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

Dwarkesh Patel 0:00 ... Grant. (piano music plays)

Grant Sanderson 0:00 You know, 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, do I want that?"

Dwarkesh Patel 0:09 (laughs)

Narrator 0:09 (laughs)

Grant Sanderson 0:09 To get more people going into math PhDs? Math academia, finance, and computer science almost certainly have an over-allocation of talent. Actually, I'm, 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 train 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.

Dwarkesh Patel 0:44 Okay, today, I have the pleasure of interviewing Grant Sanderson of the YouTube channel 3Blue1Brown. 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?

Grant Sanderson 1:05 I, to be honest, have no idea what people mean when they use the word AGI.

Dwarkesh Patel 1:09 Yeah.

Grant Sanderson 1:10 I think if you ask 10 different people, like, what they mean by it, you're gonna 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's 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 they're, now it's generally intelligent. The reason that world model doesn't really fit is it feels a lot more continuous, where, you know, GPT-4 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 wanna 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, um, the nature of how artificial intelligence can be programmed. So it's, it's generally intelligent, but maybe what people mean by, "Oh, it's not AGI," is you've got certain benchmarks where, you know, it's better than most people at some things, but it's not better at most people than others. You know, at this point, it's better than most people at math. You know, it, it's better than most people at solving AMC problems and, like, IMO problems. It's 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, it's as good as the best, and we've ticked off another domain, but I don't know. Like, i- is what you mean by AGI that you've, you've enumerated all the possible domains that something could be good at, and now it's better than humans at all of them?

Dwarkesh Patel 2:34 Or e- e- enough that it could take over substantial fractions of, you know, h- human jobs or something. Where right now, it's, it, it's impressive, but it's not gonna be even 1% of GDP. But in my mind, if it's getting gold in IMO, I mean, having seen some of those problems-

Grant Sanderson 2:50 Yeah.

Dwarkesh Patel 2:50 ... uh, from your channel, I'm, I'm thinking, "Wow, that's, that's really coming after podcasters and (laughs) video animators." I don't know.

Grant Sanderson 2:56 I don't know. That feels orthogonal, 'cause getting a gold in the IMO feels a lot more like being really, really good at Go or chess. Like, tho- those feel analogous. Well, it's, it's super creative. Like, I think anyone who... I don't, I don't know chess as well as the people who 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. Um, I think as soon as they started generating artwork, then everyone else could appreciate, "Oh, there's, there's something that deserves to be called creative here."

Dwarkesh Patel 3:23 Mm-hmm.

Grant Sanderson 3:23 And the creative side of the math, you know, I'm, I don't know how it would look when people get them to be, um, 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 ways that a lot of other domains don't.

Dwarkesh Patel 3:42 Mm-hmm.

Grant Sanderson 3:42 You could have it produce a lot of proofs in a proof-checking language-

Dwarkesh Patel 3:46 Mm-hmm.

Grant Sanderson 3:46 ... like Lean, for example, and just train on a whole bunch of those, and like-

Dwarkesh Patel 3:48 Yeah.

Grant Sanderson 3:48 ... "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 IMO level things, one of, one of two things. Either it writes a, a very good proof that you feel like is unmotivated, 'cause anyone who reads math papers has this feeling that there are, 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, you're left wanting something a little bit more.

Dwarkesh Patel 4:23 Yeah.

Grant Sanderson 4:23 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 r- what is required to replace jobs? Just, like, not really. Like, the, the, 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 (laughs) you know, some small things such that you can make connections over long periods of time and build relationships and understand where s- someone's coming from.

Dwarkesh Patel 4:51 Mm-hmm.

Grant Sanderson 4:51 And the actual problem-solving part of it, I mean, it, it's a sign that it would be a, a more helpful tool, but in the same way that, like, Mathematica can help you solve math problems much more effectively.

Dwarkesh Patel 5:01 Mm-hmm. And tell me why I should be less amazed by it, or maybe put it in a different context, but the reason I would be very impressed is that w- with chess or something, obviously this is not all that chess programs are doing, but there's a level of tree search you can do to narrow down the possibilities. And more importantly, the, i- in the math example, it seems that, w- with some of the examples you've illustrated 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-

Grant Sanderson 5:36 Mm-hmm.

Dwarkesh Patel 5:37 ... uh, in any other test. That seems different from, uh, just a chess game where you don't have to, like, w- what is the largest structure of this chess game in the same way as you do with a IMO problem.

Grant Sanderson 5:48 I think you should ask people who know r- 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 gonna be as good at Go as AlphaGo is, you're also not doing tree search, at least exclusively. It's not dependent on something as both in that, 'cause you get this combinatorial explosion-

Dwarkesh Patel 6:06 Yeah.

Grant Sanderson 6:06 ... 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-

Dwarkesh Patel 6:17 (laughs)

Grant Sanderson 6:17 ... and, like, anticipate being very impressed when you get, um, AIs that can solve these IMO problems, uh, 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 it's this wh- this historic moment analogous to chess and Go, more so than, than it's gonna be analogous to the Industrial Revolution.

Dwarkesh Patel 6:50 I'm, I'm surprised you, you wouldn't be more compelled-

Grant Sanderson 6:54 I am compelled.

Dwarkesh Patel 6:55 ... or, right, or just you, you don't think that skill of w- h- this problem is isomorphic to 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, I'm surprised you don't think that's, uh, that upstream of a lot of valuable tasks.

Grant Sanderson 7:13 I think it's a similar level of how impressed I was with, um, the, uh, Stable Diffusion type stuff where, you know, you ask it, "Give me a landscape w- of beautiful mountains but made out of, like, 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, something beyond the literal that's understood here. I'm like, "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, um, good artistic analogies and comparisons, um, 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, 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. Like, I just don't know what they mean by the word AGI and I don't think that you're gonna have something that's this measurable discreet step, much less that, um, a math tournament is gonna be an example of what that discrete step would look like.

Dwarkesh Patel 8:23 Interesting.

Dwarkesh Patel 8:24 Interesting. Um, okay. Applied mathematicians-

Grant Sanderson 8:27 Okay.

Dwarkesh Patel 8:27 ... where do we put them in society where they can have the biggest benefit? We have, like, a lot of them go into computer science and IT, and b- I'm sure there's been lots of benefits there. But where, you know, where are there parts of society where y- 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. W- where else do you think they might be useful?

Grant Sanderson 8:48 That's such a good question. In some ways, I'm like the worst person to ask about that.

Dwarkesh Patel 8:51 (laughs)

Grant Sanderson 8:51 But this isn't gonna answer your question, but instead is gonna, 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 i- like, in- inspiring people to ask that, especially students who are graduating, 'cause I think this thing happens where when you fall in love with math or some sort of technical field, by default in school, you, 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 gonna go into the field, the people that you're gaining mentorship from are academics and are researchers in the field, so it's hard not to effectively be apprenticing in that.

Dwarkesh Patel 9:37 Yeah.

Grant Sanderson 9:38 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 that are like, "Grant, you know, your 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, do I want that?"

Dwarkesh Patel 10:02 (laughs)

Grant Sanderson 10:03 "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 that, those who are doing that necessarily get shuffled into the traditional outlets like math academia. Yeah, I think you highlighted it very right. It's like math academia, finance and computer science, data science, something in there in general, um, are very common things to go to. And as a result, they almost certainly have an overallocation of talent.

Dwarkesh Patel 10:31 Mm-hmm.

Grant Sanderson 10:32 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-

Dwarkesh Patel 10:39 Yeah.

Grant Sanderson 10:39 ... where do you want people to go? Again, I'm not answering your question. I'm just asking it in other words 'cause 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, uh-

Dwarkesh Patel 10:52 Yeah.

Grant Sanderson 10:52 ... maybe one good example. Um, uh, 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, 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, you know?

Dwarkesh Patel 11:14 Mm-hmm.

Grant Sanderson 11:15 Uh, NSF grants coming with a, a requirement that 10% of your time goes towards a collaboration with another department or something like that. The thought being these, these are really good problem solvers, um, in a specific category of problems, and to just distribute that talent elsewhere might be helpful. And when I, when I, like, run this by mathematicians, sometimes there's a mixed response where they're like, "Ah, I don't know if we'd be all that useful." (laughs) 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 'cause they also have a set of problem solving abilities that's maybe generalizable?" In the back of my mind I think, "No, but the mathematicians are special." (laughs)

Dwarkesh Patel 12:11 (laughs) Yeah.

Grant Sanderson 12:11 There, there really is something general about math.

Dwarkesh Patel 12:13 Right.

Grant Sanderson 12:14 Um, 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, 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 scratched, that they were scratching by being good at school and, and getting to appreciate that, but because they stepped back and said, "What impact do I want to make on the world?" Um, 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, 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, uh, has a career in logistics or manufacturing or all of these things in such a way that I can have an in-tune feel for where is there a, uh, need for this specific kind of abstract problem-solving.

Dwarkesh Patel 13:18 It might be useful to speculate on how an undergrad or somebody who's a young math whiz might even begin to contemplate, "Here's where I can have an edge." I'm, I'm remembering actually, it just occurred to me, a former podcast guest, Lars Doucet, he was a game designer actually, and he started learning about Georgism, 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 gonna tax land, you gotta figure out what the value of land is. Like, how do you figure out the value of land? There's all these algorithms-

Grant Sanderson 13:53 Mm-hmm.

Dwarkesh Patel 13:54 ... of how you do this optimally based on neighboring land and how to average across land, and it... there's a lot of intricacies there. Um, and so he, 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. You know, just that's another example, right? Where the motivation was more philosophical, but his specialty as a technical person helped them, um, you know, h- help to make a contribution there.

Grant Sanderson 14:21 I think that's perfect. Probably the true answer is that you're not gonna give a universal thing. For any individual, it's gonna be based on where their life circumstances connect them into something, either because they... you know, he had an interest in Georgism for whatever reason. But if someone, I don't know, their, their dad runs a paper mill and they're, like, connected to the family business in that way and realize they can, like, plug themselves in a little bit more efficiently, you're, you're gonna 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-

Dwarkesh Patel 14:53 Yeah.

Grant Sanderson 14:53 ... 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.

Dwarkesh Patel 15:03 Uh, 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 it being able to count and, um, in the decimal system that even the Romans didn't have.

Grant Sanderson 15:17 Mm-hmm.

Dwarkesh Patel 15:18 How likely do you think it is that something that 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, uh, those kinds of thinking tools are basically all gone?

Grant Sanderson 15:30 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.

Dwarkesh Patel 15:40 Or it could be-

Grant Sanderson 15:40 Like, we can do commerce-

Dwarkesh Patel 15:41 Yeah, exactly.

Grant Sanderson 15:41 ... and things like that. Um, or we can think in terms of orders of magnitude-

Dwarkesh Patel 15:45 Yeah.

Grant Sanderson 15:45 ... that would have been hard to think in. We have the word "orders of magnitude"-

Dwarkesh Patel 15:48 Right.

Grant Sanderson 15:48 ... 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-

Dwarkesh Patel 15:54 Yeah.

Grant Sanderson 15:54 ... for our descendants? Um, I mean, fluency with a programming interface really can help understanding certain problems. Uh, I think when people mess around in a notebook with something and when it feels like a really good toolset, 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, uh, 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, get in the notebook, you've got to, like, find the right libraries, that there's something that feels as fluid as when you are, um, good at algebra and you're just at a whiteboard kind of noodling it out. Um, I think there's something to be said for the fact that there's still so much more value in... If, if you and I were to be like 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. Like, 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.

Dwarkesh Patel 17:34 Yeah, that's really interesting. What's up with miracle years?

Grant Sanderson 17:37 (laughs)

Dwarkesh Patel 17:37 So this is something that happens... has happened throughout science and especially with mathematicians, where they have a single year in which they make up many, if not most, of the important discoveries that they have in their career. Uh, you know, Newton, Einstein, Gauss, they all had these years. Uh, do you have some explanation of what's going on?

Grant Sanderson 17:55 What's your take? (laughs)

Dwarkesh Patel 17:57 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-

Grant Sanderson 18:06 (laughs) Every 35-year-old right now is like... (laughs) How dare you?

Dwarkesh Patel 18:10 You know, you know what I mean. (laughs)

Grant Sanderson 18:12 Yeah.

Dwarkesh Patel 18:12 They have 20 years.

Grant Sanderson 18:13 Yeah, yeah.

Dwarkesh Patel 18:14 Um, so yeah, it can't just be that. Um-... 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- you're 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- but what is your take?

Grant Sanderson 18:30 I- I don't know. I, so I agree, there's probably multiple factors, not one.

Dwarkesh Patel 18:35 Yeah.

Grant Sanderson 18:36 One thing could be that the miracle year is the like exhalation, and there's been many, many years of inhalation.

Dwarkesh Patel 18:44 Yeah.

Grant Sanderson 18:44 Where often, let's say, you know, the classic one is Einstein's, where his mir- miracle year were also some of the first papers, like kind of springing onto the scene.

Dwarkesh Patel 18:53 Yeah.

Grant Sanderson 18:53 And I would guess that a lot of the ideas were not bumping around in 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. Um, 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- 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 in some sense, like, I recognize that the well was, uh, 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- and touching something I never would have. That doesn't happen by default. I don't know if this is all the, 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, right? Uh, you know, there's- there's certain academics who made an extremely deliberate effort not to let... W- what do they call it? Is it like the curse of success or... ? There's some term for it, but I think maybe, um, James Watson, you know, 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."

Dwarkesh Patel 20:55 Yeah.

Grant Sanderson 20:55 It was much more articulate than that and he has 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.

Dwarkesh Patel 21:12 Were you a student when you started the channel?

Grant Sanderson 21:14 Technically, yeah. The very, very first video was made when I was a senior at Stanford. Um, basically, I had been like toying around with, uh, 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.

Dwarkesh Patel 21:34 Right.

Grant Sanderson 21:34 I didn't even really know what a YouTuber was. It was really just like a personal project. Um, 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.

Dwarkesh Patel 21:47 Do you think you could, you would have done it if you had, let's say, become, uh, I don't know where you're planning on doing after college. I remember you saying somewhere maybe a data scientist, but...

Grant Sanderson 21:55 Data scientist and math PhD were the two like-

Dwarkesh Patel 21:57 I see.

Grant Sanderson 21:58 ... um, 50/50 contenders basically.

Dwarkesh Patel 22:00 Is there a world in which you started doing that but then later on s- made Manim or do you think that that was only possible in a world where you had some time to kill in your senior year?

Grant Sanderson 22:10 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 Manim as the method for it because it's like so strikingly inefficient to do it that way.

Dwarkesh Patel 22:23 (laughs)

Grant Sanderson 22:23 Um, 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 I, 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- 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. Um, I don't know. Then again there's a lot of other people who make their own tools, uh, for math gifs and little illustrations and things which on the one hand feels very inefficient. Like there's, um, I don't know if people come across a math gif on Wikipedia. There's a very high probability it comes from this one individual who's just strangely prolific at producing these like creative commons visuals and, uh, he has his own like home baked thing for how he does it. Um, and then there's someone who, uh, came across on Twitter, Matt- Matt Henderson who has these completely beautiful, um, math gifs and such and again it's a very home baked thing that is 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- 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...... eh, your guess is as good as mine for those counterfactuals.

Dwarkesh Patel 24:25 Yeah. This is personally interesting to me, because I also started the podcast in college and I, it just like was off track of anything I was planning on doing otherwise. And it, this is, you know, ma- many, many orders of magnitude away from 3Blue1Brown. I don't want the audience to cr- you know, cringe in unison, but, um, I just think it's interesting, like, these kinds of projects, how often so- 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.

Grant Sanderson 24:52 Yeah. 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, I had no anticipation of, much less an expectation of 3Blue1Brown 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, uh, 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 a, as an audience, you know, a 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, they jump in with the goal of being a YouTuber. It's the, I think it's the most common, uh, desired job among the youth is to be like a TikToker or a YouTuber. Which, think of that what you will.

Dwarkesh Patel 25:46 (laughs)

Grant Sanderson 25:46 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. 'Cause one, you're probably not that good at making videos yet, and so you're competing, if it's a general, a generally applicable idea, you're competing with like all of the other communicators out there. Um, 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 gonna be disappointed. It's like outstanding when 1,000 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-

Dwarkesh Patel 26:25 Right.

Grant Sanderson 26:25 ... 'cause 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. Um, and I don't know if this is the case with your podcast when you're starting off 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.

Dwarkesh Patel 26:43 Is it surprising

Dwarkesh Patel 26:44 to you that prehistoric humans don't seem to have had, um, just basic arithmetic and numeracy? To me, with, with a sort of, I, I guess to us with 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 it's, is it that surprising, that, just the concept of numbers and...?

Grant Sanderson 27:08 Y- you're right it's so in our bones that it's hard to empathize with not having-

Dwarkesh Patel 27:12 Yeah.

Grant Sanderson 27:12 ... uh, numeracy. A lot of it's linked to if you think, okay, what's the first place that like people think about numbers-

Dwarkesh Patel 27:19 Yeah.

Grant Sanderson 27:19 ... 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 saying 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-tat structure just makes a lot more sense and actually works well and it would, it would just be obnoxious to like actually account for everything. The other loosely related idea, h- have you come across those, um, 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 you or I do? But they, you know, there's some notion of counting. You-

Dwarkesh Patel 28:00 Yeah.

Grant Sanderson 28:00 ... if you have one coconut or nine coconuts, like-

Dwarkesh Patel 28:02 Yeah.

Grant Sanderson 28:02 ... you have a sense of that. That if you ask what number is halfway between one and nine, those groups will answer three, whereas y- you or I or people in our world would probably answer five.

Dwarkesh Patel 28:14 Yeah.

Grant Sanderson 28:15 And because we think on this very linear scale, it's interesting that evidently the like natural way to think about things is logarithmically.

Dwarkesh Patel 28:21 Mm-hmm.

Grant Sanderson 28:22 Which kind of makes sense. Like, the social dynamics of as you go from solitude to a group of 10 people to a group of 100 people have roughly equal steps and increase in complexity more so than if you go from like 1 to 51 to 102.

Dwarkesh Patel 28:37 Yeah.

Grant Sanderson 28:37 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." (laughs)

Dwarkesh Patel 28:53 (laughs)

Grant Sanderson 28:53 We're like, "But that should be deep in our bones," right? So somehow it got unlearned and it 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.

Dwarkesh Patel 29:12 Yeah. I have seen the thing from Joseph Henrich where the still existing tribes where th- they're in this kind of situation. They can do numeracy and arithmetic when it's in very concrete terms, if you're talking about seeds or something.

Grant Sanderson 29:26 Mm-hmm.

Dwarkesh Patel 29:26 But the, the abstract concept of a number, uh, is not, is not available to them.

Grant Sanderson 29:32 Do you think the abstract concept of a number is useful to your life?

Dwarkesh Patel 29:37 Oh, yeah.

Grant Sanderson 29:38 Like, in what ways?

Dwarkesh Patel 29:39 It, it's almost like asking how is the concept of the alphabet useful? It, it comes up so often that it's... I mean, just like how many lights do I set up for this interview, right?

Grant Sanderson 29:48 Right. Is that the concept of an abstract number though? 'Cause it's like two people, two lights, one-to-one correspondence.

Dwarkesh Patel 29:54 (laughs)

Grant Sanderson 29:54 Like, did you leverage the abstraction of two as a object which is simultaneously irrational and a real and an integer?

Dwarkesh Patel 30:03 I see.

Grant Sanderson 30:03 And a, you know, is in the context of a group that has additive structure but also multiplication.

Dwarkesh Patel 30:09 (laughs)

Grant Sanderson 30:09 It's like, no. You just, there's light for you, light for me.

Dwarkesh Patel 30:12 Right. Yeah.

Grant Sanderson 30:13 And to be... I'm pretty sure the abstract idea of a number is important for all of us.

Dwarkesh Patel 30:16 (laughs)

Grant Sanderson 30:16 But like I don't, I don't think it's immediately obvi- It's more that it shapes the way we think. I, I'm not sure if it actually changes like the way we live.... if, if, if assuming you don't work in STEM, right? Where, like, you literally are using it all the time.

Dwarkesh Patel 30:31 Yeah. I'm, I'm trying to go through my day and think through what, what, "Where am I using them?" I mean, there's the obvious stuff like b- 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. Is, do I really use numbers that infrequently? (laughs)

Grant Sanderson 30:47 I don't know. Yeah. Th- the, many people listening are probably screaming out of their heads-

Dwarkesh Patel 30:52 (laughs)

Grant Sanderson 30:52 ... with, like, much more apt examples, uh, that... It's hard to say.

Dwarkesh Patel 30:57 Yeah, yeah. W- when a mathematician is working on a problem, what is the biggest mental constraint? Is, is it working memory? Is it processing speed? Plants are limited by nitrogen usually. Like, what is the equivalent of nitrogen for a mathematician?

Grant Sanderson 31:11 Uh, that's a fun question. I mean, so I'm not a research mathematician. I sh- I shouldn't pretend like I am, and so the right per- people to ask that question would be the research mathematicians. I wonder if you're gonna get consistent answers. As, as with so many things, there's not one. Maybe the number of available analogies to be able to draw connections. Uh, like the more, um, 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, uh, connections between different fields as, you know, functors between categories or something like that. But sometimes it's a lot more intuitive, the idea that a general flavor of an app- 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 I could take a generating function type approach." Um, maybe you have to massage it to make it work. And the, one of the reasons I, I say this is one of the tendencies that you've seen in math papers in, let's say, like the last 200 years, the typical number of authors is, like much bigger now than before.

Dwarkesh Patel 32:25 Mm-hmm.

Grant Sanderson 32:25 Your people have this, I think, misconception that math is a field with lone geniuses who are off, like, coming up with great insights, like alone next to a blackboard. And the reality is, it's a highly collaborative field. I remember one of the first, um, times that I was hearing from a mathematician as like a, 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. Um, and it's because, you know, if you're studying some very specific niche field, that the way that you make progress in that is by collaborating with other people, like, in that field or maybe adjacent to that field-

Dwarkesh Patel 32:59 Yeah.

Grant Sanderson 32:59 ... 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, 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-

Dwarkesh Patel 33:15 (laughs)

Grant Sanderson 33:15 ... that's tr- does that systematically. Um, and you could say that's a social thing, uh, you know, get, 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 a mathematician is the analogy.

Dwarkesh Patel 33:33 This actually raised an interesting question I wasn't planning on asking you, but it, 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 2,000, 3,000 years who were doing, th- you know, pretty sophisticated things, even the ancient Greeks. Not ev- it depends on what you mean by sophisticated, but, eh, eh, impressive things.

Grant Sanderson 34:03 Yeah.

Dwarkesh Patel 34:04 I don't know why linear algebra is so new given that fact.

Grant Sanderson 34:07 I wouldn't have thought of math as being, like, new in that way. If anything, especially if we're talking at the high school level, I remember there was always this sensation that it's frustrating that all of the things are actually way more than 100 years old, um, in terms of, you know, the, 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-

Dwarkesh Patel 34:29 Yeah, yeah.

Grant Sanderson 34:29 ... how proteins are formed is relatively much more modern and you might be just a couple of generations away. I, to some extent, there's a raw manpower component to it where, like, how many people did pure math-

Dwarkesh Patel 34:41 Mm-hmm.

Grant Sanderson 34:41 ... 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, or a natural philosopher. And insofar as you're doing, like, natural philosophy, one component of that is developing math, but it's, it's not the full extent of what you do. You know, even the ones who we think of as, like very, very, mm, pure mathematicians in the sense that a lot of their most famous results are pure math, like Gauss-

Dwarkesh Patel 35:06 Yeah.

Grant Sanderson 35:07 ... like actually, a lot of the work, uh, a lot of his, you know, output was also centered on very practical problems and, um, 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, the man hours that are being put into developing new theorems is probably just got this huge spike-

Dwarkesh Patel 35:29 The highest.

Grant Sanderson 35:29 ... 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, grows. Maybe it's pretty reasonable, um, that most of it is much, much more recent. Um, that would be my guess.

Dwarkesh Patel 35:44 But the, you know, I mean, some of these things seem a- actually pretty modern. Like information theory is less than 100 years old.

Grant Sanderson 35:49 That's true.

Dwarkesh Patel 35:49 And it's, you know, pretty fundamental. Like theoretically, you could've written that paper a long time ago.

Grant Sanderson 35:54 Yeah. 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-

Dwarkesh Patel 36:03 Yeah.

Grant Sanderson 36:03 ... that it's used to solve than we typically think of it. On the one hand, information theory, such a good example 'cause it's so pure-

Dwarkesh Patel 36:09 Yeah.

Grant Sanderson 36:09 ... 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 and you're at Bell Labs, that, that's, that's what prompts it. Another maybe really good example for that would be chaos theory.

Dwarkesh Patel 36:29 Mm-hmm.

Grant Sanderson 36:29 You could easily ask, "Why, why is chaos theory so recent?" Um, you could've-... you know, written the, the Lorenz equations since differential equations existed. Uh, w- 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 problems that they introduce are the problems to solve, but instead, you need them to even discover the phenomena in the first place. Like, a lot of original concepts in KS Theory came from basically running simulations or doing things that required, uh, 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-

Dwarkesh Patel 37:20 Yeah.

Grant Sanderson 37:20 ... just the nature... three body problem, like all of that kind of stuff, um, 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 in- invariably also shift like the mathematician's focus.

Dwarkesh Patel 37:44 Mm. I- I... This actually reminds me, the first day of Scott Aaronson's quantum information class, he said that, "What I'm about to describe to you could have been discovered by a mathematician before, you know, quantum physics existed, if only they'd asked the question-"

Grant Sanderson 37:59 Yeah.

Dwarkesh Patel 38:00 "...we- we're gonna do probabilities, but we're only a- allowed to use unitaries." And the rest of it, a- a... you know, you could've just discovered quantum mechanics or quantum information from there.

Grant Sanderson 38:09 I mean, the thing about math, right, 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 gonna deduce from those axioms like various different lemmas and theorems and, uh, proceed from that." And with that as the framing, you get the impression that you could have just come up with any axioms, right?

Dwarkesh Patel 38:32 (laughs)

Grant Sanderson 38:32 You 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 culls 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-

Dwarkesh Patel 38:48 Mm-hmm.

Grant Sanderson 38:48 ... is 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 b- linear algebra and probability, and that's great, but there's all sorts of other things where you could kind of try to cram-

Dwarkesh Patel 39:04 Yeah.

Grant Sanderson 39:04 ... them together and maybe get some sort of math out. And the, the question becomes, is it worth your time to do that?

Dwarkesh Patel 39:11 Mm-hmm.

Grant Sanderson 39:12 You know, knot theory is something that emerged because, I think it was Lord Kelvin had a theory that all of the, like, uh, elements on the periodic table had structures which were related to a, a knot, like, um... a, a knot being if you have a closed loop in 3D space-

Dwarkesh Patel 39:29 Mm-hmm.

Grant Sanderson 39:29 ... but maybe if you wanna 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.

Dwarkesh Patel 39:45 Mm-hmm.

Grant Sanderson 39:45 And this was started with a completely incorrect theory for what's going on at the atomic level that gives, um, atoms this very stable structure 'cause I think he, he found with smoke rings, like if you're somehow very dextrous with how you create these smoke rings, you can get them to form knots in 3D, and they're, they're very stable. It's like weirdly stable, um, in that it'll never, you know, uh, cross over itself-

Dwarkesh Patel 40:09 Mm-hmm.

Grant Sanderson 40:09 ... and so it, it has all th- 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. Um, so really you wanna ask like, what are the... 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... it's just impossible to explore it through a random walk?

Dwarkesh Patel 40:42 Wait, are you saying that Lord Kelvin's apple hitting Newton story was that he was smoking a lot of pipe and-

Grant Sanderson 40:48 (laughs)

Dwarkesh Patel 40:49 ... he needed to categorize his puffs? (laughs)

Grant Sanderson 40:51 Yeah, I- I- I can't remember if it was smoke rings out of his mouth. Th- I mean, that's a natural place. We- we should look into that.

Dwarkesh Patel 40:58 (laughs)

Grant Sanderson 40:58 I should've, uh... yeah.

Dwarkesh Patel 40:59 Okay. So you ha- you 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?

Grant Sanderson 41:20 I'm not sure I understand the question. You're saying, uh, where there's been a rise of explanatory videos-

Dwarkesh Patel 41:26 Yeah.

Grant Sanderson 41:27 ... why is there not a similar rise in like pedagogical video games?

Dwarkesh Patel 41:31 Yeah.

Grant Sanderson 41:31 And, um... well, one thing to understand... I guess games are very hard to make. (laughs) Um, it takes a lot of resources for a given game. And whenever people seem to try to do it with pedagogy as a motive, like th- they're making some game whose goal is to teach something, I- I don't play enough games so I- 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-

Dwarkesh Patel 42:00 Mm-hmm.

Grant Sanderson 42:00 ... and then the ones that are actually most effective are not, you know, as directly educational. Uh, uh, so there's, there's clearly some e-... the one, the one game that I actually have played, um, because enough of my friends told me, "Hey, you should really do this. It seems relevant to math explanation," um, in the last like decade was The Witness.

Dwarkesh Patel 42:18 Yeah.

Grant Sanderson 42:18 Have you played it?

Dwarkesh Patel 42:19 I've heard about it.

Grant Sanderson 42:20 So as, as someone who doesn't play games and then, uh, did play it, I... it's fantastic. It's absolutely well done, uh, in every possible way that you could want something to be well done. Um, and critical on that is, 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. Um, it's non-verbal.... 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 like very self-satisfying. And you can, 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. 'Cause with YouTube, there's a lot of math videos on YouTube.

Dwarkesh Patel 43:16 Yeah.

Grant Sanderson 43:16 It's okay that most of them suck, right?

Dwarkesh Patel 43:18 Right.

Grant Sanderson 43:18 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. Um, 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 asked the question on text, I don't know if I, uh, I accept the premise that there's not the same advances and innovation in the world of textual explanations. Like, um, Mathigon is a really, really good example of this that's, uh, like textbook of the future. Um, it's a, it's a website, it's basically an interactive textbook. The explanations are really good, and in so far as it doesn't have more of an impact or more of a reach as 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, um, 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, I'm quite sure that in the world of actual like written textbooks, you... I mean, there, there's so many that I like so much that I, I, 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.

Dwarkesh Patel 44:43 Should the top

Dwarkesh Patel 44:44 .1% of educators, should they be exclusively on the internet? Because it, it, 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. Uh, given the greater scale available, should, should more of them be trying to see if they can reach more people?

Grant Sanderson 45:01 I, 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-

Dwarkesh Patel 45:18 Mm-hmm.

Grant Sanderson 45:18 ... would be to put myself into more classrooms (laughs) , um, and actually, I'm, I'm quite determined at some point to like be a high school math teacher for some number of years.

Dwarkesh Patel 45:26 Huh. Really?

Grant Sanderson 45:26 I don't know when the... There's such opportunity cost, I guess. Um, I, the probab- uh, it's something I would plan on like notably later, as long as there's no other like life logistics that occupy a lot of mind share-

Dwarkesh Patel 45:39 Yeah.

Grant Sanderson 45:39 ... because everything I know about (laughs) high school teaching is like it, it just kicks your ass for the first two-

Dwarkesh Patel 45:43 Right.

Grant Sanderson 45:44 ... 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, uh, so I don't have the same empathy with what a college student's 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, uh, 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, um, because one, it makes them better at the online expo- position. But also, the other thing I might disagree with is the idea that the reach is lower, uh, 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, a social connection in a way that you, you just don't over YouTube. And I think you using the word education in a way that I would maybe sub out for the word explanation.

Dwarkesh Patel 47:09 Mm-hmm.

Grant Sanderson 47:09 Like, you want explanations to be online, but like education, the word education derives from, uh, the same root as the word educe, like to bring out. And I really like that as a bit of etymology, 'cause 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 the little bits of mentorship and encouragement along the way that requires, you know, eye contact and being there in person and being a, a true figure in their life rather than just an abstract voice behind a screen.

Dwarkesh Patel 48:00 Should... Then should we think of educators more as motivational speakers? As in the actual job of getting the content in your head is, is maybe for the textbooks or for the YouTube, but now the, 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.

Grant Sanderson 48:18 Uh, that would be a subset of it, but there's, there's more than just motivational speech that goes into it, right? There's, um, facilitation of projects or even coming up with what the-proj- projects are or recognizing what a student is interested in so that you can try to tailor a question to their specific set of interests-

Dwarkesh Patel 48:35 Yeah.

Grant Sanderson 48:35 ... or you can maybe act as the curator where, "Hey, there's a lot of online explanations for what a Poisson distribution is," and you're 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, you... be the best person that you can be-"

Dwarkesh Patel 48:58 (laughs)

Grant Sanderson 48:58 ... and, um, 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." (laughs) 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 life of the student they were talking to that then sets this, them on this whole different trajectory. Um, two, two examples of this come to mind actually. One is, uh, 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 st- consider being a math major." Which had been completely outside of her, like, purview at that time. And that, like, 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. (laughs)

Dwarkesh Patel 49:56 (laughs)

Grant Sanderson 49:56 He just pulled them and be like-

Dwarkesh Patel 49:58 Right.

Grant Sanderson 49:58 ... "Hey, you're really good at math." Um, so that's a level of impact-

Dwarkesh Patel 50:01 Yeah.

Grant Sanderson 50:01 ... that you can have in person as a, as a figure in their lives in a way that you can't over a screen. Um, then another one which was very funny, I was asking this guy why he went into th- this specific field that he did. It was a seemingly arbitrary thing in my mind, um, but I guess all, all pure math seems to be. And he said that he, for his first year of grad school, was, um, sitting in this seminar and at the end of the seminar the professor, who was this, you know, old professor, he had never met him before, they didn't have 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 thank- I think might be, um, 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 mon- months thinking about it and he comes back. And then it later came to light that the professor mistook him for someone else-

Dwarkesh Patel 50:51 (laughs)

Grant Sanderson 50:52 ... (laughs) that was-

Dwarkesh Patel 50:52 Right.

Grant Sanderson 50:52 ... someone he was supposed to be mentoring. And so he was just-

Dwarkesh Patel 50:55 (laughs)

Grant Sanderson 50:56 ... like, like the, the stereotypical image of like a doddering old math professor-

Dwarkesh Patel 51:00 Yeah.

Grant Sanderson 51:00 ... who's not very in tune with the people in his life. That was the actual situat- 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, um, and how the educator is, is right on that nexus of sensitivity who can completely swing the fences one way or another for, um, what you do. And for every one of those stories, there's gonna be an unfortunate counterbalancing story about people who were demotivated from math. I mean, I, um, I think this was seventh grade, there was this, uh, math class that I was in and I was one of those 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, I had enough ego built up, I guess, to have like a, a strong shell around things. And I guess for context, there w- I also really liked music and there was this like 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 us spend the second half of the class going over the, um, 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 'cause I was spending my whole time trying to help all of the others, and she like aft- 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."

Dwarkesh Patel 52:49 (laughs)

Grant Sanderson 52:49 And then keeps walking on. So now I was in the best possible circumstance to not let that hit hard because-

Dwarkesh Patel 52:56 Right.

Grant Sanderson 52:56 ... like one, I had the moral high ground of like, "Hey, I've just been helping all these people."

Dwarkesh Patel 53:00 Yeah.

Grant Sanderson 53:00 Like, "I understand it and I've been doing your job for you." This was my like little-

Dwarkesh Patel 53:04 Right.

Grant Sanderson 53:04 ... egotistical seventh grade brain. I, I knew that I knew this 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, 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?

Dwarkesh Patel 53:39 Yeah.

Grant Sanderson 53:39 Where there, you know, I was, I was in a robust position, it wasn't as sensitive. I was, I was gonna 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. (laughs) Or you know what I mean?

Dwarkesh Patel 53:57 Yeah.

Grant Sanderson 53:57 Like that was the other thing about it that, um, that like niggled at my brain-

Dwarkesh Patel 54:01 Yeah.

Grant Sanderson 54:02 ... when she said it. Yeah, I... All, 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, it just doesn't get at the truth that online e- explanations have nothing to do with all of that important stuff that's actually happening.

Dwarkesh Patel 54:23 Mm-hmm.

Grant Sanderson 54:23 And at best it should be like in the service of helping that, that side of things that's where the rubber meets the road.

Dwarkesh Patel 54:29 I had Tyler Cowen on the podcast and he obviously has Marginal 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-"

Grant Sanderson 54:41 Mm-hmm.

Dwarkesh Patel 54:41 "... or should we think of you as a substitute for all these economics teachers?" And y- in his mind as well, uh, he, he was more a complement to, to the functions that happened in, in the class. Um, and to your point about the initial conditions, I'm sure you remember the details of this story, but I, I just vaguely remember hearing this. Wasn't there a case where a mathematician who l- later ended up becoming famous, he arrives to a lecture late, sees... Do you, do you wanna tell the story?

Grant Sanderson 55:05 I don't remember it, uh, 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-

Dwarkesh Patel 55:14 Yeah.

Grant Sanderson 55:14 ... that the professor had written. He assumed that those two problems were homework-

Dwarkesh Patel 55:18 Yeah.

Grant Sanderson 55:18 ... 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 some of the others."

Dwarkesh Patel 55:29 (laughs)

Grant Sanderson 55:29 And the professor's like, "Oh, okay. Yeah, yeah." And just like shuffles it away.... and then a couple of 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-

Dwarkesh Patel 55:43 (laughs)

Grant Sanderson 55:43 ... 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 th- that would be more fun color to add to the story. But then, at least as the anecdote told to me however many years ago goes, the prophet then like finds the students, uh, housing and like knocks on the door, he's like, "Do you realize-"

Dwarkesh Patel 56:04 (laughs)

Grant Sanderson 56:04 ... that like, th- these were actually unsolved problems?" And, um, then he gets to basically make those his thesis.

Dwarkesh Patel 56:10 Yeah.

Grant Sanderson 56:10 So that, yeah, that idea of just being given something for completely random reasons and it, and it shifts the course of what you do.

Dwarkesh Patel 56:19 Right, right, right. Or just, uh, b- you know, it's a thing where if you know a crossword is solvable-

Grant Sanderson 56:23 Mm-hmm.

Dwarkesh Patel 56:24 ... you just like, you just keep going at it until you solve it. Or as-

Grant Sanderson 56:26 Five min- or the four-minute mile, right?

Dwarkesh Patel 56:27 Right,

Dwarkesh Patel 56:28 exactly. Yeah, exac- that's a great example. I, another valuable experience, at least one I had, was, uh, taking Aaronson'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, um, not because it increased my confidence in, uh, well, I, I didn't, I didn't have a moment where (laughs) , uh, w- I was like, "Oh wow, I'm good at this," but it was useful to know, um, you know... But podcasting is a, a, a- 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 gonna be like way above you, having that sort of awareness.

Grant Sanderson 57:07 Do you think it's fair to have a mental model that has a static G factor type quality here such that you're two sig- center 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, you know, 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?

Dwarkesh Patel 57:39 Yeah, can I tell you a story from that class actually?

Grant Sanderson 57:40 Yeah, yeah, go for it.

Dwarkesh Patel 57:41 So he's giving some proof about like how one compl... Uh, uh, actually it 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, the bounds of, um, uh, the complexity of different problems. And he explains it and he says, "You know, in 1999, I o- I, I proved 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?

Grant Sanderson 58:12 Mm-hmm.

Dwarkesh Patel 58:13 "But, uh, b- when I was a kid I was like doing Euler, that's like 2,000 years-"

Grant Sanderson 58:16 (laughs)

Dwarkesh Patel 58:16 "... in regress, now I'm six months behind." And then so later on in the day I'm like, "Wait, 1999. How old was Scott Aaronson in 1999?"

Grant Sanderson 58:24 Mm.

Dwarkesh Patel 58:24 And I think he was 18 or 19, and he was (laughs) basically proving, uh, proving f- frontier results in complexity theory that were-

Grant Sanderson 58:32 Yeah.

Dwarkesh Patel 58:32 Uh... So at, at that point you're like, (laughs) "All right, Aaronson's a special animal here." And-

Grant Sanderson 58:36 You are right, he's probably a special animal.

Dwarkesh Patel 58:39 Yeah. Uh, but it ju- it just broadly good to know, uh, just like have that sort of upper constraint on your Dunning-Kruger that you can, th- th- this exists in the world.

Grant Sanderson 58:48 I, maybe the thing that I would want to say is that whatever the scale is on which he's two standard deviations above you, le- that might not be the one scale that matters.

Dwarkesh Patel 58:57 Mm-hmm.

Grant Sanderson 58:58 And that like contributions to these fields don't always look like genius insights and that, uh, 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, the Scott Aaronsons of the world.

Dwarkesh Patel 59:16 Mm-hmm.

Grant Sanderson 59:17 Still, you are probably right about-

Dwarkesh Patel 59:19 (laughs)

Grant Sanderson 59:19 ... that truth, that there are... It's like von Neumann's another example of one of these, right?

Dwarkesh Patel 59:23 Yeah, yeah, yeah.

Dwarkesh Patel 59:25 Okay, how much does, uh, Godel's Incompleteness 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?

Grant Sanderson 59:37 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, the analogy to make, um, the paradoxical fact that it's conveying, the idea that you can't have an axiom system that is both, um, that will basically like prove all of the things that are true and which is also self-consistent, the, the contradiction that you construct out of that has the same feeling as the sentence, "This statement is a lie." Where, you know, you think about the statement, and 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 by computer?

Dwarkesh Patel 1:00:18 (laughs) Yeah, that's interesting.

Grant Sanderson 1:00:19 And so it's, 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 would be Paris Harrington theorems-

Dwarkesh Patel 1:00:35 Mm-hmm.

Grant Sanderson 1:00:35 ... where it's, it's a little pathological but it wasn't the very least a 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, it shows itself to be outside the bounds of whatever axiom system, uh, you are starting with. And it, uh, so it was, you know, shown to be unresolvable in a certain sense, but it, it was asking a... I want, 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 some times when the, um, paradoxes that are possible,

Narrator 1:01:11 (laughs)

Grant Sanderson 1:01:11 ... the Godel's im- um, possibility theorem shows, like do eke their ways in. The impression I get is, no, no mathematician is thinking about it. They're not actively worrying about it. It's not like, "Oh God, now like can I be sure that the stuff that I'm gonna show is true?"

Dwarkesh Patel 1:01:23 Yeah.

Grant Sanderson 1:01:23 For all the practical problems if it's like, you know, the Riemann hypothesis-

Dwarkesh Patel 1:01:26 Yeah.

Grant Sanderson 1:01:26 ... or twin primes, almost everyone's like, "No, there's gonna be an answer." (laughs)

Dwarkesh Patel 1:01:30 Right.

Grant Sanderson 1:01:30 Like it, may- maybe they're, they turn out to be unresolvable in, in one of these, uh, 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.

Dwarkesh Patel 1:01:45 That's, that's really interesting that something from the outside of, in, and in popularizations is, seems to be a very fundamental thing, where people, people have definitely heard about this, right?

Grant Sanderson 1:01:56 Yeah.

Dwarkesh Patel 1:01:56 It's not internally... A good a, a good analogy here is, in computer science, the halting problem.

Grant Sanderson 1:02:00 Mm-hmm.

Dwarkesh Patel 1:02:00 It's like, you take a computer science course, one of the first things you learn is a, you know, the proof of the halting problem. And, uh, i- i- it's another one of those things where y- y- you don't really need to be able to prove that every single, or y- maybe y- have that sort of program available for... Yeah. (laughs) No comment.

Grant Sanderson 1:02:19 No more comment. (laughs)

Dwarkesh Patel 1:02:21 Why are good explanations so hard to find, despite how useful they are? Um, o- obviously there's many other, other than you as well, there's many other cases of good explanations. But, uh, 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 e- explanations?

Grant Sanderson 1:02:40 (laughs)

Dwarkesh Patel 1:02:40 Is it just a really hard skill that isn't correlated with being able to come with the discovery itself? W- why are e- good explanations scarce?

Grant Sanderson 1:02:47 I think there's maybe two explanations. Um, the first, less important one, is gonna be that there's a difference between knowing something and then remembering what it's like not to know it.

Dwarkesh Patel 1:02:59 Mm-hmm.

Grant Sanderson 1:02:59 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? Um, 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, 'cause it, it shapes the way that you think so much, that remembering what it's like not to understand it. Uh, you're teaching some kid algebra, and the premise of like, a variable, they're like, "What is X?" And you're like, "Well, it's, it's, it's not necessarily anything, but it's what we're solving for." They're like, "Yeah, but what is it?" Like, trying to answer what is X is a weirdly hard thing, 'cause it is the premise that you're even starting from. Um, the more important one probably is that the best explanation depends heavily on the individual who's learning. And the perfect explanation for you, um, 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 golden nugget is like, how do you construct explanations which are as generally useful as possible, as generally appealing as possible? Um, 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.

Dwarkesh Patel 1:04:46 Yeah.

Grant Sanderson 1:04:46 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, um, motivating and approachable and clear to an extremely wide variety of different backgrounds.

Dwarkesh Patel 1:05:12 Is the process of making your videos, is that mostly you?

Grant Sanderson 1:05:15 Mm-hmm.

Dwarkesh Patel 1:05:16 Okay. Given the scale you're reaching, it seems that if it was possible, i- i- it would, 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, uh, setup are high enough that it's better to literally do y- the mundane details yourself.

Grant Sanderson 1:05:39 Yeah, I mean, uh, this could honestly just be a personal flaw. Like, I'm not good at, um, pulling people in, and I've struggled to do this effectively in the past. But a part of it is that the mun- the seemingly mundane details are sometimes just how I even think about constructing it in the first place.

Dwarkesh Patel 1:05:57 Mm-hmm.

Grant Sanderson 1:05:58 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, the editing process for me, you know, I start by laying out all of the animations and stuff that I want in a timeline, and then once I record the voiceover, the actual editing is like a day.

Dwarkesh Patel 1:06:29 Mm-hmm.

Grant Sanderson 1:06:29 And it's not, like I guess I could hire someone and gain a day back of my life, but the, 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 s- it's similar for why I maybe find it quite hard to use, um, like Copilot and some of these like-

Dwarkesh Patel 1:06:50 Mm-hmm.

Grant Sanderson 1:06:50 ... um, LLM tools and coding, where for the animation code, i- 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 and out, if I'm just using it, it feels like, oh, th- this should be the most automatable thing ever, you know?

Dwarkesh Patel 1:07:05 Yeah.

Grant Sanderson 1:07:05 It, it's, uh, it's just text. Uh, like, I should be the first YouTuber who can actually do this better 'cause the, 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, um, maybe it's just because you need a multi-modal thing that actually understands the look of the output. Like, the output isn't something that is consumable in text.

Dwarkesh Patel 1:07:26 Mm-hmm.

Grant Sanderson 1:07:26 It, 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.

Dwarkesh Patel 1:07:36 Mm-hmm.

Grant Sanderson 1:07:37 And I, 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, in the code. Um, 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 th- that feels 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 the collaborator.

Dwarkesh Patel 1:08:12 Yeah. O- on the point of working with Copilot, uh, where we kind of visualize the changes you wanted to make, there we- the paper from Microsoft Research, the Sparks of AGI paper, had an actually really interesting example where it was generating LaTex and they generated some output and they say, "Change this so that the visual that comes up and the rendering is different in this way."

Grant Sanderson 1:08:35 Mm-hmm.

Dwarkesh Patel 1:08:35 And it was actually able to do that, which was th- their evidence that it can understand the higher level of visual abstraction.

Grant Sanderson 1:08:41 Yeah.

Dwarkesh Patel 1:08:41 Um, so but I guess it can't do that for Manim?

Grant Sanderson 1:08:44 There's a couple reasons I might, um... it might not be as fair a comparison. One would be the two versions of Manim. There's, uh, 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 slight 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...

Dwarkesh Patel 1:09:14 (laughs)

Grant Sanderson 1:09:14 Like, I don't really comment it that much because it's-

Dwarkesh Patel 1:09:16 Mm-hmm.

Grant Sanderson 1:09:16 ... it's like a one and done deal.

Dwarkesh Patel 1:09:17 Yeah.

Grant Sanderson 1:09:17 It's, if it's, you know, there's... Also if I made 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 just, it's just the execution of that. Um, and you know, it's- 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- is 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-

Dwarkesh Patel 1:10:07 Mm-hmm.

Grant Sanderson 1:10:07 ... in the first