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My guest today is Grant Sanderson, the man behind one of the world’s largest math-focused YouTube channels: 3blue1brown. He has more than 3 million subscribers and his videos have been watched more than 150 million times.
Before making videos he studied math and computer science at Stanford before working at Kahn academy. On YouTube, he brings a visuals-first approach to math. Every video starts with a narrative or storyline. Then it revolves around imagery that illuminates the beauty of mathematics. Topics for his videos include linear algebra, neural networks, calculus, the math of Bitcoin, and quantum mechanics.
This episode begins with a conversation about the culture of mathematics. We talk about ideas like prime numbers, the Twin Primes conjecture, and pop culture’s role in advancing mathematics. Later in the episode, we talk about mathematical constants and the rate of progress in mathematics. Then, we close by talking about Grant’s process for writing scripts, note-taking, and researching ideas for each episode.
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Show Notes
2:14 – Why everybody loves prime numbers so much and what makes them so special.
4:56 – What was initially so interesting about math for Grant and why he didn’t end up going into a more formal researching role.
8:23 – Why Grant is getting increasingly more fed up with math that doesn’t even try to be associated to reality.
11:36 – The usefulness of “useless” knowledge and why spending an afternoon solving a math puzzle is so satisfying.
18:42 – What is driving the accelerating progress of the entire field of math.
22:19 – How Gödel’s famous theorem attacked the fundamental structure of math and changed the way mathematicians think about it.
27:31 – The unappreciated universality of math and why knowledge and interest in math by the public is higher than ever before.
31:49 – Why Grant believes that attention spans aren’t getting shorter and why the evidence is so strong.
35:43 – The importance of the principles of symmetry and creating meaningful names in math.
40:58 – Why Grant believes that distraction is key to creative work.
44:33 – Brand-building and why Grant believes it is important for anybody looking to build trust in their products.
47:40 – What videos are the hardest for Grant to produce and why.
49:31 – Building the intuition of teaching through a non-interactive medium.
54:42 – What was most unexpected to Grant about working in the field of mathematics.
1:00:19 – Where Grant gets his video ideas and how his script-writing differs from video to video.
1:05:31 – How an idea evolves from sketches and drawings into a logical coherent video.
1:07:35 – How college education in math can be improved and why it can be unnecessarily hard for students in that program.
1:11:42 – The possible implications of the collision of mathematics and computing in pure math research.
1:14:32 – The story behind some of David’s favorite quotes in Grant’s videos.
Transcript
David: I’m going to get right into it, Grant. Why do people love prime numbers so much? I talk to people who are interested in math, and you start talking to them about prime numbers, and it’s like a kid watching an airplane take off for the first time, over and over and over again. What’s going on there?
Grant: Well, they’re deeply fundamental, so in the same way that atoms build up all of the molecules in the world, you have the sense that when you’re studying numbers, just counting numbers multiplication, these are the fundamental building blocks. Often when you’re solving a problem, breaking it down in terms of primes, if you’re doing something multiplicative that’s the first step.
And yet, despite them being so fundamental and one of the first things you’d ask about, there’s just this pile of unanswered questions about them. It’s one of the few places where it doesn’t take too much digging before you find yourself at the forefront of what people even know. Just to list a handful of famous conjectures in this direction, you’ve got the twin prime conjecture. Are there infinitely many that are just two apart? The Goldbach conjecture, can you always express an even number as the sum of two prime numbers? Are there infinitely many, what are called Mersenne primes, that look like two to some power minus one?
Just a lot of things in this vein, and more often than not what you’ll find is these unsolved questions come when you ask additive questions about primes, so like twin primes. Are they two apart? You’re asking an additive question about a multiplicative structure. That mixture seems to be this perfect recipe for just injecting math right into that realm that we have no idea how to answer it, even though a lot of good work has been done on these sorts of questions. Even just within the last decade or two, yeah. Not to rant too much, but I was definitely one of those people that as a kid, first heard twin prime conjecture, was immediately captivated, right? You have that little light bulb that goes off that is born from a mixture of curiosity and naivete, which is like, ooh, what if I could be one to make progress on such a thing?
I think a lot of mathematicians in their youth have that kind of relationship with, if not prime numbers, other kinds of questions like the hailstone conjecture or such, that are easy to phrase but turn out to be very hard to solve.
David: I want to go down that rabbit hole of as a kid. Because you said something interesting there, what were the moments that got you inspired to be into math? What you said there was, it’s sort of where I want to point at this question. You said, if I could make progress. But I see you much more as a Carl Sagan, of somebody who is translating the world of math to millions of people on YouTube. You have probably done more in furthering my granted, limited understanding of math than anyone else that I follow. How much do you see yourself as sort of what was implied by that answer, of expanding the frontier, versus making the frontier that already exists legible to a certain set of people in the world?
Yeah, to be clear, I’m not a researcher, and at some point kind of, if I look back at my path, it sort of switched gears in the direction of exactly what you phrased pretty well there, which is trying to find different perspectives on things that are known, rather than trying to forge into things that are unknown. I don’t know, we can drill into why I didn’t necessarily go into research.
David: Yeah, let’s do that.
Grant: Well, I don’t know if I have a great answer. I can speculate around it, but maybe there was another part of your question there, which is just why I fell in love with the subject to begin with.
David: Yeah.
Grant: Those might actually end up being kind of related, because the pure reasons to give would be something like, oh, I just saw the beauty of numbers and these patterns and I fell in love. Really, and I think this is probably true of a lot of people, there’s probably a lot more ego than that. Because especially when you’re a young male, there’s a little bit more focus on yourself and whether you’re at the top of things than at other points in life, or other demographics out there.
My dad played a lot of math games with me when I was very little, and he clearly wanted both his sons to be curious people, as a lot of fathers do. I think that just sort of got into this positive feedback loop, where at some point, pretty young, I self-identified as being good at math. Then when you self identify that way, you like it, and then when you like it you spend more ambient time just in the back of your mind staring into space thinking about it, and that really gets this flywheel rolling.
It probably wasn’t until high school that I had a love of math that was pure, in a sense of just loving the beauty of it and letting myself get engaged with it that way, in all of the ways that nowadays I try to promote and get other people into. But I think if I look at my undergrad time, I’ve said this in other outlets before, but probably my biggest regret is not having engaged more heavily with the researchers and professors and mathematicians that I had access to, for no good reason other than shyness, I guess, mixed with maybe feeling a little bit more warm towards the computer science department than the math department, despite the fact that I was studying math.
I think honestly if I had been a little bit more engaged there, it would have been more likely that I got pulled into a traditional research route. But at the same time, I don’t know. You clearly do things on the internet. You’ve got this podcast, and you’ve got this kind of personal life investment in the idea of stuff broadcast over the internet. There’s a part of me that just had this feeling that academia, over the next couple decades, felt like a less stable life bet than trying to establish some kind of footprint on the internet.
Also just the things that bring me the most joy usually take the form of new perspective on known things, rather than that sense of being in this totally black room where you’re not entirely sure what’s going on, and it’s not even clear if things ever will become clear. That might just be a personality quirk.
David: Yeah, there was a quote that you give in your TED talk, and it is, “The mathematician does not study pure mathematics because it is useful. He studies it because he delights in it, and he delights in it because it is beautiful.” I think that speaks to a lot of what you’ve been saying here, and the question that I have for you is, how much do you think of math as a closed system in itself, that has this internal beauty, versus something that is beautiful because of its explanatory power, and because it has its boots on the ground with reality?
Grant: Personally, I like it much more when it is clear where those boots are. Over time, I think I’ve been getting increasingly fed up with math that doesn’t even try to draw that connection to reality. Mainly because if you have a desired application, or at least the threshold of, we need to be related to another field that’s not math in order to take it seriously, then you can’t move the goalposts and you can’t just have a paper that says, “Here’s something that we’re pursuing simply because a Fields medalist once asked the question.” When you can’t move the goalposts, it forces you to innovate.
Whereas if you’re just kind of leaning back on the idea of, oh, well, it’s intrinsically interesting, and something something beauty something something knot theory, then there’s no clarity on whether what you did required pushing yourself to innovate, rather than just taking what was readily findable based on intuitions that you already had and results that were already primed to be built upon.
Honestly, I do think there’s different categories of mathematicians here, where you have some that honestly do not care about applications. Either because they just love the idea of pure abstraction as a powerful thought mechanism, like people who get really into category theory without any care for application, though a lot of category theorists do care, to be clear. Then you have some that just love puzzles, like combinatorists I think probably disproportionately fall into this category, of those who just love puzzles and the kind of puzzles that math brings about, and it doesn’t really care if it’s touching the world.
But there’s just a wide swath, especially those who came at it through physics, who think of math as a tool, and the reason that it’s beautiful is because of the Eugene Wigner unreasonable effectiveness that it provides. I think the more that time goes on, the more I fall into that latter category, of really needing to see, maybe not a direct connection, but some semblance of a path towards the real world before I’m comfortable describing something that’s worthy as beautiful, as good, high quality math.
David: One of the things that I’ve been grappling with, and actually struggling with, is what is the role of studying things for the sake of studying them in my life? One of my biggest concerns, now that I’m living with a mathematician, which is awesome, because I’m realizing… it’s almost as if I didn’t see the color blue until 15 days ago when we moved in together. It’s like looking at the world, and you see something new, and it’s like, “Wait, I don’t actually have that language, have those models in my head to be able to try to understand reality.”
I’ve had a lot of these same conversations around philosophy, of trying to get into the lineage of Western though, but there’s something about it that doesn’t have that practicality to it. As you study math, how do you think about the trade off between studying math for the sake of just studying it because it’s beautiful, because you want to learn it, and then how much of your study of it is in direct pursuit of a video or a problem that you’re actively grappling with?
Grant: I think you hit the nail on the head with the very last part there. These days, almost entirely I’m viewing things through the lens of, can I use this either in a video or in something expository that is hopefully just enjoyable to consume for someone else? There’s a couple of reasons something might be enjoyable to consume. A lot of times, it’s because of the application, it’s physically relevant. But there are times when just the pure story in and of itself, like prime number puzzles are a great example of this, that you opened with. There’s something intrinsically fascinating, does not need any more direct motivation beyond that.
If anything, it’s always kind of awkward whenever people are talking about prime numbers, and almost as an obligatory thing in the footnote of their article they’re like, “By the way, prime numbers are used for cryptography,” without much more of an explanation than that. It’s this desperate grappling hook that number theorists have on the utility of their subject. But that doesn’t at all capture why they’re interested in it. Almost no one is interested in primes because of their application to cryptography. The thing that hooks them is this bizarre thing that’s going on with, why is something that’s so simple to describe so hard to draw conclusions about?
I just think of these as two different categories of videos, of expository possibilities. But everything that I think about these days ends up going through that lens. It’s not an either/or. One thing that I’ll often lean on if I’m trying to put some sort of conclusion at the end of a video that is a little bit more inside the internal bubble of math without application, is to emphasize that the kind of problem solving abilities and patterns that you’re coming to recognize, and all Of these intuitions being build through the pure puzzles, very often become relevant just as a set of problem solving muscles.
Even if you’re sitting back and you’re like, “When am I ever going to use this?” There’s not a direct answer to the this of this topic in video, solving some IMO problem or something like that. You have to acknowledge that the people who do solve these things, those who did math Olympiads when they were in high school, or did the Putnam type stuff when they were in college. There’s just this high correlation between them and people who do useful technical things for society later on. That’s obviously not a coincidence, and it’s not because they’re applying the weird esoteric problems of the IMO in their real life. Usually, there’s no direct application.
But there’s some kind of deep seated set of abstractions for what patterns we recognize, what problem solving tactics we’ve been exposed to and built the muscles for, that is hard to articulate, but nevertheless very worth thinking about and valuing, and using as a justification to spend that afternoon thinking about a puzzle that has no earthly bearing on physical reality.
I want to sort of follow this little theme, and I’m going to call it the usefulness of useless knowledge. I see the same thing with philosophy and physics. It’s as if they touch some substrate of reality that isn’t directly applicable immediately. I’ve seen programs, especially in philosophy, that are like practical philosophy programs. What’s really interesting is the second that you add that word, practical, you kind of destroy the thing that made philosophy great. It’s almost like there is a usefulness over certain time horizons. That maybe learning a certain idea that most people would consider useful is thinking about useful in terms of, it’ll be useful in the next weeks or the next month.
But I think that what you’re trying to get at here is a usefulness of a lifetime, and a usefulness of finding either meaning in your life, or being able to have a lens on the world that most people don’t have, because they didn’t have the time horizon to study some of these things.
One great example of that with philosophy, it seems like business leaders in Silicon Valley are heavily over-represented as philosophy grads. You just have to acknowledge that there’s something going on there. In much the same way that pure math enthusiasts end up heavily overrepresented in computer science, and stuff that’s heavily applied, even if that might not be the reasons they got into it.
I think one of the things that’s happening here… I used the phrase before, which is pattern recognition. Straight up pattern recognition, where there’s phenomena that happen in the world that don’t necessarily have words to them, but if you have a really good sense for convexity, let’s say. You thought about the difference of squares, and the strange idea that when you multiply two numbers that are equally distant around some midpoint, the product of them is a little bit less than the square of that midpoint. That as a loose, hard to articulate pattern, comes up in so many different ways, and maybe in a computer science landscape it looks like convex optimization, or maybe it looks like understanding geometry of complex numbers in a certain way that then gets applied to electrical engineering.
All sorts of just nuggets that happen when you really let yourself meditate on a raw pattern in and of itself, that’s one thing. Then the other is, I think there’s a social component here, where the kinds of people who are comfortable nerding out about particular kinds of math and puzzles then get drawn to each other. Then when people are together, then they exchange thoughts on all sorts of other things, and that level of collaborative socializing might not just be amplifying what pure puzzles they’re talking about, but also, oh, I’ve been dealing with this new programming problem, do you want to help me think about it? Or in the context of people who are into philosophy, they get together and have this long dinner chat over the Hegelian dialectic.
But then, as a side product, having brought together all these like minds, they find themselves chatting about something that’s more directly useful. Almost just as a common social thread to bring certain kinds of minds together, and then once they are together that’s a net productive thing. That seems to be a big part of what lies behind these correlations between the philosophers and the Silicon Valley CEOs, and the mathematicians and the programmers.
David: Yeah. I think that there’s a lot of truth to that, in terms of … Especially the Silicon Valley CEOs. That’s something that I’ve always been very interested in. I mean, one of my theories here is that a lot of these people just would have been academics, and then the actual money and the intellectual joy of solving a lot of these problems, and just the profitability of it, took a lot of those people, like Paul Graham comes to mind, outside of the world of academia and maybe brought them into the world of entrepreneurship.
But one of the questions that I wanted to ask you was, you mentioned the two primes conjecture, and we’ve made progress on that in the last 15 years. There was a major development with Fermat’s Last Theorem in the 1990s. In what sense do you think that there’s stagnation in math, or do you see an exponential improvement? How do you try to deal with that question?
Grant: it would be pretty hard to take the stance that there’s not an accelerating rate of progress. I’ve heard some cynics in that direction, and … You almost want to take it, because it’s this delightfully contrarian thing. Oh, we’ve invested 10 times as much, but we’re not making any more progress. But it just doesn’t bear out, certainly if you’re measuring it in terms of papers published. Then yeah, you’ve got this exponential increase. But a lot of papers might be frivolous, and it might not actually be making progress in the right ways.
But I think even in the right ways, one of the most important questions you can ask in math, that maybe not enough people ask, is, “What should the axioms be?” It’s not just a pursuit of what theorems are true, giving some set of axioms that we start with, but stepping back and saying, what is the right framework to even be viewing all of this? I mean, one of the biggest things that characterizes math in the last century as differentiated from those in prior centuries is viewing it through a lens of categories, and very top down in abstraction, as opposed to through the lens of … It’s not directly opposite, you can have categories and sets.
But oftentimes, either you’re thinking of a certain construct in terms of a set theoretic way, like how do you build it bottom up, versus category theoretic, which is much more leaning on, an object is defined by its relationship with other things. Now, that’s not exactly this concrete, oh, you’re going to touch the world in a certain way result. It’s not exactly something that is this named theorem, per se. It’s more just like this general perspective that’s moved its way in, and there seems to be a lot of thoughts like that, where people are asking deeper questions rather than just, how do I get to some new theorem so that I can publish as many papers as possible? Though to be clear, that is happening.
Also just in terms of looking at a scoreboard on how many unsolved problems are getting solved, there was this famous address that Hilbert gave in 1900, sort of at the turn of the century, laying out a bunch of problems that he saw as particularly important. A lot of them were maybe ill-posed and not exact, but if you just look through how many of those were solved in the last century, I think it’s fair to say we’re doing pretty well. There remain ones that aren’t answered, things like the Riemann Hypothesis is maybe the big one. But an echoing of Hilbert’s problems came in 2000 with the Clay Math Institute, putting out million dollar prizes for seven specific unsolved problems.
On the one hand, only one out of the seven has been solved so far. But on the other hand, hey, one out of seven, that ain’t bad, given that we’re just at the very beginning of this century, and they were listed as the millennium problems. They’re just very famously hard, and a lot of the partial progress made towards them, that’s what actually matters. No one cares if there’s infinitely many primes that are two apart. That’s never going to make a cryptographic algorithm more secure or anything like that. But because it necessitates creating new ideas and new math in order to even attack it, it’s I think highly likely that whatever did solve that would find itself application elsewhere. All of the partial progress being made towards these kinds of unsolved problems, that’s sort of where the meat is.
Yeah. There is that part of me that wishes I could stand and give a good justified contrarian view that oh, math is slowing down, we need to accelerate the progress engine again. But it just doesn’t bear out.
David: I’m happy to hear that. You mentioned the axioms earlier on. Is there anything in math that’s sort of like physics, where you have Newtonian physics, and you have Einsteinian physics, and they’re both axiomatic, but they don’t quite square up? It’s like there’s something in the foundation of mathematics that is nagging the field?
Grant: Oh, are you familiar with Gödel’s Theorem?
David: No, bring it on.
Grant: One of the kind of ill posed things that Hilbert set out in this 1900 speech was, giving a rigorous foundation to math. Finding the right axioms, where … Let’s just say we’re talking about facts about numbers. You want some axiom system that will describe numbers such that anything that’s true about numbers is provable. You could use those axioms, you could use first order logic, you could just go and prove something that’s true about the numbers, so that every truth is accessible in this way, and that it’s internally consistent. That you can’t prove false things.
Seems reasonable, that feels like what math should do. It’s like, yes, anything that’s true, we should be able to prove it. We shouldn’t have inconsistencies. People worked on this. One such young lad who worked on this particularly hard was Kurt Gödel, and … I’m going to get the timeline wrong. But a decade or two after this is kind of posed, Gödel comes out and is like, “Oh, turns out, not possible.” I’m sorry, what?
He’s like, “Any axiom system that you have, there’s going to be a true fact about numbers that it cannot prove, or it’s inconsistent.” Well that’s unsettling, that whatever foundation we have, there exists a truth, something that we know will be true about numbers, but it can’t be proven. It’s just this Red Queen game of, if you shift your axiom systems in another way then there’s going to be another fact that’s true but you can’t prove it.
For a while it seemed like, okay, yeah, but the way that Gödel discovered this stuff, it was this very self-referential thing. Where you start with … You know how when you say a statement like, “This statement is a lie,” and then you think about it and you’re like, is that statement true? Well, if it’s true, it’s not. If it’s not, then it is. But that’s language, that’s not rigorous math. Essentially what Gödel did was he took that self referential idea, of saying “This statement is a lie,” but you say it using numbers, in this particularly clever way. It ends up being something where you say like, “This theorem can be proven with these axioms,” but the way that you say it is with the number.
It’s very clever, but it doesn’t feel like the sort of thing you would naturally be asking about. It’s not a twin prime conjecture type thing, it’s a statement that was designed to be improvable. But then a little bit later, there was this theorem called the Paris-Harrington Conjecture, Theorem, something like that. It basically was a question that, I don’t know if it’s the most natural thing in the world, but it came out not because we were trying to come up with a self referential thing that had this paradoxical nature, just a question about discrete math. It turned out to have the same property, that it could not be proven within the scope of certain axioms, and it was a naturally arising example of the sort of things that Gödel’s Theorem said must exist among any.
That just shook the foundations of math for people even thinking that it was possible to have the correct axiom system. From my view, it kind of doesn’t matter. I think one of the biggest misconceptions about math is that it’s about absolute truths, and it’s not. What it actually is, is a means of connecting assumptions with results, and certainly any time you are trying to apply it, you feel this very viscerally, because you’re like, “Ah, well we’re doing these statistics, and according to a normal distribution, such and such should be very unlikely.” You say, “Well, a normal distribution, central limit theorem assumes that your things were independent. Were they independent?” Oh, we’ve got to question the assumptions.
There’s always this question of, were the assumptions correct in the way that we applied it? The idea that that ends up cutting in a parallel way down to the core of pure math itself, where the results are dependent on the axioms that you chose, and you never have something that is a pure capital-T Truth, it’s always a path. That feels more honest to me, and that feels good, that it mirrors the way that math ends up getting applied, rather than being some completely separate type of reasoning that happens in the pure land versus in the applied land.
David: How about progress in terms of our ability to teach math? This is of course something that you’ve been very much involved with for years, even before you really starting the YouTube channel, working at Khan Academy and looking at, how do we train people to understand math? I think that the number of people such as myself who don’t understand math as well as they should is way too high. I worry that with math in particular, that if we make fundamental new discoveries, say in rockets, we can all sit back whether it’s on Twitter or whether it’s on TV, and we can watch a rocket soar off planet Earth, and then land right where it took off, and it’s amazing. We can see the fire, we can see the fuel, we can see the smoke and the flames, and it’s grand and it’s magnificent.
My concern with math is, there’s such a small percentage of people on planet Earth who can even appreciate what’s going on at the frontier of math, and that in some way that’s hurting our ability to be inspired by the beauty of numbers.
Grant: A couple of things there. If it’s bad, let’s just agree that it’s better than it ever has been before. The ability for people to get back into math or the resources that students have available to them, or our knowledge about what it takes for people to learn, it’s definitely way better than it has been before. I think this is reflected in the fact that you have many more math literates these days than even 50 years ago, even if it’s not universal, that’s a fact.
One thing that I find quite interesting, sitting here in the land of YouTube and popularizing various different scientific subjects on YouTube, is to look at Brady Haran. Are you familiar with him?
David: I have come across him, but I haven’t watched any of his videos.
Grant: Great. He runs a number of channels. One of them is called Numberphile. It’s about math.
David: Australian guy?
Grant: Australian guy, yeah. He has a background in journalism at the BBC. But he’s just got a bunch of channels on various different subjects. Periodic videos talking about chemistry, and some that are talking about physics, some that are talking about computer science. Just really running the gamut on different topics in the same style, which is to interview experts and then to provide a good production quality surrounding that interview. It’s good stuff.
Now, if you had asked me 10 years ago, let’s say Brady’s starting these projects and you say, “Place your bets, which of these channels will be the most popular? That most people want to watch?” I’d think, well, definitely the physics one, people seem into that. That’s more tangible, same things with things like chemistry or computer science. The one about numbers and math, I don’t know, maybe that’ll be one of those backwater ones that is just for a specific esoteric audience. In reality, Numberphile is the most popular by far, and maybe that’s just because the particular people that he chose to interview in the early days of Numberphile just by happenstance are much more engaging than the ones that he happened to interview elsewhere.
But I think that it actually reflects a kind of unappreciated universality to math appeal. Where you have people who are interested in it, and just in an entertainment kind of way, just wanting to ambiently learn about it by choice as they’re scrolling around YouTube. From a lot of different backgrounds, for a lot of different reasons. Maybe it’s students, maybe it’s engineering nerds, maybe it’s science nerds. Maybe it’s people who, they’re not even into either of those but they just love patterns and Sudoku type puzzles and whatnot. You have all these different avenues that make people interested in the subject, in a way that I just really like the Brady Haran example because it’s so viscerally visible how much more popular the math version of this style ends up being than another one. That gives me a lot of hope for, I don’t know, just what the general public interest in this circle of things is.
Granted, I think it could be a lot more, and I think we could hope to see 50 years from now just an exponential increase in the number of people who are inspired in progress in math the same way that you just described rockets. But I think there’s every reason to believe we’re on that trajectory, and that it’ll continue to grow. Because through things like Khan Academy or Brilliant or all sorts of organizations out there that are aiming at having scalable quality learning accessible to whoever wants it to be there, it’s just better than it ever has been before.
David: I’m surprised you didn’t mention pop culture, because I think that that’s one of the elements here. I think that this is something that you very intimately understand, of how a story and conflict and this need to know, how you sort of create a question in your viewer’s mind, and then you get to a place where the viewer is like, “I need to know this!” Because I’m sure that you think about, oh my goodness, somebody could go watch another YouTube channel at any point. What I love about your channel is your … I guess this is going to sound crass, but I mean this in the best way. You’re kind of just a middle finger to all the people who are like, you know what? People have shorter and shorter attention spans, they’re not willing to deal with high level content.
You are proving every single day in your life that that’s just not true. I think of something like The Three Body Problem, a Chinese science fiction novel that is based off of a super simple but super confusing and absolutely wild mathematical idea.
Grant: That’s a great example. I’m right there with you on this narrative that people have shorter attention spans being kind of BS. I do think that in some avenues this plays out. This is probably why short form media like TikTok can be very addictive in a way that other things aren’t. But Netflix is typically quite long form, especially if you consider that series are a longer form than movies themselves, even if they come in more bit sized chunks. If you look at YouTube 10 years ago versus YouTube today, the average video length is much longer. There’s much more incentive around long videos, and I think that’s because it’s recognized that that’s typically what a better user experience is.
More often than not, when a video creator makes the claim that people have short attention spans and you’ve got to keep it shorter, it’s because they don’t want to make a long video. Videos take a lot of time. You’re sitting there on minute five, you’re like, “Boy, I wish I could wrap this thing up. If I have to go for another 25 minutes with this level of production quality, that’s just going to eat up my month.” But the ones who I think succeed the most are the ones where there’s real depth to the explanation. If we’re just going to point to successes of math on YouTube here, I think another nice one is Vsauce, who has videos on a number of things, but he’s got a ton of math, and they’re extremely popular. I think maybe …
I was looking at it the other day, it’s like the third most popular video on his whole channel, and this is hands down one of the biggest education channel in existence. The third most popular is about the Banach-Tarski Paradox, which is a deep and not easy to explain concept in very pure math, which is an idea of how if you take a sphere, like a solid ball, you can subdivide it into five different pieces such that when you rearrange those pieces, you get two perfect copies of the same ball. Which seems to violate very basic things like the conservation of mass, for example, but you quickly realize this is because we’re talking about a very pure mathematical sphere that’s not made of atoms, but it’s made of this uncountable infinity of points of just triplets of numbers in space. How this discrepancy between modeling something as a continuum versus modeling it as atomic allows for this very counterintuitive thing, that you can subdivide it into five pieces, rearrange those five pieces, and get two identical copies of the original thing.
Actually explaining that is quite hard. If you did it the naïve way, you’d have to give people this background in these things called freely generated groups and whatnot. Actually explaining it in a way that’s substantive but doesn’t require all of that is even harder. He does a great job, and it’s like 30 million people out there are the wiser for it, because it’s just this high quality explanation of a very real topic, and I think it satisfied a hunger that people have to actually get the real explanation. Don’t just give me this high level, divide things, get two copies of the original. I’m here, I’m waiting. I’ve got 30 minutes to spare, explain exactly what’s going on there. People are hungry for that, and I think the video creators are rewarded when they put in the effort to satiate that hunger.
David: You’re talking here about a kind of problem solving that you do, where you have a starting line of, this is an interesting idea that people need to know about and that I want to spend say, the next two weeks to two months of my life studying and explaining to people.
Then you have an end goal of, I need to figure out how to create a story that’s both engaging, compelling, and that’s going to hook a reader or a viewer in, and that’s going to basically create an image in their brain after they watch the video that they will remember what they’ve seen. That’s the challenge here. A couple of months ago, you gave some problem solving tricks. You shared nine of them, and two of them stuck out to me.
The first was leveraging symmetry, and the second one was give meaningful names. Why are those important?
Grant: I guess for very different reasons. Starting with the second one, giving meaningful names, I cannot remember whose quote this is, “Perhaps the organization of one’s thoughts is not distinct from the content of those thoughts.” That really strikes me as something that’s kind of deep, where we think of those as separate things, but the mere act of organizing might be the substance of our thoughts. There’s nothing more than that, it’s just good organization. So naming is the first step to that.
It really highlights what parts of a problem or diagram or a set of algebraic quantities are worth highlighting, and when you’re asked to name them, it’s kind of this forcing function to say, “What is meaningful about it?” What is this part of the geometry problem that I’m dealing with, or what is this right hand side of the equation representing? All of that. That, it just clarifies thoughts. The same way that writing does, when you’re forced to write an essay or write a journal entry as opposed to letting it stew around in your own head. Something different happens through the articulation.
On the symmetry front, that feels like a very deep question, where I honestly don’t know how to answer that. It’s almost bizarre how when you look for something about a problem that doesn’t change when you shift it in some way … So symmetry is most visible in geometry, but we’re not just talking about something geometric. But if you’re looking at a geometric diagram and you say, “Wow, it looks the same or it has similar properties if I look at it from two different perspectives.” That lets you create things like equalities, because maybe an aspect of it viewed through one lens is non-trivially the same as that same aspect viewed through another lens.
That, as soon as you’re starting to connect things, that gets you the progress towards whatever you’re trying to solve. I mean, like the Pythagorean theorem or something like that, where you want to say that A squared plus B squared is the same as C squared. You need to draw on equality in order to do that, finding two perspectives of the same thing is kind of how that happens.
But I mean, there’s all sorts of things that are even deeper than that. In differential equations, often you have these results around how if you have a symmetry to your setup, then you can reduce the degree of the equation for each one of those symmetries, and such. Or the way that groups are connected to physics, you had what’s known as Noether’s Theorem, that says that every symmetry to the laws of physics corresponds to a conservation law. If the laws of physics should be the same whether you’re here or two inches to the right or five light years to the right, wherever you are spatially translated, the laws of physics should be the same.
That ends up implying that there should be a kind of conservation of momentum, and rotational symmetry implies that there should be a conservation of angular momentum. In fact, every kind of symmetry corresponds to some kind of conservation. Anyone who’s done any level of physics knows that conservation laws are how you solve problems, so that’s a particular deep connection there. I don’t know if there’s a good answer for why that principle works as well as it does, but it’s worth underlining.
David: I don’t know if this is relevant or not, but I’m going to give a very Biblical answer to the give meaningful names. It’s probably not a coincidence that in the book of Genesis, God tasks with naming things. That to name something is to give people power. I mean, can you imagine … And it is to create the nature of that thing, often. I don’t know if you’ve ever suffered from naming something and having other people adopt it, and then actually not liking the name of what you named the idea. That’s happened to me a couple of times as I’ve taught my writing students, and you can’t change the name. The name itself is so deeply intertwined with the idea, and there’s something there that how you name something, just like what you said, how the organization and the content are so intimately linked.
I think that once you name a thing, you’re giving structure to that, and there’s this whole set of assumptions that are given when you take that action.
Grant: Yeah. It’s almost like if someone was dumb enough to name their YouTube channel after a piece of their physical anatomy.
David: Eye color.
Grant: You can’t change that, no matter how weirdly self centered it seems in hindsight.
David: That’s very funny. What do you think of the trade off between productivity and distraction? We usually see distraction as something that is bad, but has actually been our friend Michael Nielsen, who has changed my mind on this. He quotes Freeman Dyson talking about Robert Oppenheimer, who says, “We can see the nature of the flaw which made his life ultimately tragic. His flaw was restlessness, an inborn inability to be idle. Intervals of idleness are probably essential to creative work on the highest level. Shakespeare, we are told, was habitually idle between plays. Oppenheimer was hardly ever idle.”
What do you see as the role of idleness in your work?
Grant: Oh, it seems very important, but that sounds like a rationalization. I think it’s worth being very conscious of the role that each play, because what you just said really resonates with me, that you actually need distraction to stumble upon new insights. Have that comfort in just reading a book during the work day that you otherwise might not be reading, or letting yourself just noodle on a piece of paper or something.
But it can’t just be any distraction, right? If you’re off playing Candy Crush, I don’t think there’s any way to justify that that’s promoting creativity. But the right kinds of idleness and distraction that are not purposeful towards a given plan, I think that’s very important. But at the same time, at some point you’ve got to sit down and just get the thing done.
David: Got to get it done.
Grant: Usually the hardest part of that process is the part when you’re going to be most inclined to find yourself distracted. You’re trying to solve little micro-puzzle, you’re dealing with just a really annoying sound editing task, or whatever it is, and you’re like, “Yeah, maybe I kind of want to go read.” Those are the moments that maybe distraction isn’t the highfalutin creative inducing thing that you want it to be, and actually what it is is purely slowing you down.
Another mutual friend of ours, Andy Matuschak, I think I remember him asking about this dichotomy between … Let’s say you’re trying to write something, whether you should be going for a long walk and just sort of letting things do, versus gluing your butt to a chair and typing and just not stopping. He was asking this on Twitter, and someone gave what I thought was just an absolutely great response, that was something to the effect of, “What you’re asking about is, should I be exhaling or should I be inhaling? The question is not any one of them, it’s finding the right balance between the two.” In that way, I think idleness is like inhaling, and then just gluing yourself to the computer and getting stuff done is the exhaling, and it’s important to recognize when each one is needed.
David: I don’t really know, but I really like your point there, about how new ideas do come in a state of distraction. I’m almost wondering if productivity is sort of optimizing the system that you already know exists, whereas distraction is almost like an algorithm for randomness, where it just on average, or your median insight or moment, will be basically inconsequential. But for the small percentage of things that are consequential, they’re the only things that can take you out of the matrix that you’re locked inside of.
Grant: Easily, yeah. The best projects I’ve ever had were born in distraction, not in purposeful planning. That’s absolutely true.
David: You have a template for your work. All your videos build upon some kind of template. You must have spent a long time investing in that template. How did you create that template, and what was the thought pattern behind doing so? Was it all a one big fell swoop, where you’re just going to create it and then you’re going to use it forever? Or what was behind that?
Grant: Do I have a template? I don’t know if I’ve ever thought about things … You mean in terms of narrative construction, or in terms of visuals?
David: Visuals, visuals. You know, like the little pi with the eyes, stuff like that.
Grant: I guess in that way, the way I do animations is a little weird. It’s all programmatic, and what that will sometimes mean is, it’s nice to leverage abstraction, where if you’ve just got a scene type that you just have to tweak some attributes for, that’s just way easier than doing it ground up. In that way, a default scene is the sort of teacher-student thing where you’ve got the three blue pie creatures and the one brown one. That’s just because it’s sitting right there as a scaffold, just waiting to be filled in with content.
That doesn’t have to come from weird programmatic animation. A lot of people have assets set up like that. But it’ll also mean that during a particular project, I might set the infrastructure for the kind of thing that I want to animate. Let’s say it’s the Fourier transform, and you’ve got a particular visual that the whole video is meant to center around. You’ve got an infrastructure for it, and then everything is a variant of that core visual type.
I think a nice side effect of that is that it gives a certain stylistic consistency, and then you can connect ideas through visual similarity. This thing that we’re looking at now visually looks like this thing we were looking at 10 minutes ago, and through that the viewer’s brain can hopefully see a connection more readily than if it was more stylistically inconsistent. But I don’t actually think I’ve been very deliberate about establishing a template, and thinking what the best template is. That’s more an emergent phenomenon than anything deliberate.
David: Do you think about building a brand at all?
Grant: Brand is very important for anyone, really. That’s one of those aspects of how you run … It feels weird to call it a business, but any kind of business or something with public presence. I think it’s important to think, “What is the emotion that this evokes, when people see a few key triggers?” That’s really what a brand is.
David: What do you want?
Grant: Well, I want people to trust that they will come out understanding something and having a good experience. The hope is that if someone sees a thumbnail for a video that I do, they’re willing to do a trust fall, and they fall back into it with the next 20 minutes of their life, and that I can hopefully deliver what was a meaningful experience there. That it won’t be shallow, I guess, that if there’s a claim that something will be explained, that … I don’t know if I always follow through on this, but the goal is that someone doesn’t think that it’ll just be a high level surface thing that they would have been able to find elsewhere.
All of that gets built in. Then in order to maintain that, stylistic consistency is the other component of branding. Where whether that means just having a logo that doesn’t change too much, or having visual schematics that identify this channel as something different from other channels. Everyone has their notion of branding, whether it was deliberate or not. I think it does matter for getting this immediate mindset from the viewer that you want them to be in as they’re starting their experience.
David: When it comes to videos that you create, what math do you find to be particularly hard? Is there a subsection that is easy to other people, but someone just agonizing to you?
Grant: I always have a hard time putting together a video about probability. I actually don’t know why. It’s not because the material is harder, per se. I think it’s just that there’s so many different ways that people seem to be coming at it from, and different priors that people have, or different constructs that would be more helpful to them. There’s other topics where I think, this way of viewing it, pretty universally I think people will like. I always have the hardest time if I’m putting together a script on something about probability, thinking, “Should I say it this way? Well, someone might prefer to see it that way. Well, it kind of depends on if they’re thinking of it as someone coming in from machine learning or if they’re coming in from physics or they’re just thinking of counting problems.”
It always has this uncomfortable multiplicity. Even if the subject matter isn’t deeply challenging, expositorially it’s actually for me very challenging. Which is why I make so many promises that I don’t follow through on with respects to certain probability videos.
David: You were talking about the experience that your viewer is going to have, and how with probability, it’s hard because people come at the video from such a medley of perspectives. That leads me to my question, do you consciously get feedback? Do you have a process of getting feedback from the YouTube comments? Do you have a process from before you publish a video, like you send videos to people? How do you actually develop that intuition that is so easy for a teacher who’s just speaking to an audience in a classroom? You can tell when Jane is the back is bored, or Billy in the front is enthralled with what you’re saying. How do you build that intuition if you’re creating digitally?
Grant: Even for the teacher I think it’s hard, because unless you’re doing something one on one, it’s hard to know. Let’s say that you can tell that a student’s bored. It’s hard to know why. It might have nothing to do with your lesson, it might be because they didn’t get enough sleep. Or if it has something to do with your lesson, you don’t know what. I think one on one, either conversations or sample lessons in the ideation phase are important. I always feel good when I do that. The less I do it … I just do it more. However much I’m doing sample lessons preceding a video, it’s not enough, I think.
David: What are those sample lessons like? Are they in person? Are they virtual? Talk about those.
Grant: In person, if possible. These days, that’s harder. But even still. At the loosest, it can be if you’re just chatting on the phone with someone, though it’s harder without visual cues. But yeah, I’ve had times when I’ll just have a very basic overhead camera thing to do it virtually, and scribble something. But what you want is to kind of explain something, and then get an instinct for what excites the person, what confuses them, what perspectives resonated most, and just really iterate based on that. Like I said, I don’t think I do it enough.
David: Wow. You know, this is really interesting. I think you’re surprised by how I’m surprised by this, but I expected the answer to be no. But it’s interesting, because when I teach my writing students, we have an acronym called CRIBS. In all moments of feedback, everyone looks for things that are confusion, repeated, insightful, boring, and surprising. You said two of those and it just makes me think that there’s something fundamental about the things that you’re looking for when you’re trying to teach.
What you’re doing, when you say overhead. What is the ideal scenario, that you’re doing a 20 minute explanation, and what do you mean by overhead?
Grant: I meant that much more menially than I think you thought, just overhead camera while I’m doing a video call, that kind of thing. So that you can write down. Merely because having things that are drawn and written is much better than waving your hands in the air and asking someone to imagine.
David: One of my dreams, and someone’s going to invent this. It will be invented before I die, is I would love somebody to invent an ability to map things out in 3D as you explain things. Because I’m very handsy when I talk, because I try to sort of visualize these sort of three dimensional construction. A lot of this idea is inspired by Bret Victor, who I’m sure you know, and trying to visualize these ideas. Leads me into the next question, what do you think that you’ve taken from Bret?
Grant: That’s a good question. I mean, there’s something to be said for unabashed optimism and engaging with what a future should look like, and feeling uncomfortable with the norm. Almost, even if that’s not something that is directly applied to the same fields that he’s talking about. If it’s interactions with computers and software. Just the idea that the way that things are currently done should not be taken as static, and it’s worth thinking about, what should this be? And also looking historically to say when things shifted the most, what were the personality characteristics behind the people who made it shift the way that it did?
I remember a particularly good talk that he gave where he was presenting it as if it was the 1960s, and he was looking at the future of programming and talking about all the exciting things happening those days, from Licklider and timesharing computing, and really reimagining what programming should be, and drawing the line out. What was most compelling about that is that he’s talking about true historic things, and how if you were to draw the line out, it gets you this really wonderfully inspiring landscape for what computation can be, and what programming ended up being was a lot more linear than that. It’s this reminder that the future is ours to invent, and it won’t invent itself.
I don’t know how directly I’ve ever taken a concrete thing that he’s worked on or talked about and tried to incorporate it, but maybe I should. Maybe I should dive into the Bret rabbit hole again, because that just always leaves you more inspired than when you started.
David: You mentioned personality characteristics. What do you think some of those are? You seem to have a real appreciation for the people who came before you to create these mathematical concepts, who extended the frontier of math. I think that as much as any field, it requires a dogged commitment to your work, and a faith that things are possible. A lot of the people who have made mathematical progress were considered lunatics or outcasts by society, and lunatics to themselves at times. I can only imagine the doubt and even the self loathing of what those people must have felt, because often these problems will take 5, 10, 15 years to solve.
Grant: One common misconception about mathematicians is that they’re these lone geniuses off writing sole-authored papers in their attic. Maybe this is exacerbated based on certain prominent historical figures who are not the most social characters in the world, like Newton. But the reality of modern math is that it’s heavily collaborative. I do remember, I was in high school and I was at this university math circles type thing, where they have a mathematician come and talk, and someone just asked, “What is it about your job that you didn’t expect, that kind of surprised you about being a mathematician?” His answer was, “I travel a lot more than I thought I would.” Because you’re in some specific discipline, and you’ve got to collaborate with other people, and so a lot of his life is spent flying off to the right people to collaborate with.
I think even now, that violates what people’s image of a mathematician is. That you’re seeking out the people to work with, or that … I don’t know. A lot of people will cite Terry Tao as being one of the best living mathematicians in the world. One thing that characterizes his work is just how much of a social butterfly he is. Swoops into one area, helps there. Swoops into another, helps there. By having this super broad understanding of all the different types of math, and being able to take the pieces from one and putting it into another, that seems to be what makes him such a valuable asset. They have all sorts of characters like that. Paul Erdős is another very classic example of someone who was this itinerant … If ideas were viruses, we were describing him as the super-spreader, whose R-value is through the roof, those sort of people.
David: That’s a great question, and I’m going to throw it right back at you. What is it about your job that you didn’t expect?
Grant: Oh. Well, the job, I guess. I don’t think I would have had any…
David: The actual thing that I do of making videos about math…
Grant: If you asked me 10 years ago, “What do you think you’re going to be doing?” It’s not this. That’s for sure. Yeah, the fact that anything that I do involves this kind of stuff, right here. I’m on a podcast. This sort of public presence, or that I’ll give public talks, or that some of the things, I’ll be on camera, and having to think about entertainment in that way. That’s something I just would not have expected. It’s not really in my personality as much. I think it took me a while to be comfortable with public speaking. It takes a lot of people a while, because it’s just not a natural thing.
That, it’s fun, and I like it. But that was not expected or in any of the plans I had for what career trajectories were likely.
David: Within mathematics, what’s your favorite constant? Pi, e, something like the golden ratio?
Grant: All of those are pretty terrible, to be honest. I’m a fan of 1. Phi, everyone knows that’s overrated, that it’s just the source of weird numerological obsession that is not born in any actual facts. People look at a conch shell, they’re like, “It’s the golden spiral?” You’re like, it’s actually not, if you do the measurements. It’s a solution to a particular quadratic, it’s not all that special. Pi, very well reported how it’s a little bit awkward that we use pi instead of tau. It doesn’t really matter, it’s just like, choose your favorite circle constant and move forward with it. Pi is fine, I guess.
E, this one is I think heavily overrated because what matters is the function, this natural exponential function, and e is just the value this function happens to have at 1. It’s interesting that the value it has at 1 can sort of carry the weight of the function, and thinking of repeated multiplication with that value gets you a long way for what the function is about. But it doesn’t get you anywhere near the full breadth of what this deep exponential function does and relates to. The first time people start to get a feeling of this is when you see like, e to an imaginary number. You’re like, that doesn’t make sense. But it goes beyond that. You see raising e to the power of a matrix, or an operator. You look through quantum physics and this comes up all the time.
It actually has nothing to do with the number e, is the problem. E was this numerical residue of this function back in the land of real numbers, but it really has nothing to do with what’s going on there. That one’s not just overrated, but I think leads to a certain category of misconceptions. So yeah, as far as constants go, 1, you really can’t beat 1. Multiplicative identity, you keep adding it to itself and you get all the natural numbers. It’s this nice reduction homeland point. Just because it’s popular doesn’t mean it doesn’t deserve it. It’s not even popular, just because it’s well known doesn’t mean it deserves all of our respect and love.
David: Well if it’s a stock, why is it an undervalued stock? What is it about the nature of 1 that makes it undervalued? I saw a book recently called Zero, that came out. What’s going on with 1?
Grant: If it’s undervalued, it’s because it’s one of those backbone, core of the economy, unsexy data aggregating kind of companies-
David: Plumbing company, Grant.
Grant: Yeah! It’s not the Snapchat with the meteoric rise, it’s not the social media companies of any kinds that’s on the top of your mind because you’re a consumer. It’s instead laying the fiber or something that you just don’t necessarily think about. It’s providing oil drilling equipment. It’s core, it’s incredibly valuable. It’s not that people don’t think it’s there, it’s just that it’s not attracting the same love that some of the other pop stars of the real number line around it are attracting.
David: You mentioned writing scripts a little while ago, and I didn’t want to let go of that thread. You have two kinds of videos, problem solving videos and expository videos. How is your script writing process different for both of those?
Grant: The level of pain, for sure. Yeah. Problem solving videos are the most fun to write, and they write themselves because you know the end point and you know the start point, so you just have to draw a verbal line between the two. Expository videos, it’s like, what is the end point here, and what is the start point? What do people know about this topic, and what are we trying to say, and what are the perspectives that I can try to provide around it that wouldn’t be found in other expository works on this? Yeah, so it’s almost always the expository ones where I, if I am on top of my game, will try to do sample lessons. Because that’s where it’s valuable.
Actually, yeah. I probably almost never actually do a sample lesson on problem solving ones. I just sort of … Maybe I should, that might actually make it better. But it sort of feels a lot easier to know where everyone is. Basically because you’re bringing in something without any priors. It’s like, if I ask this question that’s a probability question on a certain geometric setup, you didn’t come in with preexisting assumptions. You just came in because you’re at the same point that everyone else is.
The challenge that that brings is that I didn’t have to motivate why we should care about this random puzzle, so maybe that’s a little bit harder, whereas exposition ones, the motivation comes pre-baked. I don’t need to tell you to care about neural networks, we can just dive right in. Whereas if I ask you, “Hey, if I choose a random semicircle on a circle, and I keep doing that, and I count how many times does it take before the whole circle is covered, where I’m doing it with uniform probability. What’s the expected number of times?” That’s a fun puzzle. I think it’s fun. It’s going to be a fun process to explain why it works, because there’s a very clever insight that will be involved.
But there’s not a great way to hook you into that, because it’s not … It’s what we were talking about earlier. It doesn’t have boots on the ground for touching reality. I would happily make a video on that problem as soon as I have a good hook, but until then it just has to kind of sit there as the script that will be easy to write one day, as soon as it has a title.
David: Interesting. There’s something that you said there, that I think that you implied, that maybe you have ideas that you’re just waiting to talk about, until you have a hook? Do you ever have a hook without an idea?
Grant: Oh, that’s a good question. None that I feel proud of, I guess. There was some shamelessly clickbaity thing I was thinking of before that could make a nice … No. Yeah, none that I feel proud of. Because usually if you have a hook, it’s because you have some sense of what you’re hooking someone into. The only thing I can think that was truly without substance, like man, if you had a video that just said, “Do not click this video.” How many people would click it? What sort of sociological narrative could you put around that to fill it with content? But there’s nothing there. To make such a video would be nothing but clickbait at that point.
David: One of the things that I think you and I both agree on is, you don’t want to start a video right at the beginning of your research process. There’s an idea that’s kind of ingrained in school. It’s like, “Okay, welcome to US History class. At the end of the semester, you’re going to write a paper on the Civil War, so now is time to start researching.” Then you know exactly what you’re going to write. You research, research, research. But that’s actually tremendously costly. What’s better to do is to, the word that you used was ambient research. You exist in this state where you’re just interested in math, and through that process, you’re researching a lot of things without even knowing it. Then one day, well, you find your hook, and it all comes together, and now it’s time to make the video. What is that process like for you?
Grant: It’s exactly what we were talking about earlier, where that’s why idleness and distraction are valuable. It’s because those are in a sense the ambient research phase. It looks like reading books or being willing to get caught into a rabbit hole of an interesting question, and not feeling guilty about part of the day washing away to that effect, and then just keeping notes, and having a spot that I can turn to on those notes at some point. It’s kind of as simple as that. Very often if you were to actually track where an idea came from for a video, the seeds of it sit years before the production process started.
David: What do those notes look like?
Grant: Sometimes it’s literal pad of paper and pencil, just scraps at the side of the computer. Sometimes it’s just a text editor document that’s open while I’m reading something. It’s probably not as organized as it should be. I’ve got a long, it’s on a list of potential video topics, and the detail section of each one will have little scraps, like bullet point of important points. Yeah, it’s honestly not terribly organized, if I’m open with you.
David: What percentage of your notes are drawing and visual, versus text and more logical?
Grant: Honestly, actually a lot more logical and text. I think there’s isolated … I’m just pawing through my notebook now like this. I guess maybe on the pen and paper it’s 50% things that are sketched and drawings. But then that’s where you’ll maybe work something out. Let’s say you want to find an intuitive explanation of something in math. Often the first step is to actually go through the non-intuitive explanation. The most recent thing that’s sitting in front of me here is trying to understand why a binomial distribution tends to look like a Gaussian with large numbers, and just kind of working out how Sterling’s approximation on factorials makes that pop out, and if there’s an elegant way that it does.
In that case, I guess the central question in mind after doing that is to look at the algebra and say, “Are there parts of this that felt like they must be true, in a way that could be visual, or could be better motivated?” Or something like that. I don’t think this particular page would ever turn into a video. But that’s I think kind of a valuable thing to do, is to just get into the nitty-gritty of working it out in the boring algebraic way, so that you can assess, does it have to be as boring as it seems?
David: Say that you have a see-saw of universities beginning to struggle, and they lose funding for mathematics departments. On one end, what’s going down is the ability for professors to wander into what you could call mathurbation, and to spend time there, and to just sort of think outside of the blue and spend time at these highfalutin to abstract ideas. But what you have on the flip side is the ability for people like you to use YouTube, and use the internet to share ideas, and to have a business model that you’re actually more in control of. What do you think hurts mathematics with that outcome, and what helps?
Grant: I guess what you want is something that is stable and scalable for all of the good thinkers out there. The university system … It’s a little bit awkward, this notion that the value comes from the teaching and printing of degrees. Kind of from the research too, but it’s heavily leaning on pumping undergrads through it. There’s a part of me that kind of hopes that industrial research fills that void a little bit more. I think it’s healthy to have people who are otherwise very theoretical have a forcing function to make it applied. If the way that the pure mathematician spends his time that’s not just the ambient thinking about stuff isn’t teaching undergrads, but is working with professionals to try to apply this material, that actually seems pretty healthy for math as a whole.
Although of course, asking yourself to teach something is a great way to come to new ways of understanding it. But the problem with that is, it will let you get more and more distant from what matters. If you actually experience an undergrad education in math, it’s not at all clear how any of this stuff is used, or why the constructs are defined the way that they are. It’s not terribly motivated, in a way that it could be. I think that might be coming about because the people who are putting together the curricula and the books and teaching it, the rest of their time is also high up in the clouds. The act of teaching undergrads still remains in the clouds, and it’s not what you described as boots on the ground.
But at the same time, I just don’t know how scalable the idea of pure math research through industrial labs actually is, and how much value there is. Because it seemed like it took something very exceptional, like Bell Labs, with this government sanctioned monopoly, in order for anything like that to actually exist. You kind of see pockets of it these days, but usually it’s statistics and machine learning research in disguise, it’s not really pure theorizing.
David: It feels like statistics is really, to go back to the stocks, it feels like a momentum stock. It feels like it’s gaining a lot of value, and popularity very fast. As opposed to say, something like calculus. In my own life, I come across the cutting edge of statistics way more than calculus, and those were the two higher level maths that I took at the end of high school. Why do you think that is?
Grant: Calculus seems to have its role in the current education system based on what engineering used to be, which is much more heavily into electrical engineering, mechanical engineering, very physics based topics, where the more physics based it is, it makes sense to have a strong foundation in calculus. Whereas the more data driven it is, then by its very nature that’s more statistical. A lot of people will say that we should push down the emphasis on calculus so as to bring up the emphasis on statistics.
It’s not really an either/or. I mean, a lot of actual stats, you do need to have some good basic calculus in order to grasp it. Calculus is fundamental for anything as soon as you’re dealing with a continuum. But it seems to be a reflection of what engineering looks like today versus a century ago, and how slow moving changes in education are.
David: You mention data, and that sort of leads me into just cutting edge technologies. If you make the assertion, which I think is generally true, that one way to almost guarantee progress in a field is to create some kind of collision, where you have the intersection of two different fields. It seems like right now, one of the things that we’re going to end up with is the intersection of a lot of computer power with mathematics. What do you think the implications of that are?
Grant: Do you mean implications for pure math research?
David: Yeah.
Grant: How pure math gets applied to this data analytics?
David: The first one, for pure math research.
Grant: I actually don’t know a ton about this, but it is the case that the field of proof checking and writing basically programming languages to formalize math such that you could write your proof in this language and have it automatically checked, that is growing and seems to be thriving, and I think could have pretty big implications. Because certainly if you get it to the point where you can actually democratize the writing of a theorem in a way that’s checked automatically, you kind of solve the crank problem, which is that if you have someone from outside academia who writes in claiming to have a new math result, it’s almost always the case that they don’t and they just don’t appreciate the nuances of why a given problem is hard.
This is the sour side, actually, of what you opened this whole podcast with, questions about prime numbers and good unsolved problems. Is that as inspiring as they are, what they end up attracting is a ton of, the common word to use is crank. Crank papers, that are just attempts to prove it that are a lot less sophisticated than what would be needed to solve it. It’s kind of a sad situation, because at the same time, every now and then there is someone from outside academia who does have a meaningful result, and you don’t want to just cast all of these out, you don’t want to be uninspiring.
But in a world where everything is so rigorously programmable that you can just say, “Oh yeah, as soon as you submit the proof and lead, and the proof checker verifies it, then yeah, we’ll give it attention, and maybe invite you over.” Then you would just have this automated non-personal process by which all of the cranks can see that oh, this is where the fault of the proof lay. And where people who aren’t cranks and actually have something to offer actually have a chance.
I think the meaningful value that will come from proof checking extends a lot beyond just the crank problem. It’s kind of setting things on a more formal foundation. It’s bringing in artificial intelligence tactics to help find new proofs or new potential avenues towards something. People talk about, “Oh, but these things won’t be creative. It’s like a very rote, exploring the decision tree of ways to prove something.” But any chess enthusiasts who follow AlphaGo seem to agree that in very rigidly defined formal environments, you can have an element of creativity that seems to be coming from the computer, and math seems like really the lowest hanging fruit for that in a non-gamified circumstance.
That promises to be pretty interesting, in the next century.
David: I want to ask you just a couple, just quick questions we’ll bump around for a couple of minutes. The first quote, and then a couple questions just about you in particular. Here’s the quote, from Angus Rogers. “Mathematics requires a small dose, not of genius, but of an imaginative freedom, which in a larger dose would be insanity.”
Grant: I like that. It seems to be hitting on this idea that insights come from exploring something that no one else thought to explore, because it really seemed like it wouldn’t yield anything. Wow, you’d have to be kind of crazy to try working with this structure, or to tweak the equation in this way, or to bring in this new feature to your diagram, or something like that. It requires doing something that other people wouldn’t think to do, and that there’s kind of a reason they wouldn’t think to do it. Too much of that is just indistinguishable from insanity, because it means completely violating the norms and priors for what will work. Yeah, that’s a very beautiful quote.
David: That’s from one of your videos!
Grant: Wait, what?
David: It’s from the series on linear algebra. You’re saying it like, “I don’t know,” and I was like … Well, I’ve got another quote for you, from one of your videos.
Grant: That’s so funny. It was four years ago, I can’t remember what I put in a video four years ago.
David: Yeah. I know. Well, it was fun. And look, that’s the benefit of creating online, right? You can create something years ago, and I can just go on and watch with a whole fresh mind. It exists forever. Okay, one more from that same series. “Unfortunately, no one can be told what the matrix is. You have to see it for yourself.”
Grant: Oh yeah. That was fun. Yeah, I just thought that was a wonderful … That one I do remember including, because the whole theme of that series was that to think about matrices, you should visualize them, and there’s a specific way to visualize them, and that you have this famous thing from cinema where Morpheus is saying, “You have to see it for yourself.” That just, I mean, come on. The parallel is too strong not to lean into. A lot of that is true, I think. This is the central theme of the whole channel. For everything else that we talked about, the actual reason I think a lot of people are engaged is because it centers on visuals, and we’re very visual thinkers.
A ton of the mass of our brains goes towards visual processing, in contrast to processing other raw sensory information. I think that it’s almost like you’re leveraging more computation, and through that it means that you’ve opened up the scope of what people can have intuitions for.
David: Oh man, that is a really interesting point. I mean, I think that there’s something even further that’s going on, is it’s the motion of visuals, and the way that math moves in your videos that creates an element of resonance.
Grant: Yeah. I mean, I guess that’s sort of … Maybe because it feels different than reading a textbook, and that’s the thing that most highlights the fact that it’s not the way that you’ve been experiencing math through school. Is the dynamic nature to it. Despite what I do for a living, I’m not entirely sold that the movement matters that much for the intuitions.
David: I think that the movement matters, I mean it obviously depends on the video. But I think that it matters the most when you just have to see how things are represented. For example, you talked about the boxes, and how the boxes hit each other, and then the boxes hitting each other equals pi. Then when you sort of rotated the sphere in that video, I think that it’s sort of unforgettable. How much do you think of yourself as teaching math versus popularizing math, versus inspiring people to learn math on their own?
Grant: Oh yeah, three different angles. I mean, so when it comes to teaching, I think to gain a new understanding of something, it has to be information flowing from inside your head to out of it, not the other way around. You can’t have someone come and pour knowledge into your head. Instead they have to evoke it out. In fact, the word ‘education’ has the same root as the word educe, to bring out. You kind of follow the etymology tree, and I think there’s no coincidence there. It’s this very Socratic idea.
In that way, the default for any video, or any lecture or any book is that it’s not teaching. But that’s a little too easy to say, because really it’s like … It is quite possible for a piece of text or for a lecture to evoke something, to bring it out of someone. If there is teaching that happens, it’s only because of that. The truth is, though, the actual value add I think lies almost entirely on the inspiration half. For getting someone to feel that math is something that they like. That they can understand, it’s worth pursuing further, that they can self-identify with, that can be part of their personality, all of that.
In order for that to work, though, you can’t just inspire by saying inspirational words around it. You have to give an actual lesson. You have to give, if not understanding, at least the illusion of understanding in a moment for that inspiration to have any kind of teeth to it. Most of the effort is still going to be around making something understandable, being as clear as you can, finding the elucidating visual, all of that. Even if the end goal and the actual mechanism of value add comes from inspiration and not from, what will this person be able to recall a few months later?
David: That’s a beautiful place to close. Grant, thank you very much.
Grant: Hey, thanks for having me on. This was fun.
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