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

Duration: 11:34 1976 transcript words EN summary

Video summary

Grant Sanderson explores the enduring question of whether mathematics is discovered or invented, proposing instead a cyclical relationship between human invention and universal discovery. He argues that while humans invent mathematical frameworks to make sense of observations from the physical world, these inventions are constrained by genuine discoveries about how reality functions. A prime example he uses is the Pythagorean theorem; historically viewed as a discovery because it was derived from observing triangles in nature, Sanderson suggests this intuition comes from interacting with physical objects. However, modern mathematics often detaches itself from physics to explore abstract concepts like string theory or general relativity that exist beyond our immediate three-dimensional comprehension. Despite these high-level abstractions involving multiple dimensions, their utility ultimately circles back to describing a fundamentally three-dimensional universe, suggesting that while the mathematical tools are invented, they must align with discovered truths about reality to be useful. The conversation delves into how we formalize abstract concepts, such as defining two-dimensional space as $\mathbb{R}^2$ and establishing specific metrics for distance. Sanderson points out a crucial nuance: if one were to define the metric differently, the Pythagorean theorem would no longer hold true in that system. This implies that our choice of mathematical structure is informed by physical discovery regarding which types of math actually apply to our world; other consistent but non-applicable metrics exist as purely invented constructs without connection to observable reality. Once a useful framework like $\mathbb{R}^2$ is established based on these discoveries, the field opens up for further invention and exploration, creating a self-sustaining loop where new inventions lead to fresh mysteries that may yield additional discoveries about what else should be invented or studied. Sanderson then addresses Richard Feynman's famous question regarding the distinction between physics and mathematics, noting that mathematicians hold diverse motivations that shape their perspective on this overlap. Some are driven by pure puzzles in areas like combinatorics, others seek applications in fields such as computer science or chaos theory, while a third group pursues extreme abstraction for its own sake, leading to developments in topology and category theory. Notable figures illustrate these divergent views; Vladimir Arnold famously considered mathematics a branch of physics motivated by differential equations, whereas many category theorists focus on the power of generality without necessarily thinking about physical applications at all. Sanderson suggests that it is somewhat surprising—and perhaps coincidental—that pure abstractions developed for intellectual satisfaction end up describing fundamental realities like quarks so accurately. This apparent simplicity and compressibility of nature's laws leads to discussions about whether reality itself is inherently simple or if our perception is limited by biological constraints similar to Noam Chomsky's arguments regarding language acquisition. Sanderson speculates that the universe might not be as clean or equation-friendly as it appears, but rather that human brains have evolved specifically to perceive and model only those parts of reality that are compressible into elegant equations like $E=mc^2$. He considers the possibility that a world governed by uncompressible laws would prevent beings from forming the coherent thoughts required to ask such questions. Ultimately, he concludes with an anthropic perspective: we can manipulate our oversimplified models effectively because they connect sufficiently well to reality, allowing us to build spaceships and understand gravity, even if those models are merely capturing a filtered slice of a potentially far more complex universe that only allows for certain kinds of intelligent life to exist within it.

Read the full video transcript

do you think math is discovered or invented so we were talking about the different kind of mathematics that could be developed by the alien species the implied question is is yeah it's math discovered or invented is you know it's 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 its 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 Ray's a discovery because there 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 person is some abstract world detached from physics but I think more and more math has become detached from you know we wouldn't even look at modern physics from string theory so 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 three dimensions maybe four dimensions know how your dimensions can be highly highly applicable I think this is a common misinterpretation the if you're asking questions about like 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 wait wait wait a minute wait a minute you're telling me you can imagine a five dimensional manifold no no that's not what I said I I'm 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 meaningful even understand a three dimensional world would be useful to have five dimensional manifold yes absolutely because of state spaces but you're saying there in some in some deep way for us humans it does it does always come back to that three-dimensional world for the useful usefulness at 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 yes I mean just to jump off of the Pythagorean theorem it feels like a discovery you've got these beautiful geometric proofs where you've got squares and you're modifying there is 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 no wait hang on a second we've defined distance so that the Pythagorean theorem is true so that 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 is 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 one of these or it's one of the others at different times it's playing a different role so then let me ask the the Richard Fineman question then along that thread is what do you think is a 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 mathema Erick's that allows you to mean just like as we were saying invent frameworks of understanding our physical wall so what do you think is the difference there and how big is it well I think of math as being the study of like 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 yeah I think you're going to get very different answers when you talk to different mathematicians because there's a wide diversity and 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 who tried 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 all right like chaos theory is a good example of something that 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 three different veins of motivation and demand 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 why is that the case that'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 what do you think the fundamentals of quarks and and the nature of reality is so compressible and too 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 no idea why is it simple I it could be the case that there's like a filter iteration I played the only things that physicists find interesting other ones little simple enough they could describe it mathematically but as soon as it's a sufficiently complex system like now 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 words 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 unlike an information theory standpoint there might be some like you you get to the lowest information component of it but I don't know it 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 it could be that it's sort of we take for granted that they're the the laws of physics for example are for the most part the same everywhere or something like that right as opposed to the sort of an alternative could be that the rules under which are the world operates is different everywhere it's like a like a deeply distributed system or just everything is just chaos like not not in a string definition of caste but meeting like just it's impossible for equations to capture for to explicitly model the world as cleanly as the physical does any we're almost taking it for granted that we can describe we can have an equation for gravity mm-hm for action at a distance we can have equations for some of these basic ways the planets moving just the low level at the atomic scale all the materials operate at the high scale how black holes operate but it doesn't it it seems like it could be there's infinite other possibilities where none of it could be compressible into such equations so it just seems beautiful it's also weird probably to the point you're making that it's very pleasant that this is true for our minds right so 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 mc-squared equations 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 it's some weird way here I don't know like I don't know what I'm talking about it or maybe the universe is actually not so compressible but the way our brain the the way our brain evolved were only able to perceive the compressible parts I mean we are so this is a sort of Chomsky argument we are just the sentence of apes over like really limited biological systems so it totally makes sense there were really limited little computers calculators that are able to perceive certain kinds of things in the actual world is much more complicated well but we can we can do pretty awesome things right like we can fly spaceships and that 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 like physics isn't too far afield simply based on what we can do and the fact that we can fly is pretty good it's great yes like and lambic on such a bit the laws were working with our are working well you