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 Three Blue One Brown,
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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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I'm Alex Friedman, spelled F R I D M A N.
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And now here's my conversation with Grant Sanderson.
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If there's intelligent life out there in the universe,
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do you think their mathematics is different than ours?
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I think it's probably very different.
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There's an obvious sense.
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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
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with the form of their existence, right?
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Do you think they have basic arithmetic?
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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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Cause 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 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 surreal 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, you know,
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model the universe and motion of planets
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with that as the backend 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 to describe
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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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have brought to the table
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is just scratching the surface,
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surface 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 the 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 a 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,
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but we'll see the number E.
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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 that implies
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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 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 a number
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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 to 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
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and be like, 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
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is actually missing the main point,
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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 tether ball 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.
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It's this 90 degree rotation.
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That's what the whole idea of like complex
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explanation 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
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and you walk, 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 of repeatedly multiplying
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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 have we were,
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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,
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I think the fact that it represents, it's pretty.
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It's not like the most beautiful thing in the world,
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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,
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perhaps the awkwardness of the notation,
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somehow still nevertheless coming together nicely.
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Cause 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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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,
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but it's not that close.
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So actually thinking of it as a function is the better idea.
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And that's a notational idea.
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And yeah, like here's the thing.
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The constant E sort of stands as this numerical representative
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of calculus, right?
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Calculus is the like study of change.
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So at the very least,
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there's a little cognitive dissonance using a constant
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to represent the science of change.
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I never thought of it that way.
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But 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
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as being much more significant 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 yeah,
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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, as 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
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of what our mind 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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Higher dimensions 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
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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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Because of state spaces.
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But you're saying 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 of 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 their 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, 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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You 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 bona fide 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 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.
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Then along that thread is,
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what do you think is the difference 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 of abstractions
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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 a set
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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, right?
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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 into the more and more abstract things
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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
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of motivation into math are gonna give you
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very different answers about what the relation
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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 like differential equations
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related things because that is the motivator
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behind the study of PDEs and things like that.
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But you'll have others who like,
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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 that,
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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 what 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 to me like Stephen Wolfram
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who thinks that sort of,
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there's incredibly simple rules 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.
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Yeah, I have no idea.
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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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other 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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they're like, yeah, 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, right?
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You know, maybe there's something where it's like,
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of course there will always be some thing 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 complicated.
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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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Like not in a strict definition of cast,
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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
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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 plan is moving just the low level
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of the atomic scale, how the materials operate
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at the high scale, how black holes operate.
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But 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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It just seems beautiful.
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It's also weird probably to the point you were 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 and have nice
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E equals empty squared equations.
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Well, I wonder would such a world with uncompressible laws
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allow for the kind of beings that can think about
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the kind of questions that you're asking?
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Like an anthropic principle coming into play
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at some weird way here.
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Like I don't know what I'm talking about.
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Well, maybe the universe is actually not so compressible,
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but the way our brain evolved
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were only able to perceive the compressible parts.
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I mean, this is a 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
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models of the world,
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but then actually twist the world 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 afield,
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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 a proof of concept that the laws
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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 that,
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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,
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you just throw a dart 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,
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like it quickly becomes the case
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that you have a limit to the layers that could exist there
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because like the resources necessary
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to simulate a universe like ours clearly is a lot.
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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 over all of those meta layers
link |
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
link |
the amount, 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 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 us, the smallness of us,
link |
the unlikeliness of everything that's on this earth
link |
coming to be, 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 |
where there are 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 to 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 |
we'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 |
Like 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 a small unit of space.
link |
That makes me think it's at least plausible
link |
that whatever the highest level of existence is
link |
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 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 and a psychology perspective.
link |
Are you psychologically comfortable
link |
with the concept of infinity?
link |
Are you okay with it?
link |
I'm pretty okay, yeah.
link |
It doesn't make any sense to me.
link |
How many possible words do you think could exist
link |
that are just like strings of letters?
link |
That's a sort of mathematical statement as beautiful
link |
and we use infinity in basically everything we do
link |
in science, math and engineering, yes.
link |
But you said exist.
link |
The question is, you said letters or words?
link |
To bring words into existence to me,
link |
you have to start saying them or writing them
link |
That's an instantiation.
link |
How many abstract words exist?
link |
Well, the idea of abstract.
link |
The idea of abstract notions and ideas.
link |
I think we should be clear on terminology.
link |
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 |
Abstraction is key 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 analog of pixels on my retina change entirely,
link |
that I can still have some coherent notion
link |
of this is Lex, I'm talking about Lex.
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.
link |
That's a kind of abstraction.
link |
It's a thing that could apply to a lot of different images
link |
that I see and it represents it in a much more compressed way
link |
and one that's much more resilient to that.
link |
I think in the same way, if I'm talking about
link |
infinity as an abstraction,
link |
I don't mean nonphysical, woo woo, 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 your face on my retina
link |
share enough common attributes
link |
that I can put this single notion to it.
link |
In that way, infinity is an abstraction,
link |
and it's very powerful.
link |
It's only through such abstractions
link |
that we can actually understand
link |
the world and logic and things.
link |
In the case of infinity, the way I think about it,
link |
the key entity is the property
link |
of always being able to add one more.
link |
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.
link |
I could always add one more
link |
that gives it this nature of infinitiveness
link |
in the same way that there are certain properties
link |
of your face that give it the lexness.
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 |
That's what our brain does.
link |
That's what intelligence 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,
link |
quote, woo, right?
link |
And because we're now, it's like,
link |
we built this stack of, you know,
link |
the only thing that's true is the stuff
link |
that's on the ground.
link |
Everything else is useful for interpreting this.
link |
And at a certain point,
link |
you might start floating into ideas
link |
that are surreal and difficult
link |
and take us into areas that are
link |
disconnected from reality
link |
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 phenomena that you're describing
link |
is overgeneralization.
link |
When you try to...
link |
Generalization, yeah.
link |
Have a concept or an idea that's so general
link |
as to apply to nothing in particular
link |
Does that map to what you're thinking of
link |
when you think of...
link |
First of all, I'm playing a little
link |
just for the fun of it.
link |
And I think our cognition,
link |
our mind is unable to visualize.
link |
So you do some incredible work
link |
with visualization and video.
link |
I think infinity is very difficult
link |
to visualize for our mind.
link |
We can delude ourselves
link |
into thinking we can visualize it.
link |
I mean, I don't...
link |
I would venture to say it's very difficult.
link |
And so there's some concepts in mathematics
link |
like maybe multiple dimensions.
link |
We could sort of talk about it.
link |
It's impossible for us to truly intuit.
link |
And it just feels dangerous to me
link |
to use these as part of our toolbox of abstractions.
link |
On behalf of your listeners,
link |
I almost fear we're getting too philosophical.
link |
But I think to that point,
link |
for any particular idea like this,
link |
there's multiple angles of attack.
link |
I think when we do visualize infinity,
link |
what we're actually doing, you write dot, dot, dot.
link |
One, two, three, four, dot, dot, dot.
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.
link |
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,
link |
no, not one, two, three, four, dot, dot, dot.
link |
One, two, three, four, five, six, seven, eight.
link |
I try to get them all in my head and you realize,
link |
oh, 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.
link |
In some ways, I get very confused
link |
often 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,
link |
yeah, 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 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,
link |
we will derive what we need in order to do the math
link |
of linear algebra or whatever it might be.
link |
I don't think that's how understanding works at all.
link |
I think how understanding works is you start
link |
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.
link |
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
link |
in a medium like video,
link |
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
link |
that that's exactly right.
link |
So the visualizations you're creating
link |
while you're sometimes talking about abstractions,
link |
the actual visualization
link |
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?
link |
the visualization itself is actually going to the bottom
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,
link |
I don't know why, I just always do,
link |
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 |
You're always going to 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
link |
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 |
a high cognitive load
link |
set of symbols that needs to be
link |
populated with examples later on,
link |
assuming someone's still with you.
link |
What is the most beautiful
link |
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
link |
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 |
Almost a moment of the discovery
link |
for you personally, almost just that
link |
leap of a ha ha moment.
link |
So something that really caught my eye.
link |
I remember when I was little,
link |
there were these like,
link |
I think the series was called like wooden books
link |
or something, these tiny little books that
link |
would have just a very short description of something
link |
on the left and then a picture on the right.
link |
I don't know who they're meant for, but maybe it's like
link |
loosely children or something like that.
link |
But it can't just be children because of some of the things
link |
on the last page of one of them,
link |
somewhere tiny in there was this little formula
link |
that on the left hand
link |
had a sum over all of the natural numbers.
link |
You know, it's like one over
link |
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
link |
had a product over all of the primes
link |
and it was a certain thing had to do with all the primes
link |
like any good young math enthusiast,
link |
I'd properly been indoctrinated with how chaotic
link |
and confusing the primes are, which they are
link |
equation where on one side you have something
link |
that's as understandable as you could possibly get
link |
the counting numbers and on the other side
link |
is all the prime numbers. It was like this.
link |
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
link |
for the zeta function as I later found out.
link |
The equation itself
link |
essentially encodes the fundamental theorem
link |
of arithmetic that every number can be expressed
link |
as a unique set of primes.
link |
I certainly don't understand this equation
link |
or this function all that well.
link |
The more I learn about it, the prettier it is.
link |
this is sort of what gets you
link |
representations of primes
link |
not in terms of primes themselves
link |
but in terms of another set of numbers
link |
that are like the non trivial zeros of the zeta function.
link |
kind of in over my head in a lot of ways
link |
as I try to get to understand it.
link |
But the more I do,
link |
it always leaves enough mystery
link |
that it remains very beautiful to me.
link |
Whenever there's a little bit of mystery
link |
just outside of the understanding
link |
the process of learning more about it,
link |
how does that come about?
link |
Just your own thought or are you
link |
Or is the process of visualization itself
link |
revealing more to you?
link |
In one time when I was just trying to understand
link |
utilizing complex functions,
link |
this is what led to a video
link |
about this function.
link |
It's titled something like visualizing the Riemann zeta function.
link |
It's one that came about because
link |
I was programming and
link |
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 |
I mean, you try to
link |
get your hands on as much reading as you can.
link |
if they have like a
link |
math background of some, like they studied some
link |
in college or something like that,
link |
like the Princeton companion to math has a really good article
link |
on analytic number theory
link |
and that itself has a whole bunch of references
link |
and, you know, anything has
link |
more references and it gives you this, like, tree
link |
to start plying 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 |
cousins of the Fourier transform,
link |
like, coming to play,
link |
and you realize, oh, it's probably pretty important
link |
to have deep intuitions of the Fourier transform,
link |
even if it's not explicitly mentioned
link |
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
link |
when I try to understand
link |
the exact relation between,
link |
like, the zeros 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 one one hundredth
link |
as well as the experts on the matter do,
link |
towards getting there is, it's fun,
link |
it's charming, and, like, to your question,
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 |
yeah, these very unpredictable patterns
link |
where these intricate
link |
properties of, like, a certain function.
link |
But at the same time, it doesn't feel like
link |
humans ever made an arbitrary choice
link |
in studying this particular thing.
link |
So, you know, it feels like
link |
you're speaking to patterns themselves
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 because
link |
this is sort of what the word
link |
contrived is meant to apply to, right?
link |
And when they're not arbitrary,
link |
it means it could be
link |
you can have a clean
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 |
I think you talked to that other civilization.
link |
They're probably also studying the zeroes
link |
of the Riemann Zeta function.
link |
There's some variant thereof
link |
clearly equivalent cousin or something like that.
link |
But that's probably on their
link |
Whenever somebody does a lot of something
link |
I'm going to ask the question
link |
that you've already been asked a lot,
link |
that you'll get more and more asked in your life.
link |
But what was your favorite
link |
Oh, favorite to create.
link |
One of my favorites is
link |
the title is Who Cares about Topology.
link |
Do you want me to pull it up?
link |
If you want, sure.
link |
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? This is not
link |
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 |
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? That's hard.
link |
But it's doable and the path to it
link |
things like looking at a
link |
torus, this surface with a single hole in it,
link |
like a donut, or looking at a mobius strip
link |
in ways that feel so much less contrived
link |
as like a little kid, learned about these surfaces
link |
and shapes, like a mobius strip and a torus.
link |
Like what you learn is,
link |
oh, this mobius strip, you take a piece of paper,
link |
put a twist, glue it together,
link |
and now you have a shape with one edge and just one side.
link |
as a student, you should think,
link |
How does that help me solve any problems?
link |
I thought math was about problem solving.
link |
So what I liked about
link |
the piece of math that this was describing
link |
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 not just doing something.
link |
It's not just playing with construction paper.
link |
And the way that it solves the problem
link |
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 is a very specific
link |
example of what you're describing.
link |
The construction here is very abstract in nature.
link |
You describe this very abstract kind of surface
link |
So then when I was finding myself,
link |
a graph that's built into OSX
link |
to draw that surface,
link |
you realize, oh man, the topology argument
link |
is very nonconstructive.
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,
link |
ah, 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 |
It'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 come about 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 what it's going to look like.
link |
And it was wonderful.
link |
It looks like a Sydney Opera House
link |
or some sort of Frank Gary design.
link |
You knew it was going to 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,
link |
it's just pleasing, right?
link |
So that was fun to make, very fun to share.
link |
I hope that it has elucidated
link |
for some people out there
link |
where these constructs of topology come from,
link |
that it's not arbitrary play
link |
with construction paper.
link |
So let's, I think this is a good,
link |
a good sort of example
link |
to talk a little bit about your process.
link |
So you have a list of ideas.
link |
So that's sort of the
link |
the curse of having
link |
having an active and brilliant mind
link |
is I'm sure you have a list
link |
that's growing faster than you can utilize.
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
link |
to write a narrative arc
link |
how do I elucidate out?
link |
How do I make this idea beautiful
link |
and clear and explain it?
link |
And then there's a set of visualizations
link |
that will be attached to it.
link |
You've talked about some of this before
link |
about sort of writing the story
link |
attaching the visualizations.
link |
Can you talk through
link |
interesting, painful,
link |
beautiful parts of that process?
link |
Well, the most painful is
link |
if you've chosen a topic that you do want to do,
link |
but then it's hard to think of
link |
I guess how to structure the script.
link |
sort of where I have been on one for like
link |
the last two or three months and I think
link |
ultimately the right resolution is just like set it aside
link |
and instead do some other things
link |
where the script comes more naturally.
link |
Because you sort of don't want to overwork
link |
that 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 judge in your head?
link |
Sort of the person,
link |
the essence that's saying
link |
this sucks or this is good.
link |
And you mentioned kind of the student
link |
you're thinking about.
link |
What is that thing?
link |
That says the perfectionist
link |
that says this thing sucks
link |
you need to work on it 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 to empathize with
link |
because that's kind of the only person I know.
link |
You don't really know anyone other than versions of yourself.
link |
So I start with the version of myself
link |
that I know who doesn't yet understand the thing.
link |
view it with fresh eyes
link |
a particular visual or a particular script.
link |
Is this motivating? Does this make sense?
link |
Which has its downsides
link |
because sometimes I find myself
link |
speaking to motivations
link |
would be interested in.
link |
I did this project on quaternions
link |
where what I really wanted
link |
was to understand what are they doing in four dimensions.
link |
Can we see what they're doing in four dimensions?
link |
had a way of thinking about it
link |
that really answered the question in my head
link |
that made me very satisfied and being able to think about
link |
concretely with a 3D visual
link |
what are they doing to a 4D sphere?
link |
And so I'm like great this is exactly what my past self
link |
would have wanted 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
link |
who want to learn about quaternions
link |
are robotics engineers
link |
or graphics programmers
link |
who want to understand how they're used
link |
to describe 3D rotations
link |
and their use case was actually a little bit different
link |
than my past self and in that way
link |
I wouldn't actually recommend that video to
link |
people who are coming at it from that angle
link |
of wanting to know hey I'm a robotics programmer
link |
how do these quaternion things
link |
position in 3D space.
link |
I would say other great resources
link |
for that if you ever find yourself
link |
wanting to say like but hang on
link |
in what sense are they acting in 4 dimensions
link |
then come back but until then
link |
it's a little different.
link |
Yeah it's interesting because
link |
you have incredible videos on neural networks
link |
for example and from my sort of
link |
perspective because I've probably
link |
I mean I looked at the
link |
it is sort of my field
link |
and I've also looked at the basic
link |
introduction of neural networks like a million
link |
times from different perspectives
link |
and it made me realize that there's a lot of ways
link |
to present it so if you were sort of
link |
you did an incredible job
link |
I mean sort of the
link |
but you could also do it
link |
differently and also incredible
link |
a beautiful presentation
link |
of a basic concept
link |
is requires sort of
link |
creativity requires genius
link |
and so on but you can take it
link |
from a bunch of different perspectives
link |
and you realize that
link |
and just as you're saying
link |
you kind of have a certain mindset
link |
a certain view but
link |
if you take a different view
link |
from a physics perspective
link |
from a neuroscience
link |
perspective talking about neural networks
link |
a robotics perspective
link |
from a pure learning theory
link |
statistics perspective so you can create
link |
totally different videos
link |
with a few actually concepts
link |
where you have taken different concepts
link |
at the Euler equation
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 |
never enough so you don't think it's
link |
the most beautiful equation in mathematics
link |
as we represent it it's one of the most hideous
link |
it involves a lot of the most hideous
link |
aspects of our notation I talked about E
link |
the fact that we use pi instead of tau
link |
call imaginary numbers imaginary
link |
and then actually wonder
link |
if we use the I because of imaginary
link |
I don't know if that's historically accurate
link |
but at least a lot of people they read the I
link |
and they think imaginary
link |
like all three of those facts it's like
link |
those are things that have added more confusion than they needed to
link |
and we're wrapping them up in one equation
link |
like boy that's just very hideous
link |
the ideas that it does
link |
tie together when you wash away the notation
link |
it's pretty it's nice
link |
mind blowing greatest thing in the universe
link |
which is maybe what I was thinking of when I said
link |
like once you understand
link |
something it doesn't have the same
link |
beauty like I feel like I understand
link |
and I feel like I understand it enough to
link |
the version that just woke up
link |
that hasn't really gotten itself dressed
link |
in the morning that's a little bit groggy and there's
link |
bags under its eyes
link |
you're past the dating
link |
stage and you're no longer dating
link |
I'm still dating the Zeta function
link |
and she's beautiful
link |
and it's that high dopamine part
link |
but maybe at some point we'll settle into
link |
the more mundane nature of the relationship
link |
where I 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
link |
that's the nice thing about mathematics
link |
as long as you don't live forever
link |
there will always be
link |
enough mystery and fun with some of the equations
link |
the rate at which questions comes up is much faster
link |
than the rate at which answers come up so
link |
if you could live forever would you
link |
you don't think mortality is the thing that makes life
link |
would your life be four times as meaningful
link |
so this goes to infinity
link |
that's really interesting so what I said is infinite
link |
not four times longer
link |
so the actual existence
link |
the existence of the end
link |
no matter the length
link |
is the thing that may
link |
for my comprehension of psychology
link |
it's such a deeply
link |
it's such a fundamental part of the human condition
link |
the fact that we're mortal
link |
the fact that things end
link |
seems to be a crucial part of what gives them
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
link |
or is it the superego
link |
the reflective self
link |
the vernici's area that puts all this stuff
link |
a small percentage of your mind
link |
is actually aware of the true
link |
motivations that drive you
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 |
like 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 construct an illusion
link |
of reality by enjoying the moment
link |
sort of the existential approach life
link |
gun to my head, I don't think that's it
link |
another sort of way to say gun to the head
link |
is sort of the deep psychological
link |
introspection of what drives us
link |
in some ways to me
link |
when I look at math, when I look at science
link |
is it kind of an escape from reality
link |
it's such a beautiful
link |
journey of discovery
link |
that it allows you to actually
link |
it allows you to achieve
link |
a kind of immortality
link |
and sort of connect yourself to the thing
link |
that is seemingly infinite
link |
that it allows you to escape
link |
the limited nature
link |
of our bodies, of our existence
link |
what else would give this podcast meaning?
link |
the fact that it will end
link |
this place closes in 40 minutes
link |
and it's so much more meaningful for it
link |
I love this room because we'll be kicked out
link |
just because you're trolling me
link |
doesn't mean I'm wrong
link |
but I take your point
link |
boy that would be a good Twitter bio
link |
you're trolling me doesn't mean I'm wrong
link |
difference in backgrounds
link |
I'm a bit Russian so we're a bit
link |
melancholic and seem to maybe
link |
assign a little too much value to
link |
suffering immortality and things like that
link |
makes for a better novel I think
link |
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 |
I'll write the script
link |
I want there to be some kind of aha moment
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 |
animate it I often change
link |
certainly the specific words
link |
but sometimes the structure itself
link |
deterministic process at that point
link |
it makes it much easier to predict when something
link |
will be done. How do you know when a script is
link |
done? For problem solving videos
link |
that's quite simple, it's once you feel
link |
like someone who didn't understand the solution now
link |
could. 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 |
just at some point you feel like
link |
a meaningful question and it answers that question
link |
What is the best way to learn math for people
link |
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 who are not really
link |
at the beginning of their journey
link |
there might be actually
link |
deep in their career
link |
type of technology taking calculus and so on
link |
but still want to sort of explore math
link |
what would be your advice
link |
instead of education at all ages?
link |
Your temptation will be to
link |
spend more time like watching lectures
link |
try to force yourself to do more problems
link |
than you naturally would.
link |
The focus time that you're spending should be
link |
on solving specific problems
link |
and seek entities that have well
link |
curated lists of problems.
link |
So going to like a textbook almost
link |
and the problems in the back of a textbook
link |
and the back of a chapter.
link |
So if you can take a little look through those
link |
questions at the end of the chapter before you read the chapter
link |
a lot of them won't make sense. Some of them might
link |
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 and then maybe
link |
take a look again and things like that.
link |
And don't consider yourself done with the chapter
link |
until you've actually worked
link |
through a couple exercises.
link |
And this is so hypocritical
link |
because I put out videos that
link |
pretty much never have associated
link |
exercises. I just view myself
link |
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
link |
questions you can work through.
link |
Just a very basic sort of
link |
just picking up getting comfortable
link |
with a very basic linear algebra or calculus
link |
Programming is actually I think a great
link |
like learn to program and like
link |
let the way that math is motivated from that
link |
through. I know a lot of people who
link |
didn't like math got into programming
link |
in some way and that's what turned them on to math.
link |
Maybe I'm biased because like I live in the Bay Area
link |
so I'm more likely to run into someone
link |
who has that phenotype
link |
willing to speculate that that is a more generalizable path.
link |
kind of in creating videos are using
link |
programming to illuminate a concept
link |
but for yourself as well.
link |
So would you recommend somebody try to
link |
make a sort of almost like
link |
try to make videos?
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
link |
numbers but it rings in the mind as
link |
being true. You remember about 10%
link |
of what you read. You remember about 20%
link |
of what you listen to. You remember
link |
about 70% of what you actively interact
link |
with in some way and then about 90%
link |
of what you teach.
link |
I think I heard again those numbers might be
link |
meaningless but they ring true don't they?
link |
I'm willing to say I
link |
learned nine times better from teaching something
link |
than reading. That might even be a low ball.
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
link |
remember that you would like
link |
to relive because it made you truly happy
link |
transformative in some
link |
fundamental way? A moment that was
link |
Or made you truly happy?
link |
Yeah, I think there's times
link |
music used to be a much bigger part of my life
link |
than it is now. Like when I was a
link |
teenager and I can think of
link |
sometimes in like playing music
link |
my brother and a friend of mine so this
link |
slightly violates the family and friends but
link |
there was music that made me happy. They were just
link |
We played a gig at a ski resort
link |
such that you take a gondola
link |
to the top and did a thing
link |
and then on the gondola right down we decided to just jam
link |
it was just like, I don't know, the gondola
link |
sort of over came over a mountain
link |
and you saw the city lights
link |
and we're just like jamming, like playing some
link |
music. I wouldn't describe that as
link |
transformative. I don't know why
link |
but that popped into my mind as a moment of
link |
in a way that wasn't associated
link |
with people I love but more with like a thing
link |
I was doing, something that was
link |
just, it was just happy and it was just like
link |
I don't think I can give you anything deeper than
link |
that though. Well as a musician
link |
myself, I'd love to see
link |
as you mentioned before
link |
music enter back into your work
link |
back into your creative work. I'd love to see that
link |
I'm certainly allowing it
link |
to enter back into mine and it's
link |
for a mathematician, for a scientist
link |
to allow music to enter their
link |
work. I think only good
link |
things can happen. Alright, I'll try to promise
link |
you a music video by 2020.
link |
By 2020? By the end of 2020.
link |
Okay, alright good. I'll give myself a longer window.
link |
we can like collaborate on a
link |
band type situation. What instruments do you play?
link |
The main instrument I play is violin
link |
but I also love to dabble around on like guitar
link |
and piano. Beautiful. Me too, guitar
link |
So in a mathematician's
link |
lament, Paul Lockhart writes
link |
the first thing to understand is that mathematics
link |
is an art. The difference between
link |
math and the other arts such as
link |
music and painting
link |
is that our culture does not recognize it
link |
as such. So I think I speak
link |
for millions of people,
link |
in saying thank you for revealing
link |
to us the art of mathematics.
link |
So thank you for everything you do
link |
and thanks for talking today. Well, thanks
link |
for saying that and thanks for having me on.
link |
Thanks for listening to this conversation
link |
with Grant Sanderson and thank you
link |
to our presenting sponsor, Cash App.
link |
You'll get $10 and $10
link |
will go to FIRST, a STEM education
link |
nonprofit that inspires hundreds of
link |
thousands of young minds to become future
link |
leaders and innovators.
link |
If you enjoy this podcast, subscribe
link |
on YouTube, give it 5 stars on Apple
link |
Podcast, support it on Patreon
link |
or connect with me on Twitter.
link |
And now, let me leave you
link |
with some words of wisdom from one of
link |
Grant's and my favorite people,
link |
Nobody ever figures
link |
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
link |
and good to see you next time.