Transcript
00:00:00 The following is a conversation with Grant Sanderson.
00:00:03 He’s a math educator and creator of 3Blue1Brown,
00:00:06 a popular YouTube channel
00:00:07 that uses programmatically animated visualizations
00:00:11 to explain concepts in linear algebra, calculus,
00:00:14 and other fields of mathematics.
00:00:16 This is the Artificial Intelligence Podcast.
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00:01:51 And now, here’s my conversation with Grant Sanderson.
00:01:56 If there’s intelligent life out there in the universe,
00:01:59 do you think their mathematics is different than ours?
00:02:03 Jumping right in.
00:02:04 I think it’s probably very different.
00:02:08 There’s an obvious sense the notation is different, right?
00:02:11 I think notation can guide what the math itself is.
00:02:14 I think it has everything to do with the form
00:02:17 of their existence, right?
00:02:19 Do you think they have basic arithmetic?
00:02:20 Sorry, I interrupted.
00:02:21 Yeah, so I think they count, right?
00:02:23 I think notions like one, two, three,
00:02:24 the natural numbers, that’s extremely, well, natural.
00:02:27 That’s almost why we put that name to it.
00:02:30 As soon as you can count,
00:02:31 you have a notion of repetition, right?
00:02:34 Because you can count by two, two times or three times.
00:02:37 And so you have this notion of repeating
00:02:39 the idea of counting, which brings you addition
00:02:42 and multiplication.
00:02:43 I think the way that we extend it to the real numbers,
00:02:47 there’s a little bit of choice in that.
00:02:49 So there’s this funny number system
00:02:50 called the servial numbers
00:02:52 that it captures the idea of continuity.
00:02:55 It’s a distinct mathematical object.
00:02:57 You could very well model the universe
00:03:00 and motion of planets with that
00:03:02 as the back end of your math, right?
00:03:04 And you still have kind of the same interface
00:03:06 with the front end of what physical laws you’re trying to,
00:03:10 or what physical phenomena you’re trying
00:03:12 to describe with math.
00:03:13 And I wonder if the little glimpses that we have
00:03:15 of what choices you can make along the way
00:03:17 based on what different mathematicians
00:03:19 I’ve brought to the table
00:03:20 is just scratching the surface
00:03:22 of what the different possibilities are
00:03:24 if you have a completely different mode of thought, right?
00:03:27 Or a mode of interacting with the universe.
00:03:29 And you think notation is a key part of the journey
00:03:32 that we’ve taken through math.
00:03:33 I think that’s the most salient part
00:03:35 that you’d notice at first.
00:03:36 I think the mode of thought is gonna influence things
00:03:38 more than like the notation itself.
00:03:40 But notation actually carries a lot of weight
00:03:42 when it comes to how we think about things,
00:03:44 more so than we usually give it credit for.
00:03:47 I would be comfortable saying.
00:03:48 Do you have a favorite or least favorite piece of notation
00:03:52 in terms of its effectiveness?
00:03:53 Yeah, yeah, well, so least favorite,
00:03:54 one that I’ve been thinking a lot about
00:03:56 that will be a video I don’t know when, but we’ll see.
00:03:59 The number e, we write the function e to the x,
00:04:02 this general exponential function
00:04:04 with a notation e to the x
00:04:06 that implies you should think about a particular number,
00:04:08 this constant of nature,
00:04:09 and you repeatedly multiply it by itself.
00:04:11 And then you say, oh, what’s e to the square root of two?
00:04:14 And you’re like, oh, well, we’ve extended the idea
00:04:15 of repeated multiplication.
00:04:17 That’s all nice, that’s all nice and well.
00:04:19 But very famously, you have like e to the pi i,
00:04:22 and you’re like, well, we’re extending the idea
00:04:24 of repeated multiplication into the complex numbers.
00:04:27 Yeah, you can think about it that way.
00:04:28 In reality, I think that it’s just the wrong way
00:04:31 of notationally representing this function,
00:04:34 the exponential function,
00:04:36 which itself could be represented
00:04:37 a number of different ways.
00:04:38 You can think about it in terms of the problem it solves,
00:04:41 a certain very simple differential equation,
00:04:43 which often yields way more insight
00:04:45 than trying to twist the idea of repeated multiplication,
00:04:48 like take its arm and put it behind its back
00:04:50 and throw it on the desk and be like,
00:04:51 you will apply to complex numbers, right?
00:04:53 That’s not, I don’t think that’s pedagogically helpful.
00:04:57 So the repeated multiplication is actually missing
00:04:59 the main point, the power of e to the x.
00:05:03 I mean, what it addresses is things where the rate
00:05:05 at which something changes depends on its own value,
00:05:10 but more specifically, it depends on it linearly.
00:05:12 So for example, if you have like a population
00:05:15 that’s growing and the rate at which it grows
00:05:16 depends on how many members of the population
00:05:18 are already there,
00:05:19 it looks like this nice exponential curve.
00:05:21 It makes sense to talk about repeated multiplication
00:05:23 because you say, how much is there after one year,
00:05:25 two years, three years, you’re multiplying by something.
00:05:27 The relationship can be a little bit different sometimes
00:05:29 where let’s say you’ve got a ball on a string,
00:05:33 like a game of tetherball going around a rope, right?
00:05:37 And you say, its velocity is always perpendicular
00:05:40 to its position.
00:05:42 That’s another way of describing its rate of change
00:05:44 is being related to where it is,
00:05:47 but it’s a different operation.
00:05:48 You’re not scaling it, it’s a rotation.
00:05:49 It’s this 90 degree rotation.
00:05:51 That’s what the whole idea of like complex exponentiation
00:05:54 is trying to capture,
00:05:55 but it’s obfuscated in the notation
00:05:57 when what it’s actually saying,
00:05:59 like if you really parse something like e to the pi i,
00:06:01 what it’s saying is choose an origin,
00:06:03 always move perpendicular to the vector
00:06:06 from that origin to you, okay?
00:06:09 Then when you walk pi times that radius,
00:06:12 you’ll be halfway around.
00:06:14 Like that’s what it’s saying.
00:06:15 It’s kind of the, you turn 90 degrees and you walk,
00:06:18 you’ll be going in a circle.
00:06:19 That’s the phenomenon that it’s describing,
00:06:22 but trying to twist the idea
00:06:24 of repeatedly multiplying a constant into that.
00:06:26 Like I can’t even think of the number of human hours
00:06:30 of like intelligent human hours that have been wasted
00:06:33 trying to parse that to their own liking and desire
00:06:36 among like scientists or electrical engineers
00:06:39 or students everywhere,
00:06:40 which if the notation were a little different
00:06:42 or the way that this whole function was introduced
00:06:45 from the get go were framed differently,
00:06:47 I think could have been avoided, right?
00:06:49 And you’re talking about
00:06:51 the most beautiful equation in mathematics,
00:06:53 but it’s still pretty mysterious, isn’t it?
00:06:55 Like you’re making it seem like it’s a notational.
00:06:58 It’s not mysterious.
00:06:59 I think the notation makes it mysterious.
00:07:01 I don’t think it’s, I think the fact that it represents,
00:07:04 it’s pretty, it’s not like the most beautiful thing
00:07:06 in the world, but it’s quite pretty.
00:07:07 The idea that if you take the linear operation
00:07:10 of a 90 degree rotation,
00:07:12 and then you do this general exponentiation thing to it,
00:07:15 that what you get are all the other kinds of rotation,
00:07:19 which is basically to say,
00:07:20 if your velocity vector is perpendicular
00:07:22 to your position vector, you walk in a circle,
00:07:25 that’s pretty.
00:07:26 It’s not the most beautiful thing in the world,
00:07:27 but it’s quite pretty.
00:07:28 The beauty of it, I think comes from perhaps
00:07:31 the awkwardness of the notation
00:07:33 somehow still nevertheless coming together nicely,
00:07:35 because you have like several disciplines coming together
00:07:38 in a single equation.
00:07:40 Well, I think.
00:07:41 In a sense, like historically speaking.
00:07:43 That’s true.
00:07:44 You’ve got, so like the number E is significant.
00:07:45 Like it shows up in probability all the time.
00:07:47 It like shows up in calculus all the time.
00:07:49 It is significant.
00:07:50 You’re seeing it sort of mated with pi,
00:07:52 this geometric constant and I,
00:07:54 like the imaginary number and such.
00:07:55 I think what’s really happening there
00:07:57 is the way that E shows up is when you have things
00:08:01 like exponential growth and decay, right?
00:08:03 It’s when this relation that something’s rate of change
00:08:06 has to itself is a simple scaling, right?
00:08:10 A similar law also describes circular motion.
00:08:14 Because we have bad notation,
00:08:16 we use the residue of how it shows up
00:08:19 in the context of self reinforcing growth,
00:08:21 like a population growing or compound interest.
00:08:23 The constant associated with that
00:08:25 is awkwardly placed into the context
00:08:27 of how rotation comes about,
00:08:29 because they both come from pretty similar equations.
00:08:32 And so what we see is the E and the pi juxtaposed
00:08:36 a little bit closer than they would be
00:08:38 with a purely natural representation, I would think.
00:08:41 Here’s how I would describe the relation between the two.
00:08:43 You’ve got a very important function we might call exp.
00:08:45 That’s like the exponential function.
00:08:47 When you plug in one,
00:08:49 you get this nice constant called E
00:08:50 that shows up in like probability and calculus.
00:08:53 If you try to move in the imaginary direction,
00:08:55 it’s periodic and the period is tau.
00:08:58 So those are these two constants
00:08:59 associated with the same central function,
00:09:02 but for kind of unrelated reasons, right?
00:09:04 And not unrelated, but like orthogonal reasons.
00:09:07 One of them is what happens
00:09:08 when you’re moving in the real direction.
00:09:09 One’s what happens when you move in the imaginary direction.
00:09:12 And like, yeah, those are related.
00:09:14 They’re not as related as the famous equation
00:09:17 seems to think it is.
00:09:18 It’s sort of putting all of the children in one bed
00:09:20 and they’d kind of like to sleep in separate beds
00:09:22 if they had the choice, but you see them all there
00:09:24 and there is a family resemblance, but it’s not that close.
00:09:28 So actually thinking of it as a function
00:09:31 is the better idea.
00:09:34 And that’s a notational idea.
00:09:36 And yeah, and like, here’s the thing.
00:09:39 The constant E sort of stands
00:09:41 as this numerical representative of calculus, right?
00:09:44 Calculus is the like study of change.
00:09:47 So at the very least there’s a little cognitive dissonance
00:09:49 using a constant to represent the science of change.
00:09:53 I never thought of it that way.
00:09:54 Yeah.
00:09:54 Right?
00:09:55 Yeah.
00:09:56 It makes sense why the notation came about that way.
00:10:00 Because this is the first way that we saw it
00:10:02 in the context of things like population growth
00:10:03 or compound interest.
00:10:04 It is nicer to think about as repeated multiplication.
00:10:07 That’s definitely nicer.
00:10:08 But it’s more that that’s the first application
00:10:11 of what turned out to be a much more general function
00:10:13 that maybe the intelligent life
00:10:15 your initial question asked about
00:10:17 would have come to recognize as being much more significant
00:10:19 than the single use case,
00:10:21 which lends itself to repeated multiplication notation.
00:10:24 But let me jump back for a second to aliens
00:10:28 and the nature of our universe.
00:10:30 Okay.
00:10:31 Do you think math is discovered or invented?
00:10:35 So we’re talking about the different kind of mathematics
00:10:37 that could be developed by the alien species.
00:10:40 The implied question is,
00:10:44 yeah, is math discovered or invented?
00:10:46 Is fundamentally everybody going to discover
00:10:49 the same principles of mathematics?
00:10:53 So the way I think about it,
00:10:54 and everyone thinks about it differently,
00:10:55 but here’s my take.
00:10:56 I think there’s a cycle at play
00:10:57 where you discover things about the universe
00:11:00 that tell you what math will be useful.
00:11:03 And that math itself is invented in a sense,
00:11:08 but of all the possible maths that you could have invented,
00:11:11 it’s discoveries about the world
00:11:12 that tell you which ones are.
00:11:14 So like a good example here is the Pythagorean theorem.
00:11:17 When you look at this,
00:11:18 do you think of that as a definition
00:11:19 or do you think of that as a discovery?
00:11:21 From the historical perspective, right, it’s a discovery
00:11:24 because they were,
00:11:25 but that’s probably because they were using physical object
00:11:29 to build their intuition.
00:11:32 And from that intuition came the mathematics.
00:11:34 So the mathematics wasn’t in some abstract world
00:11:37 detached from physics,
00:11:39 but I think more and more math has become detached from,
00:11:43 you know, when you even look at modern physics
00:11:46 from string theory to even general relativity,
00:11:49 I mean, all math behind the 20th and 21st century physics
00:11:53 kind of takes a brisk walk outside of what our mind
00:11:58 can actually even comprehend
00:12:00 in multiple dimensions, for example,
00:12:02 anything beyond three dimensions, maybe four dimensions.
00:12:05 No, no, no, no, higher dimensions
00:12:07 can be highly, highly applicable.
00:12:08 I think this is a common misinterpretation
00:12:11 that if you’re asking questions
00:12:13 about like a five dimensional manifold,
00:12:15 that the only way that that’s connected
00:12:16 to the physical world is if the physical world is itself
00:12:20 a five dimensional manifold or includes them.
00:12:22 Well, wait, wait, wait a minute, wait a minute.
00:12:25 You’re telling me you can imagine
00:12:28 a five dimensional manifold?
00:12:31 No, no, that’s not what I said.
00:12:33 I would make the claim that it is useful
00:12:35 to a three dimensional physical universe,
00:12:37 despite itself not being three dimensional.
00:12:39 So it’s useful meaning to even understand
00:12:41 a three dimensional world,
00:12:42 it’d be useful to have five dimensional manifolds.
00:12:44 Yes, absolutely, because of state spaces.
00:12:47 But you’re saying there in some deep way for us humans,
00:12:50 it does always come back to that three dimensional world
00:12:54 for the usefulness that the dimensional world
00:12:56 and therefore it starts with a discovery,
00:12:59 but then we invent the mathematics
00:13:02 that helps us make sense of the discovery in a sense.
00:13:06 Yes, I mean, just to jump off
00:13:07 of the Pythagorean theorem example,
00:13:09 it feels like a discovery.
00:13:11 You’ve got these beautiful geometric proofs
00:13:12 where you’ve got squares and you’re modifying the areas,
00:13:14 it feels like a discovery.
00:13:16 If you look at how we formalize the idea of 2D space
00:13:19 as being R2, right, all pairs of real numbers,
00:13:23 and how we define a metric on it and define distance,
00:13:25 you’re like, hang on a second,
00:13:26 we’ve defined a distance
00:13:28 so that the Pythagorean theorem is true,
00:13:30 so that suddenly it doesn’t feel that great.
00:13:32 But I think what’s going on is the thing that informed us
00:13:35 what metric to put on R2,
00:13:38 to put on our abstract representation of 2D space,
00:13:41 came from physical observations.
00:13:43 And the thing is, there’s other metrics
00:13:44 you could have put on it.
00:13:45 We could have consistent math
00:13:47 with other notions of distance,
00:13:49 it’s just that those pieces of math
00:13:50 wouldn’t be applicable to the physical world that we study
00:13:53 because they’re not the ones
00:13:54 where the Pythagorean theorem holds.
00:13:56 So we have a discovery, a genuine bonafide discovery
00:13:59 that informed the invention,
00:14:00 the invention of an abstract representation of 2D space
00:14:03 that we call R2 and things like that.
00:14:06 And then from there,
00:14:07 you just study R2 as an abstract thing
00:14:09 that brings about more ideas and inventions and mysteries
00:14:12 which themselves might yield discoveries.
00:14:14 Those discoveries might give you insight
00:14:16 as to what else would be useful to invent
00:14:19 and it kind of feeds on itself that way.
00:14:20 That’s how I think about it.
00:14:22 So it’s not an either or.
00:14:24 It’s not that math is one of these
00:14:25 or it’s one of the others.
00:14:26 At different times, it’s playing a different role.
00:14:29 So then let me ask the Richard Feynman question then,
00:14:34 along that thread,
00:14:36 is what do you think is the difference
00:14:37 between physics and math?
00:14:40 There’s a giant overlap.
00:14:43 There’s a kind of intuition that physicists have
00:14:45 about the world that’s perhaps outside of mathematics.
00:14:49 It’s this mysterious art that they seem to possess,
00:14:52 we humans generally possess.
00:14:54 And then there’s the beautiful rigor of mathematics
00:14:58 that allows you to, I mean, just like as we were saying,
00:15:02 invent frameworks of understanding our physical world.
00:15:07 So what do you think is the difference there
00:15:10 and how big is it?
00:15:11 Well, I think of math as being the study
00:15:12 of abstractions over patterns and pure patterns in logic.
00:15:16 And then physics is obviously grounded in a desire
00:15:19 to understand the world that we live in.
00:15:22 I think you’re gonna get very different answers
00:15:23 when you talk to different mathematicians
00:15:25 because there’s a wide diversity in types of mathematicians.
00:15:27 There are some who are motivated very much by pure puzzles.
00:15:30 They might be turned on by things like combinatorics.
00:15:33 And they just love the idea of building up
00:15:35 a set of problem solving tools applying to pure patterns.
00:15:40 There are some who are very physically motivated,
00:15:42 who try to invent new math or discover math in veins
00:15:48 that they know will have applications to physics
00:15:50 or sometimes computer science.
00:15:51 And that’s what drives them.
00:15:53 Like chaos theory is a good example of something
00:15:55 that’s pure math, that’s purely mathematical.
00:15:57 A lot of the statements being made,
00:15:58 but it’s heavily motivated by specific applications
00:16:02 to largely physics.
00:16:04 And then you have a type of mathematician
00:16:06 who just loves abstraction.
00:16:08 They just love pulling it to the more and more abstract
00:16:10 things, the things that feel powerful.
00:16:12 These are the ones that initially invented like topology
00:16:15 and then later on get really into category theory
00:16:17 and go on about like infinite categories and whatnot.
00:16:20 These are the ones that love to have a system
00:16:23 that can describe truths about as many things as possible.
00:16:28 People from those three different veins of motivation
00:16:31 into math are gonna give you very different answers
00:16:32 about what the relation at play here is.
00:16:34 Cause someone like Vladimir Arnold,
00:16:37 who has written a lot of great books,
00:16:40 many about like differential equations and such,
00:16:42 he would say, math is a branch of physics.
00:16:45 That’s how he would think about it.
00:16:47 And of course he was studying
00:16:48 like differential equations related things
00:16:49 because that is the motivator behind the study
00:16:52 of PDEs and things like that.
00:16:54 But you’ll have others who,
00:16:56 like especially the category theorists
00:16:58 who aren’t really thinking about physics necessarily.
00:17:01 It’s all about abstraction and the power of generality.
00:17:04 And it’s more of a happy coincidence
00:17:06 that that ends up being useful
00:17:08 for understanding the world we live in.
00:17:10 And then you can get into like, why is that the case?
00:17:12 It’s sort of surprising
00:17:14 that that which is about pure puzzles and abstraction
00:17:17 also happens to describe the very fundamentals
00:17:21 of quarks and everything else.
00:17:24 So why do you think the fundamentals of quarks
00:17:28 and the nature of reality is so compressible
00:17:33 into clean, beautiful equations
00:17:35 that are for the most part simple, relatively speaking,
00:17:39 a lot simpler than they could be?
00:17:41 So you have, we mentioned somebody like Stephen Wolfram
00:17:45 who thinks that sort of there’s incredibly simple rules
00:17:50 underlying our reality,
00:17:51 but it can create arbitrary complexity.
00:17:54 But there is simple equations.
00:17:56 What, I’m asking a million questions
00:17:59 that nobody knows the answer to, but.
00:18:01 I have no idea, why is it simple?
00:18:05 It could be the case that
00:18:07 there’s like a filter iteration at play.
00:18:08 The only things that physicists find interesting
00:18:10 are the ones that are simple enough
00:18:11 they could describe it mathematically.
00:18:13 But as soon as it’s a sufficiently complex system,
00:18:15 like, oh, that’s outside the realm of physics,
00:18:16 that’s biology or whatever have you.
00:18:19 And of course, that’s true.
00:18:21 Maybe there’s something where it’s like,
00:18:22 of course there will always be something that is simple
00:18:26 when you wash away the like non important parts
00:18:31 of whatever it is that you’re studying.
00:18:33 Just from like an information theory standpoint,
00:18:35 there might be some like,
00:18:36 you get to the lowest information component of it.
00:18:39 But I don’t know, maybe I’m just having
00:18:40 a really hard time conceiving of what it would even mean
00:18:43 for the fundamental laws to be like intrinsically
00:18:46 complicated, like some set of equations
00:18:50 that you can’t decouple from each other.
00:18:52 Well, no, it could be that sort of we take for granted
00:18:56 that the laws of physics, for example,
00:18:59 are for the most part the same everywhere
00:19:03 or something like that, right?
00:19:05 As opposed to the sort of an alternative could be
00:19:10 that the rules under which the world operates
00:19:15 is different everywhere.
00:19:17 It’s like a deeply distributed system
00:19:20 where just everything is just chaos,
00:19:23 not in a strict definition of chaos,
00:19:25 but meaning like just it’s impossible for equations
00:19:30 to capture, for to explicitly model the world
00:19:34 as cleanly as the physical does.
00:19:36 I mean, we almost take it for granted that we can describe,
00:19:39 we can have an equation for gravity,
00:19:41 for action at a distance.
00:19:42 We can have equations for some of these basic ways
00:19:45 the planet’s moving.
00:19:46 Just the low level at the atomic scale,
00:19:52 how the materials operate,
00:19:53 at the high scale, how black holes operate.
00:19:56 But it doesn’t, it seems like it could be,
00:19:59 there’s infinite other possibilities
00:20:01 where none of it could be compressible into such equations.
00:20:05 So it just seems beautiful.
00:20:06 It’s also weird, probably to the point you’re making,
00:20:10 that it’s very pleasant that this is true for our minds.
00:20:15 So it might be that our minds are biased
00:20:17 to just be looking at the parts of the universe
00:20:19 that are compressible.
00:20:21 And then we can publish papers on
00:20:23 and have nice E equals empty squared equations.
00:20:26 Right, well, I wonder would such a world
00:20:29 with uncompressible laws allow for the kind of beings
00:20:33 that can think about the kind of questions
00:20:35 that you’re asking?
00:20:37 That’s true.
00:20:38 Right, like an anthropic principle coming into play
00:20:40 in some weird way here?
00:20:42 I don’t know, like I don’t know what I’m talking about at all.
00:20:44 Maybe the universe is actually not so compressible,
00:20:47 but the way our brain, the way our brain evolved
00:20:52 we’re only able to perceive the compressible parts.
00:20:55 I mean, we are, so this is the sort of Chomsky argument.
00:20:58 We are just descendants of apes
00:20:59 over like really limited biological systems.
00:21:03 So it totally makes sense
00:21:04 that we’re really limited little computers, calculators,
00:21:08 that are able to perceive certain kinds of things
00:21:10 and the actual world is much more complicated.
00:21:13 Well, but we can do pretty awesome things, right?
00:21:16 Like we can fly spaceships
00:21:18 and we have to have some connection of reality
00:21:21 to be able to take our potentially oversimplified models
00:21:25 of the world, but then actually twist the world
00:21:27 to our will based on it.
00:21:29 So we have certain reality checks
00:21:30 that like physics isn’t too far a field
00:21:33 simply based on what we can do.
00:21:35 Yeah, the fact that we can fly is pretty good.
00:21:37 It’s great, yeah, like it’s a proof of concept
00:21:40 that the laws we’re working with are working well.
00:21:44 So I mentioned to the internet that I’m talking to you
00:21:47 and so the internet gave some questions.
00:21:50 So I apologize for these,
00:21:51 but do you think we’re living in a simulation
00:21:54 that the universe is a computer
00:21:56 or the universe is a computation running on a computer?
00:22:01 It’s conceivable.
00:22:02 What I don’t buy is, you know, you’ll have the argument
00:22:05 that, well, let’s say that it was the case
00:22:07 that you can have simulations.
00:22:09 Then the simulated world would itself
00:22:12 eventually get to a point where it’s running simulations.
00:22:15 And then the second layer down
00:22:17 would create a third layer down and on and on and on.
00:22:19 So probabilistically, you just throw a dart
00:22:21 at one of those layers,
00:22:22 we’re probably in one of the simulated layers.
00:22:24 I think if there’s some sort of limitations
00:22:27 on like the information processing
00:22:28 of whatever the physical world is,
00:22:31 like it quickly becomes the case
00:22:32 that you have a limit to the layers that could exist there
00:22:35 because like the resources necessary
00:22:38 to simulate a universe like ours clearly is a lot
00:22:41 just in terms of the number of bits at play.
00:22:43 And so then you can ask, well, what’s more plausible?
00:22:46 That there’s an unbounded capacity
00:22:49 of information processing
00:22:50 in whatever the like highest up level universe is,
00:22:53 or that there’s some bound to that capacity,
00:22:56 which then limits like the number of levels available.
00:22:58 How do you play some kind of probability distribution
00:23:00 on like what the information capacity is?
00:23:02 I have no idea.
00:23:03 But I don’t, like people almost assume
00:23:06 a certain uniform probability
00:23:08 over all of those meta layers that could conceivably exist
00:23:11 when it’s a little bit like a Pascal’s wager
00:23:15 on like you’re not giving a low enough prior
00:23:16 to the mere existence of that infinite set of layers.
00:23:20 Yeah, that’s true.
00:23:21 But it’s also very difficult to contextualize the amount.
00:23:25 So the amount of information processing power
00:23:28 required to simulate like our universe
00:23:31 seems like amazingly huge.
00:23:34 But you can always raise two to the power of that.
00:23:36 Yeah, like numbers get big.
00:23:40 And we’re easily humbled
00:23:41 by basically everything around us.
00:23:43 So it’s very difficult to kind of make sense of anything
00:23:49 actually when you look up at the sky
00:23:50 and look at the stars and the immensity of it all,
00:23:53 to make sense of the smallness of us,
00:23:57 the unlikeliness of everything
00:23:58 that’s on this earth coming to be,
00:24:02 then you could basically anything could be,
00:24:04 all laws of probability go out the window to me
00:24:09 because I guess because the amount of information
00:24:14 under which we’re operating is very low.
00:24:17 We basically know nothing about the world around us,
00:24:22 relatively speaking.
00:24:23 And so when I think about the simulation hypothesis,
00:24:26 I think it’s just fun to think about it.
00:24:29 But it’s also, I think there is a thought experiment
00:24:31 kind of interesting to think of the power of computation,
00:24:35 whether the limits of a Turing machine,
00:24:38 sort of the limits of our current computers,
00:24:41 when you start to think about artificial intelligence,
00:24:44 how far can we get with computers?
00:24:46 And that’s kind of where the simulation hypothesis
00:24:50 used with me as a thought experiment
00:24:52 is the universe just a computer?
00:24:56 Is it just a computation?
00:24:58 Is all of this just a computation?
00:25:00 And sort of the same kind of tools we apply
00:25:02 to analyzing algorithms, can that be applied?
00:25:05 If we scale further and further and further,
00:25:07 will the arbitrary power of those systems
00:25:09 start to create some interesting aspects
00:25:12 that we see in our universe?
00:25:13 Or is something fundamentally different
00:25:15 needs to be created?
00:25:17 Well, it’s interesting that in our universe,
00:25:20 it’s not arbitrarily large, the power,
00:25:22 that you can place limits on, for example,
00:25:24 how many bits of information can be stored per unit area.
00:25:27 Right, like all of the physical laws,
00:25:30 you’ve got general relativity and quantum coming together
00:25:32 to give you a certain limit on how many bits you can store
00:25:36 within a given range before it collapses into a black hole.
00:25:40 The idea that there even exists such a limit
00:25:42 is at the very least thought provoking,
00:25:44 when naively you might assume,
00:25:46 oh, well, technology could always get better and better,
00:25:49 we could get cleverer and cleverer,
00:25:50 and you could just cram as much information as you want
00:25:54 into like a small unit of space, that makes me think,
00:26:01 it’s at least plausible that whatever the highest level
00:26:06 of existence is doesn’t admit too many simulations
00:26:10 or ones that are at the scale of complexity
00:26:12 that we’re looking at.
00:26:13 Obviously, it’s just as conceivable that they do
00:26:15 and that there are many, but I guess what I’m channeling
00:26:20 is the surprise that I felt upon learning that fact,
00:26:22 that there are, that information is physical in this way.
00:26:26 There’s a finiteness to it.
00:26:27 Okay, let me just even go off on that.
00:26:29 From a mathematics perspective
00:26:31 and a psychology perspective, how do you mix,
00:26:35 are you psychologically comfortable
00:26:38 with the concept of infinity?
00:26:40 I think so.
00:26:41 Are you okay with it?
00:26:42 I’m pretty okay, yeah.
00:26:43 Are you okay?
00:26:44 No, not really, it doesn’t make any sense to me.
00:26:47 I don’t know, like how many words,
00:26:50 how many possible words do you think could exist
00:26:53 that are just like strings of letters?
00:26:55 So that’s a sort of mathematical statement as beautiful
00:26:59 and we use infinity in basically everything we do,
00:27:03 everything we do in science, math, and engineering, yes.
00:27:06 But you said exist, the question is,
00:27:12 you said letters or words?
00:27:13 I said words. Words.
00:27:16 To bring words into existence to me,
00:27:18 you have to start like saying them or like writing them
00:27:20 or like listing them.
00:27:22 That’s an instantiation.
00:27:23 Okay, how many abstract words exist?
00:27:25 Well, the idea of an abstract.
00:27:28 The idea of abstract notions and ideas.
00:27:31 I think we should be clear on terminology.
00:27:33 I mean, you think about intelligence a lot,
00:27:35 like artificial intelligence.
00:27:37 Would you not say that what it’s doing
00:27:39 is a kind of abstraction?
00:27:40 That like abstraction is key
00:27:42 to conceptualizing the universe?
00:27:45 You get this raw sensory data.
00:27:47 I need something that every time you move your face
00:27:49 a little bit and they’re not pixels,
00:27:51 but like analog of pixels on my retina changed entirely,
00:27:55 that I can still have some coherent notion of this is Lex,
00:27:58 I’m talking to Lex, right?
00:27:59 What that requires is you have a disparate set
00:28:01 of possible images hitting me
00:28:03 that are unified in a notion of Lex, right?
00:28:07 That’s a kind of abstraction.
00:28:08 It’s a thing that could apply
00:28:09 to a lot of different images that I see
00:28:12 and it represents it in a much more compressed way
00:28:15 and one that’s like much more resilient to that.
00:28:17 I think in the same way,
00:28:18 if I’m talking about infinity as an abstraction,
00:28:21 I don’t mean nonphysical woo woo,
00:28:24 like ineffable or something.
00:28:26 What I mean is it’s something that can apply
00:28:28 to a multiplicity of situations
00:28:30 that share a certain common attribute
00:28:31 in the same way that the images of like your face
00:28:33 on my retina share enough common attributes
00:28:35 that I can put the single notion to it.
00:28:37 Like in that way, infinity is an abstraction
00:28:40 and it’s very powerful and it’s only through
00:28:43 such abstractions that we can actually understand
00:28:45 like the world and logic and things.
00:28:47 And in the case of infinity,
00:28:48 the way I think about it,
00:28:49 the key entity is the property
00:28:51 of always being able to add one more.
00:28:54 Like no matter how many words you can list,
00:28:56 you just throw an A at the end of one
00:28:57 and you have another conceivable word.
00:28:59 You don’t have to think of all the words at once.
00:29:01 It’s that property, the oh, I could always add one more
00:29:04 that gives it this nature of infiniteness
00:29:08 in the same way that there’s certain like properties
00:29:09 of your face that give it the Lexness, right?
00:29:13 So like infinity should be no more worrying
00:29:16 than the I can always add one more sentiment.
00:29:19 That’s a really elegant,
00:29:21 much more elegant way than I could put it.
00:29:23 So thank you for doing that as yet another abstraction.
00:29:26 And yes, indeed, that’s what our brain does.
00:29:29 That’s what intelligent systems do.
00:29:30 That’s what programming does.
00:29:31 That’s what science does is build abstraction
00:29:34 on top of each other.
00:29:35 And yet there is at a certain point abstractions
00:29:39 that go into the quote woo, right?
00:29:42 Sort of, and because we’re now,
00:29:47 it’s like we built this stack of, you know,
00:29:52 the only thing that’s true is the stuff that’s on the ground.
00:29:54 Everything else is useful for interpreting this.
00:29:57 And at a certain point you might start floating
00:30:00 into ideas that are surreal and difficult
00:30:04 and take us into areas that are disconnected
00:30:08 from reality in a way that we could never get back.
00:30:11 What if instead of calling these abstract,
00:30:13 how different would it be in your mind
00:30:14 if we called them general?
00:30:15 And the phenomenon that you’re describing
00:30:17 is overgeneralization.
00:30:19 When you try to have a concept or an idea
00:30:21 that’s so general as to apply to nothing in particular
00:30:24 in a useful way, does that map to what you’re thinking
00:30:27 of when you think of?
00:30:28 First of all, I’m playing little just for the fun of it.
00:30:31 Devil’s advocate.
00:30:32 And I think our cognition, our mind is unable
00:30:37 to visualize.
00:30:39 So you do some incredible work with visualization and video.
00:30:42 I think infinity is very difficult to visualize
00:30:46 for our mind.
00:30:48 We can delude ourselves into thinking we can visualize it,
00:30:52 but we can’t.
00:30:54 I don’t, I mean, I don’t,
00:30:56 I would venture to say it’s very difficult.
00:30:57 And so there’s some concepts of mathematics,
00:31:00 like maybe multiple dimensions,
00:31:02 we could sort of talk about that are impossible
00:31:04 for us to truly intuit, like,
00:31:08 and it just feels dangerous to me to use these
00:31:13 as part of our toolbox of abstractions.
00:31:16 On behalf of your listeners,
00:31:17 I almost fear we’re getting too philosophical.
00:31:19 Right?
00:31:20 Heck no.
00:31:21 Heck no.
00:31:22 I think to that point for any particular idea like this,
00:31:26 there’s multiple angles of attack.
00:31:28 I think the, when we do visualize infinity,
00:31:31 what we’re actually doing, you know,
00:31:33 you write dot, dot, dot, right?
00:31:34 One, two, three, four, dot, dot, dot, right?
00:31:37 Those are symbols on the page
00:31:37 that are insinuating a certain infinity.
00:31:42 What you’re capturing with a little bit of design there
00:31:45 is the I can always add one more property, right?
00:31:49 I think I’m just as uncomfortable with you are
00:31:52 if you try to concretize it so much
00:31:56 that you have a bag of infinitely many things
00:31:58 that I actually think of, no, not one, two, three, four,
00:32:00 dot, dot, dot, one, two, three, four, five, six, seven, eight.
00:32:03 I try to get them all in my head and you realize,
00:32:05 oh, you know, your brain would literally collapse
00:32:08 into a black hole, all of that.
00:32:10 And I honestly feel this with a lot of math
00:32:12 that I try to read where I don’t think of myself
00:32:15 as like particularly good at math in some ways.
00:32:19 Like I get very confused often
00:32:21 when I am going through some of these texts.
00:32:23 And often what I’m feeling in my head is like,
00:32:25 this is just so damn abstract.
00:32:27 I just can’t wrap my head around it.
00:32:29 I just want to put something concrete to it
00:32:31 that makes me understand.
00:32:32 And I think a lot of the motivation for the channel
00:32:35 is channeling that sentiment of, yeah,
00:32:38 a lot of the things that you’re trying to read out there,
00:32:40 it’s just so hard to connect to anything
00:32:43 that you spend an hour banging your head
00:32:45 against a couple of pages and you come out
00:32:47 not really knowing anything more
00:32:49 other than some definitions maybe
00:32:51 and a certain sense of self defeat, right?
00:32:55 One of the reasons I focus so much on visualizations
00:32:58 is that I’m a big believer in,
00:33:01 I’m sorry, I’m just really hampering on
00:33:03 this idea of abstraction,
00:33:04 being clear about your layers of abstraction, right?
00:33:07 It’s always tempting to start an explanation
00:33:09 from the top to the bottom, okay?
00:33:11 You give the definition of a new theorem.
00:33:14 You’re like, this is the definition of a vector space.
00:33:16 For example, that’s how we’ll start a course.
00:33:18 These are the properties of a vector space.
00:33:20 First from these properties, we will derive what we need
00:33:23 in order to do the math of linear algebra
00:33:25 or whatever it might be.
00:33:26 I don’t think that’s how understanding works at all.
00:33:28 I think how understanding works
00:33:29 is you start at the lowest level you can get at
00:33:32 where rather than thinking about a vector space,
00:33:34 you might think of concrete vectors
00:33:36 that are just lists of numbers
00:33:37 or picturing it as like an arrow that you draw,
00:33:41 which is itself like even less abstract than numbers
00:33:44 because you’re looking at quantities,
00:33:45 like the distance of the x coordinate,
00:33:47 the distance of the y coordinate.
00:33:48 It’s as concrete as you could possibly get
00:33:50 and it has to be if you’re putting it in a visual, right?
00:33:53 It’s an actual arrow. It’s an actual vector.
00:33:56 You’re not talking about like a quote unquote vector
00:33:59 that could apply to any possible thing.
00:34:01 You have to choose one if you’re illustrating it.
00:34:03 And I think this is the power of being in a medium
00:34:05 like video or if you’re writing a textbook
00:34:08 and you force yourself to put a lot of images
00:34:10 is with every image, you’re making a choice.
00:34:13 With each choice, you’re showing a concrete example.
00:34:16 With each concrete example,
00:34:17 you’re aiding someone’s path to understanding.
00:34:19 I’m sorry to interrupt you,
00:34:21 but you just made me realize that that’s exactly right.
00:34:24 So the visualizations you’re creating
00:34:26 while you’re sometimes talking about abstractions,
00:34:29 the actual visualization is an explicit low level example.
00:34:34 Yes.
00:34:35 So there’s an actual, like in the code,
00:34:37 you have to say what the vector is,
00:34:40 what’s the direction of the arrow,
00:34:42 what’s the magnitude of the, yeah.
00:34:44 So that’s, you’re going, the visualization itself
00:34:48 is actually going to the bottom of that.
00:34:50 And I think that’s very important.
00:34:52 I also think about this a lot in writing scripts
00:34:54 where even before you get to the visuals,
00:34:57 the first instinct is to, I don’t know why,
00:35:00 I just always do, I say the abstract thing,
00:35:02 I say the general definition, the powerful thing,
00:35:05 and then I fill it in with examples later.
00:35:07 Always, it will be more compelling
00:35:08 and easier to understand when you flip that.
00:35:10 And instead, you let someone’s brain
00:35:13 do the pattern recognition.
00:35:16 You just show them a bunch of examples.
00:35:18 The brain is gonna feel a certain similarity between them.
00:35:21 Then by the time you bring in the definition,
00:35:23 or by the time you bring in the formula,
00:35:25 it’s articulating a thing that’s already in the brain
00:35:28 that was built off of looking at a bunch of examples
00:35:30 with a certain kind of similarity.
00:35:32 And what the formula does is articulate
00:35:34 what that kind of similarity is,
00:35:36 rather than being a high cognitive load set of symbols
00:35:42 that needs to be populated with examples later on,
00:35:45 assuming someone’s still with you.
00:35:48 What is the most beautiful or awe inspiring idea
00:35:51 you’ve come across in mathematics?
00:35:53 I don’t know, man.
00:35:55 Maybe it’s an idea you’ve explored in your videos,
00:35:57 maybe not.
00:35:58 What just gave you pause?
00:36:01 What’s the most beautiful idea?
00:36:03 Small or big.
00:36:04 So I think often, the things that are most beautiful
00:36:07 are the ones that you have a little bit of understanding of,
00:36:11 but certainly not an entire understanding.
00:36:14 It’s a little bit of that mystery
00:36:15 that is what makes it beautiful.
00:36:17 What was the moment of the discovery for you personally,
00:36:20 almost just that leap of aha moment?
00:36:23 So something that really caught my eye,
00:36:25 I remember when I was little, there were these,
00:36:29 I think the series was called like wooden books
00:36:31 or something, these tiny little books
00:36:32 that would have just a very short description
00:36:34 of something on the left and then a picture on the right.
00:36:36 I don’t know who they’re meant for,
00:36:37 but maybe it’s like loosely children
00:36:39 or something like that.
00:36:40 But it can’t just be children,
00:36:41 because of some of the things I was describing.
00:36:43 On the last page of one of them,
00:36:45 somewhere tiny in there was this little formula
00:36:47 that on the left hand had a sum
00:36:49 over all of the natural numbers.
00:36:51 It’s like one over one to the S plus one over two to the S
00:36:54 plus one over three to the S on and on to the infinity.
00:36:57 Then on the other side had a product over all of the primes
00:37:01 and it was a certain thing had to do with all the primes.
00:37:03 And like any good young math enthusiast,
00:37:06 I’d probably been indoctrinated with how chaotic
00:37:08 and confusing the primes are, which they are.
00:37:10 And seeing this equation where on one side
00:37:14 you have something that’s as understandable
00:37:15 as you could possibly get, the counting numbers.
00:37:18 And on the other side is all the prime numbers.
00:37:20 It was like this, whoa, they’re related like this?
00:37:23 There’s a simple description that includes
00:37:26 all the primes getting wrapped together like this.
00:37:28 This is like the Euler product for the Zeta function,
00:37:32 as I like later found out.
00:37:33 The equation itself essentially encodes
00:37:36 the fundamental theorem of arithmetic
00:37:37 that every number can be expressed
00:37:39 as a unique set of primes.
00:37:42 To me still there’s, I mean, I certainly don’t understand
00:37:44 this equation or this function all that well.
00:37:47 The more I learn about it, the prettier it is.
00:37:50 The idea that you can, this is sort of what gets you
00:37:53 representations of primes, not in terms of primes themselves,
00:37:57 but in terms of another set of numbers.
00:37:59 They’re like the non trivial zeros of the Zeta function.
00:38:01 And again, I’m very kind of in over my head
00:38:04 in a lot of ways as I like try to get to understand it.
00:38:06 But the more I do, it always leaves enough mystery
00:38:09 that it remains very beautiful to me.
00:38:11 So whenever there’s a little bit of mystery
00:38:13 just outside of the understanding that,
00:38:16 and by the way, the process of learning more about it,
00:38:19 how does that come about?
00:38:20 Just your own thought or are you reading?
00:38:23 Reading, yeah.
00:38:24 Or is the process of visualization itself
00:38:26 revealing more to you?
00:38:28 Visuals help.
00:38:29 I mean, in one time when I was just trying to understand
00:38:31 like analytic continuation and playing around
00:38:33 with visualizing complex functions,
00:38:36 this is what led to a video about this function.
00:38:39 It’s titled something like
00:38:40 Visualizing the Riemann Zeta Function.
00:38:42 It’s one that came about because I was programming
00:38:45 and tried to see what a certain thing looked like.
00:38:47 And then I looked at it and I’m like,
00:38:48 whoa, that’s elucidating.
00:38:50 And then I decided to make a video about it.
00:38:53 But I mean, you try to get your hands on
00:38:56 as much reading as you can.
00:38:58 You know, in this case, I think if anyone wants to start
00:39:01 to understand it, if they have like a math background
00:39:05 like they studied some in college or something like that,
00:39:08 like the Princeton Companion to Math
00:39:10 has a really good article on analytic number theory.
00:39:13 And that itself has a whole bunch of references
00:39:15 and you know, anything has more references
00:39:17 and it gives you this like tree to start piling through.
00:39:20 And like, you know, you try to understand,
00:39:22 I try to understand things visually as I go.
00:39:24 That’s not always possible,
00:39:26 but it’s very helpful when it does.
00:39:28 You recognize when there’s common themes,
00:39:30 like in this case, Cousins of the Fourier Transform
00:39:34 that come into play and you realize,
00:39:35 oh, it’s probably pretty important
00:39:37 to have deep intuitions of the Fourier Transform,
00:39:39 even if it’s not explicitly mentioned in like these texts.
00:39:42 And you try to get a sense of what the common players are.
00:39:45 But I’ll emphasize again, like,
00:39:47 I feel very in over my head when I try to understand
00:39:50 the exact relation between like the zeros
00:39:53 of the Riemann Zeta function
00:39:54 and how they relate to the distribution of primes.
00:39:56 I definitely understand it better than I did a year ago.
00:39:59 I definitely understand it on 100th as well as the experts
00:40:02 on the matter do, I assume.
00:40:04 But the slow path towards getting there is,
00:40:08 it’s fun, it’s charming,
00:40:09 and like to your question, very beautiful.
00:40:12 And the beauty is in the, what,
00:40:14 in the journey versus the destination?
00:40:17 Well, it’s that each thing doesn’t feel arbitrary.
00:40:19 I think that’s a big part,
00:40:20 is that you have these unpredictable,
00:40:23 no, yeah, these very unpredictable patterns
00:40:25 or these intricate properties of like a certain function.
00:40:30 But at the same time,
00:40:31 it doesn’t feel like humans ever made an arbitrary choice
00:40:33 in studying this particular thing.
00:40:35 So, you know, it feels like you’re speaking
00:40:38 to patterns themselves or nature itself.
00:40:41 That’s a big part of it.
00:40:43 I think things that are too arbitrary,
00:40:45 it’s just hard for those to feel beautiful
00:40:46 because this is sort of what the word contrived
00:40:49 is meant to apply to, right?
00:40:53 And when they’re not arbitrary means it could be,
00:40:57 you can have a clean abstraction and intuition
00:41:02 that allows you to comprehend it.
00:41:04 Well, to one of your first questions,
00:41:06 it makes you feel like if you came across
00:41:07 another intelligent civilization,
00:41:09 that they’d be studying the same thing.
00:41:12 Maybe with different notation.
00:41:13 Certainly, yeah, but yeah.
00:41:15 Like that’s what,
00:41:16 I think you talked to that other civilization,
00:41:18 they’re probably also studying the zeros
00:41:20 of the Riemann Zeta function
00:41:21 or like some variant thereof
00:41:23 that is like a clearly equivalent cousin
00:41:27 or something like that.
00:41:28 But that’s probably on their docket.
00:41:32 Whenever somebody does a lot of something amazing,
00:41:35 I’m gonna ask the question
00:41:37 that you’ve already been asked a lot
00:41:40 and that you’ll get more and more asked in your life.
00:41:43 But what was your favorite video to create?
00:41:46 Oh, favorite to create.
00:41:49 One of my favorites is,
00:41:51 the title is Who Cares About Topology?
00:41:54 You want me to pull it up or no?
00:41:55 If you want, sure, yeah.
00:41:57 It is about, well, it starts by describing
00:42:00 an unsolved problem that’s still unsolved in math
00:42:03 called the inscribed square problem.
00:42:05 You draw any loop and then you ask,
00:42:06 are there four points on that loop that make a square?
00:42:09 Totally useless, right?
00:42:10 This is not answering any physical questions.
00:42:12 It’s mostly interesting that we can’t answer that question.
00:42:14 And it seems like such a natural thing to ask.
00:42:18 Now, if you weaken it a little bit and you ask,
00:42:21 can you always find a rectangle?
00:42:22 You choose four points on this curve,
00:42:24 can you find a rectangle?
00:42:25 That’s hard, but it’s doable.
00:42:27 And the path to it involves things like looking at a torus,
00:42:32 this surface with a single hole in it, like a donut,
00:42:35 or looking at a mobius strip.
00:42:37 In ways that feel so much less contrived
00:42:39 to when I first, as like a little kid,
00:42:41 learned about these surfaces and shapes,
00:42:43 like a mobius strip and a torus.
00:42:45 Like what you learn is, oh, this mobius strip,
00:42:47 you take a piece of paper, put a twist, glue it together,
00:42:50 and now you have a shape with one edge and just one side.
00:42:53 And as a student, you should think, who cares, right?
00:42:58 Like, how does that help me solve any problems?
00:43:00 I thought math was about problem solving.
00:43:02 So what I liked about the piece of math
00:43:05 that this was describing that was in this paper
00:43:08 by a mathematician named Vaughn
00:43:10 was that it arises very naturally.
00:43:12 It’s clear what it represents.
00:43:14 It’s doing something.
00:43:15 It’s not just playing with construction paper.
00:43:17 And the way that it solves the problem is really beautiful.
00:43:21 So kind of putting all of that down
00:43:24 and concretizing it, right?
00:43:25 Like I was talking about how
00:43:27 when you have to put visuals to it,
00:43:29 it demands that what’s on screen
00:43:30 is a very specific example of what you’re describing.
00:43:33 The construction here is very abstract in nature.
00:43:35 You describe this very abstract kind of surface in 3D space.
00:43:39 So then when I was finding myself,
00:43:40 in this case, I wasn’t programming,
00:43:42 I was using a grapher that’s like built into OSX
00:43:44 for the 3D stuff to draw that surface,
00:43:48 you realize, oh man, the topology argument
00:43:50 is very non constructive.
00:43:52 I have to make a lot of,
00:43:54 you have to do a lot of extra work
00:43:55 in order to make the surface show up.
00:43:57 But then once you see it, it’s quite pretty
00:43:59 and it’s very satisfying to see a specific instance of it.
00:44:02 And you also feel like, ah,
00:44:03 I’ve actually added something
00:44:04 on top of what the original paper was doing
00:44:06 that it shows something that’s completely correct.
00:44:09 That’s a very beautiful argument,
00:44:10 but you don’t see what it looks like.
00:44:12 And I found something satisfying
00:44:14 in seeing what it looked like
00:44:16 that could only ever have come about
00:44:17 from the forcing function
00:44:19 of getting some kind of image on the screen
00:44:21 to describe the thing I was talking about.
00:44:22 So you almost weren’t able to anticipate
00:44:24 what it’s gonna look like.
00:44:25 I had no idea.
00:44:26 I had no idea.
00:44:27 And it was wonderful, right?
00:44:28 It was totally, it looks like a Sydney Opera House
00:44:30 or some sort of Frank Gehry design.
00:44:32 And it was, you knew it was gonna be something
00:44:35 and you can say various things about it.
00:44:36 Like, oh, it touches the curve itself.
00:44:39 It has a boundary that’s this curve on the 2D plane.
00:44:42 It all sits above the plane.
00:44:43 But before you actually draw it,
00:44:45 it’s very unclear what the thing will look like.
00:44:48 And to see it, it’s very, it’s just pleasing, right?
00:44:50 So that was fun to make, very fun to share.
00:44:53 I hope that it has elucidated for some people out there
00:44:58 where these constructs of topology come from,
00:45:00 that it’s not arbitrary play with construction paper.
00:45:04 So let’s, I think this is a good sort of example
00:45:07 to talk a little bit about your process.
00:45:09 You have a list of ideas.
00:45:12 So that’s sort of the curse of having an active
00:45:17 and brilliant mind is I’m sure you have a list
00:45:19 that’s growing faster than you can utilize.
00:45:22 Now I’m ahead, absolutely.
00:45:24 But there’s some sorting procedure
00:45:26 depending on mood and interest and so on.
00:45:29 But okay, so you pick an idea
00:45:32 and then you have to try to write a narrative arc
00:45:36 that sort of, how do I elucidate?
00:45:38 How do I make this idea beautiful and clear
00:45:41 and explain it?
00:45:42 And then there’s a set of visualizations
00:45:44 that will be attached to it.
00:45:46 Sort of, you’ve talked about some of this before,
00:45:48 but sort of writing the story, attaching the visualizations.
00:45:52 Can you talk through interesting, painful,
00:45:56 beautiful parts of that process?
00:45:58 Well, the most painful is if you’ve chosen a topic
00:46:02 that you do want to do, but then it’s hard to think of,
00:46:05 I guess how to structure the script.
00:46:07 This is sort of where I have been on one
00:46:10 for like the last two or three months.
00:46:12 And I think that ultimately the right resolution
00:46:13 is just like set it aside and instead do some other things
00:46:17 where the script comes more naturally.
00:46:18 Because you sort of don’t want to overwork a narrative.
00:46:23 The more you’ve thought about it,
00:46:24 the less you can empathize with the student
00:46:26 who doesn’t yet understand the thing you’re trying to teach.
00:46:28 Who is the judger in your head?
00:46:31 Sort of the person, the creature,
00:46:35 the essence that’s saying this sucks or this is good.
00:46:38 And you mentioned kind of the student you’re thinking about.
00:46:43 Can you, who is that?
00:46:44 What is that thing?
00:46:45 That says, the perfectionist that says this thing sucks.
00:46:49 You need to work on that for another two, three months.
00:46:53 I don’t know.
00:46:54 I think it’s my past self.
00:46:56 I think that’s the entity that I’m most trying
00:46:58 to empathize with is like you take who I was,
00:47:00 because that’s kind of the only person I know.
00:47:02 Like you don’t really know anyone
00:47:03 other than versions of yourself.
00:47:05 So I start with the version of myself that I know
00:47:07 who doesn’t yet understand the thing, right?
00:47:10 And then I just try to view it with fresh eyes,
00:47:15 a particular visual or a particular script.
00:47:17 Like, is this motivating?
00:47:18 Does this make sense?
00:47:20 Which has its downsides,
00:47:21 because sometimes I find myself speaking to motivations
00:47:25 that only myself would be interested in.
00:47:28 I don’t know, like I did this project on quaternions
00:47:30 where what I really wanted was to understand
00:47:33 what are they doing in four dimensions?
00:47:34 Can we see what they’re doing in four dimensions, right?
00:47:37 And I came up with a way of thinking about it
00:47:40 that really answered the question in my head
00:47:42 that made me very satisfied
00:47:43 and being able to think about concretely with a 3D visual,
00:47:46 what are they doing to a 4D sphere?
00:47:48 And so I’m like, great,
00:47:49 this is exactly what my past self would have wanted, right?
00:47:51 And I make a thing on it.
00:47:52 And I’m sure it’s what some other people wanted too.
00:47:55 But in hindsight, I think most people who wanna learn
00:47:57 about quaternions are like robotics engineers
00:48:00 or graphics programmers who want to understand
00:48:03 how they’re used to describe 3D rotations.
00:48:06 And like their use case was actually a little bit different
00:48:08 than my past self.
00:48:09 And in that way, like,
00:48:10 I wouldn’t actually recommend that video
00:48:12 to people who are coming at it from that angle
00:48:14 of wanting to know, hey, I’m a robotics programmer.
00:48:17 Like, how do these quaternion things work
00:48:20 to describe position in 3D space?
00:48:22 I would say other great resources for that.
00:48:25 If you ever find yourself wanting to say like,
00:48:27 but hang on,
00:48:28 in what sense are they acting in four dimensions?
00:48:30 Then come back.
00:48:31 But until then, that’s a little different.
00:48:34 Yeah, it’s interesting
00:48:35 because you have incredible videos on neural networks,
00:48:38 for example.
00:48:39 And from my sort of perspective,
00:48:41 because I’ve probably, I mean,
00:48:43 I looked at the,
00:48:45 is sort of my field
00:48:47 and I’ve also looked at the basic introduction
00:48:49 of neural networks like a million times
00:48:51 from different perspectives.
00:48:52 And it made me realize
00:48:53 that there’s a lot of ways to present it.
00:48:55 So you were sort of, you did an incredible job.
00:48:58 I mean, sort of the,
00:49:01 but you could also do it differently
00:49:03 and also incredible.
00:49:04 Like to create a beautiful presentation of a basic concept
00:49:11 requires sort of creativity, requires genius and so on,
00:49:16 but you can take it from a bunch of different perspectives.
00:49:18 And that video on neural networks made me realize that.
00:49:21 And just as you’re saying,
00:49:22 you kind of have a certain mindset, a certain view,
00:49:26 but from a, if you take a different view
00:49:28 from a physics perspective,
00:49:30 from a neuroscience perspective,
00:49:33 talking about neural networks
00:49:34 or from a robotics perspective,
00:49:38 or from, let’s see,
00:49:40 from a pure learning, statistics perspective.
00:49:43 So you can create totally different videos.
00:49:46 And you’ve done that with a few actually concepts
00:49:48 where you’ve have taken different cuts,
00:49:49 like at the Euler equation, right?
00:49:54 You’ve taken different views of that.
00:49:56 I think I’ve made three videos on it
00:49:58 and I definitely will make at least one more.
00:50:01 Right?
00:50:02 Never enough.
00:50:03 Never enough.
00:50:04 So you don’t think it’s the most beautiful equation
00:50:06 in mathematics?
00:50:08 Like I said, as we represent it,
00:50:10 it’s one of the most hideous.
00:50:11 It involves a lot of the most hideous aspects
00:50:13 of our notation.
00:50:14 I talked about E, the fact that we use pi instead of tau,
00:50:16 the fact that we call imaginary numbers imaginary,
00:50:20 and then, hence, I actually wonder if we use the I
00:50:23 because of imaginary.
00:50:24 I don’t know if that’s historically accurate,
00:50:26 but at least a lot of people,
00:50:27 they read the I and they think imaginary.
00:50:30 Like all three of those facts,
00:50:31 it’s like those are things that have added more confusion
00:50:33 than they needed to,
00:50:34 and we’re wrapping them up in one equation.
00:50:35 Like boy, that’s just very hideous, right?
00:50:39 The idea is that it does tie together
00:50:40 when you wash away the notation.
00:50:42 Like it’s okay, it’s pretty, it’s nice,
00:50:44 but it’s not like mind blowing greatest thing
00:50:48 in the universe,
00:50:49 which is maybe what I was thinking of when I said,
00:50:52 like once you understand something,
00:50:53 it doesn’t have the same beauty.
00:50:55 Like I feel like I understand Euler’s formula,
00:50:59 and I feel like I understand it enough
00:51:00 to sort of see the version that just woke up
00:51:05 that hasn’t really gotten itself dressed in the morning
00:51:07 that’s a little bit groggy,
00:51:08 and there’s bags under its eyes.
00:51:10 So you’re past the dating stage,
00:51:13 you’re no longer dating, right?
00:51:15 I’m still dating the Zeta function,
00:51:16 and like she’s beautiful and right,
00:51:18 and like we have fun,
00:51:20 and it’s that high dopamine part,
00:51:22 but like maybe at some point
00:51:24 we’ll settle into the more mundane nature of the relationship
00:51:26 where I like see her for who she truly is,
00:51:28 and she’ll still be beautiful in her own way,
00:51:30 but it won’t have the same romantic pizzazz, right?
00:51:33 Well, that’s the nice thing about mathematics.
00:51:35 I think as long as you don’t live forever,
00:51:38 there’ll always be enough mystery and fun
00:51:41 with some of the equations.
00:51:42 Even if you do, the rate at which questions comes up
00:51:45 is much faster than the rate at which answers come up, so.
00:51:48 If you could live forever, would you?
00:51:51 I think so, yeah.
00:51:52 So you think, you don’t think mortality
00:51:53 is the thing that makes life meaningful?
00:51:55 Would your life be four times as meaningful
00:51:58 if you died at 25?
00:52:00 So this goes to infinity.
00:52:02 I think you and I, that’s really interesting.
00:52:04 So what I said is infinite, not four times longer.
00:52:09 I said infinite.
00:52:10 So the actual existence of the finiteness,
00:52:15 the existence of the end, no matter the length,
00:52:18 is the thing that may sort of,
00:52:20 from my comprehension of psychology,
00:52:22 it’s such a deeply human,
00:52:25 it’s such a fundamental part of the human condition,
00:52:28 the fact that there is, that we’re mortal,
00:52:31 that the fact that things end,
00:52:34 it seems to be a crucial part of what gives them meaning.
00:52:37 I don’t think, at least for me,
00:52:40 it’s a very small percentage of my time
00:52:43 that mortality is salient,
00:52:45 that I’m aware of the end of my life.
00:52:47 What do you mean by me?
00:52:50 I’m trolling.
00:52:51 Is it the ego, is it the id, or is it the superego?
00:52:55 The reflective self, the Wernicke’s area
00:52:58 that puts all this stuff into words.
00:52:59 Yeah, a small percentage of your mind
00:53:02 that is actually aware of the true motivations
00:53:05 that drive you.
00:53:06 But my point is that most of my life,
00:53:08 I’m not thinking about death,
00:53:09 but I still feel very motivated to make things
00:53:12 and to interact with people,
00:53:14 experience love or things like that.
00:53:15 I’m very motivated,
00:53:16 and it’s strange that that motivation comes
00:53:19 while death is not in my mind at all.
00:53:21 And this might just be because I’m young enough
00:53:23 that it’s not salient.
00:53:24 Or it’s in your subconscious,
00:53:25 or that you’ve constructed an illusion
00:53:28 that allows you to escape the fact of your mortality
00:53:31 by enjoying the moment,
00:53:32 sort of the existential approach to life.
00:53:34 Could be.
00:53:36 Gun to my head, I don’t think that’s it.
00:53:38 Yeah, another sort of way to say gun to the head
00:53:40 is sort of the deep psychological introspection
00:53:43 of what drives us.
00:53:44 I mean, that’s, in some ways to me,
00:53:47 I mean, when I look at math, when I look at science,
00:53:49 is a kind of an escape from reality
00:53:51 in a sense that it’s so beautiful.
00:53:54 It’s such a beautiful journey of discovery
00:53:58 that it allows you to actually,
00:54:00 it sort of allows you to achieve a kind of immortality
00:54:04 of explore ideas and sort of connect yourself
00:54:09 to the thing that is seemingly infinite,
00:54:12 like the universe, right?
00:54:13 That allows you to escape the limited nature
00:54:18 of our little, of our bodies, of our existence.
00:54:24 What else would give this podcast meaning?
00:54:25 That’s right.
00:54:26 If not the fact that it will end.
00:54:28 This place closes in 40 minutes.
00:54:30 And it’s so much more meaningful for it.
00:54:33 How much more I love this room
00:54:35 because we’ll be kicked out.
00:54:38 So I understand just because you’re trolling me
00:54:42 doesn’t mean I’m wrong.
00:54:46 But I take your point.
00:54:47 I take your point.
00:54:49 Boy, that would be a good Twitter bio.
00:54:52 Just because you’re trolling me doesn’t mean I’m wrong.
00:54:54 Yeah, and sort of difference in backgrounds.
00:54:58 I’m a bit Russian, so we’re a bit melancholic
00:55:01 and seem to maybe assign a little too much value
00:55:04 to suffering and mortality and things like that.
00:55:07 Makes for a better novel, I think.
00:55:09 Oh yeah, you need some sort of existential threat
00:55:13 to drive a plot.
00:55:16 So when do you know when the video is done
00:55:18 when you’re working on it?
00:55:20 That’s pretty easy actually,
00:55:21 because I’ll write the script.
00:55:24 I want there to be some kind of aha moment in there.
00:55:27 And then hopefully the script can revolve around
00:55:28 some kind of aha moment.
00:55:30 And then from there, you’re putting visuals
00:55:32 to each sentence that exists,
00:55:34 and then you narrate it, you edit it all together.
00:55:36 So given that there’s a script,
00:55:37 the end becomes quite clear.
00:55:40 And as I animate it, I often change
00:55:45 certainly the specific words,
00:55:46 but sometimes the structure itself.
00:55:49 But it’s a very deterministic process at that point.
00:55:53 It makes it much easier to predict
00:55:54 when something will be done.
00:55:55 How do you know when a script is done?
00:55:57 It’s like, for problem solving videos,
00:55:59 that’s quite simple.
00:56:00 It’s once you feel like someone
00:56:01 who didn’t understand the solution now could.
00:56:03 For things like neural networks,
00:56:04 that was a lot harder because like you said,
00:56:06 there’s so many angles at which you could attack it.
00:56:09 And there, it’s just at some point
00:56:11 you feel like this asks a meaningful question
00:56:15 and it answers that question, right?
00:56:18 What is the best way to learn math
00:56:20 for people who might be at the beginning of that journey?
00:56:22 I think that’s a question that a lot of folks
00:56:24 kind of ask and think about.
00:56:26 And it doesn’t, even for folks
00:56:27 who are not really at the beginning of their journey,
00:56:29 like there might be actually deep in their career,
00:56:33 some type they’ve taken college
00:56:35 or taken calculus and so on,
00:56:36 but still wanna sort of explore math.
00:56:39 What would be your advice instead of education at all ages?
00:56:42 Your temptation will be to spend more time
00:56:45 like watching lectures or reading.
00:56:48 Try to force yourself to do more problems
00:56:50 than you naturally would.
00:56:52 That’s a big one.
00:56:53 Like the focus time that you’re spending
00:56:56 should be on like solving specific problems
00:56:59 and seek entities that have well curated lists of problems.
00:57:02 So go into like a textbook almost
00:57:04 and the problems in the back of a textbook kind of thing,
00:57:07 back of a chapter.
00:57:08 So if you can take a little look through those questions
00:57:10 at the end of the chapter before you read the chapter,
00:57:12 a lot of them won’t make sense.
00:57:13 Some of them might,
00:57:14 and those are the best ones to think about.
00:57:16 A lot of them won’t, but just take a quick look
00:57:18 and then read a little bit of the chapter
00:57:20 and then maybe take a look again and things like that.
00:57:22 And don’t consider yourself done with the chapter
00:57:25 until you’ve actually worked through a couple exercises.
00:57:29 And this is so hypocritical, right?
00:57:31 Cause I like put out videos
00:57:32 that pretty much never have associated exercises.
00:57:35 I just view myself as a different part of the ecosystem,
00:57:38 which means I’m kind of admitting
00:57:40 that you’re not really learning,
00:57:42 or at least this is only a partial part
00:57:44 of the learning process if you’re watching these videos.
00:57:48 I think if someone’s at the very beginning,
00:57:50 like I do think Khan Academy does a good job.
00:57:52 They have a pretty large set of questions
00:57:54 you can work through.
00:57:55 Just the very basics,
00:57:56 sort of just picking up,
00:57:58 getting comfortable with the very basic linear algebra,
00:58:01 calculus or so on, Khan Academy.
00:58:04 Programming is actually I think a great,
00:58:05 like learn to program and like let the way
00:58:08 that math is motivated from that angle push you through.
00:58:11 I know a lot of people who didn’t like math
00:58:14 got into programming in some way
00:58:15 and that’s what turned them on to math.
00:58:17 Maybe I’m biased cause like I live in the Bay area,
00:58:19 so I’m more likely to run into someone
00:58:21 who has that phenotype.
00:58:23 But I am willing to speculate
00:58:25 that that is a more generalizable path.
00:58:28 So you yourself kind of in creating the videos
00:58:30 are using programming to illuminate a concept,
00:58:32 but for yourself as well.
00:58:35 So would you recommend somebody try to make a,
00:58:37 sort of almost like try to make videos?
00:58:40 Like you do as a way to learn?
00:58:41 So one thing I’ve heard before,
00:58:43 I don’t know if this is based on any actual study.
00:58:44 This might be like a total fictional anecdote of numbers,
00:58:47 but it rings in the mind as being true.
00:58:49 You remember about 10% of what you read,
00:58:51 you remember about 20% of what you listen to,
00:58:54 you remember about 70% of what you actively interact with
00:58:57 in some way, and then about 90% of what you teach.
00:59:00 This is a thing I heard again,
00:59:02 those numbers might be meaningless,
00:59:03 but they ring true, don’t they, right?
00:59:05 I’m willing to say I learned nine times better
00:59:07 if I’m teaching something than reading.
00:59:09 That might even be a low ball, right?
00:59:11 So doing something to teach
00:59:12 or to like actively try to explain things
00:59:15 is huge for consolidating the knowledge.
00:59:17 Outside of family and friends,
00:59:19 is there a moment you can remember
00:59:22 that you would like to relive
00:59:23 because it made you truly happy
00:59:26 or it was transformative in some fundamental way?
00:59:30 A moment that was transformative.
00:59:32 Or made you truly happy?
00:59:35 Yeah, I think there’s times,
00:59:36 like music used to be a much bigger part of my life
00:59:38 than it is now, like when I was a, let’s say a teenager,
00:59:41 and I can think of some times in like playing music.
00:59:45 There was one, like my brother and a friend of mine,
00:59:48 so this slightly violates the family and friends,
00:59:50 but it was the music that made me happy.
00:59:51 They were just accompanying.
00:59:54 We like played a gig at a ski resort
00:59:57 such that you like take a gondola to the top
00:59:59 and like did a thing.
01:00:00 And then on the gondola ride down,
01:00:01 we decided to just jam a little bit.
01:00:04 And it was just like, I don’t know,
01:00:06 the gondola sort of came over a mountain
01:00:09 and you saw the city lights
01:00:10 and we’re just like jamming, like playing some music.
01:00:13 I wouldn’t describe that as transformative.
01:00:16 I don’t know why, but that popped into my mind
01:00:18 as a moment of, in a way that wasn’t associated
01:00:21 with people I love, but more with like a thing I was doing,
01:00:24 something that was just, it was just happy
01:00:26 and it was just like a great moment.
01:00:29 I don’t think I can give you anything deeper than that.
01:00:32 Well, as a musician myself, I’d love to see,
01:00:35 as you mentioned before, music enter back into your work,
01:00:38 back into your creative work.
01:00:40 I’d love to see that.
01:00:41 I’m certainly allowing it to enter back into mine.
01:00:43 And it’s a beautiful thing for a mathematician,
01:00:47 for a scientist to allow music to enter their work.
01:00:51 I think only good things can happen.
01:00:53 All right, I’ll try to promise you a music video by 2020.
01:00:57 By 2020?
01:00:58 By the end of 2020.
01:00:58 Okay, all right, good.
01:00:59 Give myself a longer window.
01:01:01 All right, maybe we can like collaborate
01:01:04 on a band type situation.
01:01:05 What instruments do you play?
01:01:07 The main instrument I play is violin,
01:01:08 but I also love to dabble around on like guitar and piano.
01:01:11 Beautiful, me too, guitar and piano.
01:01:13 So in a mathematician’s lament, Paul Lockhart writes,
01:01:18 the first thing to understand
01:01:20 is that mathematics is an art.
01:01:22 The difference between math and the other arts,
01:01:24 such as music and painting,
01:01:26 is that our culture does not recognize it as such.
01:01:29 So I think I speak for millions of people, myself included,
01:01:34 in saying thank you for revealing to us
01:01:37 the art of mathematics.
01:01:39 So thank you for everything you do
01:01:40 and thanks for talking today.
01:01:42 Wow, thanks for saying that.
01:01:43 And thanks for having me on.
01:01:45 Thanks for listening to this conversation
01:01:47 with Grant Sanderson.
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01:02:13 And now, let me leave you with some words of wisdom
01:02:16 from one of Grant’s and my favorite people, Richard Feynman.
01:02:21 Nobody ever figures out what this life is all about,
01:02:24 and it doesn’t matter.
01:02:26 Explore the world.
01:02:28 Nearly everything is really interesting
01:02:30 if you go into it deeply enough.
01:02:33 Thank you for listening, and hope to see you next time.