Dwarkesh Podcast - Grant Sanderson (@3blue1brown) — Past, present, & future of mathematics

Episode Date: October 12, 2023

I had a lot of fun chatting with Grant Sanderson (who runs the excellent 3Blue1Brown YouTube channel) about:- Whether advanced math requires AGI- What careers should mathematically talented students pursue- Why Grant plans on doing a stint as a high school teacher- Tips for self teaching- Does Godel’s incompleteness theorem actually matter- Why are good explanations so hard to find?- And much moreWatch on YouTube. Listen on Spotify, Apple Podcasts, or any other podcast platform. Full transcript here.Timestamps(0:00:00) - Does winning math competitions require AGI?(0:08:24) - Where to allocate mathematical talent?(0:17:34) - Grant’s miracle year(0:26:44) - Prehistoric humans and math(0:33:33) - Why is a lot of math so new?(0:44:44) - Future of education(0:56:28) - Math helped me realize I wasn’t that smart(0:59:25) - Does Godel’s incompleteness theorem matter?(1:05:12) - How Grant makes videos(1:10:13) - Grant’s math exposition competition(1:20:44) - Self teaching Get full access to Dwarkesh Podcast at www.dwarkesh.com/subscribe

Grant. The videos were really inspiring. Like, you're the reason I'm like going into grad school. And there's this little bell in the back of my mind that's like, do I want that? To get more people going into math PhDs. Math Academy of Finance and computer science almost certainly have an over-allocation of talent. Actually, I'm quite determined at some point to like be a high school math teacher for

some number of years. Math lends itself to synthetic data, how AlphaGo is trained. You could have it produce a lot of proofs. and just drain on a whole bunch of those. And the thing that takes, at most, 30 minutes of the teacher's time, maybe even 30 seconds, has these completely monumental rippling effects for the life of the student they were talking to that then sets them on this whole different trajectory. Okay, today I have the pleasure of interviewing Grant Sanderson of the YouTube channel, 3-Blue 1 Brown.

You all know who Grant is. I'm really excited about this one. By the time that an AI model can get gold in the International Math Olympiad, is that just AGI given the amount of creative problem solving and chain of thought required to do that? I, to be honest, have no idea what people mean when they use the word AGI. Yeah. I think if you ask 10 different people like what they mean by it, you're going to get 10 slightly different answers. And it seems like what people want to get at is a discrete change that I don't think actually exists.

Where you've got, okay, AI is up to a certain point. They're not AGI. They might be really smart, but it's not AGI. And then after some point, that's the benchmark when like now there, now it's generally intelligent. The reason that world model doesn't really fit is it feels a lot more continuous where, you know, GPD4 feels general in the sense that you have one training algorithm that applies to a very, very large set of different kinds of tasks that someone might want to be able to do. And that's cool. That's like an invention that people in the 60s might not have expected to be true for the nature of how artificial intelligence can be programmed. So it's generally intelligent, but maybe what people mean by, oh, it's not AGI's.

You've got certain benchmarks where it's better than most people at some things, but it's not better at most people than others. At this point, it's better than most people at most people at solving AMC problems and like IMO problems is just not better than the best. And so maybe at the point when it's getting golds in the IMO, that's a sign that, okay, it's as good as the best. and we've ticked off another domain, but I don't know, like, is what you mean by AGI that you've enumerated all the possible domains that something could be good at, and now it's better than humans at all of them? Or enough that it could take over substantial fractions of, you know, human jobs or something, where right now it's, it's impressive, but it's not going to be even 1% of GDP.

But in my mind, if it's getting golden IMO, I mean, having seen some of those problems from your channel. I'm thinking, wow, that's really coming after podcasters and video animators, I don't know. I don't know. That feels orthogonal because getting a gold in the IMO feels a lot more like being really, really good at Go or chess. Like those feel analogous?

Well, it's super creative. Like, I think anyone who, I don't know chess as well as the people are into it, but everything that I hear from them, the sort of moves that are made and choices have all of the air of creativity. I think as soon as they started generating artwork, then everyone else could appreciate, oh, there's something that deserves to be called creative here. And the creative side of the math, you know, I don't know how it would look when people get them to be getting golds at the IMO. But I imagine it's something that looks a little bit like how AlphaGo is trained,

where you have it like play with itself a whole bunch, you know, math lends itself to synthetic data in the ways that a lot of other domains don't. You could have it produce a lot of proofs in a proof checking language, like lean, for example, and just train on a whole bunch of those. And like, is this a valid proof? Is this not a valid proof? And then counterbalance that with English written versions of something. And so I imagine what it looks like once you get something that is solving these, I'm a level things.

One of two things. Either it writes a very good proof that you feel like is unmotivated. Because anyone who reads math papers has this feeling that there are two types. There's the ones where you morally understand why the result should be true. And then there's the ones where you're like, I can follow the steps. Why would you have come up with that? I don't know, but I guess that shows that the result is true. And you're left wanting something a little bit more.

And so you could imagine if it produces that on the gold to get a gold in the IMO. Is that the same kind of ability as what is required to replace jobs? Just like not really. Like the impediments between where it is now and replacing jobs feels like a whole different set of things. Like having a context window that is longer than some small things such that you can make a connection. over long periods of time and build relationships and understand where someone's coming from. And the actual problem solving part of it,

I mean, it's a sign that it would be a more helpful tool, but in the same way that, like, Mathematica can help you solve math problems much more effectively. Tell me why I should be less amazed by it or maybe put in a different context, but the reason I would be very impressed is that with chess or something,

obviously this is not all the chess programs are doing, but there's a level of research you can do to narrow down the possibilities. And more importantly, in the math example, it seems that with some of the examples you've elicestered it on your channel, for example, the ability to solve the problem is so dependent on coming up with the right abstraction

to think about it, coming up with ways of thinking about the problem that are not evident in the problem itself or in any other problem in any other test, that seems different from just a chess game where you don't have to like, what is the largest structure of this chess game in the same way as you do with an IMO problem.

I think you should ask people who know a lot about go and chess, and I'd be curious to hear their opinions on it, because I imagine what they would say is, if you're going to be as good at Go, as AlphaGo, as you're also not doing tree search, at least exclusively. It's not dependent on something as growth in that, because you get this commentatorial explosion, which is why people thought that game would be so much harder for so much longer.

There sort of has to be something like a higher level structure in their understanding. And then don't get me wrong, I would be super impressed and like anticipate being very impressed when you get AIs that can solve these IMO problems because you're absolutely right. There's a level of creativity involved. The only claim I'm making is that being able to do that feels distinct from the impediments between, I don't know, where we are now and the AIs take over all of our jobs or something. It just, it seems like it's going to be another one of those boxes that's this historic moment. analogous to chess and go more so than it's going to be analogous to the industrial revolution. I'm surprised you wouldn't be more compelled.

I am compelled. Right. Or just you don't think that skill of this problem is isomorphic, this completely different way of thinking about what's happening in the situation, and here's me going through the 50 steps to put all that together into this one proof. I'm surprised you don't think that's upstream of a lot of valuable tasks. I think it's a similar level of how impressed I was with the stable diffusion type stuff, where you ask it, give me a landscape of beautiful mountains but made out of quartz and gemstones or something.

And it gives you this thing, which has all of the essence of a landscape, but it's not literally a landscape. And so you realize, okay, there's something beyond the literal that's understood here. That's very impressive. And in the same way to solve one of these math problems that requires. creativity. You can't just go from the definitions. You're 100% right. You need this element of lateral thinking, which is why we find so much joy in finding the solutions ourselves or even just seeing other people get those solutions. It's exactly the kind of joy that you get out of

good artistic analogies and comparisons and mixing and matching. I'm very impressed by all of that. It's just, I think it's in the same category. And it's maybe I don't have the same opinions as a lot of other people with this hard line between there's pre-AGI and post-AGI. I just don't know what they mean by the word AGI. And I don't think that you're going to have something that's this measurable discrete step, much less that a math tournament is going to be an example of what that discrete step would look like. Interesting. Interesting. Okay. Applied mathematicians. Where do we put them in society where they can have the biggest benefit? A lot of them go into computer science and IT. And I'm sure there's been lots of benefits there. But we're, you know,

there are parts of society where you just have a whole bunch of mathematicians go in and they can make things a lot better. You know, I don't know, transportation or logistics or manufacturing. But where else do you think they might be useful? That's such a good question. In some ways, I'm like the worst person to ask about that. But this isn't going to answer your question, but instead is going to like fan the flames of why I feel it's an important question. I have actually been thinking recently about if it's worth making an out of typical video that specifically addressed it, like inspiring people to ask that, especially students who are graduating. Because I think this thing happens where when you fall in love with math or some sort of

technical field, by default in school, you study that. And when you're studying that, effectively you're going through an apprenticeship to be an expert in that or a researcher in that. You know, the structure of studying physics in a university or math in a university, even though they know that not all majors are going to go into the field. The people that you're gaining mentorship from are academics and our researchers in the field. so it's hard not to effectively be apprenticing in that. And I also have noticed that when I go and give talks at universities or things like this

and students come up after and they're like saying hi, there's a lot of them like, Grant, the videos were really inspiring like, you're the reason that I studied math or that you're the reason I'm like going into grad school. And there's this little bell in the back of my mind that's like, cool, cool, I'm amazed. I don't know if I believe I was wholly responsible for it, but like, cool to have that impact. Do I want that? Is this a good thing? to get more people going into math PhDs.

On the one hand, I unequivocally want more people to self-identify as liking math. That's very good. But those who are doing that necessarily get shuffled into the traditional outlets, like math academia, I think you highlight it very right.

It's like math academia, finance, and computer science, data science, something in there in general, are very common things to go to. And as a result, they almost certainly have an over-allocation of talent. All three of those are valuable, right? I'm not saying like those are not valuable things to go into.

But if you were playing God and like shifting around, where do you want people to go? Again, I'm not answering your question. I'm just asking it in other words because I don't really know. I think you should probably talk to the people who made that shift, of which there aren't a huge number. But like Eric Lander is maybe one good example. Jim Simons would maybe be another. Whereas people who were doing a very purely academic thing and then decided to shift to something very different. Now, I have sort of had this thought that it's very beneficial to insert some forcing function

that gets the pure mathematicians to spend some of their time in a non-pure math setting. NSF grants coming with a requirement that 10% of your time goes towards a collaboration with another department or something like that. The thought being, these are really good problem solvers in a specific category of problems and to just distribute that talent elsewhere might be helpful. And when I run this by mathematicians, sometimes there's a mixed response where they're like, I don't know if we'd be all that useful. Like there's a sense that the aesthetic of what constitutes a good math problem is by its nature

rooted in the purity of it, such that it's maybe a little elitist to assume that just because people are really, really good at solving that kind of problem, that somehow their abilities are more generalizable than other people's abilities. You know, why ask about the applied mathematicians rather than saying, like, shouldn't the applied biologists go and work in logistics and things like that? Because they also have a set of problem solving abilities that's maybe generalizable. The back of my mind, I think, no, but the mathematicians are special. There really is something general about math.

So I don't have the answers. I will say I'm actually very curious to hear from people for what they think the right answers are or from people who made that switch. Let's say they were a math major or something adjacent like computer science physics. And then they decided that they wanted to pour themselves into. something, not because that was the academic itch that they were scratching by being good at school and getting to appreciate that, but because they stepped back and said, what impact do I want to make on the world? I'm hungry for more of those stories because I think it could be very

compelling to convey those specifically to my audience who is probably on track to go into just the traditional math type fields and maybe there's room to have a little bit of influence to disperse them more effectively. But I don't know. I don't know. I don't know what more effectively looks like because at the end of the day I'm like I'm a math YouTuber right I'm not someone who has a career in logistics or manufacturing or all of these things in such way that I can have an in tune feel for where is there a need for this specific kind of abstract problem solving it might be useful to speculate on how an undergrad or somebody who's a young math was might even begin to contemplate here's where I can have an edge I'm remembering actually

it just occurred to me. A former podcast guest, Lars Doucette, he was a game designer, actually, and he started learning about Georgiaism, which is this idea that you should tax land and only land. And so he got really interested in not only writing about those ideas, but also with, well, if you're going to tax land, you had to figure out what the value of land is. How do you figure out the value of land? There's all these algorithms of how you do this optimally based on neighboring land and how to average across land. and there's a lot of intricacies there. And so he now has a startup where he just contracts with cities to implement these algorithms

to help them assess the value of their land, which makes property taxes much more feasible. That's another example, right, where the motivation was more philosophical, but his specialty as a technical person helped him, you know, help to make a contribution there. I think that's perfect. Probably the true answer is that you're not going to give a universal thing. For any individual, it's going to be based on where they're. life circumstances connect them into something, either because he had an interest in Georgeism for whatever reason. But if someone, I don't know, their dad runs a paper mill and they're

connected to the family business in that way and realize they can plug themselves in a little bit more efficiently, you're going to have this wide diversity of the ways that people are applying themselves that does not take the form of general advice given from some podcast somewhere, but instead takes the form of simply inviting people to like think critically about the question rather than following the momentum of what being good at school implies about your future? We were talking about this before the interview started, but we have a much better grasp on reality based on our mathematical tools. And I'm not talking about anything advanced.

Literally just being able to count in the decimal system that even the Romans didn't have. How likely do you think it is that something that's significant would be enjoyed by our descendants in hundreds or thousands of years? Or do you think that that kind of basic numeracy level stuff, those kinds of thinking tools are basically all gone. Just so I understand the question, right? You're talking about how having a system for numbers changes the way that we think that then lends itself to a better understanding of the world. Like we can do commerce things like that.

Or we can think in terms of orders of magnitude. It would have been hard to think. We have the word orders of magnitude in a way that is hard to write down, much less think about if you're doing Roman numerals. Like is there something analogous to that for our descendants? I mean, fluency with a programming interface really can help understanding certain problems. I think when people mess around in a notebook with something and when it feels like a really good tool set, there's a way that that has the same sensation as adopting a nice notation in that you

write something with a small number of symbols, but then you discover a lot about the implication of that. In the case of notation, it's because the rules of algebra. are very constrained. And so when you write something, you can go through an almost game-like process to see how it reduces and expands and then see something that might be non-trivial. And in the case of programming, of course, the machine is doing the crunching and you might get a plot that reveals some extra data. I think we're maybe at a phase where there's room for that to become a much more fluid process, such that rather than having these small little bits of friction, like you've got to set up the environment and you've got to like it in the notebook,

you've got to find the right libraries, that there's something that feels as fluid as when you are good at algebra and you're just at a whiteboard kind of noodling it out. I think there's something to be said for the fact that there's still so much more value in, if you and I were going to go into some math topic right now, right? You ask me something that's a terrible question for a podcast, but I'm like, oh, let's actually dig into it. The right medium to do that is still paper, I think.

I would break out some paper and we would scribble it out. And so whenever it becomes the case that the right medium to do that lends itself to simulation and to programming and all that, that feels like it would get to the point where it shifts the way that you even think about stuff. Yeah, that's really interesting. What's up with miracle years? So this is something that has happened throughout science and especially with mathematicians where they have a single year in which they wake up many, if not most of the important discoveries that they have in the career. You know, Newton, Einstein, Gauss, they all had these years. Do you have some explanation of what's going on? What's your take?

I think there's a bunch of possible explanations. It can't just be youth because youth last 10 years, not one year. So it must have something to do with... Every 35-year-old right now is like... How dare you? You know what I mean. May 20 years. So, yeah, it can't just be that.

I don't know. There's a bunch of possible things you could say. One is you're in a situation in life where you have nothing. else going for you or you just like really free for that one year and then you become successful after that year is over based on what you did yeah but what is your take i don't know i so i agree that's probably multiple factors not one one thing could be that the miracle year is the like exhalation and there's been many many years of inhalation where often let's say you know the classic one is einsteins where his miracle year were also some of the first papers like kind of springing

on to the scene. And I would guess that a lot of the ideas were not bumping around his head only in that year, but it's like many, many years of thinking about it and kind of coalescing. And so you might be in a position where you can build up all of this potential energy. And then for whatever reason, there's one time in life that lends itself to actually releasing all of that. If I try to reflect on like my own history with what I'm doing now, I think I didn't appreciate early on how much potential energy I was working from simply from being a student and in college where there's just a bunch of ways of thinking about things or empathy with new learners or just cool concepts, right? The basic concept behind a video that in fact, it was many, many years

of like all of my time having learned math before I started putting out stuff online that I was able to eat into. And then maybe it was a little later where, okay, the well never runs dry. There's always a long list of things that I want to cover. But it's a lot. In some sense, like, I recognize that the well was at risk of running dry in a way that I never thought that it could. And without being a little deliberate about devoting some of my day, not just to output and producing, but to stepping back and like learning new things and touching something I never would have. That doesn't happen by default. I don't know if this is also the case for the people who have genuine miracle years where they were like letting out all of this stuff. And then it takes a decade to build up that same level of potential energy.

The other thing, you have everything to gain and nothing to lose when you are young. So even if it's not merely youth, there's a willingness to be creative. And there's also none of the obligations that come from having found success before. You know, there's certain academics who made an extremely deliberate effort not to let. What do they call it? Is it like the curse of success? There's some term for it. But I think maybe James Watson had this standard reply to,

invitations for, you know, talks and interviews and things like that. It was basically like no to everyone because I just want to be a scientist. It was much more articulate than that. And he adds all these nine points. But that was the gist of it. And short of doing that, I think it's very easy for someone to have a lot of other things that eat into their mind share and time and all of that, that even if it's just 20 hours a week, like actually that that really interrupts a creative flow.

Were you a student when you started the channel? Technically, yeah. The very, very first video was made when I was a senior at Stanford. Basically, I had been, like, toying around with just a personal programming project in my last year of college that was the beginnings of what is now, like, the animation tool I work with. I didn't intend for it to be a thing that I would use as a math YouTuber. I didn't even really know what a YouTuber was.

It was really just, like, a personal project. And it was March of that year. I think that I published the first ever video. And so it was kind of right at that transition point. Do you think you could, you would have done it if you had, let's say, become, I don't know what you're planning on doing after college. I remember you saying somewhere, maybe a data scientist. But data scientist and math PhD were the two like, I see, 50-50 contenders, basically.

Is there a role in which he started doing that, but then later on made manum? Or do you think that that was only possible in a world where you had some time to kill in your senior year? If the goal was to make math YouTube videos, it would have been a, a wild thing to do to do it by like making manum as the method for it because it's like so strikingly inefficient to do it that way. At the very least, I probably would have built on top of an existing framework. If I had, I mean, there's so many things that I would tell my past self if I could go back in time, even if the goal was to make that. It's like certain design decisions that caused pain that could have been fixed earlier on. But if the goal was to make videos, there's just

so many good tools for making videos. I probably would have started with those. or if I wanted to script things, maybe I would have first learned after effects really effectively and then learned the scripting languages around after effects. That might have even been better for all I know. I really don't know. But I just kind of walked into it because the initial project was to make something that could illustrate certain ideas in math,

especially when it came to visualizing functions as transformations, mapping inputs to outputs as opposed to graphing. The video output was just a way of knowing that I had completed that personal project in some sense. And then it turned out to be fun because I also really enjoy teaching and really enjoy tutoring and stuff. I don't know. Then again, there's a lot of other people who make their own tools for math gifts and little illustrations and things, which on the one hand feels very inefficient. Like there's, I don't know, people come across a math gif on Wikipedia.

There's a very high probability. It comes from this one individual who just strangely prolific at producing these like creative comments visuals. and he has his own like home-baked thing for how he does it. And then there's someone who came across on Twitter, Matt Henderson, who has these completely beautiful math gifts and such. And again, it's a very home-baked thing that's like built on top of shaders, but he kind of has his own stuff there. And maybe there's something to be said for the level of ownership that you feel

once it is your own thing that just unlocks a sense of creativity and feeling like, hey, I can just describe whatever I want. because if I can't already do it, I'll just change the tool to make it able to do that. For all I know, that level of creative freedom is necessary to, like, take on a wide variety of topics. But your guess is as good as mine for those counterfactuals. Yeah. This is personally interesting to me because I also started the podcast in college, and I just, like, was off track of anything I was planning on doing otherwise. And this is, you know, many, many orders of magnitude O.A. from through the one brown.

I don't want the audience to, you know, cringe in unison. But I just think it's interesting, like these kinds of projects, how often something that, I don't know, later on ends up being successful is something that was started almost on a whim as a hobby, you know, when you're in college or something like that. I will say there's a benefit to starting it in a way that is low stakes. Like you're not banking on it growing. I had no anticipation of, much less an expectation of three blown brown growing. I think the reason I kind of kept doing it was in the fork of life if I did the math PhD and all that. I thought it might be a good idea to have a little bit of a footprint on the internet for math exposition. I was thinking of it as like a very niche thing that, you know, maybe some math

students and some people who are into math would like it, but I could sort of show the stuff as a portfolio, not as an audience, you know, size that was meaningful. And I think I was surprised by what an appetite there was for the kind of things that I was making. And in some ways maybe that's helpful because I see a lot of people who jump in, they jump in with the goal of being a YouTuber. I think it's the most common desired job among the youth is to be like a TikToker or a YouTuber, which think of that way you will. But when you jump in with that as a goal, you kind of aim for too large an audience and end up making the content which is best for no one. Because one, you're probably not that good at making videos yet. And so you're competing, if it's a generally

applicable idea, you're competing with like all of the other communicators out there. Whereas if you do something that's almost unreasonably niche and also you're not expecting it to blow up. It's like, one, you're not going to be disappointed. It's like outstanding when a thousand people view it as opposed to being disappointing. And then two, you might be making something that is the best possible version of that content for the audience who watches it because no one else is making that for them because it's too narrow a target. And that's sort of the beauty of the internet is that there's an incentive to do that.

And I don't know if this is the case with your podcast when you're starting out. where not thinking about, oh, how can I make this as big as possible, actually made it more in depth for those who were listening to it. Is it surprising to you that prehistoric humans don't seem to have had just basic arithmetic and numeracy? To me, with a sort of, I guess to us with the modern understanding, that kind of stuff seems so universally useful and so fundamental that it's shocking that it just doesn't come about naturally in the course of interacting with the world.

Do you think is that surprising that just the concept of numbers and you're right it's so in our bones that it's hard to empathize with not having numeracy a lot of it's linked to if you think okay what's the first place that like people think about numbers most people like in their daily lives it's linked to commerce and money and such so maybe some ways the question is the same as thing is it surprising that like early humanity didn't have commerce or didn't deal with money maybe when you're below Dunbar's number in your communities, like a tit for tact structure just makes a lot more sense and actually works well and it would just be obnoxious to actually account for everything. The other loosely related idea, have you come across those studies where when anthropologists will interview

tribes of people that are removed enough from normal society that they don't have the level of numeracy that your eye do? But there's some notion of counting. If you have one coconut or nine coconuts, Like you have a sense of that. That if you ask what number is halfway between one and nine, those groups will answer three, whereas you or I or people in our world will probably answer five. And because we think on this very linear scale, it's interesting that evidently the like natural way to think about things is logarithmically,

which kind of makes sense. Like the social dynamics as you go from solitude to a group of 10 people to a group of 100 people have roughly equal steps. an increase in complexity more so than if you go from like one to 51 to a 102. And I wonder if it's it's the case that by adding numeracy in some senses, we've also like lost some numeracy or lost some intuition in others where now if you ask, you know, uh, middle school teachers, what's a difficult topic to teach or for students to understand? They're like logarithms, but that should be deep in our bones, right?

So somehow it got unlearned and then needs to be maybe it's in the formal sense that it's harder to relearn it, but there's, there's maybe a sense of, like, numeracy and a sense of quantitative thinking that humans naturally do have that is hard to appreciate when it's not expressed in the same language or in the same ways. Yeah, I have seen the thing from Joseph Henrik where the still existing tribes were, they're in this kind of situation. They can do numeracy in arithmetic when it's in very concrete terms, if you talk about seeds or something, but the abstract concept of a number is not. is not available to them.

Do you think the abstract concept of a number is useful to your life? Oh, yeah. Like in what ways? It's almost like asking how is the concept of the alphabet useful? It comes up so often that it's, I mean, just like how many lights do I set up for this interview, right? Is that the concept of an abstract number, though? Because it's like two people, two lights, one-to-one correspondence. Did you leverage the abstraction of two as a object, which is,

is simultaneously a rational and a real and an integer. I see. You know, is in the context of a group that has additive structure, but also multiplicity. It's like, no, there's light for you,

light for me. Right. Yeah. I'm pretty sure the abstract idea of a number is important for all of us, but I don't think it's immediately obvious. It's more that it shapes the way we think. I'm not sure if it actually changes like the way we live. If you assume,

you don't work in STEM, right? where like you literally are using it all the time. Yeah, I'm trying to go through my day and think through where am I using them. I mean, there's the obvious stuff, like the commerce examples you mentioned where you go to a restaurant

and you're figuring out what to pay or what to tip. But that seems a very particular example. Do I really use numbers that infrequently? I don't know, yeah. Many people listening are probably screaming out of their heads with like much more apt examples. But it's hard to say. Yeah, yeah.

when a mathematician is working on a problem, what is the biggest mental constrained? Is it working memory? Is it processing speed? Plants are limited by nitrogen usually. What is the equivalent of nitrogen for a mathematician? That's a fun question. I mean, so I'm not a research mathematician. I shouldn't pretend like I am.

And so the right people to ask that question would be the research mathematicians. I wonder if you're going to get consistent answers. As with so many things, there's not one. maybe the number of available analogies to be able to draw connections, like the more exposure you've had to disparate fields such that you could maybe see, oh, a problem solving approach that was used here might be useful here. Sometimes that's literally codified in the forms of connections between different fields as, you know, functories between categories or something like that.

But sometimes it's a lot more intuitive, the idea that a general flavor of an, you know, someone's doing a combinatorics type question and they're like, oh, maybe generating functions are this useful like tool to bring to bear. And then in some completely different context of studying like prime numbers, they're like, oh, maybe it could take a generating function type approach. Maybe you have to massage it to make it work. And one of the reasons I say this is one of the tendencies that you've seen in math papers in, let's say the last 200 years, the typical number of authors is like much bigger now than

before. people have this, I think, misconception that math is a field with lone geniuses who are off, like coming up with great insights, like a loan next to a blackboard. And the reality is it's a highly collaborative field. I remember one of the first times that I was hearing from a mathematician, like a young kid and I was in this math circles event and someone was asking this person, like, what surprised you about your job? That the first thing he said was how much travel was involved, that he wasn't expecting that. And it's because, you know, if you're studying some very specific niche field, that the way that you're, you know, the way that you

that you make progress in that is by collaborating with other people in that field or maybe adjacent to that field. And there's only so many about them that probably aren't at your university. So you travel a lot to work with them. These days, a lot of that I think happens on Zoom, but conferences are still super important. And these sorts of events that bring people all under one roof, like MSRI is maybe an example of a place that does that systematically. And you could say that's a social thing, you know, get more ideas.

But I think it's maybe hitting on this idea that what you want. is exposure to as many available analogies. So the short answer to your question is like the nitrogen for mathematician is the analogy. This actually is an interesting question I wasn't planning on asking you, but it just occurred to me. Is it surprising how new a lot of mathematics is, even mathematics that is taught at the high school level, whereas with physics or biology, that's also new. But you can tell a story where we didn't have the tools to look at the cell or to inspect an electron until very recently. but we've had mathematicians for two, three thousand years

who are doing pretty sophisticated things, even the ancient Greeks. It depends on what you're being by sophisticated, but impressive things. I don't know why linear algebra is so on you given that fact. I wouldn't have thought of math as being like new in that way. If anything, especially for talking at the high school level,

I remember there's always a sensation that it's frustrating that all of the things are actually way more than 100 years old in terms of, you know, the names attached to the theorems that you're doing. like none of them are remotely modern, whereas in biology, you know, the, um, the understanding we have for like how proteins are formed is relatively much more modern and you might be just a couple of generations away. I, at some extent, there's a raw manpower component to it. We're like, how many people did pure math for most of history? For most of history, no one.

No one was a pure mathematician. They were like a mathematician plus something else. Or they were a physicist who, you know, they were a natural philosopher. And in so far as you're doing like natural philosophy. One component of that is developing math, but it's not the full extent of what you do. Even the ones who we think of as like very, very pure mathematicians in the sense that a lot of their most famous results are pure math, like Gauss. Actually, a lot of the work, a lot of his, output was also centered on very practical problems. And yeah, and maybe like since then, that's when you start to get an era of something more like pure mathematicians. And the raw number, available that you have, that man hours that are being put into developing new theorems

is probably just got this huge spike as one, the population grows, and then also the percentage of the population that has the economic freedom to do something as indulgent as academia like it grows. Maybe it's pretty reasonable, that most of it is much, much more recent. That would be my guess. But some of these things seem actually pretty modern. Like information theory is less than 100 years old. That's true. And it's, you know, pretty fundamental. Like theoretically, you could have written that paper a long time ago. That's a really good example. And maybe like this is a sign that the math that's developed is more in the service of

the world that you live in and the adjacent problems that it's used to solve than we typically think of it. On the one hand, information theory, such a good example because it's so pure that you could have asked the question. You could have defined the notion of a bit. But evidently, there wasn't a strong enough need to think in that way. Whereas when you're doing error correction or you're thinking about actual. information channels over, you know, a wire in your at Bell Labs, that's what prompts it.

Another maybe a really good example for that would be chaos theory. You could easily ask, why is chaos theory so recent? You could have, you know, written the Lorenz equations since differential equations existed. Why didn't anyone do that and understand that there was this sort of sensitivity to initial conditions? And in that case, it would maybe be the opposite, where it's not that you need the existence of computers as a problem to solve, or the problem that they introduce other problems to solve. But instead, you need them to even discover the phenomenon in the first place. Like a lot of original concepts in chaos theory came from basically running simulations

or doing things that required a massive amount of computation that simply wouldn't be done by hand. Someone could ask the question, but they wouldn't have observed the unexpected phenomenon. And there, even if it's questions that are as relevant to a pre-computer world as to a post-computer world, like the nature of weather modeling or just the nature of three body problem, like all of that kind of stuff. Somehow without the right tools for thought, it just didn't come into the mind.

And so maybe, yeah, maybe there's other things like that where those questions or pieces of technology that start to fundamentally shape everyone's life will then invariably also shift like the mathematicians focus. This actually reminds me the first day of Scott Ernst's quantum information class. he said that what I'm about to describe to you could have been discovered by a mathematician before quantum physics existed

if only they'd ask the question we're going to do probabilities but we're only allowed to use unitaries and the rest of it is you could have just discovered quantum mechanics or quantum information from there I mean the thing about math especially if you're talking about pure axiomatized math

the experience as a student as an undergrad is that you are going through a textbook and it starts with saying here's the axioms of this field. And then we're going to deduce from those axioms like various different lemmas and theorems and proceed from that. And with that as the framing, you get the impression that you could have just come up with any axioms, just make up some pile of axioms, deduce what follows from them. And the space of possible like math is unfathomably huge. And so you need some process that calls down what are the useful things to maybe pursue. And so one of the things that I think is

all too often missing in those pure math textbooks is the, basically the, basically the motivating problem. Why is it that this was the set of axioms people found to be useful and not something else? You know, the framework for quantum information theory, it's like you marry together linear algebra and probability. That's great. But there's all sorts of other things where you could kind of try to cram them together and maybe get some sort of math out. And the question becomes, is it worth your time to do that. You know, not theory is something that emerged because I think it was Lord Kelvin had a theory that all of the, like, elements on the periodic table had structures which were related to a knot, like a knot being if you have a closed loop in 3D space,

but maybe if you want to continuously deform it without it ever crossing itself, you ask the question, like, could you get back to, say, an open loop? Or if you can't get back to an open loop, what are the set of all other loops in 3D space that could be deformed into that? And you end up categorizing what all the different knots are. And this was started with a completely incorrect theory for what's going on at the atomic level that gives Adams this very stable structure. Because I think he found with smoke rings, like if you're somehow very dexterous with how you create these smoke rings, you can get them to form knots in 3D. And they're very stable.

It's like weirdly stable. It'll never, you know, cross over itself. And so it has all those properties. Now, that was irrelevant for understanding the periodic table. but it was an interesting mathematical question and people kind of ran with it. And in that case, it was an arbitrary reason that someone thought to ask the question and then some people ran with it. And frankly, it's probably fewer people who run with it than would if it turned out to be a more useful question.

So really you want to ask, like, what are the things that prompt people to ask what turns out to be a mathematical question, given that the space of what would be mathematical questions is so unfathomably huge that it's just impossible to explore it through a random walk. Wait, are you saying that Lord Kelvin's Apple hitting Newton's story was that he was smoking a lot of pipe and he needed to categorize his puffs? Yeah, I can't remember if it was smoke rings out of his mouth. I mean, that's a natural place. We should look into that. Yeah. Okay, so you've changed how you and other creators have changed how pedagogy happens via animated videos.

What would it take to do something similar for video games, text? all these other mediums. Why hasn't there been a similar sort of broad scale adoption and transformation of how teaching happens there? I'm not sure I understand the question. You're saying where there's been a rise of explanatory videos, why is there not a similar rise in like pedagogical video games? Yeah. And well, one thing to understand, I guess games are very hard to make. It takes a lot of resources for a given game. And whenever people, people seem to try to do it with pedagogy as a motive? Like they're making some game whose goal is to teach something. I don't play enough game. So I can't really speak to it the way that

well-versed game designers can. It seems to be the case that they are not fun in the way that people would want them to be fun. And then the ones that are actually most effective are not, you know, as directly educational. So there's clearly some of, the one game that I actually have played because enough of my friends told me, hey, you should really do this. It seems relevant to Nath Explanation in the last like decade was The Witness. Have you played it? I've heard about it. So as someone who doesn't play games and then did play it, it's fantastic. It's absolutely well done in every possible way that you could want something to be well done. And critical on that is the nature of how problems are solved. The reason people were recommending it to me is because the

feeling of playing the witness is a lot like the feeling of doing math. It's nonverbal. It's you come across these little puzzles where the simple mechanics of one puzzle inform you about the fundamental mechanics that become relevant to much, much harder ones such that if you do it with the right sequence, you have the feeling of epiphany in ways that are very self-satisfying.

And you come away feeling like, man, you should be able to do something like this for math. And maybe you can. And it's just that it's so hard to make a game at all that there's just not the rate of production that you would need to explore like get enough games out there that one of them hits. Because with YouTube, there's a lot of math videos on YouTube.

It's okay that most of them suck, right? It's okay because, hey, you just need enough that when someone searches for the term that they want, they get one that is good and scratches that itch. Or that, you know, they might get recommended something that is bringing a question to their mind that they wouldn't have thought about, but they become really interested once it's there. Whereas with video games, I don't know, you're also spending a lot more time as a user on each one rather than a five-minute average experience. It's like a many, many-hour average experience.

You ask the question on text. I don't know if I accept the premise that there's not the same advances and innovation in the world of textual explanations. Like Mathagon is a really, really good example of this. That's like the textbook of the future. It's a website. It's basically an interactive textbook.

The explanations are really good. And insofar as it doesn't have more of an impact or more of a reach, It's maybe just because people don't know about it or don't have an easy means of accessing something that recommends to them, like the really good innovations happening in the world of textual explanations in the way that YouTube has this recommending engine that tries its hardest to get more of these things in front of people. And I'm quite sure that in the world of actual written textbooks, I mean, there's so many that I like so much that I think it would be a disservice to talk about that medium is not like making advances in terms of, more and more thought put towards empathy to the learner and things like that. Should the top 0.1% of educators, should they be exclusively on the internet? Because it seems

like a waste if you were just a college professor or a high school professor and you were teaching 50 kids a year or something, given the greater scale available, should more of them be trying to see if they can reach more people? I think it's not a bad thing for more educators who are good at what they're doing to put their stuff online for sure. Highly encourage that, even if it's as simple as getting someone to put a camera in the back of the classroom, I don't think it would be a good idea to get those people out of the classroom. If anything, I think one of the best things that I could do for my career would be to put myself into more classrooms.

And actually, I'm quite determined at some point to be a high school math teacher for some number of years. I don't know. There's such opportunity cost, I guess. It's something I would plan on, like, notably later, as long as there's not other life logistic. that occupy a lot of mindshare because everything I know about high school teaching is like it just kicks your ass for the first two years. But I would say I, one of the most valuable things that you can have if you're trying to explain stuff online is a sense of empathy for what the possible viewers are like that that are out there. The more distance that you put between yourself

and them in terms of life circumstances, you know, I'm not a college student. So I don't have the same empathy with what a college student is like, certainly not a high school student. So I've lost that empathy. That distance just makes it more and more of an uphill battle to make the content good for them. And I think keeping people in regular touch with just what people in the classroom actively need is necessary for them to remain as good and as sharp as they are. So yes, get more of those top 0.1% to put their stuff online. But like, I would absolutely disagree with the idea of taking them out of their existing circumstances maybe for a year or two so they don't lose that sharpness, but then like put them right back in because one, it makes them better at the

online exposition. But also, the other thing I might disagree with is the idea that the reach is lower because, okay, yes, it's a smaller number of people, but you're with them for much, much more time. And you actually have the chance of influencing their trajectory through a social connection in a way that you just don't over YouTube. And I think you're using the word education. education in a way that I would maybe sub out for the word explanation. Like you want explanations to be online, but like education, the word education derives from the same root as the word educe, like to bring out. And I really like that as a bit of etymology because it reminds you that the job of an

educator is not to like take their knowledge and shove it into the heads of someone else. The job is to bring it out. That's very, very hard to do in a video. And in fact, even if you can kind of get at it by asking intriguing questions, for the most part, The video is there to answer something once someone has a question. And the teacher's job or the educator's job should be to provide the environment such that you're bringing out from your students as much as you can through inspiration, through projects, through little bits of mentorship and encouragement along the way that requires eye contact and being there in person and being a true figure in their life rather than just an abstract voice behind a screen. Then should we think of educators more as motivational speakers as in the actual job of getting the content? in your head is maybe for the textbooks or for the YouTube.

But now the job, like why we have college classes or high school classes is we have somebody who approximates Tony Robbins to, you know, get you to do the thing. That would be a subset of it. But there's more than just motivational speech that goes into it, right? There's facilitation of projects or even coming up with what the project are or recognizing what a student is interested in so that you can try to tailor a question to their specific set of interests where you can maybe act as the curator where, hey, there's a lot of online explanations for what a Poisson distribution is.

You know, like, okay, which of these is the right one that I could serve? And based on knowing you as a particular student, what might resonate, you might be in a better position to do that. All of that goes beyond being a Tony Robbins out, you know, saying like, oh, be the best person that you can be and all of that. And one one thing I might say is that any time that I'll chat with mathematicians and try to get a sense for how they got into it, what got them started. So, so often, it starts by saying, well, there was this one teacher. And that teacher did something very small.

They like pulled them aside and just said, hey, you're really good at this. Have you considered studying more? Or they give them an interesting problem. And the thing that takes, like at most 30 minutes of the teacher's time, maybe even 30 seconds, has these completely monumental rippling effects for the, the life of the student they were talking to that then sets them on this whole different trajectory. Two examples of this come to mind, actually. One is this woman who was saying she had this moment. She got pulled aside from the teacher.

And he just said, like, hey, I think you're really good at math. You should consider being a math major, which had been completely outside of her, like, per view at that time. And that changed the way she thought about it. And then later she said she learned that he, like, did that for a large number of people. He just pulled them and be like, like, hey, you're really good at that. So that's a level of impact that you can have in person as a figure in their lives in a way that you can over a screen. Then another one, which was very funny, I was asking this guy why he went into the specific field that he did.

It was a seemingly arbitrary thing in my mind, but I guess all pure math seems to be. He said that he, for his first year of grad school, was sitting in this seminar. And at the end of the seminar, the professor, who was this old professor, he had never met him before. any kind of connection. He seeks this guy out. He comes up and he says, you, I have a problem for you, a good research problem that I think might be a good place for you to start in the next couple months. And this guy was like, oh, okay. And he like gets this research problem and he spends some months thinking about it and he comes back. And then it later came to light that the professor mistook him for someone else. That was someone he was supposed to be mentoring. And so he was

just like the stereotypical image of like a dottering old math professor who's not very in tune with the people in his life, that was the actual situation. But nevertheless, that moment of accidentally giving someone a problem completely shifted the research path for him, which if nothing else shows you the sensitivity to initial conditions that takes place when you are a student and how the educator is right on that nexus of sensitivity who can completely swing the fences one way or another for what you do. And for every one of those stories, there's going to be an unfortunate counterbalancing story about people who are demotivated from math.

I mean, I think this was seventh grade. There was this math class that I was in. And I was one of the people who was like good at math and enjoyed it and would often like help the people in the class understand it and all that. And so I had enough ego built up, I guess, to have like a strong shell around things. And I guess for context, I also really liked music, and there was this, like, a concert that had happened where I had, like, a certain solo or something earlier in the day,

or earlier in that week. And there was a substitute teacher on one day. So she didn't have, like, any of the context. And she gave some lesson, and then had it been the second half of the glass going over the homework for it. All of the other students in the class were very confused. And I think I remember, like, they would kind of come to me, and I would try to offer to help them.

And the substitute was, going around the class in these circles and basically marking off a little star for how far down the homework people were, just to get a sense, are they progressing? And that was kind of her way of measuring. Are they very far? And when she got to me, I had done like none of them because I was spending my whole time trying to help all of the others. And she like, after having written a little star next to the same problem, like three different times, she said to me like, you know, sometimes music people just aren't math people. And it keeps walking on. So now I was in the best possible circumstance to not let that hit hard because like one I had the moral high ground of like hey I've just been

helping all these people like I understand it and I've been doing your job for you this was my like little egotistical seventh grade brain I knew that I knew the stuff and things like that even still even with all of that as like the armor that was put up I remember it was just this like shock to my system you know she says this thing and it just made me like strangely teary eyed or something and I can only imagine if you're in a position where you're not confident in math. And the thing that you know deep in your heart is actually you are kind of struggling with it. Just a little throwaway comment like that could completely derail the whole system in terms of your relationship with the subject. So it's another example to illustrate the sensitivity to

initial conditions, right? Where there, you know, I was, I was in a robust position. It wasn't as sensitive. I was going to love math no matter what. But you envision someone who's a little bit more on that teetering edge. And the comment one way or another, either saying you're good at this, you should consider majoring in it or saying sometimes music people aren't math. people, which isn't even true. Or you know what I mean? Like that's the other thing about it that like niggled at my brain when she said it. Yeah.

All of that is just so important for people's development that when people talk about online education as being, you know, valuable or revolutionary or anything like that, there's a part of me that sort of rolls my eyes because it just doesn't get at the truth that online explanations have nothing to do with all of that important stuff that's actually happening. And at best, it should be in the service of helping that, that side of things that's where the rubber meets the road. I had Tyler Cohen on the podcast. And he obviously has Marjoral Revolution University, these YouTube videos where he explains economics. And he had a similar answer to give.

I think I asked basically, why aren't you just replacing, or should we think of you as a substitute for all these economics teachers? And in his mind as well, he was more a compliment to the functions that happen in the class. And to your point about the initial conditions, I'm sure you remember the details of the story, but I just vaguely remember hearing this. Wasn't there a case where a mathematician who later ended up becoming famous, he arrives to a lecture late. See, do you want to tell the story? I don't remember beat for beat, but I think it was a statistics class. And he was a grad student, and he comes in, it's late. And there's two problems on the board that the professor had written.

He assumed that those two problems were homework. And so he goes home and he, like, works on them. And after a couple weeks, he goes to the professor's office and, like, turns in his homework. And he's like, I'm sorry, I'm so late. This one, it just took me a lot longer than so many others. And the professor's like, oh, okay, yeah, okay. And just, like, shuffles it away. And then a couple days later, when the prophet had the time to, like, go through and see them,

he realized that the student had fully answered these questions. What the student didn't know is that they were not homework problems written on the chalkboard. They were two unsolved problems in the field that the prophet put up as examples of what the field was striving for. I don't remember what problems they were. So that would be more fun color to add to the story. But then, at least, as the anecdote told it to me however many years ago goes, the prof then, like, finds the students' housing and, like, knocks on the doors. Like, do you realize that, like, these were actually unsolved problems?

And then he gets to basically make those his thesis. So that, yeah, that idea of just being given something for completely random reasons and it shifts the course of what you do. Right, right, right. Or just, you know, it's a thing where if you know a cross sort of solvable, you just like, you just keep going out it until you solve it. Or the four minute mile, right? Exactly.

Yeah, exactly. That's a great example. Another valuable experience, at least one I had, was taking Aronson's classes in college and realizing I am like at least two standard deviations below him. And that was actually a really valuable experience for me, not because it increased my confidence in, I didn't have a moment where I was like, oh, wow, I'm going to. good at this, but it was useful to know, you know, but podcasting is an easier thing to do, right? So then it's good to know that there's actual technical things out there where knowing that

you can get really deep into something and people are just going to be like way above you, having that sort of awareness. Do you think it's fair to have a mental model that has a static G-factor type quality here such that your two standard deviations below and that is forever the state of things? or do you think that the right mental model is something that allows for flexibility on where contributions actually come from or where intuitions come from that through many years of experience and certain kinds of problem solving maybe what seemed like a flash of insight was actually like the residue of just years of thinking about certain kinds of puzzles that he had that you maybe didn't can tell you a story from that class actually yeah go for so he's giving some proof about like how one can play it actually wasn't that he was giving a proof of that I forgot the name of the method, but it's a very important method in complexity theory that helped to prove the bounds of the complexity of different problems. And he explains it and he says, you know, in 1999,

I approved this myself, but I realized that six months before, somebody had already published a paper with this method. And I realized, I'm catching up to the frontier now, you know? But when I was a kid, I was like doing Euler, that's like 2,000 years in regress now and six months behind. And And then so later on in the day, I'm like, wait, 1999. How old was Scott Aronson in 1999? And I think it was 18 or 19. And he was basically proving frontier results in complexity theory that were. Yeah.

So at that point, you're like, all right, Aronson's a special animal here. You are right. He's probably a special animal. But it's just broadly good to know. Just like have that sort of upper constraint on your Dunning Kruger that you can, this exists in the world. Maybe the thing that I would want to say is that Whatever the scale is on which he's two standard deviations above you That might not be the one scale that matters and that like contributions to these fields

Don't always look like genius insights and that Sometimes there's there's fruit to be born from say becoming kind of an expert in two different things and then finding connections between them And like the the people who make contributions are not necessarily the Scott Aronsons of the world still you are probably a truth that there are it's like von noyman's another example of one of these yeah yeah yeah okay how much does uh gordels inclinus theorem practically matter is it is it something that comes up a lot or is it just a interesting thing to know about the bounds that uh isn't day-to-day applicable you've asked me another question where i'm not the best one to ask and i should throw that um as a caveat to begin from what i understand it really doesn't come up

I mean, the analogy to make, the paradoxical fact that it's conveying, the idea that you can't have an axiom system that is both, that will basically like prove all of the things that are true and which is also self-consistent, the contradiction that you construct out of that has the same feeling as the sentence, this statement is a lie. You know, you think about the statement. If it's false, then it must be true.

If it's true, it must be false. It's that same flavor. And you might ask, does the existence of that paradox mean that it's hard to speak English unequivocally. Yeah, that's interesting. It's so rare that you would come up with something that happens to have a bit of self-reference in it. There were one of the first times

that there was something that came up that didn't feel quite as pathological in that way if the curious listener wants to go into it, that search from would be Paris Harrington theorems, where it's a little pathological, but it wasn't at the realest question that came up that didn't seem like it was deliberately constructed to be one of these self-referential things

where, you know, it shows itself to be outside the bounds of whatever axiom system you were starting with. And it, so it was, you know, shown to be unresolvable in a certain sense. But it was asking a, I don't want to say natural, because a lot of these math questions aren't natural in the sense that most people would want. But it was asking a question where you wouldn't expect that to be true. So maybe at the edges of theory, there are sometimes when the paradoxes that are possible, if they curdles in possibility theorem shows, like do eke their ways in. The impression I get is,

no mathematician is thinking about it. They're not actively worrying about it. It's not like, oh, God, no. Like, can I be sure that the stuff that I'm going to show us to, for all the practical problems, it's like, you know, the Riemann hypothesis or twin primes,

almost everyone's like, no, there's going to be an answer. Like, maybe they turn out to be unresolvable in one of these ways. But like, there's just a strong sense that that theorem came from a pathology in a way that natural questions that people actually care about don't. That's really interesting that something from the outside and in popularizations

seems to be a very fundamental thing where people have definitely heard about this, right? It's not internally. A good analogy here is in computer science, the halting problem. It's like you take a computer science course. One of the first things you'll learn is the proof of the halting problem. And it's another one of those things where you don't really need. To be able to prove that every single, or have that sort of program available. Yeah.

No comments. No more comment. Why are good explanations so hard to find, despite how useful they are? Obviously, there's many other than you as well. There's many other cases of good explanations. But generally, it just seems like there aren't as many as there should be. Is it just a story of economics where it's nobody's incentive to be making, really spend a lot of time making good explanations? Is it just a really hard skill that isn't correlated with being able to come over with the discovery itself?

Why are a good explanation scarce? I think there's maybe two explanations. The first less important one is going to be that there's a difference between knowing something and then remembering what it's like not to know it. And the characteristic of a good explanation is that you're walking someone on a path from the feeling of not understanding up to the feeling of understanding. earlier you were asking about societies that are that lack numeracy that's such a hard brain state to put yourself in like what's it like to not even know numbers how would you start to explain

what numbers are maybe you should go from a bunch of concrete examples but like the way that you think about numbers and adding things it's just you have to really unpack a lot before you even start there and I think at higher levels of abstraction that becomes even harder because it it shapes the way that you think so much that remembering what it's like not to understand it you're teaching kid algebra and the premise of like a variable. They're like, what is X? You're like, well, it's not necessarily anything, but it's what we're solving for. Like, yeah, but what is it?

Like trying to answer what is X is a weirdly hard thing because it is the premise that you're even starting from. The more important one probably is that the best explanation depends heavily on the individual who's learning. And the perfect explanation for you often might be very different from the perfect explanation for someone else. So there's a lot of very good domain specific explanations. You know, pull up in any textbook and like chapter 12 of it is probably explaining the content in there, like, quite well, assuming that you've read chapters 1 through 11. But if you're coming in

from a cold start, it's a little bit hard. And so the real gold nugget is like, how do you construct explanations, which are as generally useful as possible, as generally appealing as possible? And that, I think, because you can't assume shared context, it becomes this challenge. And I think there's like tips and tricks along the way, but because the people that are often making explanations have a specific enough audience, it is this classroom of 30 people or it's this discipline of majors who are in their third year. All the explanations from the people who are professional explainers in some sense are so targeted that maybe it's the economic thing you're talking about. There's not, or at least until recently in history, there hasn't been the need to or the

incentive to come up with something that would be motivating and approachable and clear to an extremely wide variety of different backgrounds. Is the process of making her video, is that mostly you? Okay. Given the scale you're reaching, it seems that if it was possible, you know, just like a small increase in productivity would be worth like an entire production studio. And it's interesting to me or surprising that the transaction cost of having a production setup are high enough that it's better to literally do the mundane details yourself.

I mean, this could honestly just be a personal flaw. Like, I'm not good at pulling people in. And then I've struggled to do this effectively in the past. But a part of it is that the seemingly mundane details are sometimes just how I even think about constructing it in the first place. You know, the first thing that a lot of YouTubers will do if they can hire is hire an editor. And this will be because they film a lot of things. And so a lot of the editing process is removing the stuff that was filmed that shouldn't be in the video and just leaving the stuff that should be in the video.

And that's time consuming and it's kind of mundane. And it's probably not that relevant to what the creator should be thinking about. When you're not filming stuff, like the editing process for me, you know, I start by laying out all of the animations and stuff that I'll want in a timeline. And then once I record the voiceover, the actual editing is like a day. And it's not like, I guess I could hire someone. and gain a day back of my life. But the communication back and forth for saying what specifically I want.

Like all of the little cuts that I'm making along the way are my way of even thinking about what I want the final piece to be such that it would be hard to put it into words. It's similar for why I maybe find it quite hard to use like a copilot and some of these like LLM tools in coding

where for the animation code, you know, it can be super great if you're learning some new library and it knows about that library that you don't. But for my library that I know inside it out, if I'm just using it, it feels like, this should be the most automatable thing ever.

You know, it's just text. Like, I should be the first YouTuber who can actually do this better because the substance behind each animation is text. It's not like an editing workflow in quite the same way. But it doesn't work. And I think it's because maybe it's just because you need a multimodal thing that actually understands the look of the output. Like the output isn't something that is consumable in text.

It's something about how it looks. But at a deeper level, I can't even put into words what I want to put on the screen except to do so in code. Like, that's just the way that I'm thinking about it. And, like, if I were to try to put into English the thing that I want as a comment that then gets expanded, that task is actually harder than writing it in the code. And if it's clunky to write in code, that's a sign that I should, like, change the interface of the library such that it's less clunky to be expressive in the way that I want.

And it's in that same way where a lot of the creative process that feels, you know, that's mundane, those are just like the cogs of thought slowly turning in a way that if they weren't turning for that part, they would have to be turning during the interface of communication with a collaborator. Yeah. On the point of working with copilot, where we kind of visualize the changes you wanted to make, the paper from Microsoft research, the sparks of AGI paper had it actually a really interesting example where it was generating LATEC and they generated some output and they say, change

this so that the visual that comes up in the rendering is different in this way. And it was actually able to do that, which was their evidence that it can understand the higher level visual abstraction. So, but I guess I can't do that for Manum. There's a couple of reasons it might not be as fair comparison. One would be the two versions of Manum. There was a split where there's a community version that is, you know, by the community for the community. And then mine, the interfaces are largely similar. The rendering engines are quite different. But because of light differences in that. And, you know, it might have a tendency.

It learned from one or it saw examples from one and it's intermixing them. So stuff just doesn't quite run when there's discrepancy. I maybe shot myself in the foot where all of my code for videos, like, I don't really comment it that much because it's like a one and done deal. It's, you know, there's also the code for the core library, I could comment and document a lot better. But like the way that I'm making it feels much more like the editing flow. If you were to look at the operation history of someone in After Effects, right?

like written down. It's a little bit more like that where there's not a perfect description in English of the thing that I want to do and then the execution of that. It's just, it's just the execution of that. And, you know, it's not meant to be editable in hindsight as much because I'm just in the flow of making the scene for the one video. And, you know, maybe I could have given it a better chance to learn what it's supposed to be happening by having a really well-documented set of like, this is the input, this is the output, this is the comment describing it in English, But even then, that wouldn't hit the problem on like, I would have to articulate what the thing I want is in the first place. And the language, the program language, is just the right mode

of articulation in the first place. This is something I wanted to, I was really curious about ever since I learned about it. I watched many of the summer of math exploration prize videos. And it was shocking to me how good they, I mean, these looked like entire production studios were dedicated to making them, many of them. And it was shocking to me that you could motivate and elicit this quality of contribution given the relatively modest price pool, which I was like five winners, $1,000 each. What is your explanation of just running prices like this, why you were able to get such high-quality contributions?

Is it a price pool even relevant? Is it just about your reputation and reach? I do wonder how relevant the prize pool is. I mean, we've been thinking about this because, you know, we did it first in 2021, then we plan to continue doing it annually. And I probably, if I, like, was a mover and a shaker, could raise much more if I wanted to, like, get a big price pool there. I don't think it would change the quality of the content because the impression I get

is that people aren't fundamentally motivated by, like, winning some cash prize. Certainly, they're not investing with a, like, expected value calculation on what, if they are, that's a terrible, terrible plan. And if anything, like that might be a problem. Like, let's say it was a $100,000 prize for each of the winners. Then it would be a real problem where someone who, and people do like delusionally think that they're very likely going to be the winner, they might like actually pour a lot of their own resources into it with the expectation of gaining it.

And then that's just a messy situation. I don't want to be in with like, why wasn't mine chosen as a winner? Because the whole event is not supposed to be about winners. In fact, maybe for listeners who don't know, I should describe what you're talking about, the summer of math exposition. Actually, the history is a little bit funny because it started with an intern application where I, in 2021, wanted to have just like a couple interns to do a certain thing on my website, basically. And I put out a call for people to apply. I got 2,500 applicants. And somewhere in the application I mentioned that during the summer, in addition to like the main task I wanted them to do, you know,

know, I'd give freedom if they just wanted to do something relevant to math exposition online that was their own thing. I'd be happy to provide, you know, some mentorship or just give them the freedom to do that one day a week and give me a little pitch on what your idea would be. And as I went through, you know, all of the applications, which is a lot, I felt so bad because so often the person would have a little pitch on like what they would want to make. And in my mind, I'd think, cool, you should make that. You don't need me to like do that. Like, just spend your summer making that. Why not? And people were clearly inspired by the thought of adding something. And like I said earlier, being a YouTuber is the most common job aspiration among the youth these days. And so as a

consolation of sorts to those 99% that I had to reject for the internship, I said, oh, but what we will do, we're just going to host this thing. We'll call it the summer of math exposition where we'll give you a deadline. I'll promise to feature some people in a video. maybe I'll choose five of them to do that. And if you feel like the thing that you were going to do, like with me as your 20% project as an intern is something you're excited about, make it 100% project or 20% or whatever, whatever freedom you have, just do it anyway. And like I can give you this little carrot in the form of featuring it in a video and give you a deadline, which let's be honest is what actually makes the difference between people doing something and procrastinating on it sometimes.

Later, there was, I think it was brilliant that year, said they would be happy to put some cash. prizes in, so I said, sure, why not? I don't think the cash prize is super important, but it's nice. I mean, it shows that someone actually cared and put some real thought into doing something that wasn't just a made-up gold star, but they put some material behind saying that you were selected as a winner of this thing. But all in all, it was never supposed to be about choosing winners. It was just get more people to make stuff. And if anything, I'd actually, I love it when I see stuff from existing educators and teachers, where it's maybe not the youth who want to to be YouTubers pouring their hearts and souls into it, but it's the educator who built up a lot

of intuition over the course of their career for what constitutes a clear explanation, and they're just sharing it more broadly. So to your question on what is it that caused there to be such high production quality in some of the entries there, part of the answer might just be that tooling is so good now that individuals can actually make pretty incredible things sometimes. I misphrased if I said production quality. I just meant the whole composition. as a whole. Yeah, well, there's a selection filter too, right? Like, there were in that first year 1,200 submissions. I featured in, you know, the winning video, five of them. So of course, they're necessarily unrepresentative of the norm by the very nature of who I was choosing to feature.

But the fact that those were even in the pool, something that high quality was even in the pool. I think it hits a little bit to your miracle year point where I think what might be happening is you have people with a ton of potential energy for something that they'd kind of been thinking about making for a long time. And the hope was to give people a little push. Here's a deadline. Here's a little prize. Here's a promise that maybe if you make it, it won't just go into the void, but there's a chance that it gets exposed to more people, which I think has absolutely played out. And not for the reason that someone might expect, where I choose winners and I feature those winners and people watch them. A huge amount of viewership happens before I even begin the process.

of looking at them. And this was an accident too, where in this first year, we got 1,200 submissions. I said expect judges who are reviewing it to spend at most 10 minutes on each piece. So it could be longer, but like don't rely on someone watching it for more.

But realistically, when I'm reviewing something, I want to watch the whole piece. I absolutely do not have time to watch that mini. I've learned it takes me about two weeks of just full-time work to like watch a hundred of these pieces. and give the kind of feedback that I want.

And to manage that problem of more than we could manually review, we put together this peer review system that would basically have an algorithm feed people pairs of videos and they would just say which one is better and then it would feed them another one. And in the first two years, we just used a tool that was common for hackathons that did this. And what that did is, one,

it gave us a partially ordered list of content by quality, loosely. We didn't need it to be perfect. We just needed there to be a very high chance that the five most deserving videos were visible somewhere in that top 100, right? So there the algorithm doesn't have to be perfect. But what happened, I think, is that by having this period when a ton of people are watching the same batch of videos, so a thing I've learned about the YouTube algorithm is in theory, you would want to just use machine learning for everything, right? Like you have some massive neural network where on the input of it, it's got five billion videos or however many exist and the output it decides what seven are best to recommend to you that is completely computationally and feasible right i also think i think this is all public knowledge

um so what what you have to do instead is use some sort of proxies as a first pass to nominate a video to even be fed into the machine learning driven algorithm so that you're only feeding in like a thousand or so or something in the order of thousands like nominees so the real difference that can make if you've made a really good video between it getting to the people who would like it and not getting there. It's not the flaws in the algorithm. The algorithm is probably quite good. It's the mismatch between the proxies being used to nominate stuff to see whether it's even in the running. So one of the things used for nomination is understanding the co-watch graph where if you've watched video A and you've also watched video B and then I watch video A, you're watching both of those

gives a little link between them, or maybe you and a ton of other people watching both of them, gives a little link between such that once I watch video A, B is potentially nominated in that phase because it's recognized that there's a lot of co-watching. That's something that I'm sure still is quite challenging at scale, but is more plausible to do at scale than like running some massive neural network. And so I think what might have happened is that by having a bunch of co-watching happening on this same pool of videos, all you need is for like some of them to have decent. reach and get recommended, right? Because then that kicks off, it's like igniting a pile of kindling

where then if others are good, if they're going to give people good experiences, they get not only nominated but then recommended, which then kicks back in the feedback loop there. So that turns out to be, I think as close to a guarantee as you can get of saying, if you make something that's good, it's a good piece that will satisfy someone. They come away feeling like they learn something that they otherwise didn't know and it was well presented. If you can get it into this peer review process, it will reach people. It's not just going to be shouting into the void. And in this case, there was last year, I think over 100 videos where after the first two

weeks they had more than 10,000 views, which I know in the grand scheme, people think, like, huge numbers of views and millions out there. But like for a fresh channel talking about a niche mathematical topic, to be able to put it out and like get 10,000 people to watch it, I think is amazing. right? And the idea that that could happen for over 100 people, I think is amazing. So that had nothing to do with the prize pool, right? In that the motive might have been a hope of actually getting some reach and having some sense of a guarantee of there being some reach. Ironically, the reason to do the whole peer review system in the first place is in the service of selecting winners. And so, like, without something in the service of selecting winners, if you just said, hey, we're having a watch fest where everyone watches each other's things, somehow it wouldn't quite have the same, like, I don't know, pull that gets people into it. So I think it still makes sense to have winners

and to have some material behind those winners. It doesn't have to be much, though. And if anything, I think it might ruin it to make it too much. I'll also say it's $15,000, actually, because we give $500 to 20 different honorable mentions, at least this year. So still pretty modest in the scheme of how much money you can invest

to try to get more math lessons in the world. I watch many of the honorable mentions as well because they were just topics that were interesting to me. And I mean, it's like the thing that the president of Chicago University said at one point where he said we could discard the people we admitted and select the next thousand for our class. And there would be no difference. It was that level of, you know, you're, yeah, that level of peak. Yeah.

And I really admire, by the way, not only the education that you have provided directly with your videos, which have reached millions of people, but the fact that you were also setting up this way of getting more people to contribute and maybe get to topics that you wouldn't have. time to get to yourself. So I really admire that you're doing that. If you're self-teaching yourself a field that involves mathematics, let's say it's physics or some other thing like that, there's problems where you have to understand how do I put this in terms of a derivative and integral. And from there, it's can I solve this integral? And what would you recommend to somebody who is teaching themselves, let's say quantum mechanics? And I figured out how to put, how to

get the right mathematical equation here. Is it important for their understanding to be able to take, go from there to getting it to the end result? Or can they just say, well, I can just abstract that out. I understand the broader way to set up the problem in terms of the physics itself. I think where a lot of self-learners shoot themselves in the foot is by skipping calculations, by thinking that that's incidental to the core understanding. But actually, I think you do build a lot of intuition just by putting.

in the reps of certain calculations. Some of them maybe turn out not to be all that important. And in that case, okay, so be it. But sometimes that's what maybe shapes your sense of where the substance of a result really came from. I don't know. It might be something you realize, oh, it's because of this square root that you get this decay.

And if you didn't like really go through the exercise, you would just instead coming away thinking like such and such decays, but with other circumstances, it doesn't decay and like not really understanding what was. the core part of this, you know, high-level result that is the thing you actually want to come out remembering. Like putting in the work with the calculations is where you solidify all of those underlying intuitions. And without the forcing function of homework, we just like don't do it. So that, I think that's one thing that I learned as a big difference post-college versus during college is like post-college. It's very easy in learning stuff to just accidentally skip that.

And then it doesn't sink in as well. So I think when you're reading something, having a notebook next to you, just like having pen and paper, pencil and paper, whatever it is, should be considered part of the actual reading process. And if you are relying too much on like reading, kind of looking up and thinking in your head, you know, maybe that that's going to get something. But it's not going to be as highly leveraged as it could be.

What will be the impact of more self-teaching in terms of what kinds of personalities benefit most? So there's obviously a difference in the kind of person who benefits most in a situation where it's a college course and everybody has to do the homework, but maybe some people are better tuned for the kind of work that's placed there versus all this stuff is available for you on YouTube and then textbooks for exercises and so on. But you have to have the conscientiousness to actually go ahead and pursue it. I mean, how do you see this to the distribution of who will benefit from the more modern way in which you can get whatever you want? but you have to push yourself to get it. There's a really good book that's actually kind of relevant to some of your early questions called failure to disrupt that goes over the history of educational technology.

Trying to answer the question, you have these repeated cycles of people saying such and such technology that almost always is getting more explanations to more people promises that it'll disrupt the existing university system or disrupt the existing school system and just kind of never does. One of the things that it highlights is how stratifying these technologies will be. in that they actually are very, very good for those who are already motivated or kind of already on the top in some way. And they end up struggling the most just for those who are performing more poorly. And maybe it's because of confounding causation where the same thing that causes

someone to not do poorly in the traditional system also means that they're not going to engage as well with the plethora of tools available. I don't know if this answers your question, but I would reemphasize that what's probably most important to get people to actually learn something is not the explanation or the quality of explanations available because since the printing press, that has been not literally true because maybe access to libraries and such isn't as universal as you would want. But people had access to the explanation once they were motivated. But instead, it's going to be the social factors. Are the five best friends that you have also interested in the stuff? And do they tend to push

you up or do they tend to pull you down when it comes to learning more things? Or do you have a reason to. There's a job that you want to get or a domain that you want to enter where you just have to understand something. Is there a personal project that you're doing? The existence of compelling personal projects and encouraging friend groups probably does way, way more than the average quality of explanation online ever could because once you get someone motivated, one, they're going to learn it. It maybe makes it a more fluid process if there's good explanations versus bad ones, and it keeps you from having some people drop out of that process, which is important.

But if you're not motivating them into it in the first place, it doesn't matter if you have the most world-class explanations on every possible topic out there. It's screaming into a void effectively. And I don't know the best way to get more people to know things. I have had a thought, this is the kind of thing that could never be done in practice, but instead it's something you would write some kind of novel about, where if you want the perfect school, right, something where you can insert some students

and then you want them to get the best education that you can. What you need to do is, let's say it's a high school, you insert a lot of like really attractive high schooler plants as actors that you like get the students to develop crushes on the various plants, right? And then anything that you want to learn, the plant has to express a certain interest in it. They're like, oh, they're really interested in Charles Dickens or something. something, right? And they express this interest and then they, you know, suggest that they

would become more interested in whoever your target student is if they also read the dickens with them. They're also interested in learning the physics. Like if you get that in a way, you know, you socially engineer this setting, the effectiveness that that would have to get students to actually learn stuff is probably so many miles above anything else that we could do that nothing like that in practice could ever actually literally work, but at least viewing that as this end point of, okay, this mode of interaction would be hyper-effective at education. Is there anything that kind of gets at that, right? And so, okay, the kind of things that get at that, it would be being cognizant of your child's

peer group or something, which is something that parents, I think, very naturally do. Or, okay, it doesn't have to be a romantic crush, but it could be that there's respect for the teacher, right? It's someone that they genuinely respect and look up to, such that when they say there's an edification to come from reading Dick. that that actually lands in a way. And so taking that as a paragon and then letting everything else approximate that,

I would emphasize nothing to do with the quality of online explanations that there are out there, that at best just makes it such that, you know, you can lubricate the process once someone's sufficiently interested. You found a new replica use case. Yes. I mean, I'm not saying we should do it,

but think of how effective that would be. Okay, final question. This is something I should have followed up on earlier, but your plans to become a high school teacher, first amount of years, when are you planning on doing that, and what do you hope to get out of that? I would say no concrete plans. I would say I wouldn't want to do it in a period where, like, I also have young children, and therefore it would make sense to, like, maybe a lot of people will say this kind of thing, but there's friends of mine who would think when their child is in, like, you know, high school, that's when they would want to be a high school teacher. I think two things I would want to get out of it. One of them, as I was emphasizing,

I think you just lose touch with what it's like not to know stuff or what it's like to be a student. And so maintaining that kind of connection so that I don't become duller and dollar over time feels important. The other, I would like to live in a world where more people who are savvy with STEM spend some of their time teaching. I just think that's one of the highest leverage ways that you can think of to actually get more people to engage with math. And so I would like to encourage people to do that and call to action. Some notion of spending, maybe not your whole career, but a little bit of time in teaching. There's not as fluid a system for doing that as, say, going through a tour of service in certain

countries where everyone spends two years in the military. So shy of having a system like that for education, there's all these kind of ad hoc things where, you know, charter schools might have like an emergency credential system to like get a science teacher in. But there's a, you know, teach for America as something out there. There's enough ways that someone could spend a little bit of time that's probably not fully saturated at this point that, you know, the world would be better if more people did that. And it would be hypocritical for me to suggest that and then not to actually put my feet where my words are. Yeah. Well, I think that's a great note to leave it on, Grant. Thanks so much for

coming on the podcast. And genuinely, I mean, you're one of the people I really, really admire. But what you've done for the landscape of math education is, I mean, it's really remarkable. So this is a pleasure to talk to you. Thanks for saying that. I had a lot of fun. Hey, everybody. I hope you enjoyed that episode. As always, the most helpful thing you can do is to share the podcast. Send it to people you think might enjoy it, put it in Twitter, your group chats, et cetera, just splits the world. Appreciate you listening. I'll see you next time. Cheers. Thank you.