Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries | Lex Fridman Podcast #190

the following is a conversation with. jordan ellenberg a mathematician. at university of wisconsin and an author. who. masterfully reveals the beauty and power. of mathematics. in his 2014 book how not to be wrong. in his new book just released recently. called. shape the hidden geometry of information. biology. strategy democracy and everything else. quick mention of our sponsors. secret sauce expressvpn blinkist. and indeed check them out in the.

description to support this podcast. as a side note let me say that geometry. is what made me fall in love with. mathematics. when i was young it first showed me that. something definitive could be stated. about this. world through intuitive visual proofs. somehow. that convinced me that math is not just. abstract numbers. devoid of life but a part of life part. of this world. part of our search for meaning this is. the lex friedman podcast and here is my. conversation. with jordan ellenberg if the brain is a.

cake. it is well let's just let's go with me. on this okay okay. we'll pause it so for noam chomsky. language the universal grammar. the framework from which language. springs is like. most of the cake the delicious chocolate. center and then the rest of. cognition that we think of is built on. top. extra layers maybe the icing on the cake. maybe just.

maybe consciousness is just like a. cherry on top. where do you put in this cake. mathematical thinking. is it as fundamental as language in the. chomsky. view is it more fundamental in language. is it echoes of the same kind of. abstract framework that he's thinking. about in terms of language that they're. all. like really tightly interconnected. that's a really interesting. question you're getting me to reflect on. this question of whether the feeling of. producing mathematical output if you.

want. is like the process of you know uttering. language or producing linguistic output. i think it feels something like that and. it's certainly the case. let me put it this way it's hard to. imagine doing mathematics in a. completely non-linguistic way. it's hard to imagine doing mathematics. without. talking about mathematics and sort of. thinking in propositions but you know. maybe it's just because that's the way i. do mathematics so maybe i can't imagine. it any other way right it's a. well what about visualizing shapes.

visualizing concepts to which language. is not. obviously attachable ah that's a really. interesting question and. you know one thing it reminds me of is. one thing i talk about uh. in the book is dissection proofs these. very beautiful proofs of geometric. propositions. um there's a very famous one by bhaskara. of the the pythagorean theorem. proofs which are purely visual proofs. where you show that. two quantities are the same by taking.

the same pieces. and putting them together one way uh and. making one shape. and putting them together another way. and making a different shape and then. observing those two shapes must have the. same area because they were built out of. the same. pieces um you know there's a there's a. famous. story and it's a little bit disputed. about how accurate this is but that in. bhaskara's manuscript he sort of gives. this proof just gives the diagram and. then the. the entire uh verbal content of the. proof is he just writes under it. behold like that's it it's like.

um there's some dispute about exactly. how accurate that is but. so then that's an interesting question. um if your proof. is a diagram if your proof is a picture. or even if your proof is like a movie of. the same pieces like coming together in. two different formations to make two. different things is that. language i'm not sure i have a good. answer what do you think i think it is. i think the process of manipulating. the visual elements is the same as the. process of. manipulating the elements of language. and i think.

probably the manipulating the. aggregation the stitching stuff together. is the important part it's not the. actual specific elements it's more. more like to me language is a process. and math is a process it's not a. it's not just specific symbols it's uh. it's in action it's it's ultimately. created through action through change. and uh so you're constantly evolving. ideas. of course we kind of attach there's a. certain destination you arrive to that. you attach to and you call that a proof.

but that's not that doesn't need to end. there it's just at the end of the. chapter and then it goes on. and on and on in that kind of way but i. got to ask you about geometry and it's a. prominent. topic in your new book shape so for me. geometry is the thing just like as. you're saying made me fall in love with. mathematics when i was young. so being able to prove something. visually. just did something to my brain that. it had this it planted this hopeful seed.

that. you can understand the world like. perfectly. maybe it's an ocd thing but from a. mathematics perspective. like humans are messy the world is messy. biology is messy. your parents are yelling or making you. do stuff but. you know you can cut through all that bs. and truly understand the world through. mathematics and. nothing like geometry did that for me. for you. you did not immediately fall in love. with geometry so. uh how do you how do you think about.

geometry. why is it a special field in mathematics. and. how did you fall in love with it if you. have wow you've given me like a lot to. say and. certainly the experience that you. describe is so. typical but there's two versions of it. um you know one thing i say in the book. is that geometry is the cilantro of math. people are not neutral about it there's. people who are like who like you. are like the rest of it i could take or. leave but then at this one moment. it made sense this class made sense why.

wasn't it all like that there's other. people i can tell you because they come. and talk to me all the time. who are like i understood all the stuff. we were trying to figure out what x was. there's some mystery or trying to solve. it x is a number i figured it out but. then there was this geometry like what. was that. what happened that year like i didn't. get it i was like lost the whole year. and i didn't understand like why we even. spent the time doing that. so um but what everybody agrees on is. that it's somehow different. right there's something special about it. um. we're gonna walk around in circles a. little bit but we'll get there you asked. me. um how i fell in love with math i have a.

story about this. um when i was a. small child i don't know maybe like i. was six or seven i don't know um. i'm from the 70s i think you're from a. different decade than that but you know. in the 70s we. had them you had a cool wooden box. around your. stereo that was the look everything was. dark wood uh and the box had a bunch of. holes in it. to lift the sound out yeah um and the. holes. were in this rectangular array a six by. eight array.

um of holes and i was just kind of like. you know zoning out in the living room. as kids do. looking at this six by eight rectangular. array of holes. and if you like just by kind of like. focusing in and out just by kind of. looking at this box looking at this. rectangle. i was like well there's six rows of. eight holes each. but there's also eight columns of six. holes each. whoa so eight sixes and six eighths it's. just like the. section bruce you were just talking.

about but it's the same holes it's the. same 48 holes that's how many there are. no matter of whether you count them. as rows or count them as columns and. this was like. unbelievable to me am i allowed to cost. on your podcast i don't know if that's. uh we fcc regulated. okay it was fucking unbelievable okay. that's the last time get it in this. story merits it. so two different perspectives and the. same physical reality. exactly and it's just as you say um. you know i knew that 6 times 8 was the.

same as 8 times 6 when i knew my times. table like i knew that that was a fact. but did i really know it until that. moment that's the question. right i knew that i sort of knew that. the times table was symmetric. but i didn't know why that was the case. until that moment and in that moment i. could see like oh i didn't have to have. somebody tell me that. that's information that you can just. directly access that's a really amazing. moment and as math teachers that's. something that we're really trying to. bring. to our students and i was one of those. who did not love the kind of euclidean. geometry. ninth grade class of like prove that an.

isosceles triangle has. equal angles at the base like this kind. of thing it didn't vibe with me the way. that algebra and numbers did. um but if you go back to that moment. from my adult perspective looking back. at what happened with that rectangle i. think that is a very geometric moment in. fact that moment. exactly encapsulates the. the intertwining of algebra and geometry. this algebraic. fact that well in the instance 8 times 6. is equal to 6 times 8 but. in general that whatever two numbers you. have you multiply them one way and it's.

the same as if you multiply them in the. other order. it attaches it to this geometric fact. about a rectangle which in some sense. makes it true so you know who knows. maybe i was always faded to be an. algebraic geometer which is what i am as. a. as a researcher so that's the kind of. transformation. and you talk about symmetry in your book. what the heck is symmetry. what the heck is these kinds of. transformation on objects that uh once. you transform them they seem to be. similar. uh what do you make of it what's its use.

in mathematics or. maybe broadly in understanding our world. well it's an absolutely fundamental. concept and it starts with the word. symmetry in the way that we usually use. it when we're just like. talking english and not talking. mathematics right sort of something is. when we say something is symmetrical. we usually means it has what's called an. axis of symmetry maybe. like the left half of it looks the same. as the right half that would be like a. left-right axis of symmetry or maybe the. top half looks like the bottom half. or both right maybe there's sort of a. four-fold symmetry where the top looks. like the bottom and the left looks like.

the right. or more and that can take you in a lot. of different directions the abstract. study of what the possible combinations. of symmetries there are a subject which. is called group theory was actually. um one of my first loves in mathematics. what i thought about a lot when i was. in college but the notion of symmetry is. actually. much more general than the things that. we would call symmetry if we were. looking at like a classical. building or a painting or or something. like that. um you know nowadays in.

in math um. we could use a symmetry to to refer to. any kind of transformation of an image. or a space or an object. you know so what i talk about in in the. book is. take a figure and stretch it vertically. make it twice as. make it twice as big vertically and make. it. half as wide. that i would call a symmetry it's not a.

symmetry in the classical sense. but it's a well-defined transformation. that. has an input and an output i give you. some shape um. and it gets kind of i call this in the. book of scronch i just made it had to. make up some sort of funny sounding. name for it because it doesn't really. have um. a name um and just as you can sort of. study which kinds of objects are. symmetrical. under the operations of switching left. and right or switching top and bottom. or rotating 40 degrees or what have you.

you could study what kinds of things are. preserved by. this kind of scratch symmetry and this. kind of. more general idea of what a symmetry can. be um. let me put it this way um a fundamental. mathematical idea. in some sense i might even say the idea. that dominates contemporary mathematics. or by contemporary by the way i mean. like the last like 150 years we're on a. very long time scale. in math i don't mean like yesterday i. mean like a century or so. up till now is this idea that's a.

fundamental question of. when do we consider two things to be the. same. that might seem like a complete. triviality it's not for instance. if i have a triangle and i have a. triangle of the exact same dimensions. but it's over here. um are those the same or different well. you might say like well look there's two. different things this one's over here. this one's over there. on the other hand if you prove a theorem. about this one. it's probably still true about this one. if it has like all the same side lanes. and angles and like looks exactly the.

same. the term of art if you want it you would. say they're congruent. but one way of saying it is there's a. symmetry called translation which just. means. move everything three inches to the left. and we want. all of our theories to be translation. invariant. what that means is that if you prove a. theorem about a thing if it's over here. and then you move it three inches to the. left it would be kind of weird if all of. your theorems like didn't still. work so this question of like what are. the symmetries. and which things that you want to study. or invariant under those symmetries is.

absolutely fundamental but this is. getting a little abstract right. it's not at all abstract i think this. this this is completely central. to everything i think about in terms of. artificial intelligence i don't know if. you know about. the mnist dataset with handwritten. digits yeah. and uh you know i don't smoke much weed. or any really but it certainly feels. like it when i look at eminence and. think about this stuff which is like. what's the difference between one and. two and why are all the twos.

similar to each other what kind of. transformations. are within the category of what makes a. thing the same. and what kind of transformations are. those that make it different. and symmetries core to that in fact our. whatever the hell our brain is doing. it's really good at constructing these. arbitrary and sometimes novel which is. really important. when you look at like the iq test or. they feel novel. uh ideas of symmetry of like what like.

playing with objects we're able to see. things that are the same and not. and uh construct almost like little. geometric. theories of what makes things the same. and not and how to make. uh programs do that in ai is a total. open question. and so i kind of stared and wonder. how what kind of symmetries are enough. to solve. the mnist handwritten digit recognition.

problem and write that down. and exactly and what's so fascinating. about the work in that direction. from the point of view of a. mathematician like me and a geometer. um is that the kind of groups and of. symmetries the types of symmetries that. we know of. um are not sufficient right so in other. words like. we're just going to keep on going with. the weeds on this. the deeper the better you know a kind of. symmetry that we understand very well is. rotation. yeah right so here's what would be easy. if if if humans. if we recognize the digit as a one if it.

was like literally a rotation by some. number of degrees of some. fixed one in some typeface like palatino. or something. that would be very easy to understand. right it would be very easy to like. write a program that. could detect whether something was a. rotation. of a fixed net digit one um whatever. we're doing when you recognize the digit. one and distinguish it from the digit. two. it's not that it's not just. incorporating. uh one of the types of symmetries that.

we. understand now i would say that. i would be shocked if there was some. kind of. classical symmetry type formulation that. captured what we're doing when we tell. the difference between a two and a three. to be honest i think i think what we're. doing is actually more. complicated than that i feel like it. must be they're so simple these numbers. i mean they're. really geometric objects like we can. draw out one two three. it does seem like it's it should be.

formalizable that's why it's so. strange you think it's formalizable when. something stops being a two and starts. being a three right you can imagine. something continuously deforming from. being a two to a three. yeah but that's there is a moment. i have uh myself have written programs. that literally morph. twos and threes and so on and you watch. and there is moments that you notice. depending on the trajectory of that. transformation. that morphing that.

it uh it is a three and a two there's a. hard line. wait so if you ask people if you show. them this morph if you ask a bunch of. people do they all agree about where the. transformation. i'm questioning because i would be. surprised i think so oh my god okay we. have an empirical but here's the problem. dude. here's the problem that if i just showed. that moment that i agreed on. well that's not fair no but say i said. so. i want to move away from the agreement. because that's a fascinating uh actually. question that i want to. backtrack from because i just.

dogmatically said uh because. i could be very very wrong but the. morphing really helps that like the. change. because i mean partially because our. perception systems see this it's all. probably tied in there. somehow the change from one to the other. like seeing the video of it. allows you to pinpoint the place where. two becomes a three much better. if i just showed you one picture i think. uh. you you might you might really.

really struggle you might call a seven. like. i i think there's something uh also that. we don't often think about which is. it's not just about the static image. it's the transformation. of the image or it's not a static shape. it's the transformation of the shape. there's something in the movement that's. seems to be. not just about our perception system but. fundamental to our cognition like how we. think stuff. about stuff yeah and it's so and and you. know that's part of geometry.

too and in fact again another insight of. modern geometry is this idea that. you know maybe we would naively think. we're going to study. i don't know let's you know like. poincare we're going to study the. three-body problem we're going to study. sort of like three. objects in space moving around subject. only to the force of each other's. gravity which sounds very simple right. and if you don't know about this problem. you're probably like okay so you just. like put it in your computer and see. what they do well guess what that's like. a problem that poincare won a huge prize. for like making the first real progress. on in the 1880s. and we still don't know that much about. it.

um 150 years later i mean it's a. humongous mystery you just open the door. and we're going to. walk right in before we return to uh. symmetry. what's the uh who's ponca and what's uh. what's this conjecture that he came up. with. oh why is this such a hard problem okay. so poincare. he ends up being a major figure in the. book and i don't i didn't even really. intend for him to be such a big figure. but he's so. he's um he's first and foremost a. geometer right so he's a mathematician.

who kind of comes up. in late 19th century france. um at a time when french math was really. starting to flower actually i learned a. lot. i mean you know in math we're not really. trained on our own history when we get a. phd in math one about math so i learned. a lot. there's this whole kind of moment where. france has just been. beaten in the franco-prussian war and. they're like oh my god what did we do. wrong and they were like. we got to get strong in math like the. germans we have to be like more like the. germans so this never happens to us. again so it's very much.

it's like the sputnik moment you know. like what happens in america in the 50s. and 60s. uh with the soviet union this is. happening to france and they're trying. to kind of like. instantly like modernize that that's. fascinating. the humans and mathematics are. intricately connected to the history. of humans the cold war is. uh i think fundamental to the way people. saw. science and math in the soviet union i. don't know if that was true in the. united states but certainly was in the. soviet union. it definitely was and i would love to. hear more about how it was in the soviet.

union. i mean there's uh and we'll talk about. the the olympia. i just remember that there was this. feeling. like the world hung in a balance. and you could save the world. with the tools of science and. mathematics. was like the super power. that fuels science and so like. people were seen as you know people in.

america often idolize athletes. but ultimately the best athletes in the. world. they just throw a ball into a basket. so like there's not what people really. enjoy about sports. and i love sports is like excellence at. the highest level. but when you take that with mathematics. and science people also enjoyed. excellence in science and mathematics. and the soviet union. but there's an extra sense that that. excellence would. lead to a better world so.

that created uh all the usual things you. you think about with the olympics which. is like. extreme competitiveness right but it. also created this sense that. in the modern era in america somebody. like elon musk. whatever you think of them like jeff. bezos those folks. they inspire the possibility that one. person. or a group of smart people can change. the world like. not just be good at what they do but. actually change the world.

mathematics is at the core of that uh. and i don't know. there's a romanticism around it too like. when you read uh books about. in america people romanticize certain. things like baseball for example. there's like these beautiful poetic uh. writing about. the game of baseball the same was the. feeling with mathematics. and science in the soviet union and it. was it was in the air. everybody was forced to take high-level. mathematics courses like they you took a. lot of math. you took a lot of science and a lot of.

like really rigorous literature. like they the the level of education in. russia. this could be true in china i'm not sure. uh in a lot of countries. is uh in um whatever that's called. it's k-12 in america but like young. people education the level they were. challenged. to to learn at is incredible it's like. america falls far behind i would say. america then quickly catches up and then.

exceeds everybody else. at the like the as you start approaching. the end of high school to college. like the university system in the united. states arguably is the best in the world. but like what what we uh challenge. everybody it's not just like the good. the ace students but. everybody to learn in in the soviet. union was fascinating. i think i'm gonna pick up on something. you said i think you would love a book. called. duel at dawn by amir alexander which. i think some of the things you're. responding to what i wrote i think i. first got turned on to by amir's work.

he's a historian of math. and he writes about the story of every. east galwa which is a story that's well. known to all mathematicians this kind of. like. very very romantic figure who. he really sort of like begins the. development of this. well this theory of groups that i. mentioned earlier this general. theory of symmetries um and then dies in. a duel in his early 20s like all this. stuff. mostly unpublished it's a very very. romantic story that we all.

learn um and much of it is true but. alexander really lays out just how much. the way people. thought about math in those times in the. early 19th century. was wound up with as you say romanticism. i mean that's when the romantic. movement takes place and he really. outlines how. people were were predisposed to think. about mathematics in that way because. they thought about poetry that way and. they thought about music that way. it was the mood of the era to think. about we're reaching for the. transcendent we're sort of reaching for.

sort of direct. contact with the divine and so part of. the reason that we think of gawa that. way. was because gawa himself was a creature. of that era and he romanticized himself. yeah i mean now now you know he like. wrote lots of letters and like he was. kind of like. i mean in modern terms we would say he. was extremely emo like that's. like just we wrote all these letters. about his like floored feelings and like. the fire within him about the. mathematics and you know so he. so it's just as you say that. the math history touches human history. they're never separate because.

math is made of people yeah i mean. that's what it's. it's it's people who do it and we're. human beings doing it and we do it. within whatever. community we're in and we do it affected. by uh. the morals of the society around us so. the french the germans and the. pancreatic yes okay so back to ponca ray. so. um he's you know it's funny this book is. filled with kind of you know. mathematical characters who. often are kind of peevish or get into. feuds or sort of have like.

weird enthusiasms um because those. people are fun to write about and they. sort of like say. very salty things poincare is actually. none of this as far as i can tell. he was an extremely normal dude he. didn't get into fights with people. and everybody liked him and he was like. pretty personally modest and he had. very regular habits you know what i mean. he did math for like. four hours in the morning and four hours. in the evening and that was it like he. had his. schedule um i actually i was like i.

still am feeling like. somebody's going to tell me now the book. is out like oh didn't you know about. this like incredibly sordid episode. as far as i could tell a completely. normal guy. but um he just kind of. in many ways creates uh the geometric. world. in which we live and and you know his. first really big success. uh is this prize paper he writes for. this prize offered by the king of sweden. for the study of the three-body problem. um.

the study of what we can say about yeah. three. astronomical objects moving and what you. might think would be this very simple. way nothing's going on except. gravity uh releasing the three-body. problem why is it a problem. so the problem is to understand um when. this motion is stable and when it's not. so stable meaning they would sort of. like end up in some kind of periodic. orbital or i guess it would mean sorry. stable would mean they never sort of fly. off far apart from each other and. unstable would mean like eventually they. fly apart.

so understanding two bodies is much. easier yeah. third uh two bodies this is what. newton knew two bodies they sort of. orbit each other and some kind of uh. uh either in an ellipse which is the. stable case you know that's what. the planets do that we know um. or uh one travels on a hyperbola around. the other that's the unstable case it. sort of like zooms in from far away sort. of like whips around the. heavier thing and like zooms out um. those are basically the two options so. it's a very simple and easy to classify.

story. with three bodies just the small switch. from two to three uh. it's a complete zoo it's the first. example what we would say now is it's. the first example of what's called. chaotic dynamics. where the stable solutions and the. unstable solutions. they're kind of like wound in among each. other and a very very very tiny change. in the initial conditions can. make the long-term behavior of the. system completely different so poincare. was the first to recognize that that. phenomenon even. uh even existed what about the uh. conjecture that carries his name right.

so. he also um. was one of the pioneers of taking. geometry um which until that point. had been largely the study of two and. three-dimensional objects because that's. like. what we see right that's those are the. objects we interact with. um he developed that subject we now. called topology he called it analysis. situs he was a very. well-spoken guy with a lot of slogans. but that name did not. you can see why that name did not catch.

on so now it's called topology now. um sorry what was it called before. analysis situs. which i guess sort of roughly means like. the analysis of location or something. like that like um. it's a latin phrase. partly because he understood that even. to understand stuff that's going on. in our physical world you have to study. higher dimensional spaces. how does this how does this work and. this is kind of like where my brain went. to it because you were talking about. not just where things are but what their.

path is how they're moving when we were. talking about the path from two to three. um he understood that if you want to. study three the three bodies. moving in space well each. uh each body it has a location where it. is so it has an x coordinate a y. coordinate and a z coordinate right i. can specify a point in space by giving. you three numbers. but it also at each moment has a. velocity. so it turns out that really to. understand what's going on. you can't think of it as a point or you. could but it's. better not to think of it as a point in.

three-dimensional space that's moving. it's better to think of it as a point in. six dimensional space where the. coordinates are where is it. and what's its velocity right now that's. a higher dimensional space called. phase space and if you haven't thought. about this before i admit that it's a. little bit. mind-bending but. what he needed then was a geometry that. was flexible enough. not just to talk about two-dimensional. spaces or three-dimensional spaces but. any dimensional space so the sort of. famous first line of this paper where he. introduces analysis.

is is no one doubts nowadays that the. geometry of. n-dimensional space is an actually. existing thing right i think that. maybe that had been controversial and. he's saying like look let's face it just. because it's not physical. doesn't mean it's not there it doesn't. mean we shouldn't stop. interesting he wasn't jumping to the. physical the physical interpretation. like it does. it can be real even if it's not. perceivable to human. cognition i think i think that's right i. think. don't get me wrong poincare never strays. far from physics he's always motivated.

by physics. but the physics drove him to need to. think about. spaces of higher dimension and so he. needed a formalism that was rich enough. to enable him to do that. and once you do that that formalism is. also going to include things that are. not physical. and then you have two choices you can be. like oh well that stuff's trash. or but and this is more the. mathematicians frame of mind. if you have a formalistic framework that. like seems really good and sort of seems. to be like very elegant and work well. and it includes all the physical stuff. maybe we should think about all of it.

like maybe we should think about it. thinking maybe there's some gold to be. mined there. um and indeed like you know guess what. like before long there's relativity and. there's space time and like all of a. sudden it's like oh yeah maybe it's a. good idea we already have this geometric. apparatus like set up for like how to. think about. four-dimensional spaces like turns out. they're real after all. you know this is a a story much told. right in mathematics not just in this. context but in many i'd love to dig in a. little deeper on that actually because. i have some uh intuitions to work out.

okay my brain well i'm not a. mathematical physicist so we can work. them out together. good we'll uh we'll we'll together walk. along the path of curiosity. but pancreatic uh conjecture. what is it the point conjecture is about. curved. three-dimensional spaces so i was on my. way there i promise. um the idea is that we perceive. ourselves as living in. we don't say a three-dimensional space. we just say three-dimensional space you.

know you can go up and down you can go. left and right you can go forward and. back there's three dimensions in which. you can move. in poincare's theory there are many. possible three-dimensional spaces. in the same way that going down one. dimension to sort of. capture our intuition a little bit more. we know there are lots of different. two-dimensional surfaces right there's. a balloon and that looks one way and a. doughnut looks another way and a mobius. strip. looks a third way those are all like. two-dimensional surfaces that we can. kind of really.

uh get a global view of because we live. in three-dimensional space so we can see. a two-dimensional surface sort of. sitting in our three-dimensional space. well to see a three-dimensional space. whole we'd have to kind of have. four-dimensional eyes right which we. don't so we have to use our mathematical. lines we have to envision. um the poincare conjecture. uh says that there's a very simple way. to determine whether a three-dimensional. space. um is the standard one the one that. we're used to. um and essentially it's that it's what's.

called fundamental group. has nothing interesting in it and not. that i can actually say without saying. what the fundamental group is i can tell. you what the criterion is. this would be oh look i can even use a. visual aid so for the people watching. this on youtube you'll just see this for. the people. uh on the podcast you'll have to. visualize it so lex has been nice enough. to like. give me a surface with some interesting. topology it's a mug right here in front. of me. a mug yes i might say it's a genus one. surface but we could also say it's a mug. same thing. so if i were to draw a little circle.

on this mug oh which way should i draw. it so it's visible like here okay. yeah if i draw a little circle on this. mug imagine this to be a loop of string. i could pull that loop of string closed. on the surface of. the mug right that's definitely. something i could do i could shrink it. shrink and shrink it until it's a point. on the other hand if i draw a loop that. goes around the handle. i can kind of judge it up here and i can. judge it down there and i can sort of. slide it up and down the handle but i. can't pull it closed can't i it's. trapped. not without breaking the surface of the.

mug right now without like going inside. so um the condition of being what's. called simply connected this is. one of punk ray's inventions says that. any loop of string can be pulled shut so. it's a feature that the mug. simply does not have this is a. non-simply connected. mug and a simply connected mug would be. a cup right you would burn your hand. when you drank coffee out of it. so you're saying the universe is not a. mug. well i can't speak to the universe but. what i can say is that.

um regular old space. is not a mug regular old space if you. like sort of actually physically have. like a loop of string. you can always close your shot you're. going to pull a shit. but you know what if your piece of. string was the size of the universe like. what if your poi. your piece of string was like billions. of light years long like like how do you. actually know. i mean that's still an open question of. the shape of the universe exactly. whether it's uh i think there's a lot. there is. ideas of it being a tourist i mean. there's there's some trippy ideas and.

they're not. like weird out there controversial. there's a legitimate. at the center of uh cosmology debate. i mean i think i think somebody who. thinks that there's like some kind of. dodecahedral symmetry or i mean i. remember reading something crazy about. somebody saying that they saw the. signature of that and the. cosmic noise or what have you i mean to. make the flat earthers happy. i do believe that the current main. belief is. it's fl it's flat it's flat-ish. or something like that the shape of the.

universe is flat-ish i don't know what. the heck that means i think that. i think that has like a very i mean how. are you even supposed to think about. the shape of a thing that doesn't have. anything outside of it. i mean ah but that's exactly what. topology does topology is what's called. an intrinsic theory. that's what's so great about it this. question about the mug. you could answer it without ever leaving. the mug. right because it's a question about a. loop drawn on the surface of the mug and.

what happens if it never leaves that. surface so it's like. always there see but that's the the. difference between. then topology and say if you're like uh. trying to visualize a mug that you can't. visualize a mug while living inside the. mug. well that's true that visualization is. harder but in some sense no you're right. but if the tools of mathematics are. there. i i i don't want to fight but i think. the tools and mathematics are exactly. there to enable you to think about what. you cannot visualize. in this way let me give let's go always. to make things easier go downward.

dimension. um let's think about we live on a circle. okay. you can tell whether you live on a. circle. or a line segment because if you live in. a circle if you walk a long way in one. direction you find yourself back where. you started and if you live in a line. segment. you walk for a long enough one direction. you come to the end of the world or if. you live on a line. like a whole line an infinite line then. you walk in. one direction for a long time and like. well then there's not a sort of. terminating algorithm to figure out. whether you live on a line or a circle.

but at least you sort of. um at least you don't discover that you. live on a circle. so all of those are intrinsic things. right all of those are things that you. can figure out about your. world without leaving your world on the. other hand ready now we're going to go. from intrinsic to extrinsic why did i. not know we were going to talk about. this but why not. why not if you can't tell whether you. live in a circle. or a not like imagine like a knot. floating in three-dimensional space the. person who lives on that knot to them. it's a circle. yeah they walk a long way they come back.

to where they started now we with our. three-dimensional eyes can be like. oh this one's just a plain circle and. this one's knotted up but that's an. that's a. that has to do with how they sit in. three-dimensional space it doesn't have. to do with intrinsic features of those. people's world. we can ask you one ape to another does. it make you. how does it make you feel that you don't. know if you live in a circle. or on a knot in a knot. in inside the string that forms the knot. i'm going to even know how to say i'm.

going to be honest with you i don't know. if like. i i fear you won't like this answer but. it. does not bother me at all it does i. don't lose one minute of sleep over it. so like does it bother you that if we. look at like a mobius strip. that you don't have an obvious way of. knowing. whether you are inside of cylinder if. you live on a surface of a cylinder. or you live on the surface of a mobius. strip. no i think you can tell if you live if. which one because.

if what you do is you like tell your. friend hey stay right here i'm just. gonna go for a walk. and then you like walk for a long time. in one direction and then you come back. and you see your friend again and if. your friend is reversed then you know. you live on a mobius strip well. no because you won't see your friend. right okay. fair fair point fair point on that and. you have to believe the story is about. no i don't even know. i i i would would you even know would. you really oh no you're i know your. point is right let me try to think of it. better. let's see if i can do this may not be. correct to talk about.

cognitive beings living on a mobius. strip because. there's a lot of things taken for. granted there and we're constantly. imagining actual like. three-dimensional creatures like how it. actually. feels like to uh to live on a mobius. strip is tricky to. internalize i think that on what's. called the real projective plane which. is kind of even more sort of like messed. up version of the. mobius strip but with very similar. features this feature of kind of like. only having one side that has the.

feature that there's a loop of string. which can't be pulled closed but if you. loop. it around twice along the same path that. you can pull closed. that's extremely weird yeah. um but that would be a way you could. know without leaving your world that. something very funny is going on you. know what's extremely weird. maybe we can comment on hopefully it's. not too much of a tangent is. i remember thinking about this this. might be right.

this might be wrong but if you're if we. now. talk about a sphere and you're living. inside a sphere. that you're going to see everywhere. around you the back of your own head. that i was because like i was. this is very counterintuitive to me to. think about maybe it's wrong. but because i was thinking like earth. you know your 3d. thing on sitting on a sphere but if. you're living inside the sphere.

like you're going to see if you look. straight you're always going to see. yourself all the way around so. everywhere you look there's going to be. the back of your. head i think somehow this depends on. something of like how the physics of. light works in this scenario which i'm. sort of finding it hard to bend my. that's true the c is doing a lot of like. saying you see something's doing a lot. of work. people have thought about this i mean. this this metaphor of like what if we're. like little creatures in some sort of. smaller world like how could we. apprehend what's outside that metaphor. just comes back and back and actually i.

didn't even realize like how frequent it. is it comes up in the book a lot. i know it from a book called flatland i. don't know if you ever read this when. you were a kid. an adult you know this this uh sort of. sort of comic novel from the 19th. century about. an entire two-dimensional world. uh it's narrated by a square that's the. main character. and um the kind of strangeness that. befalls him when. you know one day he's in his house and. suddenly there's like a little circle. there and there with him. and then the circle but then the circle.

like starts getting bigger. and bigger and bigger and he's like what. the hell is going on it's like a horror. movie like for two-dimensional people. and of course what's happening is that a. sphere is entering his world and as the. sphere kind of like. moves farther and farther into the plane. it's cross-section the part of it that. he can see. to him it looks like there's like this. kind of bizarre being that's like. getting larger and larger and larger. um until it's exactly sort of halfway. through and then they have this kind of. like philosophical argument where the. sphere is like i'm a sphere i'm from the.

third dimension the square is like what. are you talking about there's no such. thing. and they have this kind of like sterile. argument where the square is not. able to kind of like follow the. mathematical reasoning of the sphere. until the sphere just kind of grabs him. and like jerks him out of the plane. and pulls him up and it's like now like. now do you see like now do you see your. whole world that you didn't understand. before so do you think. that kind of process is possible for us. humans. so we live in the three-dimensional. world maybe with the time component. four-dimensional.

and then math allows us to uh to go. high into high dimensions comfortably. and explore the world from those. perspectives. like is it possible. that the universe is uh many more. dimensions. than the ones we experience as human. beings so. if you look at uh the you know. especially in physics. theories of everything uh physics.

theories that try to unify. general relativity and quantum field. theory. they seem to go to high dimensions. to work stuff out through the tools of. mathematics. is it possible so like the two options. are one. is just a nice way to analyze a universe. but the reality is is as exactly we. perceive it it is three-dimensional. or are we just seeing are we those. flatland creatures.

they're just seeing a tiny slice of. reality. and the actual reality is many many. many more dimensions than the three. dimensions we perceive. oh i certainly think that's possible um. now how would you figure out whether it. was true or not is another question. um i suppose what you would do as with. anything else that you can't directly. perceive. is you would try to understand. what effect the presence of those extra.

dimensions. out there would have on the things we. can. perceive like what else can you do right. and in some sense. if the answer is they would have no. effect. then maybe it becomes like a little bit. of a sterile question because what. question are you even. asking right you can kind of posit. however many entities that. is it possible to intuit how to mess. with the other dimensions. while living in a three-dimensional. world i mean that seems like a very.

challenging thing to do. we the the reason flatland could be. written. is because it's coming from a. three-dimensional. writer yes but but what happens in the. book i didn't even tell you the whole. plot. what happens is the square is so excited. and so. filled with intellectual joy by the way. maybe to give the story some context. you ask like is it possible for us. humans to have this experience of being. transcendent transcendentally jerked out. of our world so we can sort of truly see.

it from above. well edwin abbott who wrote the book. certainly thought so because. edward abbott was a minister so the. whole christian subtext to this book i. had completely not grasped. reading this as a kid that it means a. very different thing right if sort of a. theologian. is saying like oh what if a higher being. could like pull you out of. this earthly world you live in so that. you can sort of see the truth and like. really see it. uh from above as it were so that's one. of the things that's going on for him. and it's a testament to his skill as a. writer that his story just works whether.

that's the framework you're coming to it. from. or not um but what happens in this book. and this part now looking at it through. a christian lens that becomes. a bit subversive is the square is so. excited about. what he's learned from the sphere and. the sphere explains them like what a. cube would be oh it's like you but. three-dimensional and the square is very. exciting and the square is like. okay i get it now so like now that you. explained to me how just by reason i can. figure out what a cube would be like. like a three-dimensional version of me. like let's figure out what a.

four-dimensional version of me would be. like. and then this fear is like what the hell. are you talking about there's no fourth. dimension that's ridiculous like. there's three dimensions like that's how. many there are i can see like i mean so. it's the sort of comic moment where the. sphere is completely unable to. uh conceptualize that there could. actually be yet another. dimension so yeah that takes the. religious allegory to like a very weird. place that i don't really like. understand theologically but. that's a nice way to talk about religion. and myth. in general as perhaps us trying to.

struggle with us meaning human. civilization trying to struggle. with ideas that are beyond our cognitive. capabilities but it's in fact not beyond. our capability it may be beyond our. cognitive capabilities. to visualize a four-dimensional cube a. tesseract as some like to call it or a. five-dimensional cube or a. six-dimensional cube. but it is not beyond our cognitive. capabilities. to figure out how many corners a. six-dimensional cube would have. that's what's so cool about us whether. we can visualize it or not we can still.

talk about it we can still reason about. it. we can still figure things out about it. that's amazing. yeah if we go back to this first of all. to the mug. but to the example you give in the book. of the straw. uh how many holes does a straw have. and you listener may uh try to answer. that in your own head. yeah i'm gonna take a drink while. everybody thinks about it a slow. sip is it uh.

zero one or two or more. more than that maybe maybe you get very. creative but. uh it's kind of interesting to uh each. uh dissecting each answer as you do in. the book is quite brilliant people. should definitely check it out. but if you could try to answer it now. like think about. all the options and why they may or may. not be right. yeah and it's one of it's one of these. questions where people on first hearing. it think it's a triviality and they're. like well the answer is obvious and then. what happens if you ever ask a group of.

people this something. wonderfully comic happens which is that. everyone's like well it's completely. obvious. and then each person realizes that half. the person the other people in the room. have a different. obvious answer for the way that they. have and then people get really heated. people are like i can't believe that you. think it has two holes or like i can't. believe that you think it has one. and then you know you really like people. really learn something about each other. and people get heated. i mean can we go through the possible. options here. is it zero one two three ten.

sure so i think you know most people. the zero holders are rare they would say. like well look. you can make a straw by taking a. rectangular piece of plastic and closing. it up the rectangular piece of plastic. doesn't have a hole in it uh i didn't. poke a hole in it. when i yeah so how can i have a hole. like it's just one thing okay. most people don't see it that way that's. like uh um. is there any truth to that kind of. conception yeah i think that would be. somebody whose account. i mean.

what i would say is you could say the. same thing. um about a bagel you could say i can. make a bagel by taking like a long. cylinder of dough which doesn't have a. hole and then smooshing the ends. together. now it's a bagel so if you're really. committed you can be like okay bagel. doesn't have a hole either but like. who are you if you say a bagel doesn't. have a hole i mean i don't know yeah so. that's almost like an engineering. definition of it. okay fair enough so what's what about.

the other options. um so you know one whole people. would say um i like how these are like. groups of people. like where we've planted our foot yes. one hole. there's books written about each belief. you know would say look there's like a. hole and it goes all the way through the. straw right there it's one region of. space that's the hole. yeah and there's one and two whole. people would say like well look there's. a hole in the top in the hole. at the bottom um i think a common thing. you see.

when people um. argue about this they would take. something like this a bottle of water. i'm holding maybe i'll open it and they. say well how many holes are there in. this and you say like well there's one. there's one hole. at the top okay what if i like poke a. hole here so that all the water. spills out well now it's a straw. yeah so if you're a one hole or i say to. you like well how many holes are. in it now there was a there was one hole. in it before and i poked a new hole in. it. and then you think there's still one.

hole even though there was. one hole and i made one more clearly not. this. is two holes yeah um and yet if you're a. two hole the one holder will say like. okay where does one hole begin in the. other whole end. yeah like what's it like and um. and in the in the book i sort of you. know in math there's two things we do. when we're faced with a problem that's. confusing us. um we can make the problem simpler. that's what we were doing a minute ago. and we were talking about high. dimensional space and i was like let's. talk about like. circles and line segments let's go down. a dimension to make it easier.

uh the other big move we have is to make. the problem harder. and try to sort of really like face up. to what are the complications so. you know what i do in the book is say. like let's stop talking about straws for. a minute and talk about pants. how many holes are there in a pair of. pants. so i think most people who say there's. two holes in the straw would say there's. three holes in a pair of pants. i guess i mean i guess we're filming. only from here i could take up no i'm. not gonna do it. you'll just have to imagine the pants. sorry yeah um. lex if you want it no okay no.

uh that's gonna be in the direction. that's the patreon-only footage. there you go so many people would say. there's three holes in the pair of pants. but you know for instance my daughter. when i asked. by the way talking to kids about this is. super fun i highly recommend it. um what did she say she said well. yeah i feel a pair of pants like just. has two holes because yes there's the. waist but that's just the two leg holes. stuck together. whoa okay two leg holes yeah okay right. i mean that's.

she's a one caller for the straw so. she's a one-holer for the straw too. and um and that really does. capture something it captures this fact. which is central to the theory of what's. called homology which is like a central. part of modern topology that. um holes whatever we may mean by them. there are somehow things which have an. arithmetic to them they're things which. can be. added like the waste like waste equals. leg plus leg. is kind of an equation but it's not an. equation about numbers it's an equation.

about some kind of. geometric some kind of topological thing. which is very strange and so. you know when i come down um. you know like a rabbi i like to kind of. like come up with these answers and. somehow like. dodge the original question and say like. you're both right my children okay so. yeah uh so for this for the. for the straw i think what a modern. mathematician would say is like. the first version would be to say like. well they're two holes but they're. really both the same hole.

well that's not quite right a better way. to say it is there's two holes. but one is the negative of the other now. what can that mean. um one way of thinking about what it. means is that if you sip something like. a milkshake through the straw. no matter what the amount of milkshake. that's flowing in one end. that same amount is flowing out the. other end. so they're not independent from each. other there's some relationship. between them in the same way that if you.

somehow could like. suck a milkshake through a pair of pants. the amount of milkshake just go with me. on this not experimenting. mom right there the amount of milkshake. that's coming in the left leg of the. pants. plus the amount of milkshake that's. coming in the right leg of the pants. is the same that's coming out the uh the. waist of the pants so just so you know i. fasted for 72 hours. yester uh the last three days so i just. broke the fast with a little bit of food. yesterday so. this is like this sounds uh food.

analogies or metaphors for this podcast. work. wonderfully because i can intensely. picture it is that your weekly routine. or just in preparation for talking about. geometry for three hours. exactly this it's hardship. to purify the mind no it's for the first. time i just wanted to try the experience. oh wow. and just to uh to pause to do things. that are out of the ordinary to pause. and to uh reflect on how grateful i am. to be just. alive and be able to do all the cool. shit that i get to do so.

did you drink water yes yes yes yes yes. water and salt so like electrolytes and. all those kinds of things. but anyway so the inflow on the top of. the pants. equals to the outflow on the bottom of. the pants. exactly so this idea that. i mean i think you know poincare really. have these i this idea this sort of. modern idea. i mean building on stuff other people. did uh betty is an important one. of this kind of modern notion of. relations between holes but the idea.

that holes really had an arithmetic. the really modern view was really emmy. nerder's idea so she kind of comes in. and sort of truly puts the subject. uh on its modern footing that we have. that we have now so. you know it's always a challenge you. know in the book i'm not gonna say i. give like. a course so that you read this chapter. and then you're like oh it's just like i. took like. a semester of algebraic apology it's not. like this and it's always a you know. it's always a challenge. writing about math because there are. some things.

that you can really do on the page and. the math is there and there's other. things which. it's too much in a book like this like. do them all the page you can only. say something about them if that makes. sense um. so you know in the book i try to do some. of both i try to do i try to. topics that are you can't really. compress. and really truly say exactly what they. are. in this amount of space um. i try to say something interesting about. them something meaningful about them so.

that readers can get the flavor. um and then in other places i really try. to get up close and personal. and really do the math and have it take. place on the page. to some degree be able to give inklings. of the beauty of the subject. yeah i mean there's you know there's a. lot of books that are like i don't quite. know how to express this well i'm still. laboring to do it but um there's a lot. of books that. are about stuff. but i want my books to not only be about.

stuff but to actually have some stuff. there on the page in the book for people. to interact with directly and not just. sort of hear me talk about distant. features about. just different distant features of it. right so. not be talking just about ideas but the. actually be. expressing the idea is there you know. somebody in the. maybe you can comment there's a guy his. youtube channel is. three blue one brown grant sanderson he. does that. masterfully well absolutely of uh. visualizing of expressing a particular.

idea and then talking about it as well. back and forth uh what do you what do. you think about grant. it's fantastic i mean the flowering of. math youtube is like such a wonderful. thing because. you know math teaching there's so many. different venues. through which we can teach people math. there's the traditional one right. well where i'm in a classroom with. you know depending on the class it could. be 30 people it could be 100 people. it could god help maybe 500 people if. it's like the big calculus lecture or.

whatever it may be. and there's sort of some but there's. some set of people of that order of. magnitude. and i'm with them for we have a long. time i'm with them for a whole semester. and i can ask them to do homework and we. talk together we have office hours that. they have one-on-one questions multiply. that's like a very high level of. engagement. but how many people am i actually. hitting at a time like not. that many right um and you can. and there's kind of an inverse. relationship where the. more and g the fewer people you're. talking to the more engagement you can. ask for the ultimate of course is like.

the mentorship relation of like a phd. advisor and a graduate student where. you spend a lot of one-on-one time. together for like you know. three to five years and the ultimate. high level of engagement. to one person um you know books. i can this can get to a lot more people. that are ever gonna sit in my. classroom and you spend like uh. however many hours it takes to read a. book uh somebody like three blue one. brown or numberphile or um.

people like vi heart i mean youtube. let's face it. has bigger reach than a book like. there's youtube videos that have many. many many more views than like. you know any hardback book like not. written by a kardashian or an obama is. gonna sell right so that's. i mean um any. and and then you know those are you know. some of them are like. longer 20 minutes long some of them are. five minutes long but they're you know. they're shorter and even so. look look like eugenia chang is a. wonderful category theorist in chicago i.

mean. she was on i think the daily show or i. mean she was on you know. she has 30 seconds but then there's like. 30 seconds to sort of say something. about math mathematics to like untold. millions of people so everywhere along. this curve isn't. is important one thing i feel like is. great right now is that people are just. broadcasting on all the channels because. we each have. our skills right somehow along the way. like i learned how to write books i had. this. kind of weird life as a writer where i. sort of spent a lot of time like. thinking about how to put. english words together into sentences.

and sentences together into paragraphs. like. at length which is this kind of like. weird specialized scale. and that's one thing but like sort of. being able to make like you know winning. good-looking eye-catching videos is like. a totally. different skill and you know probably. you know somewhere out there there's. probably sort of some. like heavy metal band that's like. teaching math through. heavy metal and like using their skills. to do that i hope there is. at any rate through music and so on yeah. but there is something to the process. i mean grant does this especially well.

which is. in order to be able to visualize. something now he writes. programs so it's programmatic. visualization so like the. the things he is basically mostly. through his uh. madam library in python everything is. drawn. through python you have to um. you have to truly understand the topic. to be able to. to visualize it in that way and not just. understand it but really kind of.

think in a very novel way it's funny. because i i've spoken with him a couple. times. i've spoken to him a lot offline as well. he really doesn't think. he's doing anything new meaning like. he sees himself as very different from. maybe like a researcher. but it feels to me. like he's creating something totally new. like that act of understanding. visualizing. is as powerful or has the same kind of.

inkling of power as does. the process of proving something you. know. it just it doesn't have that clear. destination but it's. it's pulling out an insight and creating. multiple sets of perspective. that arrive at that insight and to be. honest it's something that i think we. haven't. quite figured out how to value. inside academic mathematics in the same. way and this is a bit older that i think. we haven't quite. figured out how to value the development. of computational infrastructure you know.

we all have computers as our partners. now and people. build computers that sort of assist and. participate in our mathematics they. build those systems and that's. a kind of mathematics too but not in the. traditional form of. proving theorems and writing papers but. i think it's coming look i mean i think. you know for example the institute for. computational experimental mathematics. at brown which is like a. you know it's a nsf-funded math. institute very much part of sort of. traditional math academia they did an.

entire theme semester about visualizing. mathematics looking at the same kind of. thing that they would do for like. an up-and-coming research topic like. that's pretty cool so i think there. really is. buy-in from uh the mathematics community. to recognize that this kind of stuff is. important and counts as part of. mathematics like part of what we're. actually here to do. yeah i'm hoping to see more and more of. that from like mit faculty from faculty. from all the. the top universities in the world let me. ask you this weird question about the. fields medal which is the nobel prize in.

mathematics. do you think since we're talking about. computers there will one. day come a time. when a computer an ai system will win a. field medal. no of course that's what a human would. say why not. it's is that like you're that that's. like my captcha that's like the proof. that i'm a human. how does he want me to answer is there. something interesting to be said about.

that. yeah i mean i am tremendously interested. in. what ai can do in pure mathematics i. mean it's of course. it's a parochial interest right you're. like why am i interested in like how it. can like help feed the world or help. solve like there's problems i'm like can. i do. more math like what can i do we all have. our interests right um. but i think it is a really interesting. conceptual. question and here too i think. um it's important to be kind of. historical because it's certainly true.

that there's lots of things that we used. to call research and mathematics. that we would now call computation yeah. tasks that we've now offloaded to. machines like. you know in 1890 somebody could be like. here's my phd thesis i. computed all the invariants of this. polynomial ring under the action of some. finite group doesn't matter what those. words mean just it's like something that. in 1890. would take a person a year to do and. would be a valuable thing that you might. want to know and it's still a valuable. thing that you might want to know but.

now you type a few lines of code in. macaulay or sage. or magma and you just have it so we. don't think of that as. math anymore even though it's the same. thing what's macaulay sage and magma. oh those are computer algebra programs. so those are like sort of bespoke. systems that lots of mathematicians use. that's similar to maple and. yeah oh yeah so similar to maple and. mathematica yeah okay but a little more. specialized but yeah. it's programs that work with symbols and. allow you to do can you do proofs can. you do. kind of kind of little little leaps and.

proofs they're not really built for that. that's a whole other. story but these tools are part of the. process of mathematics now. right they are now for most. mathematicians i would say part of the. process of mathematics and so. um you know there's a story i tell in. the book which i'm fascinated by which. is. you know so far attempts. to get ais to prove interesting theorems. have not. done so well doesn't mean they can it's. actually a paper i just saw which.

has a very nice use of a neuron that. defined counter examples to conjecture. somebody said like well maybe this is. always that. yeah and you can be like well let me. sort of train an ai to sort of try. to find things where that's not true and. it actually succeeded now in this case. if you look at the things that it found. you say like okay i mean. these are not famous conjectures yes. okay so like somebody wrote this down. maybe this is so. um looking at what the ai came up with. you're like.

you know i bet if like five grad. students had thought about that problem. they wouldn't. i mean when you see it you're like okay. that is one of the things you might try. if you sort of like. put some work into it still it's pretty. awesome right but the story i tell. in the book which i'm fascinated by is. um. there is there's okay we're gonna go. back to knots. it's cool there's a knot called the. conway knot after john conway maybe. we'll talk about a very interesting. character also he has a small tangent. somebody i was supposed to talk to and. unfortunately he passed away and.

he's he's somebody i find is an. incredible mathematician incredible. human being. oh and i am sorry that you didn't get a. chance because having had the chance to. talk to him a lot when i was you know. when i was a postdoc. um yeah you missed out there's no way to. sugarcoat it i'm sorry that you didn't. get that chance. yeah it is what it is so knots. yeah so there was a question and again. it doesn't matter the technicalities of. the question but it's a question of. whether the knot is slice it has to do. with um. something about what kinds of.

three-dimensional surfaces in four. dimensions can be bounded by this knot. but never mind what it means it's some. question. uh and it's actually very hard to. compute whether or not. is slice or not um. and in particular the question of the. conway knot. whether it was slice or not was. particularly. vexed um until it was solved just a few. years ago by lisa picarillo who actually. now that i think of it was here in. austin i believe she was a grad student. at ut austin at the time i didn't even.

realize there was an austin connection. to this story until i started. telling it she is in fact i think she's. now at mit so she's basically following. you around. if i remember correctly the reverse. there's a lot of really interesting. richness to this. story one thing about it is her paper. was rather. was very short it was very short and. simple nine pages of which two were. pictures. uh very short for like a paper solving a. major conjecture. and it really makes you think about what. we mean by difficulty in mathematics. like do you say oh actually the problem. wasn't difficult because you could solve.

it so simply or do you say like well no. evidently it was difficult because like. the world's top. topologist many you know worked on it. for 20 years and nobody could solve it. so therefore it is difficult. or is it that we need sort of some new. category of things that about which. it's difficult to figure out that. they're not difficult. i mean this is the computer science. formulation but. the sort of the the journey to arrive. at the simple answer may be difficult. but once you have the answer. it will then appear simple and i mean. there might be a large category.

i hope there's a large uh set. of such solutions. because um you know. once we stand at the end of the. scientific process that we're at the. very beginning. of or at least it feels like i hope. there's just simple answers to. everything. that we'll look and it'll be simple laws. that govern the universe. simple explanation of what is. consciousness of what is. love is mortality fundamental to life.

what's the meaning of life uh. are are humans special or we're just. another sort of reflection of uh. and all that is beautiful uh in the. universe in terms of like. life forms all of it is life and just. has different. when taken from a different perspective. is all life can seem more valuable or. not but really it's all part of the same. thing. all those will have a nice like two. equations maybe one equation. why do you think you want those.

questions to have simple answers. i think just like symmetry and the. breaking of symmetry is beautiful. somehow there's something beautiful. about simplicity. i think it so it's static it's aesthetic. yeah i. or but it's aesthetic in the way that uh. happiness is an aesthetic. like uh why is that so joyful that a. simple explanation. that governs a large number of cases is. really appealing.

even when it's not like obviously we. get a huge amount of trouble with that. because oftentimes. it doesn't have to be connected with. reality or even that explanation could. be exceptionally harmful. most of like the world's history that. has. that was governed by hate and violence. had a very simple explanation at the. court. that was used to cause the violence and. the hatred so like we get into trouble. with that. but why is that so appealing and in this.

nice forms. in mathematics like you look at the. einstein papers. why are those so beautiful and why is. the andrew wiles proof of the farm. ozilized theorem. not quite so beautiful like what's. beautiful about that story. is the human struggle of like the human. story of perseverance. of the drama of not knowing if the proof. is correct and. ups and downs and all those kinds of. things that's the interesting part but. the fact that the proof is huge and. nobody understands well. from my outsider's perspective nobody.

understands what the heck it is. uh is is not as beautiful as it could. have been. i wish it was what fermat originally. said which is. you know it's it's not. it's not small enough to fit in the. margins of this page. but maybe if he had like a full page or. maybe a couple post-it notes he would. have enough to do the proof. what do you make of if we could take. another of a multitude of tangents. what do you make of fermat's last. theorem because the statement there's a.

few. theorems there's a few problems that are. deemed by the world throughout its. history to be exceptionally difficult. and. that one in particular is uh really. simple to formulate. and really hard to come up with a proof. for. and it was like taunted as simple. uh by from himself there's something. interesting to be said about. that x to the n plus y to the n equals z. to the n for n. of three or greater is there a solution.

to this. and then how do you go about proving. that like how would you. uh try to prove that and what do you. learn from. the proof that eventually emerged by. andrew wiles yeah so right to sort of. give. let me just say the background because i. don't know if everybody listening knows. the story so you know fermat. uh was an early number theorist not. really sort of an early mathematician. those. special adjacent didn't really exist. back then. he comes up in the book actually in the. context of um a different theorem of his.

that has to do with testing whether a. number is prime. or not so i write about he was one of. the ones who was salty and like he would. exchange these letters where. he and his correspondence would like try. to top each other and. vex each other with questions and stuff. like this but this particular thing. um it's called fermazl's theorem because. it's a note he wrote uh in. in his uh in his copy of the description. arithmetic eye like he wrote. here's an equation it has no solutions i.

can prove it but. the proof's like a little too long to. fit in this in the margin of this. book he was just like writing a note to. himself now let me just say historically. we know that vermont did not have a. proof of this theorem for a long time. people like. you know people were like this. mysterious proof that was lost a very. romantic story right. but fairmont later. he did prove special cases of this. theorem and. wrote about it to talk to people about. the problem uh it's very clear from the. way that he wrote where he can solve. certain examples of this type of.

equation. that he did not know how to do the whole. thing. he may have had a deep. simple intuition about this how to solve. the whole thing that he had at that. moment. without ever being able to come up with. a complete proof and that intuition may. be lost to time. maybe but i think we so but you're right. that is unknowable but i think what we. can know is that later he certainly did. not think that he had a proof that he. was concealing from people he.

yes uh he thought he didn't know how to. prove it and i also think he didn't know. how to prove it. now i understand the appeal of saying. like. wouldn't it be cool if this very simple. equation there was like a very simple. clever wonderful proof that you could do. in a page or two and. that would be great but you know what. there's lots of equations like that that. are solved by very clever methods like. that including. the special cases that female wrote. about the method of descent which is. like very wonderful and important. but in the end those are.

nice things that like you know you teach. in an undergraduate class. um and it is what it is but they're not. big. um on the other hand work on the fermat. problem that's what we like to call it. because. it's not really his theorem because we. don't think he proved it so i mean. work on the vermont problem developed. this like incredible. richness of number theory that we now. live in today like and not by the way. just wilds andrew wiles being. the first new together with richard. taylor finally proved this theorem.

but you know how you have this whole. moment that people try to prove this. theorem and they fail. and there's a famous false proof by. lemay from the 19th century. where kumar in understanding what. mistake lemay had made. in this incorrect proof basically. understands something incredible which. is that. you know a thing we know about numbers. is that um. you can factor them and you can factor. them uniquely. there's only one way to break a number. up into primes.

like if we think of a number like 12 12. is two times three. times two i had to think about it right. or it's two times two times three of. course you can reorder them right. but there's no other way to do it. there's no universe in which 12 is. something times five. or in which there's like four threes in. it nope 12 is like two twos and a three. like that. is what it is and that's such a. fundamental feature. of arithmetic that we almost think of it. like god's law you know what i mean it. has to be that way. that's a really powerful idea it's it's.

so cool. that every number is uniquely made up of. other numbers. and like made up meaning like there's. these like basic atoms. that form molecules that for that. get built on top of each other i love it. i mean when i teach you know. undergraduate number theory it's like. it's the first really deep theorem that. you prove. what's amazing is you know the fact that. you can factor a number into primes. is much easier essentially euclid knew.

it although he didn't quite put it in. that. in that way the fact that you can do it. at all what's deep is the fact that. there's only one way to do it that or. however you sort of chop the number up. you end up with the same set of. prime factors um and indeed what people. finally understood uh at the end of the. 19th century is that if you work in. number systems. slightly more general than the ones. we're used to. which it turns out are relevant for ma.

all of a sudden this stops being true. things get i mean things get more. complicated and now. because you were praising simplicity. before you were like it's so beautiful. unique factorization. uh it's so great like so when i tell you. that in more general number. systems there is no unique factorization. maybe you're like that's bad i'm like no. that's good because there's like a whole. new world of phenomena to study that you. just can't see. through the lens of the numbers that. we're used to so i'm. i'm for complication i'm highly in favor.

of complication. and every complication is like an. opportunity for new things to study. and is that the big uh kind of uh one of. the big insights for you from uh. andrew wiles is proof is there. interesting insights about. the process they use to prove that sort. of. resonates with you as a mathematician is. there an interesting concept that. emerged from it. is there interesting human aspects to. the proof. whether there's interesting human. aspects to the proof itself is an.

interesting question. certainly it has a huge amount of. richness sort of at its heart. is an argument of on of what's called. deformation theory. um which was in part. created by my my phd advisor barry mazar. can you speak to what deformation theory. is i can speak to what it's like. sure how about that what does it rhyme. with right. well the reason that barry called it. deformation theory. i think he's the one who gave it the. name um i hope i'm not wrong and saying.

this one dave. in your book you have calling different. things by the same name. as one of the things in the beautiful. map that opens the book. yes and this is a perfect example so. this is another phrase of. poincare this like incredible generator. of slogans and aphorisms he said. mathematics is the art of calling. different things by the same name. that very thing that very thing we do. right when we're like this triangle and. this triangle come on they're the same. triangle they're just in a different. place right. so in the same way um it came to be. understood that.

the kinds of objects that you study uh. when you study when you study for. maslow's theorem. and let's not even be too careful about. what these objects are i can tell you. there are gal. representations in modular forms but. saying those words. is not going to mean so much but. whatever they are they're things that. can be. deformed moved around a little bit and. um i think the inside of what andrew and. and then andrew and richard were able to.

do. was to say something like this um. deformation means. moving something just a tiny bit like an. infinitesimal amount. um if you really are good at. understanding which ways a thing can. move. in a tiny tiny tiny infinitesimal amount. in certain directions. maybe you can piece that information. together to understand the whole global. space in which it can move. and essentially their argument comes. down to showing that two of those. big global spaces are actually the same. the fabled r equals t.

part of uh part of their proof which is. at the heart of it. um and it involves this very. careful uh principle like that but that. being said. what i just said it's probably not what. you're thinking because what you're. thinking. when you think oh i have a point in. space and i move it around like a little. tiny bit. um you're using. um your notion of distance that's you. know from calculus. we know what it means for like two. points on the real line to be close.

together. so i get another thing that comes up in. the book a lot. is this fact that the notion of distance. is not given to us by god we could mean. a lot of different things by distance. and just in the english language we do. that all the time we talk about somebody. being a close relative. it doesn't mean they live next door to. you right it means something else. there's a different notion of distance. we have in mind and there are lots of. notions of distances. that you could use you know in the. natural language processing community.

and ai there might be some notion of. semantic distance or lexical distance. between two words how much do they tend. to arise in the same context that's. incredibly important. for um you know doing autocomplete and. like. machine translation and stuff like that. and it doesn't have anything to do with. are they next to each other in the. dictionary right it's a different kind. of distance. okay ready in this kind of number theory. there was a. crazy distance called the periodic. distance i didn't write about this that. much in the book because even though i. love it it's a big part of my research. life it gets a little bit into the weeds. but.

your listeners are going to hear about. it now please where. you know what a normal person says when. they say two numbers are close. they say like you know their difference. is like a small number like seven and. eight are close because their difference. is one and one's pretty small. um if we were to be what's called a two. attic number theorist. we'd say oh two numbers are close if. their difference. is a multiple of a large power of two. so like so like one and. 49 are close because their difference is.

48 and 48 is a multiple of 16 which is a. pretty large power of two. whereas whereas one and two are pretty. far away. because the difference between them is. one which is not even a multiple of a. power of 2 at all it's odd. you want to know what's really far from. 1 like. 1 and 1 64. because their difference is. a negative power of 2. 2 to the minus 6. so those points are. quite quite fast. the power of a large n would be too.

cool if that's the difference between. two numbers and they're close. yeah so two to a large power is this. multiplication. very small number and two to a negative. power. is a very big number that's two attic. okay uh. i can't even visualize that it takes. practice it takes practice if you've. ever heard of the cantor set it looks. kind of like that. so it is crazy that this is good for. anything right. i mean this just sounds like a. definition that someone would make up to. torment you.

but what's amazing is there's a general. theory of distance where you say any. definition you make that satisfies. certain axioms deserves to be called a. distance and this. see i'm starting to interrupt uh my. brain you broke my brain now awesome. uh 10 seconds ago uh because i'm also. starting to map. for the two added case to binary numbers. and sure you know because. because we romanticized those sauce. trunks oh that's exactly the right way. to think of it i was trying to. mess with number you know i was trying. to see okay which ones are close.

and then i'm starting to visualize. different binary numbers and how they. which ones are close to each other and. uh i'm not sure well i think there's. no it's very similar that's exactly the. way to think of it it's almost like. binary numbers written in reverse. right because in a in a binary expansion. two numbers are closed a number that's. small is like point zero zero zero zero. something. something that's the decimal and it. starts with a lot of zeros in the two. attic metric. a binary number is very small if it. ends with a lot of zeros and then the.

decimal point. got you so it is kind of like binary. numbers written backwards is actually i. should have. that's what i should have said lex. that's a very good metaphor. okay but so why is that why is that. interesting except for the fact that uh. it's it's it's a beautiful kind of uh. framework different kind of framework. which you think about distances. and you're talking about not just the. two attic but the generalization of that. yeah the mep and so so that because. that's the kind of deformation that. comes up. in wiles is in wiles as proof that.

defamation we're moving something a. little bit. means a little bit in this to addiction. okay no i mean it's such i mean i could. just get excited to talk about it and i. just taught this like in the fall. semester that um but it like. reformulating. why is uh. so you pick a different uh measure of. distance. over which you can talk about very tiny. changes. and then use that to then prove things. about.

the entire thing yes although you know. honestly what i would say. i mean it's true that we use it to prove. things but i would say we use it to. understand things. and then because we understand things. better then we can prove things but you. know the goal is always the. understanding the goal is not. so much to prove things the goal is not. to know what's true or false i mean this. is something i write about in the book. near the end and it's something that. it's a wonderful wonderful essay by by. bill thurston. kind of one of the great geometers of. our time who unfortunately passed away a. few years ago.

um called on proof and progress in. mathematics and he writes very. wonderfully about how. you know we're not it's not a theorem. factory where we have a. production quota i mean the point of. mathematics is to help humans understand. things. and the way we test that is that we're. proving new theorems along the way. that's the benchmark but that's not the. goal. yeah but just as a as a kind of. absolutely but as a tool. it's kind of interesting to approach a. problem by saying. how can i change the distance function.

like what the the nature of distance. because that might start to lead to. insights. for deeper understanding like if i were. to try to describe. human society by a distance two people. are close. if they love each other right and then. and then start to uh. and do a full analysis on the everybody. that lives on earth currently the seven. billion people you know and. from that perspective as opposed to the. geographic perspective of distance.

and then maybe there could be a bunch of. insights about the source of. uh violence the source of uh maybe. entrepreneurial success or invention or. economic success or different systems. of communism capitalism start to i mean. that's i guess what. economics tries to do but really saying. okay let's think outside the box about. totally new distance functions that. could unlock something. profound about the space yeah because. think about it okay here's. i mean now we're gonna talk about ai.

which you know a lot more about than i. do so. just you know start laughing. uproariously if i say something that's. completely wrong we both know very. little. relative to what we will know centuries. from now. that is that is a really good humble way. to think about it i like it okay so. let's just go for it. um okay so i think you'll agree with. this that. in some sense what's good about ai is. that. we can't test any case in advance the. whole point of ai is to make our one. point of it i guess is to make. good predictions about cases we haven't.

yet seen and in some sense that's always. going to involve some notion of distance. because it's always going to involve. somehow taking a case we haven't seen. and saying. what cases that we have seen is it close. to is it like is it somehow an. interpolation. between now when we do that in order to. talk about things being like other. things. implicitly or explicitly we're invoking. some notion of distance and boy we. better get it right. yeah right if you try to do natural. language processing and your idea about. of distance between words is.

how close they are in the dictionary. when you write them in alphabetical. order you are going to get. pretty bad translations right no the. notion of distance has to come from. somewhere else yeah that that's. essentially what neural networks are. doing this what word and bettings are. doing is. yes coming up with uh in the case of. word embeddings literally like literally. what they are doing is learning a. distance but those are super. complicated distance functions and it's. almost nice to think. maybe there's a nice transformation.

that's simple. uh sorry this there's a nice formulation. of the distance. again with the simple so you don't. let me ask you about this from an. understanding perspective there's the. richard feynman maybe attributed to him. but maybe many others. is this idea that if you can't explain. something simply that you don't. understand it. in how many cases how often. is that true do you find there's some.

profound truth in that. oh okay so you were about to ask is it. true to which i would say flatly no but. then you said. you followed that up with is there some. profound truth in it and i'm like. okay sure so there's some truth in it. but it's not true. [Laughter]. this is your mathematician answer the. truth that is in it. yeah is that learning to explain. something helps you. understand it um but real things are not.

simple. yeah a few things are most are not. um and i don't to be honest i don't i. mean i don't. we don't really know whether feynman. really said that right or something like. that is sort of. disputed but i don't think feynman could. have literally believed that. whether or not he said it and you know. he was the kind of guy i didn't know him. but i'm. reading his writing he liked to sort of. say stuff like stuff that sounded good. you know what i mean so it's it's. totally strikes me as the kind of thing. he could have said because he liked the. way. saying it made him feel but also knowing.

that he didn't like literally mean it. well i definitely have. have a lot of friends and i've talked to. a lot of physicists and they do. derive joy from believing that they can. explain stuff simply. or believing it's possible to explain. style simply. even when the explanation is not. actually that simple like i've. heard people think that the explanation. is simple and they do the explanation. and i think. it is simple but it's not capturing the. phenomena that we're discussing.

it's capturing it somehow maps in their. mind but it's it's taking as a. starting point as an assumption that. there's a deep knowledge and a deep. understanding that's. that's actually very complicated and the. simplicity is almost like a. almost like a poem about the more. complicated thing as opposed to. a distillation and i love poems but a. poem is not an explanation. well some people might disagree with. that but certainly from a mathematical. perspective no poet would disagree with. it. no poet would disagree you don't think.

there's some things that can only be. described. imprecisely i said explanation i don't. think any poem. i don't think any poet would say their. poem is an explanation they might say. it's a description they might say it's. sort of capturing sort of. well some people might say the only. truth is like music. right that the the the not the only. truth but some truth can only be. expressed through art. and i mean that's the whole thing we're. talking about religion and myth and. there's some things that uh are limited.

cognitive capabilities. and the tools of mathematics or the. tools of physics are just not going to. allow us to capture. like it's possible consciousness is one. of those things. yes that is definitely possible but i. would even say. look unconsciousness is a thing about. which we're still in the dark as to. where. whether there's an explanation we would. we would understand it as an explanation. at all by the way okay i got to give. yeah one more amazing poincare quote. because this guy just never stopped. coming out with great quotes that. um you know paul erdish another fellow.

who appears in the book and by the way. he thinks about this notion of distance. of like personal affinity kind of like. what you're talking about that kind of. social network and that. notion of distance that comes from that. so that's something that erdos did. well he thought about distances and. networks i guess he didn't probably he. didn't think about the social media that. was fascinating and that's how it. started that story of virtus number yeah. okay. but you know eredish was sort of famous. for saying and this is sort of long. lines we're saying. he talked about the book capital t. capital b.

the book and that's the book where god. keeps the right proof of every. theorem so when he saw a proof he really. liked it was like really elegant really. simple. like that's from the book that's like. you found one of the ones that's in. the book um he wasn't the religious guy. by the way he referred to god as the. supreme fascist he was like uh. but somehow he was like i don't really. believe in god but i believe in god's. book i mean it was uh. yeah um but poincare on the other hand. um and by the way there are other. members hilda hudson is one who comes up. in this book she also kind of. saw math um she's one of the people who.

sort of develops. um the disease model that we now use. that we use to sort of track pandemics. this sir model that sort of originally. comes from. her work with ronald ross but she was. also super super super devout. and she also sort of from the other side. of the religious coin was like yeah math. is how we communicate with god. she has a great all these people are. incredibly quotable she says you know. math isn't. the truth the things about mathematics. is like they're not the most important. of god thoughts. but they're the only ones that we can.

know precisely. so she's like this is the one place. where we get to sort of see what god's. thinking when we do. mathematics again not a fan of poetry or. music some people will say hendrix is. like. some some people say chapter one of that. book is mathematics and then chapter two. is like classic rock right so. like it's not clear that the i'm sorry. you just sent me off on a tangent just. imagining like eredish at a hendrick's. concert like trying to sort of figure. out if it was from. the book or not but i was what i was. coming to was justice it but one point.

said about this is he's like you know. if like this has all worked out in the. language of the divine and if a divine. being like came down and. told it to us we wouldn't be able to. understand it so it doesn't matter. so poincare was of the view that there. were things that were sort of like. inhumanly complex. and that was how they really were our. job is to figure out the things that are. not like that. that are not like that all this talk of. primes got me hungry for primes. you uh your blog post the beauty of.

bounding gaps a huge. discovery about prime numbers and what. it means for the future of math. can you tell me about prior numbers what. the heck are those what are twin primes. what are prime gaps what are bounding. gaps and primes what are all these. things. and what if anything or what exactly is. beautiful about them. yeah so you know prime numbers. are one of the things that number. theorists study the most and have.

for millennia um they are numbers. which can't be factored and then you say. like like five and then you're like wait. i can't factor five five is. five times one okay not like that that. is a factorization it absolutely is a. way of expressing five as. a product of two things but don't you. agree there's like something trivial. about it it's something you can do to. any number. it doesn't have content the way that if. i say that 12 is 6. times 2 or 35 is 7 times 5 i've really. done something to it i've broken up so. those are the kind of factorizations.

that count. and a number that doesn't have a. factorization like that is called prime. except historical side note one. which at some times in mathematical. history has been deemed to be a prime. but. currently is not and i think that's for. the best but i bring it up only because. sometimes people think that. you know these definitions are kind of. if we think about them hard enough we. can figure out which definition is true. no there's just an artifact in. mathematics so yeah one. so which definition is best for us.

for our purposes well those edge cases. are weird right so uh. so so it can't you can't be it doesn't. count when you use yourself. as a number or one as part of the. factorization. or as the entirety of the factorization. so the so you somehow get to the meat of. the number. by factorizing it and that's seems to. get to the core of all of mathematics. yeah you take any number and you. factorize it until you can factorize no.

more and what you have left is some big. pile of primes i mean by definition when. you. can't factor anymore when you when. you're done you can't. break the numbers up anymore what's left. must be prime you know 12 breaks into. two. and two and three um so these numbers. are the atoms the building blocks. of all numbers and there's a lot we know. about them but there's much more we. don't know them i'll tell you the first. few there's 2. 3 5 7 11. by the way they're all going to be odd.

from the non because if they were even i. could factor out 2 out of them. but it's not all the odd numbers 9 isn't. prime because it's 3 times 3. 15 isn't prime because it's 3 times 5. but 13 is where were we 2 3 5. 7 11 13 17 19. not 21 but 23 is etcetera etcetera okay. so you could go on. how high could you go if we were just. sitting here by the way your own brain. continuous without interruption would. you be able to go over a hundred. i think so there's always those ones. that trip people up there's. there's a famous one the groton deak.

prime 57 like sort of. alexander groddendyk the great algebraic. geometer was sort of giving. some lecture involving a choice of a. prime in general and somebody said like. can't you just choose a problem he said. okay 57 which is in fact not prime it's. three times. 19. oh damn but it was like. i promise you in some circles it's a. funny story okay um. but um there's a humor in it uh. yes i would say over a hundred i. definitely don't. remember like 107 i think i'm not sure.

okay like so is there a category of uh. like fake primes that that. are easily mistaken to be prime like 57. i wonder yeah so i would say. 57 and take a small 27 and 51. are definitely like prime offenders oh i. didn't do that on purpose. well done didn't do it on purpose anyway. they're definitely ones that people uh. or 91 is another classic seven times 13.

it really. feels kind of prime doesn't it but it is. not yeah. um but there's also by the way but. there's also an actual notion of. pseudoprime which is. which is the thing with the formal. definition which is not a psychological. thing. it is a prime which passes a primality. test. devised by fermat which is a very good. test which. if if a number fails this test it's. definitely not prime. and so there was some hope that oh maybe. if a number passes the test then it. definitely is prime that would give a. very simple criterion for formality.

unfortunately it's only perfect in one. direction. so there are numbers i want to say 341. is the smallest. uh which passed the test but are not. prime 341 is this test easily. explainable or no. uh yes actually um ready let me give you. the simplest version of it you can dress. it up a little bit but here's the basic. idea. uh i take the number the mystery number. i raised two to that power. so let's say your mystery number is six.

yeah are you sorry you asked me are you. ready no i might you're breaking my. brain again but yes. let's let's let's do it we're gonna do a. live demonstration um. let's say your number is six so i'm. going to raise 2 to the sixth power. okay so if i were working out i'd be. like that's 2 cubes squared so that's 8. times 8 so that's 64. now we're going to divide by 6 but i. don't actually care what the quotient is. only the remainder. so let's see 64 divided by 6.

is uh well it's it there's a quotient of. 10 but the remainder is 4. so you failed because the answer has to. be 2. for any prime let's do it with five. which is prime. two to the fifth is 32 divide 32 by five. uh and you get six with a remainder of. two. well the remainder of two here for seven. two to the seventh is 128. divide that by seven and let's see i. think that's seven times.

14 is that right no. seven times 18. is 126 with a remainder of two right 128. is. a multiple of seven plus two so if that. remainder is not two. then that's definitely not that it's. definitely not prime. and then if it is it's likely a prime. but not for sure. it's likely a prime but not for sure and. there's actually a beautiful geometric. proof which is in the book actually. that's like one of the most granular. parts of the book because it's such a. beautiful proof i couldn't not give it.

so you you draw a lot of like. opal and pearl necklaces and spin them. that's kind of the geometric nature of. the. of this proof of fermat's little theorem. um. so yeah so with pseudo primes there are. primes that are kind of faking if they. pass that test but there. are numbers that are faking it that pass. that test but are not actually prime. um but the point is. um there are many many many. theorems about prime numbers um. are there like there's a bunch of.

questions to ask is there an infinite. number of primes. can we say something about the gap. between primes as the. numbers grow larger and larger and. larger and so on. yeah it's a perfect example of your. desire for simplicity in all things you. know it would be really simple. if there was only finitely many primes. yes and then there would be this sim. finite set of atoms that all numbers. would be built up right. that would be very simple and good in. certain ways but it's completely false. and number theory would be totally. different if that were the case it's.

just not true. um in fact this is something else that. euclid knew so this is a very very old. fact like much before long before we had. anything like modern numbers that primes. are infinite. the primes that there are that write the. there's an infinite number of primes. so what about the gaps between the. primes right so so one thing that people. recognized. and really thought about a lot is that. the primes on average. seem to get farther and farther apart as. they get bigger and bigger in other. words it's less and less common. like i already told you of the first 10.

numbers two three five seven four of. them are prime that's a lot forty. percent. if i looked at you know ten digit. numbers. no way would forty percent of those be. prime being prime would be a lot rarer. in some sense because there's a lot more. things for them to be divisible by. that's one way of thinking of it it's. it's a lot more possible for there to be. a factorization because there's a lot of. things you can try to factor out of it. as the numbers get bigger and bigger. primarily gets rarer and rarer. and the extent to which that's the case.

that's pretty well understood but then. you can ask more fine-grained questions. and here is one. um a. twin prime is a pair of primes that are. two apart. like three and five or like 11 and 13. or like 17 and 19. and one thing we. still don't know is. are there infinitely many of those we. know on average they get farther and. farther apart but that doesn't mean. there couldn't be like occasional. folks that come close together and.

indeed. uh we think that there are and one. interesting question. i mean this is because i think you might. say like well. why how could one possibly have a right. to have an opinion about something like. that. like what you know we don't have any way. of describing a process that makes. primes like sure you can like. look at your computer and see a lot of. them but the fact that there's a lot. why is that evidence that there's. infinitely many right maybe i can go on. the computer and find 10 million well 10. million 10 million is pretty far from. infinity right so how is that how is.

that evidence there's a lot of things. there's like a lot more than 10 million. atoms that doesn't mean there's. infinitely many atoms in the universe. right i mean on most people's physical. theories there's probably not as i. understand it. okay so why would we think this. the answer is that we've that it turns. out to be like. incredibly productive and enlightening. to think about primes. as if they were random numbers as if. they were randomly distributed. according to a certain law now they're. not they're not random there's no chance. involved. it's completely deterministic whether a.

number is prime or not and yet. it just turns out to be phenomenally. useful useful in mathematics. to say even if something is. governed by a deterministic law let's. just pretend it wasn't. let's just pretend that they were. produced by some random process and see. if the behavior is roughly the same. and if it's not maybe change the random. process maybe make the randomness a. little bit different and tweak it and. see if you can find a random process. that matches the behavior we see and. then maybe you predict that. other behaviors um of the system are.

like that of the random process. and so that's kind of like it's funny. because i think when you talk to people. about the twin prime conjecture. people think you're saying wow there's. like some deep. structure there that like makes those. primes be like close together again and. again. and no it's the opposite of deep. structure what we say when we say we. believe the twin prime conjecture is. that we believe the primes are like sort. of. strewn around pretty randomly and if. they were then by chance you would. expect there to be infinitely many twin. primes and we're saying yep we expect. them to behave just like they would if.

they were random dirt. the you know the fascinating parallel. here is uh. i just got a chance to talk to sam. harris and he uses the prime numbers. as an example often i don't know if. you're familiar. with who sam is he uses that as an. example. of there being no free will. wait where did you get this well he just. uses. as an example of it might seem like this. is a random number generator. but it's all like formally defined so if.

we keep. getting more and more primes then. like that might feel like a new. discovery and that might feel like a new. experience but it's not. it was always written in the cards but. it's funny that you say that because a. lot of people think of like. randomness uh the fundamental. randomness within the nature of reality. might be the source. of something that we experience as free. will and you're saying it's like useful. to look at prime numbers as.

um as a random process. in order to prove stuff about them but. fundamentally of course it's not a. random process. well not in order to prove some stuff. about them so much as to figure out. what we expect to be true and then try. to prove that here's what you don't want. to do try really hard to prove something. that's false that makes it really hard. to prove the thing if it's false. so you certainly want to have some. heuristic ways of guessing making good. guesses about what's true so yeah. here's what i would say let's you're. going to be imaginary sam harris now yes. like. you are talking about prime numbers and.

you are like but prime numbers are. completely deterministic. and i'm saying like well but let's treat. them like a random process. and then you say but you're just saying. something that's not true they're not a. random process or deterministic and i'm. like okay great you hold to your. insistence that is honoring the process. meanwhile i'm generating insight about. the primes that you're not because i'm. willing to sort of pretend that there's. something that they're not in order to. understand what's going on yeah so it. doesn't matter what the reality is what. matters is. what's uh what framework. of thought results in the maximum number. of insights.

yeah because i feel look i'm sorry but i. feel like you have more insights about. people if you. think of them as like beings that have. wants and needs and desires and do stuff. on. purpose even if that's not true you. still understand better what's going on. by treating them in that way don't you. find look when you work on machine. learning. don't you find yourself sort of talking. about what the machine is what the. machine is trying to do. in a certain instance do you not find. yourself drawn to that language. well oh it knows this it's trying to do. that it's learning that. i'm certainly drawn to that language to.

the point where i. received quite a bit of criticisms for. it because i you know like. oh i'm on your side man so especially in. robotics. i don't know why but robotics people. don't like to. name their robots or they they certainly. don't like to gender their robots. because the moment you gender a robot. you start to anthropomorphize. if you say he or she you start to you in. your mind construct like a. um like a life story in your mind you. can't help it. this like you create like a humorous.

story to this person you start to. understand this person this robot you. start to project your own. but i think that's what we do to each. other and i think that's actually really. useful from the engineering. process especially for human robot. interaction. and yes for machine learning systems for. helping you build an intuition about a. particular problem. it's almost like asking this question. you know when a machine learning system. fails in a particular edge case. asking like what were you thinking about. like.

like asking like almost like when you're. talking about to a child. who just does something bad you're. you want to understand like what was um. how did they see the world maybe there's. a totally new maybe you're the one. that's thinking about the world. incorrectly and uh yeah that. anthropomorphization process i think is. ultimately good for insight and the same. is. i i i agree with you i tend to believe. about free will. as well let me ask you a ridiculous. question if it's okay. of course i've just recently.

most people go on like rabbit hole like. youtube. things and i went on a rabbit hole often. do of wikipedia. and i found a page on uh. finitism ultra finitism. and intuitionism or you need to i'm i. forget what it's called. yeah intuitionism intuitionism that. seemed pretty. pretty interesting i have my to-do list. to actually like look into. like is there people who like formally.

attract like real mathematicians are. trying to argue for this. but the belief there i think let's say. find nitism that. infinity is is fake. meaning um infinity might be like a. useful hack. for certain like a useful tool in. mathematics but it really. gets us into trouble because there's no. infinity. in the real world maybe i'm sort of. not expressing that uh fully correctly.

but basically saying like there's things. that are. in once you add into mathematics things. that are not. provably within the physical world. you're starting to. inject to corrupt your. framework of reason what do you think. about that. i mean i think okay so first of all i'm. not an expert and. i couldn't even tell you what the. difference is between those three terms. finetism ultrafinitism and intuitionism. although i know they're related i tend.

to associate them with the netherlands. in the 1930s. okay i'll tell you can i just quickly. comment because i read the wikipedia. page. the difference in ultra like the. ultimate sentence of the modern age can. i just comment because i read the. wikipedia page that sums up our moment. bro i'm basically an expert ultra ultra. finitism. so financialism says that the only. infinity you're allowed to have is that. the natural numbers are. infinite so like those numbers are. infinite. so like one two three four five the.

integers. are internet the ultra financialism says. nope even that infinity's fake that's. fine. i bet ultra fanatism came second i bet. it's like when there's like a hardcore. scene and then one guy is like. oh now there's a lot of people in the. scene i have to find a way to be more. hardcore than the hardcore people. go back to the emo doc yeah okay so is. there any uh. are you ever because i'm not often. uncomfortable with infinity. like psychologically i i you know i have.

i have. trouble when that sneaks in there. it's because it works so damn well i get. a little suspicious. um because it could be almost like a. crutch. or an oversimplification that's missing. something profound about reality. well so first of all okay if you say. like. is there like a serious way of doing. mathematics that doesn't. really treat infinity as a real. thing or maybe it's kind of agnostic and.

it's like i'm not really gonna make a. firm statement about whether it's a real. thing or not yeah that's called most of. the history of mathematics. right so it's only after cantor right. that we really are. sort of okay we're gonna like have a. notion of like the cardinality of an. infinite set. and like um do something that you might. call like the modern theory of infinity. um that said obviously everybody was. drawn to this notion and no not. everybody was comfortable with it look i. mean this is what happens with newton. right i mean so. newton understands that to talk about.

tangents and to talk about instantaneous. velocity. um he has to do something that we would. now call taking a limit. right the fabled dy for dx if you sort. of go back to your calculus class for. those who have taken calculus remember. this mysterious thing. and you know what is it what is it well. he'd say like. well it's like you sort of um divide the. length of this line segment. by the length of this other line segment. and then you make them a little shorter. and you divide again and then you make. them a little shorter and you divide. again and then you just keep on doing. that until they're like infinitely short.

and then you divide them again. these quantities that are like they're. not zero. but they're also smaller than any. actual number these infinite decimals. well people were queasy about it and. they weren't wrong to be queasy about it. right from a modern perspective it was. not really well formed there's this very. famous critique of newton. by bishop berkeley where he says like. what these things you define like. you know they're not zero but they're. smaller than any number are they the.

ghosts of departed quantities. that was this like ultra line. and on the one hand he was right. it wasn't really rigorously modern. standards on the other hand like newton. was out there doing calculus. and other people were not right it. worked and it worked. i think i think a sort of intuitionist. few for instance. i would say would express serious doubt. and by. it's not by the way it's not just. infinity it's like saying i think we. would express serious doubt that like. the real numbers exist now most people.

are. comfortable with the real numbers. well computer scientists with floating. point number i mean the. floating point of arithmetic that is. that's a great point actually. i think in some sense this flavor of. doing math saying. we shouldn't talk about things that we. cannot specify in a finite amount of. time there's something very. computational in flavor about that and. it's probably not a coincidence that it. becomes popular.

in the 30s and 40s which is also like. kind of like the dawn of. ideas about formal computation right you. probably know the timeline better than i. do sorry what because popular the. these ideas that maybe we should be. doing math in this more restrictive way. where. even a thing that you know because look. the origin of all this is like. you know number represents a magnitude. like the length of a line like so i mean. the idea that. there's a continuum there's sort of like. there's like. um it's pretty old but that you know.

just cause something is old doesn't mean. we can't. reject it if we want to well a lot of. the fundamental ideas in computer. science when you talk about the. complexity. of problems uh to touring himself they. rely on. an infinity as well the ideas that kind. of challenge that. the whole space of machine learning i. would say challenges that. it's almost like the engineering. approach to things like the floating. point of arithmetic the other one that. back to john conway that challenges this. idea.

i mean maybe to tying the the ideas of. deformation theory. and and uh. limits to infinity is this idea of. cellular automata. with uh john conway looking at the game. of life stephen wolfram's work. that i've been a big fan of for a while. of cellular autonomy i was i was. wondering if you have. if you have ever encountered these kinds. of objects. you ever looked at them as a. mathematician where you have very simple.

rules. of tiny little objects that when taken. as a whole create. incredible complexities but are very. difficult to analyze. very difficult to make sense of even. though the one individual object. one part it's like what you're saying. about andrew wiles like you. you can look at the deformation of a. small piece to tell you about the hole. it feels like with cellular automata. or any kind of complex systems it's. it's often very difficult to say. something about the whole thing.

even when you can precisely describe the. operation. of uh the sm the local neighborhoods. yeah i mean i love that subject i. haven't really done research in it. myself i've played around with it i'll. send you a fun blog post i wrote where i. made some cool. texture patterns from cellular. autonomous um but. um and those are really always. compelling it's like. you create simple rules and they create. some beautiful textures it doesn't make. any. actually did you see there was a great. paper i don't know if you saw this like. a machine learning paper.

yes i don't know if you want to talk. about where they were like learning the. texture is like let's try to like. reverse engineer and like learn a. cellular automaton that can reduce. texture that looks like this. from the images very cool and as you say. the thing you said is i feel the same. way when i read machine learning paper. is that what's especially interesting is. the cases where it doesn't work. like what does it do when it doesn't do. the thing that you tried to train it. yeah to do that's extremely interesting. yeah yeah that was a cool paper. so yeah so let's start with the game of. life let's start with. um or let's start with john conway so.

conway. so yeah so let's start with john conway. again just i don't know. from my outsider's perspective there's. not many mathematicians. that stand out throughout the history of. the 20th century. he's one of them i feel like he's not. sufficiently recognized. i think he's pretty recognized okay well. i mean. he was a full professor of princeton for. most of his life he was sort of. certainly at the pinnacle of. yeah but i found myself every time i. talk about conway and how excited.

i am about him i have to constantly. explain to. people who he is and that's that's. always a. sad sign to me but that's probably true. for a lot of mathematicians. i was about to say like i feel like you. have a very elevated idea of how famous. but. this is what happens when you grow up in. the soviet union you know you think the. mathematicians are like very very famous. yeah but i'm not actually so convinced. at a tiny tangent that. that shouldn't be so i mean there's uh. it's not obvious to me that that's one. of the like if if i were to analyze.

american society that uh. perhaps elevating mathematical and. scientific thinking to a little bit. higher. level would benefit the society well. both in discovering the beauty of what. it is to be human. and for actually creating cool. technology better iphones. but anyway john conway yeah and conway. is such a perfect example of somebody. whose humanity was. and his personality was like wound up. with his mathematics right so it's not. sometimes i think people. who are outside the field think of. mathematics as this kind of like. cold thing that you do separate from.

your existence as a human being no way. your personality is in there just as it. would be in like a novel you wrote or a. painting you painted or just like the. way you walk down the street like it's. in there it's you doing it. and conway was certainly a singular. personality um. i think anybody would say that. he was playful like everything was a. game to him. now what you might think i'm going to. say and it's true is that he sort of. was very playful in his way of doing.

mathematics. but it's also true it went both ways he. also sort of made mathematics. out of games he like looked like he was. a constant inventor of games with like. crazy names and. then he would sort of analyze those. games mathematically um. to the point that he and then later. collaborating with knuth like. you know created this number system the. surreal numbers. in which actually each number is a game. there's a wonderful book about this. called i mean there are his own books. and then there's like a book that he. wrote with burleigh camping guy called. winning ways.

which is such a rich source of ideas. um and he. too kind of has his own crazy number. system in which by the way there are. these. infinitesimals the ghosts of departed. quantities they're in there. now not as ghosts but as like certain. kind of two-player games. um so. you know he was a guy so i knew him when. i was a postdoc. um and i knew him at princeton and our.

research overlapped in some ways now it. was on stuff that he had worked on many. years before the stuff i was working on. kind of connected with stuff in group. theory which. somehow seems to keep coming up um. and so i often would like sort of ask. him a question i would have come upon. him in the common room and i would ask. him a question about something. and just any time you turned him on you. know what i mean you sort of asked the. question. it was just like turning a knob and. winding him up. and he would just go and you would get a.

response that was like. so rich and went so many places and. taught you so much. and usually had nothing to do with your. question yeah usually. your question was just to prompt to him. you couldn't count on actually getting. the questions. brilliant curious minds even at that age. yeah it was. definitely a huge loss uh but. on his game of life which was i think he. developed in the 70s. as almost like a side thing a fun little. experience. of life is this um it's a very simple.

algorithm it's not really a game per se. in the sense of the kinds of games that. he liked whereas people played against. each other and. um but. essentially it's a game that you play. with marking. little squares on the sheet of graph. paper and in the 70s i think he was like. literally doing it with like a pen. on graph paper you have some. configuration of squares some of the. squares in the graphic. are filled in some are not and then. there's a rule a single rule.

that tells you um at the next stage. which squares are filled in and which. squares are not. sometimes an empty square gets filled in. that's called birth sometimes a. square that's filled in gets erased. that's called death and there's rules. for which squares are born which squares. die. um it's um. the rule is very simple you can write it. on one line and then the great miracle. is. that you can start from some very. innocent looking. little small set of boxes and.

get these results of incredible richness. and of course nowadays you don't do it. on paper nowadays you're doing a. computer there's actually a great. ipad app called golly which i really. like that has like. conway's original rule and like gosh. like hundreds of other. variants and it's lightning fast so you. can just be like i want to see. 10 000 generations of this rule play out. like faster than your eye can even. follow and it's like amazing so i highly. recommend it if this is at all. intriguing to you getting golly on your. uh ios device and you can do this kind.

of process which i really enjoy doing. which is almost from like putting a. darwin hat on or. a biologist head-on and doing analysis. of. a higher level of abstraction like the. organisms that spring up. because there's different kinds of. organisms like you can think of them as. species and they interact with each. other. they can uh there's gliders they shoot. different there's like. things that can travel around there's. things that can. glider guns that can generate those. gliders they they're.

you can use the same kind of language as. you would about describing a biological. system. so it's a wonderful laboratory and it's. kind of a rebuke to someone who. doesn't think that like very very rich. complex structure can. come from very simple underlying laws. like it definitely can now. here's what's interesting if you just. picked like some random rule. you wouldn't get interesting complexity. i think that's one of the most. interesting things of these uh. one of these most interesting features.

of this whole subject that the rules. have to be tuned just. right like a sort of typical rule set. doesn't generate any kind of interesting. behavior. yeah but some do i don't. think we have a clear way of. understanding what's doing which don't i. don't know maybe stephen thinks he does. i don't know. but no no it's a giant mystery what. stephen what stephen wolfram did. is um now there's a whole interesting. aspect of the fact that he's. a little bit of an outcast in the. mathematics and physics community. because he's so focused on a particular.

his particular work. i think if you put ego aside which i. think. unfairly some people are not able to. look beyond i think his work is actually. quite brilliant. but what he did is exactly this process. of darwin-like exploration. is taking these very simple ideas and. writing a thousand page book on them. meaning like let's play around with this. thing let's see. and can we figure anything out spoiler. alert. no we can't in fact he does uh. he does a challenge uh i think it's like.

a rule 30 challenge which is quite. interesting just. simply for machine learning people for. mathematics. people is can you predict the middle. column. for his it's a it's a it's a 1d cellular. automata. can you pre generally speaking can you. predict. anything about how a particular rule. will evolve. just in the future uh very simple. just look at one particular part of the. world just zooming in on that part.

you know 100 steps ahead can you predict. something. and uh the the the challenge is to do. that kind of prediction. so far as nobody's come up with an. answer but the point is like. we can't we don't have tools or maybe. it's impossible or. i mean he has these kind of laws of. irreducibility they hear firstly but. it's poetry it's like we can't prove. these things it seems like we can't. that's the basic. uh it almost sounds like ancient. mathematics or something like that where. you. like the gods will not allow us to.

predict the cellular automata. but uh that's fascinating that we can't. i'm not sure what to make of it and. there is power to calling this. particular. set of rules game of life as conway did. because not actually exactly sure but i. think he had a sense that there's. some core ideas here that are. fundamental. to life to complex systems to the way. life emerged on earth. i'm not sure i think conway thought that.

it's something that i mean conway always. had a rather ambivalent. relationship with the game of life. because i think he. saw it as it was certainly the thing he. was most famous for in the outside world. and i think that he his view which is. correct. is that he had done things that were. much deeper mathematically than that you. know. and i think it always like grieved him a. bit that he was like the game of life. guy. when you know he proved all these. wonderful theorems and like did i mean.

created all these wonderful games like. created the serial numbers like i mean. he did i mean. he was a very tireless guy who like just. like did like an incredibly. variegated array of stuff so he was. exactly the kind of person who. you would never want to like reduce to. like one achievement you know what i. mean. let me ask about group theory. you mentioned a few times what is group. theory. what is an idea from group theory that. you find beautiful. well so i would say group theory sort of.

starts. as the general theory of symmetry is. that. you know people looked at different. kinds of things and said like. as we said like oh it could have maybe. all there is is the symmetry from left. to right like a human being right or. that's roughly bilateral bilaterally. symmetric as we say. so um so there's two symmetries and then. you're like well wait didn't i say. there's just one there's just left. to right well we always count the. symmetry of doing nothing. we always count the symmetry that's like.

there's flip and don't flip those are. the two configurations that you can be. in so there's two. um you know something like a rectangle. is bilaterally symmetric you can flip it. left to right but you can also flip it. top to bottom. so there's actually four symmetries. there's do nothing. flip it left to right and flip it top to. bottom or do both of those things. um a square. um there's even more because now you can.

rotate it. you can rotate it by 90 degrees so you. can't do that that's not a symmetry of. the rectangle if you try to rotate it 90. degrees you get a rectangle oriented in. a different way. so um a person has two symmetries a. rectangle four. a square eight different kinds of shapes. have different numbers of symmetries. um and the real observation is that. that's just not like a set of things. they can be combined you do one symmetry. then you do another. the result of that is some third.

symmetry. so a group really abstracts away this. notion of saying. um. it's just some collection of. transformations you can do to a thing. where you. combine any two of them to get a third. so you know a place where this comes up. in computer sciences and. is in sorting because the ways of. permuting a set. the ways of taking sort of some set of. things you have on the table and putting. them in a different order. shuffling a deck of cards for instance. those are the symmetries of the deck and. there's a lot of them there's not two. there's not four there's not eight think.

about how many different orders the deck. of card can be in each one of those is. the result of applying a symmetry. uh to the original deck so a shuffle is. a symmetry right you're reordering the. cards. if if i shuffle and then you shuffle the. result. is some other kind of thing you might. call it duffel. a double shuffle which is a more. complicated symmetry so group theory is. kind of. the study of the general abstract world. that encompasses. all these kinds of things but then of. course like lots of things that are way. more complicated than that.

like infinite groups of symmetries for. instance so thank you. oh yeah okay well okay ready think about. the symmetries of the line you're like. okay i can. reflect it left to right you know around. the origin. okay but i could also reflect it left to. right grabbing. somewhere else like at one or two or pi. or anywhere. or i could just slide it some distance. that's a symmetry slide it five units. over so there's clearly infinitely many. symmetries of the line.

that's an example of an infinite group. of symmetries is it possible to say. something that kind of captivates keeps. being. brought up by physicists which is gage. theory gauge symmetry. as one of the more complicated type of. symmetries is there is that. is there a easy explanation what the. heck it is is that something that comes. up. on your mind at all well i'm not a. mathematical physicist but i can say. this. it is certainly true that has been a. very. useful notion in physics to try to say.

like. what are the symmetry groups like of the. world like what are the symmetries under. which things don't change right so. we just i think we talked a little bit. earlier about it should be a basic. principle that a theorem that's true. here. is also true over there yes and same for. a physical law right i mean. if gravity is like this over here it. should also be like this over there okay. what that's saying is we think. translation in space. should be a symmetry all the laws of. physics should be unchanged. if the symmetry we have in mind is a. very simple one like translation. and so then um there becomes a question.

like what are the symmetries. of the actual world with its physical. laws. and one way of thinking isn't. oversimplification but like one way of. thinking of this big. um shift from. uh before einstein to after is that we. just changed our idea about what the. fundamental group of. symmetries were so that things like the.

lorenz contraction things like these. bizarre. relativistic phenomena or lorenz would. have said. oh to make this work we need a thing to. um. to change its shape if it's moving. yeah nearly the speed of light well. under the new frame of framework. it's much better you feel like oh no it. wasn't changing its jeep you were just. wrong about what counted as a symmetry. now that we have this new group the. so-called lorenz group now that we. understand what the symmetries really. are we see it was just an illusion that.

the the thing was changing its shape. yeah so you can then describe the. sameness of things under. this weirdness that exactly that is. general relativity for example. yeah yeah still um. i wish there was a simpler explanation. of like exactly i mean. get you know gauge symmetry is a pretty. simple. general concept about rulers being. deformed. i it's just i i uh i've actually just. personally been.

on a search not a very uh rigorous. or aggressive search but for uh. something i personally enjoy which is. taking complicated. concepts and finding the sort of minimal. example. that i can play around with especially. programmatically that's great i mean. that this is what we try to train our. students to do right i mean in class. this is exactly what. this is like best pedagogical practice i. do. hope there's simple explanation. especially like. i've uh in my sort of uh.

drunk random walk drunk walk whatever. that's called. uh sometimes stumble into the world of. topology. and like quickly like you know when you. like go to a party and you realize this. is not the right. party for me so whenever i go into. topology it's like. so much math everywhere i don't even. know what it feels like. this is me like being a hater is i think. there's way too much math like. they're two the cool kids who just want.

to have like everything is expressed. through math. because they're actually afraid to. express stuff simply through language. that's that's my hater formulation of. topology but at the same time i'm sure. that's very necessary to do sort of. rigorous. discussion but i feel like but don't you. think that's what gauge symmetry is like. i mean it's not a field i know well but. it certainly seems like yes it is like. that okay but. my problem with topology okay and even. like differential geom and differential. geometry. is like you're talking about beautiful. things.

like if they could be visualized it's. open question if. everything could be visualized but. you're talking about things that could. be. visually stunning i think. but they are hidden underneath all of. that math. like if you look at the papers that are. written in topology if you look at all. the discussions on stack exchange. they're all math dense math heavy and. the only kind of. visual things that emerge every once in. a while. is like uh something like a mobius strip.

every once in a while some kind of uh um. [Music]. simple visualizations well there's the. the vibration there's the hop vibration. or all those kinds of things. that somebody some grad student from. like 20 years ago. wrote a program in fortran to visualize. it and that's it. and it's just you know it makes me sad. because. those are visual disciplines just like. computer vision is a visual discipline. so you can provide a lot of visual. examples i wish topology.

was more excited and in love with. visualizing some of the ideas. i mean you could say that but i would. say for me a picture of the hop. vibration does nothing for me. whereas like when you're like oh it's. like about the quaternions it's like a. subgroup of the quaternions i'm like oh. so now i see what's going on like why. didn't you just say that why were you. like showing me this stupid picture. instead of telling me what you were. talking about. oh yeah yeah i'm just saying no but it. goes back to what you're saying about. teaching that like people are different. in what they'll respond to so i think.

there's no i mean. i'm very opposed to the idea that. there's one right way to explain things. i think there's a huge variation in like. you know our brains like have all these. like weird like hooks and loops and it's. like very hard to know like what's going. to latch on and it's not going to be the. same thing. for everybody so well i think. monoculture is bad right i think that's. and i think we're agreeing on that point. that like. it's good that there's like a lot of. different ways in and a lot of different. ways to describe these ideas because. different people are going to find. different things illuminating but.

that said i think there's a lot to be. discovered. when you. force little like silos of brilliant. people. to kind of find a middle ground. or like uh aggregate or. come together in a way so there's like. people that do love visual things. i mean there's a lot of disciplines. especially in computer science. that they're obsessed with visualizing. visualizing data.

visualizing neural networks i mean. neural networks themselves are. fundamentally visual there's a lot of. work in computer vision that's very. visual. and then coming together with some some. folks that were like deeply rigorous. and are like totally lost in. multi-dimensional space where it's hard. to even bring them back down to 3d. [Laughter]. they're very comfortable in this. multi-dimensional space so forcing them. to kind of work together to communicate. because it's not just about public. communication of ideas. it's also i feel like when you're forced. to do that public communication like you.

did with your book. i think deep profound ideas can be. discovered that's like applicable for. research and for science. like there's something about that. simplification or not simplification but. this distillation or condensation or. whatever the hell you call it. compression. of ideas that somehow actually. stimulates creativity. and uh i'd be excited to see more of. that in the. in in the mathematics community can you. let me make a crazy metaphor. maybe it's a little bit like the.

relation between prose and poetry right. i mean if you you might say like why do. we need anything more than prose you're. trying to convey some information so you. just like say it. um well poetry does something right it's. sort of. you might think of it as a kind of. compression of course not all poetry is. compressed like not awesome. some of it is quite baggy but like. um you are kind of. often it's compressed right a lyric poem. is often sort of like a compression of. what would take a long. time and be complicated to explain in. prose into sort of.

a different mode that it's going to hit. in a different way we talked about. poncare conjecture. there's a guy he's russian grigori. pearlman he proved poincare's conjecture. if you can comment on the proof itself. if that stands out to you something. interesting or the. human story of it which is he turned. down the fields medal. for the proof is there something.

you find inspiring or insightful about. the proof itself. or about the man yeah i mean one thing. i really like about the proof and partly. that's because it's sort of a. thing that happens again and again in. this book i mean i'm writing about. geometry and the way it sort of appears. in all these kind of real world problems. and. but it happens so often that the. geometry you think you're studying. is somehow not enough you have to go one. level higher in abstraction. and study a higher level of geometry and.

the way that plays out. is that you know poincare asks a. question about a certain kind of. three-dimensional object. is it the usual three-dimensional space. that we know or is it some kind of. exotic thing and so of course this. sounds like it's a question about the. geometry of the three-dimensional space. but no apparelment understands and by. the way in a tradition that involves. richard hamilton and many other people. like most really important mathematical. advances this doesn't happen alone it. doesn't happen in a vacuum it happens as. the culmination of a program that.

involves many people. same with wiles by the way i mean we. talked about wiles and i want to. emphasize that. starting all the way back with kumar who. i mentioned in the 19th century but. um gerhard frye and mazer and ken ribbit. and like many other people. are involved in building the other. pieces of the arch before you put the. keystone in we stand on the shoulders of. giants. yes um. so what is this idea the idea is that. well of course the geometry of the. three-dimensional. object itself is relevant but the real.

geometry you have to understand is. the geometry of the space of all. three-dimensional geometries. whoa you're going up a higher level. because when you do that you can say. now let's trace out a path. in that space yes there's a mechanism. called reachy flow and again we're. outside my research area so for all the. geometric analysts and differential. geometers out there listening to this. if i please i'm doing my best and i'm. roughly saying. so this the ritchie flow allows you to.

say like okay let's start from some. mystery three-dimensional space. which poincare would conjecture is. essentially the same thing as our. familiar three-dimensional space but we. don't know that. and now you let it flow you. sort of like let it move in its natural. path. according to some almost physical. process and ask. where it winds up and what you find is. that it always winds up. you've continuously deformed it there's. that word deformation again. and what you can prove is that the. process doesn't stop until you get to.

the usual three-dimensional space and. since you can get from the mystery thing. to the standard space by this process of. continually. changing and never kind of having any. sharp transitions. then the original shape must have been. the same as the standard. shape that's the nature of the proof now. of course it's incredibly technical i. think. as i understand it i think the hard part. is proving that. the favorite word of ai people you don't. get any singularities along the way. um but of course in this context.

singularity just means. acquiring a sharp kink it just means uh. becoming non-smooth at some point so. just saying something interesting about. uh. formal about the smooth trajectory. through this weird space. yeah but yeah so what i like about it is. that it's just one of many examples of. where. it's not about the geometry you think. it's about it's about the geometry of. all geometries so to speak and it's only. by. kind of like kind of like being jerked. out of flat land right same idea it's.

only by sort of. seeing the whole thing globally at once. that you can really make progress on. understanding like the one thing you. thought you were looking at. it's a romantic question but what do. what what do you think about him turning. down the fields medal. is uh is that just our nobel prizes and. feelings medals just. just the cherry on top of the cake and. really math itself. the process of uh curiosity of pulling. at the string of. the mystery before us that's the cake. and then.

the awards are just icing and. uh clearly i've been fasting and i'm. hungry but uh. but do you think it's um it's it's. tragic or just. just a little curiosity that he turned. on the metal. well it's interesting because on the one. hand i think it's absolutely true. that right in some kind of like vast. spiritual sense like awards are not. important like not important the way. that sort of like. understanding the universe is important.

um. on the other hand most people who are. offered that prize accept it you know. it's it is so there's something unusual. about his uh. his choice there um i i wouldn't say i. see it as. tragic i mean maybe if i don't really. feel like i have a clear picture of. of why he chose not to take it i mean. it's not he's not alone. in doing things like this people. sometimes turn down prizes for. ideological reasons. um probably more often in mathematics i. mean. i think i'm right in saying that peter.

schulze like turned down sort of some. big monetary. prize because he just you know i mean i. think he. [Music]. at some point you have plenty of money. and maybe you think it sends the wrong. message about what the point of doing. mathematics. is um i do find that there's most people. accept. you know most people are given a prize. most people take it i mean people like. to be appreciated but. like i said we're people yes not that. different from most other people. but the important reminder that that. turning down the prize. serves for me it's not that there's.

anything wrong with the prize. and there's something wonderful about. the prize i think the no. the nobel prize is trickier because so. many nobel prizes are given. first of all the nobel prize often. forgets many many of the. important people throughout history. second of all there's like these weird. rules to it there's only three people. and some projects have a huge number of. people and it's like this. it um i don't know it it doesn't. kind of highlight the way.

science is done on some of these. projects in the best possible way. but in general the prizes are great but. what this kind of teaches me and reminds. me is. sometimes in your life there'll be. moments. when the thing that you. you would really like to do society. would really like you to do. is the thing that goes against something. you believe in. whatever that is some kind of principle. and stand your ground. in the face of that it's something um.

i believe most people will have a few. moments like that in their life. maybe one moment like that and you have. to do it that's what integrity is. so like it doesn't have to make sense to. the rest of the world but to stand on. that like. to say no it's interesting because i. think do you know that he turned down. the prize. in service of some principle because i. don't know that. well yes that seems to be the inkling. but he has never made it super clear but. the the inkling is that there he had. some problems with the whole process of. mathematics that. includes awards like this hierarchies.

and. reputations and all those kinds of. things and individualism that's. fundamental to american culture. he probably because he visited the. united states quite a bit. that he probably you know. it's it's like all about experiences and. he may have had you know some parts of. academia. some pockets of academia can be less. than inspiring perhaps sometimes. because of the individual egos involved. not academia people in general smart. people with egos.

and if they if you interact with a. certain kinds of people you can become. cynical too easily. i'm one of those people that i've been. really fortunate to interact with. incredible people at mit and academia in. general but. i've met some assholes and i tend to. just kind of when i when i. run into difficult folks i just kind of. smile and send them all my love and just. kind of go. go around but for others those. experiences can be sticky. like they can become cynical about the. world. when uh folks like that exist so it's he.

he may have uh he may have become a. little bit cynical about the process of. science. well you know it's a good opportunity. let's posit that that's his reasoning. because i truly don't know. um it's an interesting opportunity to go. back to almost the very first thing we. talked about the idea of the. mathematical olympiad because of course. that is so the international. mathematical olympiad is like a. competition. for high school students solving math. problems and. in some sense it's absolutely false to. the reality of mathematics because just.

as you say. it is a contest where you win prizes. um the aim is to sort of be faster than. other people. uh and you're working on sort of canned. problems that someone already knows the. answer to like not. problems that are unknown so you know in. my own life. i think when i was in high school i was. like very motivated by those. competitions and like i went to the math. olympiad and. you won it twice and got i mean well. there's something i have to explain to. people because it says i think it says.

on wikipedia that i won. a gold medal and in the real olympics. they only give one gold medal. in each event i just have to emphasize. that the international math olympiad. is not like that the gold medal gold. medals are awarded to the top 112th of. all participants. okay so sorry to bust the legend or. anything like well you're an exceptional. performer in terms of. uh achieving high scores on the problems. and they're very difficult. so you've achieved a high level of. performance on the. in this very specialized skill and by. the way it was very it was a very cold. war activity you know when in 1987 the.

first year i went it was in havana. americans couldn't go to havana back. then it was a very complicated process. to get there. and they took the whole american team on. a field trip to the museum of american. imperialism. in havana so we could see what america. was all about. how would you recommend a person. learn math so somebody who's young. or somebody my age or somebody older. who've taken a bunch of math but wants.

to rediscover the beauty of math. and maybe integrate into their work more. so than the research space. so and and so on is there something you. could say about the. process of uh. incorporating mathematical thinking into. your life. i mean if the thing is it's in part a. journey of self-knowledge. you have to know what's going to work. for you and that's going to be. different for different people so there. are totally people who at any stage of. life. just start reading math textbooks that.

is a thing that you can do and it works. for some people and not for others. for others a gateway is you know i. always recommend like. the books of martin gardner or another. sort of person we haven't talked about. but who also. like conway embodies that spirit of play. um he wrote a column in scientific. american for decades called mathematical. recreations and there's such. joy in it and such fun and these books. the columns are collected into books and. the books are old now but for each. generation of people who discover them. they're completely.

fresh and they give a totally different. way into the subject than. reading a formal textbook which for some. people would be the right. thing to do and you know working contest. style problems too. those are bound to books like especially. like russian and bulgarian problems. right there's book after book. problems from those contexts that's. going to motivate some people um. for some people it's gonna be like. watching well-produced videos like a. totally different. format like i feel like i'm not. answering your question i'm sort of. saying there's no one answer and like. it's a journey where you figure out what.

resonates with you. for some people if the self-discovery is. trying to figure out why is it that i. want to know. okay i'll tell you a story once when i. was in grad school i was very frustrated. with my like lack of knowledge of a lot. of things. as we all are because no matter how much. we know we don't know much more and. going to grad school means just coming. face to face with like the incredible. overflowing fault of your ignorance. right so i told joe harris who was an. algebraic geometer. a professor in my department i was like. i really feel like i don't know enough. and i should just like take a year of. leave and just like read.

ega the holy textbook el amon de. geometry algebraic the elements of. algebraic geometry this like. i'm just gonna i i feel like i don't. know enough so i'm just gonna sit and. like read this like. 1500 page many volume book. um and he was like and the professor. hair was like that's a really stupid. idea. and i was like why is that a stupid idea. then i would know more algebraic. geometries like because you're not. actually gonna do it like you learn. i mean he knew me well enough to say. like you're gonna learn because you're. gonna be working on a problem and then. there's going to be a fact from ega you.

need in order to solve your problem that. you want to solve and that's how you're. going to learn it. you're not going to learn it without a. problem to bring you into it and so for. a lot of people. i think if you're like i'm trying to. understand machine learning and i'm like. i can see that there's sort of some. mathematical technology that i. don't have i think you like let. that problem that you actually care. about. drive your learning i mean one thing. i've learned from advising students you. know. math is really hard in fact anything.

that you do. right is hard um. and because it's hard like you might. sort of have some idea that somebody. else gives you oh i should learn. x y and z well if you don't actually. care you're not going to do it you might. feel like you should maybe somebody told. you you should. but i think you have to hook it to. something that you actually care about. so for a lot of people that's the way in. you have an engineering problem you're. trying. to handle you have a physics problem. you're trying to handle. you have a machine learning problem.

you're trying to handle let. that not a kind of abstract idea of what. the curriculum is. drive your mathematical learning and. also just as a brief comment. that math is hard there's a sense to. which heart is a feature not a bug. in the sense that again this maybe this. is my own learning. preference but i think it's a value to. fall in love with the process of doing. something hard. overcoming it and becoming a better. person because of it like i hate running.

i hate exercise to bring it down to like. the simplest hard. and i enjoy the part once it's done. the person i feel like for the in the. rest of the day once i've accomplished. it the actual process especially the. process of getting started. in the initial like i really i don't. feel like doing it. and i really have the way i feel about. running is the way i feel about. really anything difficult in the. intellectual space. especially mathematics but also.

just something that requires like. holding a bunch of concepts in your mind. with some uncertainty like where this. the terminology or the notation is not. very clear. and so you have to kind of hold all. those things together. and like keep pushing forward through. the frustration of really. like obviously not understanding certain. like. parts of the picture like your giant. missing parts of the picture. and still not giving up it's the same. way i feel about running. and and there's something about falling.

in love. with the feeling of after you went to. the journey of. not having a complete picture at the end. having a complete picture and then you. get to appreciate the beauty. and just remembering that it sucked for. a long time. and how great it felt when you figured. it out at least at the basic. that's not sort of research thinking. because with research you probably. also have to enjoy the dead ends. with uh with learning math from a.

textbook or from video there's a nice. you have to enjoy the dead ends but i. think you have to accept the dead ends. let me. let's put it that way well yeah. enjoy the suffering of it so i i. the way i think about it i do uh there's. an uh i don't enjoy the suffering it. pisses me off but. i accept that it's part of the process. it's interesting there's a lot of ways. to kind of deal with that dead end. um there's a guy who's the ultra. marathon runner navy seal david goggins. who kind of i mean there's a certain.

philosophy of like. most people would quit here. and so if most people would quit here. and i don't i'll have an opportunity to. discover something beautiful that others. haven't yet. so like anything any feeling that really. sucks it's like okay most people. would would just like go do something. smarter if i stick with this.

um i will discover a new garden of uh. fruit trees that i can pick okay you say. that but like what about the guy who. like wins the nathan's hot dog eating. contest every year like when he eats his. 35th hot dog he like correctly says like. okay most people would stop here. like are you like lotting that he's like. no i'm gonna eat the thirty times i am i. am. i am in the in the long arc of history. that man is onto something which brings. up. this question what advice would you give. to young people today.

thinking about their career about their. life whether it's in mathematics. uh poetry or hot dog eating cockneys. and you know i have kids so this is. actually a live issue for me right i. actually it's a it's not a photograph i. actually do have to give. advice to two young people all the time. they don't listen but i still give it. um you know one thing i often say to. students i don't think i've actually. said this to my kids yet but i say. to students a lot is you know you come.

to these decision points. and everybody is beset by self-doubt. right it's like not sure like what. they're capable of. like not sure what they're what they. really want to do. i always i sort of tell people like. often when you have a decision to make. um one of the choices is the high. self-esteem. choice and i always thought make the. high self-esteem choice make the choice. sort of take yourself out of it and like. if you didn't have those. you can probably figure out what the.

version of you feels completely. confident would do. and do that and see what happens and i. think that's often like pretty good. advice that's interesting sort of like. uh. you know like with sims you can create. characters like create. a character of yourself that lacks all. the self-doubt. right but it doesn't mean i would never. say to somebody you should just go have. high self-esteem yeah you shouldn't have. doubts no you probably should have. doubts it's okay to have them. but sometimes it's good to act in the.

way that the person who didn't have them. would act um that's a really nice way to. put it. yeah that's a that's a like from a third. person perspective. take the part of your brain that wants. to do big things what would they do. that's not afraid to do those things. what would they do. yeah that's that's really nice that's. actually a really nice way to formulate. it that's very practical advice you. should give it to your kids.

do you think there's meaning to any of. it from a mathematical perspective. this life if i were to ask you. we're talking about primes talking about. proving stuff. can we say and then the book that god. has. that mathematics allows us to arrive at. something about. in that book there's certainly a chapter. on the meaning of life in that book. do you think we humans can get to it and. maybe. if you were to write cliff notes what do. you suspect those cliff notes would say.

i mean look the way i feel is that you. know mathematics. as we've discussed like it underlies the. way we think about. constructing learning machines and. underlies physics. um it can be used i mean it does all. this stuff. and also you want the meaning of life i. mean it's like we already did a lot for. you like ask a rabbi. [Laughter]. no i mean yeah you know i wrote a lot in. the la in the last book how not to be. wrong. yeah i wrote a lot about pascal a. fascinating guy.

um who is a sort of very serious. religious mystic as well as being an. amazing. mathematician and he's well known for. pascal's wager i mean he's probably. among all mathematicians he's the ones. who's best known for this. can you actually like apply mathematics. to kind of. these transcendent questions. um but what's interesting when i really. read pascal. about what he wrote about this you know. i started to see that people often think. oh this is him saying i'm gonna use. mathematics. to sort of show you why you should.

believe in god. you know to really that's this. mathematics has the answer to this. question. um but he really doesn't say that he. almost kind of says. the opposite if you ask blaise pascal. like why do you believe in god. it's he'd be like oh because i met god. you know he had this kind of like. psychedelic experience it's like a. mystical experience where. as he tells it he just like directly. encountered god and he's like okay i. guess there's a god i met him last night. so that's. that's it that's why he believed it. didn't have to do with any kind you know.

the mathematical argument was like. um about certain reasons for behaving in. a certain way but he basically said like. look like math doesn't tell you that. god's there or not like if god's there. he'll tell you. you know you don't i love this so you ha. you have mathematics you have uh what do. you what do you have. like a ways to explore the mind let's. say psychedelics. you have like incredible technology you. also have. love and friendship and like what what.

the hell do you want to know what the. meaning of it all is just enjoy it. i don't think there's a better way to. end it jordan this was a fascinating. conversation i really love. the way you explore math in your writing. the the willingness to be specific. and clear and actually explore difficult. ideas but at the same time. stepping outside and figuring out. beautiful stuff and i love the chart. at the opening of uh your new book.

that shows the chaos the mess that is. your mind yes this is what i was trying. to keep in my head. all at once while i was writing and um. i probably should have drawn this. picture earlier in the process maybe it. would have made my organization easier i. actually drew it only at the end. and many of the things we talked about. are on this map. the connections are yet to be fully. dissected and investigated and yes. god is in the picture right on the edge. right on the edge not in the center.

thank you so much for talking it is a. huge honor that you would waste your. valuable time with me. thank you like we went to some amazing. places today this is really fun. thanks for listening to this. conversation with jordan ellenberg and. thank you to. secret sauce expressvpn blinkist. and indeed check them out in the. description to support this podcast. and now let me leave you with some words. from jordan in his book how not to be. wrong. knowing mathematics is like wearing a. pair of x-ray specs.

that reveal hidden structures underneath. the messy and chaotic surface of the. world. thank you for listening and hope to see. you next time. you.