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Grant Sanderson: 3Blue1Brown and the Beauty of Mathematics | Lex Fridman Podcast #64
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Grant Sanderson: 3Blue1Brown and the Beauty of Mathematics | Lex Fridman Podcast #64

Lex Fridman and Grant Sanderson on grant Sanderson Reveals How Notation Shapes Our Mathematical Reality.

Lex FridmanhostGrant Sandersonguest
Jan 7, 20201h 2mWatch on YouTube ↗

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  1. 0:00 – 1:32

    Show setup, guest intro, and sponsor message (Cash App + FIRST)

    1. LF

      The following is a conversation with Grant Sanderson. He's a math educator and creator of 3Blue1Brown, a popular YouTube channel that uses programmatically animated visualizations to explain concepts in linear algebra, calculus, and other fields of mathematics. This is the Artificial Intelligence Podcast. If you enjoy it, subscribe on YouTube, give us five stars on Apple Podcast, follow on Spotify, support on Patreon, or simply connect with me on Twitter @lexfridman, spelled F-R-I-D-M-A-N. I recently started doing ads at the end of the introduction. I'll do one or two minutes after introducing the episode, and never any ads in the middle that can break the flow of the conversation. I hope that works for you and doesn't hurt the listening experience. This show is presented by Cash App, the number one finance app in the App Store. I personally use Cash App to send money to friends, but you can also use it to buy, sell, and deposit bitcoin in just seconds. Cash App also has an investing feature. You can buy fractions of a stock, say $1 worth, no matter what the stock price is. Brokerage services are provided by Cash App Investing, a subsidiary of Square and member SIPC. I'm excited to be working with Cash App to support one of my favorite organizations called FIRST, best known for their FIRST Robotics & Lego competitions. They educate and inspire hundreds of thousands of students in over 110 countries, and have a perfect rating on Charity Navigator, which means the donated money is used to maximum

  2. 1:32 – 3:48

    Do aliens do math differently? Notation, cognition, and alternative number systems

    1. LF

      effectiveness. When you get Cash App from the App Store or Google Play and use code LEXPODCAST, you'll get $10, and Cash App will also donate $10 to FIRST, which, again, is an organization that I've personally seen inspire girls and boys to dream of engineering a better world. And now, here's my conversation with Grant Sanderson. If there's intelligent life out there in the universe, do you think their mathematics is different than ours?

    2. GS

      (laughs) Jumping right in. I think it's probably very different. There's an obvious sense. The notation is different, right? I think notation can guide what the math itself is. Uh, I think it has everything to do with the form of their existence, right?

    3. LF

      Do you think they have basic arithmetics? Sorry to interrupt.

    4. GS

      Yeah. So I think they count, right? I think notions like one, two, three, the natural numbers, that's extremely, well, natural. That's almost why we put that, uh, name to it. Um, as soon as you can count, you have a notion of repetition, right? 'Cause you can count by two two times or three times. And so you have this notion of repeating the idea of counting, which brings you addition and multiplication. I think the way that we extend to the real numbers, there's a little bit of choice in that. So there's this funny number system called the surreal numbers-

    5. LF

      Mm-hmm.

    6. GS

      ... that, um, it captures the idea of continuity. It's a distinct mathematical object. You could very well, you know, model u- the universe and motion of planets with that as the backend of your math, right? (laughs) And you still have kind of the same interface with the front end of what physical laws you're trying to... or what, what physical phenomena you're trying to describe with math. And I wonder if the little glimpses that we have of what choices you can make along the way based on what different mathematicians have brought to the table is just scratching the service, surface of what the different possibilities are if you have a completely different mode of thought, right? Or mode of interacting with the universe.

    7. LF

      A- and you think notation is a key part of the journey that we've taken through math.

    8. GS

      I think that's the most salient part that you'd notice at first. I think the mode of thought is gonna influence things more than, like, the notation itself. But notation actually carries a lot of weight when it comes to how we think about things, more so than we usually give it credit for, I would, I would be comfortable saying.

  3. 3:48 – 7:43

    Bad notation as a barrier: why ‘e^x’ and Euler’s formula feel mysterious

    1. LF

      Do you have a favorite or least favorite piece of notation in terms of its effectiveness?

    2. GS

      Um, yeah, yeah. Well, so least favorite, one that I've been thinking a lot about that will be a video, I don't know when, but we'll see-

    3. LF

      Yeah.

    4. GS

      ... um, uh, the number E. Uh, we write the function E to the X, this general exponential function, with the notation E to the X. That, that implies you should think about a particular number, this constant of nature, and you repeatedly multiply it by itself. And then you say, "Oh, what's E to the square root of two?" And you're like, "Oh, well, we've extended the idea of repeated multiplication." That's, that's all nice. That's all nice and well. But m- very famously, you have, like, E to the pi I, and you're like, "Well, well, we're extending the idea of repeated multiplication into the complex numbers." Yeah, you can think about it that way. In reality, I think that it's just the wrong way of, um, notationally representing this function, the exponential function, which itself could be represented a number of different ways. You can think about it in terms of the problem it solves, a certain very simple differential equation, which often yields way more insight than trying to twist the idea of repeated multiplication, like take its arm and put it behind its back and throw it on the desk and be like, "You will apply the complex numbers," right? That's not... I don't think that's pedagogically helpful. And-

    5. LF

      So the repeated multiplication is actually missing the main point, the power of E to the X?

    6. GS

      Yeah. It, I mean, what, what it addresses is things where the rate at which something changes depends on its own value, but more specifically, it depends on it linearly. So for example-

    7. LF

      Right.

    8. GS

      ... if you have, like, a population that's growing and the rate at which it grows depends on how many members of the population are already there, it looks like this nice exponential curve, it makes sense to talk about repeated multiplication 'cause you say, "How much is there after one year, two years, three years?" You're multiplying by something. The relationship can be a little bit different sometimes where, let's say, you've got, um, a ball on a string, like a, like a game of tetherball going around a rope, right? And you say, "Its velocity is always perpendicular to its position." That's another way of describing its rate of change as being related to where it is, but it's a different operation. You're not scaling it. It's a rotation. It's this 90-degree rotation. That's what the whole idea of, like, complex exponentiation is trying to capture, but it's obfuscated in the notation when what it's actually saying... Like, if you really parse something like E to the pi I, what it's saying is, "Choose an origin, always move perpendicular to the vector from that origin to you," okay? Um-... then when you walk pi times that radius, you'll be halfway around. Like, that's what it's saying.

    9. LF

      Mm-hmm.

    10. GS

      Um, it's kind of the, you turn 90 degrees and you walk, you'll be going in a circle. That's the phenomenon that it's describing, but trying to twist the idea of repeatedly multiplying a constant into that. Like, I- I- I can't even think of the number of human hours, of like intelligent human hours, that have been wasted trying to parse that to their own liking and desire among, like, scientists or electrical engineers or students everywhere. Which, if the notation were a little different or the way that this whole function was, um, introduced from the get-go were framed differently, I think could have been avoided, right?

    11. LF

      And you're talking about th- the most beautiful equation in mathematics, but it's still pretty mysterious, isn't it?

    12. GS

      No-

    13. LF

      Like, you're, you're making it seem like it's a notational...

    14. GS

      It's not mysterious. I think-

    15. LF

      (laughs)

    16. GS

      I think the notation makes it mysterious. I don't think it's... I think the fact that it represents, it's pretty. It's not like the most beautiful thing in the world, but it's, it's quite pretty. The idea that, um, if you take the linear operation of a 90-degree rotation and then you do this general exponentiation thing to it, that what you get are all the other kinds of rotation. Uh, which is basically to say, if you, if your velocity vector is perpendicular to your position vector, you walk in a circle. That's pretty. It's not the most beautiful thing in the world, but it's quite pretty.

    17. LF

      The beauty of it, I think, comes from perhaps the awkwardness of the notation somehow still nevertheless coming together nicely. 'Cause you have, like, several disciplines coming together in a single equation.

    18. GS

      Well, I think-

    19. LF

      I mean, in a sense.

    20. GS

      Yeah, well-

    21. LF

      Like, historically speaking.

  4. 7:43 – 10:26

    e, π, i (and τ): what’s really being tied together in Euler’s identity

    1. GS

      That, that's true. You've got... So, like, the number e is significant.

    2. LF

      Right.

    3. GS

      Like, it shows up in probability all the time. It, like, shows up in calculus all the time. It is significant. You're seeing it sort of mated with pi, this geometric constant, and i, like the imaginary number and such. I think what's really happening there is the, the way that e shows up is when you have things like exponential growth and decay, right? It's when this, um, relation that something's rate of change has to itself is a simple scaling, right? Um, a similar law also describes circular motion. Because we have bad notation, we use the residue of how it shows up in the context of self-reinforcing growth, like a population growing or compound interest. The constant associated with that is awkwardly placed into the context of how rotation comes about because they both come from pretty similar equations.

    4. LF

      Mm-hmm.

    5. GS

      And so what we see is the e and the pi juxtaposed, uh, a little bit closer than they would be with a purely natural representation, I would think. He- here's how I would describe the relation between the two. You've got a very important function we might call exp, that's like the exponential function.

    6. LF

      Mm-hmm.

    7. GS

      When you plug in one, you get this nice constant called e that shows up in, like, probability and calculus. If you try to move in the imaginary direction, it's periodic, and the period is tau.

    8. LF

      Mm-hmm.

    9. GS

      So, those are these two constants associated with the s- the same central function, but for kind of unrelated reasons, right?

    10. LF

      Mm-hmm.

    11. GS

      Not unrelated, but, like, orthogonal reasons. One of them is what happens when you're moving in the real direction, one's what happens when you move in the imaginary direction. And like, yeah, those are related. They're not as related as the famous equation seems to think it is. It's sort of putting all of the children in one bed, and they'd kind of like to sleep in separate beds-

    12. LF

      Mm-hmm.

    13. GS

      ... if they had the choice. But you see them all there, and, you know, there is a family resemblance-

    14. LF

      (laughs)

    15. GS

      ... but it's not that close.

    16. LF

      So, actually, thinking of it as a function is, um, is the better idea. And that's a notational idea.

    17. GS

      And, yeah. And th- like, here's the thing. The constant e sort of stands as this numerical representative of calculus, right?

    18. LF

      Yeah.

    19. GS

      Calculus is the, like, study of change.

    20. LF

      Mm-hmm.

    21. GS

      So, at the very least, there's a little cognitive dissonance using a constant-

    22. LF

      (laughs)

    23. GS

      ... to represent the science of change.

    24. LF

      I never thought of it that way. Yeah. (laughs)

    25. GS

      Right?

    26. LF

      Yeah. (laughs)

    27. GS

      It makes sense why the notation came about that way-

    28. LF

      Yes.

    29. GS

      ... because this is the first way that we saw it. Um, in the context of things like population growth or compound interest, it is nicer to think about as repeated multiplication. That's definitely nicer. But it's more that that's the s- the first application of what turned out to be a much more general function, that maybe the intelligent life your initial question asked about would have come to recognize as being much more significant than the single use case, which lends itself to repeated multiplication notation. But...

    30. LF

      Let me jump back-

  5. 10:26 – 12:05

    Is math discovered or invented? A feedback loop between world and abstraction

    1. LF

      ... for a second to aliens and the nature of our universe. Okay. Uh, do you think math is discovered or invented? So, we're talking about the different kind of mathematics that could be developed by the alien species. The implied question is, is, um, yeah, is math discovered or invented? Is, you know, is fundamentally everybody going to discover the same principles of mathematics?

    2. GS

      Uh, so the way I think about it, and everyone thinks about it differently, but here's my take. I think there's a cycle at play where you discover things about the universe that tell you what math will be useful, and that math itself is invented in a sense. But of all the possible maths that you could have invented, it's discoveries about the world that tell you which ones are. So like, a, a good example here is, um, the Pythagorean theorem.

    3. LF

      Mm-hmm.

    4. GS

      When you look at this, do you think of that as a definition or do you think of that as a discovery?

    5. LF

      From the historical perspective, right, it's a discovery.

    6. GS

      Yeah.

    7. LF

      Because they were... But th- th- that's probably because they were using physical object to build their intuition, f- a- a- and from that intuition came the mathematics. So, the, the mathematics wasn't in some abstract world detached from physics. But I think more and more, math has become detached from, you know... When you, when you even look at modern physics, from, from string theory to even general relativity, I mean, all math behind the 20th and 21st century physics kind of, uh, takes a brisk walk outside of what our mind can actually even comprehend-

    8. GS

      I-

    9. LF

      ... in multiple dimensions, for example. Anything beyond three dimensions, maybe four dimensions.

  6. 12:05 – 14:29

    Higher dimensions without higher-dimensional reality: state spaces and usefulness

    1. GS

      Oh, no, no, no. Hi- higher dimensions can be highly, highly applicable. I think this is a common misinterpretation-

    2. LF

      Mm-hmm.

    3. GS

      ... that...... if you're asking questions about, like, a five-dimensional manifold, that the only way that that's connected to the physical world is if the physical world is itself a five-dimensional manifold or includes them.

    4. LF

      W- wait, wait, wait a minute, wait a minute. You're telling me you can imagine a- a- a f- five-dimensional manifold?

    5. GS

      No, no. That's not what I said.

    6. LF

      Okay.

    7. GS

      I- I- I'm- I would make the claim that it is useful to a three-dimensional physical universe, despite itself not being three-dimensional.

    8. LF

      So, it's useful, meaning f- to even understand a three-dimensional world-

    9. GS

      Mm-hmm.

    10. LF

      ... it would be useful to have five-dimensional manifolds, potentially?

    11. GS

      Yes, absolutely, because of state spaces.

    12. LF

      But you're saying there, in some- in some deep way for us humans, it does- it does always come back to that three-dimensional world, for the usefu- usefulness of that three-dimensional world. And therefore, it starts with the discovery, but then we invent the mathematics that, um, helps us make sense of the discovery in a sense.

    13. GS

      Yes. I mean, just to jump off of the Pythagorean theorem example-

    14. LF

      Yes, yeah.

    15. GS

      ... it feels like a discovery. You've got these beautiful geometric proofs where you've got squares and you're modifying the areas. It feels like a discovery. If you look at how we formalize the idea of 2D space as being R2, right-

    16. LF

      Yeah.

    17. GS

      ... all pairs of real numbers, and how we define a metric on it and define distance, you're like, "Hang on a second. We've defined distance so that the Pythagorean theorem is true," so then suddenly, it doesn't feel that great. But I think what's going on is the thing that informed us what metric to put on R2, to put on our abstract representation of 2D space, came from physical observations. And the thing is, there's other metrics you could have put on it. We could have consistent math with other notions of distance. It's just that those pieces of math wouldn't be applicable to the physical world that we study, 'cause they're not the ones where the Pythagorean theorem holds. So, we have a discovery, a genuine bonafide discovery that informed the invention, the invention of an abstract representation of 2D space that we call R2 and things like that. And then from there, you just study R2 as an abstract thing that brings about more ideas and inventions and mysteries, which themselves might yield discoveries. Those discoveries might give you insight as to what else would be useful to invent, and it kind of feeds on itself that way. That's how I think about it. So, it's not an either/or. It's not that math is one of these or it's one of the others. At different times, it's playing a different role.

  7. 14:29 – 17:24

    Math vs physics: patterns, rigor, and different kinds of mathematicians

    1. LF

      So then, let me ask the- the Richard Feynman question then s- along that thread, is what do you think is the difference between physics and math? There's a giant overlap. The- there's a kind of intuition that physicists have about the world that's perhaps outside of mathematics. It's this mysterious art that they, uh, seem to possess, we humans generally possess. And then there's the beautiful rigor of mathematics that allows you to, uh, I mean, just like as w- as we were saying, um, invent frameworks of understanding o- our physical world. So, what do you think is the difference there, and how big is it?

    2. GS

      Well, I think of math as being the study of, like, abstractions over patterns and pure patterns in logic. And then physics is obviously grounded in a desire to understand the world that we live in.

    3. LF

      Yeah.

    4. GS

      I think you're gonna get very different answers when you talk to different mathematicians, 'cause there's a wide diversity in types of mathematicians. There are some who are motivated very much by pure puzzles. Um, they might be turned on by things like combinatorics, and they just love the idea of building up a set of problem-solving tools applying to pure patterns, right? There are some who are very physically motivated, who- who try to invent new math or discover math in veins that they know will have applications to physics or sometimes computer science, and that's what drives them, right? Like chaos theory is a good example of something that's- it's pure math, it's purely mathematical, a lot of the statements being made, but it's heavily motivated by specific applications to, largely, physics. And then you have a type of mathematician who just loves abstraction. They just love pulling it to the more and more abstract things, the things that feel powerful. These are the ones that initially invented, like, topology, and then later on get really into category theory and go on about, like, in- infinite categories and whatnot. These are the ones that love to have a system that can describe truths about as many things as possible, right?

    5. LF

      Mm-hmm.

    6. GS

      People from those three different veins of motivation into math are gonna give you very different answers about what the relation at play here is, 'cause someone like, um, Vladimir Arnold, uh, who is this- he's, uh, written a lot of great books, um, many about, like, differential equations and such. He would say math is a branch of physics.

    7. LF

      (laughs)

    8. GS

      That's how he would think about it.

    9. LF

      Mm-hmm.

    10. GS

      And of course he was studying, like, differential equations related things, because that is the motivator behind the study of PDEs and things like that. Um, but you'll have others who, like especially the category theorists, who aren't really thinking about physics necessarily. It's all about abstraction and the power of generality. And it's more of a happy coincidence that that ends up being useful for understanding the world we live in. Um, and then you can get into like, why is that the case? It's sort of surprising that that which is about pure puzzles and abstraction also happens to describe the very fundamentals of quarks and everything else.

  8. 17:24 – 21:45

    Why are physical laws so compressible? Simplicity, filtering, and the anthropic angle

    1. LF

      So, why do you think the fundamentals of quarks and- and the nature of reality is so compressible into clean, beautiful equations that are, for the most part, simple, relatively speaking? A lot simpler than they could be. So, you have, we, uh, mentioned somebody like Stephen Wolfram, who thinks that sort of there's, uh, incredibly simple rules underlying our reality, but it can create arbitrary complexity-

    2. GS

      Mm-hmm.

    3. LF

      ... but there is simple equations. What... I'm asking a million questions that nobody knows the answer to, but, uh-

    4. GS

      Yeah, I have no idea. (laughs)

    5. LF

      (laughs)

    6. GS

      Like, why is it simple?

    7. LF

      I-

    8. GS

      It could be the case that there's, like, a filter- tration at play. The only things that physicists find interesting-

    9. LF

      Hm.

    10. GS

      ... are the ones that are simple enough they could describe it mathematically.

    11. LF

      Right, yeah.

    12. GS

      But as soon as it's a sufficiently complex system, they're like, "Oh, that's outside the realm of physics."

    13. LF

      Yeah.

    14. GS

      "That's biology," or whatever have you. And, of course-

    15. LF

      That's true. Right.

    16. GS

      ... you know, maybe there's something where it's like, of course there will always be some thing that is simple when you wash away, uh...... the, like, non-important parts of whatever it is that you're studying. Just from, like, an information theory standpoint, there might be some, like, you, you get to the lowest information component of it. But I don't know, ma- maybe I'm just having a really hard time conceiving of what it would even mean for the fundamental laws to be, like, intrinsically complicated, like some, some set of equations that you can't decouple from each other. I just-

    17. LF

      Well, no. It could be, it could be that s- sort of, we take for granted that the, the, the laws of physics, for example, are, for the most part, the same everywhere.

    18. GS

      Hmm.

    19. LF

      Or something like that, right? As opposed to the, uh, sort of an alternative could be that the rules under which, uh, the world operates is different everywhere. It's like a, a d- like a deeply distributed system, where just everything is just chaos, like, um, not, not in a strict definition of chaos, but meaning, like, just it's impossible for equations to capture, for, to explicitly model the world, i- as cleanly as the p- physical does. I mean, we d- we almost take it for granted that we can describe, we can have an equation for gravity-

    20. GS

      Hmm.

    21. LF

      ... for action at a distance. We can have equations for some of these basic ways that planets move and d- j- j- just the, the, the low level, uh, the atomic scale, how the materials operate at the high scale, how black holes operate. But it doesn't, it, it seems like it could be, there's infinite other possibilities where n- none of it could be compressible into such equations. There's just, seems beautiful, and it's also weird, probably to the point you were making, that it's very pleasant that this is true-

    22. GS

      (laughs)

    23. LF

      ... for our minds.

    24. GS

      Right.

    25. LF

      So it might be that our minds are biased to just be looking at the parts of the universe that are compressible, and then we can publish papers on and have nice E equals MC squared equations.

    26. GS

      Right. Well, I wonder would such a world with incompressible laws allow for the kind of beings that can think about-

    27. LF

      Right.

    28. GS

      ... the kind of questions that you're asking?

    29. LF

      Right. That's true.

    30. GS

      Right? Like an anthropic principle coming into play at some weird way here? I don't know. Like I, I don't know what I'm talking about at all.

  9. 21:45 – 25:17

    Simulation hypothesis: infinite layers vs finite computational resources

    1. LF

      So I mentioned to the internet that I'm talking to you, and so the internet gave some questions. So I apologize for these, but, uh, do you think we're living in a simulation? That the universe is a computer or, uh, the universe is a computation running on a computer?

    2. GS

      It's conceivable. What I don't buy is, you know, you'll have the argument that, well if, let's say, that it was the case that you can have simulations, then the simulated world would itself eventually get to a point where it's running simulations.

    3. LF

      Yes.

    4. GS

      And then the, the second layer down would create-

    5. LF

      Okay.

    6. GS

      ... a third layer down, and on and on and on. So probabilistically, you just throw a dart at one of those layers, we're probably in one of the simulated layers.

    7. LF

      Mm-hmm.

    8. GS

      I think if there's some sort of limitations on like the information processing of whatever the physical world is, like it quickly becomes the case that you have a limit to the layers that could exist there. Because w- like the resources necessary to simulate a universe like ours clearly is a lot, just in terms of the number-

    9. LF

      Yeah.

    10. GS

      ... of bits at play. And so then you can ask, well, what's more plausible? That there's an unbounded capacity of information processing in whatever the like highest up level universe is, or that there's some bound to that capacity which then limits like the number of levels available? How do you place some kind of probability distribution on like what the information capacity is? I have no idea. But I, I don't imme- like people almost assume a certain uniform probability over all of those meta layers that could conceivably exist, when it's, it's a little bit like a Pascal's wager on like you're not giving a low enough prior to the mere existence of that infinite set of layers.

    11. LF

      Yeah, that's true. But it's also very difficult to contextualize the amou- so the amount of information processing power required to simulate like our universe seems like amazingly huge.

    12. GS

      But you can always raise two to the power of that-

    13. LF

      Yeah, exactly. (laughs)

    14. GS

      ... and then you have, yeah. It's a- yeah. It, like numbers get big. Um, so-

    15. LF

      And we're easily humbled by basically everything around us. So it's very difficult to, uh, to kind of make sense of anything actually when you look up at the, uh, sky and look at the stars and the immensity of it all, to make sense of us, the smallness of us, the unlikeliness of everything that's on this earth coming to be. Then you could, basically anything could be, uh, all, all laws of probability go out the window (laughs) to me because, um, I guess because the amount of information under which we're operating is very low. We basically know nothing about the world around us, relatively speaking.

    16. GS

      Right.

    17. LF

      And so, so the wha- when I think about the simulation hypothesis, I think it's just fun to think about. It, it's, uh, but it's also, I think there i- is a thought experiment, kind of interesting to think of the power of computation-... whether a, the limits of a Turing machine. Sort of, the limits of our current computers, when you start to think about artificial intelligence. How far can we get with computers?

    18. GS

      Mm-hmm.

    19. LF

      And that's kind of where the simulation hypothesis is useful to me as a thought experiment, is, is the universe just a computer? Is it just a computation? Is all of this just a computation? And sort of, the same kind of tools we apply to analyzing algorithms, can that be appli- you know, if we scale further and further and further, will the arbitrary power of those systems start to create some interesting aspects that we see in our universe? Or is something fundamentally different needs to be created?

  10. 25:17 – 26:25

    Information is physical: storage limits, black holes, and what that implies

    1. GS

      Well, it's interesting that in our universe, it's not arbitrarily large, the power, that you can place limits on, for example, how many bits of information can be stored-

    2. LF

      Right.

    3. GS

      ... per unit area. Right? Like, all of the physical laws, you've got general relativity and like quantum coming together to give you a certain limit on how many bits you can store within a- a given range before it collapses into a black hole. Like, the idea that there even exists such a limit is at the very least thought-provoking, when naively, you might assume, "Oh, well, you know, technology could always get better and better. We could get cleverer and cleverer, and you could just cram as much information as you want into like a small unit of space." Um, that makes me think it's at least plausible that whatever the highest level of existence is doesn't admit too many simulations, or ones that are at the scale of complexity that we're looking at. Obviously, it's just as conceivable that they do and that there are many. But, um, I- I- I guess what I'm channeling is the surprise that I felt upon learning that fact, that there are ... that information is physical in this way.

  11. 26:25 – 32:04

    Infinity, abstraction, and ‘overgeneralization’: making peace with the infinite

    1. LF

      Is that there's a finiteness to it. Okay. Let me just even go off on that from a mathematics perspective and a psychology perspective. How do you mix ... Are you, um, psychologically comfortable with the concept of infinity?

    2. GS

      (laughs) I think so.

    3. LF

      Are you okay with it?

    4. GS

      I'm pretty okay, yeah.

    5. LF

      (laughs)

    6. GS

      Are you okay?

    7. LF

      No, not really.

    8. GS

      (laughs)

    9. LF

      It doesn't make any sense to me.

    10. GS

      I don't know. Like, how many, how many words ... How many possible words do you think could exist that are just like strings of letters?

    11. LF

      So, y- that- that's a sort of mathematical statement that's beautiful, and we use infinity in basically everything we do m- everything we do in sci- uh, in, in science, math, and engineering, yes. But you said, "exist." My, th- the question is, uh, you said letters or words?

    12. GS

      I said words.

    13. LF

      Words. Uh, th- to bring words into existence, to me, you have to start like saying them, or like writing them, or like listing them.

    14. GS

      That's an instantiation. Okay.

    15. LF

      That's an instantiation.

    16. GS

      How many, how many abstract words exist? (laughs)

    17. LF

      The, well, the idea of a abstract.

    18. GS

      Yeah.

    19. LF

      The, the idea of abstract notions and ideas.

    20. GS

      I think we should be clear on terminology. I mean, you think about intelligence a lot.

    21. LF

      Mm-hmm.

    22. GS

      Like artificial intelligence. Would you not say that what it's doing is a kind of abstraction, that like abstraction is key to conceptualizing the universe? 'Cause you get this raw sensory data. You need, I need something that every time you move your face a little bit, and the, they're not pixels, but like analog of pixels on my retina change entirely-

    23. LF

      Yeah.

    24. GS

      ... that I can still have some coherent notion of, "This is Lex."

    25. LF

      Yes.

    26. GS

      "I'm talking to Lex."

    27. LF

      Yes.

    28. GS

      Right? What that requires is you have a, a disparate set of possible images hitting me that are unified in a notion of Lex.

    29. LF

      Yeah.

    30. GS

      Right? Um, that's a kind of abstraction. It's a thing that could apply to a lot of different images that I see, and it, it represents it in a much more compressed way, um, and one that's like much more resilient to that. I think in the same way, if I'm talking about infinity as an abstraction, I don't mean non-physical, woo-woo, it, it, like ineffable or something. What I mean is, it's something that can apply to a multiplicity of situations that share a certain common attribute, in the same way that the images of like your face on my retina share enough common attributes that I can put the single notion to it. Like, in that way, infinity is an abstraction, and it's very powerful and, and it, it's, it's only through such abstractions that we can actually understand like the world and logic and things. And in the case of infinity, the way I think about it, the key entity is the property of always being able to add one more. Or like no matter how many words you can list, you just throw an A at the end of one and you have another conceivable word.

  12. 32:04 – 35:48

    How 3Blue1Brown builds understanding: examples first, definitions later

    1. GS

      ... in my head, and you realize, oh, I... you know, your, your brain would literally collapse into a black hole, all of that. Um, and, and I, I honestly feel this with a lot of math that I try to read, where I, um... I don't think of myself as like particularly, uh, good at math, uh, uh, i- i- in some ways. Like, I get very confused often when I am going through some of these texts. Uh, and often what I'm feeling in my head is like, "This is just so damn abstract."

    2. LF

      (laughs)

    3. GS

      "I just can't wrap my head around it." I just want to put something concrete to it that makes me understand, and I think a lot of the motivation for the channel is channeling that sentiment of, yeah, a lot of the things that you're trying to read out there, it's just so hard to connect to anything that you spend an hour banging your head against a couple of pages and you come out not really knowing anything more, other than some definitions maybe and a certain sense of self-defeat, right? One of the reasons I focus so much on visualizations is that I'm a big believer in... I'm, I'm sorry. I'm just really hammering out this idea of abstraction. Being clear about your layers of abstraction.

    4. LF

      Yes.

    5. GS

      Right? It's always tempting to start an explanation from the top to the bottom, okay? You, you give the definition of a new theorem. You're like, "This is the definition of a vector space," for example. We're gonna... that's how we'll start our course. These are the properties of a vector space. You must... First, from these properties, we will derive what we need in order to do the math of linear algebra or whatever it might be. Um, I don't think that's how understanding works at all. I think how understanding works is you start at the lowest level you can get it, where rather than thinking about a vector space, you might think of concrete vectors that are just lists of numbers, or picturing it as like an arrow that you draw-

    6. LF

      Mm-hmm.

    7. GS

      ... um, which is itself like even less abstract than numbers because you're looking at quantities, like the distance of the X coordinate, the distance of the Y coordinate. It's as concrete as you could possibly get, and it has to be if you're putting it in a visual, right? Like, the, the-

    8. LF

      It's an actual arrow that-

    9. GS

      It's an actual vector.

    10. LF

      (laughs)

    11. GS

      You're not talking about like a "vector" that could apply to any possible thing. You have to choose one if you're illustrating it.

    12. LF

      Yeah.

    13. GS

      And I think this is the power of being in a, a medium like video, or if you're writing a textbook and you force yourself to put a lot of images, is with every image you're making a choice. With each choice, you're showing a concrete example. With each concrete example, you're aiding someone's path to understanding.

    14. LF

      You know, I'm sorry to in- interrupt you, but, uh, you just made me realize that that's exactly right, so the visualizations you're creating, while you're sometimes talking about abstractions, the actual visualization is an explicit low-level example.

    15. GS

      Yes.

    16. LF

      So there, there's an actual... Like, in the code, you have to say what the, what the vector is.

    17. GS

      Mm-hmm.

    18. LF

      What, what's the direction of the arrow? What's the magnitude of the, the... Uh, okay, wait... Yeah, so that's... You're going... The visualization itself is actually going to the bottom of that.

    19. GS

      I think... And I think that's very important. I, I also think about this a lot in writing scripts, where even before you get to the visuals, um, the first instinct is to... I, I don't know why. I, I just always do. I say the abstract thing, I say the general definition, the powerful thing, and then I, I fill it in with examples later. Always it will be more compelling and easier to understand when you flip that, and instead you let someone's brain do the, uh, pattern recognition. You just show them a bunch of examples. The brain is gonna feel a certain similarity between them. Then by the time you bring in the definition or by the time you bring in the formula, it's articulating a thing that's already in the brain that was built off of looking at a bunch of examples with a certain kind of similarity. And what the formula does is articulate what that kind of similarity is, rather than being a, a high cognitive load set of symbols that needs to be populated with examples later on, assuming someone's still with you.

  13. 35:48 – 41:32

    Mathematical beauty and mystery: Euler product, zeta function, and primes

    1. LF

      What is the most beautiful or awe-inspiring idea you've come across in mathematics?

    2. GS

      I don't know, man.

    3. LF

      Maybe it's an idea you've explored in your videos, maybe not. What, like, just gave you pause?

    4. GS

      What's the most beautiful idea?

    5. LF

      Small or big.

    6. GS

      So I think often the things that are most beautiful are the ones that you have like an... a little bit of understanding of, but certainly not an entire understanding. It's a little bit of that mystery that is what makes it beautiful.

    7. LF

      Almost the moment of the discovery for you personally, almost just that leap of the ah- aha moment.

    8. GS

      So something that really caught my eye, I remember when I was, um, little, there were these like, um... I, I think the series was called like Wooden Books or something, these tiny little books that would have just a very short description of something on the left and then a picture on the right. I don't know who they're meant for, but maybe it's like loosely children or something like that. But it can't just be children because of some of the things it was describing. On the last page of one of them-... somewhere tiny in there was this little formula that on the left hand had a sum over all of the natural numbers. You know, it's like 1 over 1 to the S plus 1 over 2 to the S plus 1 over 3 to the S, on and on to infinity. And then, on the other side, had a product over all of the primes, and it was a certain thing, had to do with all the primes. And like any good young math enthusiast, I'd properly been indoctrinated with how chaotic and confusing the primes are, which they are, and seeing this equation where on one side you have something that's as understandable as you could possibly get, the counting numbers-

    9. LF

      Yes.

    10. GS

      ... and on the other side is all the prime numbers, it was like this, "Whoa! They're related like this?" Th- uh, there's a, there's a simple description that includes, like, all the primes getting wrapped together like this. This is like the Euler product for zeta function, as I, like, later found out. The equation itself essentially encodes the fundamental theorem of arithmetic, that every number can be expressed as a unique set of primes.

    11. LF

      Mm-hmm.

    12. GS

      To me still, there's... I mean, I certainly don't understand this equation or this function all that well. The more I learn about it, the prettier it is. The idea that you can... Uh, this is sort of what gets you representations of primes, not in terms of primes themselves but in terms of another set of numbers-

    13. LF

      Mm-hmm.

    14. GS

      ... that are like the non-trivial zeros of the zeta function. And again, I'm very, kind of in over my head in a lot of ways as I, like, try to get to understand it. But the more I do, it's, it always leaves enough mystery that it remains very beautiful to me. And-

    15. LF

      So whenever there's, uh, a little bit of mystery just outside of the understanding that, uh... And by the way, the, the, the process of learning more about it, how does that come about? Just your own thought or are you reading?

    16. GS

      Reading, yeah. So, um-

    17. LF

      Or is the process of visualization itself revealing more to you?

    18. GS

      Visuals help. I mean, in, in, in one time when I was just trying to understand, like, analytic continuation and playing around with, um, visualizing complex functions, this is what led to a video about this function. Uh, uh, it's titled something like Visualizing the Riemann Zeta Function.

    19. LF

      Mm-hmm.

    20. GS

      It's one that came about because I was programming and tried to see what a certain thing looked like, and then I looked at it and I'm like, "Whoa, that's elucidating."

    21. LF

      (laughs)

    22. GS

      And then I decided to make a video about it. Um, but, uh, I mean, you, you, you try to, uh, get your hands on as much reading as you can. You... You know, in, in this case, I think if anyone wants to start to understand it, if they have like a, uh, a math background or some... Like, they studied some in college or something like that, um, like the Princeton Companion to Math has a really good article on analytic number theory, and that itself has a whole bunch of references. And, you know, anything has more references and it gives you this, like, tree to start pawing through. And, like, y- you know, you try to understand... I try to understand things visually as I go. It's not always possible, um, but it's very helpful when it does. You recognize when there's common themes. Like in this case, cousins of the Fourier transform, like, come into play and you realize, "Oh, it's probably pretty important to have deep intuitions of the Fourier transform," even if it's not explicitly mentioned in, like, these texts, and you try to get a sense of what the common players are. But I'll emphasize again, like, I, I feel very in over my head when I try to understand the exact relation between, like, the zeros of the Riemann zeta function and how they relate to the distribution of primes. I definitely understand it better than I did a year ago. I definitely understand it 1/100 as well as the experts on the matter do, I assume. But the slow path towards getting there is, it's fun, it's charming, and like to your question, very beautiful.

    23. LF

      And the beauty is in the, what, in the journey versus the destination?

    24. GS

      Well, it's that each, each thing doesn't feel arbitrary. I think that's a big part-

    25. LF

      Uh.

    26. GS

      ... is that you have, um, these unpredictable... No... Yeah, these very unpredictable patterns or these intricate properties of like a certain function. Um, but at the same time, it doesn't feel like humans ever made an arbitrary choice in studying this particular thing. So, you know, it feels like you're speaking to patterns themselves or nature itself. That's a big part of it. Um, I think things that are too arbitrary, it's just hard for those to feel beautiful because... And this is sort of what the word contrived is meant to apply to, right? (laughs)

    27. LF

      And, uh, the... When they're not arbitrary means it could be... You can have a clean abstraction and intuition that allows you to comprehend it?

    28. GS

      Well, to one of your first questions, it makes you feel like if you came across another intelligent civilization, that they'd be studying the same thing.

    29. LF

      (laughs)

    30. GS

      Right?

  14. 41:32 – 45:04

    Favorite video: topology made concrete via the inscribed square/rectangle problem

    1. LF

      Whenever somebody does a lot of something amazing, I'm gonna ask the question that, that you've already been asked a lot, that, uh, you'll get more and more asked in your life. But what was your favorite video to create?

    2. GS

      Oh. Favorite create? One of my favorites is, um, the title is Who Cares About Topology?

    3. LF

      (laughs)

    4. GS

      Um-

    5. LF

      Do you want me to pull it up or not?

    6. GS

      If you want, sure. Yeah. It is about... Well, it starts by describing an unsolved problem that's still unsolved in math called the inscribed square problem. You draw any loop and then you ask, are there four points on that loop that make a square? Totally useless, right? Th- this is not answering any physical questions. It's mostly interesting that we can't answer that question, and it seems like such a natural thing to ask. Um, now if you weaken it a little bit and you ask, can you always find a rectangle? You choose four points on this curve, can you find a rectangle? That's hard, but it's doable, and the path to it involves, um, things like looking at a torus, this surface with a single hole in it like a donut, or looking at a Mobius strip, um, in ways that feel so much less contrived to when I first, as like a little kid, learned about these surfaces and shapes, like a Mobius strip and a torus. Like, what you learn is, oh, this Mobius strip, you take a piece of paper, put a twist, glue it together, and now you have a shape with one edge and just one side. And-... as a student, you should think, "Who cares?" Right? Like, "How does that help me solve any problems? I thought math was about problem-solving." Um, so what I liked about the piece of math that this was describing, that was, um, in this paper by a mathematician named Vaughan, was that it arises very naturally. It's clear what it represents. It's doing something. It's not just playing with construction paper. Um, and the way that it solves the problem is really beautiful. Uh, so kind of putting all of that down, um, and concretizing it, right? Like I was talking about how when you have to put visuals to it, it demands that what's on screen is a very specific example of what you're describing. The c- the construction here is very abstract in nature. You describe this very abstract kind of surface in 3D space. So then when I was finding myself... In this case, I wasn't programming, I was using, um, Grapher. That's like built into OS X for the-

    7. LF

      Hmm.

    8. GS

      ... 3D stuff. To draw that surface you realize, "Oh, man, the topology argument is very non-constructive." I have to make a lot of... You do- you have to do a lot of extra work in order to make the surface show up. But then once you see it, it's quite pretty, and it's very satisfying to see a specific instance of it. And you also feel like, "Ah, I've actually added something on top of what the original paper was doing," that it shows something that's completely correct, um, that's a very beautiful argument, but you don't see what it looks like. And I found something satisfying in seeing what it looked like that could only ever have come about from the forcing function of getting some kind of image on the screen to describe the thing I was talking about.

    9. LF

      So, you almost weren't able to anticipate what it was gonna look like after you-

    10. GS

      I had no idea. I had no idea, and it was wonderful, right?

    11. LF

      (laughs)

    12. GS

      It was totally... It looks like the Sydney Opera House or some sort of Frank Gehry design, and it was... You, you knew it was gonna be something and you can say various things about it like, "Oh, it, it touches the curve itself. It has a boundary that's this curve on the 2D plane. It all sits above the plane." But before you actually dra- it's, it's very unclear what the thing will look like, um, and to see it, it's very, uh, it's just pleasing, right? So that was, that was fun to make, very fun to share. I, I hope that it has elucidated for some people out there where these constructs of topology come from, that it's not arbitrary play with construction paper.

  15. 45:04 – 56:18

    Creative process and perfectionism: audience, narrative, and knowing when it’s done

    1. LF

      So let's... I think this is good, uh, a good sort of example to talk a little about your process. So you have, you have a list of ideas.

    2. GS

      Mm-hmm.

    3. LF

      So the- that's sort of the, the curse of having, having an active and brilliant mind, is I'm sure you have a list that's growing faster than you can utilize. (laughs)

    4. GS

      Nail on the head. Absolutely.

    5. LF

      But there's some sorting procedure, depending on mood and interests and so on. But okay, so you pick an idea then you have to try to write a narrative arc that sort of, "How do I elucidate? How, how do I make this idea beautiful and clear and explain it?" And then there's a set of visualizations that will be attached to it. Sort of... You- you've talked about some of this before, about sort of writing the story, attaching the visualizations. Can you talk through interesting, painful, beautiful parts of that process?

    6. GS

      Well, the most painful is, um, if you've chosen a topic that you do want to do, but then it's hard to think of, I guess, how to structure the script. Um, this is sort of where I have been on one for like the last two or three months, and I think that ultimately the right resolution is just like set it aside and instead, um, do some other things where the script comes more naturally. 'Cause you sort of don't want to overwork a, a narrative, that the more you've thought about it the less you can empathize with the student who doesn't yet understand the thing you're trying to teach.

    7. LF

      Who is the judger in your head? Sort of the, the person, the c- the creature, the essence that's saying, "This sucks." Or, "This is good." And you mentioned kind of the student you're, you're thinking about.

    8. GS

      Um-

    9. LF

      Can you, uh... Who is that? What is that thing that critiz-

    10. GS

      (laughs)

    11. LF

      ... that says, that says... The perfections that says, "This thing sucks, you need to work on it for another two, three months"?

    12. GS

      I don't know. I think it's my past self. I think that's the entity that I'm most trying to empathize with, is like you take who I wa- because that's kind of the only person I know. Like, you don't really know anyone other than versions of yourself. So I start with th- the version of myself that I know who doesn't yet understand the thing, right?

    13. LF

      Ah.

    14. GS

      And then I- I just try to view, view it with fresh eyes, a particular visual or a particular script, like, "Is this motivating? Does this make sense?" Um, which has its downsides 'cause sometimes I find myself speaking to motivations that only myself would be interested in. (laughs) I don't know, like I, I did this project on quaternions where what I really wanted was to understand, what are they doing in four dimensions?

    15. LF

      Mm-hmm.

    16. GS

      Can we see what they're doing in four dimensions, right? And I came, like had a way of thinking about it that really answered the question in my head, that made me very satisfied in being able to think about concretely with a 3D visual, what are they doing to a 4D sphere. And so I'm like, "Great! This is exactly what my past self would have wanted," right? And I make a thing on it. And I'm sure it's what some other people wanted too. But in hindsight I think most people who want to learn about quaternions are like robotics engineers or graphics programmers who want to understand how they're used to describe-

    17. LF

      Yeah.

    18. GS

      ... 3D rotations. And like their use case was actually a little bit different than my past self, and in that way like I wouldn't actually recommend that video to people who are coming at it from that angle-

    19. LF

      Mm-hmm.

    20. GS

      ... of wanting to know, "Hey, I'm a robotics programmer, like how do these quaternion things work, um, to describe position in 3D space?" I would say, um, other great resources for that if you ever find yourself wanting to say like, "But hang on, in what sense are they acting in four dimensions?" Then come back.

    21. LF

      Mm-hmm.

    22. GS

      But until then, it's a little different. Um-

    23. LF

      Yeah, it's interesting 'cause, uh, sort of you have incredible videos on neural networks, for example.

    24. GS

      Mm-hmm.

    25. LF

      And from my sort of perspective, 'cause I've probably... I mean, I looked at the... I mean, this is sort of my field and, and I've also looked at the basic introduction of neural networks like a million times from different perspectives, and it made me realize that there's a lot of ways to present it. So you are sort of... You did an incredible job. I mean sort of the-

    26. GS

      Oh, thank you.

    27. LF

      ... it's, uh, but you could also do it differently and also incredible.... to create a beautiful presentation of a basic concept is, requires sort of, uh, creativity, it requires genius and so on. But you can take it from a bunch of different perspectives. And that video on neural networks made me realize that. And just as you're saying, you kind of have a certain mindset, a- a certain view, but from a, if you take a different view from a physics perspective, from a neuroscience perspective, talking about neural networks, or from a robotics perspective, or from, let's see, from a pure learning theory perspec-

    28. GS

      Statistics?

    29. LF

      ... statistics perspective. So, you, you can create totally different videos. And you've done that with a few, actually, concepts where you've, have taken different cuts. Like at the, uh, at the, at the Euler equation, right? The-

    30. GS

      Mm-hmm.

  16. 56:18 – 1:02:39

    How to learn math effectively: solve problems, program, and teach

    1. LF

      What is the best way to learn math for people who might be at the beginning of that journey? I think that's, um, that's a question that a lot of folks kind of ask and think about. And it doesn't... Even for folks who are not really at the beginning of their journey, like there might be actually, uh, deep in their career of some s- type, they've taken college, they've taken calculus and so on, but still wanna sort of explore math. What, what wo- what would be your advice in sort of education at all ages?

    2. GS

      Your temptation will be to spend more time, uh, like watching lectures or reading. Um, try to force yourself to do more problems than you naturally would. Uh, that's a big one. Um, like the, the focus time that you're spending should be on like solving specific problems and seek entities that have well-curated lists of problems.

    3. LF

      So go into like a textbook almost an-

    4. GS

      Yeah.

    5. LF

      ... and the problems in the back of a textbook-

    6. GS

      Yeah.

    7. LF

      ... kind of thing, uh, uh, uh, back of a chapter?

    8. GS

      So if you can, take a little look through those questions at the end of the chapter before you read the chapter. A lot of them won't make sense. Some of them might, and those are, those are the best ones to think about. A lot of them won't, but just, you know, take a quick look and then read a little bit of the chapter and then maybe take a look again and things like that. And don't consider yourself done with the chapter until you've actually worked through a couple exercises, right? Um, and, and this is so hypocritical, right? 'Cause I like put out videos that pretty much never have associated exercises. I just view myself as a different part of the ecosystem.

    9. LF

      Mm-hmm.

    10. GS

      Which that means I'm kind of admitting that you're not really learning, or at least this is only a partial part of the learning process if you're watching these videos. Um, I think if someone's at the very beginning, uh, like I do think Khan Academy does a good job. They have a pretty large set of questions you can work through.

    11. LF

      Just the very basics, sort of just picking, picking up, getting, getting comfortable with the very basics, linear algebra, calculus-

    12. GS

      Yep.

    13. LF

      ... what's on-

    14. GS

      Yep.

    15. LF

      ... Khan Academy.

    16. GS

      Uh, programming is actually I think a great... Like learn to program and like let the way that math is motivated from that angle push you through. Uh, I, I know a lot of people who didn't like math, got into programming in some way, and that's what turned them onto math. Maybe I'm biased 'cause like I live in the Bay Area, so I'm more likely to run into someone who has that phenotype. But I am willing to speculate that that is a more generalizable path.

    17. LF

      So you yourself kind of in creating the videos are using programming to illuminate a concept.

    18. GS

      Mm-hmm.

    19. LF

      But for yourself as well. So would you recommend somebody try to make a sort of almost like try to make videos like you do-

    20. GS

      Yeah.

    21. LF

      ... as a way to learn?

    22. GS

      Explanation is great. Well, so one thing I've heard before, I don't know if this is based on any actual study, this might be like a total fictional anecdote of numbers, but it, it rings in the mind as, as being true. You remember about 10% of what you read, you remember about 20% of what you listen to, you remember about 70% of what you actively interact with in some way, and then about 90% of what you teach.

    23. LF

      Mm-hmm.

    24. GS

      This is a thing I heard. Again, those numbers might be meaningless, but they ring true, don't they?

    25. LF

      (laughs)

    26. GS

      Right? I'm willing to say I, I learn nine times better if I'm teaching-

    27. LF

      Yes. The, the idea.

    28. GS

      ... something than reading. That might even be a low ball.

    29. LF

      Yeah.

    30. GS

      Right? Um, so, so doing something to teach or to like actively try to explain things is huge for consolidating the knowledge.

Episode duration: 1:02:45

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