Grant Sanderson: 3Blue1Brown and the Beauty of Mathematics

00:00:00.000 | The following is a conversation with Grant Sanderson.

00:00:03.080 | He's a math educator and creator of 3Blue1Brown,

00:00:06.640 | a popular YouTube channel

00:00:08.000 | that uses programmatically animated visualizations

00:00:11.020 | to explain concepts in linear algebra, calculus,

00:00:14.280 | and other fields of mathematics.

00:00:17.000 | This is the Artificial Intelligence Podcast.

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00:00:28.360 | at Lex Friedman, spelled F-R-I-D-M-A-N.

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00:01:51.440 | And now, here's my conversation with Grant Sanderson.

00:01:56.240 | If there's intelligent life out there in the universe,

00:01:59.120 | do you think their mathematics is different than ours?

00:02:02.240 | (Grant laughs)

00:02:03.280 | - Jumping right in.

00:02:04.320 | I think it's probably very different.

00:02:08.340 | There's an obvious sense the notation is different, right?

00:02:11.220 | I think notation can guide what the math itself is.

00:02:13.860 | I think it has everything to do

00:02:16.080 | with the form of their existence, right?

00:02:19.200 | - Do you think they have basic arithmetics?

00:02:20.840 | Sorry to interrupt.

00:02:21.680 | - Yeah, so I think they count, right?

00:02:23.200 | I think notions like one, two, three, the natural numbers,

00:02:25.320 | that's extremely, well, natural.

00:02:27.160 | That's almost why we put that name to it.

00:02:30.400 | As soon as you can count,

00:02:31.720 | you have a notion of repetition, right?

00:02:33.980 | 'Cause you can count by two, two times or three times.

00:02:37.280 | And so you have this notion of repeating

00:02:39.760 | the idea of counting,

00:02:40.640 | which brings you addition and multiplication.

00:02:43.760 | I think the way that we extend to the real numbers,

00:02:47.480 | there's a little bit of choice in that.

00:02:49.240 | So there's this funny number system

00:02:50.760 | called the surreal numbers

00:02:52.160 | that it captures the idea of continuity.

00:02:55.640 | It's a distinct mathematical object.

00:02:57.560 | You could very well model the universe

00:03:00.400 | and motion of planets with that

00:03:01.980 | as the back end of your math, right?

00:03:04.800 | And you still have kind of the same interface

00:03:06.480 | with the front end of what physical laws you're trying to,

00:03:10.560 | or what physical phenomena

00:03:12.080 | you're trying to describe with math.

00:03:13.760 | And I wonder if the little glimpses that we have

00:03:15.840 | of what choices you can make along the way

00:03:17.660 | based on what different mathematicians

00:03:19.280 | have brought to the table

00:03:20.400 | is just scratching the surface

00:03:22.260 | of what the different possibilities are

00:03:24.520 | if you have a completely different mode of thought, right?

00:03:27.520 | Or a mode of interacting with the universe.

00:03:29.640 | - And you think notation is a key part of the journey

00:03:32.500 | that we've taken through math.

00:03:33.840 | - I think that's the most salient part

00:03:35.080 | that you'd notice at first.

00:03:36.380 | I think the mode of thought is gonna influence things

00:03:38.840 | more than like the notation itself.

00:03:40.360 | But notation actually carries a lot of weight

00:03:42.680 | when it comes to how we think about things,

00:03:44.580 | more so than we usually give it credit for,

00:03:47.060 | I would be comfortable saying.

00:03:48.880 | - Do you have a favorite or least favorite piece of notation

00:03:52.000 | in terms of its effectiveness?

00:03:53.200 | - Yeah, yeah, well, so least favorite,

00:03:54.800 | one that I've been thinking a lot about

00:03:56.120 | that will be a video, I don't know when, but we'll see.

00:03:58.880 | The number e.

00:04:00.800 | We write the function e to the x,

00:04:02.380 | this general exponential function

00:04:04.080 | with the notation e to the x.

00:04:06.200 | That implies you should think about a particular number,

00:04:08.320 | this constant of nature,

00:04:09.360 | and you repeatedly multiply it by itself.

00:04:11.680 | And then you say, oh, what's e to the square root of two?

00:04:14.040 | And you're like, oh, well,

00:04:14.880 | we've extended the idea of repeated multiplication.

00:04:17.000 | That's all nice, that's all nice and well.

00:04:19.440 | But very famously, you have like e to the pi i,

00:04:22.600 | and you're like, well,

00:04:23.440 | we're extending the idea of repeated multiplication

00:04:25.440 | into the complex numbers.

00:04:27.040 | Yeah, you can think about it that way.

00:04:28.720 | In reality, I think that it's just the wrong way

00:04:31.800 | of notationally representing this function,

00:04:34.680 | the exponential function,

00:04:36.160 | which itself could be represented

00:04:37.780 | a number of different ways.

00:04:38.720 | You can think about it in terms of the problem it solves,

00:04:41.260 | a certain very simple differential equation,

00:04:43.200 | which often yields way more insight

00:04:45.560 | than trying to twist the idea of repeated multiplication,

00:04:48.760 | like take its arm and put it behind its back

00:04:50.720 | and throw it on the desk and be like,

00:04:51.880 | you will apply to complex numbers, right?

00:04:53.600 | That's not, I don't think that's pedagogically helpful.

00:04:56.400 | - So the repeated multiplication

00:04:58.760 | is actually missing the main point,

00:05:00.400 | the power of e to the x.

00:05:02.960 | - I mean, what it addresses is things where the rate

00:05:05.880 | at which something changes depends on its own value,

00:05:10.120 | but more specifically, it depends on it linearly.

00:05:12.460 | So for example, if you have like a population

00:05:15.060 | that's growing and the rate at which it grows

00:05:16.640 | depends on how many members of the population

00:05:18.400 | are already there,

00:05:19.240 | it looks like this nice exponential curve.

00:05:21.240 | It makes sense to talk about repeated multiplication

00:05:23.260 | 'cause you say how much is there after one year,

00:05:25.120 | two years, three years, you're multiplying by something.

00:05:27.380 | The relationship can be a little bit different sometimes

00:05:29.440 | where let's say you've got a ball on a string,

00:05:33.920 | like a game of tetherball going around a rope, right?

00:05:37.480 | And you say its velocity

00:05:39.640 | is always perpendicular to its position.

00:05:42.180 | That's another way of describing its rate of change

00:05:44.100 | as being related to where it is,

00:05:47.100 | but it's a different operation.

00:05:48.200 | You're not scaling it.

00:05:49.040 | It's a rotation.

00:05:49.880 | It's this 90 degree rotation.

00:05:51.400 | That's what the whole idea of like complex exponentiation

00:05:54.600 | is trying to capture,

00:05:55.740 | but it's obfuscated in the notation

00:05:57.660 | when what it's actually saying,

00:05:59.080 | like if you really parse something like e to the pi i,

00:06:01.440 | what it's saying is choose an origin,

00:06:03.880 | always move perpendicular to the vector

00:06:06.500 | from that origin to you, okay?

00:06:08.320 | Then when you walk pi times that radius,

00:06:12.480 | you'll be halfway around.

00:06:14.180 | Like that's what it's saying.

00:06:15.860 | It's kind of the, you turn 90 degrees and you walk,

00:06:18.620 | you'll be going in a circle.

00:06:19.940 | That's the phenomenon that it's describing,

00:06:22.460 | but trying to twist the idea

00:06:24.380 | of repeatedly multiplying a constant into that.

00:06:26.780 | Like I can't even think of the number of human hours

00:06:30.740 | of like intelligent human hours that have been wasted

00:06:33.420 | trying to parse that to their own liking and desire

00:06:36.820 | among like scientists or electrical engineers

00:06:39.500 | or students everywhere,

00:06:40.640 | which if the notation were a little different

00:06:42.740 | or the way that this whole function

00:06:44.100 | was introduced from the get-go were framed differently,

00:06:47.620 | I think could have been avoided, right?

00:06:49.980 | - And you're talking about

00:06:51.040 | the most beautiful equation in mathematics,

00:06:53.460 | but it's still pretty mysterious, isn't it?

00:06:55.080 | Like you're making it seem like it's a notation.

00:06:57.580 | - It's not mysterious.

00:06:59.780 | I think the notation makes it mysterious.

00:07:01.940 | I don't think it's,

00:07:02.940 | I think the fact that it represents, it's pretty.

00:07:05.140 | It's not like the most beautiful thing in the world,

00:07:06.660 | but it's quite pretty.

00:07:07.940 | The idea that if you take the linear operation

00:07:10.620 | of a 90 degree rotation

00:07:12.500 | and then you do this general exponentiation thing to it,

00:07:15.720 | that what you get are all the other kinds of rotation,

00:07:18.420 | which is basically to say,

00:07:20.420 | if your velocity vector is perpendicular

00:07:22.820 | to your position vector, you walk in a circle, that's pretty.

00:07:26.300 | It's not the most beautiful thing in the world,

00:07:27.780 | but it's quite pretty.

00:07:28.740 | - The beauty of it, I think,

00:07:29.740 | comes from perhaps the awkwardness of the notation

00:07:33.060 | somehow still nevertheless coming together nicely.

00:07:35.460 | 'Cause you have like several disciplines coming together

00:07:38.780 | in a single equation.

00:07:40.180 | - Well, I think-- - In a sense,

00:07:42.220 | historically speaking.

00:07:43.540 | - That's true.

00:07:44.380 | So like the number E is significant.

00:07:45.980 | It shows up in probability all the time.

00:07:47.900 | It shows up in calculus all the time.

00:07:49.380 | It is significant.

00:07:50.380 | You're seeing it sort of mated with pi,

00:07:52.300 | this geometric constant,

00:07:53.580 | and I, like the imaginary number and such.

00:07:55.820 | I think what's really happening there

00:07:57.660 | is the way that E shows up

00:08:00.320 | is when you have things like exponential growth and decay.

00:08:03.200 | It's when this relation

00:08:04.960 | that something's rate of change has to itself

00:08:07.120 | is a simple scaling, right?

00:08:08.980 | A similar law also describes circular motion.

00:08:14.080 | Because we have bad notation,

00:08:16.360 | we use the residue of how it shows up

00:08:19.120 | in the context of self-reinforcing growth,

00:08:21.080 | like a population growing or compound interest.

00:08:23.680 | The constant associated with that

00:08:25.280 | is awkwardly placed into the context

00:08:27.640 | of how rotation comes about

00:08:29.720 | because they both come from pretty similar equations.

00:08:32.280 | And so what we see is the E and the pi juxtaposed

00:08:34.840 | a little bit closer than they would be

00:08:38.280 | with a purely natural representation, I would think.

00:08:40.960 | Here's how I would describe the relation between the two.

00:08:43.360 | You've got a very important function we might call exp

00:08:45.800 | that's like the exponential function.

00:08:47.720 | When you plug in one,

00:08:49.300 | you get this nice constant called E

00:08:50.920 | that shows up in like probability and calculus.

00:08:53.300 | If you try to move in the imaginary direction,

00:08:55.320 | it's periodic and the period is tau.

00:08:58.380 | So those are these two constants

00:08:59.580 | associated with the same central function.

00:09:02.140 | But for kind of unrelated reasons, right?

00:09:04.720 | And not unrelated, but like orthogonal reasons.

00:09:07.320 | One of them is what happens

00:09:08.320 | when you're moving in the real direction.

00:09:09.640 | One's what happens when you move in the imaginary direction.

00:09:12.520 | And like, yeah, those are related.

00:09:14.160 | They're not as related as the famous equation

00:09:17.120 | seems to think it is.

00:09:18.520 | It's sort of putting all of the children in one bed

00:09:20.520 | and they'd kind of like to sleep in separate beds

00:09:22.440 | if they had the choice.

00:09:23.280 | But you see them all there and, you know,

00:09:25.840 | there is a family resemblance, but it's not that close.

00:09:29.340 | - So actually thinking of it as a function

00:09:31.740 | is the better idea.

00:09:34.720 | And that's a notational idea.

00:09:36.320 | - And yeah, and like, here's the thing.

00:09:38.440 | The constant E sort of stands

00:09:41.240 | as this numerical representative of calculus, right?

00:09:43.760 | - Yeah.

00:09:44.720 | - Calculus is the like study of change.

00:09:47.560 | So at the very least,

00:09:48.400 | there's a little cognitive dissonance

00:09:49.880 | using a constant to represent the science of change.

00:09:53.240 | - Never thought of it that way.

00:09:54.160 | Yeah.

00:09:55.000 | (laughs)

00:09:55.820 | - Right?

00:09:56.660 | - Yeah.

00:09:57.640 | - It makes sense why the notation came about that way.

00:09:59.960 | - Yes.

00:10:00.800 | - It's the first way that we saw it

00:10:02.160 | in the context of things like population growth

00:10:03.900 | or compound interest.

00:10:04.820 | It is nicer to think about as repeated multiplication.

00:10:07.100 | That's definitely nicer,

00:10:08.720 | but it's more that that's the first application

00:10:11.060 | of what turned out to be a much more general function

00:10:13.940 | that maybe the intelligent life,

00:10:15.260 | your initial question asked about,

00:10:17.200 | would have come to recognize

00:10:18.580 | as being much more significant than the single use case,

00:10:21.180 | which lends itself to repeated multiplication notation.

00:10:24.140 | - Let me jump back for a second to aliens

00:10:28.560 | and the nature of our universe.

00:10:30.100 | Okay.

00:10:30.940 | Do you think math is discovered or invented?

00:10:35.140 | So we're talking about the different kind of mathematics

00:10:37.660 | that could be developed by the alien species.

00:10:40.660 | The implied question is,

00:10:42.440 | yeah, is math discovered or invented?

00:10:46.260 | Is, you know, is fundamentally everybody going to discover

00:10:49.680 | the same principles of mathematics?

00:10:53.180 | - So the way I think about it,

00:10:54.180 | and everyone thinks about it differently,

00:10:55.420 | but here's my take.

00:10:56.460 | I think there's a cycle at play

00:10:57.960 | where you discover things about the universe

00:11:00.980 | that tell you what math will be useful.

00:11:03.980 | And that math itself is invented in a sense,

00:11:08.060 | but of all the possible maths that you could have invented,

00:11:11.420 | it's discoveries about the world

00:11:12.740 | that tell you which ones are.

00:11:14.220 | So like a good example here is the Pythagorean theorem.

00:11:17.740 | When you look at this,

00:11:18.580 | do you think of that as a definition

00:11:19.760 | or do you think of that as a discovery?

00:11:21.960 | - From the historical perspective, right,

00:11:23.540 | it's a discovery because they were,

00:11:25.120 | but that's probably because they were using physical object

00:11:29.640 | to build their intuition.

00:11:31.180 | And from that intuition came the mathematics.

00:11:34.520 | So the mathematics was in some abstract world

00:11:37.360 | detached from the physics.

00:11:39.080 | But I think more and more math has become detached from,

00:11:43.700 | you know, when you even look at modern physics,

00:11:46.640 | from string theory to even general relativity,

00:11:49.560 | I mean, all math behind the 20th and 21st century physics

00:11:53.560 | kind of takes a brisk walk outside

00:11:57.200 | of what our mind can actually even comprehend.

00:12:00.480 | In multiple dimensions, for example,

00:12:02.360 | anything beyond three dimensions, maybe four dimensions.

00:12:05.880 | - No, no, no, higher dimensions

00:12:07.260 | can be highly, highly applicable.

00:12:08.720 | I think this is a common misinterpretation

00:12:11.220 | that if you're asking questions

00:12:13.320 | about like a five-dimensional manifold,

00:12:15.240 | that the only way that that's connected

00:12:16.680 | to the physical world is if the physical world

00:12:19.100 | is itself a five-dimensional manifold or includes them.

00:12:22.960 | - Well, wait, wait, wait a minute, wait a minute.

00:12:25.240 | You're telling me you can imagine

00:12:27.200 | a five-dimensional manifold?

00:12:31.240 | - No, no, that's not what I said.

00:12:33.360 | I would make the claim that it is useful

00:12:34.980 | to a three-dimensional physical universe

00:12:37.220 | despite itself not being three-dimensional.

00:12:39.400 | - So it's useful, meaning to even understand

00:12:41.160 | a three-dimensional world, it'd be useful

00:12:43.240 | to have five-dimensional manifolds.

00:12:44.880 | - Yes, absolutely, because of state spaces.

00:12:47.160 | - But you're saying in some deep way for us humans,

00:12:50.480 | it does always come back to that three-dimensional world

00:12:54.040 | for the usefulness of the three-dimensional world.

00:12:56.600 | And therefore it starts with a discovery,

00:12:59.960 | but then we invent the mathematics

00:13:02.080 | that helps us make sense of the discovery in a sense.

00:13:06.280 | - Yes, I mean, just to jump off

00:13:07.900 | of the Pythagorean theorem example,

00:13:09.840 | it feels like a discovery.

00:13:11.220 | You've got these beautiful geometric proofs

00:13:12.920 | where you've got squares and you're modifying the areas.

00:13:14.640 | It feels like a discovery.

00:13:16.760 | If you look at how we formalize the idea of 2D space

00:13:19.640 | as being R2, all pairs of real numbers,

00:13:23.120 | and how we define a metric on it and define distance,

00:13:25.760 | you're like, "Hang on a second, we've defined distance

00:13:28.040 | "so that the Pythagorean theorem is true,

00:13:30.040 | "so that suddenly it doesn't feel that great."

00:13:32.480 | But I think what's going on is the thing that informed us

00:13:35.420 | what metric to put on R2,

00:13:38.040 | to put on our abstract representation of 2D space

00:13:41.400 | came from physical observations.

00:13:43.280 | And the thing is there's other metrics

00:13:44.640 | you could have put on it.

00:13:45.480 | We could have consistent math

00:13:47.160 | with other notions of distance.

00:13:49.160 | It's just that those pieces of math

00:13:50.840 | wouldn't be applicable to the physical world that we study

00:13:53.640 | 'cause they're not the ones

00:13:54.560 | where the Pythagorean theorem holds.

00:13:56.160 | So we have a discovery, a genuine bona fide discovery

00:13:59.040 | that informed the invention,

00:14:00.480 | the invention of an abstract representation of 2D space

00:14:03.720 | that we call R2 and things like that.

00:14:06.240 | And then from there, you just study R2 as an abstract thing

00:14:09.720 | that brings about more ideas and inventions and mysteries,

00:14:12.520 | which themselves might yield discoveries.

00:14:14.420 | Those discoveries might give you insight

00:14:17.000 | as to what else would be useful to invent.

00:14:19.360 | And it kind of feeds on itself that way.

00:14:20.960 | That's how I think about it.

00:14:22.160 | So it's not an either or.

00:14:24.100 | It's not that math is one of these or it's one of the others.

00:14:26.760 | At different times, it's playing a different role.

00:14:29.160 | - So then let me ask the Richard Feynman question

00:14:33.080 | then along that thread.

00:14:36.200 | Is what do you think is the difference

00:14:37.360 | between physics and math?

00:14:40.320 | There's a giant overlap.

00:14:42.800 | There's a kind of intuition

00:14:45.040 | that physicists have about the world

00:14:46.920 | that's perhaps outside of mathematics.

00:14:49.040 | It's this mysterious art that they seem to possess,

00:14:52.720 | we humans generally possess.

00:14:54.240 | And then there's the beautiful rigor of mathematics

00:14:58.080 | that allows you to, I mean, just like as we were saying,

00:15:01.840 | invent frameworks of understanding our physical world.

00:15:07.880 | So what do you think is the difference there

00:15:10.680 | and how big is it?

00:15:11.880 | - Well, I think of math as being the study

00:15:13.480 | of like abstractions over patterns

00:15:15.720 | and pure patterns in logic.

00:15:17.360 | And then physics is obviously grounded

00:15:19.280 | in a desire to understand the world that we live in.

00:15:22.680 | I think you're gonna get very different answers

00:15:24.040 | when you talk to different mathematicians

00:15:25.560 | 'cause there's a wide diversity in types of mathematicians.

00:15:28.120 | There are some who are motivated very much by pure puzzles.

00:15:31.480 | They might be turned on by things like combinatorics.

00:15:34.040 | And they just love the idea of building up a set

00:15:36.680 | of problem solving tools, applying to pure patterns.

00:15:40.880 | There are some who are very physically motivated,

00:15:43.320 | who try to invent new math or discover math in veins

00:15:48.320 | that they know will have applications to physics

00:15:50.960 | or sometimes computer science.

00:15:52.280 | And that's what drives them.

00:15:53.720 | Like chaos theory is a good example of something

00:15:55.560 | that's pure math, that's purely mathematical,

00:15:57.680 | a lot of the statements being made.

00:15:59.160 | But it's heavily motivated by specific applications

00:16:02.880 | to largely physics.

00:16:05.360 | And then you have a type of mathematician

00:16:06.680 | who just loves abstraction.

00:16:08.520 | They just love pulling into the more and more abstract

00:16:10.520 | things, the things that feel powerful.

00:16:12.120 | These are the ones that initially invented topology

00:16:15.200 | and then later on get really into category theory

00:16:17.520 | and go on about infinite categories and whatnot.

00:16:20.400 | These are the ones that love to have a system

00:16:23.440 | that can describe truths about as many things as possible.

00:16:27.160 | People from those three different veins of motivation

00:16:31.280 | into math are gonna give you very different answers

00:16:32.880 | about what the relation at play here is.

00:16:34.720 | 'Cause someone like Vladimir Arnold,

00:16:37.640 | who has written a lot of great books,

00:16:40.520 | many about like differential equations and such,

00:16:42.480 | he would say, "Math is a branch of physics."

00:16:45.640 | That's how he would think about it.

00:16:47.120 | And of course he was studying like differential equations

00:16:49.120 | related things, because that is the motivator

00:16:51.040 | behind the study of PDEs and things like that.

00:16:53.760 | But you'll have others who,

00:16:56.480 | like especially the category theorists,

00:16:58.240 | who aren't really thinking about physics necessarily.

00:17:01.360 | It's all about abstraction and the power of generality.

00:17:04.520 | And it's more of a happy coincidence

00:17:06.400 | that that ends up being useful

00:17:08.320 | for understanding the world we live in.

00:17:10.880 | And then you can get into like, why is that the case?

00:17:12.840 | It's sort of surprising that that which is

00:17:16.200 | about pure puzzles and abstraction

00:17:17.800 | also happens to describe the very fundamentals

00:17:21.040 | of quarks and everything else.

00:17:23.120 | - So why do you think the fundamentals of quarks

00:17:27.840 | and the nature of reality is so compressible

00:17:33.200 | into clean, beautiful equations

00:17:35.360 | that are for the most part simple, relatively speaking?

00:17:39.280 | A lot simpler than they could be.

00:17:41.680 | So you have, we mentioned somebody like Stephen Wolfram

00:17:45.360 | who thinks that sort of there's incredibly simple rules

00:17:50.360 | underlying our reality,

00:17:51.880 | but it can create arbitrary complexity.

00:17:54.920 | But there is simple equations.

00:17:56.720 | I'm asking a million questions

00:17:59.200 | that nobody knows the answer to.

00:18:00.640 | - Yeah, I have no idea.

00:18:02.040 | (laughing)

00:18:03.200 | Why is it simple?

00:18:04.800 | - It could be the case that

00:18:07.080 | there's like a filtration at play.

00:18:08.480 | The only things that physicists find interesting

00:18:10.600 | are the ones that are simple enough

00:18:11.680 | they could describe it mathematically.

00:18:13.280 | But as soon as it's a sufficiently complex system,

00:18:15.120 | like, oh, that's outside the realm of physics.

00:18:16.920 | That's biology or whatever have you.

00:18:19.320 | And of course-- - That's true.

00:18:21.600 | - Maybe there's something where it's like,

00:18:22.680 | of course there will always be some thing that is simple

00:18:26.520 | when you wash away the non-important parts

00:18:31.400 | of whatever it is that you're studying.

00:18:33.400 | Just from like an information theory standpoint,

00:18:35.160 | there might be some like,

00:18:36.640 | you get to the lowest information component of it.

00:18:39.440 | But I don't know.

00:18:40.280 | Maybe I'm just having a really hard time conceiving

00:18:41.960 | of what it would even mean for the fundamental laws

00:18:45.200 | to be intrinsically complicated.

00:18:47.380 | Like some set of equations

00:18:50.600 | that you can't decouple from each other.

00:18:52.520 | - Well, no, it could be that sort of we take for granted

00:18:56.760 | that the laws of physics, for example,

00:18:59.960 | are for the most part the same everywhere.

00:19:03.480 | Or something like that, right?

00:19:05.320 | As opposed to the sort of an alternative could be

00:19:10.320 | that the rules under which the world operates

00:19:15.400 | is different everywhere.

00:19:17.240 | It's like a deeply distributed system

00:19:20.320 | where just everything is just chaos.

00:19:22.360 | Like not in a strict definition of chaos,

00:19:25.520 | but meaning like just it's impossible

00:19:28.200 | for equations to capture,

00:19:31.720 | to explicitly model the world

00:19:34.000 | as cleanly as the physical does.

00:19:36.000 | I mean, we almost take it for granted

00:19:38.040 | that we can describe,

00:19:39.080 | we can have an equation for gravity,

00:19:41.220 | for action at a distance.

00:19:42.760 | We can have equations for some of these basic ways

00:19:45.480 | the planets move and just the low level

00:19:50.480 | at the atomic scale, how the materials operate,

00:19:54.000 | at the high scale, how black holes operate.

00:19:56.940 | But it doesn't, it seems like it could be,

00:19:59.800 | there's infinite other possibilities

00:20:01.680 | where none of it could be compressible into such equations.

00:20:05.040 | It just seems beautiful.

00:20:06.560 | It's also weird, probably to the point you were making,

00:20:10.900 | that it's very pleasant that this is true for our minds.

00:20:15.120 | So it might be that our minds are biased

00:20:17.120 | to just be looking at the parts of the universe

00:20:19.640 | that are compressible.

00:20:21.720 | And then we can publish papers on

00:20:23.700 | and have nice E equals mt squared equations.

00:20:26.440 | - Right.

00:20:27.400 | I wonder, would such a world with uncompressible laws

00:20:31.720 | allow for the kind of beings that can think about

00:20:35.160 | the kind of questions that you're asking?

00:20:37.760 | - That's true.

00:20:38.600 | - Right, like an anthropic principle coming into play

00:20:40.600 | in some weird way here?

00:20:42.560 | I don't know, I don't know what I'm talking about at all.

00:20:44.760 | - Or maybe the universe is actually not so compressible,

00:20:48.000 | but the way our brain evolved,

00:20:52.560 | we're only able to perceive the compressible parts.

00:20:55.880 | I mean, we are, so this is a sort of Chomsky argument.

00:20:58.400 | We are just descendants of apes.

00:20:59.880 | We're like really limited biological systems.

00:21:03.600 | So it totally makes sense that we're really limited

00:21:06.360 | little computers, calculators,

00:21:08.600 | that are able to perceive certain kinds of things.

00:21:10.280 | And the actual world is much more complicated.

00:21:13.160 | - Well, but we can do pretty awesome things, right?

00:21:16.680 | Like we can fly spaceships.

00:21:18.320 | And we have to have some connection of reality

00:21:21.600 | to be able to take our potentially oversimplified models

00:21:25.320 | of the world, but then actually twist the world

00:21:27.800 | to our will based on it.

00:21:29.120 | So we have certain reality checks

00:21:30.480 | that physics isn't too far afield,

00:21:33.480 | simply based on what we can do.

00:21:35.480 | - Yeah, the fact that we can fly is pretty good.

00:21:37.320 | - It's great, yeah.

00:21:38.520 | It's a pretty good proof of concept

00:21:40.600 | that the laws we're working with are working well.

00:21:44.960 | - So I mentioned to the internet that I'm talking to you,

00:21:47.800 | and so the internet gave some questions.

00:21:50.240 | So I apologize for these.

00:21:51.640 | But do you think we're living in a simulation,

00:21:54.560 | that the universe is a computer,

00:21:56.960 | or the universe is a computation running on a computer?

00:22:00.040 | - It's conceivable.

00:22:02.720 | What I don't buy is, you know, you'll have the argument

00:22:05.720 | that, well, let's say that it was the case

00:22:07.920 | that you can have simulations,

00:22:09.600 | then the simulated world would itself eventually

00:22:13.400 | get to a point where it's running simulations.

00:22:15.400 | And then the second layer down would create

00:22:17.640 | a third layer down, and on and on and on.

00:22:19.480 | So probabilistically, you just throw a dart

00:22:21.640 | at one of those layers, we're probably

00:22:22.960 | in one of the simulated layers.

00:22:25.000 | I think if there's some sort of limitations

00:22:27.120 | on the information processing

00:22:28.640 | of whatever the physical world is,

00:22:31.520 | it quickly becomes the case that you have a limit

00:22:33.920 | to the layers that could exist there,

00:22:35.680 | because the resources necessary to simulate a universe

00:22:39.200 | like ours clearly is a lot,

00:22:41.760 | just in terms of the number of bits at play.

00:22:43.760 | And so then you can ask, well, what's more plausible?

00:22:46.880 | That there's an unbounded capacity of information processing

00:22:50.440 | in whatever the highest up level universe is,

00:22:53.680 | or that there's some bound to that capacity,

00:22:56.120 | which then limits the number of levels available.

00:22:58.920 | How do you place some kind of probability distribution

00:23:00.840 | on what the information capacity is?

00:23:02.600 | I have no idea.

00:23:03.760 | But I don't, like, people almost assume

00:23:06.880 | a certain uniform probability over all of those meta layers

00:23:10.240 | that could conceivably exist when,

00:23:12.560 | it's a little bit like a Pascal's wager,

00:23:15.200 | on like, you're not giving a low enough prior

00:23:17.000 | to the mere existence of that infinite set of layers.

00:23:20.880 | - Yeah, that's true.

00:23:21.720 | But it's also very difficult to contextualize the amount.

00:23:25.080 | So the amount of information processing power

00:23:28.240 | required to simulate like our universe

00:23:31.400 | seems like amazingly huge.

00:23:34.280 | - But you can always raise two to the power of that.

00:23:37.000 | - Yeah, exactly.

00:23:38.040 | - Yeah, like numbers get big.

00:23:40.520 | - And we're easily humbled

00:23:41.800 | by basically everything around us.

00:23:43.760 | So it's very difficult to kind of make sense of anything,

00:23:48.760 | actually, when you look up at the sky

00:23:51.000 | and look at the stars and the immensity of it all,

00:23:53.560 | to make sense of the smallness of us,

00:23:57.040 | the unlikeliness of everything that's on this earth

00:24:00.640 | coming to be.

00:24:02.200 | Then you could basically, anything could be,

00:24:05.000 | all laws of probability go out the window to me.

00:24:09.120 | Because I guess, because the amount of information

00:24:14.120 | under which we're operating is very low.

00:24:17.600 | We basically know nothing about the world around us,

00:24:22.160 | relatively speaking.

00:24:23.400 | And so when I think about the simulation hypothesis,

00:24:26.640 | I think it's just fun to think about it.

00:24:29.280 | But it's also, I think there is a thought experiment

00:24:31.920 | kind of interesting to think of the power of computation,

00:24:35.240 | where there are the limits of a Turing machine.

00:24:38.960 | Sort of the limits of our current computers.

00:24:41.040 | When you start to think about artificial intelligence,

00:24:44.080 | how far can we get with computers?

00:24:45.960 | And that's kind of where the simulation hypothesis

00:24:50.800 | is useful to me as a thought experiment,

00:24:52.840 | is the universe just a computer?

00:24:55.120 | Is it just a computation?

00:24:58.600 | Is all of this just a computation?

00:25:00.520 | And sort of the same kind of tools

00:25:01.880 | we apply to analyzing algorithms, can that be applied?

00:25:04.840 | You know, if we scale further and further and further,

00:25:07.360 | will the arbitrary power of those systems

00:25:09.600 | start to create some interesting aspects

00:25:12.040 | that we see in our universe?

00:25:13.880 | Or is something fundamentally different

00:25:15.960 | needs to be created?

00:25:17.500 | - Well, it's interesting that in our universe,

00:25:20.360 | it's not arbitrarily large, the power,

00:25:22.720 | that you can place limits on, for example,

00:25:24.340 | how many bits of information can be stored per unit area.

00:25:27.720 | Right, like all of the physical laws,

00:25:30.360 | you've got general relativity and quantum coming together,

00:25:32.680 | to give you a certain limit on how many bits you can store

00:25:36.600 | within a given range before it collapses into a black hole.

00:25:40.220 | Like the idea that there even exists such a limit

00:25:42.760 | is at the very least thought provoking,

00:25:44.600 | when naively you might assume,

00:25:46.960 | oh, well, you know,

00:25:47.860 | technology could always get better and better,

00:25:49.240 | we could get cleverer and cleverer,

00:25:50.920 | and you could just cram as much information as you want

00:25:54.140 | into like a small unit of space.

00:25:56.220 | That makes me think it's at least plausible

00:26:02.960 | that whatever the highest level of existence is,

00:26:07.280 | doesn't admit too many simulations,

00:26:10.440 | or ones that are at the scale of complexity

00:26:12.200 | that we're looking at.

00:26:13.400 | Obviously, it's just as conceivable that they do,

00:26:15.360 | and that there are many,

00:26:16.540 | but I guess what I'm channeling is the surprise

00:26:20.800 | that I felt upon learning that fact,

00:26:22.560 | that there are, that information is physical in this way.

00:26:25.720 | - It's that there's a finiteness to it.

00:26:27.120 | Okay, let me just even go off on that,

00:26:29.420 | from a mathematics perspective,

00:26:31.320 | and a psychology perspective.

00:26:33.920 | How do you mix,

00:26:35.020 | are you psychologically comfortable

00:26:38.180 | with the concept of infinity?

00:26:39.740 | - I think so. - Are you okay with it?

00:26:42.200 | - I'm pretty okay, yeah.

00:26:43.680 | Are you okay?

00:26:44.560 | - No, not really, it doesn't make any sense to me.

00:26:47.160 | - I don't know, like how many words,

00:26:50.040 | how many possible words do you think could exist,

00:26:52.620 | that are just like strings of letters?

00:26:55.680 | - So that's a sort of mathematical statement

00:26:58.920 | that's beautiful,

00:26:59.760 | and we use infinity in basically everything we do,

00:27:02.360 | everything we do in science, math, and engineering, yes.

00:27:06.920 | But you said exist.

00:27:09.560 | The question is, you said letters or words?

00:27:13.400 | - I said words. - Words.

00:27:14.700 | To bring words into existence, to me,

00:27:18.200 | you have to start saying them,

00:27:19.720 | or writing them, or listing them.

00:27:22.040 | - That's an instantiation.

00:27:23.240 | Okay, how many abstract words exist?

00:27:25.800 | - Well, the idea of abstract.

00:27:28.080 | The idea of abstract notions and ideas.

00:27:31.000 | - I think we should be clear on terminology.

00:27:33.120 | I mean, you think about intelligence a lot,

00:27:35.200 | like artificial intelligence.

00:27:36.640 | Would you not say that what it's doing

00:27:39.120 | is a kind of abstraction?

00:27:40.440 | That like abstraction is key to conceptualizing the universe?

00:27:45.120 | You get this raw sensory data.

00:27:47.320 | I need something that every time

00:27:48.580 | you move your face a little bit,

00:27:50.160 | and they're not pixels, but like analog of pixels

00:27:52.880 | on my retina change entirely,

00:27:55.200 | that I can still have some coherent notion of,

00:27:57.400 | this is Lex, I'm talking to Lex, right?

00:27:59.720 | What that requires is you have a disparate set

00:28:01.740 | of possible images hitting me

00:28:03.600 | that are unified in a notion of Lex, right?

00:28:07.600 | That's a kind of abstraction.

00:28:08.680 | It's a thing that could apply

00:28:09.840 | to a lot of different images that I see,

00:28:12.400 | and it represents it in a much more compressed way,

00:28:15.280 | and one that's like much more resilient to that.

00:28:17.440 | I think in the same way,

00:28:18.360 | if I'm talking about infinity as an abstraction,

00:28:21.080 | I don't mean non-physical, woo-woo,

00:28:23.820 | like ineffable or something.

00:28:26.360 | What I mean is it's something that can apply

00:28:28.360 | to a multiplicity of situations

00:28:30.160 | that share a certain common attribute,

00:28:31.660 | in the same way that the images

00:28:32.760 | of like your face on my retina

00:28:34.600 | share enough common attributes

00:28:35.820 | that I can put this single notion to it.

00:28:37.720 | Like in that way, infinity is an abstraction,

00:28:40.780 | and it's very powerful,

00:28:41.620 | and it's only through such abstractions

00:28:44.000 | that we can actually understand

00:28:45.720 | like the world and logic and things.

00:28:47.600 | And in the case of infinity, the way I think about it,

00:28:49.400 | the key entity is the property

00:28:51.800 | of always being able to add one more.

00:28:54.120 | Like no matter how many words you can list,

00:28:56.160 | you just throw an A at the end of one,

00:28:57.760 | and you have another conceivable word.

00:28:59.800 | You don't have to think of all the words at once.

00:29:01.720 | It's that property, the oh, I could always add one more,

00:29:04.880 | that gives it this nature of infiniteness,

00:29:08.240 | in the same way that there's certain

00:29:09.160 | like properties of your face

00:29:10.280 | that give it the lexness, right?

00:29:12.760 | So like infinity should be no more worrying

00:29:16.500 | than the I can always add one more sentiment.

00:29:19.780 | - That's a really elegant,

00:29:21.640 | much more elegant way than I could put it,

00:29:23.760 | so thank you for doing that

00:29:24.840 | as yet another abstraction.

00:29:26.880 | And yes, indeed, that's what our brain does,

00:29:29.480 | that's what intelligent systems do,

00:29:30.680 | that's what programming does,

00:29:31.880 | that's what science does,

00:29:32.880 | is build abstraction on top of each other.

00:29:35.800 | And yet, there is at a certain point,

00:29:38.920 | abstractions that go into the quote, woo, right?

00:29:42.840 | (both laughing)

00:29:44.360 | Sort of, and because we're now,

00:29:47.920 | it's like we built this stack of,

00:29:51.360 | you know, the only thing that's true

00:29:53.520 | is the stuff that's on the ground,

00:29:54.720 | everything else is useful for interpreting this.

00:29:57.560 | And at a certain point,

00:29:58.680 | you might start floating into ideas

00:30:01.960 | that are surreal and difficult,

00:30:04.600 | and take us into areas that are disconnected

00:30:08.120 | from reality in a way that we could never get back.

00:30:11.080 | - What if instead of calling these abstract,

00:30:13.160 | how different would it be in your mind

00:30:14.640 | if we called them general?

00:30:15.960 | And the phenomenon that you're describing

00:30:17.400 | is overgeneralization.

00:30:19.080 | When you try to-- - Generalization, yeah.

00:30:20.480 | - Have a concept or an idea that's so general

00:30:23.000 | as to apply to nothing in particular in a useful way.

00:30:25.760 | Does that map to what you're thinking of

00:30:28.000 | when you think of-- - First of all,

00:30:29.480 | I'm playing a little just for the fun of it.

00:30:31.040 | - Yeah, I know. - I'm the devil's advocate.

00:30:32.400 | And I think our cognition, our mind,

00:30:36.320 | is unable to visualize.

00:30:39.040 | So you do some incredible work with visualization and video.

00:30:42.560 | I think infinity is very difficult to visualize

00:30:46.840 | for our mind.

00:30:48.300 | We can delude ourselves into thinking we can visualize it.

00:30:52.920 | But we can't.

00:30:54.560 | I don't, I mean, I would venture to say it's very difficult.

00:30:57.760 | And so there's some concepts in mathematics,

00:31:00.520 | like maybe multiple dimensions, we could sort of talk about,

00:31:03.560 | that are impossible for us to truly intuit.

00:31:06.800 | And it just feels dangerous to me to use these

00:31:13.160 | as part of our toolbox of abstractions.

00:31:15.760 | - On behalf of your listeners,

00:31:17.720 | I almost fear we're getting too philosophical.

00:31:19.640 | - No, heck no, heck no.

00:31:22.080 | I think to that point, for any particular idea like this,

00:31:26.760 | there's multiple angles of attack.

00:31:28.800 | I think when we do visualize infinity,

00:31:32.000 | what we're actually doing, you write dot, dot, dot.

00:31:34.800 | One, two, three, four, dot, dot, dot.

00:31:37.080 | Those are symbols on the page that are insinuating

00:31:40.000 | a certain infinity.

00:31:41.740 | What you're capturing with a little bit of design there

00:31:46.000 | is the I can always add one more property.

00:31:49.520 | I think I'm just as uncomfortable with you are

00:31:52.600 | if you try to concretize it so much

00:31:56.120 | that you have a bag of infinitely many things

00:31:59.000 | that I actually think of, no, not one, two, three, four,

00:32:00.800 | dot, dot, dot, one, two, three, four, five, six, seven,

00:32:03.280 | eight, I try to get them all in my head.

00:32:05.040 | And you realize, oh, your brain would literally collapse

00:32:08.240 | into a black hole, all of that.

00:32:10.240 | And I honestly feel this with a lot of math

00:32:12.520 | that I try to read, where I don't think of myself

00:32:15.160 | as like particularly good at math.

00:32:19.160 | In some ways, I get very confused often

00:32:21.440 | when I am going through some of these texts.

00:32:23.800 | And often what I'm feeling in my head is like,

00:32:25.760 | this is just so damn abstract.

00:32:27.920 | I just can't wrap my head around it.

00:32:29.200 | I just want to put something concrete to it

00:32:31.860 | that makes me understand.

00:32:32.980 | And I think a lot of the motivation for the channel

00:32:35.620 | is channeling that sentiment of, yeah,

00:32:38.480 | a lot of the things that you're trying to read out there,

00:32:40.980 | it's just so hard to connect to anything

00:32:43.660 | that you spend an hour banging your head

00:32:45.360 | against a couple of pages and you come out

00:32:47.260 | not really knowing anything more

00:32:49.300 | other than some definitions maybe

00:32:51.840 | and a certain sense of self-defeat.

00:32:54.560 | One of the reasons I focus so much on visualizations

00:32:58.540 | is that I'm a big believer in,

00:33:01.760 | I'm sorry, I'm just really hampering on

00:33:03.120 | this idea of abstraction,

00:33:04.360 | being clear about your layers of abstraction.

00:33:07.440 | It's always tempting to start an explanation

00:33:09.800 | from the top to the bottom.

00:33:11.160 | You give the definition of a new theorem.

00:33:14.120 | You're like, this is the definition of a vector space,

00:33:16.160 | for example, that's how we'll start a course.

00:33:18.380 | These are the properties of a vector space.

00:33:20.580 | First from these properties,

00:33:22.100 | we will derive what we need in order to do the math

00:33:24.460 | of linear algebra or whatever it might be.

00:33:27.340 | I don't think that's how understanding works at all.

00:33:29.580 | I think how understanding works

00:33:30.840 | is you start at the lowest level you can get at

00:33:33.340 | where rather than thinking about a vector space,

00:33:35.460 | you might think of concrete vectors

00:33:37.120 | that are just lists of numbers

00:33:38.760 | or picturing it as like an arrow that you draw,

00:33:41.840 | which is itself like even less abstract than numbers

00:33:45.340 | 'cause you're looking at quantities,

00:33:46.360 | like the distance of the X coordinate,

00:33:48.000 | the distance of the Y coordinate.

00:33:49.000 | It's as concrete as you could possibly get.

00:33:51.080 | And it has to be if you're putting it in a visual.

00:33:53.960 | - It's an actual arrow.

00:33:56.320 | - It's an actual vector.

00:33:57.880 | You're not talking about like a quote unquote vector

00:34:00.120 | that could apply to any possible thing.

00:34:02.000 | You have to choose one if you're illustrating it.

00:34:04.480 | And I think this is the power of being in a medium

00:34:06.920 | like video, or if you're writing a textbook

00:34:09.120 | and you force yourself to put a lot of images,

00:34:11.640 | is with every image, you're making a choice.

00:34:14.440 | With each choice, you're showing a concrete example.

00:34:17.080 | With each concrete example,

00:34:18.460 | you're aiding someone's path to understanding.

00:34:20.580 | - Yeah, I'm sorry to interrupt you,

00:34:22.260 | but you just made me realize that that's exactly right.

00:34:25.500 | So the visualizations you're creating

00:34:27.860 | while you're sometimes talking about abstractions,

00:34:30.780 | the actual visualization is an explicit low-level example.

00:34:35.340 | - Yes.

00:34:36.180 | - So there's an actual, like in the code,

00:34:38.260 | you have to say what the vector is.

00:34:41.740 | What's the direction of the arrow?

00:34:43.180 | What's the magnitude?

00:34:44.020 | The, yeah, so that's, you're going,

00:34:47.860 | the visualization itself is actually going to the bottom.

00:34:51.300 | - And I think that's very important.

00:34:52.560 | I also think about this a lot in writing scripts,

00:34:54.980 | where even before you get to the visuals,

00:34:57.380 | the first instinct is to, I don't know why,

00:35:00.220 | I just always do, I say the abstract thing,

00:35:02.540 | I say the general definition, the powerful thing,

00:35:05.020 | and then I fill it in with examples later.

00:35:07.240 | Always, it will be more compelling and easier to understand

00:35:09.620 | when you flip that.

00:35:10.740 | And instead, you let someone's brain

00:35:13.500 | do the pattern recognition.

00:35:16.260 | You just show them a bunch of examples.

00:35:18.200 | The brain is going to feel a certain similarity between them.

00:35:21.060 | Then by the time you bring in the definition,

00:35:23.540 | or by the time you bring in the formula,

00:35:25.720 | it's articulating a thing that's already in the brain

00:35:28.940 | that was built off of looking at a bunch of examples

00:35:30.900 | with a certain kind of similarity.

00:35:32.860 | And what the formula does is articulate

00:35:34.660 | what that kind of similarity is,

00:35:36.580 | rather than being a high cognitive load set of symbols

00:35:41.580 | that needs to be populated with examples later on,

00:35:45.220 | assuming someone's still with you.

00:35:46.900 | - What is the most beautiful or awe-inspiring idea

00:35:51.260 | you've come across in mathematics?

00:35:53.840 | - I don't know, man.

00:35:55.180 | - Maybe it's an idea you've explored in your videos,

00:35:57.240 | maybe not.

00:35:58.360 | What just gave you pause?

00:36:01.340 | - What's the most beautiful idea?

00:36:03.380 | - Small or big.

00:36:04.460 | - So I think often the things that are most beautiful

00:36:07.380 | are the ones that you have a little bit of understanding of,

00:36:11.820 | but certainly not an entire understanding.

00:36:14.380 | It's a little bit of that mystery

00:36:15.580 | that is what makes it beautiful.

00:36:17.340 | - What was the moment of the discovery for you personally,

00:36:20.220 | almost just that leap of aha moment?

00:36:23.540 | - So something that really caught my eye,

00:36:25.260 | I remember when I was little, there were these,

00:36:27.800 | I think the series was called wooden books or something,

00:36:31.540 | these tiny little books that would have

00:36:33.660 | just a very short description of something on the left

00:36:35.540 | and then a picture on the right.

00:36:36.940 | I don't know who they're meant for,

00:36:37.900 | but maybe it's like loosely children or something like that,

00:36:40.540 | but it can't just be children

00:36:41.460 | 'cause of some of the things it was describing.

00:36:43.180 | On the last page of one of them,

00:36:45.280 | somewhere tiny in there was this little formula

00:36:47.620 | that on the left hand had a sum

00:36:49.900 | over all of the natural numbers.

00:36:51.620 | You know, it's like one over one to the S

00:36:53.760 | plus one over two to the S plus one over three to the S

00:36:56.220 | on and on to infinity.

00:36:57.700 | Then on the other side had a product over all of the primes.

00:37:01.140 | And it was a certain thing, had to do with all the primes.

00:37:03.820 | And like any good young math enthusiast,

00:37:06.460 | I had properly been indoctrinated

00:37:07.660 | with how chaotic and confusing the primes are,

00:37:09.780 | which they are.

00:37:10.900 | And seeing this equation where on one side

00:37:14.260 | you have something that's as understandable

00:37:15.900 | as you could possibly get, the counting numbers.

00:37:18.300 | And on the other side is all the prime numbers.

00:37:20.460 | It was like this, whoa, they're related like this?

00:37:23.980 | There's a simple description that includes

00:37:26.460 | all the primes getting wrapped together like this.

00:37:28.740 | This is like the Euler product for the zeta function.

00:37:32.180 | As I like later found out, the equation itself

00:37:34.860 | essentially encodes the fundamental theorem of arithmetic

00:37:37.900 | that every number can be expressed

00:37:39.500 | as a unique set of primes.

00:37:42.100 | To me still, there's, I mean, I certainly don't understand

00:37:44.700 | this equation or this function all that well.

00:37:47.260 | The more I learn about it, the prettier it is.

00:37:50.260 | The idea that you can, this is sort of what gets you

00:37:53.340 | representations of primes, not in terms of primes themselves

00:37:57.260 | but in terms of another set of numbers

00:37:59.140 | that are like the non-trivial zeros of the zeta function.

00:38:01.980 | And again, I'm very kind of in over my head

00:38:04.260 | in a lot of ways as I like try to get to understand it.

00:38:06.660 | But the more I do, it always leaves enough mystery

00:38:09.740 | that it remains very beautiful to me.

00:38:11.700 | - So whenever there's a little bit of mystery

00:38:13.580 | just outside of the understanding that,

00:38:16.820 | and by the way, the process of learning more about it,

00:38:19.700 | how does that come about?

00:38:20.620 | Just your own thought or are you reading?

00:38:23.820 | - Reading, yeah.

00:38:24.660 | - Or is the process of visualization itself

00:38:26.540 | revealing more to you?

00:38:28.140 | - Visuals help.

00:38:28.980 | I mean, in one time when I was just trying to understand

00:38:31.300 | like analytic continuation and playing around

00:38:33.740 | with visualizing complex functions,

00:38:36.260 | this is what led to a video about this function.

00:38:39.500 | It's titled something like

00:38:40.340 | Visualizing the Riemann Zeta Function.

00:38:42.380 | It's one that came about because I was programming

00:38:45.060 | and tried to see what a certain thing looked like.

00:38:47.660 | And then I looked at it and I'm like,

00:38:48.500 | "Whoa, that's elucidating."

00:38:50.620 | And then I decided to make a video about it.

00:38:53.460 | But I mean, you try to get your hands

00:38:56.580 | on as much reading as you can.

00:38:58.140 | You, in this case, I think if anyone wants

00:39:01.420 | to start to understand it,

00:39:03.020 | if they have like a math background of some,

00:39:06.100 | like they studied some in college or something like that,

00:39:08.820 | like the Princeton Companion to Math

00:39:10.260 | has a really good article on analytic number theory.

00:39:13.060 | And that itself has a whole bunch of references

00:39:15.740 | and anything has more references

00:39:17.420 | and it gives you this like tree to start pawing through.

00:39:20.180 | And like, you try to understand,

00:39:22.180 | I try to understand things visually as I go.

00:39:24.740 | That's not always possible,

00:39:26.300 | but it's very helpful when it does.

00:39:28.340 | You recognize when there's common themes,

00:39:30.100 | like in this case, cousins of the Fourier transform

00:39:34.460 | that come into play and you realize,

00:39:35.860 | oh, it's probably pretty important

00:39:36.980 | to have deep intuitions of the Fourier transform,

00:39:39.020 | even if it's not explicitly mentioned in like these texts.

00:39:42.340 | And you try to get a sense of what the common players are.

00:39:45.260 | But I'll emphasize again, like I feel very in over my head

00:39:48.820 | when I try to understand the exact relation

00:39:52.140 | between like the zeros of the Riemann zeta function

00:39:54.620 | and how they relate to the distribution of primes.

00:39:56.940 | I definitely understand it better than I did a year ago.

00:39:59.360 | I definitely understand it 1/100 as well as the experts

00:40:02.360 | on the matter do, I assume.

00:40:04.660 | But the slow path towards getting there is,

00:40:08.260 | it's fun, it's charming.

00:40:09.580 | And like to your question, very beautiful.

00:40:12.900 | - And the beauty is in the, what,

00:40:14.820 | in the journey versus the destination?

00:40:17.100 | - Well, it's that each thing doesn't feel arbitrary.

00:40:19.420 | I think that's a big part,

00:40:20.540 | is that you have these unpredictable,

00:40:23.380 | not, yeah, these very unpredictable patterns

00:40:25.900 | or these intricate properties of like a certain function.

00:40:30.460 | But at the same time, it doesn't feel like humans

00:40:32.220 | ever made an arbitrary choice

00:40:33.780 | in studying this particular thing.

00:40:35.740 | So, it feels like you're speaking to patterns themselves

00:40:39.700 | or nature itself.

00:40:41.260 | That's a big part of it.

00:40:43.100 | I think things that are too arbitrary,

00:40:45.040 | it's just hard for those to feel beautiful

00:40:46.620 | because this is sort of what the word contrived

00:40:49.780 | is meant to apply to, right?

00:40:51.660 | - And when they're not arbitrary,

00:40:55.580 | it means it could be, you can have a clean abstraction

00:41:00.580 | and intuition that allows you to comprehend it.

00:41:04.940 | - Well, to one of your first questions,

00:41:06.220 | it makes you feel like if you came across

00:41:07.660 | another intelligent civilization,

00:41:09.700 | that they'd be studying the same thing.

00:41:12.380 | - Maybe with different notation.

00:41:13.660 | - Certainly, yeah.

00:41:14.540 | But yeah.

00:41:15.540 | Like, that's what I think,

00:41:16.780 | you talked to that other civilization,

00:41:18.740 | they're probably also studying the zeros

00:41:20.260 | of the Riemann zeta function.

00:41:21.700 | Or like some variant thereof that is like

00:41:25.700 | a clearly equivalent cousin or something like that.

00:41:28.580 | But that's probably on their docket.

00:41:31.400 | - Whenever somebody does a lot of something amazing,

00:41:35.940 | I'm gonna ask the question

00:41:37.660 | that you've already been asked a lot,

00:41:40.180 | and that you'll get more and more asked in your life.

00:41:43.340 | But what was your favorite video to create?

00:41:46.180 | - Oh, favorite to create.

00:41:49.540 | One of my favorites is,

00:41:51.300 | the title is "Who Cares About Topology?"

00:41:53.340 | - Do you want me to pull it up or no?

00:41:55.940 | - If you want, sure, yeah.

00:41:57.300 | It is about, well, it starts by describing

00:42:00.980 | an unsolved problem that's still unsolved in math

00:42:03.020 | called the inscribed square problem.

00:42:05.020 | You draw any loop, and then you ask,

00:42:06.700 | are there four points on that loop that make a square?

00:42:09.180 | Totally useless, right?

00:42:10.180 | This is not answering any physical questions.

00:42:12.460 | It's mostly interesting that we can't answer that question.

00:42:14.920 | And it seems like such a natural thing to ask.

00:42:17.220 | Now, if you weaken it a little bit and you ask,

00:42:21.180 | can you always find a rectangle?

00:42:22.580 | You choose four points on this curve.

00:42:24.280 | Can you find a rectangle?

00:42:25.620 | That's hard, but it's doable.

00:42:27.500 | And the path to it involves things like looking at a torus,

00:42:32.500 | this surface with a single hole in it, like a donut,

00:42:35.320 | or looking at a Mobius strip,

00:42:37.300 | in ways that feel so much less contrived

00:42:39.740 | to when I first, as like a little kid,

00:42:41.660 | learned about these surfaces and shapes,

00:42:43.420 | like a Mobius strip and a torus.

00:42:45.480 | Like what you learn is, oh, this Mobius strip,

00:42:47.880 | you take a piece of paper, put a twist, glue it together.

00:42:50.780 | And now you have a shape with one edge and just one side.

00:42:53.740 | And as a student, you should think, who cares, right?

00:42:58.500 | Like, how does that help me solve any problems?

00:43:00.620 | I thought math was about problem solving.

00:43:02.760 | So what I liked about the piece of math

00:43:05.660 | that this was describing that was in this paper

00:43:08.540 | by a mathematician named Vaughn

00:43:10.100 | was that it arises very naturally.

00:43:12.980 | It's clear what it represents.

00:43:14.380 | It's doing something.

00:43:15.480 | It's not just playing with construction paper.

00:43:17.840 | And the way that it solves the problem is really beautiful.

00:43:20.920 | So kind of putting all of that down

00:43:24.300 | and concretizing it, right?

00:43:25.820 | Like I was talking about how,

00:43:27.680 | when you have to put visuals to it,

00:43:29.420 | it demands that what's on screen

00:43:30.900 | is a very specific example of what you're describing.

00:43:33.620 | The construction here is very abstract in nature.

00:43:35.900 | You described this very abstract kind of surface

00:43:38.180 | in 3D space.

00:43:39.340 | So then when I was finding myself,

00:43:40.900 | in this case, I wasn't programming,

00:43:42.020 | I was using a grapher that's like built into OSX

00:43:44.540 | for the 3D stuff to draw that surface,

00:43:48.740 | you realize, oh man,

00:43:49.700 | the topology argument is very non-constructive.

00:43:52.660 | I have to make a lot of,

00:43:54.100 | you have to do a lot of extra work

00:43:55.660 | in order to make the surface show up.

00:43:57.420 | But then once you see it, it's quite pretty.

00:43:59.380 | And it's very satisfying to see a specific instance of it.

00:44:02.060 | And you also feel like, ah,

00:44:03.500 | I've actually added something

00:44:04.820 | on top of what the original paper was doing,

00:44:06.700 | that it shows something that's completely correct.

00:44:09.620 | That's a very beautiful argument,

00:44:10.860 | but you don't see what it looks like.

00:44:12.620 | And I found something satisfying

00:44:14.900 | in seeing what it looked like

00:44:16.340 | that could only ever have come about

00:44:17.940 | from the forcing function

00:44:19.180 | of getting some kind of image on the screen

00:44:21.180 | to describe the thing I was talking about.

00:44:22.020 | - So you almost weren't able to anticipate

00:44:24.220 | what it was gonna look like?

00:44:25.060 | - I had no idea.

00:44:26.140 | I had no idea.

00:44:26.980 | And it was wonderful.

00:44:28.220 | It was totally,

00:44:29.060 | it looks like a Sydney Opera House

00:44:30.340 | or some sort of Frank Gehry design.

00:44:31.980 | And it was,

00:44:33.300 | you knew it was gonna be something.

00:44:35.140 | And you can say various things about it,

00:44:36.460 | like, oh, it touches the curve itself.

00:44:39.300 | It has a boundary that's this curve on the 2D plane.

00:44:42.060 | It all sits above the plane.

00:44:43.820 | But before you actually draw it,

00:44:45.300 | it's very unclear what the thing will look like.

00:44:48.140 | And to see it, it's very,

00:44:49.740 | it's just pleasing, right?

00:44:50.740 | So that was fun to make,

00:44:51.940 | very fun to share.

00:44:53.180 | I hope that it has elucidated,

00:44:56.100 | for some people out there,

00:44:58.020 | where these constructs of topology come from,

00:45:00.100 | that it's not arbitrary play with construction paper.

00:45:03.020 | - So let's,

00:45:04.820 | I think this is a good sort of example

00:45:07.220 | to talk a little bit about your process.

00:45:09.500 | So you have a list of ideas.

00:45:12.060 | That's sort of the curse of having an active

00:45:17.500 | and brilliant mind,

00:45:18.540 | is I'm sure you have a list

00:45:19.580 | that's growing faster than you can utilize.

00:45:22.020 | - Yeah, I love the head.

00:45:23.460 | Absolutely.

00:45:24.620 | - But there's some sorting procedure,

00:45:26.900 | depending on mood and interest and so on.

00:45:29.800 | But, okay, so you pick an idea,

00:45:32.660 | then you have to try to write a narrative arc

00:45:36.140 | that's sort of, how do I elucidate,

00:45:38.820 | how do I make this idea beautiful and clear and explain it?

00:45:42.380 | And then there's a set of visualizations

00:45:44.020 | that will be attached to it.

00:45:46.140 | Sort of, you've talked about some of this before,

00:45:48.420 | but sort of writing the story,

00:45:50.580 | attaching the visualizations.

00:45:52.900 | Can you talk through interesting, painful,

00:45:56.420 | beautiful parts of that process?

00:45:58.900 | - Well, the most painful is

00:46:00.980 | if you've chosen a topic that you do want to do,

00:46:03.420 | but then it's hard to think of,

00:46:05.660 | I guess, how to structure the script.

00:46:07.500 | This is sort of where I have been on one

00:46:10.620 | for like the last two or three months.

00:46:12.220 | And I think that ultimately the right resolution

00:46:13.740 | is just like set it aside and instead do some other things

00:46:17.340 | where the script comes more naturally.

00:46:18.820 | 'Cause you sort of don't want to overwork a narrative.

00:46:23.460 | The more you've thought about it,

00:46:24.700 | the less you can empathize with the student

00:46:26.460 | who doesn't yet understand the thing you're trying to teach.

00:46:28.940 | - Who is the judger in your head?

00:46:31.860 | Sort of the person, the creature,

00:46:35.460 | the essence that's saying this sucks or this is good.

00:46:38.700 | And you mentioned kind of the student

00:46:40.700 | you're thinking about.

00:46:42.020 | Can you, who is that?

00:46:44.740 | What is that thing?

00:46:45.900 | That's Chris, that says, the perfectionist

00:46:48.500 | that says this thing sucks,

00:46:49.940 | you need to work on that for another two, three months.

00:46:53.500 | - I don't know.

00:46:54.340 | I think it's my past self.

00:46:56.180 | I think that's the entity

00:46:57.300 | that I'm most trying to empathize with.

00:46:59.420 | Is like you take who I was,

00:47:00.900 | because that's kind of the only person I know.

00:47:02.500 | Like you don't really know anyone

00:47:03.740 | other than versions of yourself.

00:47:05.500 | So I start with the version of myself that I know

00:47:07.940 | who doesn't yet understand the thing, right?

00:47:10.380 | And then I just try to view it with fresh eyes,

00:47:15.380 | a particular visual or a particular script.

00:47:17.500 | Like, is this motivating?

00:47:18.860 | Does this make sense?

00:47:20.700 | Which has its downsides.

00:47:21.580 | 'Cause sometimes I find myself speaking to motivations

00:47:25.260 | that only myself would be interested in.

00:47:28.540 | I don't know, like I did this project on quaternions.

00:47:30.880 | Where what I really wanted was to understand

00:47:33.300 | what are they doing in four dimensions?

00:47:34.940 | Can we see what they're doing in four dimensions, right?

00:47:37.580 | And I came up with a way of thinking about it

00:47:40.420 | that really answered the question in my head.

00:47:42.020 | That made me very satisfied

00:47:43.260 | in being able to think about concretely with a 3D visual,

00:47:46.660 | what are they doing to a 4D sphere?

00:47:48.700 | And so I'm like, great,

00:47:49.540 | this is exactly what my past self would have wanted, right?

00:47:51.740 | And I make a thing on it.

00:47:52.780 | And I'm sure it's what some other people wanted too.

00:47:55.140 | But in hindsight,

00:47:56.020 | I think most people who want to learn about quaternions

00:47:58.580 | are like robotics engineers or graphics programmers

00:48:01.960 | who want to understand how they're used

00:48:04.360 | to describe 3D rotations.

00:48:06.160 | And like their use case was actually a little bit different

00:48:08.280 | than my past self.

00:48:09.360 | And in that way, like,

00:48:10.600 | I wouldn't actually recommend that video

00:48:12.080 | to people who are coming at it from that angle

00:48:14.960 | of wanting to know, hey, I'm a robotics programmer.

00:48:17.480 | Like how do these quaternion things work

00:48:20.280 | to describe position in 3D space?

00:48:22.560 | I would say other great resources for that.

00:48:25.680 | If you ever find yourself wanting to say like,

00:48:27.840 | but hang on, in what sense are they acting in four dimensions

00:48:30.860 | then come back.

00:48:31.860 | But until then, that's a little different.

00:48:34.460 | - Yeah, it's interesting

00:48:35.300 | 'cause you have incredible videos on neural networks,

00:48:38.780 | for example.

00:48:39.820 | And from my sort of perspective,

00:48:41.060 | 'cause I've probably, I mean,

00:48:43.660 | I looked at the,

00:48:44.700 | it's sort of my field

00:48:46.940 | and I've also looked at the basic introduction

00:48:49.260 | of neural networks like a million times

00:48:51.060 | from different perspectives.

00:48:52.420 | And it made me realize

00:48:53.460 | that there's a lot of ways to present it.

00:48:55.740 | So you are sort of, you did an incredible job.

00:48:58.920 | I mean, sort of the,

00:49:00.080 | but you could also do it differently and also incredible.

00:49:04.880 | Like to create a beautiful presentation of a basic concept

00:49:09.880 | is, requires sort of creativity, requires genius and so on,

00:49:16.040 | but you can take it from a bunch of different perspectives.

00:49:18.600 | And that video on neural networks made me realize that.

00:49:21.560 | And just as you're saying,

00:49:22.920 | you're kind of have a certain mindset, a certain view,

00:49:26.240 | but from a, if you take a different view

00:49:28.940 | from a physics perspective,

00:49:30.640 | from a neuroscience perspective,

00:49:33.320 | talking about neural networks,

00:49:34.400 | or from a robotics perspective,

00:49:38.360 | or from, let's see,

00:49:40.400 | from a pure learning theory, - Statistics.

00:49:42.120 | - statistics perspective.

00:49:43.260 | So you can create totally different videos.

00:49:46.320 | And you've done that with a few actually concepts

00:49:48.200 | where you've have taken different cuts,

00:49:49.880 | like at the, at the, at the Euler equation, right?

00:49:54.120 | The, you've taken different views of that.

00:49:56.920 | - I think I've made three videos on it

00:49:58.680 | and I definitely will make at least one more.

00:50:00.920 | - Never enough.

00:50:03.080 | - Never enough.

00:50:04.080 | - So you don't think it's the most beautiful equation

00:50:06.160 | in mathematics?

00:50:07.200 | - Like I said, as we represent it,

00:50:10.020 | it's one of the most hideous.

00:50:11.400 | It involves a lot of the most hideous aspects

00:50:13.280 | of our notation.

00:50:14.120 | I talked about E, the fact that we use pi instead of tau,

00:50:16.920 | the fact that we call imaginary numbers imaginary,

00:50:20.640 | and then, hence,

00:50:22.080 | actually wonder if we use the I because of imaginary.

00:50:24.840 | I don't know if that's historically accurate,

00:50:26.560 | but at least a lot of people,

00:50:27.680 | they read the I and they think imaginary.

00:50:30.280 | Like all three of those facts,

00:50:31.400 | it's like, those are things that have added more confusion

00:50:33.680 | than they needed to,

00:50:34.520 | and we're wrapping them up in one equation.

00:50:35.800 | Like, boy, that's just very hideous, right?

00:50:39.040 | The ideas that it does tie together

00:50:40.840 | when you wash away the notation,

00:50:42.160 | like it's okay, it's pretty, it's nice,

00:50:44.800 | but it's not like, mind-blowing,

00:50:47.800 | greatest thing in the universe,

00:50:49.480 | which is maybe what I was thinking of when I said,

00:50:51.980 | like once you understand something,

00:50:53.280 | it doesn't have the same beauty.

00:50:55.940 | Like I feel like I understand Euler's formula,

00:50:58.960 | and I feel like I understand it enough to sort of see

00:51:02.520 | the version that just woke up,

00:51:04.980 | that hasn't really gotten itself dressed in the morning,

00:51:07.520 | that's a little bit groggy,

00:51:08.600 | and there's bags under its eyes.

00:51:10.080 | It's like it's the real-- - So you're past

00:51:11.320 | the dating stage, and you're now married.

00:51:13.880 | - I'm no longer dating, right?

00:51:15.040 | I'm still dating the zeta function,

00:51:16.760 | and like she's beautiful, and right,

00:51:18.920 | and like we have fun, and it's that high dopamine part,

00:51:22.660 | but like maybe at some point we'll settle

00:51:24.400 | into the more mundane nature of the relationship

00:51:26.840 | where I like see her for who she truly is,

00:51:28.480 | and she'll still be beautiful in her own way,

00:51:30.220 | but it won't have the same romantic pizzazz, right?

00:51:33.760 | - Well, that's the nice thing about mathematics.

00:51:35.520 | I think as long as you don't live forever,

00:51:38.480 | there'll always be enough mystery and fun

00:51:41.840 | with some of the equations.

00:51:42.920 | Even if you do, the rate at which questions comes up

00:51:45.480 | is much faster than the rate at which answers come up.

00:51:48.080 | - If you could live forever, would you?

00:51:50.040 | - I think so, yeah.

00:51:52.160 | - So you don't think mortality's the thing

00:51:54.120 | that makes life meaningful?

00:51:55.980 | - Would your life be four times as meaningful

00:51:58.240 | if you died at 25?

00:52:00.440 | - So this goes to infinity.

00:52:02.200 | I think you and I, that's really interesting.

00:52:04.700 | So what I said is infinite, not four times longer.

00:52:09.360 | I said infinite.

00:52:10.400 | So the actual existence of the finiteness,

00:52:15.240 | the existence of the end, no matter the length,

00:52:18.060 | is the thing that may sort of,

00:52:20.740 | from my comprehension of psychology,

00:52:22.400 | it's such a deeply human,

00:52:25.640 | it's such a fundamental part of the human condition,

00:52:28.600 | the fact that there is, that we're mortal,

00:52:31.160 | that the fact that things end,

00:52:34.500 | it seems to be a crucial part of what gives them meaning.

00:52:37.920 | - I don't think, at least for me,

00:52:39.700 | like it's a very small percentage of my time

00:52:43.040 | that mortality is salient,

00:52:45.200 | that I'm like aware of the end of my life.

00:52:47.200 | - What do you mean by me?

00:52:48.440 | I'm trolling.

00:52:51.280 | Is it the ego, is it the id, or is it the super ego?

00:52:54.080 | - The reflective self, the Wernicke's area

00:52:58.160 | that puts all this stuff into words.

00:52:59.880 | - Yeah, a small percentage of your mind

00:53:02.520 | that is actually aware of the true motivations

00:53:05.780 | that drive you.

00:53:06.960 | - But my point is that most of my life,

00:53:08.360 | I'm not thinking about death.

00:53:09.600 | But I still feel very motivated to make things

00:53:12.020 | and to interact with people,

00:53:13.860 | like experience love or things like that.

00:53:15.400 | I'm very motivated.

00:53:16.680 | And it's strange that that motivation comes

00:53:19.400 | while death is not in my mind at all.

00:53:21.600 | And this might just be because I'm young enough

00:53:23.600 | that it's not salient.

00:53:24.680 | - Or it's in your subconscious,

00:53:25.960 | or that you constructed an illusion

00:53:28.100 | that allows you to escape the fact of your mortality

00:53:31.180 | by enjoying the moment,

00:53:32.760 | sort of the existential approach to life.

00:53:34.600 | - Could be.

00:53:36.160 | Gun to my head, I don't think that's it.

00:53:38.080 | - Yeah, another sort of way to say gun to the head

00:53:40.640 | is the deep psychological introspection of what drives us.

00:53:44.160 | I mean, that's in some ways to me,

00:53:46.960 | I mean, when I look at math, when I look at science,

00:53:49.040 | is a kind of an escape from reality

00:53:51.560 | in a sense that it's so beautiful.

00:53:54.360 | It's such a beautiful journey of discovery

00:53:58.720 | that it allows you to actually,

00:54:00.800 | it sort of allows you to achieve a kind of immortality

00:54:04.720 | of explore ideas and sort of connect yourself

00:54:09.680 | to the thing that is seemingly infinite,

00:54:12.360 | like the universe, right?

00:54:13.960 | That allows you to escape the limited nature

00:54:18.600 | of our little, of our bodies, of our existence.

00:54:23.600 | - What else would give this podcast meaning?

00:54:25.960 | - That's right.

00:54:26.800 | - If not the fact that it will end.

00:54:28.000 | - This place closes in 40 minutes.

00:54:30.920 | - And it's so much more meaningful for it.

00:54:34.200 | How much more I love this room because we'll be kicked out.

00:54:37.200 | - So I understand just because you're trolling me

00:54:42.280 | doesn't mean I'm wrong.

00:54:43.600 | But I take your point, I take your point.

00:54:49.000 | - Boy, that would be a good Twitter bio.

00:54:51.000 | Just because you're trolling me doesn't mean I'm wrong.

00:54:54.320 | - Yeah, and sort of difference in backgrounds.

00:54:58.560 | I'm a bit Russian, so we're a bit melancholic

00:55:01.520 | and seem to maybe assign a little too much value

00:55:04.360 | to suffering and mortality and things like that.

00:55:07.040 | Makes for a better novel, I think.

00:55:09.880 | - Oh yeah, you need some sort of existential threat

00:55:13.400 | to drive a plot.

00:55:15.320 | - So when do you know when the video is done

00:55:18.520 | when you're working on it?

00:55:19.800 | - That's pretty easy, actually,

00:55:21.560 | because I'll write the script.

00:55:24.400 | I want there to be some kind of aha moment in there,

00:55:27.140 | and then hopefully the script can revolve around

00:55:28.760 | some kind of aha moment.

00:55:30.280 | And then from there, you're putting visuals

00:55:32.400 | to each sentence that exists,

00:55:34.040 | and then you narrate it, you edit it all together.

00:55:36.360 | So given that there's a script, the end becomes quite clear.

00:55:39.760 | And as I animate it, I often change

00:55:44.320 | certainly the specific words,

00:55:46.800 | but sometimes the structure itself.

00:55:49.320 | But it's a very deterministic process at that point.

00:55:53.240 | It makes it much easier to predict

00:55:54.440 | when something will be done.

00:55:55.860 | How do you know when a script is done?

00:55:57.080 | It's like, for problem-solving videos,

00:55:59.120 | that's quite simple.

00:56:00.440 | It's once you feel like someone

00:56:01.480 | who didn't understand the solution now could.

00:56:03.580 | For things like neural networks, that was a lot harder,

00:56:05.480 | because like you said, there's so many angles

00:56:07.640 | at which you could attack it.

00:56:09.580 | And there, it's just at some point you feel like

00:56:12.480 | this asks a meaningful question,

00:56:15.680 | and it answers that question.

00:56:17.080 | - What is the best way to learn math

00:56:20.280 | for people who might be at the beginning of that journey?

00:56:22.400 | I think that's a question that a lot of folks

00:56:24.840 | kind of ask and think about.

00:56:26.200 | And it doesn't, even for folks who are not really

00:56:28.620 | at the beginning of their journey,

00:56:29.900 | like they might be actually deep in their career,

00:56:33.920 | some type, they've taken college,

00:56:35.720 | they've taken calculus and so on,

00:56:36.900 | but still wanna sort of explore math.

00:56:39.080 | What would be your advice in sort of education at all ages?

00:56:42.860 | - Your temptation will be to spend more time

00:56:45.960 | watching lectures or reading.

00:56:48.160 | Try to force yourself to do more problems

00:56:50.560 | than you naturally would.

00:56:52.160 | That's a big one.

00:56:53.780 | Like the focus time that you're spending

00:56:56.040 | should be on solving specific problems

00:56:59.020 | and seek entities that have well curated lists of problems.

00:57:02.380 | - So go into like a textbook almost,

00:57:04.260 | and the problems in the back of a text,

00:57:06.020 | kind of thing, back of a chapter.

00:57:08.100 | - So if you can, take a little look through those questions

00:57:10.580 | at the end of the chapter before you read the chapter.

00:57:12.500 | A lot of them won't make sense.

00:57:13.620 | Some of them might, and those are the best ones

00:57:15.820 | to think about.

00:57:16.660 | A lot of them won't, but just take a quick look

00:57:18.940 | and then read a little bit of the chapter

00:57:20.140 | and then maybe take a look again and things like that.

00:57:22.460 | And don't consider yourself done with the chapter

00:57:25.140 | until you've actually worked through a couple exercises.

00:57:28.180 | And this is so hypocritical, right?

00:57:31.220 | 'Cause I like put out videos that pretty much never

00:57:33.980 | have associated exercises.

00:57:35.980 | I just view myself as a different part of the ecosystem,

00:57:38.680 | which means I'm kind of admitting

00:57:40.780 | that you're not really learning,

00:57:42.900 | or at least this is only a partial part

00:57:44.740 | of the learning process if you're watching these videos.

00:57:47.440 | I think if someone's at the very beginning,

00:57:50.380 | like I do think Khan Academy does a good job.

00:57:52.220 | They have a pretty large set of questions

00:57:54.900 | you can work through.

00:57:55.880 | - Just the very basics, sort of just picking up,

00:57:58.820 | getting comfortable with the very basic linear algebra

00:58:01.220 | or calculus or something, Khan Academy.

00:58:04.100 | - Programming is actually, I think, a great,

00:58:05.940 | like learn to program and let the way that math

00:58:08.820 | is motivated from that angle push you through.

00:58:11.780 | I know a lot of people who didn't like math

00:58:14.340 | got into programming in some way,

00:58:15.540 | and that's what turned them on to math.

00:58:17.260 | Maybe I'm biased 'cause I live in the Bay Area,

00:58:19.180 | so I'm more likely to run into someone

00:58:21.060 | who has that phenotype, but I am willing to speculate

00:58:25.780 | that that is a more generalizable path.

00:58:28.140 | - So you yourself kind of in creating the videos

00:58:30.100 | are using programming to illuminate a concept,

00:58:32.980 | but for yourself as well.

00:58:35.060 | So would you recommend somebody try to make a,

00:58:37.980 | sort of almost like try to make videos?

00:58:40.260 | Like you do as a way to learn?

00:58:41.860 | - So one thing I've heard before,

00:58:43.140 | I don't know if this is based on any actual study.

00:58:44.740 | This might be like a total fictional anecdote of numbers,

00:58:47.240 | but it rings in the mind as being true.

00:58:49.760 | You remember about 10% of what you read.

00:58:51.860 | You remember about 20% of what you listen to.

00:58:54.420 | You remember about 70% of what you actively interact with

00:58:57.340 | in some way, and then about 90% of what you teach.

00:59:00.500 | This is a thing I heard again,

00:59:02.100 | those numbers might be meaningless,

00:59:03.500 | but they ring true, don't they?

00:59:05.500 | Right, I'm willing to say I learned nine times better

00:59:07.900 | if I'm teaching something than reading.

00:59:09.220 | That might even be a low ball, right?

00:59:11.660 | So doing something to teach or to like actively try

00:59:14.500 | to explain things is huge for consolidating the knowledge.

00:59:17.820 | - Outside of family and friends,

00:59:19.720 | is there a moment you can remember

00:59:22.480 | that you would like to relive

00:59:23.800 | because it made you truly happy

00:59:26.240 | or it was transformative in some fundamental way?

00:59:30.240 | - A moment that was transformative?

00:59:32.760 | - Or made you truly happy?

00:59:35.120 | - Yeah, I think there's times,

00:59:36.840 | like music used to be a much bigger part of my life

00:59:38.780 | than it is now.

00:59:39.620 | Like when I was a, let's say a teenager.

00:59:41.680 | And I can think of sometimes in like playing music,

00:59:45.460 | there was one, my brother and a friend of mine,

00:59:48.160 | so this slightly violates the family and friends,

00:59:50.160 | but there was a music that made me happy.

00:59:51.840 | They were just accompanying.

00:59:54.440 | We had like played a gig at a ski resort,

00:59:57.480 | such that you like take a gondola to the top

00:59:59.320 | and like did a thing.

01:00:00.680 | Then on the gondola ride down,

01:00:01.840 | we decided to just jam a little bit.

01:00:04.180 | And it was just like, I don't know,

01:00:06.280 | the gondola sort of over, came over a mountain

01:00:09.040 | and you saw the city lights

01:00:10.740 | and we're just like jamming, like playing some music.

01:00:13.920 | I wouldn't describe that as transformative.

01:00:16.320 | I don't know why, but that popped into my mind

01:00:18.040 | as a moment of, in a way that wasn't associated

01:00:21.180 | with people I love, but more with like a thing I was doing,

01:00:24.180 | something that was just, it was just happy.

01:00:26.160 | And it was just like a great moment.

01:00:28.000 | I don't think I can give you anything deeper

01:00:30.800 | than that though.

01:00:32.080 | - Well, as a musician myself, I'd love to see,

01:00:34.840 | as you mentioned before, music enter back into your work,

01:00:38.800 | back into your creative work.

01:00:40.080 | I'd love to see that.

01:00:41.320 | I'm certainly allowing it to enter back into mine

01:00:43.880 | and it's a beautiful thing for a mathematician,

01:00:47.880 | for a scientist to allow music to enter their work.

01:00:51.500 | I think only good things can happen.

01:00:53.960 | - All right, I'll try to promise you a music video by 2020.

01:00:57.280 | - By 2020? - By the end of 2020.

01:00:58.840 | - Okay, all right, good.

01:00:59.680 | - I'll give myself a longer window.

01:01:01.440 | - All right, maybe we can like collaborate

01:01:04.520 | on a band type situation.

01:01:05.720 | What instruments do you play?

01:01:07.080 | - The main instrument I play is violin,

01:01:08.640 | but I also love to dabble around on like guitar and piano.

01:01:11.720 | - Beautiful, me too, guitar and piano.

01:01:14.620 | So in "The Mathematician's Lament,"

01:01:17.180 | Paul Lockhart writes, "The first thing to understand

01:01:20.100 | "is that mathematics is an art.

01:01:22.060 | "The difference between math and the other arts,

01:01:24.140 | "such as music and painting,

01:01:26.740 | "is that our culture does not recognize it as such."

01:01:30.000 | So I think I speak for millions of people,

01:01:32.660 | myself included, in saying thank you

01:01:35.500 | for revealing to us the art of mathematics.

01:01:39.720 | So thank you for everything you do

01:01:40.980 | and thanks for talking today.

01:01:42.340 | - Wow, thanks for saying that

01:01:43.300 | and thanks for having me on.

01:01:45.380 | - Thanks for listening to this conversation

01:01:47.100 | with Grant Sanderson.

01:01:48.340 | And thank you to our presenting sponsor, Cash App.

01:01:51.700 | Download it, use code LEXPODCAST,

01:01:54.820 | you'll get $10 and $10 will go to FIRST,

01:01:57.740 | a STEM education nonprofit

01:01:59.460 | that inspires hundreds of thousands of young minds

01:02:01.980 | to become future leaders and innovators.

01:02:04.940 | If you enjoy this podcast, subscribe on YouTube,

01:02:07.600 | give it five stars on Apple Podcast,

01:02:09.460 | support it on Patreon, or connect with me on Twitter.

01:02:13.140 | And now, let me leave you with some words of wisdom

01:02:16.060 | from one of Grant's and my favorite people, Richard Feynman.

01:02:20.280 | "Nobody ever figures out what this life is all about,

01:02:24.700 | "and it doesn't matter.

01:02:26.400 | "Explore the world.

01:02:28.460 | "Nearly everything is really interesting

01:02:30.560 | "if you go into it deeply enough."

01:02:33.300 | Thank you for listening and hope to see you next time.

01:02:36.560 | (upbeat music)

01:02:39.140 | (upbeat music)

01:02:41.720 | [BLANK_AUDIO]

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

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