Grant Sanderson (3Blue1Brown): Is Math Discovered or Invented?

00:00:00.000 | - Do you think math is discovered or invented?

00:00:04.640 | So we're talking about the different kind of mathematics

00:00:07.160 | that could be developed by the alien species.

00:00:10.160 | The implied question is, yeah,

00:00:14.040 | is math discovered or invented?

00:00:15.760 | Is fundamentally everybody going to discover

00:00:19.180 | the same principles of mathematics?

00:00:22.680 | - So the way I think about it,

00:00:23.700 | and everyone thinks about it differently,

00:00:24.920 | but here's my take.

00:00:25.960 | I think there's a cycle at play

00:00:27.460 | where you discover things about the universe

00:00:30.520 | that tell you what math will be useful.

00:00:33.520 | And that math itself is invented in a sense,

00:00:37.600 | but of all the possible maths that you could have invented,

00:00:40.960 | it's discoveries about the world

00:00:42.280 | that tell you which ones are.

00:00:43.740 | So like a good example here is the Pythagorean theorem.

00:00:47.280 | When you look at this, do you think of that as a definition

00:00:49.280 | or do you think of that as a discovery?

00:00:51.480 | - From the historical perspective, right, it's a discovery

00:00:53.840 | because they were,

00:00:54.680 | but that's probably because they were using physical object

00:00:59.160 | to build their intuition.

00:01:00.700 | And from that intuition came the mathematics.

00:01:04.040 | So the mathematics was in some abstract world

00:01:06.880 | detached from the physics.

00:01:08.600 | But I think more and more math has become detached from,

00:01:12.800 | when you even look at modern physics,

00:01:16.160 | from string theory to even general relativity,

00:01:19.080 | I mean, all math behind the 20th and 21st century physics,

00:01:23.080 | kind of takes a brisk walk outside

00:01:26.720 | of what our mind can actually even comprehend.

00:01:30.000 | In multiple dimensions, for example,

00:01:31.880 | anything beyond three dimensions, maybe four dimensions.

00:01:35.400 | - No, no, no, higher dimensions

00:01:36.800 | can be highly, highly applicable.

00:01:38.260 | I think this is a common misinterpretation

00:01:40.760 | that if you're asking questions

00:01:42.840 | about like a five-dimensional manifold,

00:01:44.760 | that the only way that that's connected

00:01:46.220 | to the physical world is if the physical world

00:01:48.640 | is itself a five-dimensional manifold or includes them.

00:01:52.480 | - Well, wait, wait, wait a minute, wait a minute.

00:01:54.760 | You're telling me you can imagine

00:01:56.720 | a five-dimensional manifold?

00:02:00.760 | - No, no, that's not what I said.

00:02:02.880 | I would make the claim that it is useful

00:02:04.520 | to a three-dimensional physical universe,

00:02:06.740 | despite itself not being three-dimensional.

00:02:08.940 | - So it's useful, meaning to even understand

00:02:10.720 | a three-dimensional world, it'd be useful

00:02:12.800 | to have five-dimensional manifolds.

00:02:14.440 | - Yes, absolutely, because of state spaces.

00:02:16.680 | - But you're saying there, in some deep way,

00:02:19.300 | for us humans, it does always come back

00:02:21.960 | to that three-dimensional world,

00:02:23.560 | for the usefulness of the three-dimensional world,

00:02:26.120 | and therefore it starts with a discovery,

00:02:29.480 | but then we invent the mathematics

00:02:31.560 | that helps us make sense of the discovery, in a sense.

00:02:35.760 | - Yes, I mean, just to jump off

00:02:37.400 | of the Pythagorean theorem example,

00:02:39.340 | it feels like a discovery.

00:02:40.720 | You've got these beautiful geometric proofs

00:02:42.400 | where you've got squares and you're modifying the areas.

00:02:44.140 | It feels like a discovery.

00:02:46.240 | If you look at how we formalize the idea of 2D space

00:02:49.120 | as being R2, all pairs of real numbers,

00:02:52.600 | and how we define a metric on it and define distance,

00:02:55.240 | you're like, "Hang on a second, we've defined distance

00:02:57.520 | "so that the Pythagorean theorem is true,

00:02:59.520 | "so that suddenly it doesn't feel that great."

00:03:01.960 | But I think what's going on is the thing that informed us

00:03:04.920 | what metric to put on R2, to put on our abstract

00:03:09.120 | representation of 2D space, came from physical observations.

00:03:12.800 | And the thing is, there's other metrics

00:03:14.120 | you could have put on it.

00:03:14.960 | We could have consistent math with other notions of distance,

00:03:18.640 | it's just that those pieces of math wouldn't be applicable

00:03:21.360 | to the physical world that we study,

00:03:23.140 | 'cause they're not the ones

00:03:24.080 | where the Pythagorean theorem holds.

00:03:25.680 | So we have a discovery, a genuine bonafide discovery

00:03:28.540 | that informed the invention,

00:03:30.000 | the invention of an abstract representation of 2D space

00:03:33.200 | that we call R2 and things like that.

00:03:35.720 | And then from there, you just study R2 as an abstract thing

00:03:39.220 | that brings about more ideas and inventions and mysteries,

00:03:42.000 | which themselves might yield discoveries.

00:03:43.920 | Those discoveries might give you insight

00:03:46.480 | as to what else would be useful to invent,

00:03:48.880 | and it kind of feeds on itself that way.

00:03:50.480 | That's how I think about it.

00:03:51.680 | So it's not an either/or.

00:03:53.640 | It's not that math is one of these or it's one of the others.

00:03:56.280 | At different times, it's playing a different role.

00:03:58.680 | - So then let me ask the Richard Feynman question,

00:04:02.600 | then, along that thread.

00:04:05.720 | Is what do you think is the difference

00:04:06.880 | between physics and math?

00:04:09.820 | There's a giant overlap.

00:04:12.320 | There's a kind of intuition that physicists have

00:04:15.460 | about the world that's perhaps outside of mathematics.

00:04:18.560 | It's this mysterious art that they seem to possess,

00:04:22.200 | we humans generally possess.

00:04:23.680 | And then there's the beautiful rigor of mathematics

00:04:27.560 | that allows you to, I mean, just like as we were saying,

00:04:31.340 | invent frameworks of understanding our physical world.

00:04:37.360 | So what do you think is the difference there,

00:04:40.200 | and how big is it?

00:04:41.380 | - Well, I think of math as being the study

00:04:43.000 | of abstractions over patterns and pure patterns in logic.

00:04:46.880 | And then physics is obviously grounded

00:04:48.820 | in a desire to understand the world that we live in.

00:04:52.200 | I think you're gonna get very different answers

00:04:53.560 | when you talk to different mathematicians,

00:04:55.100 | 'cause there's a wide diversity in types of mathematicians.

00:04:57.640 | There are some who are motivated very much by pure puzzles.

00:05:01.000 | They might be turned on by things like combinatorics.

00:05:03.600 | And they just love the idea of building up a set

00:05:06.220 | of problem-solving tools applying to pure patterns.

00:05:10.420 | There are some who are very physically motivated,

00:05:12.840 | who try to invent new math or discover math in veins

00:05:17.840 | that they know will have applications to physics

00:05:20.520 | or sometimes computer science.

00:05:21.840 | And that's what drives them.

00:05:23.240 | Like chaos theory is a good example of something

00:05:25.080 | that's pure math, that's purely mathematical,

00:05:27.200 | a lot of the statements being made.

00:05:28.720 | But it's heavily motivated by specific applications

00:05:32.400 | to largely physics.

00:05:34.880 | And then you have a type of mathematician

00:05:36.220 | who just loves abstraction.

00:05:38.060 | They just love pulling it to the more

00:05:39.440 | and more abstract things, the things that feel powerful.

00:05:41.680 | These are the ones that initially invented topology

00:05:44.760 | and then later on get really into category theory

00:05:47.060 | and go on about infinite categories and whatnot.

00:05:49.940 | These are the ones that love to have a system

00:05:53.000 | that can describe truths about as many things as possible.

00:05:56.720 | People from those three different veins

00:06:00.320 | of motivation into math are gonna give you

00:06:01.680 | very different answers about what the relation

00:06:03.240 | at play here is.

00:06:04.280 | 'Cause someone like Vladimir Arnold,

00:06:07.200 | who has written a lot of great books,

00:06:10.100 | many about differential equations and such,

00:06:12.040 | he would say, "Math is a branch of physics."

00:06:15.220 | That's how he would think about it.

00:06:16.720 | And of course he was studying differential equations

00:06:18.720 | related things because that is the motivator

00:06:20.600 | behind the study of PDEs and things like that.

00:06:23.340 | But you'll have others who,

00:06:26.040 | like especially the category theorists,

00:06:27.800 | who aren't really thinking about physics necessarily.

00:06:30.920 | It's all about abstraction and the power of generality.

00:06:34.080 | And it's more of a happy coincidence

00:06:35.980 | that that ends up being useful

00:06:37.880 | for understanding the world we live in.

00:06:40.420 | And then you can get into like, why is that the case?

00:06:42.380 | It's sort of surprising that that which is

00:06:45.740 | about pure puzzles and abstraction

00:06:47.380 | also happens to describe the very fundamentals

00:06:50.580 | of quarks and everything else.

00:06:52.680 | - So why do you think the fundamentals of quarks

00:06:57.380 | and the nature of reality is so compressible

00:07:02.780 | into clean, beautiful equations

00:07:04.860 | that are for the most part simple, relatively speaking?

00:07:08.820 | A lot simpler than they could be.

00:07:11.220 | So you have, we mentioned somebody like Stephen Wolfram

00:07:14.900 | who thinks that sort of there's incredibly simple rules

00:07:19.900 | underlying our reality,

00:07:21.420 | but it can create arbitrary complexity.

00:07:24.460 | But there is simple equations.

00:07:26.280 | What, I'm asking a million questions

00:07:28.740 | that nobody knows the answer to.

00:07:30.180 | - Yeah, I have no idea.

00:07:31.580 | (laughing)

00:07:32.740 | Why is it simple?

00:07:34.780 | - It could be the case that

00:07:36.620 | there's like a filtration at play.

00:07:38.020 | The only things that physicists find interesting

00:07:40.140 | are the ones that are simple enough

00:07:41.220 | they could describe it mathematically.

00:07:42.820 | But as soon as it's a sufficiently complex system,

00:07:44.660 | like, oh, that's outside the realm of physics.

00:07:46.460 | That's biology or whatever have you.

00:07:48.860 | And of course-- - That's true.

00:07:50.980 | - You know, maybe there's something where it's like,

00:07:52.220 | of course there will always be some thing that is simple

00:07:56.060 | when you wash away the like non-important parts

00:08:00.900 | of whatever it is that you're studying.

00:08:02.940 | Just from like an information theory standpoint,

00:08:04.680 | there might be some like,

00:08:06.180 | you get to the lowest information component of it.

00:08:08.960 | But I don't know, maybe I'm just having

00:08:10.420 | a really hard time conceiving of what it would even mean

00:08:12.500 | for the fundamental laws to be like intrinsically complicated

00:08:16.860 | like some set of equations

00:08:20.140 | that you can't decouple from each other.

00:08:22.060 | - Well, no, it could be that sort of we take for granted

00:08:26.300 | that the laws of physics, for example,

00:08:29.500 | are for the most part the same everywhere

00:08:33.000 | or something like that, right?

00:08:34.860 | As opposed to the sort of an alternative could be

00:08:39.860 | that the rules under which the world operates

00:08:44.940 | is different everywhere.

00:08:46.780 | It's like a deeply distributed system

00:08:49.860 | where just everything is just chaos.

00:08:51.900 | Like not in a strict definition of chaos,

00:08:55.060 | but meaning like just it's impossible for equations

00:08:59.940 | to capture, for to explicitly model the world

00:09:03.560 | as cleanly as the physical does.

00:09:05.560 | I mean, we almost take it for granted that we can describe,

00:09:08.640 | we can have an equation for gravity,

00:09:10.800 | for action at a distance.

00:09:12.340 | We can have equations for some of these basic ways

00:09:15.040 | the planets move and just the low level

00:09:20.040 | at the atomic scale, how the materials operate

00:09:23.560 | at the high scale, how black holes operate.

00:09:26.520 | But it doesn't, it seems like it could be,

00:09:29.360 | there's infinite other possibilities

00:09:31.240 | where none of it could be compressible into such equations.

00:09:34.600 | It just seems beautiful.

00:09:36.080 | It's also weird, probably to the point you were making,

00:09:40.440 | that it's very pleasant that this is true for our minds.

00:09:44.680 | So it might be that our minds are biased

00:09:46.680 | to just be looking at the parts of the universe

00:09:49.180 | that are compressible.

00:09:51.280 | And then we can publish papers on

00:09:53.240 | and have nice E equals mc squared equations.

00:09:56.000 | - Right.

00:09:56.840 | - So I wonder, would such a world with uncompressible laws

00:10:01.280 | allow for the kind of beings that can think about

00:10:04.720 | the kind of questions that you're asking?

00:10:07.360 | - That's true.

00:10:08.180 | - Right, like an anthropic principle coming into play

00:10:10.160 | in some weird way here?

00:10:12.160 | I don't know, I don't know what I'm talking about at all.

00:10:14.360 | - Or maybe the universe is actually not so compressible,

00:10:17.580 | but the way our brain evolved,

00:10:22.120 | we're only able to perceive the compressible parts.

00:10:25.440 | I mean, we are, so this is a sort of Chomsky argument.

00:10:27.960 | We are just descendants of apes.

00:10:29.440 | We're like really limited biological systems.

00:10:33.160 | So it totally makes sense that we're really limited

00:10:35.960 | little computers, calculators,

00:10:38.160 | that are able to perceive certain kinds of things.

00:10:39.840 | And the actual world is much more complicated.

00:10:42.720 | - Well, but we can do pretty awesome things, right?

00:10:46.240 | Like we can fly spaceships.

00:10:47.880 | And we have to have some connection of reality

00:10:51.160 | to be able to take our potentially oversimplified models

00:10:54.880 | of the world, but then actually twist the world

00:10:57.360 | to our will based on it.

00:10:58.680 | So we have certain reality checks

00:11:00.000 | that physics isn't too far afield,

00:11:03.000 | simply based on what we can do.

00:11:05.040 | - Yeah, the fact that we can fly is pretty good.

00:11:06.880 | - It's great, yeah.

00:11:08.000 | It's a pretty good proof of concept

00:11:10.120 | that the laws we're working with are working well.

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

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