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

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The following is a conversation with Grant Sanderson.

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He's a math educator and creator of 3Blue1Brown,

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a popular YouTube channel

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that uses programmatically animated visualizations

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to explain concepts in linear algebra, calculus,

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and other fields of mathematics.

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This is the Artificial Intelligence Podcast.

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at Lex Friedman, spelled F R I D M A N.

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And now, here's my conversation with Grant Sanderson.

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If there's intelligent life out there in the universe,

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do you think their mathematics is different than ours?

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I think it's probably very different.

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There's an obvious sense the notation is different, right?

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I think notation can guide what the math itself is.

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I think it has everything to do with the form

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of their existence, right?

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Do you think they have basic arithmetic?

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Sorry, I interrupted.

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Yeah, so I think they count, right?

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I think notions like one, two, three,

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the natural numbers, that's extremely, well, natural.

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That's almost why we put that name to it.

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As soon as you can count,

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you have a notion of repetition, right?

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Because you can count by two, two times or three times.

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And so you have this notion of repeating

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the idea of counting, which brings you addition

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and multiplication.

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I think the way that we extend it to the real numbers,

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there's a little bit of choice in that.

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So there's this funny number system

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called the servial numbers

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that it captures the idea of continuity.

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It's a distinct mathematical object.

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You could very well model the universe

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and motion of planets with that

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as the back end of your math, right?

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And you still have kind of the same interface

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with the front end of what physical laws you're trying to,

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or what physical phenomena you're trying

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to describe with math.

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And I wonder if the little glimpses that we have

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of what choices you can make along the way

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based on what different mathematicians

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I've brought to the table

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is just scratching the surface

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of what the different possibilities are

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if you have a completely different mode of thought, right?

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Or a mode of interacting with the universe.

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And you think notation is a key part of the journey

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that we've taken through math.

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I think that's the most salient part

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that you'd notice at first.

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I think the mode of thought is gonna influence things

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more than like the notation itself.

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But notation actually carries a lot of weight

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when it comes to how we think about things,

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more so than we usually give it credit for.

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I would be comfortable saying.

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Do you have a favorite or least favorite piece of notation

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in terms of its effectiveness?

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Yeah, yeah, well, so least favorite,

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one that I've been thinking a lot about

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that will be a video I don't know when, but we'll see.

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The number e, we write the function e to the x,

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this general exponential function

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with a notation e to the x

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that implies you should think about a particular number,

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this constant of nature,

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and you repeatedly multiply it by itself.

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And then you say, oh, what's e to the square root of two?

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And you're like, oh, well, we've extended the idea

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of repeated multiplication.

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That's all nice, that's all nice and well.

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But very famously, you have like e to the pi i,

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and you're like, well, we're extending the idea

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of repeated multiplication into the complex numbers.

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Yeah, you can think about it that way.

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In reality, I think that it's just the wrong way

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of notationally representing this function,

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the exponential function,

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which itself could be represented

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a number of different ways.

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You can think about it in terms of the problem it solves,

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a certain very simple differential equation,

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which often yields way more insight

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than trying to twist the idea of repeated multiplication,

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like take its arm and put it behind its back

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and throw it on the desk and be like,

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you will apply to complex numbers, right?

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That's not, I don't think that's pedagogically helpful.

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So the repeated multiplication is actually missing

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the main point, the power of e to the x.

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I mean, what it addresses is things where the rate

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at which something changes depends on its own value,

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but more specifically, it depends on it linearly.

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So for example, if you have like a population

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that's growing and the rate at which it grows

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depends on how many members of the population

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are already there,

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it looks like this nice exponential curve.

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It makes sense to talk about repeated multiplication

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because you say, how much is there after one year,

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two years, three years, you're multiplying by something.

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The relationship can be a little bit different sometimes

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where let's say you've got a ball on a string,

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like a game of tetherball going around a rope, right?

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And you say, its velocity is always perpendicular

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That's another way of describing its rate of change

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is being related to where it is,

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but it's a different operation.

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You're not scaling it, it's a rotation.

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It's this 90 degree rotation.

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That's what the whole idea of like complex exponentiation

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is trying to capture,

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but it's obfuscated in the notation

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when what it's actually saying,

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like if you really parse something like e to the pi i,

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what it's saying is choose an origin,

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always move perpendicular to the vector

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from that origin to you, okay?

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Then when you walk pi times that radius,

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you'll be halfway around.

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Like that's what it's saying.

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It's kind of the, you turn 90 degrees and you walk,

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you'll be going in a circle.

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That's the phenomenon that it's describing,

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but trying to twist the idea

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of repeatedly multiplying a constant into that.

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Like I can't even think of the number of human hours

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of like intelligent human hours that have been wasted

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trying to parse that to their own liking and desire

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among like scientists or electrical engineers

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or students everywhere,

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which if the notation were a little different

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or the way that this whole function was introduced

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from the get go were framed differently,

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I think could have been avoided, right?

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And you're talking about

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the most beautiful equation in mathematics,

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but it's still pretty mysterious, isn't it?

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

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It's not mysterious.

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I think the notation makes it mysterious.

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I don't think it's, I think the fact that it represents,

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it's pretty, it's not like the most beautiful thing

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in the world, but it's quite pretty.

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The idea that if you take the linear operation

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of a 90 degree rotation,

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and then you do this general exponentiation thing to it,

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that what you get are all the other kinds of rotation,

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which is basically to say,

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if your velocity vector is perpendicular

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to your position vector, you walk in a circle,

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It's not the most beautiful thing in the world,

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but it's quite pretty.

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The beauty of it, I think comes from perhaps

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the awkwardness of the notation

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somehow still nevertheless coming together nicely,

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because you have like several disciplines coming together

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in a single equation.

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In a sense, like historically speaking.

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You've got, so like the number E is significant.

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Like it shows up in probability all the time.

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It like shows up in calculus all the time.

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It is significant.

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You're seeing it sort of mated with pi,

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this geometric constant and I,

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like the imaginary number and such.

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I think what's really happening there

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is the way that E shows up is when you have things

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like exponential growth and decay, right?

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It's when this relation that something's rate of change

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has to itself is a simple scaling, right?

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A similar law also describes circular motion.

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Because we have bad notation,

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we use the residue of how it shows up

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in the context of self reinforcing growth,

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like a population growing or compound interest.

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The constant associated with that

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is awkwardly placed into the context

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of how rotation comes about,

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because they both come from pretty similar equations.

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And so what we see is the E and the pi juxtaposed

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a little bit closer than they would be

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with a purely natural representation, I would think.

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Here's how I would describe the relation between the two.

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You've got a very important function we might call exp.

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That's like the exponential function.

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When you plug in one,

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you get this nice constant called E

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that shows up in like probability and calculus.

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If you try to move in the imaginary direction,

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it's periodic and the period is tau.

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So those are these two constants

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associated with the same central function,

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but for kind of unrelated reasons, right?

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And not unrelated, but like orthogonal reasons.

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One of them is what happens

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when you're moving in the real direction.

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One's what happens when you move in the imaginary direction.

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And like, yeah, those are related.

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They're not as related as the famous equation

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seems to think it is.

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It's sort of putting all of the children in one bed

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and they'd kind of like to sleep in separate beds

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if they had the choice, but you see them all there

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and there is a family resemblance, but it's not that close.

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So actually thinking of it as a function

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is the better idea.

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And that's a notational idea.

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And yeah, and like, here's the thing.

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The constant E sort of stands

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as this numerical representative of calculus, right?

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Calculus is the like study of change.

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So at the very least there's a little cognitive dissonance

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using a constant to represent the science of change.

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I never thought of it that way.

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It makes sense why the notation came about that way.

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Because this is the first way that we saw it

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in the context of things like population growth

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or compound interest.

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It is nicer to think about as repeated multiplication.

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That's definitely nicer.

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But it's more that that's the first application

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of what turned out to be a much more general function

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that maybe the intelligent life

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your initial question asked about

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would have come to recognize as being much more significant

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than the single use case,

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which lends itself to repeated multiplication notation.

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But let me jump back for a second to aliens

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and the nature of our universe.

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Do you think math is discovered or invented?

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So we're talking about the different kind of mathematics

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that could be developed by the alien species.

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The implied question is,

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yeah, is math discovered or invented?

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Is fundamentally everybody going to discover

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the same principles of mathematics?

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So the way I think about it,

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and everyone thinks about it differently,

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but here's my take.

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I think there's a cycle at play

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where you discover things about the universe

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that tell you what math will be useful.

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And that math itself is invented in a sense,

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but of all the possible maths that you could have invented,

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it's discoveries about the world

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that tell you which ones are.

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So like a good example here is the Pythagorean theorem.

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When you look at this,

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do you think of that as a definition

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or do you think of that as a discovery?

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

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because they were,

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but that's probably because they were using physical object

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to build their intuition.

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And from that intuition came the mathematics.

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So the mathematics wasn't in some abstract world

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detached from physics,

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but I think more and more math has become detached from,

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you know, when you even look at modern physics

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from string theory to even general relativity,

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I mean, all math behind the 20th and 21st century physics

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kind of takes a brisk walk outside of what our mind

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can actually even comprehend

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in multiple dimensions, for example,

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anything beyond three dimensions, maybe four dimensions.

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No, no, no, no, higher dimensions

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can be highly, highly applicable.

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I think this is a common misinterpretation

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that if you're asking questions

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about like a five dimensional manifold,

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that the only way that that's connected

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to the physical world is if the physical world is itself

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a five dimensional manifold or includes them.

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Well, wait, wait, wait a minute, wait a minute.

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You're telling me you can imagine

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a five dimensional manifold?

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

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I would make the claim that it is useful

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to a three dimensional physical universe,

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despite itself not being three dimensional.

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So it's useful meaning to even understand

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a three dimensional world,

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it'd be useful to have five dimensional manifolds.

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Yes, absolutely, because of state spaces.

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But you're saying there in some deep way for us humans,

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it does always come back to that three dimensional world

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for the usefulness that the dimensional world

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and therefore it starts with a discovery,

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but then we invent the mathematics

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that helps us make sense of the discovery in a sense.

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Yes, I mean, just to jump off

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of the Pythagorean theorem example,

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it feels like a discovery.

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You've got these beautiful geometric proofs

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where you've got squares and you're modifying the areas,

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it feels like a discovery.

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If you look at how we formalize the idea of 2D space

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as being R2, right, all pairs of real numbers,

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and how we define a metric on it and define distance,

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you're like, hang on a second,

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we've defined a distance

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so that the Pythagorean theorem is true,

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so that suddenly it doesn't feel that great.

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But I think what's going on is the thing that informed us

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what metric to put on R2,

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to put on our abstract representation of 2D space,

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came from physical observations.

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And the thing is, there's other metrics

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you could have put on it.

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We could have consistent math

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with other notions of distance,

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it's just that those pieces of math

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wouldn't be applicable to the physical world that we study

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because they're not the ones

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where the Pythagorean theorem holds.

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So we have a discovery, a genuine bonafide discovery

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that informed the invention,

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the invention of an abstract representation of 2D space

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that we call R2 and things like that.

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And then from there,

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you just study R2 as an abstract thing

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that brings about more ideas and inventions and mysteries

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which themselves might yield discoveries.

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Those discoveries might give you insight

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as to what else would be useful to invent

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and it kind of feeds on itself that way.

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That's how I think about it.

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So it's not an either or.

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It's not that math is one of these

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or it's one of the others.

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At different times, it's playing a different role.

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So then let me ask the Richard Feynman question then,

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along that thread,

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is what do you think is the difference

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between physics and math?

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There's a giant overlap.

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There's a kind of intuition that physicists have

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about the world that's perhaps outside of mathematics.

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It's this mysterious art that they seem to possess,

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we humans generally possess.

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And then there's the beautiful rigor of mathematics

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that allows you to, I mean, just like as we were saying,

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invent frameworks of understanding our physical world.

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So what do you think is the difference there

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and how big is it?

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Well, I think of math as being the study

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of abstractions over patterns and pure patterns in logic.

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And then physics is obviously grounded in a desire

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to understand the world that we live in.

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I think you're gonna get very different answers

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when you talk to different mathematicians

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because there's a wide diversity in types of mathematicians.

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There are some who are motivated very much by pure puzzles.

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They might be turned on by things like combinatorics.

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And they just love the idea of building up

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a set of problem solving tools applying to pure patterns.

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There are some who are very physically motivated,

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who try to invent new math or discover math in veins

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that they know will have applications to physics

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or sometimes computer science.

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And that's what drives them.

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Like chaos theory is a good example of something

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that's pure math, that's purely mathematical.

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A lot of the statements being made,

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but it's heavily motivated by specific applications

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to largely physics.

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And then you have a type of mathematician

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who just loves abstraction.

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They just love pulling it to the more and more abstract

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things, the things that feel powerful.

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These are the ones that initially invented like topology

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and then later on get really into category theory

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and go on about like infinite categories and whatnot.

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These are the ones that love to have a system

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that can describe truths about as many things as possible.

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People from those three different veins of motivation

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into math are gonna give you very different answers

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about what the relation at play here is.

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Cause someone like Vladimir Arnold,

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who has written a lot of great books,

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many about like differential equations and such,

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he would say, math is a branch of physics.

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That's how he would think about it.

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And of course he was studying

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like differential equations related things

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because that is the motivator behind the study

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of PDEs and things like that.

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But you'll have others who,

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like especially the category theorists

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who aren't really thinking about physics necessarily.

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It's all about abstraction and the power of generality.

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And it's more of a happy coincidence

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that that ends up being useful

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for understanding the world we live in.

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And then you can get into like, why is that the case?

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It's sort of surprising

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that that which is about pure puzzles and abstraction

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also happens to describe the very fundamentals

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of quarks and everything else.

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So why do you think the fundamentals of quarks

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and the nature of reality is so compressible

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into clean, beautiful equations

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that are for the most part simple, relatively speaking,

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a lot simpler than they could be?

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So you have, we mentioned somebody like Stephen Wolfram

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who thinks that sort of there's incredibly simple rules

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underlying our reality,

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but it can create arbitrary complexity.

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But there is simple equations.

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What, I'm asking a million questions

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that nobody knows the answer to, but.

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I have no idea, why is it simple?

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It could be the case that

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there's like a filter iteration at play.

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The only things that physicists find interesting

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are the ones that are simple enough

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they could describe it mathematically.

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But as soon as it's a sufficiently complex system,

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like, oh, that's outside the realm of physics,

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that's biology or whatever have you.

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And of course, that's true.

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Maybe there's something where it's like,

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of course there will always be something that is simple

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when you wash away the like non important parts

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of whatever it is that you're studying.

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Just from like an information theory standpoint,

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there might be some like,

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you get to the lowest information component of it.

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But I don't know, maybe I'm just having

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a really hard time conceiving of what it would even mean

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for the fundamental laws to be like intrinsically

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complicated, like some set of equations

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that you can't decouple from each other.

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Well, no, it could be that sort of we take for granted

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that the laws of physics, for example,

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are for the most part the same everywhere

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or something like that, right?

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As opposed to the sort of an alternative could be

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that the rules under which the world operates

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is different everywhere.

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It's like a deeply distributed system

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where just everything is just chaos,

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not in a strict definition of chaos,

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but meaning like just it's impossible for equations

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to capture, for to explicitly model the world

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as cleanly as the physical does.

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I mean, we almost take it for granted that we can describe,

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we can have an equation for gravity,

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for action at a distance.

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We can have equations for some of these basic ways

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the planet's moving.

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Just the low level at the atomic scale,

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how the materials operate,

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at the high scale, how black holes operate.

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But it doesn't, it seems like it could be,

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there's infinite other possibilities

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where none of it could be compressible into such equations.

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So it just seems beautiful.

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It's also weird, probably to the point you're making,

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that it's very pleasant that this is true for our minds.

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So it might be that our minds are biased

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to just be looking at the parts of the universe

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that are compressible.

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And then we can publish papers on

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and have nice E equals empty squared equations.

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Right, well, I wonder would such a world

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with uncompressible laws allow for the kind of beings

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that can think about the kind of questions

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that you're asking?

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Right, like an anthropic principle coming into play

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in some weird way here?

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I don't know, like I don't know what I'm talking about at all.

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Maybe the universe is actually not so compressible,

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but the way our brain, the way our brain evolved

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we're only able to perceive the compressible parts.

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I mean, we are, so this is the sort of Chomsky argument.

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We are just descendants of apes

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over like really limited biological systems.

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So it totally makes sense

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that we're really limited little computers, calculators,

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that are able to perceive certain kinds of things

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and the actual world is much more complicated.

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Well, but we can do pretty awesome things, right?

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Like we can fly spaceships

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and we have to have some connection of reality

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to be able to take our potentially oversimplified models

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of the world, but then actually twist the world

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to our will based on it.

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So we have certain reality checks

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that like physics isn't too far a field

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simply based on what we can do.

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Yeah, the fact that we can fly is pretty good.

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It's great, yeah, like it's a proof of concept

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that the laws we're working with are working well.

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So I mentioned to the internet that I'm talking to you

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and so the internet gave some questions.

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So I apologize for these,

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but do you think we're living in a simulation

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that the universe is a computer

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or the universe is a computation running on a computer?

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What I don't buy is, you know, you'll have the argument

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that, well, let's say that it was the case

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that you can have simulations.

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Then the simulated world would itself

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eventually get to a point where it's running simulations.

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And then the second layer down

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would create a third layer down and on and on and on.

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So probabilistically, you just throw a dart

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at one of those layers,

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we're probably in one of the simulated layers.

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I think if there's some sort of limitations

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on like the information processing

link |

of whatever the physical world is,

link |

like it quickly becomes the case

link |

that you have a limit to the layers that could exist there

link |

because like the resources necessary

link |

to simulate a universe like ours clearly is a lot

link |

just in terms of the number of bits at play.

link |

And so then you can ask, well, what's more plausible?

link |

That there's an unbounded capacity

link |

of information processing

link |

in whatever the like highest up level universe is,

link |

or that there's some bound to that capacity,

link |

which then limits like the number of levels available.

link |

How do you play some kind of probability distribution

link |

on like what the information capacity is?

link |

But I don't, like people almost assume

link |

a certain uniform probability

link |

over all of those meta layers that could conceivably exist

link |

when it's a little bit like a Pascal's wager

link |

on like you're not giving a low enough prior

link |

to the mere existence of that infinite set of layers.

link |

Yeah, that's true.

link |

But it's also very difficult to contextualize the amount.

link |

So the amount of information processing power

link |

required to simulate like our universe

link |

seems like amazingly huge.

link |

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

link |

Yeah, like numbers get big.

link |

And we're easily humbled

link |

by basically everything around us.

link |

So it's very difficult to kind of make sense of anything

link |

actually when you look up at the sky

link |

and look at the stars and the immensity of it all,

link |

to make sense of the smallness of us,

link |

the unlikeliness of everything

link |

that's on this earth coming to be,

link |

then you could basically anything could be,

link |

all laws of probability go out the window to me

link |

because I guess because the amount of information

link |

under which we're operating is very low.

link |

We basically know nothing about the world around us,

link |

relatively speaking.

link |

And so when I think about the simulation hypothesis,

link |

I think it's just fun to think about it.

link |

But it's also, I think there is a thought experiment

link |

kind of interesting to think of the power of computation,

link |

whether the limits of a Turing machine,

link |

sort of the limits of our current computers,

link |

when you start to think about artificial intelligence,

link |

how far can we get with computers?

link |

And that's kind of where the simulation hypothesis

link |

used with me as a thought experiment

link |

is the universe just a computer?

link |

Is it just a computation?

link |

Is all of this just a computation?

link |

And sort of the same kind of tools we apply

link |

to analyzing algorithms, can that be applied?

link |

If we scale further and further and further,

link |

will the arbitrary power of those systems

link |

start to create some interesting aspects

link |

that we see in our universe?

link |

Or is something fundamentally different

link |

needs to be created?

link |

Well, it's interesting that in our universe,

link |

it's not arbitrarily large, the power,

link |

that you can place limits on, for example,

link |

how many bits of information can be stored per unit area.

link |

Right, like all of the physical laws,

link |

you've got general relativity and quantum coming together

link |

to give you a certain limit on how many bits you can store

link |

within a given range before it collapses into a black hole.

link |

The idea that there even exists such a limit

link |

is at the very least thought provoking,

link |

when naively you might assume,

link |

oh, well, technology could always get better and better,

link |

we could get cleverer and cleverer,

link |

and you could just cram as much information as you want

link |

into like a small unit of space, that makes me think,

link |

it's at least plausible that whatever the highest level

link |

of existence is doesn't admit too many simulations

link |

or ones that are at the scale of complexity

link |

that we're looking at.

link |

Obviously, it's just as conceivable that they do

link |

and that there are many, but I guess what I'm channeling

link |

is the surprise that I felt upon learning that fact,

link |

that there are, that information is physical in this way.

link |

There's a finiteness to it.

link |

Okay, let me just even go off on that.

link |

From a mathematics perspective

link |

and a psychology perspective, how do you mix,

link |

are you psychologically comfortable

link |

with the concept of infinity?

link |

Are you okay with it?

link |

I'm pretty okay, yeah.

link |

No, not really, it doesn't make any sense to me.

link |

I don't know, like how many words,

link |

how many possible words do you think could exist

link |

that are just like strings of letters?

link |

So that's a sort of mathematical statement as beautiful

link |

and we use infinity in basically everything we do,

link |

everything we do in science, math, and engineering, yes.

link |

But you said exist, the question is,

link |

you said letters or words?

link |

I said words. Words.

link |

To bring words into existence to me,

link |

you have to start like saying them or like writing them

link |

or like listing them.

link |

That's an instantiation.

link |

Okay, how many abstract words exist?

link |

Well, the idea of an abstract.

link |

The idea of abstract notions and ideas.

link |

I think we should be clear on terminology.

link |

I mean, you think about intelligence a lot,

link |

like artificial intelligence.

link |

Would you not say that what it's doing

link |

is a kind of abstraction?

link |

That like abstraction is key

link |

to conceptualizing the universe?

link |

You get this raw sensory data.

link |

I need something that every time you move your face

link |

a little bit and they're not pixels,

link |

but like analog of pixels on my retina changed entirely,

link |

that I can still have some coherent notion of this is Lex,

link |

I'm talking to Lex, right?

link |

What that requires is you have a disparate set

link |

of possible images hitting me

link |

that are unified in a notion of Lex, right?

link |

That's a kind of abstraction.

link |

It's a thing that could apply

link |

to a lot of different images that I see

link |

and it represents it in a much more compressed way

link |

and one that's like much more resilient to that.

link |

I think in the same way,

link |

if I'm talking about infinity as an abstraction,

link |

I don't mean nonphysical woo woo,

link |

like ineffable or something.

link |

What I mean is it's something that can apply

link |

to a multiplicity of situations

link |

that share a certain common attribute

link |

in the same way that the images of like your face

link |

on my retina share enough common attributes

link |

that I can put the single notion to it.

link |

Like in that way, infinity is an abstraction

link |

and it's very powerful and it's only through

link |

such abstractions that we can actually understand

link |

like the world and logic and things.

link |

And in the case of infinity,

link |

the way I think about it,

link |

the key entity is the property

link |

of always being able to add one more.

link |

Like no matter how many words you can list,

link |

you just throw an A at the end of one

link |

and you have another conceivable word.

link |

You don't have to think of all the words at once.

link |

It's that property, the oh, I could always add one more

link |

that gives it this nature of infiniteness

link |

in the same way that there's certain like properties

link |

of your face that give it the Lexness, right?

link |

So like infinity should be no more worrying

link |

than the I can always add one more sentiment.

link |

That's a really elegant,

link |

much more elegant way than I could put it.

link |

So thank you for doing that as yet another abstraction.

link |

And yes, indeed, that's what our brain does.

link |

That's what intelligent systems do.

link |

That's what programming does.

link |

That's what science does is build abstraction

link |

on top of each other.

link |

And yet there is at a certain point abstractions

link |

that go into the quote woo, right?

link |

Sort of, and because we're now,

link |

it's like we built this stack of, you know,

link |

the only thing that's true is the stuff that's on the ground.

link |

Everything else is useful for interpreting this.

link |

And at a certain point you might start floating

link |

into ideas that are surreal and difficult

link |

and take us into areas that are disconnected

link |

from reality in a way that we could never get back.

link |

What if instead of calling these abstract,

link |

how different would it be in your mind

link |

if we called them general?

link |

And the phenomenon that you're describing

link |

is overgeneralization.

link |

When you try to have a concept or an idea

link |

that's so general as to apply to nothing in particular

link |

in a useful way, does that map to what you're thinking

link |

of when you think of?

link |

First of all, I'm playing little just for the fun of it.

link |

And I think our cognition, our mind is unable

link |

So you do some incredible work with visualization and video.

link |

I think infinity is very difficult to visualize

link |

We can delude ourselves into thinking we can visualize it,

link |

I don't, I mean, I don't,

link |

I would venture to say it's very difficult.

link |

And so there's some concepts of mathematics,

link |

like maybe multiple dimensions,

link |

we could sort of talk about that are impossible

link |

for us to truly intuit, like,

link |

and it just feels dangerous to me to use these

link |

as part of our toolbox of abstractions.

link |

On behalf of your listeners,

link |

I almost fear we're getting too philosophical.

link |

I think to that point for any particular idea like this,

link |

there's multiple angles of attack.

link |

I think the, when we do visualize infinity,

link |

what we're actually doing, you know,

link |

you write dot, dot, dot, right?

link |

One, two, three, four, dot, dot, dot, right?

link |

Those are symbols on the page

link |

that are insinuating a certain infinity.

link |

What you're capturing with a little bit of design there

link |

is the I can always add one more property, right?

link |

I think I'm just as uncomfortable with you are

link |

if you try to concretize it so much

link |

that you have a bag of infinitely many things

link |

that I actually think of, no, not one, two, three, four,

link |

dot, dot, dot, one, two, three, four, five, six, seven, eight.

link |

I try to get them all in my head and you realize,

link |

oh, you know, your brain would literally collapse

link |

into a black hole, all of that.

link |

And I honestly feel this with a lot of math

link |

that I try to read where I don't think of myself

link |

as like particularly good at math in some ways.

link |

Like I get very confused often

link |

when I am going through some of these texts.

link |

And often what I'm feeling in my head is like,

link |

this is just so damn abstract.

link |

I just can't wrap my head around it.

link |

I just want to put something concrete to it

link |

that makes me understand.

link |

And I think a lot of the motivation for the channel

link |

is channeling that sentiment of, yeah,

link |

a lot of the things that you're trying to read out there,

link |

it's just so hard to connect to anything

link |

that you spend an hour banging your head

link |

against a couple of pages and you come out

link |

not really knowing anything more

link |

other than some definitions maybe

link |

and a certain sense of self defeat, right?

link |

One of the reasons I focus so much on visualizations

link |

is that I'm a big believer in,

link |

I'm sorry, I'm just really hampering on

link |

this idea of abstraction,

link |

being clear about your layers of abstraction, right?

link |

It's always tempting to start an explanation

link |

from the top to the bottom, okay?

link |

You give the definition of a new theorem.

link |

You're like, this is the definition of a vector space.

link |

For example, that's how we'll start a course.

link |

These are the properties of a vector space.

link |

First from these properties, we will derive what we need

link |

in order to do the math of linear algebra

link |

or whatever it might be.

link |

I don't think that's how understanding works at all.

link |

I think how understanding works

link |

is you start at the lowest level you can get at

link |

where rather than thinking about a vector space,

link |

you might think of concrete vectors

link |

that are just lists of numbers

link |

or picturing it as like an arrow that you draw,

link |

which is itself like even less abstract than numbers

link |

because you're looking at quantities,

link |

like the distance of the x coordinate,

link |

the distance of the y coordinate.

link |

It's as concrete as you could possibly get

link |

and it has to be if you're putting it in a visual, right?

link |

It's an actual arrow. It's an actual vector.

link |

You're not talking about like a quote unquote vector

link |

that could apply to any possible thing.

link |

You have to choose one if you're illustrating it.

link |

And I think this is the power of being in a medium

link |

like video or if you're writing a textbook

link |

and you force yourself to put a lot of images

link |

is with every image, you're making a choice.

link |

With each choice, you're showing a concrete example.

link |

With each concrete example,

link |

you're aiding someone's path to understanding.

link |

I'm sorry to interrupt you,

link |

but you just made me realize that that's exactly right.

link |

So the visualizations you're creating

link |

while you're sometimes talking about abstractions,

link |

the actual visualization is an explicit low level example.

link |

So there's an actual, like in the code,

link |

you have to say what the vector is,

link |

what's the direction of the arrow,

link |

what's the magnitude of the, yeah.

link |

So that's, you're going, the visualization itself

link |

is actually going to the bottom of that.

link |

And I think that's very important.

link |

I also think about this a lot in writing scripts

link |

where even before you get to the visuals,

link |

the first instinct is to, I don't know why,

link |

I just always do, I say the abstract thing,

link |

I say the general definition, the powerful thing,

link |

and then I fill it in with examples later.

link |

Always, it will be more compelling

link |

and easier to understand when you flip that.

link |

And instead, you let someone's brain

link |

do the pattern recognition.

link |

You just show them a bunch of examples.

link |

The brain is gonna feel a certain similarity between them.

link |

Then by the time you bring in the definition,

link |

or by the time you bring in the formula,

link |

it's articulating a thing that's already in the brain

link |

that was built off of looking at a bunch of examples

link |

with a certain kind of similarity.

link |

And what the formula does is articulate

link |

what that kind of similarity is,

link |

rather than being a high cognitive load set of symbols

link |

that needs to be populated with examples later on,

link |

assuming someone's still with you.

link |

What is the most beautiful or awe inspiring idea

link |

you've come across in mathematics?

link |

I don't know, man.

link |

Maybe it's an idea you've explored in your videos,

link |

What just gave you pause?

link |

What's the most beautiful idea?

link |

So I think often, the things that are most beautiful

link |

are the ones that you have a little bit of understanding of,

link |

but certainly not an entire understanding.

link |

It's a little bit of that mystery

link |

that is what makes it beautiful.

link |

What was the moment of the discovery for you personally,

link |

almost just that leap of aha moment?

link |

So something that really caught my eye,

link |

I remember when I was little, there were these,

link |

I think the series was called like wooden books

link |

or something, these tiny little books

link |

that would have just a very short description

link |

of something on the left and then a picture on the right.

link |

I don't know who they're meant for,

link |

but maybe it's like loosely children

link |

or something like that.

link |

But it can't just be children,

link |

because of some of the things I was describing.

link |

On the last page of one of them,

link |

somewhere tiny in there was this little formula

link |

that on the left hand had a sum

link |

over all of the natural numbers.

link |

It's like one over one to the S plus one over two to the S

link |

plus one over three to the S on and on to the infinity.

link |

Then on the other side had a product over all of the primes

link |

and it was a certain thing had to do with all the primes.

link |

And like any good young math enthusiast,

link |

I'd probably been indoctrinated with how chaotic

link |

and confusing the primes are, which they are.

link |

And seeing this equation where on one side

link |

you have something that's as understandable

link |

as you could possibly get, the counting numbers.

link |

And on the other side is all the prime numbers.

link |

It was like this, whoa, they're related like this?

link |

There's a simple description that includes

link |

all the primes getting wrapped together like this.

link |

This is like the Euler product for the Zeta function,

link |

as I like later found out.

link |

The equation itself essentially encodes

link |

the fundamental theorem of arithmetic

link |

that every number can be expressed

link |

as a unique set of primes.

link |

To me still there's, I mean, I certainly don't understand

link |

this equation or this function all that well.

link |

The more I learn about it, the prettier it is.

link |

The idea that you can, this is sort of what gets you

link |

representations of primes, not in terms of primes themselves,

link |

but in terms of another set of numbers.

link |

They're like the non trivial zeros of the Zeta function.

link |

And again, I'm very kind of in over my head

link |

in a lot of ways as I like try to get to understand it.

link |

But the more I do, it always leaves enough mystery

link |

that it remains very beautiful to me.

link |

So whenever there's a little bit of mystery

link |

just outside of the understanding that,

link |

and by the way, the process of learning more about it,

link |

how does that come about?

link |

Just your own thought or are you reading?

link |

Or is the process of visualization itself

link |

revealing more to you?

link |

I mean, in one time when I was just trying to understand

link |

like analytic continuation and playing around

link |

with visualizing complex functions,

link |

this is what led to a video about this function.

link |

It's titled something like

link |

Visualizing the Riemann Zeta Function.

link |

It's one that came about because I was programming

link |

and tried to see what a certain thing looked like.

link |

And then I looked at it and I'm like,

link |

whoa, that's elucidating.

link |

And then I decided to make a video about it.

link |

But I mean, you try to get your hands on

link |

as much reading as you can.

link |

You know, in this case, I think if anyone wants to start

link |

to understand it, if they have like a math background

link |

like they studied some in college or something like that,

link |

like the Princeton Companion to Math

link |

has a really good article on analytic number theory.

link |

And that itself has a whole bunch of references

link |

and you know, anything has more references

link |

and it gives you this like tree to start piling through.

link |

And like, you know, you try to understand,

link |

I try to understand things visually as I go.

link |

That's not always possible,

link |

but it's very helpful when it does.

link |

You recognize when there's common themes,

link |

like in this case, Cousins of the Fourier Transform

link |

that come into play and you realize,

link |

oh, it's probably pretty important

link |

to have deep intuitions of the Fourier Transform,

link |

even if it's not explicitly mentioned in like these texts.

link |

And you try to get a sense of what the common players are.

link |

But I'll emphasize again, like,

link |

I feel very in over my head when I try to understand

link |

the exact relation between like the zeros

link |

of the Riemann Zeta function

link |

and how they relate to the distribution of primes.

link |

I definitely understand it better than I did a year ago.

link |

I definitely understand it on 100th as well as the experts

link |

on the matter do, I assume.

link |

But the slow path towards getting there is,

link |

it's fun, it's charming,

link |

and like to your question, very beautiful.

link |

And the beauty is in the, what,

link |

in the journey versus the destination?

link |

Well, it's that each thing doesn't feel arbitrary.

link |

I think that's a big part,

link |

is that you have these unpredictable,

link |

no, yeah, these very unpredictable patterns

link |

or these intricate properties of like a certain function.

link |

But at the same time,

link |

it doesn't feel like humans ever made an arbitrary choice

link |

in studying this particular thing.

link |

So, you know, it feels like you're speaking

link |

to patterns themselves or nature itself.

link |

That's a big part of it.

link |

I think things that are too arbitrary,

link |

it's just hard for those to feel beautiful

link |

because this is sort of what the word contrived

link |

is meant to apply to, right?

link |

And when they're not arbitrary means it could be,

link |

you can have a clean abstraction and intuition

link |

that allows you to comprehend it.

link |

Well, to one of your first questions,

link |

it makes you feel like if you came across

link |

another intelligent civilization,

link |

that they'd be studying the same thing.

link |

Maybe with different notation.

link |

Certainly, yeah, but yeah.

link |

I think you talked to that other civilization,

link |

they're probably also studying the zeros

link |

of the Riemann Zeta function

link |

or like some variant thereof

link |

that is like a clearly equivalent cousin

link |

or something like that.

link |

But that's probably on their docket.

link |

Whenever somebody does a lot of something amazing,

link |

I'm gonna ask the question

link |

that you've already been asked a lot

link |

and that you'll get more and more asked in your life.

link |

But what was your favorite video to create?

link |

Oh, favorite to create.

link |

One of my favorites is,

link |

the title is Who Cares About Topology?

link |

You want me to pull it up or no?

link |

If you want, sure, yeah.

link |

It is about, well, it starts by describing

link |

an unsolved problem that's still unsolved in math

link |

called the inscribed square problem.

link |

You draw any loop and then you ask,

link |

are there four points on that loop that make a square?

link |

Totally useless, right?

link |

This is not answering any physical questions.

link |

It's mostly interesting that we can't answer that question.

link |

And it seems like such a natural thing to ask.

link |

Now, if you weaken it a little bit and you ask,

link |

can you always find a rectangle?

link |

You choose four points on this curve,

link |

can you find a rectangle?

link |

That's hard, but it's doable.

link |

And the path to it involves things like looking at a torus,

link |

this surface with a single hole in it, like a donut,

link |

or looking at a mobius strip.

link |

In ways that feel so much less contrived

link |

to when I first, as like a little kid,

link |

learned about these surfaces and shapes,

link |

like a mobius strip and a torus.

link |

Like what you learn is, oh, this mobius strip,

link |

you take a piece of paper, put a twist, glue it together,

link |

and now you have a shape with one edge and just one side.

link |

And as a student, you should think, who cares, right?

link |

Like, how does that help me solve any problems?

link |

I thought math was about problem solving.

link |

So what I liked about the piece of math

link |

that this was describing that was in this paper

link |

by a mathematician named Vaughn

link |

was that it arises very naturally.

link |

It's clear what it represents.

link |

It's doing something.

link |

It's not just playing with construction paper.

link |

And the way that it solves the problem is really beautiful.

link |

So kind of putting all of that down

link |

and concretizing it, right?

link |

Like I was talking about how

link |

when you have to put visuals to it,

link |

it demands that what's on screen

link |

is a very specific example of what you're describing.

link |

The construction here is very abstract in nature.

link |

You describe this very abstract kind of surface in 3D space.

link |

So then when I was finding myself,

link |

in this case, I wasn't programming,

link |

I was using a grapher that's like built into OSX

link |

for the 3D stuff to draw that surface,

link |

you realize, oh man, the topology argument

link |

is very non constructive.

link |

I have to make a lot of,

link |

you have to do a lot of extra work

link |

in order to make the surface show up.

link |

But then once you see it, it's quite pretty

link |

and it's very satisfying to see a specific instance of it.

link |

And you also feel like, ah,

link |

I've actually added something

link |

on top of what the original paper was doing

link |

that it shows something that's completely correct.

link |

That's a very beautiful argument,

link |

but you don't see what it looks like.

link |

And I found something satisfying

link |

in seeing what it looked like

link |

that could only ever have come about

link |

from the forcing function

link |

of getting some kind of image on the screen

link |

to describe the thing I was talking about.

link |

So you almost weren't able to anticipate

link |

what it's gonna look like.

link |

And it was wonderful, right?

link |

It was totally, it looks like a Sydney Opera House

link |

or some sort of Frank Gehry design.

link |

And it was, you knew it was gonna be something

link |

and you can say various things about it.

link |

Like, oh, it touches the curve itself.

link |

It has a boundary that's this curve on the 2D plane.

link |

It all sits above the plane.

link |

But before you actually draw it,

link |

it's very unclear what the thing will look like.

link |

And to see it, it's very, it's just pleasing, right?

link |

So that was fun to make, very fun to share.

link |

I hope that it has elucidated for some people out there

link |

where these constructs of topology come from,

link |

that it's not arbitrary play with construction paper.

link |

So let's, I think this is a good sort of example

link |

to talk a little bit about your process.

link |

You have a list of ideas.

link |

So that's sort of the curse of having an active

link |

and brilliant mind is I'm sure you have a list

link |

that's growing faster than you can utilize.

link |

Now I'm ahead, absolutely.

link |

But there's some sorting procedure

link |

depending on mood and interest and so on.

link |

But okay, so you pick an idea

link |

and then you have to try to write a narrative arc

link |

that sort of, how do I elucidate?

link |

How do I make this idea beautiful and clear

link |

And then there's a set of visualizations

link |

that will be attached to it.

link |

Sort of, you've talked about some of this before,

link |

but sort of writing the story, attaching the visualizations.

link |

Can you talk through interesting, painful,

link |

beautiful parts of that process?

link |

Well, the most painful is if you've chosen a topic

link |

that you do want to do, but then it's hard to think of,

link |

I guess how to structure the script.

link |

This is sort of where I have been on one

link |

for like the last two or three months.

link |

And I think that ultimately the right resolution

link |

is just like set it aside and instead do some other things

link |

where the script comes more naturally.

link |

Because you sort of don't want to overwork a narrative.

link |

The more you've thought about it,

link |

the less you can empathize with the student

link |

who doesn't yet understand the thing you're trying to teach.

link |

Who is the judger in your head?

link |

Sort of the person, the creature,

link |

the essence that's saying this sucks or this is good.

link |

And you mentioned kind of the student you're thinking about.

link |

Can you, who is that?

link |

What is that thing?

link |

That says, the perfectionist that says this thing sucks.

link |

You need to work on that for another two, three months.

link |

I think it's my past self.

link |

I think that's the entity that I'm most trying

link |

to empathize with is like you take who I was,

link |

because that's kind of the only person I know.

link |

Like you don't really know anyone

link |

other than versions of yourself.

link |

So I start with the version of myself that I know

link |

who doesn't yet understand the thing, right?

link |

And then I just try to view it with fresh eyes,

link |

a particular visual or a particular script.

link |

Like, is this motivating?

link |

Does this make sense?

link |

Which has its downsides,

link |

because sometimes I find myself speaking to motivations

link |

that only myself would be interested in.

link |

I don't know, like I did this project on quaternions

link |

where what I really wanted was to understand

link |

what are they doing in four dimensions?

link |

Can we see what they're doing in four dimensions, right?

link |

And I came up with a way of thinking about it

link |

that really answered the question in my head

link |

that made me very satisfied

link |

and being able to think about concretely with a 3D visual,

link |

what are they doing to a 4D sphere?

link |

And so I'm like, great,

link |

this is exactly what my past self would have wanted, right?

link |

And I make a thing on it.

link |

And I'm sure it's what some other people wanted too.

link |

But in hindsight, I think most people who wanna learn

link |

about quaternions are like robotics engineers

link |

or graphics programmers who want to understand

link |

how they're used to describe 3D rotations.

link |

And like their use case was actually a little bit different

link |

than my past self.

link |

And in that way, like,

link |

I wouldn't actually recommend that video

link |

to people who are coming at it from that angle

link |

of wanting to know, hey, I'm a robotics programmer.

link |

Like, how do these quaternion things work

link |

to describe position in 3D space?

link |

I would say other great resources for that.

link |

If you ever find yourself wanting to say like,

link |

in what sense are they acting in four dimensions?

link |

But until then, that's a little different.

link |

Yeah, it's interesting

link |

because you have incredible videos on neural networks,

link |

And from my sort of perspective,

link |

because I've probably, I mean,

link |

is sort of my field

link |

and I've also looked at the basic introduction

link |

of neural networks like a million times

link |

from different perspectives.

link |

And it made me realize

link |

that there's a lot of ways to present it.

link |

So you were sort of, you did an incredible job.

link |

I mean, sort of the,

link |

but you could also do it differently

link |

and also incredible.

link |

Like to create a beautiful presentation of a basic concept

link |

requires sort of creativity, requires genius and so on,

link |

but you can take it from a bunch of different perspectives.

link |

And that video on neural networks made me realize that.

link |

And just as you're saying,

link |

you kind of have a certain mindset, a certain view,

link |

but from a, if you take a different view

link |

from a physics perspective,

link |

from a neuroscience perspective,

link |

talking about neural networks

link |

or from a robotics perspective,

link |

or from, let's see,

link |

from a pure learning, statistics perspective.

link |

So you can create totally different videos.

link |

And you've done that with a few actually concepts

link |

where you've have taken different cuts,

link |

like at the Euler equation, right?

link |

You've taken different views of that.

link |

I think I've made three videos on it

link |

and I definitely will make at least one more.

link |

So you don't think it's the most beautiful equation

link |

Like I said, as we represent it,

link |

it's one of the most hideous.

link |

It involves a lot of the most hideous aspects

link |

I talked about E, the fact that we use pi instead of tau,

link |

the fact that we call imaginary numbers imaginary,

link |

and then, hence, I actually wonder if we use the I

link |

because of imaginary.

link |

I don't know if that's historically accurate,

link |

but at least a lot of people,

link |

they read the I and they think imaginary.

link |

Like all three of those facts,

link |

it's like those are things that have added more confusion

link |

than they needed to,

link |

and we're wrapping them up in one equation.

link |

Like boy, that's just very hideous, right?

link |

The idea is that it does tie together

link |

when you wash away the notation.

link |

Like it's okay, it's pretty, it's nice,

link |

but it's not like mind blowing greatest thing

link |

which is maybe what I was thinking of when I said,

link |

like once you understand something,

link |

it doesn't have the same beauty.

link |

Like I feel like I understand Euler's formula,

link |

and I feel like I understand it enough

link |

to sort of see the version that just woke up

link |

that hasn't really gotten itself dressed in the morning

link |

that's a little bit groggy,

link |

and there's bags under its eyes.

link |

So you're past the dating stage,

link |

you're no longer dating, right?

link |

I'm still dating the Zeta function,

link |

and like she's beautiful and right,

link |

and like we have fun,

link |

and it's that high dopamine part,

link |

but like maybe at some point

link |

we'll settle into the more mundane nature of the relationship

link |

where I like see her for who she truly is,

link |

and she'll still be beautiful in her own way,

link |

but it won't have the same romantic pizzazz, right?

link |

Well, that's the nice thing about mathematics.

link |

I think as long as you don't live forever,

link |

there'll always be enough mystery and fun

link |

with some of the equations.

link |

Even if you do, the rate at which questions comes up

link |

is much faster than the rate at which answers come up, so.

link |

If you could live forever, would you?

link |

So you think, you don't think mortality

link |

is the thing that makes life meaningful?

link |

Would your life be four times as meaningful

link |

if you died at 25?

link |

So this goes to infinity.

link |

I think you and I, that's really interesting.

link |

So what I said is infinite, not four times longer.

link |

So the actual existence of the finiteness,

link |

the existence of the end, no matter the length,

link |

is the thing that may sort of,

link |

from my comprehension of psychology,

link |

it's such a deeply human,

link |

it's such a fundamental part of the human condition,

link |

the fact that there is, that we're mortal,

link |

that the fact that things end,

link |

it seems to be a crucial part of what gives them meaning.

link |

I don't think, at least for me,

link |

it's a very small percentage of my time

link |

that mortality is salient,

link |

that I'm aware of the end of my life.

link |

What do you mean by me?

link |

Is it the ego, is it the id, or is it the superego?

link |

The reflective self, the Wernicke's area

link |

that puts all this stuff into words.

link |

Yeah, a small percentage of your mind

link |

that is actually aware of the true motivations

link |

But my point is that most of my life,

link |

I'm not thinking about death,

link |

but I still feel very motivated to make things

link |

and to interact with people,

link |

experience love or things like that.

link |

I'm very motivated,

link |

and it's strange that that motivation comes

link |

while death is not in my mind at all.

link |

And this might just be because I'm young enough

link |

that it's not salient.

link |

Or it's in your subconscious,

link |

or that you've constructed an illusion

link |

that allows you to escape the fact of your mortality

link |

by enjoying the moment,

link |

sort of the existential approach to life.

link |

Gun to my head, I don't think that's it.

link |

Yeah, another sort of way to say gun to the head

link |

is sort of the deep psychological introspection

link |

of what drives us.

link |

I mean, that's, in some ways to me,

link |

I mean, when I look at math, when I look at science,

link |

is a kind of an escape from reality

link |

in a sense that it's so beautiful.

link |

It's such a beautiful journey of discovery

link |

that it allows you to actually,

link |

it sort of allows you to achieve a kind of immortality

link |

of explore ideas and sort of connect yourself

link |

to the thing that is seemingly infinite,

link |

like the universe, right?

link |

That allows you to escape the limited nature

link |

of our little, of our bodies, of our existence.

link |

What else would give this podcast meaning?

link |

If not the fact that it will end.

link |

This place closes in 40 minutes.

link |

And it's so much more meaningful for it.

link |

How much more I love this room

link |

because we'll be kicked out.

link |

So I understand just because you're trolling me

link |

doesn't mean I'm wrong.

link |

But I take your point.

link |

I take your point.

link |

Boy, that would be a good Twitter bio.

link |

Just because you're trolling me doesn't mean I'm wrong.

link |

Yeah, and sort of difference in backgrounds.

link |

I'm a bit Russian, so we're a bit melancholic

link |

and seem to maybe assign a little too much value

link |

to suffering and mortality and things like that.

link |

Makes for a better novel, I think.

link |

Oh yeah, you need some sort of existential threat

link |

So when do you know when the video is done

link |

when you're working on it?

link |

That's pretty easy actually,

link |

because I'll write the script.

link |

I want there to be some kind of aha moment in there.

link |

And then hopefully the script can revolve around

link |

some kind of aha moment.

link |

And then from there, you're putting visuals

link |

to each sentence that exists,

link |

and then you narrate it, you edit it all together.

link |

So given that there's a script,

link |

the end becomes quite clear.

link |

And as I animate it, I often change

link |

certainly the specific words,

link |

but sometimes the structure itself.

link |

But it's a very deterministic process at that point.

link |

It makes it much easier to predict

link |

when something will be done.

link |

How do you know when a script is done?

link |

It's like, for problem solving videos,

link |

that's quite simple.

link |

It's once you feel like someone

link |

who didn't understand the solution now could.

link |

For things like neural networks,

link |

that was a lot harder because like you said,

link |

there's so many angles at which you could attack it.

link |

And there, it's just at some point

link |

you feel like this asks a meaningful question

link |

and it answers that question, right?

link |

What is the best way to learn math

link |

for people who might be at the beginning of that journey?

link |

I think that's a question that a lot of folks

link |

kind of ask and think about.

link |

And it doesn't, even for folks

link |

who are not really at the beginning of their journey,

link |

like there might be actually deep in their career,

link |

some type they've taken college

link |

or taken calculus and so on,

link |

but still wanna sort of explore math.

link |

What would be your advice instead of education at all ages?

link |

Your temptation will be to spend more time

link |

like watching lectures or reading.

link |

Try to force yourself to do more problems

link |

than you naturally would.

link |

Like the focus time that you're spending

link |

should be on like solving specific problems

link |

and seek entities that have well curated lists of problems.

link |

So go into like a textbook almost

link |

and the problems in the back of a textbook kind of thing,

link |

back of a chapter.

link |

So if you can take a little look through those questions

link |

at the end of the chapter before you read the chapter,

link |

a lot of them won't make sense.

link |

Some of them might,

link |

and those are the best ones to think about.

link |

A lot of them won't, but just take a quick look

link |

and then read a little bit of the chapter

link |

and then maybe take a look again and things like that.

link |

And don't consider yourself done with the chapter

link |

until you've actually worked through a couple exercises.

link |

And this is so hypocritical, right?

link |

Cause I like put out videos

link |

that pretty much never have associated exercises.

link |

I just view myself as a different part of the ecosystem,

link |

which means I'm kind of admitting

link |

that you're not really learning,

link |

or at least this is only a partial part

link |

of the learning process if you're watching these videos.

link |

I think if someone's at the very beginning,

link |

like I do think Khan Academy does a good job.

link |

They have a pretty large set of questions

link |

you can work through.

link |

Just the very basics,

link |

sort of just picking up,

link |

getting comfortable with the very basic linear algebra,

link |

calculus or so on, Khan Academy.

link |

Programming is actually I think a great,

link |

like learn to program and like let the way

link |

that math is motivated from that angle push you through.

link |

I know a lot of people who didn't like math

link |

got into programming in some way

link |

and that's what turned them on to math.

link |

Maybe I'm biased cause like I live in the Bay area,

link |

so I'm more likely to run into someone

link |

who has that phenotype.

link |

But I am willing to speculate

link |

that that is a more generalizable path.

link |

So you yourself kind of in creating the videos

link |

are using programming to illuminate a concept,

link |

but for yourself as well.

link |

So would you recommend somebody try to make a,

link |

sort of almost like try to make videos?

link |

Like you do as a way to learn?

link |

So one thing I've heard before,

link |

I don't know if this is based on any actual study.

link |

This might be like a total fictional anecdote of numbers,

link |

but it rings in the mind as being true.

link |

You remember about 10% of what you read,

link |

you remember about 20% of what you listen to,

link |

you remember about 70% of what you actively interact with

link |

in some way, and then about 90% of what you teach.

link |

This is a thing I heard again,

link |

those numbers might be meaningless,

link |

but they ring true, don't they, right?

link |

I'm willing to say I learned nine times better

link |

if I'm teaching something than reading.

link |

That might even be a low ball, right?

link |

So doing something to teach

link |

or to like actively try to explain things

link |

is huge for consolidating the knowledge.

link |

Outside of family and friends,

link |

is there a moment you can remember

link |

that you would like to relive

link |

because it made you truly happy

link |

or it was transformative in some fundamental way?

link |

A moment that was transformative.

link |

Or made you truly happy?

link |

Yeah, I think there's times,

link |

like music used to be a much bigger part of my life

link |

than it is now, like when I was a, let's say a teenager,

link |

and I can think of some times in like playing music.

link |

There was one, like my brother and a friend of mine,

link |

so this slightly violates the family and friends,

link |

but it was the music that made me happy.

link |

They were just accompanying.

link |

We like played a gig at a ski resort

link |

such that you like take a gondola to the top

link |

and like did a thing.

link |

And then on the gondola ride down,

link |

we decided to just jam a little bit.

link |

And it was just like, I don't know,

link |

the gondola sort of came over a mountain

link |

and you saw the city lights

link |

and we're just like jamming, like playing some music.

link |

I wouldn't describe that as transformative.

link |

I don't know why, but that popped into my mind

link |

as a moment of, in a way that wasn't associated

link |

with people I love, but more with like a thing I was doing,

link |

something that was just, it was just happy

link |

and it was just like a great moment.

link |

I don't think I can give you anything deeper than that.

link |

Well, as a musician myself, I'd love to see,

link |

as you mentioned before, music enter back into your work,

link |

back into your creative work.

link |

I'd love to see that.

link |

I'm certainly allowing it to enter back into mine.

link |

And it's a beautiful thing for a mathematician,

link |

for a scientist to allow music to enter their work.

link |

I think only good things can happen.

link |

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

link |

By the end of 2020.

link |

Okay, all right, good.

link |

Give myself a longer window.

link |

All right, maybe we can like collaborate

link |

on a band type situation.

link |

What instruments do you play?

link |

The main instrument I play is violin,

link |

but I also love to dabble around on like guitar and piano.

link |

Beautiful, me too, guitar and piano.

link |

So in a mathematician's lament, Paul Lockhart writes,

link |

the first thing to understand

link |

is that mathematics is an art.

link |

The difference between math and the other arts,

link |

such as music and painting,

link |

is that our culture does not recognize it as such.

link |

So I think I speak for millions of people, myself included,

link |

in saying thank you for revealing to us

link |

the art of mathematics.

link |

So thank you for everything you do

link |

and thanks for talking today.

link |

Wow, thanks for saying that.

link |

And thanks for having me on.

link |

Thanks for listening to this conversation

link |

with Grant Sanderson.

link |

And thank you to our presenting sponsor, Cash App.

link |

Download it, use code LEXPodcast.

link |

You'll get $10 and $10 will go to FIRST,

link |

a STEM education nonprofit that inspires

link |

hundreds of thousands of young minds

link |

to become future leaders and innovators.

link |

If you enjoy this podcast, subscribe on YouTube,

link |

give it five stars on Apple Podcast,

link |

support on Patreon, or connect with me on Twitter.

link |

And now, let me leave you with some words of wisdom

link |

from one of Grant's and my favorite people, Richard Feynman.

link |

Nobody ever figures out what this life is all about,

link |

and it doesn't matter.

link |

Explore the world.

link |

Nearly everything is really interesting

link |

if you go into it deeply enough.

link |

Thank you for listening, and hope to see you next time.