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!