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.
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.
David: If you're willing, as both of you do, say, look there's some kind of ideas, truths, in this case mathematical ones, which are woven into the fabric of the universe, could there be other kinds of truths? Like moral truths or aesthetic truths?
RP: Other kinds of truths? Well there certainly could be, yes, I'm not saying that mathematical truth is the whole of truth. That would be too arrogant a statement to make. We don't know of any other area which is so successful in describing the physical world. Now you see...
David: But could there be different truths altogether?
RP: Well, you see, there could be.
David: What's your feeling about that? Do you think the only truths that are in the world are maths?
RP: I think the trouble is the word truth. You can have things that people have a greater respect for. They might be great works of music, of architecture or painting, all sorts of things which have a value of their own, which you can't say you could reduce to mathematics, and I wouldn't necessarily want to do that.
Ard: Could there be other kinds of truths that you in effect discover rather than create?
RP: Well you might say... You see, I'm not against, I mean we're going to get sort of Platonism in some form. You see, the Platonic ideals, truth is only one of them, and you would say truth, beauty, if you like, and morality.
Ard: And goodness?
RP: The true, the good and the beautiful. Now I would be quite happy to give some kind of reality to all these things. Now the only thing that mathematics has to say in a clear way, I suppose, is the truth part, and it's talking about necessary truths. Well it's not talking contingent truths. We're not talking about something which might be here or might be there. We're talking about things which by their very nature are true or false.
It's also clear that there are inter-relations between, in particular, beauty and mathematics, and people very often talk in terms of a beautiful result. And it's certainly the case that if you have two alternatives where you worry about which is true, it's a better bet to think that the one which is more beautiful is more likely to be true. But this is always a very subtle issue. You might find there's a deeper reason that you hadn't realised before which makes the other one actually beautiful in a deep sense that you hadn't appreciated before.
Ard: So beauty is a guide to truth?
RP: I think beauty is a clear guide to truth.
Ard: But sometimes beauty… it's hard to be sure you've perceived it correctly.
RP: Yes, and beauty is, of course, a very personal thing, and people may have different views.
David: Has it been a guide for you in your work?
RP: Yes, certainly. I am definitely sympathetic to all three of the Platonic ideals. The truth one, which I'm taking as the pure, necessary truth, I think that's an absolute thing. And when it comes to beauty, well, you see, I would say there is a great subjectivity to beauty, and there's no doubt about that. But I would say there's a kernel of truth to all that which is independent of people. And I really sort of argue that great music can be great in itself, not just because people appreciate it.
David: And the moral?
RP: And then the moral, I would see even more so, probably. But, you see, this is an interesting question, because one of the things I spent a lot of time worrying about has been the issue of consciousness. And so I have these three worlds in a sense: you have the mathematical world, and then the physical world here. And the laws of physics seemed to be governed by mathematics, but it's only a part of the mathematical world, as far as we know, which governs the laws of physics. And it's only a part of the physical world, as far as we know, which has conscious experiences.
But if you are worrying about the other Platonic values, you see, do they have absolute existence as well? And the moral one seems to depend ‒ I'm not sure I think that this is entirely so ‒ but it seems to depend on the existence of consciousness. I mean, if there were no conscious beings around, the notion of morality somehow seems to evaporate. It has to do with conscious beings.
So I would say that truth and beauty are tied up together, and it's certainly a good guide to truth. In mathematics it's certainly true. And I would say also that the issue of consciousness is connected with this, because in order to understand what's going on in the mathematical world, I would argue that you need consciousness.
Ard: In some sense what you're saying is consciousness is needed to probe mathematical truth?
Ard: But conscious may also needed to probe goodness or moral truth?
RP: Well it’s all tied up with moral truth, because morality is tied up with consciousness.
Ard: Do you think these things like cruelty being wrong is something that we discover is true… like mathematical… like 1+1=2 is true?
RP: Yes, well I guess, in a sense, I am trying to take a view like that. It's a Platonic kind of view, but I think you have to be jolly careful with these things because there are always many different sides, and there is a danger, you know, if you're setting yourself up as saying this is the truth. Not just this is the truth, but this is right; this is what people should do.
Ard: But you might say, well, okay, some kinds of truths are…we know that they're there, but they're hard to access in a way that we're sure about, so we're more careful about them. Whereas if somebody comes around and says 1+1 does not equal 2, we can be fairly authoritarian and tell them that they're wrong.
David: I certainly wouldn't cross the bridge that they built.
Ard: That's right, that's a good point. Whereas I think the point is people worry about morals…that we're going to start imposing in the same way that when I mark an exam somebody doesn't get what 1+1 equals and says 1+1=3, I'll say it's wrong. They worry that if I think that moral truths are also there to be discovered that I'm going to do the same thing to them. But the fact is that morals, like beauty, there's a kernel there. There's definitely something about it which is much harder to put your finger on, but I think it's important to say it's more than just something that we can't quite put our finger on; it’s something that’s actually out there.
RP: Yes, I think would say that.
David: Why do you say that? I mean is it important to you? Why would you be led to think those ideas, to think those things?
RP: Very hard to know why one thinks something.
David: Because a lot of people might listen to you and say, well, look, wait a minute, I thought he was a mathematician and a scientist, and now he's veering off into these other things?
RP: That’s why I don’t like to talk about them. I think the danger there is it's much easier to get it wrong. You see in mathematics, that’s the whole beauty of the subject, if you like, that you can see who's right and who's wrong. And that is a big, important thing. Now that doesn't spread much into other areas, I mean it does to a degree. You can in physics, or in the real world, or in geography, you could say, okay, there's a continent over there, and you can go and find it, or not. You see that's… I mean there are...
David: But then why do you…? Since it is dangerous to have these ideas, where does this feeling or intuition come from? Do you think for you, such that you think, well, yes I just, I do think that there could be a moral...?
RP: It is true ‒ I don't think it is just a question of what works best in society. I mean you certainly get the view with people, often, that right and wrong is just a question of what makes society work.
David: It's a matter of fashion almost.
RP: It's a matter of fashion or a matter of…
RP: Convenience. It's a society that goes with…chugs away in a nicely oiled fashion.
David: But do you have a feeling it's...
RP: There is something much deeper.
David ...that some kernel of it is woven...?
RP: Yes, I do, yes.
David: But do you, how...?
RP: Without being religious, you see, I suppose that's what's a bit unusual, because you get people who would have a religion, and which they strongly believe in, and which they would argue is why they hold these beliefs.
David: But that's not the case for you?
RP: That's not what I would say, no. But if you… I mean, do you say Platonism is a religion of some sort? I don't.
David: Some people we're talking to say there's only one kind of knowledge which counts as knowledge ‒ which gets you to the truth ‒ and that's scientific or mathematical knowledge, and everything else ‒ art, poetry ‒ it's just story-telling, it's fantasy, it's entertainment. This is what one person said. Do you hold with that?
RP: Well it's difficult to say what I think because it's an unresolved issue. I think there are things which have to do with our feelings and so on which are very real and which certainly have no room in the physics that we know today. But I'm regarding the physics that we know today as a shadow. That’s maybe not quite the right term. I'm not sure of what's really going on, and we certainly don't almost know what's going on.
Scientific knowledge has the advantage that you can test things, and you can see when things are wrong and when they're right. To an extent you have to be jolly careful with that, too. But they, to a good extent, can tell you when things are right and when they're wrong. Now, that's not true of many things where we have to form our judgements. We have to form aesthetic judgements, often, about things, and are these things reducible to scientific things?
This is a question of reductionism, I suppose, and the view is that you… I suppose reductionism means more than one thing. It means, partly, if you have a thing and you want to know how it works, you say, well, if you know all the parts work, then that will tell you how it works. So you work out how all the parts work. And then, as science has gone, you say, well, you've got molecules, and then we've got atoms, and then we've got particles, and you've got protons and neutrons, and then we've got quarks, and then we've got things which might constitute them, and then you're going smaller and smaller and smaller and smaller.
However, the view of reductionism there is if you know how the little things work, you know how the big things work. But I don't think that's reliable, because particularly it has to do with where quantum mechanics goes wrong. That's big things: things which on the scale of normal understanding of quantum mechanics would be huge.
David: So you think that complete reductionism just can't be right?
RP: It's misleading, I suppose, partly because quantum mechanics itself is not a reductionist theory in a sense. There are these quantum entanglements. You can have experiments which tell you things that vast distances apart are not independent of each other: those things are entangled. This is a basic part of quantum mechanics pointed out by Schrödinger, and these things are now experiments showing this has happened. You can have many, many hundreds of kilometres apart...
David: And does that tell you that reductionism can't be the whole story?
RP: One view is that it is all determined by the little things. So, you know, that's a good old fashioned Laplacian universe thing. It's all determined by the little things, and therefore deterministically determined by the little things. But, yet, there are questions you ask about the big things which somehow are new kinds of questions which you don't see if you just study the little things. Now, I'm saying something more than that: there are things which start to come in on a big scale which are different from what the laws of the little things are. I'm not saying we won't have a theory of that, maybe, I don't know when ‒ it might be soon; it might not be very soon.
David: Does the realm of ideas…? Is that a realm that's not completely governed by the rules of the little things? Do ideas cause things to happen? Are they real? Do they have power in addition to the bumping of molecules?
RP: Well, of course, they certainly…ideas certainly have power, but whether those ideas… You see some people say, well, they're just the little things bumping each other, and they bump each other in a particular way which happens to take off. That would be a kind of view.
David: Do you believe that?
RP: I think that's not the way I would look at things completely. You see it's all to do with this consciousness issue, and I think something else comes in which is outside the science that we presently know. It doesn't mean it's outside science. So you see, when you're asking me is it all science, well, science is limited at the moment, because it only deals with certain areas which don't include that.
David: Yes, but you could say it's going to be natural rather than supernatural. You don't have to say that there's something supernatural. You could say it's going to be explicable by a kind of science which maybe we don't have yet?
RP: I guess I would say that, yes. But then it's hard to know because if we don't have it yet, you don't know what it's like.
David: Alright, thank you.