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

1. 0:00 – 15:00

   The following is a…

   1. LFLex Fridman0:00

      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 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. GSGrant Sanderson2:02

      (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. LFLex Fridman2:19

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

   4. GSGrant Sanderson2:21

      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. LFLex Fridman2:52

      Mm-hmm.

   6. GSGrant Sanderson2:52

      ... 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. LFLex Fridman3:29

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

   8. GSGrant Sanderson3:33

      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.

   9. LFLex Fridman3:48

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

   10. GSGrant Sanderson3:53

       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-

   11. LFLex Fridman3:58

       Yeah.

   12. GSGrant Sanderson3:58

       ... 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-

   13. LFLex Fridman4:57

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

   14. GSGrant Sanderson5:02

       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-

   15. LFLex Fridman5:13

       Right.

   16. GSGrant Sanderson5:13

       ... 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.

   17. LFLex Fridman6:15

       Mm-hmm.

   18. GSGrant Sanderson6:15

       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?

   19. LFLex Fridman6:50

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

   20. GSGrant Sanderson6:55

       No-

   21. LFLex Fridman6:55

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

   22. GSGrant Sanderson6:58

       It's not mysterious. I think-

   23. LFLex Fridman7:00

       (laughs)

   24. GSGrant Sanderson7:00

       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.

   25. LFLex Fridman7:28

       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.

   26. GSGrant Sanderson7:40

       Well, I think-

   27. LFLex Fridman7:41

       I mean, in a sense.

   28. GSGrant Sanderson7:41

       Yeah, well-

   29. LFLex Fridman7:42

       Like, historically speaking.

   30. GSGrant Sanderson7:43

       That, that's true. You've got... So, like, the number e is significant.
2. 15:00 – 30:00

   Well, I think of…

   1. LFLex Fridman15:00

      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. GSGrant Sanderson15:11

      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. LFLex Fridman15:22

      Yeah.

   4. GSGrant Sanderson15:22

      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. LFLex Fridman16:28

      Mm-hmm.

   6. GSGrant Sanderson16:28

      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. LFLex Fridman16:45

      (laughs)

   8. GSGrant Sanderson16:45

      That's how he would think about it.

   9. LFLex Fridman16:46

      Mm-hmm.

   10. GSGrant Sanderson16:47

       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.

   11. LFLex Fridman17:24

       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-

   12. GSGrant Sanderson17:54

       Mm-hmm.

   13. LFLex Fridman17:54

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

   14. GSGrant Sanderson18:01

       Yeah, I have no idea. (laughs)

   15. LFLex Fridman18:03

       (laughs)

   16. GSGrant Sanderson18:03

       Like, why is it simple?

   17. LFLex Fridman18:04

       I-

   18. GSGrant Sanderson18:05

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

   19. LFLex Fridman18:10

       Hm.

   20. GSGrant Sanderson18:10

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

   21. LFLex Fridman18:12

       Right, yeah.

   22. GSGrant Sanderson18:13

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

   23. LFLex Fridman18:16

       Yeah.

   24. GSGrant Sanderson18:16

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

   25. LFLex Fridman18:20

       That's true. Right.

   26. GSGrant Sanderson18:21

       ... 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-

   27. LFLex Fridman18:52

       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.

   28. GSGrant Sanderson19:03

       Hmm.

   29. LFLex Fridman19:03

       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-

   30. GSGrant Sanderson19:40

       Hmm.
3. 30:00 – 45:00

   (laughs) …

   1. LFLex Fridman30:00

      into, um-

   2. GSGrant Sanderson30:01

      (laughs)

   3. LFLex Fridman30:01

      ... ideas that are, uh, surreal and difficult, and, and take us into areas that are disconnected from reality in, in a way that we could never get back.

   4. GSGrant Sanderson30:11

      What if instead of calling these abstract, how, how different would it be in your mind if we called them general? And the phenomena that you're describing is overgeneralization, when you try to-

   5. LFLex Fridman30:19

      Overgeneralization, yeah.

   6. GSGrant Sanderson30:20

      ... have a concept or an idea that's so general as to apply to nothing in particular in a useful way.

   7. LFLex Fridman30:26

      (sighs)

   8. GSGrant Sanderson30:26

      Does that map to what you're thinking of when you think of-

   9. LFLex Fridman30:28

      No. First of all, I'm, I'm playing little just for the fun of it-

   10. GSGrant Sanderson30:31

       Yeah, I know.

   11. LFLex Fridman30:31

       ... the devil's advocate. And, uh, I, I think our cognition, our mind, is unable to visualize. So, you do some incredible work with visualization and video. I think-... infinity is very difficult to visualize for our mind. We can delude ourselves-

   12. GSGrant Sanderson30:50

       Mm-hmm.

   13. LFLex Fridman30:50

       ... into thinking we can visualize it.

   14. GSGrant Sanderson30:52

       Mm-hmm.

   15. LFLex Fridman30:53

       But w- we can't. I don't... I mean, it's... I, I don't... I would venture to say it's very difficult. And so there's some concepts in mathematics, like maybe multiple dimensions, we could sort of talk about it-

   16. GSGrant Sanderson31:03

       Mm-hmm.

   17. LFLex Fridman31:03

       ... that are impossible for us to truly in- intuit, like... And it just feels dangerous to me to use these as part of our toolbox of abstractions.

   18. GSGrant Sanderson31:16

       On behalf of your listeners, I almost fear we're getting too philosophical, right?

   19. LFLex Fridman31:19

       No. Heck, heck no. Heck no.

   20. GSGrant Sanderson31:21

       (laughs) But I, I think, to that point, for any particular idea-

   21. LFLex Fridman31:26

       Yeah.

   22. GSGrant Sanderson31:26

       ... like this, there's multiple angles of attack. I think the... when we do visualize infinity, what we're actually doi-... you know, you write, "Dot, dot, dot," right?

   23. LFLex Fridman31:34

       Yeah.

   24. GSGrant Sanderson31:34

       "One, two, three, four, dot, dot, dot," right? That's... those are symbols on the page that are insinuating a certain infinity. Um, what you're capturing, with a little bit of design there, is the "I can always add one more" property, right?

   25. LFLex Fridman31:49

       Yes.

   26. GSGrant Sanderson31:49

       I think, um, I'm, I'm just as uncomfortable with you are if you try to, um, concretize it so much that you have a bag of infinitely many things, that I actually think of, "No, not one, two, three, four, dot, dot, dot. One, two, three, four, five, six, seven, eight." I try to get them all-

   27. LFLex Fridman32:04

       (laughs)

   28. GSGrant Sanderson32:04

       ... 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."

   29. LFLex Fridman32:27

       (laughs)

   30. GSGrant Sanderson32:27

       "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. 45:00 – 1:00:00

   So let's... I think…

   1. GSGrant Sanderson45:00

      that it's not arbitrary play with construction paper.

   2. LFLex Fridman45:04

      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.

   3. GSGrant Sanderson45:12

      Mm-hmm.

   4. LFLex Fridman45:12

      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)

   5. GSGrant Sanderson45:22

      Nail on the head. Absolutely.

   6. LFLex Fridman45:24

      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?

   7. GSGrant Sanderson45:58

      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.

   8. LFLex Fridman46:29

      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.

   9. GSGrant Sanderson46:42

      Um-

   10. LFLex Fridman46:43

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

   11. GSGrant Sanderson46:47

       (laughs)

   12. LFLex Fridman46:47

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

   13. GSGrant Sanderson46:53

       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?

   14. LFLex Fridman47:10

       Ah.

   15. GSGrant Sanderson47:10

       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?

   16. LFLex Fridman47:34

       Mm-hmm.

   17. GSGrant Sanderson47:34

       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-

   18. LFLex Fridman48:04

       Yeah.

   19. GSGrant Sanderson48:04

       ... 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-

   20. LFLex Fridman48:14

       Mm-hmm.

   21. GSGrant Sanderson48:14

       ... 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.

   22. LFLex Fridman48:31

       Mm-hmm.

   23. GSGrant Sanderson48:32

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

   24. LFLex Fridman48:34

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

   25. GSGrant Sanderson48:39

       Mm-hmm.

   26. LFLex Fridman48:39

       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-

   27. GSGrant Sanderson49:00

       Oh, thank you.

   28. LFLex Fridman49:00

       ... 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-

   29. GSGrant Sanderson49:42

       Statistics?

   30. LFLex Fridman49:42

       ... 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-
5. 1:00:00 – 1:02:39

   (laughs) Well, as a…

   1. GSGrant Sanderson1:00:00

      thing. And then on the gondola ride down, we decided to just jam a little bit and it was just like, I don't know, the, the gondola sort of over, came over a mountain and you saw the, the city lights and we're just like jamming, like playing some music. I wouldn't describe that as transformative. I don't know why, but that popped into my mind as a moment of, uh, in a way that wasn't associated with people I love, but more with like a thing I was doing, um, something that was just, it was just happy and it was just like a, a great moment. Um, I, I don't think I can give you anything deeper than that though.

   2. LFLex Fridman1:00:31

      (laughs) Well, as a musician myself, I'd love to see, uh, th- as you mentioned before, music enter back into your work, uh-

   3. GSGrant Sanderson1:00:39

      Mm-hmm.

   4. LFLex Fridman1:00:39

      ... back into your creative work. I'd love to see that. I'm certainly allowing it to enter back into mine and it's, it's, uh, it's a beautiful thing for a mathematician, for scientists to allow music to enter their work. I think only good things can happen.

   5. GSGrant Sanderson1:00:53

      All right. I'll, I'll try to promise you a music video by 2020. (laughs)

   6. LFLex Fridman1:00:57

      Twen- b- by 2020?

   7. GSGrant Sanderson1:00:57

      By the end of 2020.

   8. LFLex Fridman1:00:58

      Okay. All right, good.

   9. GSGrant Sanderson1:00:59

      I'll give, give myself a longer window. (laughs)

   10. LFLex Fridman1:01:01

       All right. Maybe we can, uh, like collaborate on a band type situation. What instruments do you play?

   11. GSGrant Sanderson1:01:07

       The main instrument I play is violin, but I also love to dabble around on like guitar and piano.

   12. LFLex Fridman1:01:11

       Beautiful. Me too, guitar and piano. So in, in A Mathematician's Lament, Paul Lockhart writes, "The first thing to understand is that mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such." So I think I speak for millions of people, myself included, in saying thank you for revealing to us the art of mathematics. So thank you for everything you do and thanks for talking today.

   13. GSGrant Sanderson1:01:42

       Wow. Thanks for saying that and thanks for having me on.

   14. LFLex Fridman1:01:45

       Thanks for listening to this conversation with Grant Sanderson, and thank you to our presenting sponsor, Cash App. Download it, use code LEXPODCAST, you'll get $10 and $10 will go to FIRST, a STEM education nonprofit that inspires hundreds of thousands of young minds to become future leaders and innovators. If you enjoy this podcast, subscribe on YouTube, give it five stars on Apple Podcasts, support it on Patreon, or connect with me on Twitter. And now let me leave you with some words of wisdom from one of Grant's and my favorite people, Richard Feynman. "Nobody ever figures out what this life is all about, and it doesn't matter. Explore the world. Nearly everything is really interesting if you go into it deeply enough." Thank you for listening and hope to see you next time.

Episode duration: 1:02:45