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.