Grant Sanderson: 3Blue1Brown and the Beauty of Mathematics | Lex Fridman Podcast #64
the following is a conversation with grant Sanderson he's a math educator and creator of three blue one brown a popular YouTube channel that uses programmatically animated visualizations to explain concepts and linear algebra calculus and other fields of mathematics this is the artificial intelligence podcast if you enjoy it subscribe on YouTube give it five stars an apple podcast follow on Spotify support on patreon or simply connect with me on Twitter and Lex Friedman spelled Fri D ma n I recently started doing ads at the end of the introduction I'll do one or two minutes after introducing the episode and never any ads in the middle that can break the flow of the conversation I hope that works for you and doesn't hurt the listening experience this show is presented by cash app the number one finance app in the App Store I personally use cash app to send money to friends but you can also use it to buy sell and deposit Bitcoin in just seconds cash app also has an investing feature you 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obvious sense the notation is different right I think notation can guide what the math itself is I think it has everything to do with the form of their existence right do you think they have basic arithmetic sorry interrupt yeah so I think they count right I think notions like 1 2 3 the natural numbers that's extremely well natural that's almost why we put that name to it as soon as you can count you have a notion of repetition right because you can count by two two times or three times and so you have this notion of repeating the idea of counting which brings you addition and multiplication I think the way that we extend to the real numbers there's a little bit of choice in that so there's this funny number system called the serial numbers that it captures the idea of continuity it's a distinct mathematical object you could very well you know model the universe and motion of planets with that as the backend of your math right and you still have kind of the same interface with the front end of what physical laws you're trying to or what physical phenomena you're trying to describe with math and I wonder if the little glimpses that we have of what choices you can make along the way based on what different mathematicians have brought to the table is just scratching the surface surface of what the different possibilities are if you have a completely different mode of thought right or a mode of interacting with the universe and you think notation is the key part of the journey that we've taken through math I think that's the most salient part that you'd notice at first I think the mode of thought is going to influence things more than like the notation itself but notation actually carries a lot of weight when it comes to how we think about things more so than we usually give it credit for I would I would be comfortable saying give a favor or least favorite piece of notation in terms of its effectiveness yes yeah well so at least favorite one that I've been thinking a lot about that will be a video I don't know when but we'll see the number e we write the function e to the X this general exponential function with a notation e to the X that implies you should think about a particular number this constant of nature you repeatedly multiply it by itself and then you say well what's e to the square root of two and you're like oh well we've extended the idea of repeated multiplication that's and that's all nice that's all nice and well but very famously you have like e to the PI you're like well we're extending the idea of repeated multiplication into the complex numbers yeah you can think about it that way in reality I think that it's just the wrong way of notationally representing this function the exponential function which itself could be represented a number of different ways you can think about it in terms of the problem it solves a certain very simple differential equation which often yields way more insight than trying to twist to the idea of repeated multiplication like take its arm and put it behind its back and throw it on the desk and be like you will apply to complex numbers right that's not I don't think that's pedagogically helpful and so the repeater multiplication is actually missing the main point the power of e to the S I mean what it addresses is things where the rate at which something changes depends on its own value but more specifically it depends on it linearly so for example if you have like a population that's growing and the rate at which it grows depends on how many members of the population are already there it looks like this nice exponential curve it makes sense to talk about repeated multiplication because you say how much is there after one year two years three years you're multiplying by something the relationship can be a little bit different sometimes where let's say you've got a ball on a string like a like a game of tetherball going around a rope right and you say it's velocity is always perpendicular to its position that's another way of describing its rate of change is being related to where it is but it's a different operation you're not scaling it it's a rotation it's this 90 degree rotation that's what the whole idea of like complex exponentiation is trying to capture but it's obfuscated in the notation when what it's actually saying like if you really parse something like e to the PI I what it's saying is choose an origin always move perpendicular to the vector from that origin to you okay then when you walk PI times that radius you'll be halfway around like that's what it's saying it's kind of the u-turn 90 degrees and you walk you'll be going in a circle that's the phenomenon that it's describing but trying to twist the idea of repeatedly multiplying a constant into that like I I can't even think of the number of human hours of like intelligent human hours that have been wasted trying to parse that to their own liking and desire among like scientists or electrical engineers if students have we were which if the notation were a little different or the way that this whole function was introduced from the get-go were framed differently I think could have been avoided right and you're talking about the most beautiful equation in mathematics but it's still pretty mysterious isn't it like you're making it seem like it's a notational it's not mysterious I think I think the notation makes it mysterious I don't think it's I think the fact that it represents it's pretty it's not like the most beautiful thing in the world but it's quite pretty the idea that if you take the linear operation of a 90 degree rotation and then you do this general exponentiation thing to it that what you get are all the other kinds of rotation which is basically to say if you if your velocity vector is perpendicular to your position vector you walk in a circle that's pretty it's not the most beautiful thing in the world but it's quite pretty the beauty of it I think comes from perhaps the awkwardness of the notation somehow still nevertheless coming together nicely because you have like several disciplines coming together in a single equation well in a sense like historically speaking but that's true you've got so like the number E is significant like it shows up in probability all the time it like shows up in calculus all the time it is significant you're seeing is sort of mated with PI this geometric constant and I like the imaginary number and such I think what's really happening there is the the way that a shows up is when you have things like exponential growth and decay right it's when this relation that that something's rate of change has to itself is a simple scaling right a similar law also describes circular motion because we have bad notation we use the residue of how it shows up in the context of self-reinforcing growth like a population growing or compound interest the constant associated with that is awkwardly placed into the context of how rotation comes about because they both come from pretty similar equations and so what we see is the e and the pi juxtaposed a little bit closer than they would be with a purely natural representation I would think here's how I would describe the relation between the two you've got a very important function we might call X that's like the exponential function when you plug in one you get this nice constant called EE that shows up in like probability and calculus if you try to move in the imaginary direction it's periodic and the period is tau so those are these two constants associated with this the same central function but for kind of unrelated reasons right and not unrelated but like orthogonal reasons one of them is what happens when you're moving in the real direction one's what happens when you move in the imaginary direction and like yeah those are related they're not as related as the famous equation seems to think it is it's sort of putting all of the children in one bed and they kind of like to sleep in separate beds if they have the choice but you see them all there and you know there is a family resemblance but it's not that close so actually think of it as a function is uh this is the better idea and that's a notational idea and yeah and like here's the thing the constant e sort of stands is this numerical representative of calculus right yeah calculus is the like study of change mm-hmm so it's very at least there's a little cognitive dissonance using a constant to represent the science of change never thought of it that way yeah yeah it makes sense why the notation came about that way yes because this is the first way that we saw it um in the context of things like population growth or compound interest it is nicer to think about as repeated multiplication that's definitely nicer but it's more that that's the first application of what turned out to be a much more general function that maybe the intelligent life your initial question asked about would have come to recognize as being much more significant than the single use case which lends itself to repeated multiplication notation but let me jump back for a second to aliens and the nature of our universe okay 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 is yeah it's math discovered or invented is you know 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 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 was some abstract world detached from physics but I think more and more math has become detached from you know we when you 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 no higher 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 it would be useful to have five dimensional manifolds 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 that 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 okay 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 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 rigour of mathematics that allows you to I mean just like what 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 well I think of math as being the study of like abstractions over patterns and pure 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 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 into math are going to give you very different answers about what the relation at play here is because someone like flightmare 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 well 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 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 all right you know maybe there's something what'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 some like an information theory standpoint there might be some like you 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 it could be that 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 strict definition of cast but meaning 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 take it for granted that we can describe we can have an equation for gravity for action in a distance we can have equations for some of these basic ways the planets moving just the 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 equation so it just seems beautiful it's also weird probably to the point you're making that it's very pleasant that this is just 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 at 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 totally make 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 the laws were working with our are working well so I mentioned to the internet that I'm talking to you and so the internet gave some questions so I apologize for these but do you think we're living in a simulation that the universe is a computer or the universe is the computation running a computer it's conceivable what I don't buy is you know you'll have the argument that well let's say that it was the case that you can have simulations then the simulated world would itself eventually get to a point where it's running simulations yes and then the the second layer down would create a third layer down and on and on and on so probabilistically you just throw a dart at one of those layers we're probably in one of the simulated layers I think if there's some sort of limitations unlike the information processing of whatever the physical world is like it quickly becomes the case that you have a limit to the layers that could exist there because like the resources necessary to simulate a universe like ours clearly is a lot just in terms of the number of bits at play and so then you can ask well what's more plausible that there's an unbounded capacity of information processing in whatever they like highest up level universe is or that there's some bound to that capacity which then limits like the number of levels available how do you play some kind of probability distribution on like what the information capacity is I have no idea but I I don't mean like people almost assume a certain uniform probability over all of those metal layers that could conceivably exist when it's a little bit like a Pascal's wager on like you're not giving a low enough prior to the mere existence of that infinite set of layers yeah that's true but it's also very difficult to contextualize the amount so the amount of information processing power required to simulate like our universe seems like amazingly huge what you can always raise two to the power of that exactly yeah like numbers bit big and we're easily humbled but basically everything around us so it's very difficult to to kind of make sense of anything actually when you look up at the sky and look at the stars in the immensity of it all to make sense of us the smallness of us the unlikeliness of everything that's on this earth coming to be then you could basically anything could be all laws of probability go out the window to me because I guess because the amount of information under which we're operating is very low we basically know nothing about the world around us relatively speaking and so so when I think about the simulation hypothesis I think is just fun to think about it but it's also I think there is a thought experiment kind of interesting to think of the power of computation where there are the limits of a Turing machine sort of the limits of our current computers when you start to think about artificial intelligence how far can we get with computers and that's kind of where the simulation hypothesis useless me as a thought experiment is is the universe just the computer is it just the computation is all of this just the computation and so the same kind of tools we apply to analyzing algorithms can that be applied you know if we scale further and further and further well the arbitrary power of those systems start to create some interesting aspects that we see in our universe or something fundamentally different needs to be created well it's interesting that in our universe it's not arbitrarily large the power that you can place limits on for example how many bits of information can be stored per unit area right like all of the physical laws we've got general relativity and like quantum coming together to give you a certain limit on how many bits you can store within a given range before it collapses into a black hole like the idea that there even exists such a limit is that the very least thought-provoking when naively you might assume oh well you know technology could always get better and better we could get cleverer and cleverer and you could just cram as much information as you want into like a small unit of space that makes me think it's at least plausible that whatever the highest level of existence is doesn't admit too many simulations or ones that are at the scale of complexity that we're looking at obviously it's just as conceivable that they do and that there are many but I I guess what I'm channeling is the surprise that I felt learning that fact that there are the information is physical in this way is that there's a finance to it okay let me just even go off on that from mathematics perspective and the psychology perspective how do you mix are you psychologically comfortable with the concept of infinity I think so are you okay with it I'm pretty okay yeah okay no not really it doesn't make any sense to me I don't know like how many how many words how many possible words do you think could exist that are just like strings of letters so that that's a sort of mathematical statement as beautiful and we use infinity basically everything we do everything we do inside in science math and engineering yes but you said exist my the question is well you said letters of words I said words words the to bring words into existence to me you have to start like saying them or like writing them or like listing them that's an instantiation okay combination how many abstract words exist it was the idea of abstract yeah the the idea of abstract notions and ideas I think we should be clear around terminology I mean you think about intelligence a lot like artificial intelligence would you not say that what it's doing is a kind of abstraction they're like abstraction is key to conceptualizing the universe you get this raw sensory data you need I need something that every time you move your face a little bit and the they're not pixels but like analog of pixels on my retina change entirely yeah that I can still have some coherent notion of this is Lex and planet Lex yes right what that requires is you have a disparate set of possible images hitting me that are unified in a notion of Lex yeah right that's a kind of abstraction it's a thing that could apply to a lot of different images that I see and it represents it in a much more compressed way and one that's like much more resilient to that I think in the same way if I'm talking about infinity as an abstraction I don't mean non-physical woowoo it like ineffable or something what I mean is it something that can apply to a multiplicity of situations that share certain common attribute in the same way that the images of like your face on my retina Sharon common attributes that I can put the single notion to it like in that way infinity is an abstraction and it's very powerful and and it's a it's only through such abstraction that we can actually understand like the world and logic and things and in the case of infinity the way I think about it the key entity is the property of always being able to add one more like no matter how many words you can list you just throw an A at the end of one and you have another conceivable word you don't have to think of all the words at once it's that property the oh I could always add one more that gives it this nature of infinite enos in the same way that they're certain like properties of your face that give it the Lexx miss right so like infinity should be no more worrying than the I can always add one more sentiment that's a really elegant much more elegant way than I could put it so thank you for doing that as yet another abstraction and yes indeed that's what our brain does that's what intelligence systems do this what programming does that's what science does is build abstraction on top of each other and yet there is at a certain point abstractions that go into the quote whoo right sort of and because we're now it's like it's like we built this stack of you know the the only thing that's true is the stuff that's on the ground everything else is useful for interpreting this and at a certain point you might start floating into ideas that are surreal and difficult and and take us into areas that are disconnected from reality in a way that we could never give back what if instead of calling these abstract how different would it be in your mind if we call them general and the phenomenon that you're describing is over generalization when you try them channelization yeah have a concept or an idea that's so general as to apply to nothing in particular in a useful way does that map to what you're thinking of when you think of first of all I'm playing a little just for the fun of it yeah and that devil's advocate and uh I I think our cognition our mind is unable to visualize so you do some incredible work with visualization and video I think infinity is very difficult to visualize for our mind we can delude ourselves into thinking we can visualize it but we can't I don't that means I don't I would venture to say it's very difficult and so there's some concepts of mathematics like maybe multiple dimensions we could sort of talk about that are impossible for us to truly into it like and it just feels dangerous to me to use these as part of our toolbox of abstractions on behalf of your listeners I almost fear we're getting too philosophical oh no I I think to that point for any particular idea like this there's multiple angles of attack I think the when we do visualize infinity what we're actually doing you know you write dot dot dot one two three four dot dot right that's those are symbols on the page that are insinuating a certain infinity what you're capturing with a little bit of design there is the I can always add one more property right I think I'm just as uncomfortable with you are if you try to concretize it so much that you have a bag of infinitely many things that I actually think of no not one two three four dot dot dot one two three four five six seven eight I try to get them all and I had and you realize oh I you know your your brain would literally collapse into a black hole all of that and and I honestly feel this with a lot of math that I tried to read where I I don't think of myself as like particularly good at math in some ways like I get very confused often when I am going through some of these texts and often when I'm feeling my head is like this is just so damn have strict I just can't wrap my head around I just wanted to put something concrete to it that makes me understand and I think a lot of the motivation for the channel is channeling that sentiment of yeah a lot of the things that you're trying to read out there it's just so hard to connect to anything that you spend an hour banging your head against a couple of pages and you come out not really knowing anything more other than some definitions maybe and a certain sense of self defeat right one of the reasons I focus so much on visualizations is that I'm a big believer in I'm sorry I'm just really hampering out this idea of abstraction being clear about your layers of abstraction yes right it's always tempting to start an explanation from the top to the bottom yeh you you give the definition of a new theorem you're like this is the definition of a vector space for example we're gonna that's how well start of course these are the properties of a vector space Yuma first from these properties we will derive what we need in order to do the math of linear algebra or whatever it might be I don't think that's how understanding works at all I think how understanding works is you start at the lowest level you can get it where rather than thinking about a vector space you might think of concrete vectors that are just lists of numbers or picturing it as like an arrow that you draw which is itself like even less abstract the numbers because you're looking at quantities like the distance of the x-coordinate the distance of the y-coordinate it's as concrete as you could possibly get and it has to be if you're putting it in a visual right like that it's an actual arrow it's an actual vector you're not talking about like a quote-unquote vector that could apply to any possible thing you have to choose one if you're illustrating it and I think this is the power of being in a medium like video or if you're writing a textbook and you force yourself to put a lot of images is with every image you're making a choice with each choice you're showing a concrete example with each concrete example you're eating someone's path to understanding you know I'm sorry to interrupt you but you just made me realize that that's exactly right so the visualization is you're creating while you're sometimes talking about abstractions the actual visualization is a explicit low-level example yes so there there's an actual like in the code you have to say what the what the vector is what's the direction of the arrow what's the magnitude of the yeah so that's you're going the visualization itself is actually going to the bottom I think and I think that's very important I also think about this a lot in writing scripts where even before you get to the visuals the first instinct is to I don't know why I just always do I say the abstract thing I say the general definition the powerful thing and then I fill it in with examples later always it will be more compelling and easier to understand when you flip that and instead you let someone's brain do the pattern recognition you just show them a bunch of examples the brain is gonna feel a certain similarity between them then by the time you bring in the definition or by the time you bring in the formula its articulating a thing that's already in the brain that was built off of looking at a bunch of examples with a certain kind of similarity and what the formula does is articulate what that kind of similarity is rather than being a high cognitive load set of symbols that needs to be populated with examples later on assuming someone still with you what is the most beautiful or on inspiring idea you've come across in mathematics I don't know maybe it's an ID you've explored in your videos maybe not what mike just gave you pause it's the most beautiful idea small or big so I think often the things that are most beautiful are the ones that you have like a little bit of understanding of but certainly not an entire understanding it's a little bit of that mystery that is what makes it beautiful almost a moment of the discovery for you personally almost just that leap of haha moment so something that really caught my eye I remember when I was little there were these I come I think the series was called like wooden books or something these tiny little books that would have just a very short description of something on the left and then a picture on the right I don't know who they're meant for but maybe it's like loosely children or something like that but it can't just be children because of some of the things I was describing on the last page of one of them some were tiny in there was this little formula the on the left hand had a sum over all of the natural numbers you know it's like 1 over 1 to the s plus 1 over 2 to the s plus 1 over 3 to the s on and on to the infinity then on the other side had a product over all of the primes and it was a certain thing I had to do with all the primes and like any good young math enthusiast I'd probably been indoctrinated with how chaotic and confusing the primes are which they are and seeing this equation where on one site you have something that's as understandable as you could possibly get the counting numbers yes and on the other side is all the prime numbers it was like this whoa they're related like this there's there's a simple description that includes like all the primes getting wrapped together this this is like the Euler product for zeta function as I like later found out the equation itself essentially encodes the fundamental theorem of arithmetic that every number can be expressed as a unique set of primes to me still there's I mean I certainly don't understand this equation or this function all that well the more I learn about it the prettier it is the idea that you can this is sort of what gets you representations of primes not in terms of Prime's themselves but in terms of another set of numbers they're like the non-trivial zeros of the zeta function and again I'm very kind of in over my head in a lot of ways as I like try to get to understand it but the more I do its it always leaves enough mystery that it remains very beautiful to me so whenever there's a little bit of mystery just outside of the understanding that and by the way the process of learning more about it how does that come about just your own thought or are you reading reading yes or is the visualization itself revealing more to you visuals help I mean in one time when I was just trying to understand like analytic continuation and playing around with visualizing complex functions this is what led to a video about this function it's titled something like visualizing the riemann zeta function it's one that came about because i was programming and tried to see what a certain thing looked like and then i looked at it like well that's elucidating and then i decided to make a video about it but i mean you try to get your hands on as much reading as you can you you know in this case i think if anyone wants to start to understand it if they have like a math background of some like they studied some in college or something like that like the princeton companion to math has a really good article on analytic number theory and that itself has a whole bunch of references and you know anything has more references and it gives you this like tree to start pawing through and like you know you try to understand I try to understand things visually as I go that's not always possible but it's very helpful when it does you recognize when there's common themes like in this case cousins of the Fourier transform I come into play and you realize oh it's probably pretty important to have deep intuitions of the fourier transform even if it's not explicitly mentioned in like these texts and you try to get a sense of what the common players are but I'll emphasize again like I feel very in over my head when I try to understand the exact relation between like the zeros of the Riemann zeta function and how they relate to the distribution of primes I definitely understand it better than I did a year ago I definitely understand it 1/100 as well as the experts on the matter do I assume but the slow path towards getting theirs it's fun it's charming and like to your question very beautiful and the beauty is in the what in the journey versus the destination well it's that each each thing doesn't feel arbitrary I think that's a big part is that you have these unpredictable none yeah these very unpredictable patterns were these intricate properties of like a certain function but at the same time it doesn't feel like humans ever made an arbitrary choice in studying this particular thing so you know it feels like you're speaking to patterns themselves or nature itself that's a big part of it I think things that are too arbitrary it's just hard for those to feel beautiful because and this is sort of what the word contrived is meant to apply to right and the one they're not arbitrary means it could be you can have a clean abstraction an intuition that allows you to comprehend it well to one of your first questions it makes you feel like if you came across another intelligent civilization that they'd be studying the same thing it may be with different notation but personally yeah but yeah like that's what I think you talk to that other civilization they're probably also studying the zeros of the Riemann zeta function or it's like some variant thereof that is like a clearly equivalent cousin or something like that but that's probably on their on their docket whenever somebody does a lot of something amazing I'm gonna ask the question that that you've already been asked a lot that you'll get more and more asked in your life but what was your favorite video to create Oh favorite to create one of my favorites is the title is who cares about topology you want me to pull it up or not if you want sure yeah it is about well it starts by describing an unsolved problem that still unsolved in math called the inscribed square problem you draw any loop and then you ask are there four points on that loop that make a square totally useless right this is not answering any physical questions it's mostly interesting that we can't answer that question and it seems like such a natural thing to ask now if you weaken it a little bit and you ask can you always find a rectangle you choose four points on this curve can you find a rectangle that's hard but it's doable and the path to it involves things like looking at a torus this surface with a single hole in it like a donut we're looking at a mobius strip in ways that feel so much less contrived to when I first as like a little kid learned about these surfaces and shapes like a mobius strip and a torus like what you learn is oh this mobius strip you take a piece of paper put a twist glue it together and now you have a shape with one edge and just one side and as a student you should think who cares right like how does that help me solve any problems I thought math was about problem solving so what I liked about the piece of math that this was describing that was in this paper by a mathematician named Vaughan was that it arises very naturally it's clear what it represents it's doing something it's not just playing with construction paper and the way that it solves the problem is really beautiful so kind of putting all of that down and concretizing it right like I was talking about how when you have to put visuals to it it demands that what's on screen is a very specific example of what you're describing the cut the construction here is very abstract in nature you describe this very abstract kind of surface in 3d space so then when I was finding myself in this case I wasn't programming I was using Grapher that's like built into OSX for the 3d stuff to draw that surface you realize oh man the topology argument is very non constructive I have to make a lot of you have to do a lot of extra work in order to make the surface show up but then once you see it it's quite pretty very satisfying to see a specific instance of it and you also feel like I've actually added something on top of what the original paper was doing that it shows something that's completely correct it's a very beautiful argument but you don't see what it looks like and I found something satisfying and seeing what it looked like that could only ever have come about from the forcing function of getting some kind of image on the screen to describe the thing I was talking abut your most weren't able to anticipate what its gonna look like I don't know idea I had no idea and it was wonderful right it was totally it looks like a Sydney Opera House or some sort of Frank Gehry design and it was you knew it was gonna be something and you can say various things about it like oh it it touches the curve itself it has a boundary that's this curve on the 2d plane it all sits above the plane but before you actually draw it so it's very unclear what the thing will look like and to see it it's very it's just pleasing right so that was that was fun to make very fun to share I hope that it has elucidated for some people out there where these constructs of topology come from that it's not arbitrary play with construction paper so that's I think this is good a good sort of example to talk a little bit about your process so you have you have a list of ideas so that sort of the the curse of having having an active and brilliant mind is I'm sure you have a list that's growing faster than you can utilize and but there's some sorting procedure depending on mood and interest and so on but okay so pick an idea then you have to try to write a narrative arc it's sort of how do I elucidate how how do I make this idea beautiful and clear and explain it and then there's a set of visualizations that'll be attached to it sort of you've talked about some of this before but sort of writing the story attaching the visualizations can you talk through interesting painful beautiful parts of that process well the most painful is if you've chosen a topic that you do want to do but then it's hard to think of I guess how to structure the script this is sort of where I have been on one for like the last two or three months and I think the ultimately the right resolution just like set it aside and instead do some other things where the script comes more naturally because you sort of don't want to overwork a narrative that the more you've thought about it the less you can empathize with the student who doesn't yet understand the thing you're trying to teach who is the judger in your head sort of the person the creature the essence that's saying this sucks er this is good and you mentioned kind of the student you're you're thinking about um what can you uh who is that what is that thing that's Chris that is the perfections that says this thing sucks you need to work on it in front of the two three months I don't know I think it's my past self I think that's the entity that I'm most trying to empathize with is like you take who I was because it's kind of the only person I know like you don't really know anyone other than versions of yourself so I start with the version of myself that I know who doesn't yet understand the thing right and then I just try to view it with fresh eyes a particular visual or a particular script like is this motivating does this make sense which has its downsides because sometimes I find myself speaking to motivations that only myself would be interested in I don't like it I did this project on quaternions where what I really wanted was to understand what are they doing in four dimensions can we see what they're doing in four dimensions right and I can way of thinking about it that really answered the question in my head that maybe very satisfied and being able to think about concretely with a 3d visual what are they doing to a 4d sphere and some like great this is exactly what my past self would have wanted right and I make a thing on it and I'm sure it's what some other people wanted to it but in hindsight I think most people who want to learn about quaternions are like robotics engineers or graphics programmers who want to understand how they're used to describe 3d rotations and like their use case was actually a little bit different than my past self and in that way like I wouldn't actually recommend that video to people who are coming at it from that angle of wanting to know hey I'm a robotics program or like how do these quarter neon things work to describe position in 3d space I would say other great resources for that if you ever find yourself wanting to say but hang on in what sense are they acting in four dimensions then come back but until then it's a little different yeah it's interesting because so you have incredible videos on your networks for example and for my certain perspectives have probably I mean I looked at the is served my field and I've also looked at the basic introduction of neural networks like a million times from different perspectives and it made me realize that there's a lot of ways to present it so you were sort of you did an incredible job I mean sort of the but you could also do it differently and also incredible like to create a beautiful presentation of a basic concept is requires sort of creativity requires genius and so on but you can take it from a bunch of different perspectives in that video and you'll know which mean you realize that and just as you're saying you kind of have a certain mindset a certain view but from if you take a different view from a physics perspective from a neuroscience perspective talking about neural networks or from robotics perspective or from let's see from a pure learning statistics perspective so you you can create totally different videos and you've done that with a few actually concepts where you've have taken different costs like at the at the at the Euler equation right the you've taken different views of that I think I've made three videos on it and I definitely will make at least one more never enough never enough so you don't think it's the most beautiful equation in mathematics no like I said as we represent it it's one of the most hideous it involves a lot of the most hideous aspects of our notation I talked about II the fact that we use PI instead of tau the fact that we call imaginary numbers imaginary and then and actually wonder if we use the I because of imaginary I don't know if that's historically accurate but at least a lot of people they read the eye and they think imaginary like all three of those facts it's like those are things that have added more confusion than they needed to and we're wrapping them up in one equation like boy that's just very hideous right the idea is that it does tie together when you want away the notation look it's okay it's pretty it's nice but it's not like mind-blowing greatest thing in the universe which is maybe what I was thinking of when I said like once you understand something it doesn't have the same beauty like I feel like I understand Euler's formula and I feel like I understand it enough to sort of see the version that just woke up it hasn't really gotten itself dressed in the morning that's a little bit groggy and there's bags under its eyes so years like it's their past yeah the the the dating stage you know we're no longer dating right instead of dating Bizet des function this head like she's beautiful and right and like we have fun and it's that that high dopamine part for like maybe at some point will settle into the more mundane nature the relationship where I like see her for who she truly is and she'll still be beautiful in her own way but it won't have the same romantic pizzazz right well that's the nice thing about mathematics I think as long as you don't live forever there will always be enough mystery and fun with some of the equations even if you do the rate at which questions comes up is much faster than the rate at which answers come up so if you could live forever would you I think so yeah do you think you don't think mortality is the thing that makes life meaningful would your life be four times as meaningful if you died at 25 so this goes to infinity I think you and I that's really interesting so what I said is infinite not not four times longer mm-hmm I said infinite so the the actual existence of the finiteness the existence of the end no matter the length is the thing that may sort of from my comprehension of psychology it's such a deeply human it's such a fundamental part of the human condition the fact that there is that we're mortal that the the fact that things and they see it seems to be a crucial part of what gives them meaning I don't think at least for me like it's a very small percentage of my time that mortality is salient that I'm like aware of the end of my life what do you mean by me I'm trolling is it the ego is that the aid there's that the super-ego is a so you're the reflective self the Verna Keys area that puts all this stuff into words yeah a small percentage of your mind that is actually aware of the true motivations that drive you but my point is that most of my life I'm not thinking about death but I still feel very motivated to like make things and to like interact with people like experienced love or things like that I'm very motivated and I it's strange that that motivation comes while death is not in my mind at all and this might just be because I'm young enough that it's not salient force in your subconscious or that you were instructed an illusion that allows you to escape the fact of your mortality by enjoying the moment so to the existential approach life would be gun to my head I don't think that's it yeah another another sort of would say gun to the head it's the deep psychological introspection of what drives us I mean that's uh in some ways to me I mean when I look at math when I look at science is the kind of an escape from reality in a sense that it's so beautiful it's such a beautiful journey of discovery that it allows you to actually is it allows you to achieve a kind of immortality of explore ideas and sort of connect yourself to the thing that is seemingly infinite like the universe right that allows you to escape the the limited nature of our little of our bodies of our existence what else would give this podcast meaning that's right if not the fact that it will end this place closes in in 40 minutes and it's so much more meaningful for it how much more I love this room because we'll be kicked out so I understand just because you're trolling me doesn't mean I'm wrong but I take your point I take your point boy that would be a good Twitter bio just because you're trolling me doesn't mean I'm wrong yeah and and sort of difference in backgrounds I'm a bit Russian so we're a bit melancholic and it seemed to maybe assign a little too much value just suffering immortality and things like that make makes for a better novel I think oh yeah you need you need some sort of existential threat yeah to drive a plot so when do you know when the video is done when you're working on it that's pretty easy actually because I'm you know I'll write the script I want there to be some kind of aha moment in there and then hopefully the script can revolve around some kind of aha moment and then from there you know you're putting visuals to each sentence that exists and then you narrate it you edit it all together so given that there's a script the the end becomes quite clear and you know you're as I as I animated I often change the certainly the specific words but sometimes the structure itself but it's a very deterministic process at that point it makes it much easier to predict when something will be done how do you know when a script is done it's like for problem-solving videos that's quite simple it's it's once you feel like someone who didn't understand the solution now could for things like neural networks that was a lot harder because like you said there's so many angles at which you could attack it and there it's it's just at some point you feel like this this asks a meaningful question and it answers that question right what is the best way to learn math for people who might be at the beginning of that journey I think that's a it's a question that a lot of folks kind of ask and think about and it doesn't even for folks who are not really at the beginning of their journey like there might be actually is deep in their career some type they've taken college a taking calculus and so on but still wanna sort of explore math well what would be your advice instead of education at all ages your temptation will be to spend more time like watching lectures or reading try to force yourself to do more problems than you naturally would that's a big one like the the focus time that you're spending should be on like solving specific problems and seek entities that have well curated lists of problems so go into like a textbook almost in and the problems in the back of a text and back of a chapter so if you can take a little look through those questions at the end of the chapter before you read the chapter a lot of them won't make sense some of them might and those are those are the best ones to think about a lot of them won't but just you know take a quick look and then read a little bit of the chapter and maybe take a look again and things like that and don't consider yourself done with the chapter until you've actually worked through a couple exercises right and and this is so hypocritical right because I like put out videos that pretty much never have associated exercises I just view myself as a different part of the ecosystem which means I'm kind of admitting that you're not really learning or at least this is only a partial part of the learning process if you're watching these videos I think if someone's at the very beginning like I do think Khan Academy does a good job they have a pretty large set of questions you can work through just a very basic sort of just picking picking out getting getting comfortable is a very basically algebra calculus fun Khan Academy programming is actually I think a great like learned to program and like let the way the math is motivated from that angle push you through I know a lot of people who didn't like math got into programming in some way and that's what turned them on to math maybe I'm biased cuz like I live in the Bay Area so I'm more likely to run into someone who has that phenotype but I am willing to speculate that that is a more generalizable path so you yourself kind of in creating the videos are using programming to illuminate a concept but for yourself as well so would you recommend somebody try to make a sort of almost like try to make videos like you do what's the one thing I've heard before I don't know if this is based on any actual study this might be like a total fictional anecdote of numbers but it rings in the mind as being true you remember about 10 percent of what you read you remember about 20% of what you listen to remember about 70% of what you actively interact with in some way and then about 90% of what you teach this is a thing I heard again those numbers might be meaningless but they bring true don't they right I'm willing to say I learned the nine times better than reading that might even be a lowball yeah right so so doing something to teach or to like actively try to explain things is huge for consolidating the knowledge outside of family and friends is there a moment you can remember that you would like to relive because it made you truly happy or it was transformative in some fundamental way a moment that was transformed or made you truly happy yeah I think there's times like music used to be a much bigger part of my life than it is now like when I was a let's say a teenager and I can think of sometimes in like playing music there was one way at my command my brother and a friend of mine this slightly violates the family and friends but there was a music that made me happy they were just company um we like played a gig at a ski resort such that you like take a gondola to the top and like did a thing then on the gondola ride down we decided to just jam a little bit and it was just like I don't know the the gondola sort of over came over a mountain and you saw the the city lights and were just like jamming like playing some music I wouldn't describe that nice transformative I don't know why but that popped into my mind as a moment of in a way that wasn't associated with people I love but more with like a thing I was doing something that was just it was just happy and it was just like it a great moment I don't think I can give you anything deeper than that though well as a musician myself I'd love to see as you mentioned before music enter back into your work and back into your creative work I'd love to see that I'm certainly allowing you to enter back into mine and it's it's a it's a beautiful thing for mathematician for scientists to allow music to enter their work I think only good things can happen all right I'll try to promise you a music video by 2021 but by 20 by the end of 2020 I give myself a longer window all right maybe we can like collaborate on a band type situation what instruments do you play the main instrument I play is violin but I also love to devil around on like guitar and piano for me to eat our own piano so in in a mathematicians limit Paul Lockhart writes the first thing to understand is that mathematics is an art the difference between math and the other arts such as music and painting is that our culture does not recognize it as such so I think I speak for millions of people myself included in saying thank you for revealing to us the art of mathematics so thank you for everything you do and thanks for talking today well thanks for saying that and thanks for having me on thanks for listening to this conversation of grants Anderson and thank you to our presenting sponsor cash app downloaded use code Lex podcasts you'll get ten dollars and $10 will go to first a stem education nonprofit inspires hundreds of thousands of young minds to become future leaders and innovators if you enjoy this podcast subscribe on youtube give it five stars an apple podcast supported on patreon or connect with me on Twitter and now let me leave you with some words of wisdom from one of grants and my favorite people Richard Fineman nobody ever figures out what this life is all about and it doesn't matter explore the world nearly everything is really interesting if you go into it deeply enough thank you for listening and hope to see you next time you