back to indexGrant 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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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
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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.