Mathematics: a part of the universe

Mathematics

Why does mathematics explain the workings of the universe so well?

Roger Penrose

Is maths discovered or invented?

'I certainly belong to the school of thought that it's discovery. Some of these have features which reveal deep truths which you had no conception of before.'

Transcript

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.

Alex Rosenberg

Mathematics: a problem for scientism

‘Mathematics is the major problem on the research programme of scientism. It’s not just a fly in the ointment: it’s a big project on the agenda of naturalistic philosophy.'

Transcript

MATHEMATICS: A PROBLEM FOR SCIENTISM

Ard: So there are thoughts that are quite constrained, like mathematical thoughts. So I say 1+1=2, but the fact that 1+1=2 is independent of whether or not my brain does those things, right? So the mathematics is true, regardless of whether bosons or fermions exist. Isn’t that right?

AR: Yes. And there, I think, you have the major problem on the research programme of scientism.

Ard: Okay.

AR: The scientistic world view, I think, has very good answers to a huge range of the questions that really trouble human beings when they can’t get to sleep at night, and they’re looking up at the ceiling and wondering about themselves and their place in the universe. The domain in which we have the most trouble is not a domain that most people are interested in: it’s the nature of mathematics and our knowledge of mathematical truths. As you just said, it looks very much like mathematical truths are true, independent of anybody ever having thought them.

Many people are inclined to think that mathematics is just ideas in the head, but it can’t be, for a lot of reasons, and we’ve recognised this ever since Plato. And the one thing that we scientistic philosophers don’t yet have a good account of is how we can have knowledge of mathematics, because we think that knowledge is a causal process that involves an interaction between us and the objects of knowledge. And two, and equals, and prime number, these are abstract objects that do not exist in space and time and that therefore we can’t have a causal connection to. And so our knowledge of mathematics is a deep mystery.

Ard: Doesn’t it trouble you that you need mathematics so much to do science?

AR: Yes.

Ard: So, it’s not just…

AR: When I said it’s a deep mystery, I meant it’s not just a fly in the ointment. It’s a big project on the agenda of naturalistic philosophy. We need an answer to this question. Now, I’ve been wrapping my head around this problem, a bit. Not as much as the philosophers of mathematics, who are much more deeply steeped in the difficulties of the subject. But the interesting thing is, as often the case in science, what really troubles the people who are at the frontiers of the discipline is so arcane and so alien to the interests of most people that it’s even hard to keep them awake long enough to explain the problem.

Ard: So would you say…?

AR: But it’s a big problem.

Ard: So it’s not just a few clouds on the horizon?

AR: Well, I’m inclined to say that it is a cloud on the horizon – a cloud no bigger than your hand.

Ard: Okay.

AR: But you know how these problems have a tendency to grow, the way the cloud no bigger than your hand ends up being a thunderstorm.

Ard: Yes.

AR: Now I don’t think it’s going to become a thunderstorm.

AR: But it’s a problem.

David: But…

Ard: It’s a very interesting problem also.

AR: A friend of mine once said that this is a problem on which two millennia of geniuses have laboured with great intensity and made no progress at all.

David: So let’s say when they did wrap their heads round it, they discovered, ‘You know what. It is the way it seemed. There are certain ideas which just exist in the universe, and we somehow can find them…’

AR: Right. So there’s an agen…

David: But that wouldn’t undermine science, would it? I mean, I can’t imagine how it would.

AR: Well, I don’t think it would. There’s an agenda of problems that face the philosophy of science, metaphysics, epistemology, which are generated by the sciences, questions that the sciences, at least not yet, can’t answer. And philosophy is in many ways the guardian of those questions.

Now, there are some philosophers who hold, and some theologians who hold, that these questions will never be answered by the sciences, and therefore they provide good grounds for supposing that science is somehow incomplete and that there are truths of a non-scientific kind that, in competition with scientific truths, somehow may win out. And among these there might well be religious truths.

Now here’s the thing: when I weigh the philosophical puzzles that remain, like the nature of our knowledge of mathematics, against what science has accomplished over the 400 years since Galileo and Newton, when I weigh those in the balance, it seems to me that the balance is so heavily tipped towards science and its accomplishments, by way of explaining and enabling us to understand nature and ourselves, that I cannot take these puzzles seriously.

Now, at the end of the 20th, beginning of the 21st century, there’s still a package of problems that the sciences can’t yet answer, of which I think, as I said, the nature of our knowledge of mathematical truths is one. Do I think that science will never answer them? No, that’s what scientism consists in. It’s the prediction that eventually we’re going to successfully answer these questions.

Ard: So to summarise your argument: what you’re saying is science has shown us so many advances that we should essentially trust it to answer all questions.

AR: The inductive evidence favours that conclusion more strongly than the conclusion that there’s some domain of questions – real questions as opposed to pseudo questions – to which it can give no answer.

Ard: Great.

David: Why don’t you believe that, then?

Ard: I think that the clouds on the horizon, like mathematics, but I think also moral knowledge is true. I think science’s power derives precisely from its limitations to certain types of questions. I think there are many questions… I think we actually both agree… There are questions…

AR: But you cannot allow the fact that science would give disobliging answers to those questions to rule them as out of bounds for science.

Gregory Chaitin

The joy of mathematical discovery

‘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.’

Transcript

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.

Peter Atkins

The mathematical universe

‘It is the deep truth about the nature of the universe that it is a mathematical place, and by mathematical I mean logical and ultimately simple.’

Transcript

The mathematical universe


David: Is mathematics… Are there truths which just are true? Are mathematical truths just there? Or is it a human construction, do you think?

PA: There must be truths that are in mathematics that we humans are gradually digging out of all the clutter that shrouds. So I think the ultimate truth is mathematical.

David: So in other words…

Ard: Are we discovering those truths?

PA: We’re discovering, yes.

David: Right. You see, that is really interesting. So in some sense there’s the stuff of the universe which, as a chemist, you understand its rules.

PA: Yes.

David: But then there’s this other kind of truth in the universe. Because those truths… How would they exist? They’re not matter, but you’re saying they’re in the universe: we’re not making them up.

PA: The truth that the angles of a triangle add up to 180o is not really a truth, because there is no triangle that is in absolutely flat space. So what we try to do is to distil from everyday experience models that approximate what we’re really trying to talk about. And try to identify the truths that those models represent.

David: Okay, but there are these Truths, with a capital T, in the universe. Somehow it’s woven into the fabric of the universe in some way?

PA: Yes. 1+1=2 is certainly a universal Truth.

David: With a capital T?

PA: Yes.

David: So it’s not just something that we’re making up, like…?

PA: Well we made it up and it’s found to be true.

David: Ooh.

PA: Ooh, I’m not sure whether that’s true.

David: I can’t help you there and you would know this! But I’m trying to get at whether you think there are some transcendent Truths, and whether they are mathematical. Whether mathematical truths are in some sense a transcendent Truth. It’s not a social construct that we just agreed on.

PA: Well the purest kind of truth is mathematical.

David: Right.

PA: And the purest kind of truth, I suppose, within mathematics is the properties of the integers.

David: Right, okay.

PA: And I think the whole of mathematics has emerged from us forcing on the integers things that they weren’t intended to do: we turn them into fractions; we use them, somehow, to generate irrational numbers. So we take the square root of 2, for example, and suddenly we’ve got funny things that the integers have led us to do. And so it’s really the exploration of integers that has…

David: Right. But you make that sound like this is just something that we’re forcing on the world because we feel like it, and yet the stories that you [Ard] tell, and other mathematicians and physicists tell, is that sometimes you’ll have a scientist who’s just working with abstract numbers and equations and then… What was the fellow you told us about?

Ard: Paul Dirac.

David: Right, the Dirac story, where he’s just dealing with equations and numbers – these peculiar numbers that you’re talking about – and then says, ‘Do you know what. There has to be antimatter.’ And no-one had even heard of antimatter. And he said: ‘And not only that, but I can tell you… We’ll call it the positron and it’ll have these properties and be the opposite of the electron.’

PA: Yes.

David: And then it turns out he was right.

PA: Yes.

David: Out of just, as you said, forcing numbers to do unnatural things, it matched perfectly on to the universe, which suggests that it’s not an arbitrary thing we’re just forcing numbers to do, but somehow it is uncovering the actual…

PA: Or conversely, that the universe is not just a random collection of entities.

David: But more than that, because it could not be random; it might not be a random collection of entities, but how does…? It could be non-random in any number of ways. It turned out to fit perfectly with mathematics.

PA: Yeah.

David: Which is odd, is it not?

PA: Yes, I mean…

David: I mean, really catastrophically odd?

PA: It is the deep truth about the nature of the universe that it is a mathematical place; and by mathematics I mean logical and ultimately simple. And so, I think, the real message that is coming through is that the universe is ultimately, and by ultimately I mean, ultimately ultimately, a very simple place.

David: You think there will be a theory of everything? You know, the fabled Theory of Everything which once you unpack it…?

PA: Yes. The universe, in a sense, is either zero or one, but conglomerated in such an extraordinary way that we can have this conversation.

David: Okay, if we’re willing to entertain the notion that the universe could have certain truths woven into it, mathematical ones, why would it not be possible that there could be other kinds of truths woven into it: aesthetic truths or moral truths? It’s a peculiar question.

PA: I mean, it would be wrong for a scientist to say that any idea is absolute nonsense, so I won’t go that far. You have to leave the door ajar for any possibility.

David: For the feeble-minded!

PA: So the possibility that the universe is not a mathematical place, but is ultimately a moral place, is something to entertain. There is no evidence for it, but it would be improper to deny the possibility of anything.

David: Yes. I mean, I’m not saying I think it is a moral… that there are moral truths. But I can’t in all honesty say if I’m willing to accept that mathematical truths are in the universe, I don't know by what authority I would say other kind of truths can’t be. I don't know how I would justify that.

PA: It depends what you mean by truths, doesn’t it, really?

Ard: Largely very philosophical?

David: Ah, yes!

PA: But I’m talking about truths. You’re talking about different levels of knowledge. I mean, ultimately, you could argue – and this might not be true – you could argue that mathematics is the lowest level of the foundation of all knowledge. You could say that poetry and aesthetic delight is right at the top. And there are all these intermediate steps, some of which are absolutely true. Mathematics is possibly absolutely true. 1+1 is probably absolutely true. And mathematics is simply an elaboration of 1+1=2.

David: And then poetic truth is just going to be largely entertainment?

PA: Just entertainment.

Ard: Not much more?

David: You can’t say that! You cannot say that! Do you really believe that, or are you just doing that to get a…?

PA: I’m prepared to believe it for a few milliseconds.

David: Okay!

Gregory Chaitin

Is there a mathematics of life?

‘What I see biology is all about is tremendous creativity, and I see that connected with mathematical creativity.'

Transcript

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.

George Ellis

The eternal truths of mathematics

‘Do we invent mathematics or do we find mathematics? I’m an unashamed mathematical Platonist: we discover mathematics.’

Transcript

THE ETERNAL TRUTHS OF MATHEMATICS

David: So this is something that Ard and I were discussing earlier. Are you saying that when I ask you what does two plus two equal, and you say four, it’s always seemed to me the reductionistic argument – when they say, well, consciousness doesn’t exist – is that somehow you come up with the answer four because you were forced to because electrons just got into that state? Whereas I’ve always thought that the reason you say the answer is four is because of the logic of mathematics. So, in other words, it’s the logic of mathematics which is pushing the electrons around, not the other way, where the electrons are forcing you to have a thought.

GE: No, you’re quite right. That’s exactly the way it is.

David: Okay, so he does agree with us. Because we were discussing this earlier, and then we thought, crikey, maybe we’ve both really misunderstood it.

GE: Do you want me to open this up to an even more mind-boggling place?

David: Go on then.

GE: Okay. Where does the logic of mathematics come from?

David: Oh dear.

GE: This is the old question: do we invent mathematics or do we find mathematics? And I’m an unashamed mathematical Platonist: we discover mathematics. Two plus two is four is too simple. Let’s take something more interesting like the fact that the square root of two is irrational. Now the square root of two is irrational no matter whether you’re an Ancient Greek or someone here or someone on Mars. The square root of two is irrational. It’s a timeless, eternal, unchanging mathematical truth. In other words it’s a Platonic kind of statement.

The ontology is the mathematics exists and is there and is unchanging. The fact that the square root of two is irrational is an eternal unchanging truth. What we understand about it is a historically contingent thing, and we didn’t know that 10,000 years ago and we do know it now.

David: But the thing which is true was always true?

GE: The thing which is true is always true and has been true since the beginning of the universe.

David: Right, so in other words it was true when there were only dinosaurs around, and it’s still true.

GE: It was true at the start of the Big Bang. It was true before, when there was just hot gas and nothing else.

Ard: I mean, if you think about it that way, it’s really hard to believe that wouldn’t be the case.

David: Except that if people, physicists, would say look, ‘I’ve got bosons and I’ve got quarks, you know, what is the particle that carries the idea?’ That’s what...

GE: Yeah, but physicists have great trouble telling you this famous question. Why does mathematics underlie physics? The famous thing that Galileo said that the nature of the universe is written in mathematics. And Wigner and Penrose and other people have pondered, why is it that physics can be written in mathematical terms? And that’s a deep philosophical question for which we don’t have a proper answer.

Ard: So the unreasonable effectiveness of mathematics?

GE: The unreasonable effectiveness of mathematics, yes.

Gregory Chaitin

Randomness and creativity

'I’m trying to get to the concentrated essence of the mystery: the mystery is creativity, and I think that’s deeply meaningful.'

Transcript

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.

Denis Noble

Mathematics and music

'I get a different sound because I’ve divided the string exactly into two. I jump an octave – it’s beautiful. You get lovely bell-shaped sounds out of that. I mean, if you don’t think that’s beautiful, I don't know what is!'

Transcript

Mathematics AND Music

David: The laws of harmony. This link between music and mathematics. It’s a completely different subject, but…

DN: Yes.

David: If you would do that basic job of explaining that there is this sort of Pythagorean deep link between mathematics and music.

DN: Yes, very much so, because I play the guitar, so I sort of experience this almost every day when I practise or perform. And I know, as a matter of fact, that if I stop the string at this point, and I do something like that, to get what we call a harmonic, I get a totally different sound from what I would get if I tapped the string just at that point normally.

And I get a different sound because I’ve divided the string exactly into two. I jump an octave – it’s beautiful. You get lovely bell-shaped sounds out of that. I mean, if you don’t think that’s beautiful, I don't know what is! And, as you say, it comes out of the equations of the vibrations of that string.

DN: Those are both E’s. That one is a high E because I’m dividing the string by exactly 50 percent of its length. I can play it in a different way to get what are called octave harmonics which are using that halving principle in a way that produces a very different kind of note.

David: And that’s Pythagoras: that’s the mathematics.

DN: Absolutely so, yes. And you can do the same of course by dividing up into thirds, fifths and so on which is how you get chords that are harmonic and pleasing to the ear.

David: Can you show us a third or a fifth.

DN: For example, if I do here, those two notes sound nice together. So do those. Those all sound nice, because they are related by the mathematics of the way in which the string is working.

David: So that’s a link. That’s your mathematics and beauty then.

DN: Exactly so, and you can put a whole chord together with those, you see. It’s the relationships that enable all of that to happen: it’s not the individual notes. I can play individual notes, nothing’s there. Together you get a totally different flavour and you can only say that that is beautiful, but the individual notes on their own are not.

David: Is it stretching it then to say that the music emerges out of the individual notes when you put them together.

DN: Not only that, but it emerges when you put them together in relation to each other, just as you put hydrogen and oxygen together to form water. It’s also the case that the way I play this in every concert is different, so it depends on your mood and so many other things. The way in which I will play that will be different each time: it will never be the same.

David: And yet the music is the same, the score.

DN: The notes, the chords are the same. What’s written in the score is the same, but a performer never performs it exactly the same way.

David: Does this metaphor then of music… it seems like it’s helped you to formulate your ideas. It’s a powerful metaphor for you?

DN: For me it’s an extraordinary powerful metaphor, because after all we are dealing here with some things you can give a nice, reductionist, scientific account, which is if I do that [plays two notes], I get an octave, and if I do the thirds and the fifths, I get what I get. So all of that looks nice and reductionist: the individual notes. But it is really only when I put it together that I get something beautiful.

Ard: And the beauty is in those variations.

DN: Exactly so. In fact that beauty comes out of what is not in the score. Precisely. It’s a creative process, and I think the evolutionary process has been a creative process that is built on each novelty as it’s arisen to build meaning into the process of how it has all changed and evolved. So you don’t have to ask the question, where does the meaning all come from? You can ask that question if you want, and if you are of a particular disposition of a particular metaphysical kind you will do that, but you don’t have to ask that question in order to see the meaning. The meaning is there, obvious, just as in the heartbeat – I’m pretty clear about the purpose of the heartbeat. As a physiologist I couldn’t be anything other than clear.

Marcelo Gleiser

Mathematics: the language of God?

‘I think that mathematics is a human invention. It’s a product of how we evolved in this very specific planet to make sense of things and to survive.’

Transcript

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.

Denis Noble

Mathematics and the body

‘There is a sense in which it exists, because mathematicians discover, they don’t invent. Go back to my rhythm in the heart. If you want to describe that, one of the ways is to use in your mathematics the square root of minus one.’

Transcript

MATHEMATICS AND THE BODY

David: We’ve been talking about the notion that some ideas are just there, like mathematics is an idea.

DN: Ah, that’s very interesting, mathematics. Where is that?

David: Yes, thank you. Precisely!

DN: Where is it?

Ard: I just think, ‘Where is it?’ is the wrong question.

DN: Exactly. It’s the wrong question.

David: The wrong metaphor?

DN: The where question is the wrong question.

David: Okay.

DN: There is a sense in which it exists, because mathematicians discover, they don’t invent.

David: Discover?

Ard: We sense very deep in our bones that we discover. We were talking about Paul Dirac earlier who used mathematics to look at the electron, and what happens if the electron goes fast, and from the mathematics came anti-matter: that’s something that you discover, it’s not something that you…

DN: Exactly so.

David: But then this notion of discovering makes me think, in my metaphorical misery over here, that you discovered this realm of ideas. Where are these ideas?

DN: Okay. Realm is okay, provided we don’t think it is a where.

David: No, okay.

DN: I cannot go to the realm where ideas are.

David: Okay. But exist in what way then? Because conkers exist as conkers and plants exist as plants, in a reductionistic, material world.

DN: But the square root of minus one does not exist.

David: Yes! How does that exist?

DN: Now, you see, that’s a lovely example, because go back to my rhythm in the heart: if you want to describe that, interestingly enough, one of the ways of describing that is to use in your mathematics the square root of minus one. Now the square root of minus one clearly does not exist, but the idea has a fantastic utility in mathematics. So I think you’ve got to see those ideas as tools.

Ard: Yeah, it’s a nice example. Because the square root of minus one…

David: It’s an imaginary number.

Ard: .It’s an imaginary number, and yet in oscillations it’s incredibly helpful and useful.

DN: Absolutely, yes.

Ard: And, actually, it has a kind of explanatory power that allows us to peer much more deeply into the universe than we would have been able to without it. And yet, there it is.

David: But you can see my problem, though, can’t you? How do ideas exist? Because they don’t exist in the same way that reductionists like.

DN: Indeed so. But doesn’t that, as it were, induce a sense of humility?

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