Grant Sanderson: 3Blue1Brown and the Beauty of Mathematics
数学音乐与艺术技术与编程物理与宇宙学心理与人性
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🔑 关键词
mathdonbeautifulitselffunctionuniversecertainmathematicsnotationnumbersdoesnphysicalabstractphysicsabstractionputtalkingwordsdoingapp
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🎙️ 完整对话（1576 条）
Lex Fridman (00:00.000)
The following is a conversation with Grant Sanderson.
以下是与格兰特·桑德森的对话。
Lex Fridman (00:03.060)
He's a math educator and creator of 3Blue1Brown,
他是一名数学教育家和 3Blue1Brown 的创建者，
Lex Fridman (00:06.620)
a popular YouTube channel
热门 YouTube 频道
Lex Fridman (00:07.980)
that uses programmatically animated visualizations
使用以编程方式动画可视化
Lex Fridman (00:11.020)
to explain concepts in linear algebra, calculus,
解释线性代数、微积分中的概念，
Lex Fridman (00:14.260)
and other fields of mathematics.
以及其他数学领域。
Lex Fridman (00:16.980)
This is the Artificial Intelligence Podcast.
这是人工智能播客。
Grant Sanderson (00:19.820)
If you enjoy it, subscribe on YouTube,
如果您喜欢，请在 YouTube 上订阅，
Lex Fridman (00:22.100)
give us five stars on Apple Podcast,
在 Apple Podcast 上给我们五颗星，
Grant Sanderson (00:23.960)
follow on Spotify, support on Patreon,
关注 Spotify，支持 Patreon，
Lex Fridman (00:26.340)
or simply connect with me on Twitter
或者直接在 Twitter 上与我联系
Grant Sanderson (00:28.340)
at Lex Friedman, spelled F R I D M A N.
在 Lex Friedman，拼写为 F R I D M A N。
Lex Fridman (00:32.180)
I recently started doing ads
我最近开始做广告
Grant Sanderson (00:34.020)
at the end of the introduction.
在介绍的最后。
Lex Fridman (00:35.700)
I'll do one or two minutes after introducing the episode
我会在介绍剧集后做一两分钟
Lex Fridman (00:38.500)
and never any ads in the middle
中间没有任何广告
Lex Fridman (00:40.020)
that can break the flow of the conversation.
这可能会破坏谈话的流畅性。
Grant Sanderson (00:42.160)
I hope that works for you
我希望这对你有用
Lex Fridman (00:43.580)
and doesn't hurt the listening experience.
并且不会影响聆听体验。
Grant Sanderson (00:47.300)
This show is presented by Cash App,
本节目由Cash App 呈现，
Lex Fridman (00:49.660)
the number one finance app in the App Store.
Grant Sanderson (00:52.020)
I personally use Cash App to send money to friends,
Lex Fridman (00:54.820)
but you can also use it to buy, sell,
Lex Fridman (00:56.520)
and deposit Bitcoin in just seconds.
Lex Fridman (00:58.940)
Cash App also has an investing feature.
Grant Sanderson (01:01.500)
You can buy fractions of a stock, say $1 worth,
Lex Fridman (01:04.540)
no matter what the stock price is.
Grant Sanderson (01:06.780)
Broker services are provided by Cash App Investing,
Lex Fridman (01:09.580)
a subsidiary of Square, and member SIPC.
Grant Sanderson (01:13.460)
I'm excited to be working with Cash App
Lex Fridman (01:15.460)
to support one of my favorite organizations called FIRST,
Grant Sanderson (01:18.580)
best known for their FIRST Robotics and LEGO competitions.
Lex Fridman (01:22.020)
They educate and inspire hundreds of thousands of students
Grant Sanderson (01:25.340)
in over 110 countries
Lex Fridman (01:27.060)
and have a perfect rating on Charity Navigator,
Grant Sanderson (01:29.680)
which means the donated money is used
Lex Fridman (01:31.660)
to maximum effectiveness.
Grant Sanderson (01:34.020)
When you get Cash App from the App Store at Google Play
Lex Fridman (01:36.900)
and use code LEXPODCAST, you'll get $10,
Lex Fridman (01:40.140)
and Cash App will also donate $10 to FIRST,
Lex Fridman (01:42.820)
which again is an organization
Grant Sanderson (01:44.920)
that I've personally seen inspire girls and boys
Lex Fridman (01:47.640)
to dream of engineering a better world.
Lex Fridman (01:51.400)
And now, here's my conversation with Grant Sanderson.
Lex Fridman (01:56.220)
If there's intelligent life out there in the universe,
Lex Fridman (01:59.100)
do you think their mathematics is different than ours?
Lex Fridman (02:03.260)
Jumping right in.
Grant Sanderson (02:04.300)
I think it's probably very different.
Lex Fridman (02:08.340)
There's an obvious sense the notation is different, right?
Grant Sanderson (02:11.220)
I think notation can guide what the math itself is.
Lex Fridman (02:14.500)
I think it has everything to do with the form
Lex Fridman (02:17.020)
of their existence, right?
Lex Fridman (02:19.180)
Do you think they have basic arithmetic?
Grant Sanderson (02:20.760)
Sorry, I interrupted.
Lex Fridman (02:21.600)
Yeah, so I think they count, right?
Grant Sanderson (02:23.220)
I think notions like one, two, three,
Lex Fridman (02:24.580)
the natural numbers, that's extremely, well, natural.
Grant Sanderson (02:27.200)
That's almost why we put that name to it.
Lex Fridman (02:30.460)
As soon as you can count,
Lex Fridman (02:31.760)
you have a notion of repetition, right?
Lex Fridman (02:34.020)
Because you can count by two, two times or three times.
Lex Fridman (02:37.340)
And so you have this notion of repeating
Lex Fridman (02:39.800)
the idea of counting, which brings you addition
Lex Fridman (02:42.180)
and multiplication.
Lex Fridman (02:43.780)
I think the way that we extend it to the real numbers,
Grant Sanderson (02:47.540)
there's a little bit of choice in that.
Lex Fridman (02:49.280)
So there's this funny number system
Grant Sanderson (02:50.820)
called the servial numbers
Lex Fridman (02:52.220)
that it captures the idea of continuity.
Grant Sanderson (02:55.660)
It's a distinct mathematical object.
Lex Fridman (02:57.580)
You could very well model the universe
Lex Fridman (03:00.420)
and motion of planets with that
Lex Fridman (03:02.020)
as the back end of your math, right?
Lex Fridman (03:04.820)
And you still have kind of the same interface
Lex Fridman (03:06.500)
with the front end of what physical laws you're trying to,
Grant Sanderson (03:10.620)
or what physical phenomena you're trying
Lex Fridman (03:12.420)
to describe with math.
Lex Fridman (03:13.780)
And I wonder if the little glimpses that we have
Lex Fridman (03:15.900)
of what choices you can make along the way
Grant Sanderson (03:17.700)
based on what different mathematicians
Lex Fridman (03:19.320)
I've brought to the table
Grant Sanderson (03:20.440)
is just scratching the surface
Lex Fridman (03:22.300)
of what the different possibilities are
Lex Fridman (03:24.540)
if you have a completely different mode of thought, right?
Lex Fridman (03:27.540)
Or a mode of interacting with the universe.
Lex Fridman (03:29.700)
And you think notation is a key part of the journey
Lex Fridman (03:32.540)
that we've taken through math.
Grant Sanderson (03:33.860)
I think that's the most salient part
Lex Fridman (03:35.140)
that you'd notice at first.
Grant Sanderson (03:36.420)
I think the mode of thought is gonna influence things
Lex Fridman (03:38.880)
more than like the notation itself.
Lex Fridman (03:40.400)
But notation actually carries a lot of weight
Lex Fridman (03:42.700)
when it comes to how we think about things,
Grant Sanderson (03:44.620)
more so than we usually give it credit for.
Lex Fridman (03:47.100)
I would be comfortable saying.
Lex Fridman (03:48.900)
Do you have a favorite or least favorite piece of notation
Lex Fridman (03:52.020)
in terms of its effectiveness?
Grant Sanderson (03:53.260)
Yeah, yeah, well, so least favorite,
Lex Fridman (03:54.860)
one that I've been thinking a lot about
Grant Sanderson (03:56.140)
that will be a video I don't know when, but we'll see.
Lex Fridman (03:59.900)
The number e, we write the function e to the x,
Grant Sanderson (04:02.420)
this general exponential function
Lex Fridman (04:04.140)
with a notation e to the x
Grant Sanderson (04:06.260)
that implies you should think about a particular number,
Lex Fridman (04:08.380)
this constant of nature,
Lex Fridman (04:09.380)
and you repeatedly multiply it by itself.
Lex Fridman (04:11.740)
And then you say, oh, what's e to the square root of two?
Lex Fridman (04:14.100)
And you're like, oh, well, we've extended the idea
Lex Fridman (04:15.700)
of repeated multiplication.
Grant Sanderson (04:17.020)
That's all nice, that's all nice and well.
Lex Fridman (04:19.460)
But very famously, you have like e to the pi i,
Lex Fridman (04:22.660)
and you're like, well, we're extending the idea
Lex Fridman (04:24.420)
of repeated multiplication into the complex numbers.
Grant Sanderson (04:27.060)
Yeah, you can think about it that way.
Lex Fridman (04:28.780)
In reality, I think that it's just the wrong way
Grant Sanderson (04:31.820)
of notationally representing this function,
Lex Fridman (04:34.740)
the exponential function,
Grant Sanderson (04:36.180)
which itself could be represented
Lex Fridman (04:37.820)
a number of different ways.
Grant Sanderson (04:38.780)
You can think about it in terms of the problem it solves,
Lex Fridman (04:41.300)
a certain very simple differential equation,
Grant Sanderson (04:43.260)
which often yields way more insight
Lex Fridman (04:45.620)
than trying to twist the idea of repeated multiplication,
Grant Sanderson (04:48.820)
like take its arm and put it behind its back
Lex Fridman (04:50.740)
and throw it on the desk and be like,
Lex Fridman (04:51.940)
you will apply to complex numbers, right?
Lex Fridman (04:53.620)
That's not, I don't think that's pedagogically helpful.
Lex Fridman (04:57.420)
So the repeated multiplication is actually missing
Lex Fridman (04:59.740)
the main point, the power of e to the x.
Grant Sanderson (05:03.020)
I mean, what it addresses is things where the rate
Lex Fridman (05:05.900)
at which something changes depends on its own value,
Lex Fridman (05:10.140)
but more specifically, it depends on it linearly.
Lex Fridman (05:12.500)
So for example, if you have like a population
Grant Sanderson (05:15.100)
that's growing and the rate at which it grows
Lex Fridman (05:16.660)
depends on how many members of the population
Grant Sanderson (05:18.420)
are already there,
Lex Fridman (05:19.260)
it looks like this nice exponential curve.
Grant Sanderson (05:21.300)
It makes sense to talk about repeated multiplication
Lex Fridman (05:23.300)
because you say, how much is there after one year,
Grant Sanderson (05:25.140)
two years, three years, you're multiplying by something.
Lex Fridman (05:27.420)
The relationship can be a little bit different sometimes
Grant Sanderson (05:29.460)
where let's say you've got a ball on a string,
Lex Fridman (05:33.940)
like a game of tetherball going around a rope, right?
Lex Fridman (05:37.540)
And you say, its velocity is always perpendicular
Lex Fridman (05:40.780)
to its position.
Grant Sanderson (05:42.220)
That's another way of describing its rate of change
Lex Fridman (05:44.140)
is being related to where it is,
Lex Fridman (05:47.100)
but it's a different operation.
Lex Fridman (05:48.220)
You're not scaling it, it's a rotation.
Grant Sanderson (05:49.740)
It's this 90 degree rotation.
Lex Fridman (05:51.420)
That's what the whole idea of like complex exponentiation
Grant Sanderson (05:54.580)
is trying to capture,
Lex Fridman (05:55.740)
but it's obfuscated in the notation
Grant Sanderson (05:57.660)
when what it's actually saying,
Lex Fridman (05:59.060)
like if you really parse something like e to the pi i,
Lex Fridman (06:01.460)
what it's saying is choose an origin,
Lex Fridman (06:03.900)
always move perpendicular to the vector
Lex Fridman (06:06.500)
from that origin to you, okay?
Lex Fridman (06:09.380)
Then when you walk pi times that radius,
Grant Sanderson (06:12.460)
you'll be halfway around.
Lex Fridman (06:14.180)
Like that's what it's saying.
Grant Sanderson (06:15.860)
It's kind of the, you turn 90 degrees and you walk,
Lex Fridman (06:18.620)
you'll be going in a circle.
Grant Sanderson (06:19.940)
That's the phenomenon that it's describing,
Lex Fridman (06:22.460)
but trying to twist the idea
Grant Sanderson (06:24.380)
of repeatedly multiplying a constant into that.
Lex Fridman (06:26.820)
Like I can't even think of the number of human hours
Grant Sanderson (06:30.740)
of like intelligent human hours that have been wasted
Lex Fridman (06:33.420)
trying to parse that to their own liking and desire
Grant Sanderson (06:36.820)
among like scientists or electrical engineers
Lex Fridman (06:39.500)
or students everywhere,
Grant Sanderson (06:40.660)
which if the notation were a little different
Lex Fridman (06:42.740)
or the way that this whole function was introduced
Grant Sanderson (06:45.620)
from the get go were framed differently,
Lex Fridman (06:47.620)
I think could have been avoided, right?
Lex Fridman (06:49.980)
And you're talking about
Lex Fridman (06:51.020)
the most beautiful equation in mathematics,
Lex Fridman (06:53.420)
but it's still pretty mysterious, isn't it?
Lex Fridman (06:55.100)
Like you're making it seem like it's a notational.
Grant Sanderson (06:58.660)
It's not mysterious.
Lex Fridman (06:59.740)
I think the notation makes it mysterious.
Grant Sanderson (07:01.940)
I don't think it's, I think the fact that it represents,
Lex Fridman (07:04.620)
it's pretty, it's not like the most beautiful thing
Grant Sanderson (07:06.340)
in the world, but it's quite pretty.
Lex Fridman (07:07.940)
The idea that if you take the linear operation
Grant Sanderson (07:10.620)
of a 90 degree rotation,
Lex Fridman (07:12.500)
and then you do this general exponentiation thing to it,
Grant Sanderson (07:15.700)
that what you get are all the other kinds of rotation,
Lex Fridman (07:19.460)
which is basically to say,
Grant Sanderson (07:20.420)
if your velocity vector is perpendicular
Lex Fridman (07:22.820)
to your position vector, you walk in a circle,
Grant Sanderson (07:25.380)
that's pretty.
Lex Fridman (07:26.340)
It's not the most beautiful thing in the world,
Lex Fridman (07:27.780)
but it's quite pretty.
Lex Fridman (07:28.740)
The beauty of it, I think comes from perhaps
Grant Sanderson (07:31.220)
the awkwardness of the notation
Lex Fridman (07:33.060)
somehow still nevertheless coming together nicely,
Grant Sanderson (07:35.460)
because you have like several disciplines coming together
Lex Fridman (07:38.780)
in a single equation.
Grant Sanderson (07:40.180)
Well, I think.
Lex Fridman (07:41.020)
In a sense, like historically speaking.
Grant Sanderson (07:43.500)
That's true.
Lex Fridman (07:44.340)
You've got, so like the number E is significant.
Grant Sanderson (07:45.980)
Like it shows up in probability all the time.
Lex Fridman (07:47.860)
It like shows up in calculus all the time.
Grant Sanderson (07:49.380)
It is significant.
Lex Fridman (07:50.380)
You're seeing it sort of mated with pi,
Grant Sanderson (07:52.300)
this geometric constant and I,
Lex Fridman (07:54.060)
like the imaginary number and such.
Grant Sanderson (07:55.820)
I think what's really happening there
Lex Fridman (07:57.660)
is the way that E shows up is when you have things
Lex Fridman (08:01.140)
like exponential growth and decay, right?
Lex Fridman (08:03.180)
It's when this relation that something's rate of change
Lex Fridman (08:06.100)
has to itself is a simple scaling, right?
Lex Fridman (08:10.500)
A similar law also describes circular motion.
Grant Sanderson (08:14.100)
Because we have bad notation,
Lex Fridman (08:16.340)
we use the residue of how it shows up
Grant Sanderson (08:19.100)
in the context of self reinforcing growth,
Lex Fridman (08:21.060)
like a population growing or compound interest.
Grant Sanderson (08:23.660)
The constant associated with that
Lex Fridman (08:25.300)
is awkwardly placed into the context
Grant Sanderson (08:27.660)
of how rotation comes about,
Lex Fridman (08:29.740)
because they both come from pretty similar equations.
Lex Fridman (08:32.300)
And so what we see is the E and the pi juxtaposed
Lex Fridman (08:36.220)
a little bit closer than they would be
Grant Sanderson (08:38.340)
with a purely natural representation, I would think.
Lex Fridman (08:41.020)
Here's how I would describe the relation between the two.
Grant Sanderson (08:43.380)
You've got a very important function we might call exp.
Lex Fridman (08:45.820)
That's like the exponential function.
Grant Sanderson (08:47.780)
When you plug in one,
Lex Fridman (08:49.340)
you get this nice constant called E
Grant Sanderson (08:50.980)
that shows up in like probability and calculus.
Lex Fridman (08:53.340)
If you try to move in the imaginary direction,
Grant Sanderson (08:55.340)
it's periodic and the period is tau.
Lex Fridman (08:58.420)
So those are these two constants
Grant Sanderson (08:59.620)
associated with the same central function,
Lex Fridman (09:02.180)
but for kind of unrelated reasons, right?
Lex Fridman (09:04.780)
And not unrelated, but like orthogonal reasons.
Lex Fridman (09:07.380)
One of them is what happens
Grant Sanderson (09:08.340)
when you're moving in the real direction.
Lex Fridman (09:09.700)
One's what happens when you move in the imaginary direction.
Lex Fridman (09:12.580)
And like, yeah, those are related.
Lex Fridman (09:14.180)
They're not as related as the famous equation
Grant Sanderson (09:17.180)
seems to think it is.
Lex Fridman (09:18.540)
It's sort of putting all of the children in one bed
Lex Fridman (09:20.540)
and they'd kind of like to sleep in separate beds
Lex Fridman (09:22.500)
if they had the choice, but you see them all there
Lex Fridman (09:24.580)
and there is a family resemblance, but it's not that close.
Lex Fridman (09:28.380)
So actually thinking of it as a function
Grant Sanderson (09:31.740)
is the better idea.
Lex Fridman (09:34.740)
And that's a notational idea.
Lex Fridman (09:36.340)
And yeah, and like, here's the thing.
Lex Fridman (09:39.340)
The constant E sort of stands
Lex Fridman (09:41.220)
as this numerical representative of calculus, right?
Lex Fridman (09:44.700)
Calculus is the like study of change.
Lex Fridman (09:47.580)
So at the very least there's a little cognitive dissonance
Lex Fridman (09:49.860)
using a constant to represent the science of change.
Grant Sanderson (09:53.260)
I never thought of it that way.
Lex Fridman (09:54.140)
Yeah.
Lex Fridman (09:54.980)
Right?
Lex Fridman (09:55.820)
Yeah.
Grant Sanderson (09:56.660)
It makes sense why the notation came about that way.
Lex Fridman (10:00.180)
Because this is the first way that we saw it
Grant Sanderson (10:02.100)
in the context of things like population growth
Lex Fridman (10:03.860)
or compound interest.
Grant Sanderson (10:04.780)
It is nicer to think about as repeated multiplication.
Lex Fridman (10:07.060)
That's definitely nicer.
Lex Fridman (10:08.700)
But it's more that that's the first application
Lex Fridman (10:11.020)
of what turned out to be a much more general function
Grant Sanderson (10:13.900)
that maybe the intelligent life
Lex Fridman (10:15.220)
your initial question asked about
Grant Sanderson (10:17.180)
would have come to recognize as being much more significant
Lex Fridman (10:19.780)
than the single use case,
Grant Sanderson (10:21.140)
which lends itself to repeated multiplication notation.
Lex Fridman (10:24.300)
But let me jump back for a second to aliens
Lex Fridman (10:28.500)
and the nature of our universe.
Lex Fridman (10:30.060)
Okay.
Lex Fridman (10:31.940)
Do you think math is discovered or invented?
Lex Fridman (10:35.100)
So we're talking about the different kind of mathematics
Grant Sanderson (10:37.620)
that could be developed by the alien species.
Lex Fridman (10:40.620)
The implied question is,
Lex Fridman (10:44.300)
yeah, is math discovered or invented?
Lex Fridman (10:46.220)
Is fundamentally everybody going to discover
Lex Fridman (10:49.660)
the same principles of mathematics?
Lex Fridman (10:53.140)
So the way I think about it,
Lex Fridman (10:54.140)
and everyone thinks about it differently,
Lex Fridman (10:55.380)
but here's my take.
Grant Sanderson (10:56.420)
I think there's a cycle at play
Lex Fridman (10:57.940)
where you discover things about the universe
Grant Sanderson (11:00.980)
that tell you what math will be useful.
Lex Fridman (11:03.980)
And that math itself is invented in a sense,
Lex Fridman (11:08.060)
but of all the possible maths that you could have invented,
Lex Fridman (11:11.420)
it's discoveries about the world
Grant Sanderson (11:12.740)
that tell you which ones are.
Lex Fridman (11:14.220)
So like a good example here is the Pythagorean theorem.
Grant Sanderson (11:17.780)
When you look at this,
Lex Fridman (11:18.620)
do you think of that as a definition
Lex Fridman (11:19.780)
or do you think of that as a discovery?
Lex Fridman (11:21.940)
From the historical perspective, right, it's a discovery
Grant Sanderson (11:24.300)
because they were,
Lex Fridman (11:25.140)
but that's probably because they were using physical object
Grant Sanderson (11:29.620)
to build their intuition.
Lex Fridman (11:32.260)
And from that intuition came the mathematics.
Lex Fridman (11:34.500)
So the mathematics wasn't in some abstract world
Lex Fridman (11:37.340)
detached from physics,
Lex Fridman (11:39.060)
but I think more and more math has become detached from,
Lex Fridman (11:43.700)
you know, when you even look at modern physics
Grant Sanderson (11:46.140)
from string theory to even general relativity,
Lex Fridman (11:49.580)
I mean, all math behind the 20th and 21st century physics
Grant Sanderson (11:53.580)
kind of takes a brisk walk outside of what our mind
Lex Fridman (11:58.460)
can actually even comprehend
Grant Sanderson (12:00.500)
in multiple dimensions, for example,
Lex Fridman (12:02.380)
anything beyond three dimensions, maybe four dimensions.
Grant Sanderson (12:05.940)
No, no, no, no, higher dimensions
Lex Fridman (12:07.300)
can be highly, highly applicable.
Grant Sanderson (12:08.780)
I think this is a common misinterpretation
Lex Fridman (12:11.260)
that if you're asking questions
Grant Sanderson (12:13.380)
about like a five dimensional manifold,
Lex Fridman (12:15.260)
that the only way that that's connected
Grant Sanderson (12:16.740)
to the physical world is if the physical world is itself
Lex Fridman (12:20.220)
a five dimensional manifold or includes them.
Grant Sanderson (12:22.980)
Well, wait, wait, wait a minute, wait a minute.
Lex Fridman (12:25.260)
You're telling me you can imagine
Lex Fridman (12:28.940)
a five dimensional manifold?
Lex Fridman (12:31.260)
No, no, that's not what I said.
Grant Sanderson (12:33.380)
I would make the claim that it is useful
Lex Fridman (12:35.020)
to a three dimensional physical universe,
Grant Sanderson (12:37.260)
despite itself not being three dimensional.
Lex Fridman (12:39.460)
So it's useful meaning to even understand
Grant Sanderson (12:41.220)
a three dimensional world,
Lex Fridman (12:42.740)
it'd be useful to have five dimensional manifolds.
Grant Sanderson (12:44.940)
Yes, absolutely, because of state spaces.
Lex Fridman (12:47.180)
But you're saying there in some deep way for us humans,
Grant Sanderson (12:50.540)
it does always come back to that three dimensional world
Lex Fridman (12:54.060)
for the usefulness that the dimensional world
Lex Fridman (12:56.620)
and therefore it starts with a discovery,
Lex Fridman (12:59.980)
but then we invent the mathematics
Grant Sanderson (13:02.060)
that helps us make sense of the discovery in a sense.
Lex Fridman (13:06.260)
Yes, I mean, just to jump off
Grant Sanderson (13:07.900)
of the Pythagorean theorem example,
Lex Fridman (13:09.820)
it feels like a discovery.
Grant Sanderson (13:11.220)
You've got these beautiful geometric proofs
Lex Fridman (13:12.900)
where you've got squares and you're modifying the areas,
Grant Sanderson (13:14.620)
it feels like a discovery.
Lex Fridman (13:16.740)
If you look at how we formalize the idea of 2D space
Grant Sanderson (13:19.620)
as being R2, right, all pairs of real numbers,
Lex Fridman (13:23.100)
and how we define a metric on it and define distance,
Grant Sanderson (13:25.740)
you're like, hang on a second,
Lex Fridman (13:26.700)
we've defined a distance
Lex Fridman (13:28.100)
so that the Pythagorean theorem is true,
Lex Fridman (13:30.100)
so that suddenly it doesn't feel that great.
Lex Fridman (13:32.460)
But I think what's going on is the thing that informed us
Lex Fridman (13:35.420)
what metric to put on R2,
Grant Sanderson (13:38.020)
to put on our abstract representation of 2D space,
Lex Fridman (13:41.380)
came from physical observations.
Lex Fridman (13:43.300)
And the thing is, there's other metrics
Lex Fridman (13:44.660)
you could have put on it.
Grant Sanderson (13:45.500)
We could have consistent math
Lex Fridman (13:47.180)
with other notions of distance,
Grant Sanderson (13:49.140)
it's just that those pieces of math
Lex Fridman (13:50.860)
wouldn't be applicable to the physical world that we study
Grant Sanderson (13:53.660)
because they're not the ones
Lex Fridman (13:54.580)
where the Pythagorean theorem holds.
Lex Fridman (13:56.180)
So we have a discovery, a genuine bonafide discovery
Lex Fridman (13:59.060)
that informed the invention,
Grant Sanderson (14:00.500)
the invention of an abstract representation of 2D space
Lex Fridman (14:03.700)
that we call R2 and things like that.
Lex Fridman (14:06.220)
And then from there,
Lex Fridman (14:07.300)
you just study R2 as an abstract thing
Grant Sanderson (14:09.740)
that brings about more ideas and inventions and mysteries
Lex Fridman (14:12.500)
which themselves might yield discoveries.
Grant Sanderson (14:14.420)
Those discoveries might give you insight
Lex Fridman (14:16.980)
as to what else would be useful to invent
Lex Fridman (14:19.380)
and it kind of feeds on itself that way.
Lex Fridman (14:20.980)
That's how I think about it.
Lex Fridman (14:22.180)
So it's not an either or.
Lex Fridman (14:24.140)
It's not that math is one of these
Grant Sanderson (14:25.380)
or it's one of the others.
Lex Fridman (14:26.780)
At different times, it's playing a different role.
Lex Fridman (14:29.180)
So then let me ask the Richard Feynman question then,
Lex Fridman (14:34.460)
along that thread,
Grant Sanderson (14:36.220)
is what do you think is the difference
Lex Fridman (14:37.380)
between physics and math?
Grant Sanderson (14:40.300)
There's a giant overlap.
Lex Fridman (14:43.060)
There's a kind of intuition that physicists have
Grant Sanderson (14:45.940)
about the world that's perhaps outside of mathematics.
Lex Fridman (14:49.020)
It's this mysterious art that they seem to possess,
Grant Sanderson (14:52.700)
we humans generally possess.
Lex Fridman (14:54.220)
And then there's the beautiful rigor of mathematics
Grant Sanderson (14:58.060)
that allows you to, I mean, just like as we were saying,
Lex Fridman (15:02.860)
invent frameworks of understanding our physical world.
Lex Fridman (15:07.860)
So what do you think is the difference there
Lex Fridman (15:10.180)
and how big is it?
Grant Sanderson (15:11.340)
Well, I think of math as being the study
Lex Fridman (15:12.980)
of abstractions over patterns and pure patterns in logic.
Lex Fridman (15:16.860)
And then physics is obviously grounded in a desire
Lex Fridman (15:19.300)
to understand the world that we live in.
Grant Sanderson (15:22.180)
I think you're gonna get very different answers
Lex Fridman (15:23.540)
when you talk to different mathematicians
Grant Sanderson (15:25.060)
because there's a wide diversity in types of mathematicians.
Lex Fridman (15:27.620)
There are some who are motivated very much by pure puzzles.
Grant Sanderson (15:30.980)
They might be turned on by things like combinatorics.
Lex Fridman (15:33.540)
And they just love the idea of building up
Grant Sanderson (15:35.860)
a set of problem solving tools applying to pure patterns.
Lex Fridman (15:40.420)
There are some who are very physically motivated,
Grant Sanderson (15:42.820)
who try to invent new math or discover math in veins
Lex Fridman (15:48.180)
that they know will have applications to physics
Grant Sanderson (15:50.500)
or sometimes computer science.
Lex Fridman (15:51.860)
And that's what drives them.
Grant Sanderson (15:53.340)
Like chaos theory is a good example of something
Lex Fridman (15:55.100)
that's pure math, that's purely mathematical.
Grant Sanderson (15:57.180)
A lot of the statements being made,
Lex Fridman (15:58.700)
but it's heavily motivated by specific applications
Grant Sanderson (16:02.420)
to largely physics.
Lex Fridman (16:04.860)
And then you have a type of mathematician
Grant Sanderson (16:06.660)
who just loves abstraction.
Lex Fridman (16:08.500)
They just love pulling it to the more and more abstract
Grant Sanderson (16:10.500)
things, the things that feel powerful.
Lex Fridman (16:12.100)
These are the ones that initially invented like topology
Lex Fridman (16:15.220)
and then later on get really into category theory
Lex Fridman (16:17.500)
and go on about like infinite categories and whatnot.
Grant Sanderson (16:20.380)
These are the ones that love to have a system
Lex Fridman (16:23.460)
that can describe truths about as many things as possible.
Grant Sanderson (16:28.700)
People from those three different veins of motivation
Lex Fridman (16:31.300)
into math are gonna give you very different answers
Grant Sanderson (16:32.860)
about what the relation at play here is.
Lex Fridman (16:34.740)
Cause someone like Vladimir Arnold,
Grant Sanderson (16:37.660)
who has written a lot of great books,
Lex Fridman (16:40.540)
many about like differential equations and such,
Grant Sanderson (16:42.500)
he would say, math is a branch of physics.
Lex Fridman (16:45.700)
That's how he would think about it.
Lex Fridman (16:47.180)
And of course he was studying
Lex Fridman (16:48.220)
like differential equations related things
Grant Sanderson (16:49.780)
because that is the motivator behind the study
Lex Fridman (16:52.260)
of PDEs and things like that.
Lex Fridman (16:54.820)
But you'll have others who,
Lex Fridman (16:56.500)
like especially the category theorists
Grant Sanderson (16:58.260)
who aren't really thinking about physics necessarily.
Lex Fridman (17:01.380)
It's all about abstraction and the power of generality.
Lex Fridman (17:04.540)
And it's more of a happy coincidence
Lex Fridman (17:06.460)
that that ends up being useful
Grant Sanderson (17:08.340)
for understanding the world we live in.
Lex Fridman (17:10.900)
And then you can get into like, why is that the case?
Grant Sanderson (17:12.860)
It's sort of surprising
Lex Fridman (17:14.140)
that that which is about pure puzzles and abstraction
Grant Sanderson (17:17.820)
also happens to describe the very fundamentals
Lex Fridman (17:21.060)
of quarks and everything else.
Lex Fridman (17:24.180)
So why do you think the fundamentals of quarks
Lex Fridman (17:28.820)
and the nature of reality is so compressible
Grant Sanderson (17:33.260)
into clean, beautiful equations
Lex Fridman (17:35.380)
that are for the most part simple, relatively speaking,
Lex Fridman (17:39.300)
a lot simpler than they could be?
Lex Fridman (17:41.740)
So you have, we mentioned somebody like Stephen Wolfram
Grant Sanderson (17:45.380)
who thinks that sort of there's incredibly simple rules
Lex Fridman (17:50.540)
underlying our reality,
Lex Fridman (17:51.940)
but it can create arbitrary complexity.
Lex Fridman (17:54.940)
But there is simple equations.
Grant Sanderson (17:56.780)
What, I'm asking a million questions
Lex Fridman (17:59.220)
that nobody knows the answer to, but.
Lex Fridman (18:01.060)
I have no idea, why is it simple?
Lex Fridman (18:05.300)
It could be the case that
Grant Sanderson (18:07.100)
there's like a filter iteration at play.
Lex Fridman (18:08.500)
The only things that physicists find interesting
Grant Sanderson (18:10.660)
are the ones that are simple enough
Lex Fridman (18:11.700)
they could describe it mathematically.
Lex Fridman (18:13.340)
But as soon as it's a sufficiently complex system,
Lex Fridman (18:15.180)
like, oh, that's outside the realm of physics,
Grant Sanderson (18:16.940)
that's biology or whatever have you.
Lex Fridman (18:19.340)
And of course, that's true.
Grant Sanderson (18:21.660)
Maybe there's something where it's like,
Lex Fridman (18:22.740)
of course there will always be something that is simple
Grant Sanderson (18:26.540)
when you wash away the like non important parts
Lex Fridman (18:31.420)
of whatever it is that you're studying.
Grant Sanderson (18:33.420)
Just from like an information theory standpoint,
Lex Fridman (18:35.180)
there might be some like,
Grant Sanderson (18:36.660)
you get to the lowest information component of it.
Lex Fridman (18:39.460)
But I don't know, maybe I'm just having
Grant Sanderson (18:40.900)
a really hard time conceiving of what it would even mean
Lex Fridman (18:43.020)
for the fundamental laws to be like intrinsically
Grant Sanderson (18:46.580)
complicated, like some set of equations
Lex Fridman (18:50.660)
that you can't decouple from each other.
Grant Sanderson (18:52.580)
Well, no, it could be that sort of we take for granted
Lex Fridman (18:56.820)
that the laws of physics, for example,
Grant Sanderson (18:59.980)
are for the most part the same everywhere
Lex Fridman (19:03.500)
or something like that, right?
Grant Sanderson (19:05.340)
As opposed to the sort of an alternative could be
Lex Fridman (19:10.620)
that the rules under which the world operates
Grant Sanderson (19:15.420)
is different everywhere.
Lex Fridman (19:17.260)
It's like a deeply distributed system
Grant Sanderson (19:20.340)
where just everything is just chaos,
Lex Fridman (19:23.380)
not in a strict definition of chaos,
Lex Fridman (19:25.540)
but meaning like just it's impossible for equations
Lex Fridman (19:30.420)
to capture, for to explicitly model the world
Grant Sanderson (19:34.020)
as cleanly as the physical does.
Lex Fridman (19:36.020)
I mean, we almost take it for granted that we can describe,
Grant Sanderson (19:39.100)
we can have an equation for gravity,
Lex Fridman (19:41.260)
for action at a distance.
Grant Sanderson (19:42.780)
We can have equations for some of these basic ways
Lex Fridman (19:45.500)
the planet's moving.
Grant Sanderson (19:46.580)
Just the low level at the atomic scale,
Lex Fridman (19:52.060)
how the materials operate,
Grant Sanderson (19:53.980)
at the high scale, how black holes operate.
Lex Fridman (19:56.940)
But it doesn't, it seems like it could be,
Grant Sanderson (19:59.820)
there's infinite other possibilities
Lex Fridman (1:00:00.660)
And then on the gondola ride down,
Grant Sanderson (1:00:01.820)
we decided to just jam a little bit.
Lex Fridman (1:00:04.180)
And it was just like, I don't know,
Grant Sanderson (1:00:06.280)
the gondola sort of came over a mountain
Lex Fridman (1:00:09.040)
and you saw the city lights
Lex Fridman (1:00:10.740)
and we're just like jamming, like playing some music.
Lex Fridman (1:00:13.920)
I wouldn't describe that as transformative.
Grant Sanderson (1:00:16.320)
I don't know why, but that popped into my mind
Lex Fridman (1:00:18.040)
as a moment of, in a way that wasn't associated
Grant Sanderson (1:00:21.200)
with people I love, but more with like a thing I was doing,
Lex Fridman (1:00:24.200)
something that was just, it was just happy
Lex Fridman (1:00:26.160)
and it was just like a great moment.
Lex Fridman (1:00:29.280)
I don't think I can give you anything deeper than that.
Grant Sanderson (1:00:32.080)
Well, as a musician myself, I'd love to see,
Lex Fridman (1:00:35.720)
as you mentioned before, music enter back into your work,
Grant Sanderson (1:00:38.800)
back into your creative work.
Lex Fridman (1:00:40.080)
I'd love to see that.
Grant Sanderson (1:00:41.320)
I'm certainly allowing it to enter back into mine.
Lex Fridman (1:00:43.880)
And it's a beautiful thing for a mathematician,
Grant Sanderson (1:00:47.840)
for a scientist to allow music to enter their work.
Lex Fridman (1:00:51.440)
I think only good things can happen.
Grant Sanderson (1:00:53.920)
All right, I'll try to promise you a music video by 2020.
Lex Fridman (1:00:57.200)
By 2020?
Grant Sanderson (1:00:58.040)
By the end of 2020.
Lex Fridman (1:00:58.860)
Okay, all right, good.
Grant Sanderson (1:00:59.700)
Give myself a longer window.
Lex Fridman (1:01:01.400)
All right, maybe we can like collaborate
Grant Sanderson (1:01:04.480)
on a band type situation.
Lex Fridman (1:01:05.680)
What instruments do you play?
Grant Sanderson (1:01:07.020)
The main instrument I play is violin,
Lex Fridman (1:01:08.600)
but I also love to dabble around on like guitar and piano.
Grant Sanderson (1:01:11.680)
Beautiful, me too, guitar and piano.
Lex Fridman (1:01:13.540)
So in a mathematician's lament, Paul Lockhart writes,
Grant Sanderson (1:01:18.640)
the first thing to understand
Lex Fridman (1:01:20.080)
is that mathematics is an art.
Grant Sanderson (1:01:22.040)
The difference between math and the other arts,
Lex Fridman (1:01:24.080)
such as music and painting,
Grant Sanderson (1:01:26.740)
is that our culture does not recognize it as such.
Lex Fridman (1:01:29.920)
So I think I speak for millions of people, myself included,
Grant Sanderson (1:01:34.420)
in saying thank you for revealing to us
Lex Fridman (1:01:37.460)
the art of mathematics.
Lex Fridman (1:01:39.640)
So thank you for everything you do
Lex Fridman (1:01:40.920)
and thanks for talking today.
Grant Sanderson (1:01:42.260)
Wow, thanks for saying that.
Lex Fridman (1:01:43.240)
And thanks for having me on.
Grant Sanderson (1:01:45.400)
Thanks for listening to this conversation
Lex Fridman (1:01:47.120)
with Grant Sanderson.
Lex Fridman (1:01:48.340)
And thank you to our presenting sponsor, Cash App.
Lex Fridman (1:01:51.720)
Download it, use code LEXPodcast.
Grant Sanderson (1:01:54.840)
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Lex Fridman (1:01:57.720)
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Grant Sanderson (1:02:00.040)
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Lex Fridman (1:02:01.960)
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Grant Sanderson (1:02:04.960)
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Lex Fridman (1:02:07.600)
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Grant Sanderson (1:02:09.440)
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Lex Fridman (1:02:13.440)
And now, let me leave you with some words of wisdom
Grant Sanderson (1:02:16.080)
from one of Grant's and my favorite people, Richard Feynman.
Lex Fridman (1:02:21.440)
Nobody ever figures out what this life is all about,
Lex Fridman (1:02:24.800)
and it doesn't matter.
Lex Fridman (1:02:26.480)
Explore the world.
Grant Sanderson (1:02:28.540)
Nearly everything is really interesting
Lex Fridman (1:02:30.640)
if you go into it deeply enough.
Grant Sanderson (1:02:33.300)
Thank you for listening, and hope to see you next time.
Lex Fridman (20:01.700)
where none of it could be compressible into such equations.
Lex Fridman (20:05.020)
So it just seems beautiful.
Lex Fridman (20:06.540)
It's also weird, probably to the point you're making,
Grant Sanderson (20:10.900)
that it's very pleasant that this is true for our minds.
Lex Fridman (20:15.140)
So it might be that our minds are biased
Grant Sanderson (20:17.140)
to just be looking at the parts of the universe
Lex Fridman (20:19.660)
that are compressible.
Lex Fridman (20:21.740)
And then we can publish papers on
Lex Fridman (20:23.740)
and have nice E equals empty squared equations.
Grant Sanderson (20:26.460)
Right, well, I wonder would such a world
Lex Fridman (20:29.260)
with uncompressible laws allow for the kind of beings
Grant Sanderson (20:33.580)
that can think about the kind of questions
Lex Fridman (20:35.940)
that you're asking?
Grant Sanderson (20:37.780)
That's true.
Lex Fridman (20:38.620)
Right, like an anthropic principle coming into play
Lex Fridman (20:40.620)
in some weird way here?
Lex Fridman (20:42.580)
I don't know, like I don't know what I'm talking about at all.
Grant Sanderson (20:44.780)
Maybe the universe is actually not so compressible,
Lex Fridman (20:47.980)
but the way our brain, the way our brain evolved
Grant Sanderson (20:52.540)
we're only able to perceive the compressible parts.
Lex Fridman (20:55.860)
I mean, we are, so this is the sort of Chomsky argument.
Grant Sanderson (20:58.380)
We are just descendants of apes
Lex Fridman (20:59.860)
over like really limited biological systems.
Lex Fridman (21:03.580)
So it totally makes sense
Lex Fridman (21:04.660)
that we're really limited little computers, calculators,
Grant Sanderson (21:08.580)
that are able to perceive certain kinds of things
Lex Fridman (21:10.220)
and the actual world is much more complicated.
Lex Fridman (21:13.100)
Well, but we can do pretty awesome things, right?
Lex Fridman (21:16.620)
Like we can fly spaceships
Lex Fridman (21:18.260)
and we have to have some connection of reality
Lex Fridman (21:21.540)
to be able to take our potentially oversimplified models
Grant Sanderson (21:25.260)
of the world, but then actually twist the world
Lex Fridman (21:27.740)
to our will based on it.
Lex Fridman (21:29.060)
So we have certain reality checks
Lex Fridman (21:30.380)
that like physics isn't too far a field
Grant Sanderson (21:33.380)
simply based on what we can do.
Lex Fridman (21:35.420)
Yeah, the fact that we can fly is pretty good.
Grant Sanderson (21:37.260)
It's great, yeah, like it's a proof of concept
Lex Fridman (21:40.500)
that the laws we're working with are working well.
Lex Fridman (21:44.900)
So I mentioned to the internet that I'm talking to you
Lex Fridman (21:47.740)
and so the internet gave some questions.
Lex Fridman (21:50.180)
So I apologize for these,
Lex Fridman (21:51.580)
but do you think we're living in a simulation
Grant Sanderson (21:54.540)
that the universe is a computer
Lex Fridman (21:56.940)
or the universe is a computation running on a computer?
Grant Sanderson (22:01.260)
It's conceivable.
Lex Fridman (22:02.700)
What I don't buy is, you know, you'll have the argument
Grant Sanderson (22:05.700)
that, well, let's say that it was the case
Lex Fridman (22:07.900)
that you can have simulations.
Grant Sanderson (22:09.540)
Then the simulated world would itself
Lex Fridman (22:12.580)
eventually get to a point where it's running simulations.
Lex Fridman (22:15.380)
And then the second layer down
Lex Fridman (22:17.220)
would create a third layer down and on and on and on.
Lex Fridman (22:19.420)
So probabilistically, you just throw a dart
Lex Fridman (22:21.620)
at one of those layers,
Grant Sanderson (22:22.460)
we're probably in one of the simulated layers.
Lex Fridman (22:24.940)
I think if there's some sort of limitations
Grant Sanderson (22:27.060)
on like the information processing
Lex Fridman (22:28.580)
of whatever the physical world is,
Grant Sanderson (22:31.420)
like it quickly becomes the case
Lex Fridman (22:32.700)
that you have a limit to the layers that could exist there
Grant Sanderson (22:35.620)
because like the resources necessary
Lex Fridman (22:38.100)
to simulate a universe like ours clearly is a lot
Grant Sanderson (22:41.660)
just in terms of the number of bits at play.
Lex Fridman (22:43.700)
And so then you can ask, well, what's more plausible?
Grant Sanderson (22:46.820)
That there's an unbounded capacity
Lex Fridman (22:49.140)
of information processing
Grant Sanderson (22:50.340)
in whatever the like highest up level universe is,
Lex Fridman (22:53.620)
or that there's some bound to that capacity,
Grant Sanderson (22:56.060)
which then limits like the number of levels available.
Lex Fridman (22:58.860)
How do you play some kind of probability distribution
Lex Fridman (23:00.820)
on like what the information capacity is?
Lex Fridman (23:02.580)
I have no idea.
Lex Fridman (23:03.740)
But I don't, like people almost assume
Lex Fridman (23:06.860)
a certain uniform probability
Grant Sanderson (23:08.340)
over all of those meta layers that could conceivably exist
Lex Fridman (23:11.900)
when it's a little bit like a Pascal's wager
Grant Sanderson (23:15.180)
on like you're not giving a low enough prior
Lex Fridman (23:16.980)
to the mere existence of that infinite set of layers.
Grant Sanderson (23:20.900)
Yeah, that's true.
Lex Fridman (23:21.740)
But it's also very difficult to contextualize the amount.
Lex Fridman (23:25.060)
So the amount of information processing power
Lex Fridman (23:28.260)
required to simulate like our universe
Grant Sanderson (23:31.380)
seems like amazingly huge.
Lex Fridman (23:34.260)
But you can always raise two to the power of that.
Grant Sanderson (23:36.940)
Yeah, like numbers get big.
Lex Fridman (23:40.460)
And we're easily humbled
Grant Sanderson (23:41.780)
by basically everything around us.
Lex Fridman (23:43.700)
So it's very difficult to kind of make sense of anything
Grant Sanderson (23:49.300)
actually when you look up at the sky
Lex Fridman (23:50.980)
and look at the stars and the immensity of it all,
Grant Sanderson (23:53.500)
to make sense of the smallness of us,
Lex Fridman (23:57.020)
the unlikeliness of everything
Grant Sanderson (23:58.780)
that's on this earth coming to be,
Lex Fridman (24:02.180)
then you could basically anything could be,
Grant Sanderson (24:04.940)
all laws of probability go out the window to me
Lex Fridman (24:09.100)
because I guess because the amount of information
Grant Sanderson (24:14.140)
under which we're operating is very low.
Lex Fridman (24:17.540)
We basically know nothing about the world around us,
Grant Sanderson (24:22.100)
relatively speaking.
Lex Fridman (24:23.380)
And so when I think about the simulation hypothesis,
Grant Sanderson (24:26.580)
I think it's just fun to think about it.
Lex Fridman (24:29.220)
But it's also, I think there is a thought experiment
Grant Sanderson (24:31.860)
kind of interesting to think of the power of computation,
Lex Fridman (24:35.220)
whether the limits of a Turing machine,
Grant Sanderson (24:38.940)
sort of the limits of our current computers,
Lex Fridman (24:41.020)
when you start to think about artificial intelligence,
Lex Fridman (24:44.060)
how far can we get with computers?
Lex Fridman (24:46.820)
And that's kind of where the simulation hypothesis
Grant Sanderson (24:50.820)
used with me as a thought experiment
Lex Fridman (24:52.820)
is the universe just a computer?
Lex Fridman (24:56.660)
Is it just a computation?
Lex Fridman (24:58.620)
Is all of this just a computation?
Lex Fridman (25:00.500)
And sort of the same kind of tools we apply
Lex Fridman (25:02.340)
to analyzing algorithms, can that be applied?
Grant Sanderson (25:05.100)
If we scale further and further and further,
Lex Fridman (25:07.340)
will the arbitrary power of those systems
Grant Sanderson (25:09.620)
start to create some interesting aspects
Lex Fridman (25:12.020)
that we see in our universe?
Grant Sanderson (25:13.880)
Or is something fundamentally different
Lex Fridman (25:15.980)
needs to be created?
Grant Sanderson (25:17.500)
Well, it's interesting that in our universe,
Lex Fridman (25:20.380)
it's not arbitrarily large, the power,
Grant Sanderson (25:22.700)
that you can place limits on, for example,
Lex Fridman (25:24.340)
how many bits of information can be stored per unit area.
Grant Sanderson (25:27.300)
Right, like all of the physical laws,
Lex Fridman (25:30.360)
you've got general relativity and quantum coming together
Grant Sanderson (25:32.680)
to give you a certain limit on how many bits you can store
Lex Fridman (25:36.600)
within a given range before it collapses into a black hole.
Grant Sanderson (25:40.440)
The idea that there even exists such a limit
Lex Fridman (25:42.760)
is at the very least thought provoking,
Grant Sanderson (25:44.600)
when naively you might assume,
Lex Fridman (25:46.960)
oh, well, technology could always get better and better,
Grant Sanderson (25:49.240)
we could get cleverer and cleverer,
Lex Fridman (25:50.920)
and you could just cram as much information as you want
Grant Sanderson (25:54.140)
into like a small unit of space, that makes me think,
Lex Fridman (26:01.040)
it's at least plausible that whatever the highest level
Grant Sanderson (26:06.080)
of existence is doesn't admit too many simulations
Lex Fridman (26:10.440)
or ones that are at the scale of complexity
Grant Sanderson (26:12.200)
that we're looking at.
Lex Fridman (26:13.400)
Obviously, it's just as conceivable that they do
Lex Fridman (26:15.360)
and that there are many, but I guess what I'm channeling
Lex Fridman (26:20.080)
is the surprise that I felt upon learning that fact,
Grant Sanderson (26:22.560)
that there are, that information is physical in this way.
Lex Fridman (26:26.120)
There's a finiteness to it.
Grant Sanderson (26:27.120)
Okay, let me just even go off on that.
Lex Fridman (26:29.420)
From a mathematics perspective
Lex Fridman (26:31.320)
and a psychology perspective, how do you mix,
Lex Fridman (26:35.040)
are you psychologically comfortable
Lex Fridman (26:38.200)
with the concept of infinity?
Lex Fridman (26:40.800)
I think so.
Lex Fridman (26:41.640)
Are you okay with it?
Lex Fridman (26:42.460)
I'm pretty okay, yeah.
Lex Fridman (26:43.680)
Are you okay?
Lex Fridman (26:44.560)
No, not really, it doesn't make any sense to me.
Grant Sanderson (26:47.180)
I don't know, like how many words,
Lex Fridman (26:50.040)
how many possible words do you think could exist
Lex Fridman (26:53.480)
that are just like strings of letters?
Lex Fridman (26:55.700)
So that's a sort of mathematical statement as beautiful
Lex Fridman (26:59.600)
and we use infinity in basically everything we do,
Lex Fridman (27:03.520)
everything we do in science, math, and engineering, yes.
Lex Fridman (27:06.880)
But you said exist, the question is,
Lex Fridman (27:12.040)
you said letters or words?
Grant Sanderson (27:13.360)
I said words. Words.
Lex Fridman (27:16.440)
To bring words into existence to me,
Grant Sanderson (27:18.200)
you have to start like saying them or like writing them
Lex Fridman (27:20.920)
or like listing them.
Grant Sanderson (27:22.040)
That's an instantiation.
Lex Fridman (27:23.240)
Okay, how many abstract words exist?
Grant Sanderson (27:25.800)
Well, the idea of an abstract.
Lex Fridman (27:28.080)
The idea of abstract notions and ideas.
Grant Sanderson (27:31.000)
I think we should be clear on terminology.
Lex Fridman (27:33.120)
I mean, you think about intelligence a lot,
Grant Sanderson (27:35.200)
like artificial intelligence.
Lex Fridman (27:37.860)
Would you not say that what it's doing
Lex Fridman (27:39.120)
is a kind of abstraction?
Lex Fridman (27:40.440)
That like abstraction is key
Lex Fridman (27:42.280)
to conceptualizing the universe?
Lex Fridman (27:45.120)
You get this raw sensory data.
Grant Sanderson (27:47.300)
I need something that every time you move your face
Lex Fridman (27:49.240)
a little bit and they're not pixels,
Lex Fridman (27:51.620)
but like analog of pixels on my retina changed entirely,
Lex Fridman (27:55.200)
that I can still have some coherent notion of this is Lex,
Lex Fridman (27:58.040)
I'm talking to Lex, right?
Lex Fridman (27:59.720)
What that requires is you have a disparate set
Grant Sanderson (28:01.720)
of possible images hitting me
Lex Fridman (28:03.580)
that are unified in a notion of Lex, right?
Grant Sanderson (28:07.580)
That's a kind of abstraction.
Lex Fridman (28:08.680)
It's a thing that could apply
Grant Sanderson (28:09.820)
to a lot of different images that I see
Lex Fridman (28:12.400)
and it represents it in a much more compressed way
Lex Fridman (28:15.280)
and one that's like much more resilient to that.
Lex Fridman (28:17.440)
I think in the same way,
Grant Sanderson (28:18.360)
if I'm talking about infinity as an abstraction,
Lex Fridman (28:21.080)
I don't mean nonphysical woo woo,
Grant Sanderson (28:24.960)
like ineffable or something.
Lex Fridman (28:26.360)
What I mean is it's something that can apply
Grant Sanderson (28:28.360)
to a multiplicity of situations
Lex Fridman (28:30.160)
that share a certain common attribute
Grant Sanderson (28:31.660)
in the same way that the images of like your face
Lex Fridman (28:33.720)
on my retina share enough common attributes
Grant Sanderson (28:35.820)
that I can put the single notion to it.
Lex Fridman (28:37.700)
Like in that way, infinity is an abstraction
Lex Fridman (28:40.780)
and it's very powerful and it's only through
Lex Fridman (28:43.260)
such abstractions that we can actually understand
Grant Sanderson (28:45.680)
like the world and logic and things.
Lex Fridman (28:47.560)
And in the case of infinity,
Grant Sanderson (28:48.560)
the way I think about it,
Lex Fridman (28:49.400)
the key entity is the property
Grant Sanderson (28:51.760)
of always being able to add one more.
Lex Fridman (28:54.080)
Like no matter how many words you can list,
Grant Sanderson (28:56.120)
you just throw an A at the end of one
Lex Fridman (28:57.720)
and you have another conceivable word.
Grant Sanderson (28:59.760)
You don't have to think of all the words at once.
Lex Fridman (29:01.680)
It's that property, the oh, I could always add one more
Grant Sanderson (29:04.840)
that gives it this nature of infiniteness
Lex Fridman (29:08.200)
in the same way that there's certain like properties
Lex Fridman (29:09.760)
of your face that give it the Lexness, right?
Lex Fridman (29:13.720)
So like infinity should be no more worrying
Grant Sanderson (29:16.480)
than the I can always add one more sentiment.
Lex Fridman (29:19.760)
That's a really elegant,
Grant Sanderson (29:21.600)
much more elegant way than I could put it.
Lex Fridman (29:23.720)
So thank you for doing that as yet another abstraction.
Lex Fridman (29:26.840)
And yes, indeed, that's what our brain does.
Lex Fridman (29:29.440)
That's what intelligent systems do.
Grant Sanderson (29:30.640)
That's what programming does.
Lex Fridman (29:31.840)
That's what science does is build abstraction
Grant Sanderson (29:34.680)
on top of each other.
Lex Fridman (29:35.760)
And yet there is at a certain point abstractions
Lex Fridman (29:39.560)
that go into the quote woo, right?
Lex Fridman (29:42.800)
Sort of, and because we're now,
Grant Sanderson (29:47.920)
it's like we built this stack of, you know,
Lex Fridman (29:52.460)
the only thing that's true is the stuff that's on the ground.
Grant Sanderson (29:54.680)
Everything else is useful for interpreting this.
Lex Fridman (29:57.480)
And at a certain point you might start floating
Grant Sanderson (30:00.000)
into ideas that are surreal and difficult
Lex Fridman (30:04.600)
and take us into areas that are disconnected
Grant Sanderson (30:08.120)
from reality in a way that we could never get back.
Lex Fridman (30:11.080)
What if instead of calling these abstract,
Lex Fridman (30:13.160)
how different would it be in your mind
Lex Fridman (30:14.640)
if we called them general?
Lex Fridman (30:15.960)
And the phenomenon that you're describing
Lex Fridman (30:17.400)
is overgeneralization.
Grant Sanderson (30:19.080)
When you try to have a concept or an idea
Lex Fridman (30:21.720)
that's so general as to apply to nothing in particular
Grant Sanderson (30:24.760)
in a useful way, does that map to what you're thinking
Lex Fridman (30:27.960)
of when you think of?
Grant Sanderson (30:28.800)
First of all, I'm playing little just for the fun of it.
Lex Fridman (30:31.480)
Devil's advocate.
Lex Fridman (30:32.400)
And I think our cognition, our mind is unable
Lex Fridman (30:37.400)
to visualize.
Lex Fridman (30:39.000)
So you do some incredible work with visualization and video.
Lex Fridman (30:42.520)
I think infinity is very difficult to visualize
Grant Sanderson (30:46.800)
for our mind.
Lex Fridman (30:48.240)
We can delude ourselves into thinking we can visualize it,
Lex Fridman (30:52.880)
but we can't.
Lex Fridman (30:54.480)
I don't, I mean, I don't,
Grant Sanderson (30:56.120)
I would venture to say it's very difficult.
Lex Fridman (30:57.680)
And so there's some concepts of mathematics,
Grant Sanderson (31:00.440)
like maybe multiple dimensions,
Lex Fridman (31:02.040)
we could sort of talk about that are impossible
Grant Sanderson (31:04.720)
for us to truly intuit, like,
Lex Fridman (31:08.160)
and it just feels dangerous to me to use these
Grant Sanderson (31:13.120)
as part of our toolbox of abstractions.
Lex Fridman (31:16.680)
On behalf of your listeners,
Grant Sanderson (31:17.680)
I almost fear we're getting too philosophical.
Lex Fridman (31:19.600)
Right?
Grant Sanderson (31:20.440)
Heck no.
Lex Fridman (31:21.280)
Heck no.
Grant Sanderson (31:22.120)
I think to that point for any particular idea like this,
Lex Fridman (31:26.720)
there's multiple angles of attack.
Grant Sanderson (31:28.720)
I think the, when we do visualize infinity,
Lex Fridman (31:31.920)
what we're actually doing, you know,
Lex Fridman (31:33.160)
you write dot, dot, dot, right?
Lex Fridman (31:34.720)
One, two, three, four, dot, dot, dot, right?
Grant Sanderson (31:37.000)
Those are symbols on the page
Lex Fridman (31:37.960)
that are insinuating a certain infinity.
Lex Fridman (31:42.880)
What you're capturing with a little bit of design there
Lex Fridman (31:45.920)
is the I can always add one more property, right?
Grant Sanderson (31:49.400)
I think I'm just as uncomfortable with you are
Lex Fridman (31:52.480)
if you try to concretize it so much
Grant Sanderson (31:56.040)
that you have a bag of infinitely many things
Lex Fridman (31:58.880)
that I actually think of, no, not one, two, three, four,
Grant Sanderson (32:00.720)
dot, dot, dot, one, two, three, four, five, six, seven, eight.
Lex Fridman (32:03.360)
I try to get them all in my head and you realize,
Grant Sanderson (32:05.600)
oh, you know, your brain would literally collapse
Lex Fridman (32:08.120)
into a black hole, all of that.
Lex Fridman (32:10.120)
And I honestly feel this with a lot of math
Lex Fridman (32:12.440)
that I try to read where I don't think of myself
Grant Sanderson (32:15.040)
as like particularly good at math in some ways.
Lex Fridman (32:19.880)
Like I get very confused often
Grant Sanderson (32:21.360)
when I am going through some of these texts.
Lex Fridman (32:23.720)
And often what I'm feeling in my head is like,
Grant Sanderson (32:25.680)
this is just so damn abstract.
Lex Fridman (32:27.840)
I just can't wrap my head around it.
Grant Sanderson (32:29.160)
I just want to put something concrete to it
Lex Fridman (32:31.800)
that makes me understand.
Lex Fridman (32:32.920)
And I think a lot of the motivation for the channel
Lex Fridman (32:35.560)
is channeling that sentiment of, yeah,
Grant Sanderson (32:38.400)
a lot of the things that you're trying to read out there,
Lex Fridman (32:40.920)
it's just so hard to connect to anything
Grant Sanderson (32:43.600)
that you spend an hour banging your head
Lex Fridman (32:45.280)
against a couple of pages and you come out
Grant Sanderson (32:47.200)
not really knowing anything more
Lex Fridman (32:49.240)
other than some definitions maybe
Lex Fridman (32:51.760)
and a certain sense of self defeat, right?
Lex Fridman (32:55.520)
One of the reasons I focus so much on visualizations
Grant Sanderson (32:58.480)
is that I'm a big believer in,
Lex Fridman (33:01.720)
I'm sorry, I'm just really hampering on
Grant Sanderson (33:03.080)
this idea of abstraction,
Lex Fridman (33:04.320)
being clear about your layers of abstraction, right?
Grant Sanderson (33:07.400)
It's always tempting to start an explanation
Lex Fridman (33:09.760)
from the top to the bottom, okay?
Grant Sanderson (33:11.920)
You give the definition of a new theorem.
Lex Fridman (33:14.080)
You're like, this is the definition of a vector space.
Grant Sanderson (33:16.120)
For example, that's how we'll start a course.
Lex Fridman (33:18.320)
These are the properties of a vector space.
Grant Sanderson (33:20.520)
First from these properties, we will derive what we need
Lex Fridman (33:23.480)
in order to do the math of linear algebra
Grant Sanderson (33:25.120)
or whatever it might be.
Lex Fridman (33:26.320)
I don't think that's how understanding works at all.
Grant Sanderson (33:28.520)
I think how understanding works
Lex Fridman (33:29.880)
is you start at the lowest level you can get at
Grant Sanderson (33:32.440)
where rather than thinking about a vector space,
Lex Fridman (33:34.600)
you might think of concrete vectors
Grant Sanderson (33:36.240)
that are just lists of numbers
Lex Fridman (33:37.880)
or picturing it as like an arrow that you draw,
Grant Sanderson (33:41.920)
which is itself like even less abstract than numbers
Lex Fridman (33:44.400)
because you're looking at quantities,
Grant Sanderson (33:45.440)
like the distance of the x coordinate,
Lex Fridman (33:47.000)
the distance of the y coordinate.
Grant Sanderson (33:48.080)
It's as concrete as you could possibly get
Lex Fridman (33:50.120)
and it has to be if you're putting it in a visual, right?
Grant Sanderson (33:53.640)
It's an actual arrow. It's an actual vector.
Lex Fridman (33:56.880)
You're not talking about like a quote unquote vector
Grant Sanderson (33:59.120)
that could apply to any possible thing.
Lex Fridman (34:01.080)
You have to choose one if you're illustrating it.
Lex Fridman (34:03.480)
And I think this is the power of being in a medium
Lex Fridman (34:05.920)
like video or if you're writing a textbook
Lex Fridman (34:08.160)
and you force yourself to put a lot of images
Lex Fridman (34:10.760)
is with every image, you're making a choice.
Grant Sanderson (34:13.520)
With each choice, you're showing a concrete example.
Lex Fridman (34:16.240)
With each concrete example,
Grant Sanderson (34:17.560)
you're aiding someone's path to understanding.
Lex Fridman (34:19.680)
I'm sorry to interrupt you,
Lex Fridman (34:21.360)
but you just made me realize that that's exactly right.
Lex Fridman (34:24.600)
So the visualizations you're creating
Grant Sanderson (34:26.880)
while you're sometimes talking about abstractions,
Lex Fridman (34:29.840)
the actual visualization is an explicit low level example.
Grant Sanderson (34:34.560)
Yes.
Lex Fridman (34:35.400)
So there's an actual, like in the code,
Grant Sanderson (34:37.400)
you have to say what the vector is,
Lex Fridman (34:40.600)
what's the direction of the arrow,
Grant Sanderson (34:42.200)
what's the magnitude of the, yeah.
Lex Fridman (34:44.960)
So that's, you're going, the visualization itself
Grant Sanderson (34:48.200)
is actually going to the bottom of that.
Lex Fridman (34:50.200)
And I think that's very important.
Grant Sanderson (34:52.560)
I also think about this a lot in writing scripts
Lex Fridman (34:54.960)
where even before you get to the visuals,
Grant Sanderson (34:57.360)
the first instinct is to, I don't know why,
Lex Fridman (35:00.200)
I just always do, I say the abstract thing,
Grant Sanderson (35:02.560)
I say the general definition, the powerful thing,
Lex Fridman (35:05.040)
and then I fill it in with examples later.
Grant Sanderson (35:07.240)
Always, it will be more compelling
Lex Fridman (35:08.680)
and easier to understand when you flip that.
Lex Fridman (35:10.720)
And instead, you let someone's brain
Lex Fridman (35:13.480)
do the pattern recognition.
Grant Sanderson (35:16.240)
You just show them a bunch of examples.
Lex Fridman (35:18.200)
The brain is gonna feel a certain similarity between them.
Grant Sanderson (35:21.080)
Then by the time you bring in the definition,
Lex Fridman (35:23.560)
or by the time you bring in the formula,
Grant Sanderson (35:25.720)
it's articulating a thing that's already in the brain
Lex Fridman (35:28.920)
that was built off of looking at a bunch of examples
Grant Sanderson (35:30.880)
with a certain kind of similarity.
Lex Fridman (35:32.840)
And what the formula does is articulate
Lex Fridman (35:34.680)
what that kind of similarity is,
Lex Fridman (35:36.560)
rather than being a high cognitive load set of symbols
Grant Sanderson (35:42.200)
that needs to be populated with examples later on,
Lex Fridman (35:45.200)
assuming someone's still with you.
Lex Fridman (35:48.640)
What is the most beautiful or awe inspiring idea
Lex Fridman (35:51.280)
you've come across in mathematics?
Grant Sanderson (35:53.840)
I don't know, man.
Lex Fridman (35:55.160)
Maybe it's an idea you've explored in your videos,
Grant Sanderson (35:57.240)
maybe not.
Lex Fridman (35:58.360)
What just gave you pause?
Lex Fridman (36:01.280)
What's the most beautiful idea?
Lex Fridman (36:03.360)
Small or big.
Lex Fridman (36:04.440)
So I think often, the things that are most beautiful
Lex Fridman (36:07.360)
are the ones that you have a little bit of understanding of,
Lex Fridman (36:11.800)
but certainly not an entire understanding.
Lex Fridman (36:14.360)
It's a little bit of that mystery
Grant Sanderson (36:15.560)
that is what makes it beautiful.
Lex Fridman (36:17.280)
What was the moment of the discovery for you personally,
Lex Fridman (36:20.200)
almost just that leap of aha moment?
Lex Fridman (36:23.480)
So something that really caught my eye,
Grant Sanderson (36:25.240)
I remember when I was little, there were these,
Lex Fridman (36:29.320)
I think the series was called like wooden books
Grant Sanderson (36:31.120)
or something, these tiny little books
Lex Fridman (36:32.400)
that would have just a very short description
Grant Sanderson (36:34.680)
of something on the left and then a picture on the right.
Lex Fridman (36:36.920)
I don't know who they're meant for,
Lex Fridman (36:37.880)
but maybe it's like loosely children
Lex Fridman (36:39.640)
or something like that.
Lex Fridman (36:40.520)
But it can't just be children,
Lex Fridman (36:41.440)
because of some of the things I was describing.
Grant Sanderson (36:43.160)
On the last page of one of them,
Lex Fridman (36:45.280)
somewhere tiny in there was this little formula
Grant Sanderson (36:47.600)
that on the left hand had a sum
Lex Fridman (36:49.880)
over all of the natural numbers.
Grant Sanderson (36:51.840)
It's like one over one to the S plus one over two to the S
Lex Fridman (36:54.760)
plus one over three to the S on and on to the infinity.
Grant Sanderson (36:57.680)
Then on the other side had a product over all of the primes
Lex Fridman (37:01.120)
and it was a certain thing had to do with all the primes.
Lex Fridman (37:03.820)
And like any good young math enthusiast,
Lex Fridman (37:06.480)
I'd probably been indoctrinated with how chaotic
Lex Fridman (37:08.400)
and confusing the primes are, which they are.
Lex Fridman (37:10.920)
And seeing this equation where on one side
Grant Sanderson (37:14.280)
you have something that's as understandable
Lex Fridman (37:15.900)
as you could possibly get, the counting numbers.
Lex Fridman (37:18.280)
And on the other side is all the prime numbers.
Lex Fridman (37:20.480)
It was like this, whoa, they're related like this?
Grant Sanderson (37:23.960)
There's a simple description that includes
Lex Fridman (37:26.480)
all the primes getting wrapped together like this.
Grant Sanderson (37:28.760)
This is like the Euler product for the Zeta function,
Lex Fridman (37:32.200)
as I like later found out.
Grant Sanderson (37:33.800)
The equation itself essentially encodes
Lex Fridman (37:36.120)
the fundamental theorem of arithmetic
Grant Sanderson (37:37.900)
that every number can be expressed
Lex Fridman (37:39.480)
as a unique set of primes.
Grant Sanderson (37:42.080)
To me still there's, I mean, I certainly don't understand
Lex Fridman (37:44.700)
this equation or this function all that well.
Grant Sanderson (37:47.280)
The more I learn about it, the prettier it is.
Lex Fridman (37:50.280)
The idea that you can, this is sort of what gets you
Grant Sanderson (37:53.360)
representations of primes, not in terms of primes themselves,
Lex Fridman (37:57.240)
but in terms of another set of numbers.
Grant Sanderson (37:59.160)
They're like the non trivial zeros of the Zeta function.
Lex Fridman (38:01.960)
And again, I'm very kind of in over my head
Grant Sanderson (38:04.280)
in a lot of ways as I like try to get to understand it.
Lex Fridman (38:06.660)
But the more I do, it always leaves enough mystery
Grant Sanderson (38:09.720)
that it remains very beautiful to me.
Lex Fridman (38:11.640)
So whenever there's a little bit of mystery
Grant Sanderson (38:13.560)
just outside of the understanding that,
Lex Fridman (38:16.780)
and by the way, the process of learning more about it,
Lex Fridman (38:19.680)
how does that come about?
Lex Fridman (38:20.580)
Just your own thought or are you reading?
Grant Sanderson (38:23.800)
Reading, yeah.
Lex Fridman (38:24.640)
Or is the process of visualization itself
Lex Fridman (38:26.480)
revealing more to you?
Lex Fridman (38:28.160)
Visuals help.
Grant Sanderson (38:29.000)
I mean, in one time when I was just trying to understand
Lex Fridman (38:31.320)
like analytic continuation and playing around
Grant Sanderson (38:33.760)
with visualizing complex functions,
Lex Fridman (38:36.240)
this is what led to a video about this function.
Grant Sanderson (38:39.500)
It's titled something like
Lex Fridman (38:40.340)
Visualizing the Riemann Zeta Function.
Grant Sanderson (38:42.360)
It's one that came about because I was programming
Lex Fridman (38:45.040)
and tried to see what a certain thing looked like.
Lex Fridman (38:47.680)
And then I looked at it and I'm like,
Lex Fridman (38:48.520)
whoa, that's elucidating.
Lex Fridman (38:50.640)
And then I decided to make a video about it.
Lex Fridman (38:53.460)
But I mean, you try to get your hands on
Grant Sanderson (38:56.680)
as much reading as you can.
Lex Fridman (38:58.120)
You know, in this case, I think if anyone wants to start
Grant Sanderson (39:01.800)
to understand it, if they have like a math background
Lex Fridman (39:05.800)
like they studied some in college or something like that,
Grant Sanderson (39:08.840)
like the Princeton Companion to Math
Lex Fridman (39:10.280)
has a really good article on analytic number theory.
Lex Fridman (39:13.040)
And that itself has a whole bunch of references
Lex Fridman (39:15.720)
and you know, anything has more references
Lex Fridman (39:17.420)
and it gives you this like tree to start piling through.
Lex Fridman (39:20.160)
And like, you know, you try to understand,
Grant Sanderson (39:22.160)
I try to understand things visually as I go.
Lex Fridman (39:24.720)
That's not always possible,
Lex Fridman (39:26.320)
but it's very helpful when it does.
Lex Fridman (39:28.340)
You recognize when there's common themes,
Grant Sanderson (39:30.100)
like in this case, Cousins of the Fourier Transform
Lex Fridman (39:34.460)
that come into play and you realize,
Grant Sanderson (39:35.880)
oh, it's probably pretty important
Lex Fridman (39:37.000)
to have deep intuitions of the Fourier Transform,
Grant Sanderson (39:39.040)
even if it's not explicitly mentioned in like these texts.
Lex Fridman (39:42.360)
And you try to get a sense of what the common players are.
Lex Fridman (39:45.240)
But I'll emphasize again, like,
Lex Fridman (39:47.220)
I feel very in over my head when I try to understand
Grant Sanderson (39:50.520)
the exact relation between like the zeros
Lex Fridman (39:53.520)
of the Riemann Zeta function
Lex Fridman (39:54.600)
and how they relate to the distribution of primes.
Lex Fridman (39:56.940)
I definitely understand it better than I did a year ago.
Grant Sanderson (39:59.360)
I definitely understand it on 100th as well as the experts
Lex Fridman (40:02.360)
on the matter do, I assume.
Lex Fridman (40:04.680)
But the slow path towards getting there is,
Lex Fridman (40:08.280)
it's fun, it's charming,
Lex Fridman (40:09.600)
and like to your question, very beautiful.
Lex Fridman (40:12.900)
And the beauty is in the, what,
Lex Fridman (40:14.800)
in the journey versus the destination?
Lex Fridman (40:17.080)
Well, it's that each thing doesn't feel arbitrary.
Grant Sanderson (40:19.420)
I think that's a big part,
Lex Fridman (40:20.560)
is that you have these unpredictable,
Grant Sanderson (40:23.360)
no, yeah, these very unpredictable patterns
Lex Fridman (40:25.880)
or these intricate properties of like a certain function.
Lex Fridman (40:30.480)
But at the same time,
Lex Fridman (40:31.320)
it doesn't feel like humans ever made an arbitrary choice
Grant Sanderson (40:33.800)
in studying this particular thing.
Lex Fridman (40:35.760)
So, you know, it feels like you're speaking
Grant Sanderson (40:38.440)
to patterns themselves or nature itself.
Lex Fridman (40:41.280)
That's a big part of it.
Grant Sanderson (40:43.160)
I think things that are too arbitrary,
Lex Fridman (40:45.080)
it's just hard for those to feel beautiful
Grant Sanderson (40:46.640)
because this is sort of what the word contrived
Lex Fridman (40:49.800)
is meant to apply to, right?
Lex Fridman (40:53.420)
And when they're not arbitrary means it could be,
Lex Fridman (40:57.640)
you can have a clean abstraction and intuition
Grant Sanderson (41:02.940)
that allows you to comprehend it.
Lex Fridman (41:04.940)
Well, to one of your first questions,
Grant Sanderson (41:06.220)
it makes you feel like if you came across
Lex Fridman (41:07.640)
another intelligent civilization,
Grant Sanderson (41:09.680)
that they'd be studying the same thing.
Lex Fridman (41:12.360)
Maybe with different notation.
Grant Sanderson (41:13.680)
Certainly, yeah, but yeah.
Lex Fridman (41:15.520)
Like that's what,
Grant Sanderson (41:16.360)
I think you talked to that other civilization,
Lex Fridman (41:18.740)
they're probably also studying the zeros
Grant Sanderson (41:20.280)
of the Riemann Zeta function
Lex Fridman (41:21.680)
or like some variant thereof
Grant Sanderson (41:23.840)
that is like a clearly equivalent cousin
Lex Fridman (41:27.520)
or something like that.
Lex Fridman (41:28.560)
But that's probably on their docket.
Lex Fridman (41:32.480)
Whenever somebody does a lot of something amazing,
Grant Sanderson (41:35.920)
I'm gonna ask the question
Lex Fridman (41:37.640)
that you've already been asked a lot
Lex Fridman (41:40.160)
and that you'll get more and more asked in your life.
Lex Fridman (41:43.320)
But what was your favorite video to create?
Grant Sanderson (41:46.160)
Oh, favorite to create.
Lex Fridman (41:49.520)
One of my favorites is,
Lex Fridman (41:51.280)
the title is Who Cares About Topology?
Lex Fridman (41:54.220)
You want me to pull it up or no?
Grant Sanderson (41:55.920)
If you want, sure, yeah.
Lex Fridman (41:57.300)
It is about, well, it starts by describing
Grant Sanderson (42:00.960)
an unsolved problem that's still unsolved in math
Lex Fridman (42:03.040)
called the inscribed square problem.
Grant Sanderson (42:05.020)
You draw any loop and then you ask,
Lex Fridman (42:06.720)
are there four points on that loop that make a square?
Lex Fridman (42:09.180)
Totally useless, right?
Lex Fridman (42:10.320)
This is not answering any physical questions.
Grant Sanderson (42:12.480)
It's mostly interesting that we can't answer that question.
Lex Fridman (42:14.920)
And it seems like such a natural thing to ask.
Grant Sanderson (42:18.360)
Now, if you weaken it a little bit and you ask,
Lex Fridman (42:21.160)
can you always find a rectangle?
Grant Sanderson (42:22.580)
You choose four points on this curve,
Lex Fridman (42:24.280)
can you find a rectangle?
Grant Sanderson (42:25.640)
That's hard, but it's doable.
Lex Fridman (42:27.520)
And the path to it involves things like looking at a torus,
Grant Sanderson (42:32.960)
this surface with a single hole in it, like a donut,
Lex Fridman (42:35.320)
or looking at a mobius strip.
Grant Sanderson (42:37.300)
In ways that feel so much less contrived
Lex Fridman (42:39.760)
to when I first, as like a little kid,
Grant Sanderson (42:41.680)
learned about these surfaces and shapes,
Lex Fridman (42:43.400)
like a mobius strip and a torus.
Grant Sanderson (42:45.460)
Like what you learn is, oh, this mobius strip,
Lex Fridman (42:47.840)
you take a piece of paper, put a twist, glue it together,
Lex Fridman (42:50.760)
and now you have a shape with one edge and just one side.
Lex Fridman (42:53.720)
And as a student, you should think, who cares, right?
Lex Fridman (42:58.480)
Like, how does that help me solve any problems?
Lex Fridman (43:00.600)
I thought math was about problem solving.
Lex Fridman (43:02.720)
So what I liked about the piece of math
Lex Fridman (43:05.640)
that this was describing that was in this paper
Grant Sanderson (43:08.500)
by a mathematician named Vaughn
Lex Fridman (43:10.080)
was that it arises very naturally.
Grant Sanderson (43:12.960)
It's clear what it represents.
Lex Fridman (43:14.360)
It's doing something.
Grant Sanderson (43:15.440)
It's not just playing with construction paper.
Lex Fridman (43:17.800)
And the way that it solves the problem is really beautiful.
Lex Fridman (43:21.800)
So kind of putting all of that down
Lex Fridman (43:24.280)
and concretizing it, right?
Grant Sanderson (43:25.840)
Like I was talking about how
Lex Fridman (43:27.680)
when you have to put visuals to it,
Grant Sanderson (43:29.400)
it demands that what's on screen
Lex Fridman (43:30.920)
is a very specific example of what you're describing.
Grant Sanderson (43:33.320)
The construction here is very abstract in nature.
Lex Fridman (43:35.920)
You describe this very abstract kind of surface in 3D space.
Lex Fridman (43:39.320)
So then when I was finding myself,
Lex Fridman (43:40.920)
in this case, I wasn't programming,
Grant Sanderson (43:42.040)
I was using a grapher that's like built into OSX
Lex Fridman (43:44.560)
for the 3D stuff to draw that surface,
Grant Sanderson (43:48.780)
you realize, oh man, the topology argument
Lex Fridman (43:50.800)
is very non constructive.
Grant Sanderson (43:52.680)
I have to make a lot of,
Lex Fridman (43:54.160)
you have to do a lot of extra work
Grant Sanderson (43:55.680)
in order to make the surface show up.
Lex Fridman (43:57.440)
But then once you see it, it's quite pretty
Lex Fridman (43:59.440)
and it's very satisfying to see a specific instance of it.
Lex Fridman (44:02.120)
And you also feel like, ah,
Grant Sanderson (44:03.540)
I've actually added something
Lex Fridman (44:04.840)
on top of what the original paper was doing
Grant Sanderson (44:06.720)
that it shows something that's completely correct.
Lex Fridman (44:09.640)
That's a very beautiful argument,
Lex Fridman (44:10.880)
but you don't see what it looks like.
Lex Fridman (44:12.680)
And I found something satisfying
Grant Sanderson (44:14.960)
in seeing what it looked like
Lex Fridman (44:16.400)
that could only ever have come about
Grant Sanderson (44:17.960)
from the forcing function
Lex Fridman (44:19.240)
of getting some kind of image on the screen
Grant Sanderson (44:21.200)
to describe the thing I was talking about.
Lex Fridman (44:22.560)
So you almost weren't able to anticipate
Lex Fridman (44:24.260)
what it's gonna look like.
Lex Fridman (44:25.100)
I had no idea.
Grant Sanderson (44:26.160)
I had no idea.
Lex Fridman (44:27.000)
And it was wonderful, right?
Grant Sanderson (44:28.280)
It was totally, it looks like a Sydney Opera House
Lex Fridman (44:30.360)
or some sort of Frank Gehry design.
Lex Fridman (44:32.040)
And it was, you knew it was gonna be something
Lex Fridman (44:35.200)
and you can say various things about it.
Grant Sanderson (44:36.480)
Like, oh, it touches the curve itself.
Lex Fridman (44:39.320)
It has a boundary that's this curve on the 2D plane.
Grant Sanderson (44:42.080)
It all sits above the plane.
Lex Fridman (44:43.840)
But before you actually draw it,
Grant Sanderson (44:45.320)
it's very unclear what the thing will look like.
Lex Fridman (44:48.160)
And to see it, it's very, it's just pleasing, right?
Lex Fridman (44:50.760)
So that was fun to make, very fun to share.
Lex Fridman (44:53.240)
I hope that it has elucidated for some people out there
Grant Sanderson (44:58.040)
where these constructs of topology come from,
Lex Fridman (45:00.120)
that it's not arbitrary play with construction paper.
Lex Fridman (45:04.040)
So let's, I think this is a good sort of example
Lex Fridman (45:07.240)
to talk a little bit about your process.
Grant Sanderson (45:09.640)
You have a list of ideas.
Lex Fridman (45:12.760)
So that's sort of the curse of having an active
Lex Fridman (45:17.480)
and brilliant mind is I'm sure you have a list
Lex Fridman (45:19.600)
that's growing faster than you can utilize.
Grant Sanderson (45:22.000)
Now I'm ahead, absolutely.
Lex Fridman (45:24.560)
But there's some sorting procedure
Grant Sanderson (45:26.880)
depending on mood and interest and so on.
Lex Fridman (45:29.800)
But okay, so you pick an idea
Lex Fridman (45:32.640)
and then you have to try to write a narrative arc
Lex Fridman (45:36.160)
that sort of, how do I elucidate?
Lex Fridman (45:38.800)
How do I make this idea beautiful and clear
Lex Fridman (45:41.320)
and explain it?
Lex Fridman (45:42.240)
And then there's a set of visualizations
Lex Fridman (45:44.040)
that will be attached to it.
Grant Sanderson (45:46.160)
Sort of, you've talked about some of this before,
Lex Fridman (45:48.440)
but sort of writing the story, attaching the visualizations.
Lex Fridman (45:52.920)
Can you talk through interesting, painful,
Lex Fridman (45:56.440)
beautiful parts of that process?
Grant Sanderson (45:58.880)
Well, the most painful is if you've chosen a topic
Lex Fridman (46:02.000)
that you do want to do, but then it's hard to think of,
Grant Sanderson (46:05.680)
I guess how to structure the script.
Lex Fridman (46:07.480)
This is sort of where I have been on one
Grant Sanderson (46:10.640)
for like the last two or three months.
Lex Fridman (46:12.240)
And I think that ultimately the right resolution
Grant Sanderson (46:13.720)
is just like set it aside and instead do some other things
Lex Fridman (46:17.360)
where the script comes more naturally.
Grant Sanderson (46:18.840)
Because you sort of don't want to overwork a narrative.
Lex Fridman (46:23.480)
The more you've thought about it,
Grant Sanderson (46:24.700)
the less you can empathize with the student
Lex Fridman (46:26.480)
who doesn't yet understand the thing you're trying to teach.
Lex Fridman (46:28.940)
Who is the judger in your head?
Lex Fridman (46:31.860)
Sort of the person, the creature,
Grant Sanderson (46:35.480)
the essence that's saying this sucks or this is good.
Lex Fridman (46:38.680)
And you mentioned kind of the student you're thinking about.
Lex Fridman (46:43.000)
Can you, who is that?
Lex Fridman (46:44.740)
What is that thing?
Grant Sanderson (46:45.880)
That says, the perfectionist that says this thing sucks.
Lex Fridman (46:49.960)
You need to work on that for another two, three months.
Grant Sanderson (46:53.520)
I don't know.
Lex Fridman (46:54.360)
I think it's my past self.
Grant Sanderson (46:56.160)
I think that's the entity that I'm most trying
Lex Fridman (46:58.280)
to empathize with is like you take who I was,
Grant Sanderson (47:00.880)
because that's kind of the only person I know.
Lex Fridman (47:02.480)
Like you don't really know anyone
Grant Sanderson (47:03.720)
other than versions of yourself.
Lex Fridman (47:05.480)
So I start with the version of myself that I know
Lex Fridman (47:07.880)
who doesn't yet understand the thing, right?
Lex Fridman (47:10.320)
And then I just try to view it with fresh eyes,
Grant Sanderson (47:15.560)
a particular visual or a particular script.
Lex Fridman (47:17.480)
Like, is this motivating?
Lex Fridman (47:18.840)
Does this make sense?
Lex Fridman (47:20.680)
Which has its downsides,
Grant Sanderson (47:21.560)
because sometimes I find myself speaking to motivations
Lex Fridman (47:25.200)
that only myself would be interested in.
Grant Sanderson (47:28.520)
I don't know, like I did this project on quaternions
Lex Fridman (47:30.840)
where what I really wanted was to understand
Lex Fridman (47:33.280)
what are they doing in four dimensions?
Lex Fridman (47:34.920)
Can we see what they're doing in four dimensions, right?
Lex Fridman (47:37.560)
And I came up with a way of thinking about it
Lex Fridman (47:40.400)
that really answered the question in my head
Grant Sanderson (47:42.000)
that made me very satisfied
Lex Fridman (47:43.280)
and being able to think about concretely with a 3D visual,
Lex Fridman (47:46.640)
what are they doing to a 4D sphere?
Lex Fridman (47:48.720)
And so I'm like, great,
Lex Fridman (47:49.560)
this is exactly what my past self would have wanted, right?
Lex Fridman (47:51.720)
And I make a thing on it.
Lex Fridman (47:52.760)
And I'm sure it's what some other people wanted too.
Lex Fridman (47:55.140)
But in hindsight, I think most people who wanna learn
Grant Sanderson (47:57.400)
about quaternions are like robotics engineers
Lex Fridman (48:00.460)
or graphics programmers who want to understand
Lex Fridman (48:03.320)
how they're used to describe 3D rotations.
Lex Fridman (48:06.160)
And like their use case was actually a little bit different
Grant Sanderson (48:08.280)
than my past self.
Lex Fridman (48:09.360)
And in that way, like,
Grant Sanderson (48:10.600)
I wouldn't actually recommend that video
Lex Fridman (48:12.100)
to people who are coming at it from that angle
Grant Sanderson (48:14.920)
of wanting to know, hey, I'm a robotics programmer.
Lex Fridman (48:17.460)
Like, how do these quaternion things work
Lex Fridman (48:20.260)
to describe position in 3D space?
Lex Fridman (48:22.560)
I would say other great resources for that.
Grant Sanderson (48:25.660)
If you ever find yourself wanting to say like,
Lex Fridman (48:27.840)
but hang on,
Lex Fridman (48:28.880)
in what sense are they acting in four dimensions?
Lex Fridman (48:30.880)
Then come back.
Lex Fridman (48:31.900)
But until then, that's a little different.
Lex Fridman (48:34.520)
Yeah, it's interesting
Grant Sanderson (48:35.360)
because you have incredible videos on neural networks,
Lex Fridman (48:38.800)
for example.
Lex Fridman (48:39.840)
And from my sort of perspective,
Lex Fridman (48:41.080)
because I've probably, I mean,
Grant Sanderson (48:43.720)
I looked at the,
Lex Fridman (48:45.520)
is sort of my field
Lex Fridman (48:47.120)
and I've also looked at the basic introduction
Lex Fridman (48:49.320)
of neural networks like a million times
Grant Sanderson (48:51.100)
from different perspectives.
Lex Fridman (48:52.480)
And it made me realize
Grant Sanderson (48:53.480)
that there's a lot of ways to present it.
Lex Fridman (48:55.800)
So you were sort of, you did an incredible job.
Grant Sanderson (48:58.960)
I mean, sort of the,
Lex Fridman (49:01.560)
but you could also do it differently
Lex Fridman (49:03.440)
and also incredible.
Lex Fridman (49:04.900)
Like to create a beautiful presentation of a basic concept
Grant Sanderson (49:11.800)
requires sort of creativity, requires genius and so on,
Lex Fridman (49:16.080)
but you can take it from a bunch of different perspectives.
Lex Fridman (49:18.620)
And that video on neural networks made me realize that.
Lex Fridman (49:21.440)
And just as you're saying,
Grant Sanderson (49:22.960)
you kind of have a certain mindset, a certain view,
Lex Fridman (49:26.280)
but from a, if you take a different view
Grant Sanderson (49:28.980)
from a physics perspective,
Lex Fridman (49:30.680)
from a neuroscience perspective,
Grant Sanderson (49:33.380)
talking about neural networks
Lex Fridman (49:34.440)
or from a robotics perspective,
Grant Sanderson (49:38.420)
or from, let's see,
Lex Fridman (49:40.440)
from a pure learning, statistics perspective.
Lex Fridman (49:43.300)
So you can create totally different videos.
Lex Fridman (49:46.360)
And you've done that with a few actually concepts
Grant Sanderson (49:48.240)
where you've have taken different cuts,
Lex Fridman (49:49.920)
like at the Euler equation, right?
Grant Sanderson (49:54.840)
You've taken different views of that.
Lex Fridman (49:56.960)
I think I've made three videos on it
Lex Fridman (49:58.720)
and I definitely will make at least one more.
Lex Fridman (50:01.040)
Right?
Grant Sanderson (50:02.040)
Never enough.
Lex Fridman (50:03.080)
Never enough.
Lex Fridman (50:04.120)
So you don't think it's the most beautiful equation
Lex Fridman (50:06.200)
in mathematics?
Grant Sanderson (50:08.280)
Like I said, as we represent it,
Lex Fridman (50:10.040)
it's one of the most hideous.
Grant Sanderson (50:11.440)
It involves a lot of the most hideous aspects
Lex Fridman (50:13.320)
of our notation.
Grant Sanderson (50:14.160)
I talked about E, the fact that we use pi instead of tau,
Lex Fridman (50:16.960)
the fact that we call imaginary numbers imaginary,
Lex Fridman (50:20.680)
and then, hence, I actually wonder if we use the I
Lex Fridman (50:23.420)
because of imaginary.
Grant Sanderson (50:24.880)
I don't know if that's historically accurate,
Lex Fridman (50:26.600)
but at least a lot of people,
Grant Sanderson (50:27.960)
they read the I and they think imaginary.
Lex Fridman (50:30.300)
Like all three of those facts,
Grant Sanderson (50:31.440)
it's like those are things that have added more confusion
Lex Fridman (50:33.720)
than they needed to,
Lex Fridman (50:34.560)
and we're wrapping them up in one equation.
Lex Fridman (50:35.840)
Like boy, that's just very hideous, right?
Grant Sanderson (50:39.080)
The idea is that it does tie together
Lex Fridman (50:40.880)
when you wash away the notation.
Grant Sanderson (50:42.160)
Like it's okay, it's pretty, it's nice,
Lex Fridman (50:44.820)
but it's not like mind blowing greatest thing
Grant Sanderson (50:48.320)
in the universe,
Lex Fridman (50:49.520)
which is maybe what I was thinking of when I said,
Grant Sanderson (50:52.020)
like once you understand something,
Lex Fridman (50:53.320)
it doesn't have the same beauty.
Grant Sanderson (50:55.960)
Like I feel like I understand Euler's formula,
Lex Fridman (50:59.000)
and I feel like I understand it enough
Grant Sanderson (51:00.840)
to sort of see the version that just woke up
Lex Fridman (51:05.000)
that hasn't really gotten itself dressed in the morning
Grant Sanderson (51:07.560)
that's a little bit groggy,
Lex Fridman (51:08.640)
and there's bags under its eyes.
Lex Fridman (51:10.120)
So you're past the dating stage,
Lex Fridman (51:13.320)
you're no longer dating, right?
Grant Sanderson (51:15.040)
I'm still dating the Zeta function,
Lex Fridman (51:16.760)
and like she's beautiful and right,
Lex Fridman (51:18.920)
and like we have fun,
Lex Fridman (51:20.100)
and it's that high dopamine part,
Lex Fridman (51:22.640)
but like maybe at some point
Lex Fridman (51:24.000)
we'll settle into the more mundane nature of the relationship
Grant Sanderson (51:26.840)
where I like see her for who she truly is,
Lex Fridman (51:28.480)
and she'll still be beautiful in her own way,
Lex Fridman (51:30.220)
but it won't have the same romantic pizzazz, right?
Lex Fridman (51:33.720)
Well, that's the nice thing about mathematics.
Grant Sanderson (51:35.520)
I think as long as you don't live forever,
Lex Fridman (51:38.500)
there'll always be enough mystery and fun
Grant Sanderson (51:41.840)
with some of the equations.
Lex Fridman (51:42.920)
Even if you do, the rate at which questions comes up
Grant Sanderson (51:45.480)
is much faster than the rate at which answers come up, so.
Lex Fridman (51:48.080)
If you could live forever, would you?
Grant Sanderson (51:51.160)
I think so, yeah.
Lex Fridman (51:52.160)
So you think, you don't think mortality
Lex Fridman (51:53.760)
is the thing that makes life meaningful?
Lex Fridman (51:55.980)
Would your life be four times as meaningful
Lex Fridman (51:58.240)
if you died at 25?
Lex Fridman (52:00.440)
So this goes to infinity.
Grant Sanderson (52:02.200)
I think you and I, that's really interesting.
Lex Fridman (52:04.700)
So what I said is infinite, not four times longer.
Grant Sanderson (52:09.400)
I said infinite.
Lex Fridman (52:10.360)
So the actual existence of the finiteness,
Grant Sanderson (52:15.200)
the existence of the end, no matter the length,
Lex Fridman (52:18.020)
is the thing that may sort of,
Grant Sanderson (52:20.680)
from my comprehension of psychology,
Lex Fridman (52:22.380)
it's such a deeply human,
Grant Sanderson (52:25.600)
it's such a fundamental part of the human condition,
Lex Fridman (52:28.560)
the fact that there is, that we're mortal,
Grant Sanderson (52:31.120)
that the fact that things end,
Lex Fridman (52:34.480)
it seems to be a crucial part of what gives them meaning.
Grant Sanderson (52:37.880)
I don't think, at least for me,
Lex Fridman (52:40.560)
it's a very small percentage of my time
Grant Sanderson (52:43.040)
that mortality is salient,
Lex Fridman (52:45.200)
that I'm aware of the end of my life.
Lex Fridman (52:47.200)
What do you mean by me?
Lex Fridman (52:50.080)
I'm trolling.
Lex Fridman (52:51.280)
Is it the ego, is it the id, or is it the superego?
Lex Fridman (52:55.840)
The reflective self, the Wernicke's area
Grant Sanderson (52:58.160)
that puts all this stuff into words.
Lex Fridman (52:59.860)
Yeah, a small percentage of your mind
Grant Sanderson (53:02.520)
that is actually aware of the true motivations
Lex Fridman (53:05.800)
that drive you.
Lex Fridman (53:06.960)
But my point is that most of my life,
Lex Fridman (53:08.360)
I'm not thinking about death,
Lex Fridman (53:09.600)
but I still feel very motivated to make things
Lex Fridman (53:12.000)
and to interact with people,
Grant Sanderson (53:14.000)
experience love or things like that.
Lex Fridman (53:15.400)
I'm very motivated,
Lex Fridman (53:16.680)
and it's strange that that motivation comes
Lex Fridman (53:19.400)
while death is not in my mind at all.
Lex Fridman (53:21.600)
And this might just be because I'm young enough
Lex Fridman (53:23.600)
that it's not salient.
Grant Sanderson (53:24.720)
Or it's in your subconscious,
Lex Fridman (53:25.960)
or that you've constructed an illusion
Grant Sanderson (53:28.100)
that allows you to escape the fact of your mortality
Lex Fridman (53:31.180)
by enjoying the moment,
Grant Sanderson (53:32.760)
sort of the existential approach to life.
Lex Fridman (53:34.600)
Could be.
Grant Sanderson (53:36.160)
Gun to my head, I don't think that's it.
Lex Fridman (53:38.120)
Yeah, another sort of way to say gun to the head
Grant Sanderson (53:40.680)
is sort of the deep psychological introspection
Lex Fridman (53:43.400)
of what drives us.
Grant Sanderson (53:44.240)
I mean, that's, in some ways to me,
Lex Fridman (53:47.000)
I mean, when I look at math, when I look at science,
Grant Sanderson (53:49.080)
is a kind of an escape from reality
Lex Fridman (53:51.600)
in a sense that it's so beautiful.
Grant Sanderson (53:54.400)
It's such a beautiful journey of discovery
Lex Fridman (53:58.740)
that it allows you to actually,
Grant Sanderson (54:00.820)
it sort of allows you to achieve a kind of immortality
Lex Fridman (54:04.760)
of explore ideas and sort of connect yourself
Grant Sanderson (54:09.680)
to the thing that is seemingly infinite,
Lex Fridman (54:12.360)
like the universe, right?
Grant Sanderson (54:13.960)
That allows you to escape the limited nature
Lex Fridman (54:18.600)
of our little, of our bodies, of our existence.
Lex Fridman (54:24.040)
What else would give this podcast meaning?
Lex Fridman (54:25.960)
That's right.
Grant Sanderson (54:26.800)
If not the fact that it will end.
Lex Fridman (54:28.000)
This place closes in 40 minutes.
Lex Fridman (54:30.920)
And it's so much more meaningful for it.
Lex Fridman (54:33.280)
How much more I love this room
Grant Sanderson (54:35.480)
because we'll be kicked out.
Lex Fridman (54:38.200)
So I understand just because you're trolling me
Grant Sanderson (54:42.280)
doesn't mean I'm wrong.
Lex Fridman (54:46.200)
But I take your point.
Grant Sanderson (54:47.840)
I take your point.
Lex Fridman (54:49.000)
Boy, that would be a good Twitter bio.
Grant Sanderson (54:52.120)
Just because you're trolling me doesn't mean I'm wrong.
Lex Fridman (54:54.320)
Yeah, and sort of difference in backgrounds.
Grant Sanderson (54:58.560)
I'm a bit Russian, so we're a bit melancholic
Lex Fridman (55:01.520)
and seem to maybe assign a little too much value
Grant Sanderson (55:04.320)
to suffering and mortality and things like that.
Lex Fridman (55:07.360)
Makes for a better novel, I think.
Grant Sanderson (55:09.840)
Oh yeah, you need some sort of existential threat
Lex Fridman (55:13.400)
to drive a plot.
Lex Fridman (55:16.400)
So when do you know when the video is done
Lex Fridman (55:18.520)
when you're working on it?
Grant Sanderson (55:20.680)
That's pretty easy actually,
Lex Fridman (55:21.560)
because I'll write the script.
Grant Sanderson (55:24.400)
I want there to be some kind of aha moment in there.
Lex Fridman (55:27.120)
And then hopefully the script can revolve around
Grant Sanderson (55:28.760)
some kind of aha moment.
Lex Fridman (55:30.280)
And then from there, you're putting visuals
Grant Sanderson (55:32.400)
to each sentence that exists,
Lex Fridman (55:34.040)
and then you narrate it, you edit it all together.
Lex Fridman (55:36.360)
So given that there's a script,
Lex Fridman (55:37.760)
the end becomes quite clear.
Lex Fridman (55:40.680)
And as I animate it, I often change
Lex Fridman (55:45.560)
certainly the specific words,
Lex Fridman (55:46.800)
but sometimes the structure itself.
Lex Fridman (55:49.320)
But it's a very deterministic process at that point.
Grant Sanderson (55:53.240)
It makes it much easier to predict
Lex Fridman (55:54.440)
when something will be done.
Lex Fridman (55:55.840)
How do you know when a script is done?
Lex Fridman (55:57.080)
It's like, for problem solving videos,
Grant Sanderson (55:59.120)
that's quite simple.
Lex Fridman (56:00.400)
It's once you feel like someone
Grant Sanderson (56:01.440)
who didn't understand the solution now could.
Lex Fridman (56:03.560)
For things like neural networks,
Grant Sanderson (56:04.680)
that was a lot harder because like you said,
Lex Fridman (56:06.720)
there's so many angles at which you could attack it.
Lex Fridman (56:09.560)
And there, it's just at some point
Lex Fridman (56:11.720)
you feel like this asks a meaningful question
Lex Fridman (56:15.680)
and it answers that question, right?
Lex Fridman (56:18.360)
What is the best way to learn math
Lex Fridman (56:20.280)
for people who might be at the beginning of that journey?
Lex Fridman (56:22.400)
I think that's a question that a lot of folks
Grant Sanderson (56:24.840)
kind of ask and think about.
Lex Fridman (56:26.200)
And it doesn't, even for folks
Grant Sanderson (56:27.800)
who are not really at the beginning of their journey,
Lex Fridman (56:29.920)
like there might be actually deep in their career,
Grant Sanderson (56:33.920)
some type they've taken college
Lex Fridman (56:35.720)
or taken calculus and so on,
Lex Fridman (56:36.920)
but still wanna sort of explore math.
Lex Fridman (56:39.080)
What would be your advice instead of education at all ages?
Grant Sanderson (56:42.880)
Your temptation will be to spend more time
Lex Fridman (56:45.840)
like watching lectures or reading.
Grant Sanderson (56:48.160)
Try to force yourself to do more problems
Lex Fridman (56:50.560)
than you naturally would.
Grant Sanderson (56:52.160)
That's a big one.
Lex Fridman (56:53.800)
Like the focus time that you're spending
Grant Sanderson (56:56.040)
should be on like solving specific problems
Lex Fridman (56:59.000)
and seek entities that have well curated lists of problems.
Lex Fridman (57:02.360)
So go into like a textbook almost
Lex Fridman (57:04.240)
and the problems in the back of a textbook kind of thing,
Grant Sanderson (57:07.160)
back of a chapter.
Lex Fridman (57:08.080)
So if you can take a little look through those questions
Grant Sanderson (57:10.560)
at the end of the chapter before you read the chapter,
Lex Fridman (57:12.480)
a lot of them won't make sense.
Grant Sanderson (57:13.600)
Some of them might,
Lex Fridman (57:14.560)
and those are the best ones to think about.
Grant Sanderson (57:16.600)
A lot of them won't, but just take a quick look
Lex Fridman (57:18.920)
and then read a little bit of the chapter
Lex Fridman (57:20.120)
and then maybe take a look again and things like that.
Lex Fridman (57:22.400)
And don't consider yourself done with the chapter
Grant Sanderson (57:25.120)
until you've actually worked through a couple exercises.
Lex Fridman (57:29.800)
And this is so hypocritical, right?
Grant Sanderson (57:31.160)
Cause I like put out videos
Lex Fridman (57:32.400)
that pretty much never have associated exercises.
Grant Sanderson (57:35.920)
I just view myself as a different part of the ecosystem,
Lex Fridman (57:38.640)
which means I'm kind of admitting
Grant Sanderson (57:40.760)
that you're not really learning,
Lex Fridman (57:42.880)
or at least this is only a partial part
Grant Sanderson (57:44.720)
of the learning process if you're watching these videos.
Lex Fridman (57:48.600)
I think if someone's at the very beginning,
Grant Sanderson (57:50.360)
like I do think Khan Academy does a good job.
Lex Fridman (57:52.200)
They have a pretty large set of questions
Grant Sanderson (57:54.840)
you can work through.
Lex Fridman (57:55.840)
Just the very basics,
Grant Sanderson (57:56.840)
sort of just picking up,
Lex Fridman (57:58.760)
getting comfortable with the very basic linear algebra,
Grant Sanderson (58:01.200)
calculus or so on, Khan Academy.
Lex Fridman (58:04.040)
Programming is actually I think a great,
Grant Sanderson (58:05.880)
like learn to program and like let the way
Lex Fridman (58:08.440)
that math is motivated from that angle push you through.
Grant Sanderson (58:11.760)
I know a lot of people who didn't like math
Lex Fridman (58:14.280)
got into programming in some way
Lex Fridman (58:15.520)
and that's what turned them on to math.
Lex Fridman (58:17.240)
Maybe I'm biased cause like I live in the Bay area,
Lex Fridman (58:19.160)
so I'm more likely to run into someone
Lex Fridman (58:21.000)
who has that phenotype.
Lex Fridman (58:23.320)
But I am willing to speculate
Lex Fridman (58:25.760)
that that is a more generalizable path.
Lex Fridman (58:28.080)
So you yourself kind of in creating the videos
Lex Fridman (58:30.080)
are using programming to illuminate a concept,
Lex Fridman (58:32.960)
but for yourself as well.
Lex Fridman (58:35.000)
So would you recommend somebody try to make a,
Lex Fridman (58:37.920)
sort of almost like try to make videos?
Lex Fridman (58:40.240)
Like you do as a way to learn?
Lex Fridman (58:41.840)
So one thing I've heard before,
Lex Fridman (58:43.120)
I don't know if this is based on any actual study.
Grant Sanderson (58:44.680)
This might be like a total fictional anecdote of numbers,
Lex Fridman (58:47.200)
but it rings in the mind as being true.
Grant Sanderson (58:49.720)
You remember about 10% of what you read,
Lex Fridman (58:51.820)
you remember about 20% of what you listen to,
Grant Sanderson (58:54.400)
you remember about 70% of what you actively interact with
Lex Fridman (58:57.280)
in some way, and then about 90% of what you teach.
Grant Sanderson (59:00.480)
This is a thing I heard again,
Lex Fridman (59:02.080)
those numbers might be meaningless,
Lex Fridman (59:03.460)
but they ring true, don't they, right?
Lex Fridman (59:05.840)
I'm willing to say I learned nine times better
Grant Sanderson (59:07.880)
if I'm teaching something than reading.
Lex Fridman (59:09.200)
That might even be a low ball, right?
Lex Fridman (59:11.640)
So doing something to teach
Lex Fridman (59:12.960)
or to like actively try to explain things
Grant Sanderson (59:15.180)
is huge for consolidating the knowledge.
Lex Fridman (59:17.800)
Outside of family and friends,
Grant Sanderson (59:19.640)
is there a moment you can remember
Lex Fridman (59:22.400)
that you would like to relive
Grant Sanderson (59:23.720)
because it made you truly happy
Lex Fridman (59:26.160)
or it was transformative in some fundamental way?
Grant Sanderson (59:30.160)
A moment that was transformative.
Lex Fridman (59:32.680)
Or made you truly happy?
Grant Sanderson (59:35.040)
Yeah, I think there's times,
Lex Fridman (59:36.760)
like music used to be a much bigger part of my life
Grant Sanderson (59:38.720)
than it is now, like when I was a, let's say a teenager,
Lex Fridman (59:41.600)
and I can think of some times in like playing music.
Grant Sanderson (59:45.400)
There was one, like my brother and a friend of mine,
Lex Fridman (59:48.160)
so this slightly violates the family and friends,
Lex Fridman (59:50.160)
but it was the music that made me happy.
Lex Fridman (59:51.800)
They were just accompanying.
Grant Sanderson (59:54.440)
We like played a gig at a ski resort
Lex Fridman (59:57.480)
such that you like take a gondola to the top
Lex Fridman (59:59.320)
and like did a thing.
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