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Sean Carroll is a theoretical physicist at Caltech and Santa Fe Institute specializing in quantum mechanics, arrow of time, cosmology, and gravitation. He is the author of several popular books and is a host of a great podcast called Mindscape.

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What's the difference between math and physics to you?

To me, you know, very, very roughly, math is about the logical structure of all possible worlds and physics is about our actual world.

And it just feels like our actual world is a gray area when you start talking about interpretations of quantum mechanics or no.

I'm certainly using the word world in the broadest sense, all of reality.

I don't think that there's every possible thing going on in reality.

I think that there are rules, whether it's the Schrodinger equation or whatever.

So I think that there's a sensible notion of the set of all possible worlds and we live in 1 of them.

The world that we're talking about might be a multiverse, might be many worlds of quantum mechanics, might be much bigger than the world of our everyday experience, but it's still 1 physically contiguous world in some sense.

But so if you look at the overlap of math and physics, It feels like when physics tries to reach for understanding of our world, it uses the tools of math to sort of reach beyond the limit of our current understanding.

What do you make of that process of sort of using math to, so you start maybe with intuition or you might start with the math and then build up an intuition, or but this kind of reaching into the darkness, into the mystery of the world with math?

Well, I think I would put it a little bit differently.

I think we have theories, theories of the physical world, which we then extrapolate and ask, what do we conclude if we take these seriously well beyond where we've actually tested them?

It is separately true that math is really, really useful when we construct physical theories.

And famously, Eugene Wigner asked about the unreasonable success of mathematics and physics.

I think that's a little bit wrong because anything that could happen, any other theory of physics that wasn't the real world but some other world, you could always describe it mathematically.

The surprising thing is not that math works, but that the math is so simple and easy that you can write it down on a t-shirt, right?

That's an enormous compression of information that seems to be valid in the real world.

So that's an interesting fact about our world, which maybe we could hope to explain or just take as a brute fact, I don't know.

But once you have that, you know, there's this indelible relationship between math and physics.

But philosophically, I do want to separate them.

What we extrapolate, we don't extrapolate math because there's a whole bunch of wrong math, you know, that doesn't apply to our world, right?

We extrapolate the physical theory that we best think explains our world.


Speaker 0: What's the difference between math and physics to you?

Speaker 1: To me, you know, very, very roughly, math is about the logical structure of all possible worlds and physics is about our actual world.

Speaker 0: And it just feels like our actual world is a gray area when you start talking about interpretations of quantum mechanics or no.

Speaker 1: I'm certainly using the word world in the broadest sense, all of reality. So I think that reality is specific. I don't think that there's every possible thing going on in reality. I think that there are rules, whether it's the Schrodinger equation or whatever. So I think that there's a sensible notion of the set of all possible worlds and we live in 1 of them.

The world that we're talking about might be a multiverse, might be many worlds of quantum mechanics, might be much bigger than the world of our everyday experience, but it's still 1 physically contiguous world in some sense.

Speaker 0: But so if you look at the overlap of math and physics, It feels like when physics tries to reach for understanding of our world, it uses the tools of math to sort of reach beyond the limit of our current understanding. What do you make of that process of sort of using math to, so you start maybe with intuition or you might start with the math and then build up an intuition, or but this kind of reaching into the darkness, into the mystery of the world with math?

Speaker 1: Well, I think I would put it a little bit differently. I think we have theories, theories of the physical world, which we then extrapolate and ask, what do we conclude if we take these seriously well beyond where we've actually tested them? It is separately true that math is really, really useful when we construct physical theories. And famously, Eugene Wigner asked about the unreasonable success of mathematics and physics. I think that's a little bit wrong because anything that could happen, any other theory of physics that wasn't the real world but some other world, you could always describe it mathematically.

It's just that it might be a mess. The surprising thing is not that math works, but that the math is so simple and easy that you can write it down on a t-shirt, right? I mean, that's what is amazing. That's an enormous compression of information that seems to be valid in the real world. So that's an interesting fact about our world, which maybe we could hope to explain or just take as a brute fact, I don't know.

But once you have that, you know, there's this indelible relationship between math and physics. But philosophically, I do want to separate them. What we extrapolate, we don't extrapolate math because there's a whole bunch of wrong math, you know, that doesn't apply to our world, right? We extrapolate the physical theory that we best think explains our world.

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Sean Carroll: Difference Between Math and Physics | AI Podcast Clips