Roger, how accurately does math describe the physical world? Well, it is extraordinarily precise. But I think people often find it puzzling that something abstract like mathematics could really describe reality as we understand it. I mean, reality you think of something like a chair or something, you know, something made of glass. solid stuff.
And then you say, well, what's our best scientific understanding of what that is? Well, you say it's made of fibers and cells and so on. And these are made of molecules, and those molecules are made of atoms, and those atoms are made out of neutrons. nuclei and electrons going around.
And then you say, well, what's a nucleus? Then you say, well, it's a protons and neutrons, and they're held together by things called gluons, and their neutrons and protons are made of things called quarks and so on. And then you say, well, what is an electron and what's a quark?
And at that stage, the best you can do is to describe some mathematical structure. You say they're things that satisfy the Dirac equation or something like that, which you can't understand what that means without mathematics. I mean, the mathematical description of real life is not a mathematical description.
It's a mathematical description of reality is where we're always led and these equations are fantastically accurate. Feynman had a very good description he said it you can just it describes the distance between New York and Los Angeles to an accuracy of less than the thickness of a human hair. So that's pretty precise. I mean Newton's theory already had a precision of something like one part in ten to the seven so that's ten million.
Wow. And then Einstein comes along and produces a theory which is now known to have a precision something like 10 to the power 14. So in a sense, this is telling us that our picture of physical reality depends on something which is more precise, at least in our understanding of it, than how we think about the world. And this precision really dates back to the ancient Greeks, the time of Pythagoras and later, where they developed the mathematical ideas as a field of study. study, stimulated to some degree by physical reality. But they developed this mathematical scheme purely as a study on its own.
And ever since then, mathematics has been a subject which you can study for its own sake. It has its own life, in a sense. And certainly mathematicians view it this way.
It's something out there which seems to have a reality independent of the ordinary kind of reality like things like chairs and so on, which what we normally think of as real. It's sometimes referred to as real. referred to as a platonic world, a platonic reality.
And sometimes people have a lot of trouble thinking of that as real. I mean, philosophers worry about that and so on. What would that mean, a platonic reality?
It's a different kind of reality from the reality of the physical world. I mean, I tend to think of there being different ways of looking at reality. There's the reality of our mental experience, which, okay, interrelates with the physical reality. But so then does the mathematical reality of this platonic world. which gives reality to these notions.
So if you like mathematical facts, like there is no largest prime number, it's something independent of ourselves. It's always been true. Didn't somehow become true as soon as somebody saw how to prove it. It's always been true.
And it would have been true if nobody ever pre-improved it. Exactly, yes. In a sense, that had to be so, because if the physical world depended so precisely on these mathematical laws, I couldn't have known what to do in a certain sense if the mathematics hadn't already been there. I mean, it's not us that imposes this on the world. It's out there.
Sometimes people think that, you know, maybe the reason we have good mathematical laws of physics is that's the best way we can come to understand the world. But it's something more than that. It really is out there in the world. I like to think of mathematics as a bit like geology or archaeology, where you're really exploring something out there in the world and you're finding beautiful things or things.
which have been there, in fact, for ages and ages and ages, and you're revealing them for the first time. Some of which you never dreamed of. I've never dreamed of some of them. That's absolutely right.