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Do you think math is discovered or invented?

So we're talking about the different kind of mathematics that could be developed by the alien species.

The implied question is, yeah, is math discovered or invented?

Is fundamentally everybody going to discover the same principles of mathematics?

So the way I think about it, and everyone thinks about it differently, but here's my take.

I think there's a cycle at play where you discover things about the universe that tell you what math will be useful.

And that math itself is invented in a sense, but of all the possible maths that you could have invented, it's discoveries about the world that tell you which ones are.

So like a good example here is the Pythagorean theorem.

When you look at this, do you think of that as a definition or do you think of that as a discovery?

From the historical perspective, right, it's a discovery because they were, but that's probably because they were using physical object to build their intuition.

And from that intuition came the mathematics.

So the mathematics was in some abstract world detached from physics, but I think more and more math has become detached from, you know, when you even look at modern physics, from string theory to even general relativity, I mean, all math behind the 20th and 21st century physics kind of takes a brisk walk outside of what our mind can actually even comprehend.

In multiple dimensions, for example, Anything beyond 3 dimensions, maybe 4 dimensions.

No, no, no, higher dimensions can be highly, highly applicable.

I think this is a common misinterpretation that if you're asking questions about a five-dimensional manifold, that the only way that that's connected to the physical world is if the physical world is itself a five-dimensional manifold or includes them.

Well, wait, wait, wait a minute, wait a minute.

You're telling me you can imagine a five-dimensional manifold?

I would make the claim that it is useful to a three-dimensional physical universe, despite itself not being three-dimensional.

So it's useful, meaning to even understand a three-dimensional world, it'd be useful to have five-dimensional manifolds.

Yes, absolutely, because of state spaces.

But you're saying there in some deep way for us humans, it does always come back to that 3 dimensional world for the usefulness of the dimensional world and therefore it starts with a discovery, but then we invent the mathematics that helps us make sense of the discovery in a sense?

I mean, just to jump off of the Pythagorean theorem example, it feels like a discovery.

You've got these beautiful geometric proofs where you've got squares and you're modifying the areas.

If you look at how we formalize the idea of 2D space as being R2, right?

All pairs of real numbers and how we define a metric on it and define distance.

We've defined distance so that the Pythagorean theorem is true.

So then suddenly it doesn't feel that great.

But I think what's going on is the thing that informed us what metric to put on R2, to put on our abstract representation of 2D space, came from physical observations.

And the thing is there's other metrics you could have put on it.

We could have consistent math with other notions of distance, it's just that those pieces of math wouldn't be applicable to the physical world that we study because they're not the ones where the Pythagorean theorem holds.

So we have a discovery, a genuine bona fide discovery that informed the invention, the invention of an abstract representation of 2D space that we call R2 and things like that.

And then from there, you just study R2 as an abstract thing that brings about more ideas and inventions and mysteries, which themselves might yield discoveries.

Those discoveries might give you insight as to what else would be useful to invent.

So it's not an either or, it's not that math is 1 of these or it's 1 of the others at different times it's playing a different role.

So then let me ask the Richard Feynman question then along that thread is what do you think is the difference between physics and math?

There's a kind of intuition that physicists have about the world that's perhaps outside of mathematics?

It's this mysterious art that they seem to possess, we humans generally possess.

And then there's the beautiful rigor of mathematics that allows you to, I mean, just like as we were saying, invent frameworks of understanding our physical world.

So what do you think is the difference there?

Well, I think of math as being the study of abstractions over patterns and pure patterns in logic.

And then physics is obviously grounded in a desire to understand the world that we live in.

I think you're going to get very different answers when you talk to different mathematicians because there's a wide diversity in types of mathematicians.

There are some who are motivated very much by pure puzzles.

They might be turned on by things like combinatorics, and they just love the idea of building up a set of problem-solving tools, applying to pure patterns, right?

There are some who are very physically motivated, who try to invent new math or discover math in veins that they know will have applications to physics or sometimes computer science.

Like chaos theory is a good example of something that's, it's pure math.

A lot of the statements being made, but it's heavily motivated by specific applications to largely physics.

And then you have a type of mathematician who just loves abstraction.

They just love pulling into the more and more abstract things, the things that feel powerful.

These are the ones that initially invented like topology and then later on get really into category theory and go on about like infinite categories and whatnot.

These are the ones that love to have a system that can describe truths about as many things as possible, right?

People from those 3 different veins of motivation into math are going to give you very different answers about what the relation at play here is.

Because someone like Vladimir Arnold, who is this, he's written a lot of great books, many about like differential equations and such, he would say math is a branch of physics.

And of course he was studying like differential equations related things because that is the motivator behind the study of PDEs and things like that.

But you'll have others who, like especially the category theorists, who aren't really thinking about physics necessarily.

It's all about abstraction and the power of generality.

And it's more of a happy coincidence that that ends up being useful for understanding the world we live in.

And then you can get into like, why is that the case?

It's sort of surprising that that which is about pure puzzles and abstraction, also happens to describe the very fundamentals of quarks and everything else.

So why do you think the fundamentals of quarks

and the nature of reality is so compressible into clean, beautiful equations that are for the most part simple, relatively speaking.

So you have, we mentioned somebody like Stephen Wolfram who thinks that sort of there's incredibly simple rules underlying our reality, but it can create arbitrary complexity.

What, I'm asking a million questions that nobody knows the answer to.

It could be the case that there's like a filtration at play.

The only things that physicists find interesting are the ones that are simple enough they could describe it mathematically.

But as soon as it's a sufficiently complex system, like, oh, that's outside the realm of physics, that's biology or whatever have you.

You know, maybe there's something where it's like, of course there will always be some thing that is simple when you wash away, the like non-important parts of whatever it is that you're studying.

Just from like an information theory standpoint, there might be some, like you get to the lowest information component of it, but I don't know, maybe I'm just having a really hard time conceiving of what it would even mean for the fundamental laws to be like intrinsically complicated, like some, some set of equations that you can't decouple from each other?

Well, no, it could be that, if we take for granted that the laws of physics, for example, are for the most part the same everywhere, or something like that, right?

As opposed to, sort of an alternative could be that the rules under which the world operates is different everywhere.

It's like a deeply distributed system where everything is just chaos.

Not in a strict definition of chaos, but meaning like just it's impossible for equations to capture, to explicitly model the world as cleanly as the physical does.

I mean, we almost take it for granted that we can describe, we can have an equation for gravity, for action at a distance.

We can have equations for some of these basic ways the planets move and just the low level at the atomic scale, how the materials operate, at the high scale, how black holes operate, but it doesn't, it seems like it could be, there's infinite other possibilities where none of it could be compressible into such equations.

It's also weird, probably to the point you were making, that it's very pleasant that this is true for our minds.

it might be that our minds are biased to just be looking at the parts of the universe that are compressible.

And then we can publish papers on and have nice E equals empty squared equations.

Well, I wonder, would such a world with uncompressible laws allow for the kind of beings that can think about the kind of questions that you're asking?

Right, like an anthropic principle coming into play in some weird way here?

I don't know, I don't know what I'm talking about at all.

Or maybe the universe is actually not so compressible, but the way our brain evolved, we're only able to perceive the compressible parts.

I mean, we are, so this is a sort of Chomsky argument.

We're like really limited biological systems.

So it totally makes sense that we're really limited little computers, calculators, that are able to perceive certain kinds of things and the actual world is much more complicated.

Well, but we can do pretty awesome things, right?

Like we can fly spaceships, and we have to have some connection of reality to be able to take our potentially oversimplified models of the world, but then actually twist the world to our will based on it.

So we have certain reality checks that physics isn't too far afield, simply based on what we can do.

Yeah, the fact that we can fly is pretty good.

proof of concept that the laws we're working with are working well.


Speaker 0: Do you think math is discovered or invented? So we're talking about the different kind of mathematics that could be developed by the alien species. The implied question is, yeah, is math discovered or invented? Is fundamentally everybody going to discover the same principles of mathematics?

Speaker 1: So the way I think about it, and everyone thinks about it differently, but here's my take. I think there's a cycle at play where you discover things about the universe that tell you what math will be useful. And that math itself is invented in a sense, but of all the possible maths that you could have invented, it's discoveries about the world that tell you which ones are. So like a good example here is the Pythagorean theorem. When you look at this, do you think of that as a definition or do you think of that as a discovery?

Speaker 0: From the historical perspective, right, it's a discovery because they were, but that's probably because they were using physical object to build their intuition. And from that intuition came the mathematics. So the mathematics was in some abstract world detached from physics, but I think more and more math has become detached from, you know, when you even look at modern physics, from string theory to even general relativity, I mean, all math behind the 20th and 21st century physics kind of takes a brisk walk outside of what our mind can actually even comprehend. In multiple dimensions, for example, Anything beyond 3 dimensions, maybe 4 dimensions.

Speaker 1: No, no, no, higher dimensions can be highly, highly applicable. I think this is a common misinterpretation that if you're asking questions about a five-dimensional manifold, that the only way that that's connected to the physical world is if the physical world is itself a five-dimensional manifold or includes them.

Speaker 0: Well, wait, wait, wait a minute, wait a minute. You're telling me you can imagine a five-dimensional manifold?

Speaker 1: No, no, that's not what I said. I would make the claim that it is useful to a three-dimensional physical universe, despite itself not being three-dimensional.

Speaker 0: So it's useful, meaning to even understand a three-dimensional world, it'd be useful to have five-dimensional manifolds.

Speaker 1: Yes, absolutely, because of state spaces.

Speaker 0: But you're saying there in some deep way for us humans, it does always come back to that 3 dimensional world for the usefulness of the dimensional world and therefore it starts with a discovery, but then we invent the mathematics that helps us make sense of the discovery in a sense?

Speaker 1: Yes. I mean, just to jump off of the Pythagorean theorem example, it feels like a discovery. You've got these beautiful geometric proofs where you've got squares and you're modifying the areas. It feels like a discovery. If you look at how we formalize the idea of 2D space as being R2, right?

All pairs of real numbers and how we define a metric on it and define distance. You're like, hang on a second. We've defined distance so that the Pythagorean theorem is true. So then suddenly it doesn't feel that great. But I think what's going on is the thing that informed us what metric to put on R2, to put on our abstract representation of 2D space, came from physical observations.

And the thing is there's other metrics you could have put on it. We could have consistent math with other notions of distance, it's just that those pieces of math wouldn't be applicable to the physical world that we study because they're not the ones where the Pythagorean theorem holds. So we have a discovery, a genuine bona fide discovery that informed the invention, the invention of an abstract representation of 2D space that we call R2 and things like that. And then from there, you just study R2 as an abstract thing that brings about more ideas and inventions and mysteries, which themselves might yield discoveries. Those discoveries might give you insight as to what else would be useful to invent.

And it kind of feeds on itself that way. That's how I think about it. So it's not an either or, it's not that math is 1 of these or it's 1 of the others at different times it's playing a different role.

Speaker 0: So then let me ask the Richard Feynman question then along that thread is what do you think is the difference between physics and math? There's a giant overlap. There's a kind of intuition that physicists have about the world that's perhaps outside of mathematics? It's this mysterious art that they seem to possess, we humans generally possess. And then there's the beautiful rigor of mathematics that allows you to, I mean, just like as we were saying, invent frameworks of understanding our physical world.

So what do you think is the difference there? And how big is it?

Speaker 1: Well, I think of math as being the study of abstractions over patterns and pure patterns in logic. And then physics is obviously grounded in a desire to understand the world that we live in. I think you're going to get very different answers when you talk to different mathematicians because there's a wide diversity in types of mathematicians. There are some who are motivated very much by pure puzzles. They might be turned on by things like combinatorics, and they just love the idea of building up a set of problem-solving tools, applying to pure patterns, right?

There are some who are very physically motivated, who try to invent new math or discover math in veins that they know will have applications to physics or sometimes computer science. And that's what drives them, right? Like chaos theory is a good example of something that's, it's pure math. That's purely mathematical. A lot of the statements being made, but it's heavily motivated by specific applications to largely physics.

And then you have a type of mathematician who just loves abstraction. They just love pulling into the more and more abstract things, the things that feel powerful. These are the ones that initially invented like topology and then later on get really into category theory and go on about like infinite categories and whatnot. These are the ones that love to have a system that can describe truths about as many things as possible, right? People from those 3 different veins of motivation into math are going to give you very different answers about what the relation at play here is.

Because someone like Vladimir Arnold, who is this, he's written a lot of great books, many about like differential equations and such, he would say math is a branch of physics. That's how he would think about it. And of course he was studying like differential equations related things because that is the motivator behind the study of PDEs and things like that. But you'll have others who, like especially the category theorists, who aren't really thinking about physics necessarily. It's all about abstraction and the power of generality.

And it's more of a happy coincidence that that ends up being useful for understanding the world we live in. And then you can get into like, why is that the case? It's sort of surprising that that which is about pure puzzles and abstraction, also happens to describe the very fundamentals of quarks and everything else. So why do you think the fundamentals of quarks

Speaker 0: and the nature of reality is so compressible into clean, beautiful equations that are for the most part simple, relatively speaking. A lot simpler than they could be. So you have, we mentioned somebody like Stephen Wolfram who thinks that sort of there's incredibly simple rules underlying our reality, but it can create arbitrary complexity. But there is simple equations. What, I'm asking a million questions that nobody knows the answer to.

But- I

Speaker 1: have no idea. Like, Why is it simple? It could be the case that there's like a filtration at play. The only things that physicists find interesting are the ones that are simple enough they could describe it mathematically. But as soon as it's a sufficiently complex system, like, oh, that's outside the realm of physics, that's biology or whatever have you.

And of course that's true. Right. You know, maybe there's something where it's like, of course there will always be some thing that is simple when you wash away, the like non-important parts of whatever it is that you're studying. Just from like an information theory standpoint, there might be some, like you get to the lowest information component of it, but I don't know, maybe I'm just having a really hard time conceiving of what it would even mean for the fundamental laws to be like intrinsically complicated, like some, some set of equations that you can't decouple from each other?

Speaker 0: Well, no, it could be that, if we take for granted that the laws of physics, for example, are for the most part the same everywhere, or something like that, right? As opposed to, sort of an alternative could be that the rules under which the world operates is different everywhere. It's like a deeply distributed system where everything is just chaos. Not in a strict definition of chaos, but meaning like just it's impossible for equations to capture, to explicitly model the world as cleanly as the physical does. I mean, we almost take it for granted that we can describe, we can have an equation for gravity, for action at a distance.

We can have equations for some of these basic ways the planets move and just the low level at the atomic scale, how the materials operate, at the high scale, how black holes operate, but it doesn't, it seems like it could be, there's infinite other possibilities where none of it could be compressible into such equations. It just seems beautiful. It's also weird, probably to the point you were making, that it's very pleasant that this is true for our minds.

Speaker 1: So

Speaker 0: it might be that our minds are biased to just be looking at the parts of the universe that are compressible. And then we can publish papers on and have nice E equals empty squared equations.

Speaker 1: Right. Well, I wonder, would such a world with uncompressible laws allow for the kind of beings that can think about the kind of questions that you're asking? That's true. Right, like an anthropic principle coming into play in some weird way here? I don't know, I don't know what I'm talking about at all.

Speaker 0: Or maybe the universe is actually not so compressible, but the way our brain evolved, we're only able to perceive the compressible parts. I mean, we are, so this is a sort of Chomsky argument. We are just descendants of apes. We're like really limited biological systems. So it totally makes sense that we're really limited little computers, calculators, that are able to perceive certain kinds of things and the actual world is much more complicated.

Speaker 1: Well, but we can do pretty awesome things, right? Like we can fly spaceships, and we have to have some connection of reality to be able to take our potentially oversimplified models of the world, but then actually twist the world to our will based on it. So we have certain reality checks that physics isn't too far afield, simply based on what we can do.

Speaker 0: Yeah, the fact that we can fly is pretty good.

Speaker 1: It's great, yeah.

Speaker 0: And land. It's a pretty good

Speaker 1: proof of concept that the laws we're working with are working well. You

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Grant Sanderson (3Blue1Brown): Is Math Discovered or Invented? | AI Podcast Clips