Ard: Four forces…

GC: Yeah, there’s general relativity, there’s quantum mechanics, there’s electromagnetism and there’s Newtonian physics. And these are… are tremendous landmarks. They’re like crystals. They’re places where you can see that the universe actually… that mathematics is relevant.

Mathematics is not relevant to everything. For example, I don’t think it’s relevant to biology, and in Proving Darwin I tried to think, what is possible? What kind of mathematical theories might be possible as fundamental theories for biology? So what I see biology is all about is tremendous creativity: where new ideas come from, biological ideas. And I see that connected with mathematical creativity.

I don’t believe that there is a direct mathematical biology, like there is… mathematical physics is deeply mathematical. I don’t think that biology will ever be that way. So I propose creating a theoretical biology at one remove from real biology. It’s a toy model working with a model of biology which I call meta-biology. And that’s the idea. Instead of studying what happens when you… the evolution of making random changes in DNA software, DNA programs. What happens if you make random mutations in computer software and subject it to selection? So this is much simpler than real biology and has some mathematical ideas and is mathematically tractable. I don't know how far this idea will go, but that’s the kind of thing which I think might be possible in biology.

Understanding it is a patchwork quilt. No one idea is enough to understand biology. Population genetics is splendid, but creativity has sort of gotten lost in all of this. And I wanted to bring the focus back to that, which for me is the central issue.

Ard: There’s a very famous quote by a Dutchman, Hugo de Vries, who said, ‘Natural selection explains the survival of the fittest, but doesn’t explain the arrival of the fittest.’

David: Yes, that’s nice.

Ard: Where do new species come from? Where does biological creativity come from? And that’s in the variation. And that’s what your book is trying to do.

GC: That’s what I’m trying to do.

Ard: You’re trying to say, where does that creativity come from in the first place?

GC: I’m trying to propose a possible framework where this question can at least be asked, because in conventional population genetics, it can’t even be formulated.

David: Which leaves out the very engine of natural selection.

GC: Well, it depends if you’re interested in micro-evolution, which is sort of small variations, or if you’re interested in taking a broad perspective on the sweep of things and the major transitions in evolution, which is like when you go from single-cellular to multi-cellular, when you go from plants to animals, where the brain comes from. Those are major transitions in evolution, and those are much more mysterious.

There’s this tremendous outburst of creativity that takes place in a relatively short period where you find nature seems to be trying out every possible body plan. It’s just fabulous, and it just seems to be too fast, perhaps.

But if you think of genomes, entire genomes as software, a small change in a program can make an enormous change in what it does, in its output. So from that point of view, I think abrupt change doesn’t seem quite so mysterious.

Ard: People often think about mathematics, and even science, as just being a closed machine. And so what science is telling us is that we’re just machines, and if we’re just machines then we’re predictable.

GC: I hate… I detest that point of view.

Ard: You detest that point of view because you think it’s wrong?

GC: Well, look at Euler. Look at Ramanujan. Look at… Let’s take my favourite mathematician, Cantor: his theory of infinite sets which is really mathematical theology.

GC: So the stuff I’m working on is like a lower-level version of this. It’s a little more down to earth than Cantor’s infinities, and greater and greater infinities. But the whole question – the whole issue in both cases – is creativity, an open system.

David: You contrast one view of mathematics with another. And we’ve talked about one kind of science: the let’s just take it all apart and reduce it and claim that we’re machines. Is there something at stake, do you think, for you? For us?

GC: Well, there’s definitely something at stake. I want to save the human soul. One has to save the human race. If we think of ourselves… maybe we are machines. What do I know? But I think the idea of thinking that we’re just machines is a destructive idea. Even if we are machines, I think it’s better to think we’re not machines. I’m not saying that we shouldn’t do science to understand as much as possible how the brain works, how the body works, but if the human brain is a machine, it’s quite an amazing machine. It’s a machine on a level so different from our current machines.

David: So if we imagine that we are a machine, we need to be very careful about what kind of machine we imagine ourselves to be?

GC: This is a radically different kind of machine. And it’s a concept of a machine that will end up being totally un-machine-like. So in the end the machine and the sacred may meet.

RANDOMNESS AND CREATIVITY

David: In the book you said, ‘Darwin replaced God with randomness…’

GC: That’s right.

David: ‘…but randomness is lovely.’ Would you tell us that, because that…

GC: Well, that was one of the reasons that people initially rejected Darwin’s theory. One of the things they didn’t like is randomness. The idea that we’re sort of a random product – that there’s no purpose – makes everything meaningless. Now, randomness and atheism have become the new religion.

Ard: Do you think randomness is…? Do you think they’ve misinterpreted randomness?

GC: Yeah. Randomness does not mean everything is meaningless. Randomness is, sort of… You’re looking at creativity in its primordial state.

You see one of the characteristics of randomness is unpredictability. Now, something is unpredictable if you couldn’t predict it in advance: that’s creativity. So, in other words, randomness and creativity are practically different names for the same thing. Something that isn’t random is something you can predict, which means that it’s not creative. You’re sticking within your current system of concepts.

David: In my mind, some randomness is just, ‘Well, we can’t figure it out now,’ and some randomness is, ‘You will never…’ What do you mean?

GC: Yeah, I’m on the, ‘You’ll never’. It’s something that can’t be compressed. The technical definition is that a string of zeros and ones, a finite sequence of zeros and ones, is random if there’s no compact theory for it. If there’s no way to compress it into a program that’s much smaller in bits that generates it. There’s no theory… no concise theory for it.

Ard: Is that like saying, very crudely, if I put it on my computer and tried to compress it, I can’t compress it?

GC: Right.

Ard: It’s random.

GC: Yeah, but it means no computer could compress it. It’s not just one computer. Another way to put it is that there is no concise theory: it has to be comprehended or apprehended as a thing in itself, to use Kantian terminology. There is no theory for it: the only theory is to write it out bit by bit. There is no more compressed, compact way to give it structure than just to write it out bit by bit. There is no simple theory for it. It would be experimental data for which there is no simple theory. The only thing you can say is, ‘It was zero, then it was one, then it was zero…’

David: Right, so randomness, when you talk about it, is…

GC: Lack of law, lack of structure, lack of…

David: It’s genuine randomness. It’s not something that we will figure out, we will be able to predict later. This is… You just won’t.

GC: That’s correct.

David: Well, can you tell me about that, because if I’ve understood you right, then you’re saying that this is where creativity comes from.

GC: The question of creativity… The problem of creativity is, can you have a mathematical theory of creativity? Well it can’t be a theory that will give you a mechanical procedure for being creative because then it’s not creative. So a mathematical theory of creativity has to be indirect. Creativity is by definition uncomputable. If we knew how to do it, it wouldn’t be creative.

When you have maximum creativity, it looks random because it’s totally unpredictable from what you knew before.

GC: That means that you’re really being creative. So randomness is the extreme of creativity, really. They’re…

Ard: They’re connected together.

GC: And if you can calculate… if you can calculate something, then it’s not creative because you’re working within your existing system. So there’s this paradoxical aspect. A mathematical theory of creativity is a more abstract kind of mathematics where you can prove theorems about creativity – you can describe it – maybe you can show it’s highly probable, but it won’t give you a way to mechanically produce creativity, which is the kind of thing that instrumental mathematics normally does.

Ard: You can also say what creativity is not.

GC: Yes, that’s very important too. So the more we can say what it’s not, we begin to see the complement to what it is.

Ard: It’s interesting, there’s a very famous tradition in theology which says that you can’t speak about God, you can only say about what God is not. And so there’s something interesting there with creativity. You can’t nail down creativity, but you can say what it isn’t.

GC: Well, isn’t God pure creativity?

Ard: Apophatic theology is what it’s called. You have an apophatic theory of creativity. You’re speaking about it in an indirect way, but you can never grasp it. If you could grasp it, then you wouldn’t have it. That’s what you’re saying.

GC: That’s right.

Ard: You’re saying, if you could nail it down, then it wouldn’t be creativity.

GC: It wouldn’t be creative by definition.

Ard: And in the same way a theologian would say if you could nail down God, then that’s by definition not God.

GC: That wouldn’t be God, that would be a limited being.

Ard: Yeah, exactly.

David: And it’s lovely that you’re doing it in mathematics, because most people think mathematics is doing the opposite. It’s…

GC: It trivialises things. It makes them like arithmetic – boring, uninteresting, meaningless. So this is a different kind of mathematics: the mathematics of creativity. Math as an open system, not a closed system.

I’m trying to get to the concentrated essence of the mystery: the mystery is creativity, and I think that’s deeply meaningful. I mean, in Brooksonian terms the universe wants to create us. The universe wants to create mind. The universe maybe wants to get closer to God, or maybe the universe is God and it’s trying to increase its level of perception, its level of understanding.

Albrecht von Müller, he goes to an extreme that I like, which is the idea that the whole universe is actually creative: that perhaps the ontology is not fixed in static. Perhaps it’s actually plastic, even at the fundamental level. That would be much more fun. I don't know if this universe has that property, but that might be an interesting universe that is fundamentally creative.

So I’m in favour of creativity. I’m in favour of rocking the boat. I’m in favour of new ideas. And new ideas will always be fought.

IS MATHEMATICS THE LANGUAGE OF GOD?

David: So how come you described yourself once as a Platonist? What does that mean?

MG: The notion of being a Platonist, and the fact that even Plato wasn’t as much a Platonist as the ones who followed him, this whole idea of Platonism simply means that there is truth in mathematics. That nature is mathematical. That if you want to understand the hidden code of nature, you have to do it through mathematics, and only through mathematics.

David: And did you used to believe that? Was that your philosophy?

MG: Yes, because it’s a very compelling idea. Because when you say 2+2=4, there’s some finality to that statement. You can hold that and say, ‘I know that,’ and that gives you a sense of safety, of security.

So you don’t want flakiness when you’re trying to pursue the quest of understanding everything. You want the most profound kind of truth that you can find. So if math can give you that, you embrace it with all your might.

David: And Ard has, I think!

Ard: Yeah…

MG: And lots of mathematicians and lots of physicists believe that. They believe that the fundamental core of nature is essentially pure mathematics.

David: And you’re saying you’ve moved away from that?

Ard: You don’t believe it anymore?

MG: No, I don’t believe that anymore. I think that mathematics is a human invention.

Ard: Oh!

MG: It’s a product of how we evolved in this very specific planet to make sense of things and to survive. So there are certain things in mathematics which are definitely true. So if you’re an intelligence that can count, you know, one, two, three, four, then you can develop the sense of a set of integer numbers, and from there you can go on and do other things. But there may be intelligences that do not count.

There’s this famous mathematician from England called Michael Atiyah who had this image of this intelligent blob that lived at the bottom of the ocean. It was dark. It didn’t move. It didn’t have to collect any food. The food just came to it from above, and all it sensed was the flow of currents. So this intelligence created a super-sophisticated hydrodynamics – the physics of fluids and how they move about – but it would not count because there was nothing to count, unless it could hear its own heart beat or something like that.

So the fact that you’re intelligent does not mean that you have to create the integers. It really depends on the context in which you evolved.

Ard: Would you say that’s a lack that it had?

David: I’m not convinced by that, because surely the prime numbers would still be prime, even though this creature hadn’t discovered…

MG: What if there are no primes? There are no numbers?

Ard: If you don’t experience numbers, you may or may not be able to count with them, but that doesn’t mean that they don’t exist.

MG: It exists to whom? I mean to that blob? It exists, and that’s all that matters, and it would know a lot about plasticity and shapes and the form of shapes, continuous…

David: That’s quite radical, isn’t it?

Ard: My dog doesn’t know about prime numbers, but that doesn’t mean they don’t exist.

MG: Right.

Ard: It’s because it lacks the intelligence, or maybe the experience of them.

MG: Where do they exist?

Ard: Who knows?

David: I asked this question and they laughed at me.

Ard: I think they’re non-empirical realities, but what’s wrong with that?

David: What’s a non-empirical reality?

Ard: Something you don’t need to measure in order to know that it’s true.

Ard: For example, in physics, Sir Roger Penrose gave us the idea of complex numbers, or imaginary numbers.

MG: Mm-hm.

Ard: So it’s very strange…  it surely looks like an invented thing – the square root of minus one – but it turns out that it had all this surprising richness to it that eventually allowed us to formulate quantum mechanics in that language. So, the argument is that you have something that seems very abstract and made up, but then it turns out to have a life of its own, and not only does it have a life of its own, but it ends up being able to describe things about the physical world that you didn’t know about when you came up with it. That’s very surprising.

MG: It’s surprising to a certain extent.

Ard: Okay.

MG: Because physicists are really good at picking the bits of mathematics that are useful.

Ard: Sure.

MG: There are all sorts of mathematics that are completely useless. To physics, they’re not picked. So I wouldn’t jump to conclusions like that, because, I think, at the very bottom of this question is the following question: what comes before, mind or reality?

Ard: What does that mean? Mind or reality?

MG: So the people that say that mathematics is the language, is the code of nature, they’re basically saying that there is mind before everything else. There is sort of almost like some sort of metaphysical plan to reality, which is that mathematics is the fundamental blueprint of everything that exists, and we’re just discovering that stuff because it’s just there. We’re just plucking the fruit.

David: A lot of mathematicians we’ve talked to have said that.

MG: Yeah, the pure mathematicians.

David: ‘We’re not making this up. We are discovering it.’ And you don’t think that?

Ard: You don’t think it’s true?

MG: Well, I don’t know it’s true, but I don’t agree with that. I think that first comes reality. Not just reality, but the parts of reality that we can observe. Then our minds try to make sense of what’s going on, creating concepts which are useful to us.

So, for example, if you go way before mathematics, if you were a hunter-gatherer in some forest and you couldn’t distinguish between the brush and a panther, you die.

So, clearly pattern recognition was incredibly important to the survival of humanity way before there was mathematics. So our brains were moulded in such a way to favour certain things over other things so that we could thrive in an environment which was very hostile. One of them was the notion of pattern recognition. The other one was the notion of ordering. So you wanted to order things – you order space; you order time – so we created this mathematics and then the science based on this mathematics because it was very useful to us.

And so, to me, what we are trying to do, is we are trying to use our minds to describe the portions of reality that we can. And then, of course, as we evolve as a species and we learn more about reality, we pluck more math and more math and more math to do that job. And so there is a very productive, symbiotic relationship between our minds and reality.

But the notion that there is a grand plan in nature which is mathematical, and we’re just trying to uncover that, sounds to me very crypto-religious. It’s very much like some sort of medieval cult and that God was this supreme mathematician, and the job of scientists is essentially to uncover the Truth – with a capital T – which is reading the mind of God, so to speak.

Ard: And you don’t like that?

MG: I don’t like that.

David: I love the idea that you’re part of a medieval cult! This is great! This day is getting better and better.

Ard: That’s why you want to be careful around me late at night! But you don’t like it because you think it smacks too much of religion?

MG: I don’t think we need that. I don’t think that we need that in order to make sense of things…

David: Even if we, as you were saying, create the beginnings of mathematics, isn’t it that once you’ve created it, there are a whole load of consequences which flow from that first stuff? In some sort of theoretical sense, all of those consequences of what you started with, they’re already there, and you are now going to discover them.

So there is a process of discovery. In other words, we’re not free to make up the next bit of mathematics any old way. The next bit of mathematics is already decided because of the few things we invented at the beginning. So in that sense we really are. So you could forgive people for thinking, ‘My god! That was there before I got there. It was waiting for me.’ Because, in some sense, it really was. It didn’t need to be put there by God, did it? It was just a consequence of the first few ideas that we had.

MG: Could you argue the same way about music?

David: Precisely. I would say yes.

MG: Because, you know, music is… once you have the notes, everything else follows. So any species that is intelligent enough to understand that there are musical scales can come up with all the symphonies that… because they’re just waiting there to be discovered, right? And I think they are not. They are just being created by this very clever neuronal network that we have in our heads that allows us to do these wonderful things. I think… Let me put it another way…

David: Okay, go on.

MG: I think that to say that everything is out there…

David: I’m not saying everything.

MG: But it’s taking away from how amazing humans are.