Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries
数学音乐与艺术生物与进化心理与人性哲学与宗教
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donmathematicsbookmathdimensionalnumbersinterestingspacesimpledoesnprimecalledproofsymmetrysayingdoingstuffdidntalktrue
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🎙️ 完整对话（3960 条）
Lex Fridman (00:00.000)
The following is a conversation with Jordan Ellenberg,
以下是与乔丹·艾伦伯格的对话，
Lex Fridman (00:02.960)
a mathematician at University of Wisconsin
威斯康星大学数学家
Lex Fridman (00:05.560)
and an author who masterfully reveals the beauty
以及一位巧妙地揭示美的作家
Lex Fridman (00:08.900)
and power of mathematics in his 2014 book,
以及他 2014 年书中数学的力量，
Lex Fridman (00:12.180)
How Not To Be Wrong, and his new book,
如何不犯错误，以及他的新书，
Jordan Ellenberg (00:15.000)
just released recently, called Shape,
最近刚刚发布，名为 Shape，
Lex Fridman (00:17.540)
The Hidden Geometry of Information, Biology,
信息的隐藏几何，生物学，
Jordan Ellenberg (00:20.440)
Strategy, Democracy, and Everything Else.
战略、民主和其他一切。
Lex Fridman (00:23.400)
Quick mention of our sponsors,
快速提及我们的赞助商，
Jordan Ellenberg (00:25.360)
Secret Sauce, ExpressVPN, Blinkist, and Indeed.
Secret Sauce、ExpressVPN、Blinkist 和 Indeed。
Lex Fridman (00:29.840)
Check them out in the description to support this podcast.
在说明中查看它们以支持此播客。
Jordan Ellenberg (00:33.160)
As a side note, let me say that geometry
作为旁注，让我说一下几何
Lex Fridman (00:35.020)
is what made me fall in love with mathematics
是什么让我爱上了数学
Jordan Ellenberg (00:37.440)
when I was young.
当我年轻的时候。
Lex Fridman (00:38.560)
It first showed me that something definitive
它首先向我展示了一些确定的东西
Jordan Ellenberg (00:41.100)
could be stated about this world
可以描述这个世界
Lex Fridman (00:42.960)
through intuitive visual proofs.
通过直观的视觉证明。
Jordan Ellenberg (00:45.440)
Somehow, that convinced me that math
不知何故，这让我相信数学
Lex Fridman (00:47.840)
is not just abstract numbers devoid of life,
不仅仅是没有生命的抽象数字，
Lex Fridman (00:50.600)
but a part of life, part of this world,
但生活的一部分，这个世界的一部分，
Lex Fridman (00:53.380)
part of our search for meaning.
Jordan Ellenberg (00:55.720)
This is the Lex Friedman podcast,
Lex Fridman (00:57.680)
and here is my conversation with Jordan Ellenberg.
Jordan Ellenberg (01:01.860)
If the brain is a cake.
Lex Fridman (01:03.880)
It is?
Lex Fridman (01:05.040)
Well, let's just go with me on this, okay?
Lex Fridman (01:07.080)
Okay, we'll pause it.
Lex Fridman (01:08.160)
So for Noam Chomsky, language,
Lex Fridman (01:12.680)
the universal grammar, the framework
Jordan Ellenberg (01:16.000)
from which language springs is like most of the cake,
Lex Fridman (01:19.960)
the delicious chocolate center,
Lex Fridman (01:21.880)
and then the rest of cognition that we think of
Lex Fridman (01:25.040)
is built on top, extra layers,
Jordan Ellenberg (01:27.760)
maybe the icing on the cake,
Lex Fridman (01:28.940)
maybe consciousness is just like a cherry on top.
Lex Fridman (01:34.100)
Where do you put in this cake mathematical thinking?
Lex Fridman (01:37.980)
Is it as fundamental as language?
Lex Fridman (01:40.800)
In the Chomsky view, is it more fundamental than language?
Lex Fridman (01:44.600)
Is it echoes of the same kind of abstract framework
Jordan Ellenberg (01:47.760)
that he's thinking about in terms of language
Lex Fridman (01:49.480)
that they're all really tightly interconnected?
Jordan Ellenberg (01:52.980)
That's a really interesting question.
Lex Fridman (01:54.880)
You're getting me to reflect on this question
Jordan Ellenberg (01:56.560)
of whether the feeling of producing mathematical output,
Lex Fridman (02:00.780)
if you want, is like the process of uttering language
Jordan Ellenberg (02:04.400)
or producing linguistic output.
Lex Fridman (02:07.080)
I think it feels something like that,
Lex Fridman (02:09.000)
and it's certainly the case.
Lex Fridman (02:10.640)
Let me put it this way.
Jordan Ellenberg (02:11.480)
It's hard to imagine doing mathematics
Lex Fridman (02:14.480)
in a completely nonlinguistic way.
Jordan Ellenberg (02:17.400)
It's hard to imagine doing mathematics
Lex Fridman (02:19.680)
without talking about mathematics
Lex Fridman (02:22.040)
and sort of thinking in propositions.
Lex Fridman (02:23.800)
But maybe it's just because that's the way I do mathematics,
Lex Fridman (02:26.800)
and maybe I can't imagine it any other way, right?
Lex Fridman (02:29.600)
Well, what about visualizing shapes,
Jordan Ellenberg (02:32.760)
visualizing concepts to which language
Lex Fridman (02:35.680)
is not obviously attachable?
Jordan Ellenberg (02:38.240)
Ah, that's a really interesting question.
Lex Fridman (02:40.120)
And one thing it reminds me of is one thing I talk about
Jordan Ellenberg (02:43.560)
in the book is dissection proofs,
Lex Fridman (02:45.800)
these very beautiful proofs of geometric propositions.
Jordan Ellenberg (02:48.760)
There's a very famous one by Baskara
Lex Fridman (02:50.520)
of the Pythagorean theorem, proofs which are purely visual,
Jordan Ellenberg (02:56.560)
proofs where you show that two quantities are the same
Lex Fridman (03:00.260)
by taking the same pieces and putting them together one way
Lex Fridman (03:04.360)
and making one shape and putting them together another way
Lex Fridman (03:07.400)
and making a different shape,
Lex Fridman (03:08.520)
and then observing that those two shapes
Lex Fridman (03:09.760)
must have the same area
Jordan Ellenberg (03:10.640)
because they were built out of the same pieces.
Lex Fridman (03:14.520)
There's a famous story,
Lex Fridman (03:16.120)
and it's a little bit disputed about how accurate this is,
Lex Fridman (03:19.300)
but that in Baskara's manuscript,
Jordan Ellenberg (03:20.680)
he sort of gives this proof, just gives the diagram,
Lex Fridman (03:22.620)
and then the entire verbal content of the proof
Jordan Ellenberg (03:26.160)
is he just writes under it, behold.
Lex Fridman (03:28.400)
Like that's it.
Lex Fridman (03:29.240)
And it's like, there's some dispute
Lex Fridman (03:32.280)
about exactly how accurate that is.
Lex Fridman (03:33.680)
But so then there's an interesting question.
Lex Fridman (03:36.520)
If your proof is a diagram, if your proof is a picture,
Jordan Ellenberg (03:39.920)
or even if your proof is like a movie of the same pieces
Lex Fridman (03:42.280)
like coming together in two different formations
Lex Fridman (03:43.940)
to make two different things, is that language?
Lex Fridman (03:45.760)
I'm not sure I have a good answer.
Lex Fridman (03:46.600)
What do you think?
Lex Fridman (03:47.680)
I think it is. I think the process
Jordan Ellenberg (03:51.080)
of manipulating the visual elements
Lex Fridman (03:55.000)
is the same as the process
Jordan Ellenberg (03:56.520)
of manipulating the elements of language.
Lex Fridman (03:59.480)
And I think probably the manipulating, the aggregation,
Jordan Ellenberg (04:02.960)
the stitching stuff together is the important part.
Lex Fridman (04:05.800)
It's not the actual specific elements.
Jordan Ellenberg (04:07.680)
It's more like, to me, language is a process
Lex Fridman (04:10.520)
and math is a process.
Jordan Ellenberg (04:11.780)
It's not just specific symbols.
Lex Fridman (04:15.040)
It's in action.
Jordan Ellenberg (04:19.080)
It's ultimately created through action, through change.
Lex Fridman (04:23.680)
And so you're constantly evolving ideas.
Jordan Ellenberg (04:26.200)
Of course, we kind of attach,
Lex Fridman (04:27.880)
there's a certain destination you arrive to
Jordan Ellenberg (04:29.760)
that you attach to and you call that a proof,
Lex Fridman (04:32.080)
but that's not, that doesn't need to end there.
Jordan Ellenberg (04:34.600)
It's just at the end of the chapter
Lex Fridman (04:36.000)
and then it goes on and on and on in that kind of way.
Lex Fridman (04:39.160)
But I gotta ask you about geometry
Lex Fridman (04:40.720)
and it's a prominent topic in your new book, Shape.
Lex Fridman (04:44.840)
So for me, geometry is the thing,
Lex Fridman (04:48.020)
just like as you're saying,
Jordan Ellenberg (04:49.560)
made me fall in love with mathematics when I was young.
Lex Fridman (04:53.340)
So being able to prove something visually
Jordan Ellenberg (04:56.760)
just did something to my brain that it had this,
Lex Fridman (05:01.000)
it planted this hopeful seed
Jordan Ellenberg (05:02.600)
that you can understand the world, like perfectly.
Lex Fridman (05:07.320)
Maybe it's an OCD thing,
Lex Fridman (05:08.580)
but from a mathematics perspective,
Lex Fridman (05:10.440)
like humans are messy, the world is messy, biology is messy.
Jordan Ellenberg (05:14.480)
Your parents are yelling or making you do stuff,
Lex Fridman (05:17.120)
but you can cut through all that BS
Lex Fridman (05:19.560)
and truly understand the world through mathematics
Lex Fridman (05:22.000)
and nothing like geometry did that for me.
Jordan Ellenberg (05:25.320)
For you, you did not immediately fall in love
Lex Fridman (05:28.240)
with geometry, so how do you think about geometry?
Lex Fridman (05:33.760)
Why is it a special field in mathematics?
Lex Fridman (05:36.740)
And how did you fall in love with it if you have?
Jordan Ellenberg (05:39.920)
Wow, you've given me like a lot to say.
Lex Fridman (05:41.760)
And certainly the experience that you describe
Jordan Ellenberg (05:44.320)
is so typical, but there's two versions of it.
Lex Fridman (05:48.400)
One thing I say in the book
Jordan Ellenberg (05:49.240)
is that geometry is the cilantro of math.
Lex Fridman (05:51.720)
People are not neutral about it.
Jordan Ellenberg (05:52.880)
There's people who like you are like,
Lex Fridman (05:55.560)
the rest of it I could take or leave,
Lex Fridman (05:56.840)
but then at this one moment, it made sense.
Lex Fridman (05:59.200)
This class made sense, why wasn't it all like that?
Jordan Ellenberg (06:01.400)
There's other people, I can tell you,
Lex Fridman (06:02.880)
because they come and talk to me all the time,
Jordan Ellenberg (06:04.740)
who are like, I understood all the stuff
Lex Fridman (06:06.760)
where you're trying to figure out what X was,
Jordan Ellenberg (06:08.140)
there's some mystery you're trying to solve it,
Lex Fridman (06:09.460)
X is a number, I figured it out.
Lex Fridman (06:10.680)
But then there was this geometry, like what was that?
Lex Fridman (06:12.960)
What happened that year? Like I didn't get it.
Jordan Ellenberg (06:14.360)
I was like lost the whole year
Lex Fridman (06:15.480)
and I didn't understand like why we even
Jordan Ellenberg (06:17.080)
spent the time doing that.
Lex Fridman (06:18.160)
So, but what everybody agrees on
Lex Fridman (06:20.440)
is that it's somehow different, right?
Lex Fridman (06:22.360)
There's something special about it.
Jordan Ellenberg (06:25.800)
We're gonna walk around in circles a little bit,
Lex Fridman (06:27.120)
but we'll get there.
Jordan Ellenberg (06:27.960)
You asked me how I fell in love with math.
Lex Fridman (06:32.440)
I have a story about this.
Jordan Ellenberg (06:36.800)
When I was a small child, I don't know,
Lex Fridman (06:39.040)
maybe like I was six or seven, I don't know.
Jordan Ellenberg (06:42.040)
I'm from the 70s.
Lex Fridman (06:42.880)
I think you're from a different decade than that.
Lex Fridman (06:44.440)
But in the 70s, we had a cool wooden box
Lex Fridman (06:48.560)
around your stereo.
Jordan Ellenberg (06:49.520)
That was the look, everything was dark wood.
Lex Fridman (06:51.460)
And the box had a bunch of holes in it
Jordan Ellenberg (06:53.560)
to let the sound out.
Lex Fridman (06:56.320)
And the holes were in this rectangular array,
Jordan Ellenberg (06:58.960)
a six by eight array of holes.
Lex Fridman (07:02.480)
And I was just kind of like zoning out
Jordan Ellenberg (07:04.280)
in the living room as kids do,
Lex Fridman (07:06.040)
looking at this six by eight rectangular array of holes.
Lex Fridman (07:09.800)
And if you like, just by kind of like focusing in and out,
Lex Fridman (07:12.720)
just by kind of looking at this box,
Jordan Ellenberg (07:14.480)
looking at this rectangle, I was like,
Lex Fridman (07:17.080)
well, there's six rows of eight holes each,
Lex Fridman (07:21.900)
but there's also eight columns of six holes each.
Lex Fridman (07:25.640)
Whoa.
Lex Fridman (07:26.480)
So eight sixes and six eights.
Lex Fridman (07:29.040)
It's just like the dissection proofs
Jordan Ellenberg (07:30.080)
we were just talking about, but it's the same holes.
Lex Fridman (07:32.320)
It's the same 48 holes.
Jordan Ellenberg (07:33.480)
That's how many there are,
Lex Fridman (07:34.320)
no matter whether you count them as rows
Jordan Ellenberg (07:36.800)
or count them as columns.
Lex Fridman (07:38.000)
And this was like unbelievable to me.
Lex Fridman (07:41.760)
Am I allowed to cuss on your podcast?
Lex Fridman (07:43.000)
I don't know if that's, are we FCC regulated?
Jordan Ellenberg (07:45.240)
Okay, it was fucking unbelievable.
Lex Fridman (07:47.080)
Okay, that's the last time.
Jordan Ellenberg (07:48.000)
Get it in there.
Lex Fridman (07:48.840)
This story merits it.
Lex Fridman (07:49.660)
So two different perspectives in the same physical reality.
Lex Fridman (07:54.240)
Exactly.
Lex Fridman (07:55.280)
And it's just as you say.
Lex Fridman (07:58.520)
I knew that six times eight was the same as eight times six.
Jordan Ellenberg (08:01.440)
I knew my times tables.
Lex Fridman (08:02.640)
I knew that that was a fact.
Lex Fridman (08:04.720)
But did I really know it until that moment?
Lex Fridman (08:06.600)
That's the question, right?
Jordan Ellenberg (08:08.640)
I sort of knew that the times table was symmetric,
Lex Fridman (08:11.640)
but I didn't know why that was the case until that moment.
Lex Fridman (08:13.960)
And in that moment I could see like,
Lex Fridman (08:15.160)
oh, I didn't have to have somebody tell me that.
Jordan Ellenberg (08:17.700)
That's information that you can just directly access.
Lex Fridman (08:20.200)
That's a really amazing moment.
Lex Fridman (08:21.600)
And as math teachers, that's something
Lex Fridman (08:22.920)
that we're really trying to bring to our students.
Lex Fridman (08:25.560)
And I was one of those who did not love
Lex Fridman (08:27.560)
the kind of Euclidean geometry ninth grade class
Jordan Ellenberg (08:30.540)
of like prove that an isosceles triangle
Lex Fridman (08:33.080)
has equal angles at the base, like this kind of thing.
Jordan Ellenberg (08:35.520)
It didn't vibe with me the way that algebra and numbers did.
Lex Fridman (08:39.020)
But if you go back to that moment,
Jordan Ellenberg (08:40.720)
from my adult perspective,
Lex Fridman (08:41.860)
looking back at what happened with that rectangle,
Jordan Ellenberg (08:43.620)
I think that is a very geometric moment.
Lex Fridman (08:45.460)
In fact, that moment exactly encapsulates
Jordan Ellenberg (08:49.920)
the intertwining of algebra and geometry.
Lex Fridman (08:53.060)
This algebraic fact that, well, in the instance,
Jordan Ellenberg (08:55.880)
eight times six is equal to six times eight.
Lex Fridman (08:57.880)
But in general, that whatever two numbers you have,
Jordan Ellenberg (09:00.160)
you multiply them one way.
Lex Fridman (09:01.320)
And it's the same as if you multiply them
Jordan Ellenberg (09:02.760)
in the other order.
Lex Fridman (09:03.700)
It attaches it to this geometric fact about a rectangle,
Jordan Ellenberg (09:07.500)
which in some sense makes it true.
Lex Fridman (09:09.540)
So, who knows, maybe I was always fated
Jordan Ellenberg (09:11.380)
to be an algebraic geometer,
Lex Fridman (09:12.380)
which is what I am as a researcher.
Lex Fridman (09:15.060)
So that's the kind of transformation.
Lex Fridman (09:17.340)
And you talk about symmetry in your book.
Lex Fridman (09:20.340)
What the heck is symmetry?
Lex Fridman (09:22.940)
What the heck is these kinds of transformation on objects
Lex Fridman (09:26.260)
that once you transform them, they seem to be similar?
Lex Fridman (09:29.480)
What do you make of it?
Jordan Ellenberg (09:30.500)
What's its use in mathematics
Lex Fridman (09:32.180)
or maybe broadly in understanding our world?
Jordan Ellenberg (09:35.340)
Well, it's an absolutely fundamental concept.
Lex Fridman (09:37.380)
And it starts with the word symmetry
Jordan Ellenberg (09:39.740)
in the way that we usually use it
Lex Fridman (09:41.220)
when we're just like talking English
Lex Fridman (09:42.620)
and not talking mathematics, right?
Lex Fridman (09:43.780)
Sort of something is, when we say something is symmetrical,
Jordan Ellenberg (09:46.940)
we usually means it has what's called an axis of symmetry.
Lex Fridman (09:49.820)
Maybe like the left half of it
Jordan Ellenberg (09:51.260)
looks the same as the right half.
Lex Fridman (09:52.580)
That would be like a left, right axis of symmetry.
Jordan Ellenberg (09:55.020)
Or maybe the top half looks like the bottom half or both.
Lex Fridman (09:57.820)
Maybe there's sort of a fourfold symmetry
Jordan Ellenberg (09:59.320)
where the top looks like the bottom
Lex Fridman (10:00.160)
and the left looks like the right or more.
Lex Fridman (10:03.620)
And that can take you in a lot of different directions.
Lex Fridman (10:06.380)
The abstract study of what the possible combinations
Jordan Ellenberg (10:09.480)
of symmetries there are,
Lex Fridman (10:10.480)
a subject which is called group theory
Jordan Ellenberg (10:11.820)
was actually one of my first loves in mathematics
Lex Fridman (10:14.940)
when I thought about a lot when I was in college.
Lex Fridman (10:17.600)
But the notion of symmetry is actually much more general
Lex Fridman (10:21.300)
than the things that we would call symmetry
Jordan Ellenberg (10:23.100)
if we were looking at like a classical building
Lex Fridman (10:25.480)
or a painting or something like that.
Jordan Ellenberg (10:30.300)
Nowadays in math,
Lex Fridman (10:35.520)
we could use a symmetry to refer to
Jordan Ellenberg (10:38.740)
any kind of transformation of an image
Lex Fridman (10:41.580)
or a space or an object.
Lex Fridman (10:43.900)
So what I talk about in the book is
Lex Fridman (10:48.040)
take a figure and stretch it vertically,
Jordan Ellenberg (10:50.580)
make it twice as big vertically
Lex Fridman (10:53.940)
and make it half as wide.
Jordan Ellenberg (10:58.920)
That I would call a symmetry.
Lex Fridman (11:00.100)
It's not a symmetry in the classical sense,
Lex Fridman (11:03.300)
but it's a well defined transformation
Lex Fridman (11:05.740)
that has an input and an output.
Jordan Ellenberg (11:07.620)
I give you some shape and it gets kind of,
Lex Fridman (11:10.960)
I call this in the book a scrunch.
Jordan Ellenberg (11:12.180)
I just had to make up some sort of funny sounding name
Lex Fridman (11:14.460)
for it because it doesn't really have a name.
Lex Fridman (11:20.120)
And just as you can sort of study
Lex Fridman (11:21.740)
which kinds of objects are symmetrical
Jordan Ellenberg (11:23.820)
under the operations of switching left and right
Lex Fridman (11:26.020)
or switching top and bottom
Jordan Ellenberg (11:27.540)
or rotating 40 degrees or what have you,
Lex Fridman (11:31.580)
you could study what kinds of things are preserved
Jordan Ellenberg (11:33.900)
by this kind of scrunch symmetry.
Lex Fridman (11:36.360)
And this kind of more general idea
Jordan Ellenberg (11:39.500)
of what a symmetry can be.
Lex Fridman (11:42.520)
Let me put it this way.
Jordan Ellenberg (11:44.860)
A fundamental mathematical idea,
Lex Fridman (11:47.300)
in some sense, I might even say the idea
Jordan Ellenberg (11:49.020)
that dominates contemporary mathematics.
Lex Fridman (11:51.380)
Or by contemporary, by the way,
Jordan Ellenberg (11:52.380)
I mean like the last like 150 years.
Lex Fridman (11:54.260)
We're on a very long time scale in math.
Jordan Ellenberg (11:56.220)
I don't mean like yesterday.
Lex Fridman (11:57.060)
I mean like a century or so up till now.
Jordan Ellenberg (12:00.960)
Is this idea that it's a fundamental question
Lex Fridman (12:02.580)
of when do we consider two things to be the same?
Jordan Ellenberg (12:07.140)
That might seem like a complete triviality.
Lex Fridman (12:08.740)
It's not.
Jordan Ellenberg (12:10.020)
For instance, if I have a triangle
Lex Fridman (12:13.200)
and I have a triangle of the exact same dimensions,
Lex Fridman (12:14.940)
but it's over here, are those the same or different?
Lex Fridman (12:19.340)
Well, you might say, well, look,
Jordan Ellenberg (12:20.280)
there's two different things.
Lex Fridman (12:21.120)
This one's over here, this one's over there.
Jordan Ellenberg (12:22.420)
On the other hand, if you prove a theorem about this one,
Lex Fridman (12:25.780)
it's probably still true about this one
Jordan Ellenberg (12:27.700)
if it has like all the same side lanes and angles
Lex Fridman (12:29.860)
and like looks exactly the same.
Jordan Ellenberg (12:31.460)
The term of art, if you want it,
Lex Fridman (12:32.660)
you would say they're congruent.
Lex Fridman (12:34.740)
But one way of saying it is there's a symmetry
Lex Fridman (12:36.840)
called translation, which just means
Jordan Ellenberg (12:38.620)
move everything three inches to the left.
Lex Fridman (12:40.780)
And we want all of our theories
Jordan Ellenberg (12:43.100)
to be translation invariant.
Lex Fridman (12:45.460)
What that means is that if you prove a theorem
Jordan Ellenberg (12:46.940)
about a thing that's over here,
Lex Fridman (12:48.880)
and then you move it three inches to the left,
Jordan Ellenberg (12:51.660)
it would be kind of weird if all of your theorems
Lex Fridman (12:53.460)
like didn't still work.
Lex Fridman (12:55.820)
So this question of like, what are the symmetries
Lex Fridman (12:58.620)
and which things that you want to study
Jordan Ellenberg (12:59.940)
are invariant under those symmetries
Lex Fridman (13:01.460)
is absolutely fundamental.
Lex Fridman (13:02.380)
Boy, this is getting a little abstract, right?
Lex Fridman (13:04.020)
It's not at all abstract.
Jordan Ellenberg (13:05.160)
I think this is completely central
Lex Fridman (13:08.380)
to everything I think about
Jordan Ellenberg (13:09.760)
in terms of artificial intelligence.
Lex Fridman (13:11.380)
I don't know if you know about the MNIST dataset,
Jordan Ellenberg (13:13.540)
what's handwritten digits.
Lex Fridman (13:15.140)
And you know, I don't smoke much weed or any really,
Lex Fridman (13:21.680)
but it certainly feels like it when I look at MNIST
Lex Fridman (13:24.240)
and think about this stuff, which is like,
Lex Fridman (13:26.360)
what's the difference between one and two?
Lex Fridman (13:28.600)
And why are all the twos similar to each other?
Lex Fridman (13:32.200)
What kind of transformations are within the category
Lex Fridman (13:37.200)
of what makes a thing the same?
Lex Fridman (13:39.160)
And what kind of transformations
Lex Fridman (13:40.720)
are those that make it different?
Lex Fridman (13:42.520)
And symmetries core to that.
Lex Fridman (13:44.020)
In fact, whatever the hell our brain is doing,
Jordan Ellenberg (13:46.760)
it's really good at constructing these arbitrary
Lex Fridman (13:50.440)
and sometimes novel, which is really important
Jordan Ellenberg (13:53.160)
when you look at like the IQ test or they feel novel,
Lex Fridman (13:58.000)
ideas of symmetry of like playing with objects,
Jordan Ellenberg (14:02.880)
we're able to see things that are the same and not
Lex Fridman (14:07.020)
and construct almost like little geometric theories
Jordan Ellenberg (14:11.720)
of what makes things the same and not
Lex Fridman (14:13.400)
and how to make programs do that in AI
Jordan Ellenberg (14:17.400)
is a total open question.
Lex Fridman (14:19.120)
And so I kind of stared and wonder
Jordan Ellenberg (14:22.960)
how, what kind of symmetries are enough to solve
Lex Fridman (14:27.040)
the MNIST handwritten digit recognition problem
Lex Fridman (14:30.880)
and write that down.
Lex Fridman (14:32.320)
And exactly, and what's so fascinating
Jordan Ellenberg (14:33.880)
about the work in that direction
Lex Fridman (14:35.440)
from the point of view of a mathematician like me
Lex Fridman (14:38.320)
and a geometer is that the kind of groups of symmetries,
Lex Fridman (14:42.560)
the types of symmetries that we know of are not sufficient.
Lex Fridman (14:45.960)
So in other words, like we're just gonna keep on going
Lex Fridman (14:48.680)
into the weeds on this.
Jordan Ellenberg (14:51.320)
The deeper, the better.
Lex Fridman (14:53.680)
A kind of symmetry that we understand very well
Jordan Ellenberg (14:55.320)
is rotation.
Lex Fridman (14:56.720)
So here's what would be easy.
Jordan Ellenberg (14:57.920)
If humans, if we recognize the digit as a one,
Lex Fridman (15:01.980)
if it was like literally a rotation
Jordan Ellenberg (15:03.640)
by some number of degrees or some fixed one
Lex Fridman (15:07.220)
in some typeface like Palatino or something,
Jordan Ellenberg (15:10.440)
that would be very easy to understand.
Lex Fridman (15:12.080)
It would be very easy to like write a program
Jordan Ellenberg (15:13.960)
that could detect whether something was a rotation
Lex Fridman (15:17.400)
of a fixed digit one.
Jordan Ellenberg (15:20.680)
Whatever we're doing when you recognize the digit one
Lex Fridman (15:22.640)
and distinguish it from the digit two, it's not that.
Jordan Ellenberg (15:25.920)
It's not just incorporating one of the types of symmetries
Lex Fridman (15:30.680)
that we understand.
Jordan Ellenberg (15:32.120)
Now, I would say that I would be shocked
Lex Fridman (15:36.640)
if there was some kind of classical symmetry type formulation
Jordan Ellenberg (15:40.640)
that captured what we're doing
Lex Fridman (15:43.360)
when we tell the difference between a two and a three.
Jordan Ellenberg (15:45.600)
To be honest, I think what we're doing
Lex Fridman (15:48.040)
is actually more complicated than that.
Jordan Ellenberg (15:50.240)
I feel like it must be.
Lex Fridman (15:52.280)
They're so simple, these numbers.
Jordan Ellenberg (15:53.680)
I mean, they're really geometric objects.
Lex Fridman (15:55.840)
Like we can draw out one, two, three.
Jordan Ellenberg (15:58.600)
It does seem like it should be formalizable.
Lex Fridman (16:01.160)
That's why it's so strange.
Lex Fridman (16:03.280)
Do you think it's formalizable
Lex Fridman (16:04.220)
when something stops being a two and starts being a three?
Jordan Ellenberg (16:06.840)
Right, you can imagine something continuously deforming
Lex Fridman (16:09.000)
from being a two to a three.
Jordan Ellenberg (16:11.120)
Yeah, but that's, there is a moment.
Lex Fridman (16:15.440)
Like I have myself written programs
Jordan Ellenberg (16:17.620)
that literally morph twos and threes and so on.
Lex Fridman (16:20.760)
And you watch, and there is moments that you notice
Jordan Ellenberg (16:23.960)
depending on the trajectory of that transformation,
Lex Fridman (16:26.920)
that morphing, that it is a three and a two.
Jordan Ellenberg (16:32.480)
There's a hard line.
Lex Fridman (16:33.600)
Wait, so if you ask people, if you showed them this morph,
Jordan Ellenberg (16:36.360)
if you ask a bunch of people,
Lex Fridman (16:37.360)
do they all agree about where the transition happened?
Jordan Ellenberg (16:39.800)
Because I would be surprised.
Lex Fridman (16:40.760)
I think so.
Jordan Ellenberg (16:41.600)
Oh my God, okay, we have an empirical dispute.
Lex Fridman (16:42.920)
But here's the problem.
Jordan Ellenberg (16:44.600)
Here's the problem, that if I just showed that moment
Lex Fridman (16:48.240)
that I agreed on.
Jordan Ellenberg (16:50.680)
Well, that's not fair.
Lex Fridman (16:51.720)
No, but say I said,
Lex Fridman (16:53.480)
so I want to move away from the agreement
Lex Fridman (16:55.080)
because that's a fascinating actually question
Jordan Ellenberg (16:57.320)
that I want to backtrack from because I just dogmatically
Lex Fridman (17:02.400)
said, because I could be very, very wrong.
Lex Fridman (17:04.840)
But the morphing really helps that like the change,
Lex Fridman (17:09.960)
because I mean, partially it's because our perception
Jordan Ellenberg (17:11.880)
systems, see this, it's all probably tied in there.
Lex Fridman (17:15.040)
Somehow the change from one to the other,
Jordan Ellenberg (17:18.000)
like seeing the video of it allows you to pinpoint
Lex Fridman (17:21.000)
the place where a two becomes a three much better.
Jordan Ellenberg (17:23.680)
If I just showed you one picture,
Lex Fridman (17:26.000)
I think you might really, really struggle.
Jordan Ellenberg (17:31.000)
You might call a seven.
Lex Fridman (17:32.120)
I think there's something also that we don't often
Jordan Ellenberg (17:38.280)
think about, which is it's not just about the static image,
Lex Fridman (17:41.600)
it's the transformation of the image,
Jordan Ellenberg (17:43.960)
or it's not a static shape,
Lex Fridman (17:45.560)
it's the transformation of the shape.
Jordan Ellenberg (17:47.600)
There's something in the movement that seems to be
Lex Fridman (17:51.560)
not just about our perception system,
Lex Fridman (17:53.320)
but fundamental to our cognition,
Lex Fridman (17:55.040)
like how we think about stuff.
Jordan Ellenberg (17:57.720)
Yeah, and that's part of geometry too.
Lex Fridman (18:00.360)
And in fact, again, another insight of modern geometry
Jordan Ellenberg (18:03.200)
is this idea that maybe we would naively think
Lex Fridman (18:06.040)
we're gonna study, I don't know,
Jordan Ellenberg (18:08.320)
like Poincare, we're gonna study the three body problem.
Lex Fridman (18:10.400)
We're gonna study sort of like three objects in space
Jordan Ellenberg (18:13.440)
moving around subject only to the force
Lex Fridman (18:15.240)
of each other's gravity, which sounds very simple, right?
Lex Fridman (18:17.560)
And if you don't know about this problem,
Lex Fridman (18:18.720)
you're probably like, okay, so you just like put it
Jordan Ellenberg (18:20.040)
in your computer and see what they do.
Lex Fridman (18:21.240)
Well, guess what?
Jordan Ellenberg (18:22.080)
That's like a problem that Poincare won a huge prize for
Lex Fridman (18:25.080)
like making the first real progress on in the 1880s.
Lex Fridman (18:27.360)
And we still don't know that much about it 150 years later.
Lex Fridman (18:32.200)
I mean, it's a humongous mystery.
Jordan Ellenberg (18:34.800)
You just opened the door and we're gonna walk right in
Lex Fridman (18:38.000)
before we return to symmetry.
Jordan Ellenberg (18:40.840)
What's the, who's Poincare and what's this conjecture
Lex Fridman (18:44.840)
that he came up with?
Lex Fridman (18:46.680)
Why is it such a hard problem?
Lex Fridman (18:48.560)
Okay, so Poincare, he ends up being a major figure
Jordan Ellenberg (18:52.120)
in the book and I didn't even really intend for him
Lex Fridman (18:54.200)
to be such a big figure, but he's first and foremost
Lex Fridman (18:59.280)
a geometer, right?
Lex Fridman (19:00.120)
So he's a mathematician who kind of comes up
Jordan Ellenberg (19:02.600)
in late 19th century France at a time when French math
Lex Fridman (19:07.880)
is really starting to flower.
Jordan Ellenberg (19:09.360)
Actually, I learned a lot.
Lex Fridman (19:10.200)
I mean, in math, we're not really trained
Jordan Ellenberg (19:11.640)
on our own history.
Lex Fridman (19:12.680)
We got a PhD in math, learned about math.
Lex Fridman (19:14.240)
So I learned a lot.
Lex Fridman (19:15.200)
There's this whole kind of moment where France
Jordan Ellenberg (19:18.600)
has just been beaten in the Franco Prussian war.
Lex Fridman (19:22.040)
And they're like, oh my God, what did we do wrong?
Lex Fridman (19:23.840)
And they were like, we gotta get strong in math
Lex Fridman (19:26.440)
like the Germans.
Jordan Ellenberg (19:27.280)
We have to be like more like the Germans.
Lex Fridman (19:28.440)
So this never happens to us again.
Lex Fridman (19:29.880)
So it's very much, it's like the Sputnik moment,
Lex Fridman (19:31.960)
like what happens in America in the 50s and 60s
Jordan Ellenberg (19:34.600)
with the Soviet Union.
Lex Fridman (19:35.440)
This is happening to France and they're trying
Jordan Ellenberg (19:37.280)
to kind of like instantly like modernize.
Lex Fridman (19:40.360)
That's fascinating that the humans and mathematics
Jordan Ellenberg (19:43.120)
are intricately connected to the history of humans.
Lex Fridman (19:46.800)
The Cold War is I think fundamental to the way people
Jordan Ellenberg (19:51.800)
saw science and math in the Soviet Union.
Lex Fridman (19:55.160)
I don't know if that was true in the United States,
Lex Fridman (19:56.720)
but certainly it was in the Soviet Union.
Lex Fridman (19:58.520)
It definitely was, and I would love to hear more
Jordan Ellenberg (1:00:04.380)
It's funny because I've spoken with him a couple of times,
Lex Fridman (1:00:07.540)
spoken to him a lot offline as well.
Jordan Ellenberg (1:00:09.860)
He really doesn't think he's doing anything new,
Lex Fridman (1:00:14.100)
meaning like he sees himself as very different
Jordan Ellenberg (1:00:17.380)
from maybe like a researcher,
Lex Fridman (1:00:20.460)
but it feels to me like he's creating something totally new.
Jordan Ellenberg (1:00:26.400)
Like that act of understanding and visualizing
Lex Fridman (1:00:29.340)
is as powerful or has the same kind of inkling of power
Jordan Ellenberg (1:00:33.380)
as does the process of proving something.
Lex Fridman (1:00:36.980)
It doesn't have that clear destination,
Lex Fridman (1:00:39.940)
but it's pulling out an insight
Lex Fridman (1:00:42.180)
and creating multiple sets of perspective
Jordan Ellenberg (1:00:44.900)
that arrive at that insight.
Lex Fridman (1:00:46.980)
And to be honest, it's something that I think
Jordan Ellenberg (1:00:49.220)
we haven't quite figured out how to value
Lex Fridman (1:00:53.340)
inside academic mathematics in the same way,
Lex Fridman (1:00:55.380)
and this is a bit older,
Lex Fridman (1:00:56.220)
that I think we haven't quite figured out
Lex Fridman (1:00:57.780)
how to value the development
Lex Fridman (1:00:59.460)
of computational infrastructure.
Jordan Ellenberg (1:01:01.020)
We all have computers as our partners now
Lex Fridman (1:01:02.900)
and people build computers that sort of assist
Lex Fridman (1:01:07.860)
and participate in our mathematics.
Lex Fridman (1:01:09.300)
They build those systems
Lex Fridman (1:01:10.480)
and that's a kind of mathematics too,
Lex Fridman (1:01:12.600)
but not in the traditional form
Jordan Ellenberg (1:01:14.020)
of proving theorems and writing papers.
Lex Fridman (1:01:16.420)
But I think it's coming.
Jordan Ellenberg (1:01:17.260)
Look, I mean, I think, for example,
Lex Fridman (1:01:20.380)
the Institute for Computational Experimental Mathematics
Jordan Ellenberg (1:01:23.520)
at Brown, which is like, it's a NSF funded math institute,
Lex Fridman (1:01:27.820)
very much part of sort of traditional math academia.
Jordan Ellenberg (1:01:29.860)
They did an entire theme semester
Lex Fridman (1:01:31.720)
about visualizing mathematics,
Jordan Ellenberg (1:01:33.180)
looking at the same kind of thing that they would do
Lex Fridman (1:01:34.620)
for like an up and coming research topic.
Jordan Ellenberg (1:01:37.780)
Like that's pretty cool.
Lex Fridman (1:01:38.620)
So I think there really is buy in
Jordan Ellenberg (1:01:40.260)
from the mathematics community
Lex Fridman (1:01:43.380)
to recognize that this kind of stuff is important
Lex Fridman (1:01:45.420)
and counts as part of mathematics,
Lex Fridman (1:01:47.540)
like part of what we're actually here to do.
Jordan Ellenberg (1:01:50.560)
Yeah, I'm hoping to see more and more of that
Lex Fridman (1:01:52.020)
from like MIT faculty, from faculty,
Jordan Ellenberg (1:01:54.420)
from all the top universities in the world.
Lex Fridman (1:01:57.540)
Let me ask you this weird question about the Fields Medal,
Jordan Ellenberg (1:02:00.020)
which is the Nobel Prize in Mathematics.
Lex Fridman (1:02:02.960)
Do you think, since we're talking about computers,
Jordan Ellenberg (1:02:05.540)
there will one day come a time when a computer,
Lex Fridman (1:02:11.860)
an AI system will win a Fields Medal?
Jordan Ellenberg (1:02:16.060)
No.
Lex Fridman (1:02:16.900)
Of course, that's what a human would say.
Lex Fridman (1:02:19.740)
Why not?
Lex Fridman (1:02:20.580)
Is that like, that's like my captcha?
Lex Fridman (1:02:23.380)
That's like the proof that I'm a human?
Lex Fridman (1:02:24.500)
Is that like the lie that I know?
Jordan Ellenberg (1:02:25.340)
Yeah.
Lex Fridman (1:02:26.820)
What is, how does he want me to answer?
Lex Fridman (1:02:28.940)
Is there something interesting to be said about that?
Lex Fridman (1:02:31.980)
Yeah, I mean, I am tremendously interested
Jordan Ellenberg (1:02:34.620)
in what AI can do in pure mathematics.
Lex Fridman (1:02:37.820)
I mean, it's, of course, it's a parochial interest, right?
Jordan Ellenberg (1:02:40.500)
You're like, why am I interested in like,
Lex Fridman (1:02:41.700)
how it can like help feed the world
Lex Fridman (1:02:43.140)
or help solve like real social problems?
Lex Fridman (1:02:44.700)
I'm like, can it do more math?
Lex Fridman (1:02:46.140)
Like, what can I do?
Lex Fridman (1:02:47.540)
We all have our interests, right?
Lex Fridman (1:02:49.580)
But I think it is a really interesting conceptual question.
Lex Fridman (1:02:53.700)
And here too, I think it's important to be kind of historical
Jordan Ellenberg (1:02:59.820)
because it's certainly true that there's lots of things
Lex Fridman (1:03:02.260)
that we used to call research in mathematics
Jordan Ellenberg (1:03:04.940)
that we would now call computation.
Lex Fridman (1:03:07.380)
Tasks that we've now offloaded to machines.
Jordan Ellenberg (1:03:09.620)
Like, you know, in 1890, somebody could be like,
Lex Fridman (1:03:12.580)
here's my PhD thesis.
Jordan Ellenberg (1:03:13.780)
I computed all the invariants of this polynomial ring
Lex Fridman (1:03:18.180)
under the action of some finite group.
Jordan Ellenberg (1:03:19.900)
Doesn't matter what those words mean,
Lex Fridman (1:03:21.380)
just it's like some thing that in 1890
Jordan Ellenberg (1:03:24.020)
would take a person a year to do
Lex Fridman (1:03:26.060)
and would be a valuable thing that you might wanna know.
Lex Fridman (1:03:28.060)
And it's still a valuable thing that you might wanna know,
Lex Fridman (1:03:29.940)
but now you type a few lines of code
Jordan Ellenberg (1:03:32.780)
in Macaulay or Sage or Magma and you just have it.
Lex Fridman (1:03:37.740)
So we don't think of that as math anymore,
Jordan Ellenberg (1:03:40.260)
even though it's the same thing.
Lex Fridman (1:03:41.700)
What's Macaulay, Sage and Magma?
Jordan Ellenberg (1:03:43.420)
Oh, those are computer algebra programs.
Lex Fridman (1:03:45.060)
So those are like sort of bespoke systems
Jordan Ellenberg (1:03:46.900)
that lots of mathematicians use.
Lex Fridman (1:03:48.140)
That's similar to Maple and...
Jordan Ellenberg (1:03:49.620)
Yeah, oh yeah, so it's similar to Maple and Mathematica,
Lex Fridman (1:03:51.580)
yeah, but a little more specialized, but yeah.
Jordan Ellenberg (1:03:54.700)
It's programs that work with symbols
Lex Fridman (1:03:56.620)
and allow you to do, can you do proofs?
Lex Fridman (1:03:58.180)
Can you do kind of little leaps and proofs?
Lex Fridman (1:04:01.060)
They're not really built for that.
Lex Fridman (1:04:02.420)
And that's a whole other story.
Lex Fridman (1:04:04.780)
But these tools are part of the process of mathematics now.
Jordan Ellenberg (1:04:07.300)
Right, they are now for most mathematicians, I would say,
Lex Fridman (1:04:09.940)
part of the process of mathematics.
Lex Fridman (1:04:11.620)
And so, you know, there's a story I tell in the book,
Lex Fridman (1:04:14.740)
which I'm fascinated by, which is, you know,
Lex Fridman (1:04:17.740)
so far, attempts to get AIs
Lex Fridman (1:04:22.540)
to prove interesting theorems have not done so well.
Jordan Ellenberg (1:04:27.300)
It doesn't mean they can.
Lex Fridman (1:04:28.140)
There's actually a paper I just saw,
Jordan Ellenberg (1:04:29.740)
which has a very nice use of a neural net
Lex Fridman (1:04:32.460)
to find counter examples to conjecture.
Jordan Ellenberg (1:04:34.620)
Somebody said like, well, maybe this is always that.
Lex Fridman (1:04:37.220)
And you can be like, well, let me sort of train an AI
Jordan Ellenberg (1:04:39.300)
to sort of try to find things where that's not true.
Lex Fridman (1:04:43.180)
And it actually succeeded.
Jordan Ellenberg (1:04:44.020)
Now, in this case, if you look at the things that it found,
Lex Fridman (1:04:48.180)
you say like, okay, I mean, these are not famous conjectures.
Jordan Ellenberg (1:04:53.180)
Okay, so like somebody wrote this down, maybe this is so.
Lex Fridman (1:04:58.220)
Looking at what the AI came up with, you're like,
Jordan Ellenberg (1:05:00.900)
you know, I bet if like five grad students
Lex Fridman (1:05:03.700)
had thought about that problem,
Jordan Ellenberg (1:05:04.660)
they wouldn't have come up with that.
Lex Fridman (1:05:05.500)
I mean, when you see it, you're like,
Jordan Ellenberg (1:05:06.820)
okay, that is one of the things you might try
Lex Fridman (1:05:08.380)
if you sort of like put some work into it.
Jordan Ellenberg (1:05:10.500)
Still, it's pretty awesome.
Lex Fridman (1:05:12.620)
But the story I tell in the book, which I'm fascinated by
Jordan Ellenberg (1:05:15.740)
is there is, okay, we're gonna go back to knots.
Lex Fridman (1:05:21.940)
There's a knot called the Conway knot.
Jordan Ellenberg (1:05:23.940)
After John Conway, maybe we'll talk about
Lex Fridman (1:05:25.380)
a very interesting character also.
Jordan Ellenberg (1:05:26.420)
Yeah, it's a small tangent.
Lex Fridman (1:05:28.140)
Somebody I was supposed to talk to
Lex Fridman (1:05:29.420)
and unfortunately he passed away
Lex Fridman (1:05:30.780)
and he's somebody I find as an incredible mathematician,
Jordan Ellenberg (1:05:35.220)
incredible human being.
Lex Fridman (1:05:36.220)
Oh, and I am sorry that you didn't get a chance
Jordan Ellenberg (1:05:38.300)
because having had the chance to talk to him a lot
Lex Fridman (1:05:40.300)
when I was a postdoc, yeah, you missed out.
Jordan Ellenberg (1:05:44.140)
There's no way to sugarcoat it.
Lex Fridman (1:05:45.140)
I'm sorry that you didn't get that chance.
Jordan Ellenberg (1:05:46.620)
Yeah, it is what it is.
Lex Fridman (1:05:47.900)
So knots.
Jordan Ellenberg (1:05:50.100)
Yeah, so there was a question and again,
Lex Fridman (1:05:52.580)
it doesn't matter the technicalities of the question,
Lex Fridman (1:05:54.380)
but it's a question of whether the knot is slice.
Lex Fridman (1:05:56.300)
It has to do with something about what kinds
Jordan Ellenberg (1:05:59.980)
of three dimensional surfaces and four dimensions
Lex Fridman (1:06:02.340)
can be bounded by this knot.
Lex Fridman (1:06:03.460)
But nevermind what it means, it's some question.
Lex Fridman (1:06:06.180)
And it's actually very hard to compute
Jordan Ellenberg (1:06:08.300)
whether a knot is slice or not.
Lex Fridman (1:06:12.860)
And in particular, the question of the Conway knot,
Jordan Ellenberg (1:06:16.940)
whether it was slice or not, was particularly vexed
Lex Fridman (1:06:23.060)
until it was solved just a few years ago
Jordan Ellenberg (1:06:24.620)
by Lisa Piccarillo, who actually,
Lex Fridman (1:06:26.260)
now that I think of it, was here in Austin.
Jordan Ellenberg (1:06:27.700)
I believe she was a grad student at UT Austin at the time.
Lex Fridman (1:06:29.980)
I didn't even realize there was an Austin connection
Jordan Ellenberg (1:06:31.540)
to this story until I started telling it.
Lex Fridman (1:06:34.100)
In fact, I think she's now at MIT,
Lex Fridman (1:06:35.780)
so she's basically following you around.
Lex Fridman (1:06:38.140)
If I remember correctly.
Jordan Ellenberg (1:06:38.980)
The reverse.
Lex Fridman (1:06:39.820)
There's a lot of really interesting richness to this story.
Jordan Ellenberg (1:06:42.700)
One thing about it is her paper was rather,
Lex Fridman (1:06:45.620)
was very short, it was very short and simple.
Jordan Ellenberg (1:06:48.140)
Nine pages of which two were pictures.
Lex Fridman (1:06:51.660)
Very short for like a paper solving a major conjecture.
Lex Fridman (1:06:54.620)
And it really makes you think about what we mean
Lex Fridman (1:06:55.900)
by difficulty in mathematics.
Jordan Ellenberg (1:06:57.300)
Like, do you say, oh, actually the problem wasn't difficult
Lex Fridman (1:06:59.460)
because you could solve it so simply?
Jordan Ellenberg (1:07:00.860)
Or do you say like, well, no, evidently it was difficult
Lex Fridman (1:07:03.300)
because like the world's top topologists,
Jordan Ellenberg (1:07:05.180)
many, you know, worked on it for 20 years
Lex Fridman (1:07:06.580)
and nobody could solve it, so therefore it is difficult.
Jordan Ellenberg (1:07:08.540)
Or is it that we need sort of some new category
Lex Fridman (1:07:10.460)
of things that about which it's difficult
Lex Fridman (1:07:12.820)
to figure out that they're not difficult?
Lex Fridman (1:07:15.700)
I mean, this is the computer science formulation,
Lex Fridman (1:07:18.660)
but the sort of the journey to arrive
Lex Fridman (1:07:22.740)
at the simple answer may be difficult,
Lex Fridman (1:07:24.700)
but once you have the answer, it will then appear simple.
Lex Fridman (1:07:28.700)
And I mean, there might be a large category.
Jordan Ellenberg (1:07:30.620)
I hope there's a large set of such solutions,
Lex Fridman (1:07:37.380)
because, you know, once we stand
Jordan Ellenberg (1:07:41.460)
at the end of the scientific process
Lex Fridman (1:07:43.380)
that we're at the very beginning of,
Jordan Ellenberg (1:07:46.100)
or at least it feels like,
Lex Fridman (1:07:47.540)
I hope there's just simple answers to everything
Jordan Ellenberg (1:07:50.140)
that we'll look and it'll be simple laws
Lex Fridman (1:07:53.580)
that govern the universe,
Jordan Ellenberg (1:07:55.100)
simple explanation of what is consciousness,
Lex Fridman (1:07:58.020)
what is love, is mortality fundamental to life,
Jordan Ellenberg (1:08:02.340)
what's the meaning of life, are humans special
Lex Fridman (1:08:07.100)
or we're just another sort of reflection
Jordan Ellenberg (1:08:09.100)
of all that is beautiful in the universe
Lex Fridman (1:08:13.740)
in terms of like life forms, all of it is life
Lex Fridman (1:08:16.180)
and just has different,
Lex Fridman (1:08:18.380)
when taken from a different perspective
Jordan Ellenberg (1:08:19.900)
is all life can seem more valuable or not,
Lex Fridman (1:08:22.460)
but really it's all part of the same thing.
Jordan Ellenberg (1:08:24.180)
All those will have a nice, like two equations,
Lex Fridman (1:08:26.500)
maybe one equation, but.
Lex Fridman (1:08:28.100)
Why do you think you want those questions
Lex Fridman (1:08:30.820)
to have simple answers?
Jordan Ellenberg (1:08:32.740)
I think just like symmetry
Lex Fridman (1:08:35.700)
and the breaking of symmetry is beautiful somehow.
Jordan Ellenberg (1:08:39.020)
There's something beautiful about simplicity.
Lex Fridman (1:08:41.420)
I think it, what is that?
Lex Fridman (1:08:42.260)
So it's aesthetic.
Lex Fridman (1:08:43.420)
It's aesthetic, yeah.
Jordan Ellenberg (1:08:45.140)
Or, but it's aesthetic in the way
Lex Fridman (1:08:47.100)
that happiness is an aesthetic.
Jordan Ellenberg (1:08:49.660)
Like, why is that so joyful
Lex Fridman (1:08:53.660)
that a simple explanation that governs
Lex Fridman (1:08:57.700)
a large number of cases is really appealing?
Lex Fridman (1:09:01.940)
Even when it's not, like obviously we get
Jordan Ellenberg (1:09:05.820)
a huge amount of trouble with that
Lex Fridman (1:09:07.340)
because oftentimes it doesn't have to be connected
Jordan Ellenberg (1:09:11.460)
with reality or even that explanation
Lex Fridman (1:09:13.500)
could be exceptionally harmful.
Jordan Ellenberg (1:09:15.580)
Most of like the world's history that has,
Lex Fridman (1:09:18.860)
that was governed by hate and violence
Jordan Ellenberg (1:09:21.020)
had a very simple explanation at the core
Lex Fridman (1:09:23.980)
that was used to cause the violence and the hatred.
Lex Fridman (1:09:26.980)
So like we get into trouble with that,
Lex Fridman (1:09:28.780)
but why is that so appealing?
Lex Fridman (1:09:30.420)
And in this nice forms in mathematics,
Lex Fridman (1:09:33.820)
like you look at the Einstein papers,
Lex Fridman (1:09:36.580)
why are those so beautiful?
Lex Fridman (1:09:38.020)
And why is the Andrew Wiles proof
Lex Fridman (1:09:40.180)
of the Fermat's last theorem not quite so beautiful?
Lex Fridman (1:09:43.220)
Like what's beautiful about that story
Jordan Ellenberg (1:09:45.620)
is the human struggle of like the human story
Lex Fridman (1:09:48.900)
of perseverance, of the drama,
Jordan Ellenberg (1:09:51.700)
of not knowing if the proof is correct
Lex Fridman (1:09:53.900)
and ups and downs and all of those kinds of things.
Jordan Ellenberg (1:09:56.060)
That's the interesting part.
Lex Fridman (1:09:57.220)
But the fact that the proof is huge
Lex Fridman (1:09:58.660)
and nobody understands, well,
Lex Fridman (1:10:00.020)
from my outsider's perspective,
Jordan Ellenberg (1:10:01.220)
nobody understands what the heck it is,
Lex Fridman (1:10:04.620)
is not as beautiful as it could have been.
Jordan Ellenberg (1:10:06.700)
I wish it was what Fermat originally said,
Lex Fridman (1:10:09.220)
which is, you know, it's not,
Jordan Ellenberg (1:10:13.740)
it's not small enough to fit in the margins of this page,
Lex Fridman (1:10:17.220)
but maybe if he had like a full page
Jordan Ellenberg (1:10:19.300)
or maybe a couple of post it notes,
Lex Fridman (1:10:20.820)
he would have enough to do the proof.
Lex Fridman (1:10:22.940)
What do you make of,
Lex Fridman (1:10:23.860)
if we could take another of a multitude of tangents,
Lex Fridman (1:10:27.740)
what do you make of Fermat's last theorem?
Lex Fridman (1:10:29.260)
Because the statement, there's a few theorems,
Jordan Ellenberg (1:10:31.660)
there's a few problems that are deemed by the world
Lex Fridman (1:10:35.540)
throughout its history to be exceptionally difficult.
Lex Fridman (1:10:37.780)
And that one in particular is really simple to formulate
Lex Fridman (1:10:42.380)
and really hard to come up with a proof for.
Lex Fridman (1:10:46.260)
And it was like taunted as simple by Fermat himself.
Lex Fridman (1:10:51.340)
Is there something interesting to be said about
Jordan Ellenberg (1:10:53.900)
that X to the N plus Y to the N equals Z to the N
Lex Fridman (1:10:57.700)
for N of three or greater, is there a solution to this?
Lex Fridman (1:11:02.540)
And then how do you go about proving that?
Lex Fridman (1:11:04.300)
Like, how would you try to prove that?
Lex Fridman (1:11:08.180)
And what do you learn from the proof
Lex Fridman (1:11:09.980)
that eventually emerged by Andrew Wiles?
Jordan Ellenberg (1:11:12.100)
Yeah, so right, so to give,
Lex Fridman (1:11:13.460)
let me just say the background,
Jordan Ellenberg (1:11:14.380)
because I don't know if everybody listening knows the story.
Lex Fridman (1:11:17.020)
So, you know, Fermat was an early number theorist,
Jordan Ellenberg (1:11:21.940)
at least sort of an early mathematician,
Lex Fridman (1:11:23.100)
those special adjacent didn't really exist back then.
Jordan Ellenberg (1:11:27.340)
He comes up in the book actually,
Lex Fridman (1:11:28.660)
in the context of a different theorem of his
Jordan Ellenberg (1:11:31.460)
that has to do with testing,
Lex Fridman (1:11:32.620)
whether a number is prime or not.
Lex Fridman (1:11:34.620)
So I write about, he was one of the ones who was salty
Lex Fridman (1:11:37.380)
and like, he would exchange these letters
Jordan Ellenberg (1:11:39.460)
where he and his correspondents would like
Lex Fridman (1:11:41.100)
try to top each other and vex each other with questions
Lex Fridman (1:11:44.060)
and stuff like this.
Lex Fridman (1:11:44.900)
But this particular thing,
Jordan Ellenberg (1:11:47.900)
it's called Fermat's Last Theorem because it's a note
Lex Fridman (1:11:50.780)
he wrote in his copy of the Disquisitiones Arithmetic I.
Jordan Ellenberg (1:11:57.340)
Like he wrote, here's an equation, it has no solutions.
Lex Fridman (1:12:00.820)
I can prove it, but the proof's like a little too long
Jordan Ellenberg (1:12:03.300)
to fit in the margin of this book.
Lex Fridman (1:12:05.500)
He was just like writing a note to himself.
Jordan Ellenberg (1:12:07.060)
Now, let me just say historically,
Lex Fridman (1:12:08.540)
we know that Fermat did not have a proof of this theorem.
Jordan Ellenberg (1:12:11.540)
For a long time, people were like this mysterious proof
Lex Fridman (1:12:15.420)
that was lost, a very romantic story, right?
Lex Fridman (1:12:17.220)
But a fair amount later,
Lex Fridman (1:12:21.300)
he did prove special cases of this theorem
Lex Fridman (1:12:24.260)
and wrote about it, talked to people about the problem.
Lex Fridman (1:12:27.300)
It's very clear from the way that he wrote
Jordan Ellenberg (1:12:29.060)
where he can solve certain examples
Lex Fridman (1:12:30.700)
of this type of equation
Jordan Ellenberg (1:12:32.100)
that he did not know how to do the whole thing.
Lex Fridman (1:12:35.700)
He may have had a deep, simple intuition
Jordan Ellenberg (1:12:39.860)
about how to solve the whole thing
Lex Fridman (1:12:41.740)
that he had at that moment
Jordan Ellenberg (1:12:43.780)
without ever being able to come up with a complete proof.
Lex Fridman (1:12:47.020)
And that intuition maybe lost the time.
Jordan Ellenberg (1:12:50.420)
Maybe, but you're right, that is unknowable.
Lex Fridman (1:12:54.500)
But I think what we can know is that later,
Jordan Ellenberg (1:12:56.940)
he certainly did not think that he had a proof
Lex Fridman (1:12:59.100)
that he was concealing from people.
Jordan Ellenberg (1:13:00.620)
He thought he didn't know how to prove it,
Lex Fridman (1:13:04.340)
and I also think he didn't know how to prove it.
Jordan Ellenberg (1:13:06.380)
Now, I understand the appeal of saying like,
Lex Fridman (1:13:10.180)
wouldn't it be cool if this very simple equation
Jordan Ellenberg (1:13:12.500)
there was like a very simple, clever, wonderful proof
Lex Fridman (1:13:16.020)
that you could do in a page or two.
Lex Fridman (1:13:17.340)
And that would be great, but you know what?
Lex Fridman (1:13:18.980)
There's lots of equations like that
Jordan Ellenberg (1:13:20.340)
that are solved by very clever methods like that,
Lex Fridman (1:13:22.180)
including the special cases that Fermat wrote about,
Jordan Ellenberg (1:13:24.340)
the method of descent,
Lex Fridman (1:13:25.180)
which is like very wonderful and important.
Lex Fridman (1:13:26.860)
But in the end, those are nice things
Lex Fridman (1:13:31.700)
that like you teach in an undergraduate class,
Lex Fridman (1:13:34.780)
and it is what it is,
Lex Fridman (1:13:35.860)
but they're not big.
Jordan Ellenberg (1:13:38.660)
On the other hand, work on the Fermat problem,
Lex Fridman (1:13:41.580)
that's what we like to call it
Jordan Ellenberg (1:13:42.420)
because it's not really his theorem
Lex Fridman (1:13:44.100)
because we don't think he proved it.
Jordan Ellenberg (1:13:45.220)
So, I mean, work on the Fermat problem
Lex Fridman (1:13:49.180)
developed this like incredible richness of number theory
Jordan Ellenberg (1:13:52.340)
that we now live in today.
Lex Fridman (1:13:54.780)
Like, and not, by the way,
Jordan Ellenberg (1:13:56.060)
just Wiles, Andrew Wiles being the person
Lex Fridman (1:13:58.660)
who, together with Richard Taylor,
Jordan Ellenberg (1:13:59.660)
finally proved this theorem.
Lex Fridman (1:14:01.700)
But you know how you have this whole moment
Jordan Ellenberg (1:14:03.220)
that people try to prove this theorem
Lex Fridman (1:14:05.380)
and they fail,
Lex Fridman (1:14:06.540)
and there's a famous false proof by LeMay
Lex Fridman (1:14:08.780)
from the 19th century,
Jordan Ellenberg (1:14:10.420)
where Kummer, in understanding what mistake LeMay had made
Lex Fridman (1:14:14.460)
in this incorrect proof,
Jordan Ellenberg (1:14:16.300)
basically understands something incredible,
Lex Fridman (1:14:18.340)
which is that a thing we know about numbers
Jordan Ellenberg (1:14:20.940)
is that you can factor them
Lex Fridman (1:14:24.500)
and you can factor them uniquely.
Jordan Ellenberg (1:14:26.940)
There's only one way to break a number up into primes.
Lex Fridman (1:14:30.300)
Like if we think of a number like 12,
Jordan Ellenberg (1:14:32.260)
12 is two times three times two.
Lex Fridman (1:14:35.500)
I had to think about it.
Jordan Ellenberg (1:14:38.420)
Or it's two times two times three,
Lex Fridman (1:14:39.700)
of course you can reorder them.
Lex Fridman (1:14:41.580)
But there's no other way to do it.
Lex Fridman (1:14:43.220)
There's no universe in which 12 is something times five,
Jordan Ellenberg (1:14:46.140)
or in which there's like four threes in it.
Lex Fridman (1:14:47.500)
Nope, 12 is like two twos and a three.
Jordan Ellenberg (1:14:49.140)
Like that is what it is.
Lex Fridman (1:14:50.700)
And that's such a fundamental feature of arithmetic
Jordan Ellenberg (1:14:54.820)
that we almost think of it like God's law.
Lex Fridman (1:14:56.540)
You know what I mean?
Jordan Ellenberg (1:14:57.380)
It has to be that way.
Lex Fridman (1:14:58.220)
That's a really powerful idea.
Jordan Ellenberg (1:15:00.020)
It's so cool that every number
Lex Fridman (1:15:02.540)
is uniquely made up of other numbers.
Lex Fridman (1:15:05.620)
And like made up meaning like there's these like basic atoms
Lex Fridman (1:15:10.900)
that form molecules that get built on top of each other.
Jordan Ellenberg (1:15:15.380)
I love it.
Lex Fridman (1:15:16.220)
I mean, when I teach undergraduate number theory,
Jordan Ellenberg (1:15:18.060)
it's like, it's the first really deep theorem
Lex Fridman (1:15:22.180)
that you prove.
Jordan Ellenberg (1:15:23.540)
What's amazing is the fact
Lex Fridman (1:15:25.300)
that you can factor a number into primes is much easier.
Jordan Ellenberg (1:15:28.980)
Essentially Euclid knew it,
Lex Fridman (1:15:30.340)
although he didn't quite put it in that way.
Jordan Ellenberg (1:15:33.700)
The fact that you can do it at all.
Lex Fridman (1:15:34.860)
What's deep is the fact that there's only one way to do it
Jordan Ellenberg (1:15:38.820)
or however you sort of chop the number up,
Lex Fridman (1:15:40.620)
you end up with the same set of prime factors.
Lex Fridman (1:15:44.820)
And indeed what people finally understood
Lex Fridman (1:15:49.300)
at the end of the 19th century is that
Jordan Ellenberg (1:15:51.900)
if you work in number systems slightly more general
Lex Fridman (1:15:54.620)
than the ones we're used to,
Jordan Ellenberg (1:15:56.100)
which it turns out are relevant to Fermat,
Lex Fridman (1:16:01.220)
all of a sudden this stops being true.
Jordan Ellenberg (1:16:04.320)
Things get, I mean, things get more complicated
Lex Fridman (1:16:07.980)
and now because you were praising simplicity before
Jordan Ellenberg (1:16:10.060)
you were like, it's so beautiful, unique factorization.
Lex Fridman (1:16:12.700)
It's so great.
Jordan Ellenberg (1:16:13.740)
Like, so when I tell you
Lex Fridman (1:16:14.900)
that in more general number systems,
Jordan Ellenberg (1:16:16.700)
there is no unique factorization.
Lex Fridman (1:16:18.380)
Maybe you're like, that's bad.
Jordan Ellenberg (1:16:19.360)
I'm like, no, that's good
Lex Fridman (1:16:20.260)
because there's like a whole new world of phenomena
Jordan Ellenberg (1:16:22.580)
to study that you just can't see
Lex Fridman (1:16:24.380)
through the lens of the numbers that we're used to.
Lex Fridman (1:16:26.980)
So I'm for complication.
Lex Fridman (1:16:29.940)
I'm highly in favor of complication
Jordan Ellenberg (1:16:32.380)
because every complication is like an opportunity
Lex Fridman (1:16:34.620)
for new things to study.
Lex Fridman (1:16:35.900)
And is that the big kind of one of the big insights
Lex Fridman (1:16:40.180)
for you from Andrew Wiles's proof?
Jordan Ellenberg (1:16:42.900)
Is there interesting insights about the process
Lex Fridman (1:16:46.300)
that you used to prove that sort of resonates
Lex Fridman (1:16:49.580)
with you as a mathematician?
Lex Fridman (1:16:51.380)
Is there an interesting concept that emerged from it?
Lex Fridman (1:16:54.420)
Is there interesting human aspects to the proof?
Lex Fridman (1:16:57.980)
Whether there's interesting human aspects
Jordan Ellenberg (1:16:59.860)
to the proof itself is an interesting question.
Lex Fridman (1:17:02.640)
Certainly it has a huge amount of richness.
Jordan Ellenberg (1:17:05.520)
Sort of at its heart is an argument
Lex Fridman (1:17:07.680)
of what's called deformation theory,
Jordan Ellenberg (1:17:12.420)
which was in part created by my PhD advisor, Barry Mazer.
Lex Fridman (1:17:18.200)
Can you speak to what deformation theory is?
Jordan Ellenberg (1:17:20.180)
I can speak to what it's like.
Lex Fridman (1:17:21.940)
How about that?
Lex Fridman (1:17:22.940)
What does it rhyme with?
Lex Fridman (1:17:24.660)
Right, well, the reason that Barry called it
Jordan Ellenberg (1:17:27.340)
deformation theory, I think he's the one
Lex Fridman (1:17:29.460)
who gave it the name.
Jordan Ellenberg (1:17:30.860)
I hope I'm not wrong in saying it's a name.
Lex Fridman (1:17:32.340)
In your book, you have calling different things
Jordan Ellenberg (1:17:35.140)
by the same name as one of the things
Lex Fridman (1:17:37.860)
in the beautiful map that opens the book.
Jordan Ellenberg (1:17:40.380)
Yes, and this is a perfect example.
Lex Fridman (1:17:42.040)
So this is another phrase of Poincare,
Jordan Ellenberg (1:17:44.100)
this like incredible generator of slogans and aphorisms.
Lex Fridman (1:17:46.780)
He said, mathematics is the art
Jordan Ellenberg (1:17:47.900)
of calling different things by the same name.
Lex Fridman (1:17:49.860)
That very thing we do, right?
Jordan Ellenberg (1:17:52.500)
When we're like this triangle and this triangle,
Lex Fridman (1:17:53.980)
come on, they're the same triangle,
Lex Fridman (1:17:55.040)
they're just in a different place, right?
Lex Fridman (1:17:56.500)
So in the same way, it came to be understood
Jordan Ellenberg (1:18:00.420)
that the kinds of objects that you study
Lex Fridman (1:18:06.900)
when you study Fermat's Last Theorem,
Lex Fridman (1:18:10.180)
and let's not even be too careful
Lex Fridman (1:18:12.100)
about what these objects are.
Jordan Ellenberg (1:18:13.700)
I can tell you there are gaol representations
Lex Fridman (1:18:15.780)
in modular forms, but saying those words
Jordan Ellenberg (1:18:18.420)
is not gonna mean so much.
Lex Fridman (1:18:19.700)
But whatever they are, they're things that can be deformed,
Jordan Ellenberg (1:18:23.740)
moved around a little bit.
Lex Fridman (1:18:25.940)
And I think the insight of what Andrew
Lex Fridman (1:18:28.460)
and then Andrew and Richard were able to do
Lex Fridman (1:18:31.300)
was to say something like this.
Jordan Ellenberg (1:18:33.700)
A deformation means moving something just a tiny bit,
Lex Fridman (1:18:36.680)
like an infinitesimal amount.
Jordan Ellenberg (1:18:39.380)
If you really are good at understanding
Lex Fridman (1:18:41.460)
which ways a thing can move in a tiny, tiny, tiny,
Jordan Ellenberg (1:18:44.700)
infinitesimal amount in certain directions,
Lex Fridman (1:18:46.760)
maybe you can piece that information together
Jordan Ellenberg (1:18:49.260)
to understand the whole global space in which it can move.
Lex Fridman (1:18:52.500)
And essentially, their argument comes down
Jordan Ellenberg (1:18:54.420)
to showing that two of those big global spaces
Lex Fridman (1:18:57.320)
are actually the same, the fabled R equals T,
Jordan Ellenberg (1:19:00.060)
part of their proof, which is at the heart of it.
Lex Fridman (1:19:05.220)
And it involves this very careful principle like that.
Lex Fridman (1:19:09.540)
But that being said, what I just said,
Lex Fridman (1:19:12.900)
it's probably not what you're thinking,
Jordan Ellenberg (1:19:14.620)
because what you're thinking when you think,
Lex Fridman (1:19:16.300)
oh, I have a point in space and I move it around
Jordan Ellenberg (1:19:18.500)
like a little tiny bit,
Lex Fridman (1:19:22.220)
you're using your notion of distance
Jordan Ellenberg (1:19:26.720)
that's from calculus.
Lex Fridman (1:19:28.300)
We know what it means for like two points
Jordan Ellenberg (1:19:29.540)
on the real line to be close together.
Lex Fridman (1:19:32.960)
So yet another thing that comes up in the book a lot
Jordan Ellenberg (1:19:37.080)
is this fact that the notion of distance
Lex Fridman (1:19:41.180)
is not given to us by God.
Jordan Ellenberg (1:19:42.620)
We could mean a lot of different things by distance.
Lex Fridman (1:19:44.620)
And just in the English language, we do that all the time.
Jordan Ellenberg (1:19:46.500)
We talk about somebody being a close relative.
Lex Fridman (1:19:49.020)
It doesn't mean they live next door to you, right?
Jordan Ellenberg (1:19:51.060)
It means something else.
Lex Fridman (1:19:52.780)
There's a different notion of distance we have in mind.
Lex Fridman (1:19:54.840)
And there are lots of notions of distances
Lex Fridman (1:19:57.500)
that you could use.
Jordan Ellenberg (1:19:58.820)
In the natural language processing community and AI,
Lex Fridman (1:20:01.540)
there might be some notion of semantic distance
Jordan Ellenberg (1:20:04.120)
or lexical distance between two words.
Lex Fridman (1:20:06.340)
How much do they tend to arise in the same context?
Jordan Ellenberg (1:20:08.740)
That's incredibly important for doing autocomplete
Lex Fridman (1:20:13.440)
and like machine translation and stuff like that.
Lex Fridman (1:20:15.440)
And it doesn't have anything to do with
Lex Fridman (1:20:16.380)
are they next to each other in the dictionary, right?
Jordan Ellenberg (1:20:17.940)
It's a different kind of distance.
Lex Fridman (1:20:19.260)
Okay, ready?
Jordan Ellenberg (1:20:20.100)
In this kind of number theory,
Lex Fridman (1:20:21.840)
there was a crazy distance called the peatic distance.
Jordan Ellenberg (1:20:25.100)
I didn't write about this that much in the book
Lex Fridman (1:20:26.740)
because even though I love it
Lex Fridman (1:20:27.580)
and it's a big part of my research life,
Lex Fridman (1:20:28.620)
it gets a little bit into the weeds,
Lex Fridman (1:20:29.740)
but your listeners are gonna hear about it now.
Lex Fridman (1:20:32.500)
Please.
Lex Fridman (1:20:34.340)
What a normal person says
Lex Fridman (1:20:35.900)
when they say two numbers are close,
Jordan Ellenberg (1:20:37.700)
they say like their difference is like a small number,
Lex Fridman (1:20:40.220)
like seven and eight are close
Jordan Ellenberg (1:20:41.660)
because their difference is one and one's pretty small.
Lex Fridman (1:20:44.300)
If we were to be what's called a two attic number theorist,
Jordan Ellenberg (1:20:48.580)
we'd say, oh, two numbers are close
Lex Fridman (1:20:50.900)
if their difference is a multiple of a large power of two.
Lex Fridman (1:20:55.660)
So like one and 49 are close
Lex Fridman (1:21:00.940)
because their difference is 48
Lex Fridman (1:21:02.980)
and 48 is a multiple of 16,
Lex Fridman (1:21:04.820)
which is a pretty large power of two.
Jordan Ellenberg (1:21:06.700)
Whereas one and two are pretty far away
Lex Fridman (1:21:09.700)
because the difference between them is one,
Jordan Ellenberg (1:21:12.460)
which is not even a multiple of a power of two at all.
Lex Fridman (1:21:14.260)
That's odd.
Lex Fridman (1:21:15.620)
You wanna know what's really far from one?
Lex Fridman (1:21:17.700)
Like one and 1 64th
Jordan Ellenberg (1:21:21.620)
because their difference is a negative power of two,
Lex Fridman (1:21:24.700)
two to the minus six.
Lex Fridman (1:21:25.660)
So those points are quite, quite far away.
Lex Fridman (1:21:28.220)
Two to the power of a large N would be two,
Jordan Ellenberg (1:21:33.740)
if that's the difference between two numbers
Lex Fridman (1:21:35.620)
then they're close.
Jordan Ellenberg (1:21:37.140)
Yeah, so two to a large power is in this metric
Lex Fridman (1:21:40.140)
a very small number
Lex Fridman (1:21:41.660)
and two to a negative power is a very big number.
Lex Fridman (1:21:44.820)
That's two attic.
Jordan Ellenberg (1:21:45.660)
Okay, I can't even visualize that.
Lex Fridman (1:21:48.700)
It takes practice.
Jordan Ellenberg (1:21:49.740)
It takes practice.
Lex Fridman (1:21:50.580)
If you've ever heard of the Cantor set,
Jordan Ellenberg (1:21:51.860)
it looks kind of like that.
Lex Fridman (1:21:54.100)
So it is crazy that this is good for anything, right?
Jordan Ellenberg (1:21:57.300)
I mean, this just sounds like a definition
Lex Fridman (1:21:58.860)
that someone would make up to torment you.
Lex Fridman (1:22:00.660)
But what's amazing is there's a general theory of distance
Lex Fridman (1:22:05.580)
where you say any definition you make
Jordan Ellenberg (1:22:08.380)
to satisfy certain axioms deserves to be called a distance
Lex Fridman (1:22:11.300)
and this.
Jordan Ellenberg (1:22:12.140)
See, I'm sorry to interrupt.
Lex Fridman (1:22:13.860)
My brain, you broke my brain.
Jordan Ellenberg (1:22:15.460)
Awesome.
Lex Fridman (1:22:16.540)
10 seconds ago.
Jordan Ellenberg (1:22:18.100)
Cause I'm also starting to map for the two attic case
Lex Fridman (1:22:21.500)
to binary numbers.
Lex Fridman (1:22:23.100)
And you know, cause we romanticize those.
Lex Fridman (1:22:25.300)
So I was trying to.
Jordan Ellenberg (1:22:26.140)
Oh, that's exactly the right way to think of it.
Lex Fridman (1:22:27.260)
I was trying to mess with number,
Jordan Ellenberg (1:22:29.500)
I was trying to see, okay, which ones are close.
Lex Fridman (1:22:31.740)
And then I'm starting to visualize
Jordan Ellenberg (1:22:33.020)
different binary numbers and how they,
Lex Fridman (1:22:35.620)
which ones are close to each other.
Lex Fridman (1:22:37.300)
And I'm not sure.
Lex Fridman (1:22:38.700)
Well, I think there's a.
Jordan Ellenberg (1:22:39.540)
No, no, it's very similar.
Lex Fridman (1:22:40.580)
That's exactly the right way to think of it.
Jordan Ellenberg (1:22:41.980)
It's almost like binary numbers written in reverse.
Lex Fridman (1:22:44.580)
Because in a binary expansion, two numbers are close.
Jordan Ellenberg (1:22:47.420)
A number that's small is like 0.0000 something.
Lex Fridman (1:22:50.860)
Something that's the decimal
Lex Fridman (1:22:51.700)
and it starts with a lot of zeros.
Lex Fridman (1:22:53.220)
In the two attic metric, a binary number is very small
Jordan Ellenberg (1:22:56.860)
if it ends with a lot of zeros and then the decimal point.
Lex Fridman (1:23:01.700)
Gotcha.
Lex Fridman (1:23:02.540)
So it is kind of like binary numbers written backwards
Lex Fridman (1:23:04.060)
is actually, I should have said,
Jordan Ellenberg (1:23:05.100)
that's what I should have said, Lex.
Lex Fridman (1:23:07.420)
That's a very good metaphor.
Jordan Ellenberg (1:23:08.780)
Okay, but so why is that interesting
Lex Fridman (1:23:12.020)
except for the fact that it's a beautiful kind of framework,
Jordan Ellenberg (1:23:18.380)
different kind of framework
Lex Fridman (1:23:19.580)
of which to think about distances.
Lex Fridman (1:23:20.940)
And you're talking about not just the two attic,
Lex Fridman (1:23:23.060)
but the generalization of that.
Lex Fridman (1:23:24.220)
Why is that interesting?
Lex Fridman (1:23:25.060)
Yeah, the NEP.
Lex Fridman (1:23:25.900)
And so that, because that's the kind of deformation
Lex Fridman (1:23:27.580)
that comes up in Wiles's proof,
Jordan Ellenberg (1:23:31.700)
that deformation where moving something a little bit
Lex Fridman (1:23:34.300)
means a little bit in this two attic sense.
Jordan Ellenberg (1:23:36.980)
Trippy, okay.
Lex Fridman (1:23:38.060)
No, I mean, it's such a,
Jordan Ellenberg (1:23:38.900)
I mean, I just get excited talking about it
Lex Fridman (1:23:40.020)
and I just taught this like in the fall semester that.
Lex Fridman (1:23:43.980)
But it like reformulating, why is,
Lex Fridman (1:23:49.380)
so you pick a different measure of distance
Jordan Ellenberg (1:23:53.740)
over which you can talk about very tiny changes
Lex Fridman (1:23:56.980)
and then use that to then prove things
Jordan Ellenberg (1:23:59.660)
about the entire thing.
Lex Fridman (1:24:02.300)
Yes, although, honestly, what I would say,
Jordan Ellenberg (1:24:05.060)
I mean, it's true that we use it to prove things,
Lex Fridman (1:24:07.340)
but I would say we use it to understand things.
Lex Fridman (1:24:09.660)
And then because we understand things better,
Lex Fridman (1:24:11.540)
then we can prove things.
Lex Fridman (1:24:12.620)
But the goal is always the understanding.
Lex Fridman (1:24:14.300)
The goal is not so much to prove things.
Jordan Ellenberg (1:24:16.900)
The goal is not to know what's true or false.
Lex Fridman (1:24:18.820)
I mean, this is something I write about
Jordan Ellenberg (1:24:19.860)
in the book, Near the End.
Lex Fridman (1:24:20.700)
And it's something that,
Jordan Ellenberg (1:24:21.540)
it's a wonderful, wonderful essay by Bill Thurston,
Lex Fridman (1:24:25.460)
kind of one of the great geometers of our time,
Jordan Ellenberg (1:24:27.100)
who unfortunately passed away a few years ago,
Lex Fridman (1:24:29.700)
called on proof and progress in mathematics.
Lex Fridman (1:24:32.900)
And he writes very wonderfully about how,
Lex Fridman (1:24:35.380)
we're not, it's not a theorem factory
Jordan Ellenberg (1:24:38.100)
where you have a production quota.
Lex Fridman (1:24:39.820)
I mean, the point of mathematics
Jordan Ellenberg (1:24:40.940)
is to help humans understand things.
Lex Fridman (1:24:43.580)
And the way we test that
Jordan Ellenberg (1:24:45.300)
is that we're proving new theorems along the way.
Lex Fridman (1:24:46.900)
That's the benchmark, but that's not the goal.
Jordan Ellenberg (1:24:49.180)
Yeah, but just as a kind of, absolutely,
Lex Fridman (1:24:51.580)
but as a tool, it's kind of interesting
Jordan Ellenberg (1:24:54.100)
to approach a problem by saying,
Lex Fridman (1:24:56.660)
how can I change the distance function?
Jordan Ellenberg (1:24:59.780)
Like what, the nature of distance,
Lex Fridman (1:25:03.700)
because that might start to lead to insights
Jordan Ellenberg (1:25:07.060)
for deeper understanding.
Lex Fridman (1:25:08.420)
Like if I were to try to describe human society
Jordan Ellenberg (1:25:12.580)
by a distance, two people are close
Lex Fridman (1:25:15.500)
if they love each other.
Jordan Ellenberg (1:25:17.140)
Right.
Lex Fridman (1:25:17.980)
And then start to do a full analysis
Jordan Ellenberg (1:25:21.060)
on the everybody that lives on earth currently,
Lex Fridman (1:25:23.820)
the 7 billion people.
Lex Fridman (1:25:25.820)
And from that perspective,
Lex Fridman (1:25:27.700)
as opposed to the geographic perspective of distance.
Lex Fridman (1:25:30.860)
And then maybe there could be a bunch of insights
Lex Fridman (1:25:32.980)
about the source of violence,
Jordan Ellenberg (1:25:35.580)
the source of maybe entrepreneurial success
Lex Fridman (1:25:39.260)
or invention or economic success or different systems,
Jordan Ellenberg (1:25:42.980)
communism, capitalism start to,
Lex Fridman (1:25:44.660)
I mean, that's, I guess what economics tries to do,
Lex Fridman (1:25:47.460)
but really saying, okay, let's think outside the box
Lex Fridman (1:25:50.500)
about totally new distance functions
Jordan Ellenberg (1:25:52.820)
that could unlock something profound about the space.
Lex Fridman (1:25:57.220)
Yeah, because think about it.
Jordan Ellenberg (1:25:58.060)
Okay, here's, I mean, now we're gonna talk about AI,
Lex Fridman (1:26:01.180)
which you know a lot more about than I do.
Lex Fridman (1:26:02.980)
So just start laughing uproariously
Lex Fridman (1:26:05.820)
if I say something that's completely wrong.
Jordan Ellenberg (1:26:07.060)
We both know very little relative
Lex Fridman (1:26:09.860)
to what we will know centuries from now.
Jordan Ellenberg (1:26:12.620)
That is a really good humble way to think about it.
Lex Fridman (1:26:15.700)
I like it.
Jordan Ellenberg (1:26:16.540)
Okay, so let's just go for it.
Lex Fridman (1:26:18.340)
Okay, so I think you'll agree with this,
Jordan Ellenberg (1:26:20.500)
that in some sense, what's good about AI
Lex Fridman (1:26:23.020)
is that we can't test any case in advance,
Jordan Ellenberg (1:26:26.340)
the whole point of AI is to make,
Lex Fridman (1:26:27.820)
or one point of it, I guess, is to make good predictions
Jordan Ellenberg (1:26:30.540)
about cases we haven't yet seen.
Lex Fridman (1:26:32.620)
And in some sense, that's always gonna involve
Jordan Ellenberg (1:26:34.820)
some notion of distance,
Lex Fridman (1:26:35.980)
because it's always gonna involve
Jordan Ellenberg (1:26:37.860)
somehow taking a case we haven't seen
Lex Fridman (1:26:40.060)
and saying what cases that we have seen is it close to,
Jordan Ellenberg (1:26:43.860)
is it like, is it somehow an interpolation between.
Lex Fridman (1:26:47.820)
Now, when we do that,
Jordan Ellenberg (1:26:49.140)
in order to talk about things being like other things,
Lex Fridman (1:26:52.020)
implicitly or explicitly,
Jordan Ellenberg (1:26:53.460)
we're invoking some notion of distance,
Lex Fridman (1:26:55.580)
and boy, we better get it right.
Jordan Ellenberg (1:26:57.620)
If you try to do natural language processing
Lex Fridman (1:26:59.220)
and your idea of distance between words
Jordan Ellenberg (1:27:01.220)
is how close they are in the dictionary,
Lex Fridman (1:27:03.180)
when you write them in alphabetical order,
Lex Fridman (1:27:04.460)
you are gonna get pretty bad translations, right?
Lex Fridman (1:27:08.180)
No, the notion of distance has to come from somewhere else.
Jordan Ellenberg (1:27:11.580)
Yeah, that's essentially what neural networks are doing,
Lex Fridman (1:27:14.180)
that's what word embeddings are doing is coming up with.
Jordan Ellenberg (1:27:17.340)
In the case of word embeddings, literally,
Lex Fridman (1:27:18.980)
literally what they are doing is learning a distance.
Lex Fridman (1:27:21.020)
But those are super complicated distance functions,
Lex Fridman (1:27:23.620)
and it's almost nice to think
Jordan Ellenberg (1:27:26.220)
maybe there's a nice transformation that's simple.
Lex Fridman (1:27:31.500)
Sorry, there's a nice formulation of the distance.
Jordan Ellenberg (1:27:34.460)
Again with the simple.
Lex Fridman (1:27:36.540)
So you don't, let me ask you about this.
Jordan Ellenberg (1:27:41.380)
From an understanding perspective,
Lex Fridman (1:27:43.380)
there's the Richard Feynman, maybe attributed to him,
Lex Fridman (1:27:45.620)
but maybe many others,
Lex Fridman (1:27:48.780)
is this idea that if you can't explain something simply
Jordan Ellenberg (1:27:52.460)
that you don't understand it.
Lex Fridman (1:27:56.380)
In how many cases, how often is that true?
Lex Fridman (1:28:00.700)
Do you find there's some profound truth in that?
Lex Fridman (1:28:05.580)
Oh, okay, so you were about to ask, is it true?
Jordan Ellenberg (1:28:07.700)
To which I would say flatly, no.
Lex Fridman (1:28:09.300)
But then you said, you followed that up with,
Lex Fridman (1:28:11.260)
is there some profound truth in it?
Lex Fridman (1:28:13.220)
And I'm like, okay, sure.
Lex Fridman (1:28:14.140)
So there's some truth in it.
Lex Fridman (1:28:15.420)
It's not true. But it's not true.
Jordan Ellenberg (1:28:16.740)
It's just not.
Lex Fridman (1:28:17.740)
That's such a mathematician answer.
Jordan Ellenberg (1:28:22.820)
The truth that is in it is that learning
Lex Fridman (1:28:25.740)
to explain something helps you understand it.
Lex Fridman (1:28:29.980)
But real things are not simple.
Lex Fridman (1:28:33.460)
A few things are, most are not.
Lex Fridman (1:28:36.660)
And to be honest, we don't really know
Lex Fridman (1:28:40.140)
whether Feynman really said that right
Jordan Ellenberg (1:28:41.300)
or something like that is sort of disputed.
Lex Fridman (1:28:43.060)
But I don't think Feynman could have literally believed that
Jordan Ellenberg (1:28:46.180)
whether or not he said it.
Lex Fridman (1:28:47.220)
And he was the kind of guy, I didn't know him,
Lex Fridman (1:28:49.620)
but I've been reading his writing,
Lex Fridman (1:28:51.380)
he liked to sort of say stuff, like stuff that sounded good.
Lex Fridman (1:28:55.020)
You know what I mean?
Lex Fridman (1:28:55.860)
So it's totally strikes me as the kind of thing
Jordan Ellenberg (1:28:57.640)
he could have said because he liked the way saying it
Lex Fridman (1:29:00.200)
made him feel, but also knowing
Jordan Ellenberg (1:29:02.980)
that he didn't like literally mean it.
Lex Fridman (1:29:04.500)
Well, I definitely have a lot of friends
Lex Fridman (1:29:07.780)
and I've talked to a lot of physicists
Lex Fridman (1:29:09.540)
and they do derive joy from believing
Jordan Ellenberg (1:29:12.740)
that they can explain stuff simply
Lex Fridman (1:29:14.540)
or believing it's possible to explain stuff simply,
Jordan Ellenberg (1:29:17.820)
even when the explanation is not actually that simple.
Lex Fridman (1:29:20.180)
Like I've heard people think that the explanation is simple
Lex Fridman (1:29:23.940)
and they do the explanation.
Lex Fridman (1:29:25.060)
And I think it is simple,
Lex Fridman (1:29:27.600)
but it's not capturing the phenomena that we're discussing.
Lex Fridman (1:29:30.580)
It's capturing, it's somehow maps in their mind,
Lex Fridman (1:29:33.060)
but it's taking as a starting point,
Lex Fridman (1:29:35.980)
as an assumption that there's a deep knowledge
Lex Fridman (1:29:38.180)
and a deep understanding that's actually very complicated.
Lex Fridman (1:29:41.780)
And the simplicity is almost like a poem
Jordan Ellenberg (1:29:45.220)
about the more complicated thing
Lex Fridman (1:29:46.820)
as opposed to a distillation.
Lex Fridman (1:29:48.700)
And I love poems, but a poem is not an explanation.
Lex Fridman (1:29:51.860)
Well, some people might disagree with that,
Lex Fridman (1:29:55.540)
but certainly from a mathematical perspective.
Lex Fridman (1:29:57.460)
No poet would disagree with it.
Jordan Ellenberg (1:29:59.580)
No poet would disagree.
Lex Fridman (1:30:01.220)
You don't think there's some things
Lex Fridman (1:30:02.760)
that can only be described imprecisely?
Lex Fridman (1:30:06.520)
As an explanation.
Jordan Ellenberg (1:30:07.500)
I don't think any poet would say their poem
Lex Fridman (1:30:09.700)
is an explanation.
Jordan Ellenberg (1:30:10.540)
They might say it's a description.
Lex Fridman (1:30:11.820)
They might say it's sort of capturing sort of.
Jordan Ellenberg (1:30:14.440)
Well, some people might say the only truth is like music.
Lex Fridman (1:30:20.060)
Not the only truth,
Lex Fridman (1:30:20.940)
but some truths can only be expressed through art.
Lex Fridman (1:30:24.820)
And I mean, that's the whole thing
Jordan Ellenberg (1:30:26.380)
we're talking about religion and myth.
Lex Fridman (1:30:27.700)
And there's some things
Jordan Ellenberg (1:30:28.880)
that are limited cognitive capabilities
Lex Fridman (1:30:32.340)
and the tools of mathematics or the tools of physics
Jordan Ellenberg (1:30:35.180)
are just not going to allow us to capture.
Lex Fridman (1:30:37.340)
Like it's possible consciousness is one of those things.
Jordan Ellenberg (1:30:39.900)
And.
Lex Fridman (1:30:42.740)
Yes, that is definitely possible.
Lex Fridman (1:30:44.600)
But I would even say,
Lex Fridman (1:30:46.100)
look, I mean, consciousness is a thing about
Jordan Ellenberg (1:30:47.500)
which we're still in the dark
Lex Fridman (1:30:48.440)
as to whether there's an explanation
Jordan Ellenberg (1:30:50.420)
we would understand it as an explanation at all.
Lex Fridman (1:30:53.620)
By the way, okay.
Jordan Ellenberg (1:30:54.440)
I got to give yet one more amazing Poincare quote
Lex Fridman (1:30:56.340)
because this guy just never stopped coming up
Jordan Ellenberg (1:30:57.700)
with great quotes that,
Lex Fridman (1:31:00.660)
Paul Erdős, another fellow who appears in the book.
Lex Fridman (1:31:02.820)
And by the way,
Lex Fridman (1:31:03.660)
he thinks about this notion of distance
Jordan Ellenberg (1:31:05.520)
of like personal affinity,
Lex Fridman (1:31:07.540)
kind of like what you're talking about,
Jordan Ellenberg (1:31:08.540)
the kind of social network and that notion of distance
Lex Fridman (1:31:11.260)
that comes from that.
Lex Fridman (1:31:12.100)
So that's something that Paul Erdős.
Lex Fridman (1:31:13.300)
Erdős did?
Jordan Ellenberg (1:31:14.340)
Well, he thought about distances and networks.
Lex Fridman (1:31:16.020)
I guess he didn't probably,
Jordan Ellenberg (1:31:16.840)
he didn't think about the social network.
Lex Fridman (1:31:17.680)
Oh, that's fascinating.
Lex Fridman (1:31:18.520)
And that's how it started that story of Erdős number.
Lex Fridman (1:31:20.100)
Yeah, okay.
Jordan Ellenberg (1:31:20.940)
It's hard to distract.
Lex Fridman (1:31:22.700)
But you know, Erdős was sort of famous for saying,
Lex Fridman (1:31:25.100)
and this is sort of long lines we're saying,
Lex Fridman (1:31:26.860)
he talked about the book,
Jordan Ellenberg (1:31:28.420)
capital T, capital B, the book.
Lex Fridman (1:31:31.340)
And that's the book where God keeps the right proof
Jordan Ellenberg (1:31:33.460)
of every theorem.
Lex Fridman (1:31:34.740)
So when he saw a proof he really liked,
Jordan Ellenberg (1:31:36.380)
it was like really elegant, really simple.
Lex Fridman (1:31:38.020)
Like that's from the book.
Jordan Ellenberg (1:31:39.100)
That's like you found one of the ones that's in the book.
Lex Fridman (1:31:43.180)
He wasn't a religious guy, by the way.
Jordan Ellenberg (1:31:44.680)
He referred to God as the supreme fascist.
Lex Fridman (1:31:46.900)
He was like, but somehow he was like,
Jordan Ellenberg (1:31:48.800)
I don't really believe in God,
Lex Fridman (1:31:49.720)
but I believe in God's book.
Jordan Ellenberg (1:31:50.740)
I mean, it was,
Lex Fridman (1:31:53.300)
but Poincare on the other hand,
Lex Fridman (1:31:55.980)
and by the way, there were other managers.
Lex Fridman (1:31:57.020)
Hilda Hudson is one who comes up in this book.
Jordan Ellenberg (1:31:58.700)
She also kind of saw math.
Lex Fridman (1:32:01.800)
She's one of the people who sort of develops
Jordan Ellenberg (1:32:05.300)
the disease model that we now use,
Lex Fridman (1:32:06.880)
that we use to sort of track pandemics,
Jordan Ellenberg (1:32:08.380)
this SIR model that sort of originally comes
Lex Fridman (1:32:10.380)
from her work with Ronald Ross.
Lex Fridman (1:32:11.940)
But she was also super, super, super devout.
Lex Fridman (1:32:14.500)
And she also sort of on the other side
Jordan Ellenberg (1:32:17.380)
of the religious coin was like,
Lex Fridman (1:32:18.320)
yeah, math is how we communicate with God.
Jordan Ellenberg (1:32:20.460)
She has a great,
Lex Fridman (1:32:21.300)
all these people are incredibly quotable.
Jordan Ellenberg (1:32:22.560)
She says, you know, math is,
Lex Fridman (1:32:24.680)
the truth, the things about mathematics,
Jordan Ellenberg (1:32:26.620)
she's like, they're not the most important of God thoughts,
Lex Fridman (1:32:29.460)
but they're the only ones that we can know precisely.
Lex Fridman (1:32:32.620)
So she's like, this is the one place
Lex Fridman (1:32:34.020)
where we get to sort of see what God's thinking
Jordan Ellenberg (1:32:35.460)
when we do mathematics.
Lex Fridman (1:32:37.380)
Again, not a fan of poetry or music.
Jordan Ellenberg (1:32:39.160)
Some people will say Hendrix is like,
Lex Fridman (1:32:41.020)
some people say chapter one of that book is mathematics,
Lex Fridman (1:32:44.340)
and then chapter two is like classic rock.
Lex Fridman (1:32:46.860)
Right?
Lex Fridman (1:32:48.580)
So like, it's not clear that the...
Lex Fridman (1:32:51.380)
I'm sorry, you just sent me off on a tangent,
Jordan Ellenberg (1:32:52.740)
just imagining like Erdos at a Hendrix concert,
Lex Fridman (1:32:54.940)
like trying to figure out if it was from the book or not.
Lex Fridman (1:32:59.740)
What I was coming to was just to say,
Lex Fridman (1:33:00.980)
but what Poincaré said about this is he's like,
Jordan Ellenberg (1:33:03.100)
you know, if like, this is all worked out
Lex Fridman (1:33:07.400)
in the language of the divine,
Lex Fridman (1:33:08.460)
and if a divine being like came down and told it to us,
Lex Fridman (1:33:12.460)
we wouldn't be able to understand it, so it doesn't matter.
Lex Fridman (1:33:15.020)
So Poincaré was of the view that there were things
Lex Fridman (1:33:17.400)
that were sort of like inhumanly complex,
Lex Fridman (1:33:19.340)
and that was how they really were.
Lex Fridman (1:33:21.060)
Our job is to figure out the things that are not like that.
Jordan Ellenberg (1:33:23.780)
That are not like that.
Lex Fridman (1:33:25.600)
All this talk of primes got me hungry for primes.
Jordan Ellenberg (1:33:29.380)
You wrote a blog post, The Beauty of Bounding Gaps,
Lex Fridman (1:33:32.580)
a huge discovery about prime numbers
Lex Fridman (1:33:35.260)
and what it means for the future of math.
Lex Fridman (1:33:39.140)
Can you tell me about prime numbers?
Lex Fridman (1:33:40.820)
What the heck are those?
Lex Fridman (1:33:41.860)
What are twin primes?
Lex Fridman (1:33:42.820)
What are prime gaps?
Lex Fridman (1:33:43.740)
What are bounding gaps and primes?
Lex Fridman (1:33:46.760)
What are all these things?
Lex Fridman (1:33:47.820)
And what, if anything,
Lex Fridman (1:33:49.820)
or what exactly is beautiful about them?
Lex Fridman (1:33:52.100)
Yeah, so, you know, prime numbers are one of the things
Jordan Ellenberg (1:33:57.100)
that number theorists study the most and have for millennia.
Lex Fridman (1:34:02.820)
They are numbers which can't be factored.
Lex Fridman (1:34:06.220)
And then you say, like, five.
Lex Fridman (1:34:08.140)
And then you're like, wait, I can factor five.
Jordan Ellenberg (1:34:09.780)
Five is five times one.
Lex Fridman (1:34:11.820)
Okay, not like that.
Jordan Ellenberg (1:34:13.500)
That is a factorization.
Lex Fridman (1:34:14.540)
It absolutely is a way of expressing five
Jordan Ellenberg (1:34:16.700)
as a product of two things.
Lex Fridman (1:34:18.380)
But don't you agree there's like something trivial about it?
Jordan Ellenberg (1:34:20.900)
It's something you could do to any number.
Lex Fridman (1:34:22.300)
It doesn't have content the way that if I say
Jordan Ellenberg (1:34:24.340)
that 12 is six times two or 35 is seven times five,
Lex Fridman (1:34:27.640)
I've really done something to it.
Jordan Ellenberg (1:34:28.960)
I've broken up.
Lex Fridman (1:34:29.800)
So those are the kind of factorizations that count.
Lex Fridman (1:34:31.700)
And a number that doesn't have a factorization like that
Lex Fridman (1:34:34.460)
is called prime, except, historical side note,
Jordan Ellenberg (1:34:38.100)
one, which at some times in mathematical history
Lex Fridman (1:34:42.440)
has been deemed to be a prime, but currently is not.
Lex Fridman (1:34:46.040)
And I think that's for the best.
Lex Fridman (1:34:47.140)
But I bring it up only because sometimes people think that,
Jordan Ellenberg (1:34:49.580)
you know, these definitions are kind of,
Lex Fridman (1:34:52.220)
if we think about them hard enough,
Jordan Ellenberg (1:34:53.500)
we can figure out which definition is true.
Lex Fridman (1:34:56.780)
No.
Jordan Ellenberg (1:34:57.620)
There's just an artifact of mathematics.
Lex Fridman (1:34:58.820)
So it's a question of which definition is best for us,
Jordan Ellenberg (1:35:03.460)
for our purposes.
Lex Fridman (1:35:04.300)
Well, those edge cases are weird, right?
Lex Fridman (1:35:06.020)
So it can't be, it doesn't count when you use yourself
Lex Fridman (1:35:11.700)
as a number or one as part of the factorization
Jordan Ellenberg (1:35:15.000)
or as the entirety of the factorization.
Lex Fridman (1:35:19.320)
So you somehow get to the meat of the number
Jordan Ellenberg (1:35:22.920)
by factorizing it.
Lex Fridman (1:35:24.180)
And that seems to get to the core of all of mathematics.
Jordan Ellenberg (1:35:27.420)
Yeah, you take any number and you factorize it
Lex Fridman (1:35:29.940)
until you can factorize no more.
Lex Fridman (1:35:31.440)
And what you have left is some big pile of primes.
Lex Fridman (1:35:33.900)
I mean, by definition, when you can't factor anymore,
Jordan Ellenberg (1:35:36.380)
when you're done, when you can't break the numbers up
Lex Fridman (1:35:39.020)
anymore, what's left must be prime.
Jordan Ellenberg (1:35:40.900)
You know, 12 breaks into two and two and three.
Lex Fridman (1:35:45.580)
So these numbers are the atoms, the building blocks
Jordan Ellenberg (1:35:48.220)
of all numbers.
Lex Fridman (1:35:50.760)
And there's a lot we know about them,
Jordan Ellenberg (1:35:52.180)
or there's much more that we don't know about them.
Lex Fridman (1:35:53.420)
I'll tell you the first few.
Jordan Ellenberg (1:35:54.340)
There's two, three, five, seven, 11.
Lex Fridman (1:35:59.140)
By the way, they're all gonna be odd from then on
Jordan Ellenberg (1:36:00.780)
because if they were even, I could factor out
Lex Fridman (1:36:02.020)
two out of them.
Lex Fridman (1:36:03.060)
But it's not all the odd numbers.
Lex Fridman (1:36:04.300)
Nine isn't prime because it's three times three.
Jordan Ellenberg (1:36:06.440)
15 isn't prime because it's three times five,
Lex Fridman (1:36:08.180)
but 13 is.
Lex Fridman (1:36:09.000)
Where were we?
Lex Fridman (1:36:09.840)
Two, three, five, seven, 11, 13, 17, 19.
Jordan Ellenberg (1:36:13.820)
Not 21, but 23 is, et cetera, et cetera.
Lex Fridman (1:36:15.940)
Okay, so you could go on.
Lex Fridman (1:36:17.060)
How high could you go if we were just sitting here?
Lex Fridman (1:36:19.580)
By the way, your own brain.
Jordan Ellenberg (1:36:20.980)
If continuous, without interruption,
Lex Fridman (1:36:23.980)
would you be able to go over 100?
Jordan Ellenberg (1:36:25.980)
I think so.
Lex Fridman (1:36:27.140)
There's always those ones that trip people up.
Jordan Ellenberg (1:36:29.100)
There's a famous one, the Grotendeek prime 57,
Lex Fridman (1:36:31.780)
like sort of Alexander Grotendeek,
Jordan Ellenberg (1:36:33.380)
the great algebraic geometer was sort of giving
Lex Fridman (1:36:35.740)
some lecture involving a choice of a prime in general.
Lex Fridman (1:36:38.740)
And somebody said, can't you just choose a prime?
Lex Fridman (1:36:41.500)
And he said, okay, 57, which is in fact not prime.
Jordan Ellenberg (1:36:43.540)
It's three times 19.
Lex Fridman (1:36:45.800)
Oh, damn.
Lex Fridman (1:36:46.640)
But it was like, I promise you in some circles
Lex Fridman (1:36:49.300)
it's a funny story.
Lex Fridman (1:36:50.140)
But there's a humor in it.
Lex Fridman (1:36:55.740)
Yes, I would say over 100, I definitely don't remember.
Jordan Ellenberg (1:36:59.220)
Like 107, I think, I'm not sure.
Lex Fridman (1:37:02.100)
Okay, like, I mean.
Lex Fridman (1:37:03.460)
So is there a category of like fake primes
Lex Fridman (1:37:08.900)
that are easily mistaken to be prime?
Jordan Ellenberg (1:37:12.900)
Like 57, I wonder.
Lex Fridman (1:37:14.660)
Yeah, so I would say 57 and 51 are definitely
Jordan Ellenberg (1:37:20.740)
like prime offenders.
Lex Fridman (1:37:21.900)
Oh, I didn't do that on purpose.
Jordan Ellenberg (1:37:23.060)
Oh, well done.
Lex Fridman (1:37:24.300)
Didn't do it on purpose.
Jordan Ellenberg (1:37:25.340)
Anyway, they're definitely ones that people,
Lex Fridman (1:37:28.180)
or 91 is another classic, seven times 13.
Lex Fridman (1:37:30.700)
It really feels kind of prime, doesn't it?
Lex Fridman (1:37:32.900)
But it is not.
Jordan Ellenberg (1:37:34.020)
Yeah.
Lex Fridman (1:37:35.820)
But there's also, by the way,
Lex Fridman (1:37:36.900)
but there's also an actual notion of pseudo prime,
Lex Fridman (1:37:39.600)
which is a thing with a formal definition,
Jordan Ellenberg (1:37:41.460)
which is not a psychological thing.
Lex Fridman (1:37:43.380)
It is a prime which passes a primality test
Jordan Ellenberg (1:37:47.540)
devised by Fermat, which is a very good test,
Lex Fridman (1:37:50.380)
which if a number fails this test,
Jordan Ellenberg (1:37:52.540)
it's definitely not prime.
Lex Fridman (1:37:54.580)
And so there was some hope that,
Jordan Ellenberg (1:37:55.600)
oh, maybe if a number passes the test,
Lex Fridman (1:37:57.280)
then it definitely is prime.
Jordan Ellenberg (1:37:58.420)
That would give a very simple criterion for primality.
Lex Fridman (1:38:00.660)
Unfortunately, it's only perfect in one direction.
Lex Fridman (1:38:03.980)
So there are numbers, I want to say 341 is the smallest,
Lex Fridman (1:38:09.800)
which pass the test but are not prime, 341.
Lex Fridman (1:38:12.380)
Is this test easily explainable or no?
Lex Fridman (1:38:14.780)
Yes, actually.
Jordan Ellenberg (1:38:16.820)
Ready, let me give you the simplest version of it.
Lex Fridman (1:38:18.300)
You can dress it up a little bit, but here's the basic idea.
Jordan Ellenberg (1:38:22.660)
I take the number, the mystery number,
Lex Fridman (1:38:25.180)
I raise two to that power.
Lex Fridman (1:38:29.540)
So let's say your mystery number is six.
Lex Fridman (1:38:32.780)
Are you sorry you asked me?
Lex Fridman (1:38:33.900)
Are you ready?
Lex Fridman (1:38:34.740)
No, you're breaking my brain again, but yes.
Jordan Ellenberg (1:38:37.140)
Let's do it.
Lex Fridman (1:38:38.220)
We're going to do a live demonstration.
Jordan Ellenberg (1:38:40.220)
Let's say your number is six.
Lex Fridman (1:38:43.340)
So I'm going to raise two to the sixth power.
Jordan Ellenberg (1:38:45.980)
Okay, so if I were working on it,
Lex Fridman (1:38:46.820)
I'd be like that's two cubes squared,
Lex Fridman (1:38:48.660)
so that's eight times eight, so that's 64.
Lex Fridman (1:38:51.680)
Now we're going to divide by six,
Lex Fridman (1:38:53.520)
but I don't actually care what the quotient is,
Lex Fridman (1:38:54.980)
only the remainder.
Lex Fridman (1:38:57.300)
So let's see, 64 divided by six is,
Lex Fridman (1:39:01.420)
well, there's a quotient of 10, but the remainder is four.
Lex Fridman (1:39:05.460)
So you failed because the answer has to be two.
Lex Fridman (1:39:08.640)
For any prime, let's do it with five, which is prime.
Jordan Ellenberg (1:39:13.260)
Two to the fifth is 32.
Lex Fridman (1:39:15.580)
Divide 32 by five, and you get six with a remainder of two.
Jordan Ellenberg (1:39:23.100)
With a remainder of two, yeah.
Lex Fridman (1:39:24.220)
For seven, two to the seventh is 128.
Jordan Ellenberg (1:39:26.700)
Divide that by seven, and let's see,
Lex Fridman (1:39:29.480)
I think that's seven times 14, is that right?
Jordan Ellenberg (1:39:32.340)
No.
Lex Fridman (1:39:33.160)
Seven times 18 is 126 with a remainder of two, right?
Jordan Ellenberg (1:39:40.760)
128 is a multiple of seven plus two.
Lex Fridman (1:39:43.360)
So if that remainder is not two,
Jordan Ellenberg (1:39:46.520)
then it's definitely not prime.
Lex Fridman (1:39:49.480)
And then if it is, it's likely a prime, but not for sure.
Jordan Ellenberg (1:39:53.320)
It's likely a prime, but not for sure.
Lex Fridman (1:39:54.660)
And there's actually a beautiful geometric proof
Jordan Ellenberg (1:39:56.280)
which is in the book, actually.
Lex Fridman (1:39:57.240)
That's like one of the most granular parts of the book
Jordan Ellenberg (1:39:58.720)
because it's such a beautiful proof, I couldn't not give it.
Lex Fridman (1:40:00.440)
So you draw a lot of like opal and pearl necklaces
Lex Fridman (1:40:05.400)
and spin them.
Lex Fridman (1:40:06.240)
That's kind of the geometric nature
Jordan Ellenberg (1:40:07.440)
of this proof of Fermat's Little Theorem.
Lex Fridman (1:40:11.920)
So yeah, so with pseudo primes,
Jordan Ellenberg (1:40:13.680)
there are primes that are kind of faking it.
Lex Fridman (1:40:14.760)
They pass that test, but there are numbers
Jordan Ellenberg (1:40:16.560)
that are faking it that pass that test,
Lex Fridman (1:40:17.960)
but are not actually prime.
Lex Fridman (1:40:20.680)
But the point is, there are many, many,
Lex Fridman (1:40:26.000)
many theorems about prime numbers.
Jordan Ellenberg (1:40:28.900)
There's a bunch of questions to ask.
Lex Fridman (1:40:32.100)
Is there an infinite number of primes?
Jordan Ellenberg (1:40:34.660)
Can we say something about the gap between primes
Lex Fridman (1:40:37.460)
as the numbers grow larger and larger and larger and so on?
Jordan Ellenberg (1:40:40.940)
Yeah, it's a perfect example of your desire
Lex Fridman (1:40:43.200)
for simplicity in all things.
Lex Fridman (1:40:44.620)
You know what would be really simple?
Lex Fridman (1:40:46.300)
If there was only finitely many primes
Lex Fridman (1:40:48.780)
and then there would be this finite set of atoms
Lex Fridman (1:40:51.540)
that all numbers would be built up.
Jordan Ellenberg (1:40:53.860)
That would be very simple and good in certain ways,
Lex Fridman (1:40:56.860)
but it's completely false.
Lex Fridman (1:40:58.860)
And number theory would be totally different
Lex Fridman (1:41:00.220)
if that were the case.
Jordan Ellenberg (1:41:01.040)
It's just not true.
Lex Fridman (1:41:03.180)
In fact, this is something else that Euclid knew.
Lex Fridman (1:41:04.700)
So this is a very, very old fact,
Lex Fridman (1:41:07.540)
like much before, long before we've had anything
Jordan Ellenberg (1:41:10.340)
like modern number theory.
Lex Fridman (1:41:11.180)
The primes are infinite.
Jordan Ellenberg (1:41:12.140)
The primes that there are, right.
Lex Fridman (1:41:14.020)
There's an infinite number of primes.
Lex Fridman (1:41:15.460)
So what about the gaps between the primes?
Lex Fridman (1:41:17.740)
Right, so one thing that people recognized
Lex Fridman (1:41:20.460)
and really thought about a lot is that the primes,
Lex Fridman (1:41:22.220)
on average, seem to get farther and farther apart
Jordan Ellenberg (1:41:25.840)
as they get bigger and bigger.
Lex Fridman (1:41:27.020)
In other words, it's less and less common.
Jordan Ellenberg (1:41:29.140)
Like I already told you of the first 10 numbers,
Lex Fridman (1:41:31.140)
two, three, five, seven, four of them are prime.
Jordan Ellenberg (1:41:32.940)
That's a lot, 40%.
Lex Fridman (1:41:34.700)
If I looked at 10 digit numbers,
Jordan Ellenberg (1:41:38.540)
no way would 40% of those be prime.
Lex Fridman (1:41:40.620)
Being prime would be a lot rarer.
Jordan Ellenberg (1:41:42.020)
In some sense, because there's a lot more things
Lex Fridman (1:41:43.940)
for them to be divisible by.
Jordan Ellenberg (1:41:45.860)
That's one way of thinking of it.
Lex Fridman (1:41:47.140)
It's a lot more possible for there to be a factorization
Jordan Ellenberg (1:41:49.420)
because there's a lot of things
Lex Fridman (1:41:50.380)
you can try to factor out of it.
Jordan Ellenberg (1:41:52.140)
As the numbers get bigger and bigger,
Lex Fridman (1:41:53.420)
primality gets rarer and rarer, and the extent
Jordan Ellenberg (1:41:58.820)
to which that's the case, that's pretty well understood.
Lex Fridman (1:42:01.700)
But then you can ask more fine grained questions,
Lex Fridman (1:42:03.840)
and here is one.
Lex Fridman (1:42:07.900)
A twin prime is a pair of primes that are two apart,
Jordan Ellenberg (1:42:11.740)
like three and five, or like 11 and 13, or like 17 and 19.
Lex Fridman (1:42:17.260)
And one thing we still don't know
Lex Fridman (1:42:18.900)
is are there infinitely many of those?
Lex Fridman (1:42:21.960)
We know on average, they get farther and farther apart,
Lex Fridman (1:42:24.100)
but that doesn't mean there couldn't be occasional folks
Lex Fridman (1:42:28.140)
that come close together.
Lex Fridman (1:42:30.180)
And indeed, we think that there are.
Lex Fridman (1:42:33.820)
And one interesting question, I mean, this is,
Jordan Ellenberg (1:42:37.300)
because I think you might say,
Lex Fridman (1:42:38.140)
well, how could one possibly have a right
Lex Fridman (1:42:41.060)
to have an opinion about something like that?
Lex Fridman (1:42:44.020)
We don't have any way of describing a process
Jordan Ellenberg (1:42:46.380)
that makes primes.
Lex Fridman (1:42:49.540)
Sure, you can look at your computer
Lex Fridman (1:42:51.480)
and see a lot of them, but the fact that there's a lot,
Lex Fridman (1:42:53.860)
why is that evidence that there's infinitely many, right?
Jordan Ellenberg (1:42:55.940)
Maybe I can go on the computer and find 10 million.
Lex Fridman (1:42:57.660)
Well, 10 million is pretty far from infinity, right?
Lex Fridman (1:42:59.940)
So how is that evidence?
Lex Fridman (1:43:01.620)
There's a lot of things.
Jordan Ellenberg (1:43:02.520)
There's like a lot more than 10 million atoms.
Lex Fridman (1:43:04.180)
That doesn't mean there's infinitely many atoms
Lex Fridman (1:43:05.500)
in the universe, right?
Lex Fridman (1:43:06.340)
I mean, on most people's physical theories,
Jordan Ellenberg (1:43:07.740)
there's probably not, as I understand it.
Lex Fridman (1:43:10.180)
Okay, so why would we think this?
Jordan Ellenberg (1:43:13.240)
The answer is that it turns out to be like incredibly
Lex Fridman (1:43:17.460)
productive and enlightening to think about primes
Jordan Ellenberg (1:43:21.840)
as if they were random numbers,
Lex Fridman (1:43:23.260)
as if they were randomly distributed
Jordan Ellenberg (1:43:24.900)
according to a certain law.
Lex Fridman (1:43:26.060)
Now they're not, they're not random.
Jordan Ellenberg (1:43:27.740)
There's no chance involved.
Lex Fridman (1:43:28.740)
There it's completely deterministic
Jordan Ellenberg (1:43:30.140)
whether a number is prime or not.
Lex Fridman (1:43:31.620)
And yet it just turns out to be phenomenally useful
Jordan Ellenberg (1:43:35.420)
in mathematics to say,
Lex Fridman (1:43:38.100)
even if something is governed by a deterministic law,
Jordan Ellenberg (1:43:41.740)
let's just pretend it wasn't.
Lex Fridman (1:43:43.100)
Let's just pretend that they were produced
Jordan Ellenberg (1:43:44.460)
by some random process and see if the behavior
Lex Fridman (1:43:46.560)
is roughly the same.
Lex Fridman (1:43:47.940)
And if it's not, maybe change the random process,
Lex Fridman (1:43:49.620)
maybe make the randomness a little bit different
Lex Fridman (1:43:51.100)
and tweak it and see if you can find a random process
Lex Fridman (1:43:53.820)
that matches the behavior we see.
Lex Fridman (1:43:55.380)
And then maybe you predict that other behaviors
Lex Fridman (1:44:00.140)
of the system are like that of the random process.
Lex Fridman (1:44:02.900)
And so that's kind of like, it's funny
Lex Fridman (1:44:04.060)
because I think when you talk to people
Jordan Ellenberg (1:44:05.260)
at the twin prime conjecture,
Lex Fridman (1:44:07.420)
people think you're saying,
Jordan Ellenberg (1:44:09.940)
wow, there's like some deep structure there
Lex Fridman (1:44:12.420)
that like makes those primes be like close together
Jordan Ellenberg (1:44:15.180)
again and again.
Lex Fridman (1:44:16.020)
And no, it's the opposite of deep structure.
Lex Fridman (1:44:18.300)
What we say when we say we believe the twin prime conjecture
Lex Fridman (1:44:20.860)
is that we believe the primes are like sort of
Jordan Ellenberg (1:44:22.860)
strewn around pretty randomly.
Lex Fridman (1:44:24.580)
And if they were, then by chance,
Jordan Ellenberg (1:44:26.100)
you would expect there to be infinitely many twin primes.
Lex Fridman (1:44:28.180)
And we're saying, yeah, we expect them to behave
Jordan Ellenberg (1:44:29.660)
just like they would if they were random dirt.
Lex Fridman (1:44:33.140)
The fascinating parallel here is,
Jordan Ellenberg (1:44:36.180)
I just got a chance to talk to Sam Harris
Lex Fridman (1:44:38.420)
and he uses the prime numbers as an example.
Jordan Ellenberg (1:44:41.300)
Often, I don't know if you're familiar with who Sam is.
Lex Fridman (1:44:44.940)
He uses that as an example of there being no free will.
Lex Fridman (1:44:50.380)
Wait, where does he get this?
Lex Fridman (1:44:52.380)
Well, he just uses as an example of,
Jordan Ellenberg (1:44:54.820)
it might seem like this is a random number generator,
Lex Fridman (1:44:58.460)
but it's all like formally defined.
Lex Fridman (1:45:01.780)
So if we keep getting more and more primes,
Lex Fridman (1:45:05.120)
then like that might feel like a new discovery
Lex Fridman (1:45:09.180)
and that might feel like a new experience, but it's not.
Lex Fridman (1:45:12.160)
It was always written in the cards.
Lex Fridman (1:45:14.340)
But it's funny that you say that
Lex Fridman (1:45:15.700)
because a lot of people think of like randomness,
Jordan Ellenberg (1:45:19.580)
the fundamental randomness within the nature of reality
Lex Fridman (1:45:23.420)
might be the source of something
Jordan Ellenberg (1:45:25.900)
that we experience as free will.
Lex Fridman (1:45:27.780)
And you're saying it's like useful to look at prime numbers
Jordan Ellenberg (1:45:30.180)
as a random process in order to prove stuff about them.
Lex Fridman (1:45:35.620)
But fundamentally, of course, it's not a random process.
Jordan Ellenberg (1:45:38.820)
Well, not in order to prove some stuff about them
Lex Fridman (1:45:40.900)
so much as to figure out what we expect to be true
Lex Fridman (1:45:43.740)
and then try to prove that.
Lex Fridman (1:45:44.940)
Because here's what you don't want to do.
Jordan Ellenberg (1:45:45.900)
Try really hard to prove something that's false.
Lex Fridman (1:45:48.340)
That makes it really hard to prove the thing if it's false.
Lex Fridman (1:45:51.100)
So you certainly want to have some heuristic ways
Lex Fridman (1:45:53.020)
of guessing, making good guesses about what's true.
Lex Fridman (1:45:55.120)
So yeah, here's what I would say.
Lex Fridman (1:45:56.660)
You're going to be imaginary Sam Harris now.
Jordan Ellenberg (1:45:58.740)
Like you are talking about prime numbers
Lex Fridman (1:46:00.980)
and you are like,
Lex Fridman (1:46:01.860)
but prime numbers are completely deterministic.
Lex Fridman (1:46:04.000)
And I'm saying like,
Jordan Ellenberg (1:46:04.840)
well, but let's treat them like a random process.
Lex Fridman (1:46:06.940)
And then you say,
Lex Fridman (1:46:08.240)
but you're just saying something that's not true.
Lex Fridman (1:46:09.580)
They're not a random process, they're deterministic.
Lex Fridman (1:46:10.900)
And I'm like, okay, great.
Lex Fridman (1:46:11.860)
You hold to your insistence that it's not a random process.
Jordan Ellenberg (1:46:13.940)
Meanwhile, I'm generating insight about the primes
Lex Fridman (1:46:15.860)
that you're not because I'm willing to sort of pretend
Jordan Ellenberg (1:46:17.740)
that there's something that they're not
Lex Fridman (1:46:18.660)
in order to understand what's going on.
Jordan Ellenberg (1:46:20.440)
Yeah, so it doesn't matter what the reality is.
Lex Fridman (1:46:22.920)
What matters is what framework of thought
Jordan Ellenberg (1:46:28.220)
results in the maximum number of insights.
Lex Fridman (1:46:30.780)
Yeah, because I feel, look, I'm sorry,
Lex Fridman (1:46:32.380)
but I feel like you have more insights about people.
Lex Fridman (1:46:34.180)
If you think of them as like beings that have wants
Lex Fridman (1:46:37.980)
and needs and desires and do stuff on purpose,
Lex Fridman (1:46:40.860)
even if that's not true,
Jordan Ellenberg (1:46:41.820)
you still understand better what's going on
Lex Fridman (1:46:43.500)
by treating them in that way.
Jordan Ellenberg (1:46:44.620)
Don't you find, look, when you work on machine learning,
Lex Fridman (1:46:46.480)
don't you find yourself sort of talking
Jordan Ellenberg (1:46:48.020)
about what the machine is trying to do
Lex Fridman (1:46:51.500)
in a certain instance?
Lex Fridman (1:46:52.780)
Do you not find yourself drawn to that language?
Lex Fridman (1:46:54.940)
Well, it knows this, it's trying to do that,
Jordan Ellenberg (1:46:57.700)
it's learning that.
Lex Fridman (1:46:58.980)
I'm certainly drawn to that language
Jordan Ellenberg (1:47:00.980)
to the point where I receive quite a bit of criticisms
Lex Fridman (1:47:03.380)
for it because I, you know, like.
Jordan Ellenberg (1:47:05.420)
Oh, I'm on your side, man.
Lex Fridman (1:47:07.020)
So especially in robotics, I don't know why,
Lex Fridman (1:47:09.740)
but robotics people don't like to name their robots.
Lex Fridman (1:47:14.260)
They certainly don't like to gender their robots
Jordan Ellenberg (1:47:17.020)
because the moment you gender a robot,
Lex Fridman (1:47:18.780)
you start to anthropomorphize.
Jordan Ellenberg (1:47:20.580)
If you say he or she, you start to,
Lex Fridman (1:47:22.940)
in your mind, construct like a life story.
Jordan Ellenberg (1:47:27.340)
In your mind, you can't help it.
Lex Fridman (1:47:29.020)
There's like, you create like a humorous story
Jordan Ellenberg (1:47:31.520)
to this person.
Lex Fridman (1:47:32.360)
You start to, this person, this robot,
Jordan Ellenberg (1:47:35.580)
you start to project your own.
Lex Fridman (1:47:37.300)
But I think that's what we do to each other.
Lex Fridman (1:47:38.780)
And I think that's actually really useful
Lex Fridman (1:47:40.500)
for the engineering process,
Jordan Ellenberg (1:47:42.620)
especially for human robot interaction.
Lex Fridman (1:47:44.580)
And yes, for machine learning systems,
Jordan Ellenberg (1:47:46.620)
for helping you build an intuition
Lex Fridman (1:47:48.020)
about a particular problem.
Jordan Ellenberg (1:47:49.900)
It's almost like asking this question,
Lex Fridman (1:47:53.060)
you know, when a machine learning system fails
Jordan Ellenberg (1:47:55.940)
in a particular edge case, asking like,
Lex Fridman (1:47:57.960)
what were you thinking about?
Jordan Ellenberg (1:47:59.820)
Like, like asking, like almost like
Lex Fridman (1:48:02.020)
when you're talking about to a child
Jordan Ellenberg (1:48:04.540)
who just did something bad, you want to understand
Lex Fridman (1:48:08.580)
like what was, how did they see the world?
Jordan Ellenberg (1:48:12.060)
Maybe there's a totally new, maybe you're the one
Lex Fridman (1:48:13.980)
that's thinking about the world incorrectly.
Lex Fridman (1:48:16.820)
And yeah, that anthropomorphization process,
Lex Fridman (1:48:19.900)
I think is ultimately good for insight.
Lex Fridman (1:48:21.380)
And the same is, I agree with you.
Lex Fridman (1:48:23.620)
I tend to believe about free will as well.
Jordan Ellenberg (1:48:26.660)
Let me ask you a ridiculous question, if it's okay.
Lex Fridman (1:48:28.900)
Of course.
Jordan Ellenberg (1:48:30.260)
I've just recently, most people go on like rabbit hole,
Lex Fridman (1:48:34.420)
like YouTube things.
Lex Fridman (1:48:35.660)
And I went on a rabbit hole often do of Wikipedia.
Lex Fridman (1:48:39.820)
And I found a page on
Jordan Ellenberg (1:48:43.860)
finiteism, ultra finiteism and intuitionism
Lex Fridman (1:48:49.100)
or into, I forget what it's called.
Jordan Ellenberg (1:48:51.180)
Yeah, intuitionism.
Lex Fridman (1:48:52.140)
Intuitionism.
Jordan Ellenberg (1:48:53.740)
That seemed pretty, pretty interesting.
Lex Fridman (1:48:55.580)
I have it on my to do list actually like look into
Jordan Ellenberg (1:48:58.420)
like, is there people who like formally attract,
Lex Fridman (1:49:00.820)
like real mathematicians are trying to argue for this.
Lex Fridman (1:49:03.600)
But the belief there, I think, let's say finiteism
Lex Fridman (1:49:07.500)
that infinity is fake.
Jordan Ellenberg (1:49:11.860)
Meaning, infinity might be like a useful hack
Lex Fridman (1:49:16.740)
for certain, like a useful tool in mathematics,
Lex Fridman (1:49:18.860)
but it really gets us into trouble
Lex Fridman (1:49:22.460)
because there's no infinity in the real world.
Jordan Ellenberg (1:49:26.660)
Maybe I'm sort of not expressing that fully correctly,
Lex Fridman (1:49:30.980)
but basically saying like there's things
Jordan Ellenberg (1:49:32.780)
that once you add into mathematics,
Lex Fridman (1:49:37.020)
things that are not provably within the physical world,
Jordan Ellenberg (1:49:41.020)
you're starting to inject to corrupt your framework
Lex Fridman (1:49:45.800)
of reason.
Lex Fridman (1:49:47.620)
What do you think about that?
Lex Fridman (1:49:49.180)
I mean, I think, okay, so first of all, I'm not an expert
Lex Fridman (1:49:51.660)
and I couldn't even tell you what the difference is
Lex Fridman (1:49:54.780)
between those three terms, finiteism, ultra finiteism
Lex Fridman (1:49:58.340)
and intuitionism, although I know they're related
Lex Fridman (1:49:59.940)
and I tend to associate them with the Netherlands
Jordan Ellenberg (1:50:01.620)
in the 1930s.
Lex Fridman (1:50:02.620)
Okay, I'll tell you, can I just quickly comment
Jordan Ellenberg (1:50:04.860)
because I read the Wikipedia page.
Lex Fridman (1:50:06.860)
The difference in ultra.
Jordan Ellenberg (1:50:07.700)
That's like the ultimate sentence of the modern age.
Lex Fridman (1:50:10.480)
Can I just comment because I read the Wikipedia page.
Jordan Ellenberg (1:50:12.620)
That sums up our moment.
Lex Fridman (1:50:14.660)
Bro, I'm basically an expert.
Jordan Ellenberg (1:50:17.540)
Ultra finiteism.
Lex Fridman (1:50:19.700)
So, finiteism says that the only infinity
Jordan Ellenberg (1:50:22.860)
you're allowed to have is that the natural numbers
Lex Fridman (1:50:25.220)
are infinite.
Jordan Ellenberg (1:50:27.020)
So, like those numbers are infinite.
Lex Fridman (1:50:29.240)
So, like one, two, three, four, five,
Jordan Ellenberg (1:50:32.200)
the integers are infinite.
Lex Fridman (1:50:35.460)
The ultra finiteism says, nope, even that infinity is fake.
Jordan Ellenberg (1:50:41.480)
I'll bet ultra finiteism came second.
Lex Fridman (1:50:43.120)
I'll bet it's like when there's like a hardcore scene
Lex Fridman (1:50:44.740)
and then one guy's like, oh, now there's a lot of people
Lex Fridman (1:50:47.180)
in the scene.
Jordan Ellenberg (1:50:48.020)
I have to find a way to be more hardcore
Lex Fridman (1:50:49.180)
than the hardcore people.
Jordan Ellenberg (1:50:50.180)
It's all back to the emo, Doc.
Lex Fridman (1:50:52.460)
Okay, so is there any, are you ever,
Jordan Ellenberg (1:50:54.780)
because I'm often uncomfortable with infinity,
Lex Fridman (1:50:58.020)
like psychologically.
Jordan Ellenberg (1:50:59.700)
I have trouble when that sneaks in there.
Lex Fridman (1:51:04.620)
It's because it works so damn well,
Jordan Ellenberg (1:51:06.660)
I get a little suspicious,
Lex Fridman (1:51:09.340)
because it could be almost like a crutch
Jordan Ellenberg (1:51:12.580)
or an oversimplification that's missing something profound
Lex Fridman (1:51:15.540)
about reality.
Jordan Ellenberg (1:51:17.500)
Well, so first of all, okay, if you say like,
Lex Fridman (1:51:20.720)
is there like a serious way of doing mathematics
Jordan Ellenberg (1:51:24.900)
that doesn't really treat infinity as a real thing
Lex Fridman (1:51:29.300)
or maybe it's kind of agnostic
Lex Fridman (1:51:30.600)
and it's like, I'm not really gonna make a firm statement
Lex Fridman (1:51:32.660)
about whether it's a real thing or not.
Jordan Ellenberg (1:51:33.980)
Yeah, that's called most of the history of mathematics.
Lex Fridman (1:51:36.620)
So it's only after Cantor that we really are sort of,
Jordan Ellenberg (1:51:41.520)
okay, we're gonna like have a notion
Lex Fridman (1:51:43.920)
of like the cardinality of an infinite set
Lex Fridman (1:51:45.660)
and like do something that you might call
Lex Fridman (1:51:49.100)
like the modern theory of infinity.
Jordan Ellenberg (1:51:51.340)
That said, obviously everybody was drawn to this notion
Lex Fridman (1:51:54.100)
and no, not everybody was comfortable with it.
Jordan Ellenberg (1:51:55.800)
Look, I mean, this is what happens with Newton.
Lex Fridman (1:51:57.700)
I mean, so Newton understands that to talk about tangents
Lex Fridman (1:52:01.380)
and to talk about instantaneous velocity,
Lex Fridman (1:52:04.580)
he has to do something that we would now call
Lex Fridman (1:52:06.620)
taking a limit, right?
Lex Fridman (1:52:08.700)
The fabled dy over dx, if you sort of go back
Jordan Ellenberg (1:52:11.260)
to your calculus class, for those who have taken calculus
Lex Fridman (1:52:13.100)
and remember this mysterious thing.
Lex Fridman (1:52:14.860)
And you know, what is it?
Lex Fridman (1:52:17.360)
What is it?
Jordan Ellenberg (1:52:18.200)
Well, he'd say like, well, it's like,
Lex Fridman (1:52:19.740)
you sort of divide the length of this line segment
Jordan Ellenberg (1:52:24.060)
by the length of this other line segment.
Lex Fridman (1:52:25.300)
And then you make them a little shorter
Lex Fridman (1:52:26.340)
and you divide again.
Lex Fridman (1:52:27.180)
And then you make them a little shorter
Lex Fridman (1:52:28.100)
and you divide again.
Lex Fridman (1:52:28.940)
And then you just keep on doing that
Jordan Ellenberg (1:52:29.780)
until they're like infinitely short
Lex Fridman (1:52:30.780)
and then you divide them again.
Jordan Ellenberg (1:52:32.520)
These quantities that are like, they're not zero,
Lex Fridman (1:52:36.360)
but they're also smaller than any actual number,
Jordan Ellenberg (1:52:42.020)
these infinitesimals.
Lex Fridman (1:52:43.420)
Well, people were queasy about it
Lex Fridman (1:52:46.380)
and they weren't wrong to be queasy about it, right?
Lex Fridman (1:52:48.180)
From a modern perspective, it was not really well formed.
Jordan Ellenberg (1:52:50.100)
There's this very famous critique of Newton
Lex Fridman (1:52:52.300)
by Bishop Berkeley, where he says like,
Lex Fridman (1:52:54.500)
what these things you define, like, you know,
Lex Fridman (1:52:57.820)
they're not zero, but they're smaller than any number.
Lex Fridman (1:53:00.260)
Are they the ghosts of departed quantities?
Lex Fridman (1:53:02.420)
That was this like ultra burn of Newton.
Lex Fridman (1:53:06.860)
And on the one hand, he was right.
Lex Fridman (1:53:10.040)
It wasn't really rigorous by modern standards.
Jordan Ellenberg (1:53:11.740)
On the other hand, like Newton was out there doing calculus
Lex Fridman (1:53:14.380)
and other people were not, right?
Jordan Ellenberg (1:53:15.380)
It works, it works.
Lex Fridman (1:53:17.380)
I think a sort of intuitionist view, for instance,
Jordan Ellenberg (1:53:20.660)
I would say would express serious doubt.
Lex Fridman (1:53:23.620)
And by the way, it's not just infinity.
Jordan Ellenberg (1:53:25.940)
It's like saying, I think we would express serious doubt
Lex Fridman (1:53:28.100)
that like the real numbers exist.
Jordan Ellenberg (1:53:31.320)
Now, most people are comfortable with the real numbers.
Lex Fridman (1:53:36.820)
Well, computer scientists with floating point number,
Jordan Ellenberg (1:53:39.220)
I mean, floating point arithmetic.
Lex Fridman (1:53:42.740)
That's a great point, actually.
Jordan Ellenberg (1:53:44.720)
I think in some sense, this flavor of doing math,
Lex Fridman (1:53:48.420)
saying we shouldn't talk about things
Jordan Ellenberg (1:53:51.220)
that we cannot specify in a finite amount of time,
Lex Fridman (1:53:53.620)
there's something very computational in flavor about that.
Lex Fridman (1:53:55.980)
And it's probably not a coincidence
Lex Fridman (1:53:57.580)
that it becomes popular in the 30s and 40s,
Jordan Ellenberg (1:54:01.740)
which is also like kind of like the dawn of ideas
Lex Fridman (1:54:04.980)
about formal computation, right?
Jordan Ellenberg (1:54:06.180)
You probably know the timeline better than I do.
Lex Fridman (1:54:07.940)
Sorry, what becomes popular?
Jordan Ellenberg (1:54:09.620)
These ideas that maybe we should be doing math
Lex Fridman (1:54:12.200)
in this more restrictive way where even a thing that,
Jordan Ellenberg (1:54:16.140)
because look, the origin of all this is like,
Lex Fridman (1:54:18.540)
number represents a magnitude, like the length of a line.
Lex Fridman (1:54:22.580)
So I mean, the idea that there's a continuum,
Lex Fridman (1:54:26.060)
there's sort of like, it's pretty old,
Lex Fridman (1:54:30.580)
but just because something is old
Lex Fridman (1:54:31.900)
doesn't mean we can't reject it if we want to.
Jordan Ellenberg (1:54:34.220)
Well, a lot of the fundamental ideas in computer science,
Lex Fridman (1:54:36.580)
when you talk about the complexity of problems,
Jordan Ellenberg (1:54:41.380)
to Turing himself, they rely on an infinity as well.
Lex Fridman (1:54:45.060)
The ideas that kind of challenge that,
Jordan Ellenberg (1:54:47.540)
the whole space of machine learning,
Lex Fridman (1:54:48.780)
I would say, challenges that.
Jordan Ellenberg (1:54:51.000)
It's almost like the engineering approach to things,
Lex Fridman (1:54:53.020)
like the floating point arithmetic.
Jordan Ellenberg (1:54:54.660)
The other one that, back to John Conway,
Lex Fridman (1:54:57.340)
that challenges this idea,
Jordan Ellenberg (1:55:00.780)
I mean, maybe to tie in the ideas of deformation theory
Lex Fridman (1:55:06.540)
and limits to infinity is this idea of cellular automata
Jordan Ellenberg (1:55:13.980)
with John Conway looking at the game of life,
Lex Fridman (1:55:17.340)
Stephen Wolfram's work,
Jordan Ellenberg (1:55:19.340)
that I've been a big fan of for a while, cellular automata.
Lex Fridman (1:55:22.580)
I was wondering if you have,
Jordan Ellenberg (1:55:23.780)
if you have ever encountered these kinds of objects,
Lex Fridman (1:55:26.900)
you ever looked at them as a mathematician,
Jordan Ellenberg (1:55:29.320)
where you have very simple rules of tiny little objects
Lex Fridman (1:55:34.840)
that when taken as a whole create incredible complexities,
Lex Fridman (1:55:37.980)
but are very difficult to analyze,
Lex Fridman (1:55:39.820)
very difficult to make sense of,
Jordan Ellenberg (1:55:41.980)
even though the one individual object, one part,
Lex Fridman (1:55:45.120)
it's like what we were saying about Andrew Wiles,
Jordan Ellenberg (1:55:47.540)
you can look at the deformation of a small piece
Lex Fridman (1:55:49.780)
to tell you about the whole.
Jordan Ellenberg (1:55:51.340)
It feels like with cellular automata
Lex Fridman (1:55:54.460)
or any kind of complex systems,
Jordan Ellenberg (1:55:57.340)
it's often very difficult to say something
Lex Fridman (1:55:59.820)
about the whole thing,
Jordan Ellenberg (1:56:01.620)
even when you can precisely describe the operation
Lex Fridman (1:56:05.100)
of the local neighborhoods.
Jordan Ellenberg (1:56:09.380)
Yeah, I mean, I love that subject.
Lex Fridman (1:56:10.980)
I haven't really done research on it myself.
Jordan Ellenberg (1:56:12.660)
I've played around with it.
Lex Fridman (1:56:13.540)
I'll send you a fun blog post I wrote
Jordan Ellenberg (1:56:15.060)
where I made some cool texture patterns
Lex Fridman (1:56:17.340)
from cellular automata that I, but.
Lex Fridman (1:56:20.980)
And those are really always compelling
Lex Fridman (1:56:22.460)
is like you create simple rules
Lex Fridman (1:56:24.140)
and they create some beautiful textures.
Lex Fridman (1:56:25.820)
It doesn't make any sense.
Jordan Ellenberg (1:56:26.660)
Actually, did you see, there was a great paper.
Lex Fridman (1:56:28.020)
I don't know if you saw this,
Jordan Ellenberg (1:56:28.980)
like a machine learning paper.
Lex Fridman (1:56:30.640)
Yes.
Jordan Ellenberg (1:56:31.480)
I don't know if you saw the one I'm talking about
Lex Fridman (1:56:32.300)
where they were like learning the texture
Jordan Ellenberg (1:56:33.300)
as like let's try to like reverse engineer
Lex Fridman (1:56:35.660)
and like learn a cellular automaton
Jordan Ellenberg (1:56:37.220)
that can reduce texture that looks like this
Lex Fridman (1:56:39.340)
from the images.
Jordan Ellenberg (1:56:40.340)
Very cool.
Lex Fridman (1:56:41.300)
And as you say, the thing you said is I feel the same way
Jordan Ellenberg (1:56:44.760)
when I read machine learning paper
Lex Fridman (1:56:45.980)
is that what's especially interesting
Jordan Ellenberg (1:56:47.660)
is the cases where it doesn't work.
Lex Fridman (1:56:49.540)
Like what does it do when it doesn't do the thing
Lex Fridman (1:56:51.260)
that you tried to train it to do?
Lex Fridman (1:56:53.380)
That's extremely interesting.
Jordan Ellenberg (1:56:54.480)
Yeah, yeah, that was a cool paper.
Lex Fridman (1:56:56.100)
So yeah, so let's start with the game of life.
Jordan Ellenberg (1:56:58.340)
Let's start with, or let's start with John Conway.
Lex Fridman (1:57:02.300)
So Conway.
Lex Fridman (1:57:03.620)
So yeah, so let's start with John Conway again.
Lex Fridman (1:57:06.060)
Just, I don't know, from my outsider's perspective,
Jordan Ellenberg (1:57:08.620)
there's not many mathematicians that stand out
Lex Fridman (1:57:11.500)
throughout the history of the 20th century.
Lex Fridman (1:57:13.800)
And he's one of them.
Lex Fridman (1:57:15.100)
I feel like he's not sufficiently recognized.
Jordan Ellenberg (1:57:18.180)
I think he's pretty recognized.
Lex Fridman (1:57:20.120)
Okay, well.
Jordan Ellenberg (1:57:21.120)
I mean, he was a full professor at Princeton
Lex Fridman (1:57:24.360)
for most of his life.
Jordan Ellenberg (1:57:25.200)
He was sort of certainly at the pinnacle of.
Lex Fridman (1:57:27.100)
Yeah, but I found myself every time I talk about Conway
Lex Fridman (1:57:30.180)
and how excited I am about him,
Lex Fridman (1:57:33.140)
I have to constantly explain to people who he is.
Lex Fridman (1:57:36.660)
And that's always a sad sign to me.
Lex Fridman (1:57:39.540)
But that's probably true for a lot of mathematicians.
Jordan Ellenberg (1:57:41.540)
I was about to say,
Lex Fridman (1:57:42.380)
I feel like you have a very elevated idea of how famous.
Jordan Ellenberg (1:57:44.940)
This is what happens when you grow up in the Soviet Union
Lex Fridman (1:57:46.740)
or you think the mathematicians are like very, very famous.
Jordan Ellenberg (1:57:49.860)
Yeah, but I'm not actually so convinced at a tiny tangent
Lex Fridman (1:57:53.100)
that that shouldn't be so.
Jordan Ellenberg (1:57:54.660)
I mean, there's, it's not obvious to me
Lex Fridman (1:57:57.640)
that that's one of the,
Jordan Ellenberg (1:57:59.160)
like if I were to analyze American society,
Lex Fridman (1:58:01.540)
that perhaps elevating mathematical and scientific thinking
Jordan Ellenberg (1:58:05.060)
to a little bit higher level would benefit the society.
Lex Fridman (1:58:08.740)
Well, both in discovering the beauty of what it is
Jordan Ellenberg (1:58:11.300)
to be human and for actually creating cool technology,
Lex Fridman (1:58:15.020)
better iPhones.
Lex Fridman (1:58:16.240)
But anyway, John Conway.
Lex Fridman (1:58:18.140)
Yeah, and Conway is such a perfect example
Jordan Ellenberg (1:58:20.000)
of somebody whose humanity was,
Lex Fridman (1:58:22.060)
and his personality was like wound up
Lex Fridman (1:58:24.020)
with his mathematics, right?
Lex Fridman (1:58:25.020)
And so it's not, sometimes I think people
Jordan Ellenberg (1:58:26.780)
who are outside the field think of mathematics
Lex Fridman (1:58:28.620)
as this kind of like cold thing that you do
Jordan Ellenberg (1:58:31.220)
separate from your existence as a human being.
Lex Fridman (1:58:33.100)
No way, your personality is in there,
Jordan Ellenberg (1:58:34.780)
just as it would be in like a novel you wrote
Lex Fridman (1:58:37.140)
or a painting you painted
Jordan Ellenberg (1:58:38.220)
or just like the way you walk down the street.
Lex Fridman (1:58:40.100)
Like it's in there, it's you doing it.
Lex Fridman (1:58:41.780)
And Conway was certainly a singular personality.
Lex Fridman (1:58:46.240)
I think anybody would say that he was playful,
Jordan Ellenberg (1:58:50.980)
like everything was a game to him.
Lex Fridman (1:58:54.240)
Now, what you might think I'm gonna say,
Lex Fridman (1:58:56.580)
and it's true is that he sort of was very playful
Lex Fridman (1:58:59.260)
in his way of doing mathematics,
Lex Fridman (1:59:01.780)
but it's also true, it went both ways.
Lex Fridman (1:59:03.700)
He also sort of made mathematics out of games.
Jordan Ellenberg (1:59:06.220)
He like looked at, he was a constant inventor of games
Lex Fridman (1:59:08.880)
or like crazy names.
Lex Fridman (1:59:10.080)
And then he would sort of analyze those games mathematically
Lex Fridman (1:59:15.220)
to the point that he,
Lex Fridman (1:59:16.300)
and then later collaborating with Knuth like,
Lex Fridman (1:59:19.120)
created this number system, the serial numbers
Jordan Ellenberg (1:59:22.420)
in which actually each number is a game.
Lex Fridman (1:59:25.200)
There's a wonderful book about this called,
Jordan Ellenberg (1:59:26.640)
I mean, there are his own books.
Lex Fridman (1:59:27.620)
And then there's like a book that he wrote
Jordan Ellenberg (1:59:28.780)
with Berlekamp and Guy called Winning Ways,
Lex Fridman (1:59:31.180)
which is such a rich source of ideas.
Lex Fridman (1:59:35.260)
And he too kind of has his own crazy number system
Lex Fridman (1:59:41.720)
in which by the way, there are these infinitesimals,
Jordan Ellenberg (1:59:44.240)
the ghosts of departed quantities.
Lex Fridman (1:59:45.640)
They're in there now, not as ghosts,
Lex Fridman (1:59:47.900)
but as like certain kind of two player games.
Lex Fridman (1:59:53.620)
So, he was a guy, so I knew him when I was a postdoc
Jordan Ellenberg (20:00.000)
about how it was in the Soviet Union.
Lex Fridman (20:01.560)
I mean, there was, and we'll talk about the Olympiad.
Jordan Ellenberg (20:04.960)
I just remember that there was this feeling
Lex Fridman (20:09.720)
like the world hung in a balance
Lex Fridman (20:14.320)
and you could save the world with the tools of science.
Lex Fridman (20:19.320)
And mathematics was like the superpower that fuels science.
Lex Fridman (20:26.520)
And so like people were seen as, you know,
Lex Fridman (20:30.240)
people in America often idolize athletes,
Lex Fridman (20:32.880)
but ultimately the best athletes in the world,
Lex Fridman (20:36.920)
they just throw a ball into a basket.
Lex Fridman (20:40.000)
So like there's not, what people really enjoy about sports,
Lex Fridman (20:44.240)
I love sports, is like excellence at the highest level.
Lex Fridman (20:48.640)
But when you take that with mathematics and science,
Lex Fridman (20:51.320)
people also enjoyed excellence in science and mathematics
Jordan Ellenberg (20:54.280)
in the Soviet Union, but there's an extra sense
Lex Fridman (20:56.880)
that that excellence would lead to a better world.
Lex Fridman (21:01.400)
So that created all the usual things you think about
Lex Fridman (21:07.360)
with the Olympics, which is like extreme competitiveness.
Lex Fridman (21:12.160)
But it also created this sense that in the modern era
Lex Fridman (21:15.120)
in America, somebody like Elon Musk, whatever you think
Jordan Ellenberg (21:19.400)
of him, like Jeff Bezos, those folks,
Lex Fridman (21:21.480)
they inspire the possibility that one person
Jordan Ellenberg (21:24.480)
or a group of smart people can change the world.
Lex Fridman (21:27.040)
Like not just be good at what they do,
Lex Fridman (21:29.040)
but actually change the world.
Lex Fridman (21:30.640)
Mathematics was at the core of that.
Jordan Ellenberg (21:33.320)
I don't know, there's a romanticism around it too.
Lex Fridman (21:36.040)
Like when you read books about in America,
Jordan Ellenberg (21:39.480)
people romanticize certain things like baseball, for example.
Lex Fridman (21:42.640)
There's like these beautiful poetic writing
Jordan Ellenberg (21:45.680)
about the game of baseball.
Lex Fridman (21:47.400)
The same was the feeling with mathematics and science
Jordan Ellenberg (21:50.640)
in the Soviet Union, and it was in the air.
Lex Fridman (21:53.160)
Everybody was forced to take high level mathematics courses.
Jordan Ellenberg (21:57.280)
Like you took a lot of math, you took a lot of science
Lex Fridman (22:00.480)
and a lot of like really rigorous literature.
Jordan Ellenberg (22:03.240)
Like the level of education in Russia,
Lex Fridman (22:06.560)
this could be true in China, I'm not sure,
Jordan Ellenberg (22:09.200)
in a lot of countries is in whatever that's called,
Lex Fridman (22:14.120)
it's K to 12 in America, but like young people education.
Jordan Ellenberg (22:18.760)
The level they were challenged to learn at is incredible.
Lex Fridman (22:23.360)
It's like America falls far behind, I would say.
Jordan Ellenberg (22:27.960)
America then quickly catches up
Lex Fridman (22:29.880)
and then exceeds everybody else as you start approaching
Jordan Ellenberg (22:33.880)
the end of high school to college.
Lex Fridman (22:35.360)
Like the university system in the United States
Jordan Ellenberg (22:37.040)
arguably is the best in the world.
Lex Fridman (22:39.280)
But like what we challenge everybody,
Jordan Ellenberg (22:44.200)
it's not just like the good, the A students,
Lex Fridman (22:46.560)
but everybody to learn in the Soviet Union was fascinating.
Jordan Ellenberg (22:50.200)
I think I'm gonna pick up on something you said.
Lex Fridman (22:52.080)
I think you would love a book called
Jordan Ellenberg (22:53.800)
Dual at Dawn by Amir Alexander,
Lex Fridman (22:56.360)
which I think some of the things you're responding to
Lex Fridman (22:58.440)
and what I wrote, I think I first got turned on to
Lex Fridman (23:01.040)
by Amir's work, he's a historian of math.
Lex Fridman (23:02.880)
And he writes about the story of Everest to Galois,
Lex Fridman (23:06.040)
which is a story that's well known to all mathematicians,
Jordan Ellenberg (23:08.320)
this kind of like very, very romantic figure
Lex Fridman (23:12.880)
who he really sort of like begins the development of this
Jordan Ellenberg (23:18.040)
or this theory of groups that I mentioned earlier,
Lex Fridman (23:20.120)
this general theory of symmetries
Lex Fridman (23:23.480)
and then dies in a duel in his early 20s,
Lex Fridman (23:25.520)
like all this stuff, mostly unpublished.
Jordan Ellenberg (23:28.400)
It's a very, very romantic story that we all learn.
Lex Fridman (23:32.400)
And much of it is true,
Lex Fridman (23:33.440)
but Alexander really lays out just how much
Lex Fridman (23:37.600)
the way people thought about math in those times
Jordan Ellenberg (23:40.480)
in the early 19th century was wound up with,
Lex Fridman (23:43.200)
as you say, romanticism.
Jordan Ellenberg (23:44.480)
I mean, that's when the romantic movement takes place
Lex Fridman (23:47.160)
and he really outlines how people were predisposed
Jordan Ellenberg (23:51.200)
to think about mathematics in that way
Lex Fridman (23:52.800)
because they thought about poetry that way
Lex Fridman (23:54.240)
and they thought about music that way.
Lex Fridman (23:55.680)
It was the mood of the era to think about
Jordan Ellenberg (23:58.240)
we're reaching for the transcendent,
Lex Fridman (23:59.920)
we're sort of reaching for sort of direct contact
Jordan Ellenberg (24:02.000)
with the divine.
Lex Fridman (24:02.840)
And part of the reason that we think of Gawa that way
Jordan Ellenberg (24:06.040)
was because Gawa himself was a creature of that era
Lex Fridman (24:08.680)
and he romanticized himself.
Jordan Ellenberg (24:10.600)
I mean, now we know he wrote lots of letters
Lex Fridman (24:12.640)
and he was kind of like, I mean, in modern terms,
Jordan Ellenberg (24:14.880)
we would say he was extremely emo.
Lex Fridman (24:16.520)
Like we wrote all these letters
Jordan Ellenberg (24:19.800)
about his like florid feelings
Lex Fridman (24:21.320)
and like the fire within him about the mathematics.
Lex Fridman (24:23.280)
And so it's just as you say
Lex Fridman (24:26.280)
that the math history touches human history.
Jordan Ellenberg (24:29.600)
They're never separate because math is made of people.
Lex Fridman (24:32.720)
I mean, that's what, it's people who do it
Lex Fridman (24:35.560)
and we're human beings doing it
Lex Fridman (24:36.840)
and we do it within whatever community we're in
Lex Fridman (24:39.120)
and we do it affected by the mores
Lex Fridman (24:42.640)
of the society around us.
Lex Fridman (24:44.080)
So the French, the Germans and Poincare.
Lex Fridman (24:47.360)
Yes, okay, so back to Poincare.
Lex Fridman (24:48.880)
So he's, you know, it's funny.
Lex Fridman (24:52.520)
This book is filled with kind of mathematical characters
Jordan Ellenberg (24:55.880)
who often are kind of peevish or get into feuds
Lex Fridman (25:00.080)
or sort of have like weird enthusiasms
Jordan Ellenberg (25:03.840)
because those people are fun to write about
Lex Fridman (25:05.160)
and they sort of like say very salty things.
Jordan Ellenberg (25:07.440)
Poincare is actually none of this.
Lex Fridman (25:09.560)
As far as I can tell, he was an extremely normal dude
Jordan Ellenberg (25:12.440)
who didn't get into fights with people
Lex Fridman (25:15.240)
and everybody liked him
Lex Fridman (25:16.280)
and he was like pretty personally modest
Lex Fridman (25:18.040)
and he had very regular habits.
Lex Fridman (25:20.240)
You know what I mean?
Lex Fridman (25:21.080)
He did math for like four hours in the morning
Lex Fridman (25:23.760)
and four hours in the evening and that was it.
Lex Fridman (25:25.640)
Like he had his schedule.
Jordan Ellenberg (25:28.200)
I actually, it was like, I still am feeling like
Lex Fridman (25:31.640)
somebody's gonna tell me now that the book is out,
Jordan Ellenberg (25:33.360)
like, oh, didn't you know about this
Lex Fridman (25:34.720)
like incredibly sordid episode?
Jordan Ellenberg (25:37.000)
As far as I could tell, a completely normal guy.
Lex Fridman (25:39.920)
But he just kind of, in many ways,
Jordan Ellenberg (25:44.280)
creates the geometric world in which we live
Lex Fridman (25:47.760)
and his first really big success is this prize paper
Jordan Ellenberg (25:53.360)
he writes for this prize offered by the King of Sweden
Lex Fridman (25:55.960)
for the study of the three body problem.
Jordan Ellenberg (26:01.080)
The study of what we can say about, yeah,
Lex Fridman (26:04.240)
three astronomical objects moving
Jordan Ellenberg (26:07.280)
in what you might think would be this very simple way.
Lex Fridman (26:09.080)
Nothing's going on except gravity.
Lex Fridman (26:12.240)
So what's the three body problem?
Lex Fridman (26:13.640)
Why is it a problem?
Lex Fridman (26:15.000)
So the problem is to understand
Lex Fridman (26:16.800)
when this motion is stable and when it's not.
Lex Fridman (26:20.000)
So stable meaning they would sort of like end up
Lex Fridman (26:21.840)
in some kind of periodic orbit.
Jordan Ellenberg (26:23.600)
Or I guess it would mean, sorry,
Lex Fridman (26:25.400)
stable would mean they never sort of fly off
Jordan Ellenberg (26:26.960)
far apart from each other.
Lex Fridman (26:28.040)
And unstable would mean like eventually they fly apart.
Lex Fridman (26:30.160)
So understanding two bodies is much easier.
Lex Fridman (26:32.880)
Yes, exactly.
Jordan Ellenberg (26:33.720)
When you have the third wheel is always a problem.
Lex Fridman (26:36.480)
This is what Newton knew.
Jordan Ellenberg (26:37.320)
Two bodies, they sort of orbit each other
Lex Fridman (26:38.760)
in some kind of either in an ellipse,
Jordan Ellenberg (26:41.280)
which is the stable case.
Lex Fridman (26:42.240)
You know, that's what the planets do that we know.
Jordan Ellenberg (26:46.400)
Or one travels on a hyperbola around the other.
Lex Fridman (26:49.400)
That's the unstable case.
Jordan Ellenberg (26:50.320)
It sort of like zooms in from far away,
Lex Fridman (26:51.920)
sort of like whips around the heavier thing
Lex Fridman (26:54.280)
and like zooms out.
Lex Fridman (26:56.720)
Those are basically the two options.
Lex Fridman (26:58.120)
So it's a very simple and easy to classify story.
Lex Fridman (27:00.840)
With three bodies, just the small switch from two to three,
Jordan Ellenberg (27:04.160)
it's a complete zoo.
Lex Fridman (27:05.200)
It's the first, what we would say now
Jordan Ellenberg (27:07.000)
is it's the first example of what's called chaotic dynamics,
Lex Fridman (27:09.920)
where the stable solutions and the unstable solutions,
Jordan Ellenberg (27:13.000)
they're kind of like wound in among each other.
Lex Fridman (27:14.480)
And a very, very, very tiny change in the initial conditions
Jordan Ellenberg (27:17.640)
can make the longterm behavior of the system
Lex Fridman (27:20.200)
completely different.
Lex Fridman (27:21.200)
So Poincare was the first to recognize
Lex Fridman (27:22.960)
that that phenomenon even existed.
Lex Fridman (27:27.000)
What about the conjecture that carries his name?
Lex Fridman (27:31.120)
Right, so he also was one of the pioneers
Jordan Ellenberg (27:36.880)
of taking geometry, which until that point
Lex Fridman (27:41.440)
had been largely the study of two
Lex Fridman (27:44.080)
and three dimensional objects,
Lex Fridman (27:45.240)
because that's like what we see, right?
Jordan Ellenberg (27:47.480)
That's those are the objects we interact with.
Lex Fridman (27:49.680)
He developed the subject we now called topology.
Jordan Ellenberg (27:53.560)
He called it analysis situs.
Lex Fridman (27:55.320)
He was a very well spoken guy with a lot of slogans,
Lex Fridman (27:57.840)
but that name did not,
Lex Fridman (27:59.600)
you can see why that name did not catch on.
Lex Fridman (28:01.120)
So now it's called topology now.
Lex Fridman (28:05.080)
Sorry, what was it called before?
Jordan Ellenberg (28:06.280)
Analysis situs, which I guess sort of roughly means
Lex Fridman (28:09.360)
like the analysis of location or something like that.
Jordan Ellenberg (28:11.680)
Like it's a Latin phrase.
Lex Fridman (28:14.160)
Partly because he understood that even to understand
Jordan Ellenberg (28:19.800)
stuff that's going on in our physical world,
Lex Fridman (28:22.480)
you have to study higher dimensional spaces.
Lex Fridman (28:24.400)
How does this work?
Lex Fridman (28:25.520)
And this is kind of like where my brain went to it
Jordan Ellenberg (28:27.480)
because you were talking about not just where things are,
Lex Fridman (28:29.880)
but what their path is, how they're moving
Jordan Ellenberg (28:31.720)
when we were talking about the path from two to three.
Lex Fridman (28:34.840)
He understood that if you wanna study
Jordan Ellenberg (28:36.240)
three bodies moving in space,
Lex Fridman (28:39.600)
well, each body, it has a location where it is.
Lex Fridman (28:44.040)
So it has an X coordinate, a Y coordinate,
Lex Fridman (28:45.920)
a Z coordinate, right?
Jordan Ellenberg (28:46.760)
I can specify a point in space by giving you three numbers,
Lex Fridman (28:49.440)
but it also at each moment has a velocity.
Lex Fridman (28:53.400)
So it turns out that really to understand what's going on,
Lex Fridman (28:56.520)
you can't think of it as a point or you could,
Lex Fridman (28:58.920)
but it's better not to think of it as a point
Lex Fridman (29:01.040)
in three dimensional space that's moving.
Jordan Ellenberg (29:03.280)
It's better to think of it as a point
Lex Fridman (29:04.440)
in six dimensional space where the coordinates
Jordan Ellenberg (29:06.360)
are where is it and what's its velocity right now.
Lex Fridman (29:09.320)
That's a higher dimensional space called phase space.
Lex Fridman (29:11.800)
And if you haven't thought about this before,
Lex Fridman (29:13.200)
I admit that it's a little bit mind bending,
Lex Fridman (29:15.920)
but what he needed then was a geometry
Lex Fridman (29:20.720)
that was flexible enough,
Jordan Ellenberg (29:22.720)
not just to talk about two dimensional spaces
Lex Fridman (29:24.520)
or three dimensional spaces, but any dimensional space.
Lex Fridman (29:27.440)
So the sort of famous first line of this paper
Lex Fridman (29:29.320)
where he introduces analysis of Cetus
Jordan Ellenberg (29:30.800)
is no one doubts nowadays that the geometry
Lex Fridman (29:34.280)
of n dimensional space is an actually existing thing, right?
Jordan Ellenberg (29:37.720)
I think that maybe that had been controversial.
Lex Fridman (29:39.600)
And he's saying like, look, let's face it,
Jordan Ellenberg (29:41.360)
just because it's not physical doesn't mean it's not there.
Lex Fridman (29:44.040)
It doesn't mean we shouldn't study it.
Jordan Ellenberg (29:46.000)
Interesting.
Lex Fridman (29:46.920)
He wasn't jumping to the physical interpretation.
Jordan Ellenberg (29:49.760)
Like it can be real,
Lex Fridman (29:51.640)
even if it's not perceivable to the human cognition.
Jordan Ellenberg (29:55.720)
I think that's right.
Lex Fridman (29:56.880)
I think, don't get me wrong,
Jordan Ellenberg (29:58.400)
Poincare never strays far from physics.
Lex Fridman (2:00:00.280)
and I knew him at Princeton
Lex Fridman (2:00:01.280)
and our research overlapped in some ways.
Lex Fridman (2:00:03.620)
Now it was on stuff that he had worked on many years before.
Jordan Ellenberg (2:00:05.880)
The stuff I was working on kind of connected
Lex Fridman (2:00:07.400)
with stuff in group theory,
Jordan Ellenberg (2:00:08.280)
which somehow seems to keep coming up.
Lex Fridman (2:00:13.880)
And so I often would like sort of ask him a question.
Jordan Ellenberg (2:00:16.040)
I would sort of come upon him in the common room
Lex Fridman (2:00:17.680)
and I would ask him a question about something.
Lex Fridman (2:00:19.080)
And just anytime you turned him on, you know what I mean?
Lex Fridman (2:00:23.780)
You sort of asked the question,
Jordan Ellenberg (2:00:25.240)
it was just like turning a knob and winding him up
Lex Fridman (2:00:28.280)
and he would just go and you would get a response
Jordan Ellenberg (2:00:31.040)
that was like so rich and went so many places
Lex Fridman (2:00:35.240)
and taught you so much.
Lex Fridman (2:00:37.360)
And usually had nothing to do with your question.
Lex Fridman (2:00:40.160)
Usually your question was just a prompt to him.
Jordan Ellenberg (2:00:43.080)
You couldn't count on actually getting the question answered.
Lex Fridman (2:00:44.760)
Yeah, those brilliant, curious minds even at that age.
Jordan Ellenberg (2:00:47.400)
Yeah, it was definitely a huge loss.
Lex Fridman (2:00:51.920)
But on his game of life,
Jordan Ellenberg (2:00:54.680)
which was I think he developed in the 70s
Lex Fridman (2:00:56.960)
as almost like a side thing, a fun little experiment.
Jordan Ellenberg (2:00:59.720)
His game of life is this, it's a very simple algorithm.
Lex Fridman (2:01:05.200)
It's not really a game per se
Jordan Ellenberg (2:01:07.800)
in the sense of the kinds of games that he liked
Lex Fridman (2:01:09.520)
where people played against each other.
Lex Fridman (2:01:12.780)
But essentially it's a game that you play
Lex Fridman (2:01:16.560)
with marking little squares on the sheet of graph paper.
Lex Fridman (2:01:20.400)
And in the 70s, I think he was like literally doing it
Lex Fridman (2:01:22.360)
with like a pen on graph paper.
Jordan Ellenberg (2:01:24.240)
You have some configuration of squares.
Lex Fridman (2:01:26.040)
Some of the squares in the graph paper are filled in,
Jordan Ellenberg (2:01:28.280)
some are not.
Lex Fridman (2:01:29.120)
And there's a rule, a single rule that tells you
Jordan Ellenberg (2:01:33.360)
at the next stage, which squares are filled in
Lex Fridman (2:01:36.480)
and which squares are not.
Jordan Ellenberg (2:01:38.120)
Sometimes an empty square gets filled in,
Lex Fridman (2:01:39.720)
that's called birth.
Jordan Ellenberg (2:01:40.560)
Sometimes a square that's filled in gets erased,
Lex Fridman (2:01:43.000)
that's called death.
Lex Fridman (2:01:43.960)
And there's rules for which squares are born
Lex Fridman (2:01:45.880)
and which squares die.
Jordan Ellenberg (2:01:50.960)
The rule is very simple.
Lex Fridman (2:01:51.920)
You can write it on one line.
Lex Fridman (2:01:53.640)
And then the great miracle is that you can start
Lex Fridman (2:01:56.240)
from some very innocent looking little small set of boxes
Lex Fridman (2:02:00.320)
and get these results of incredible richness.
Lex Fridman (2:02:04.160)
And of course, nowadays you don't do it on paper.
Jordan Ellenberg (2:02:05.680)
Nowadays you do it in a computer.
Lex Fridman (2:02:07.000)
There's actually a great iPad app called Golly,
Jordan Ellenberg (2:02:09.320)
which I really like that has like Conway's original rule
Lex Fridman (2:02:12.800)
and like, gosh, like hundreds of other variants
Lex Fridman (2:02:15.600)
and it's a lightning fast.
Lex Fridman (2:02:16.820)
So you can just be like,
Jordan Ellenberg (2:02:17.660)
I wanna see 10,000 generations of this rule play out
Lex Fridman (2:02:21.400)
like faster than your eye can even follow.
Lex Fridman (2:02:23.000)
And it's like amazing.
Lex Fridman (2:02:24.040)
So I highly recommend it if this is at all intriguing to you
Jordan Ellenberg (2:02:26.360)
getting Golly on your iOS device.
Lex Fridman (2:02:29.400)
And you can do this kind of process,
Jordan Ellenberg (2:02:30.740)
which I really enjoy doing,
Lex Fridman (2:02:32.080)
which is almost from like putting a Darwin hat on
Jordan Ellenberg (2:02:35.080)
or a biologist hat on and doing analysis
Lex Fridman (2:02:38.600)
of a higher level of abstraction,
Jordan Ellenberg (2:02:41.500)
like the organisms that spring up.
Lex Fridman (2:02:43.520)
Cause there's different kinds of organisms.
Jordan Ellenberg (2:02:45.160)
Like you can think of them as species
Lex Fridman (2:02:46.880)
and they interact with each other.
Jordan Ellenberg (2:02:48.640)
They can, there's gliders, they shoot different,
Lex Fridman (2:02:51.040)
there's like things that can travel around.
Jordan Ellenberg (2:02:54.320)
There's things that can,
Lex Fridman (2:02:55.920)
glider guns that can generate those gliders.
Jordan Ellenberg (2:02:59.520)
You can use the same kind of language
Lex Fridman (2:03:01.800)
as you would about describing a biological system.
Lex Fridman (2:03:04.600)
So it's a wonderful laboratory
Lex Fridman (2:03:06.240)
and it's kind of a rebuke to someone
Jordan Ellenberg (2:03:07.920)
who doesn't think that like very, very rich,
Lex Fridman (2:03:10.960)
complex structure can come from very simple underlying laws.
Jordan Ellenberg (2:03:16.440)
Like it definitely can.
Lex Fridman (2:03:18.880)
Now, here's what's interesting.
Jordan Ellenberg (2:03:20.600)
If you just pick like some random rule,
Lex Fridman (2:03:24.640)
you wouldn't get interesting complexity.
Jordan Ellenberg (2:03:26.200)
I think that's one of the most interesting things
Lex Fridman (2:03:28.360)
of these, one of these most interesting features
Jordan Ellenberg (2:03:31.440)
of this whole subject,
Lex Fridman (2:03:32.280)
that the rules have to be tuned just right.
Jordan Ellenberg (2:03:34.100)
Like a sort of typical rule set
Lex Fridman (2:03:36.060)
doesn't generate any kind of interesting behavior.
Lex Fridman (2:03:38.760)
But some do.
Lex Fridman (2:03:40.660)
And I don't think we have a clear way of understanding
Jordan Ellenberg (2:03:44.560)
which do and which don't.
Lex Fridman (2:03:45.400)
Maybe Steven thinks he does, I don't know.
Jordan Ellenberg (2:03:47.320)
No, no, it's a giant mystery where Steven Wolfram did is,
Lex Fridman (2:03:53.960)
now there's a whole interesting aspect to the fact
Jordan Ellenberg (2:03:56.000)
that he's a little bit of an outcast
Lex Fridman (2:03:57.640)
in the mathematics and physics community
Jordan Ellenberg (2:03:59.920)
because he's so focused on a particular,
Lex Fridman (2:04:02.640)
his particular work.
Jordan Ellenberg (2:04:03.960)
I think if you put ego aside,
Lex Fridman (2:04:05.800)
which I think unfairly some people
Jordan Ellenberg (2:04:08.640)
are not able to look beyond,
Lex Fridman (2:04:09.940)
I think his work is actually quite brilliant.
Lex Fridman (2:04:11.880)
But what he did is exactly this process
Lex Fridman (2:04:13.840)
of Darwin like exploration.
Jordan Ellenberg (2:04:15.920)
He's taking these very simple ideas
Lex Fridman (2:04:17.400)
and writing a thousand page book on them,
Jordan Ellenberg (2:04:19.880)
meaning like, let's play around with this thing.
Lex Fridman (2:04:22.280)
Let's see.
Lex Fridman (2:04:23.480)
And can we figure anything out?
Lex Fridman (2:04:25.480)
Spoiler alert, no, we can't.
Jordan Ellenberg (2:04:28.400)
In fact, he does a challenge.
Lex Fridman (2:04:31.040)
I think it's like rule 30 challenge,
Jordan Ellenberg (2:04:33.240)
which is quite interesting,
Lex Fridman (2:04:34.160)
just simply for machine learning people,
Jordan Ellenberg (2:04:36.400)
for mathematics people,
Lex Fridman (2:04:39.420)
is can you predict the middle column?
Jordan Ellenberg (2:04:41.800)
For his, it's a 1D cellular automata.
Lex Fridman (2:04:45.980)
Can you, generally speaking,
Lex Fridman (2:04:48.240)
can you predict anything about
Lex Fridman (2:04:50.280)
how a particular rule will evolve just in the future?
Jordan Ellenberg (2:04:55.480)
Very simple.
Lex Fridman (2:04:56.320)
Just looking at one particular part of the world,
Jordan Ellenberg (2:04:59.040)
just zooming in on that part,
Lex Fridman (2:05:02.040)
100 steps ahead, can you predict something?
Lex Fridman (2:05:04.720)
And the challenge is to do that kind of prediction
Lex Fridman (2:05:08.800)
so far as nobody's come up with an answer.
Lex Fridman (2:05:10.340)
But the point is like, we can't.
Lex Fridman (2:05:13.520)
We don't have tools or maybe it's impossible or,
Jordan Ellenberg (2:05:16.880)
I mean, he has these kind of laws of irreducibility
Lex Fridman (2:05:19.960)
that he refers to, but it's poetry.
Jordan Ellenberg (2:05:21.520)
It's like, we can't prove these things.
Lex Fridman (2:05:22.880)
It seems like we can't.
Jordan Ellenberg (2:05:24.420)
That's the basic.
Lex Fridman (2:05:26.280)
It almost sounds like ancient mathematics
Jordan Ellenberg (2:05:28.520)
or something like that, where you're like,
Lex Fridman (2:05:30.060)
the gods will not allow us to predict the cellular automata.
Lex Fridman (2:05:34.320)
But that's fascinating that we can't.
Lex Fridman (2:05:37.840)
I'm not sure what to make of it.
Lex Fridman (2:05:39.120)
And there's power to calling this particular set of rules
Lex Fridman (2:05:43.000)
game of life as Conway did, because not exactly sure,
Lex Fridman (2:05:47.880)
but I think he had a sense that there's some core ideas here
Lex Fridman (2:05:51.480)
that are fundamental to life, to complex systems,
Jordan Ellenberg (2:05:55.800)
to the way life emerge on earth.
Lex Fridman (2:05:59.320)
I'm not sure I think Conway thought that.
Jordan Ellenberg (2:06:01.720)
It's something that, I mean, Conway always had
Lex Fridman (2:06:03.200)
a rather ambivalent relationship with the game of life
Jordan Ellenberg (2:06:05.880)
because I think he saw it as,
Lex Fridman (2:06:11.120)
it was certainly the thing he was most famous for
Jordan Ellenberg (2:06:12.960)
in the outside world.
Lex Fridman (2:06:14.680)
And I think that he, his view, which is correct,
Jordan Ellenberg (2:06:18.640)
is that he had done things
Lex Fridman (2:06:19.600)
that were much deeper mathematically than that.
Lex Fridman (2:06:22.120)
And I think it always aggrieved him a bit
Lex Fridman (2:06:24.260)
that he was the game of life guy
Jordan Ellenberg (2:06:26.200)
when he proved all these wonderful theorems
Lex Fridman (2:06:28.640)
and created all these wonderful games,
Jordan Ellenberg (2:06:32.080)
created the serial numbers.
Lex Fridman (2:06:33.360)
I mean, he was a very tireless guy
Jordan Ellenberg (2:06:36.520)
who just did an incredibly variegated array of stuff.
Lex Fridman (2:06:40.800)
So he was exactly the kind of person
Jordan Ellenberg (2:06:42.480)
who you would never want to reduce to one achievement.
Lex Fridman (2:06:45.600)
You know what I mean?
Jordan Ellenberg (2:06:46.920)
Let me ask you about group theory.
Lex Fridman (2:06:50.400)
You mentioned it a few times.
Lex Fridman (2:06:51.800)
What is group theory?
Lex Fridman (2:06:53.440)
What is an idea from group theory that you find beautiful?
Jordan Ellenberg (2:06:58.600)
Well, so I would say group theory sort of starts
Lex Fridman (2:07:01.960)
as the general theory of symmetries,
Jordan Ellenberg (2:07:04.660)
that people looked at different kinds of things
Lex Fridman (2:07:08.280)
and said, as we said, oh, it could have,
Jordan Ellenberg (2:07:12.960)
maybe all there is is symmetry from left to right,
Lex Fridman (2:07:16.440)
like a human being, right?
Jordan Ellenberg (2:07:17.760)
That's roughly bilaterally symmetric, as we say.
Lex Fridman (2:07:21.320)
So there's two symmetries.
Lex Fridman (2:07:23.840)
And then you're like, well, wait, didn't I say
Lex Fridman (2:07:24.920)
there's just one, there's just left to right?
Jordan Ellenberg (2:07:26.720)
Well, we always count the symmetry of doing nothing.
Lex Fridman (2:07:30.080)
We always count the symmetry
Jordan Ellenberg (2:07:31.100)
that's like there's flip and don't flip.
Lex Fridman (2:07:33.080)
Those are the two configurations that you can be in.
Lex Fridman (2:07:35.200)
So there's two.
Lex Fridman (2:07:37.600)
You know, something like a rectangle
Jordan Ellenberg (2:07:40.240)
is bilaterally symmetric.
Lex Fridman (2:07:41.560)
You can flip it left to right,
Lex Fridman (2:07:42.600)
but you can also flip it top to bottom.
Lex Fridman (2:07:45.880)
So there's actually four symmetries.
Jordan Ellenberg (2:07:47.680)
There's do nothing, flip it left to right
Lex Fridman (2:07:50.320)
and flip it top to bottom or do both of those things.
Lex Fridman (2:07:52.960)
And then a square, there's even more,
Lex Fridman (2:07:59.700)
because now you can rotate it.
Jordan Ellenberg (2:08:01.700)
You can rotate it by 90 degrees.
Lex Fridman (2:08:03.060)
So you can't do that.
Jordan Ellenberg (2:08:03.900)
That's not a symmetry of the rectangle.
Lex Fridman (2:08:04.940)
If you try to rotate it 90 degrees,
Jordan Ellenberg (2:08:06.180)
you get a rectangle oriented in a different way.
Lex Fridman (2:08:08.880)
So a person has two symmetries,
Jordan Ellenberg (2:08:11.880)
a rectangle four, a square eight,
Lex Fridman (2:08:14.420)
different kinds of shapes
Jordan Ellenberg (2:08:15.420)
have different numbers of symmetries.
Lex Fridman (2:08:18.860)
And the real observation is that
Jordan Ellenberg (2:08:19.940)
that's just not like a set of things, they can be combined.
Lex Fridman (2:08:25.060)
You do one symmetry, then you do another.
Jordan Ellenberg (2:08:27.700)
The result of that is some third symmetry.
Lex Fridman (2:08:31.020)
So a group really abstracts away this notion of saying,
Jordan Ellenberg (2:08:38.780)
it's just some collection of transformations
Lex Fridman (2:08:41.180)
you can do to a thing
Jordan Ellenberg (2:08:42.060)
where you combine any two of them to get a third.
Lex Fridman (2:08:44.380)
So, you know, a place where this comes up
Jordan Ellenberg (2:08:45.620)
in computer science is in sorting,
Lex Fridman (2:08:48.260)
because the ways of permuting a set,
Jordan Ellenberg (2:08:50.500)
the ways of taking sort of some set of things
Lex Fridman (2:08:52.340)
you have on the table
Lex Fridman (2:08:53.180)
and putting them in a different order,
Lex Fridman (2:08:54.260)
shuffling a deck of cards, for instance,
Jordan Ellenberg (2:08:56.100)
those are the symmetries of the deck.
Lex Fridman (2:08:57.580)
And there's a lot of them.
Jordan Ellenberg (2:08:58.420)
There's not two, there's not four, there's not eight.
Lex Fridman (2:09:00.140)
Think about how many different orders
Jordan Ellenberg (2:09:01.560)
the deck of card can be in.
Lex Fridman (2:09:02.620)
Each one of those is the result of applying a symmetry
Jordan Ellenberg (2:09:06.820)
to the original deck.
Lex Fridman (2:09:07.660)
So a shuffle is a symmetry, right?
Jordan Ellenberg (2:09:09.060)
You're reordering the cards.
Lex Fridman (2:09:10.620)
If I shuffle and then you shuffle,
Jordan Ellenberg (2:09:12.940)
the result is some other kind of thing.
Lex Fridman (2:09:16.020)
You might call it a double shuffle,
Jordan Ellenberg (2:09:17.780)
which is a more complicated symmetry.
Lex Fridman (2:09:19.980)
So group theory is kind of the study
Jordan Ellenberg (2:09:22.180)
of the general abstract world
Lex Fridman (2:09:24.460)
that encompasses all these kinds of things.
Lex Fridman (2:09:27.020)
But then of course, like lots of things
Lex Fridman (2:09:29.380)
that are way more complicated than that.
Jordan Ellenberg (2:09:31.780)
Like infinite groups of symmetries, for instance.
Lex Fridman (2:09:33.540)
So they can be infinite, huh?
Jordan Ellenberg (2:09:35.100)
Oh yeah.
Lex Fridman (2:09:35.940)
Okay.
Lex Fridman (2:09:36.780)
Well, okay, ready?
Lex Fridman (2:09:37.620)
Think about the symmetries of the line.
Jordan Ellenberg (2:09:41.180)
You're like, okay, I can reflect it left to right,
Lex Fridman (2:09:45.020)
you know, around the origin.
Jordan Ellenberg (2:09:46.820)
Okay, but I could also reflect it left to right,
Lex Fridman (2:09:49.580)
grabbing somewhere else, like at one or two
Jordan Ellenberg (2:09:52.180)
or pi or anywhere.
Lex Fridman (2:09:54.620)
Or I could just slide it some distance.
Jordan Ellenberg (2:09:56.440)
That's a symmetry.
Lex Fridman (2:09:57.340)
Slide it five units over.
Lex Fridman (2:09:58.540)
So there's clearly infinitely many symmetries of the line.
Lex Fridman (2:10:01.220)
That's an example of an infinite group of symmetries.
Jordan Ellenberg (2:10:03.500)
Is it possible to say something that kind of captivates,
Lex Fridman (2:10:06.940)
keeps being brought up by physicists,
Jordan Ellenberg (2:10:09.420)
which is gauge theory, gauge symmetry,
Lex Fridman (2:10:12.640)
as one of the more complicated type of symmetries?
Lex Fridman (2:10:14.900)
Is there an easy explanation of what the heck it is?
Lex Fridman (2:10:18.380)
Is that something that comes up on your mind at all?
Jordan Ellenberg (2:10:21.860)
Well, I'm not a mathematical physicist,
Lex Fridman (2:10:23.380)
but I can say this.
Jordan Ellenberg (2:10:24.380)
It is certainly true that it has been a very useful notion
Lex Fridman (2:10:29.460)
in physics to try to say like,
Lex Fridman (2:10:31.860)
what are the symmetry groups of the world?
Lex Fridman (2:10:34.580)
Like what are the symmetries
Lex Fridman (2:10:35.660)
under which things don't change, right?
Lex Fridman (2:10:36.980)
So we just, I think we talked a little bit earlier
Jordan Ellenberg (2:10:39.220)
about it should be a basic principle
Lex Fridman (2:10:40.700)
that a theorem that's true here is also true over there.
Lex Fridman (2:10:44.180)
And same for a physical law, right?
Lex Fridman (2:10:45.700)
I mean, if gravity is like this over here,
Jordan Ellenberg (2:10:47.660)
it should also be like this over there.
Lex Fridman (2:10:49.140)
Okay, what that's saying is we think translation in space
Jordan Ellenberg (2:10:52.660)
should be a symmetry.
Lex Fridman (2:10:54.020)
All the laws of physics should be unchanged
Jordan Ellenberg (2:10:56.540)
if the symmetry we have in mind
Lex Fridman (2:10:57.860)
is a very simple one like translation.
Lex Fridman (2:10:59.700)
And so then there becomes a question,
Lex Fridman (2:11:03.820)
like what are the symmetries of the actual world
Lex Fridman (2:11:07.980)
with its physical laws?
Lex Fridman (2:11:09.820)
And one way of thinking, this isn't oversimplification,
Lex Fridman (2:11:12.900)
but like one way of thinking of this big shift
Lex Fridman (2:11:18.420)
from before Einstein to after
Jordan Ellenberg (2:11:22.420)
is that we just changed our idea
Lex Fridman (2:11:25.300)
about what the fundamental group of symmetries were.
Lex Fridman (2:11:29.780)
So that things like the Lorenz contraction,
Lex Fridman (2:11:31.820)
things like these bizarre relativistic phenomenon
Jordan Ellenberg (2:11:34.340)
or Lorenz would have said, oh, to make this work,
Lex Fridman (2:11:37.700)
we need a thing to change its shape
Jordan Ellenberg (2:11:44.580)
if it's moving nearly the speed of light.
Lex Fridman (2:11:47.460)
Well, under the new framework, it's much better.
Jordan Ellenberg (2:11:50.260)
You say, oh, no, it wasn't changing its shape.
Lex Fridman (2:11:51.700)
You were just wrong about what counted as a symmetry.
Jordan Ellenberg (2:11:54.420)
Now that we have this new group,
Lex Fridman (2:11:55.420)
the so called Lorenz group,
Jordan Ellenberg (2:11:57.380)
now that we understand what the symmetries really are,
Lex Fridman (2:11:59.220)
we see it was just an illusion
Jordan Ellenberg (2:12:00.340)
that the thing was changing its shape.
Lex Fridman (2:12:02.940)
Yeah, so you can then describe the sameness of things
Jordan Ellenberg (2:12:05.780)
under this weirdness that is general relativity,
Lex Fridman (2:12:08.820)
for example.
Jordan Ellenberg (2:12:10.940)
Yeah, yeah, still, I wish there was a simpler explanation
Lex Fridman (2:12:16.020)
of like exact, I mean, gauge symmetries,
Jordan Ellenberg (2:12:19.820)
pretty simple general concept about rulers being deformed.
Lex Fridman (2:12:26.260)
I've actually just personally been on a search,
Jordan Ellenberg (2:12:31.500)
not a very rigorous or aggressive search,
Lex Fridman (2:12:34.740)
but for something I personally enjoy,
Jordan Ellenberg (2:12:37.980)
which is taking complicated concepts
Lex Fridman (2:12:40.980)
and finding the sort of minimal example
Jordan Ellenberg (2:12:44.780)
that I can play around with, especially programmatically.
Lex Fridman (2:12:47.580)
That's great, I mean,
Lex Fridman (2:12:48.420)
this is what we try to train our students to do, right?
Lex Fridman (2:12:50.220)
I mean, in class, this is exactly what,
Jordan Ellenberg (2:12:52.620)
this is like best pedagogical practice.
Lex Fridman (2:12:54.620)
I do hope there's simple explanation,
Jordan Ellenberg (2:12:57.380)
especially like I've in my sort of drunk random walk,
Lex Fridman (2:13:02.380)
drunk walk, whatever that's called,
Jordan Ellenberg (2:13:04.580)
sometimes stumble into the world of topology
Lex Fridman (2:13:08.580)
and like quickly, like, you know when you go into a party
Lex Fridman (2:13:11.420)
and you realize this is not the right party for me?
Lex Fridman (2:13:14.260)
It's, so whenever I go into topology,
Jordan Ellenberg (2:13:16.900)
it's like so much math everywhere.
Lex Fridman (2:13:20.420)
I don't even know what, it feels like this is me
Jordan Ellenberg (2:13:23.100)
like being a hater, I think there's way too much math.
Lex Fridman (2:13:25.900)
Like there are two, the cool kids who just want to have,
Jordan Ellenberg (2:13:29.220)
like everything is expressed through math.
Lex Fridman (2:13:31.060)
Because they're actually afraid to express stuff
Jordan Ellenberg (2:13:33.100)
simply through language.
Lex Fridman (2:13:34.860)
That's my hater formulation of topology.
Lex Fridman (2:13:37.580)
But at the same time, I'm sure that's very necessary
Lex Fridman (2:13:39.620)
to do sort of rigorous discussion.
Lex Fridman (2:13:41.300)
But I feel like.
Lex Fridman (2:13:42.620)
But don't you think that's what gauge symmetry is like?
Jordan Ellenberg (2:13:44.300)
I mean, it's not a field I know well,
Lex Fridman (2:13:45.420)
but it certainly seems like.
Jordan Ellenberg (2:13:46.500)
Yes, it is like that.
Lex Fridman (2:13:47.820)
But my problem with topology, okay,
Lex Fridman (2:13:50.620)
and even like differential geometry is like,
Lex Fridman (2:13:55.140)
you're talking about beautiful things.
Jordan Ellenberg (2:13:59.060)
Like if they could be visualized, it's open question
Lex Fridman (2:14:02.060)
if everything could be visualized,
Lex Fridman (2:14:03.900)
but you're talking about things
Lex Fridman (2:14:05.020)
that can be visually stunning, I think.
Lex Fridman (2:14:09.180)
But they are hidden underneath all of that math.
Lex Fridman (2:14:13.900)
Like if you look at the papers that are written
Jordan Ellenberg (2:14:16.380)
in topology, if you look at all the discussions
Lex Fridman (2:14:18.580)
on Stack Exchange, they're all math dense, math heavy.
Lex Fridman (2:14:22.140)
And the only kind of visual things
Lex Fridman (2:14:25.420)
that emerge every once in a while,
Jordan Ellenberg (2:14:27.540)
is like something like a Mobius strip.
Lex Fridman (2:14:30.980)
Every once in a while, some kind of simple visualizations.
Jordan Ellenberg (2:14:33.980)
Every once in a while, some kind of simple visualizations.
Lex Fridman (2:14:36.980)
Every once in a while, some kind of simple visualizations.
Jordan Ellenberg (2:14:37.460)
Well, there's the vibration, there's the hop vibration
Lex Fridman (2:14:40.260)
or all those kinds of things that somebody,
Jordan Ellenberg (2:14:42.500)
some grad student from like 20 years ago
Lex Fridman (2:14:45.180)
wrote a program in Fortran to visualize it, and that's it.
Lex Fridman (2:14:48.500)
And it's just, you know, it's makes me sad
Lex Fridman (2:14:51.060)
because those are visual disciplines.
Jordan Ellenberg (2:14:53.460)
Just like computer vision is a visual discipline.
Lex Fridman (2:14:56.460)
So you can provide a lot of visual examples.
Jordan Ellenberg (2:14:59.700)
I wish topology was more excited
Lex Fridman (2:15:03.380)
and in love with visualizing some of the ideas.
Jordan Ellenberg (2:15:07.220)
I mean, you could say that, but I would say for me,
Lex Fridman (2:15:09.060)
a picture of the hop vibration does nothing for me.
Jordan Ellenberg (2:15:11.940)
Whereas like when you're like, oh,
Lex Fridman (2:15:13.540)
it's like about the quaternions.
Jordan Ellenberg (2:15:14.740)
It's like a subgroup of the quaternions.
Lex Fridman (2:15:16.100)
And I'm like, oh, so now I see what's going on.
Lex Fridman (2:15:17.860)
Like, why didn't you just say that?
Lex Fridman (2:15:18.900)
Why were you like showing me this stupid picture
Lex Fridman (2:15:20.580)
instead of telling me what you were talking about?
Lex Fridman (2:15:22.460)
Oh, yeah, yeah.
Jordan Ellenberg (2:15:25.020)
I'm just saying, no, but it goes back
Lex Fridman (2:15:26.460)
to what you were saying about teaching
Jordan Ellenberg (2:15:27.380)
that like people are different in what they'll respond to.
Lex Fridman (2:15:29.780)
So I think there's no, I mean, I'm very opposed
Jordan Ellenberg (2:15:32.100)
to the idea that there's a one right way to explain things.
Lex Fridman (2:15:34.420)
I think there's like a huge variation in like, you know,
Jordan Ellenberg (2:15:37.260)
our brains like have all these like weird like hooks
Lex Fridman (2:15:40.300)
and loops and it's like very hard to know
Jordan Ellenberg (2:15:42.100)
like what's gonna latch on
Lex Fridman (2:15:43.300)
and it's not gonna be the same thing for everybody.
Lex Fridman (2:15:46.140)
So I think monoculture is bad, right?
Lex Fridman (2:15:49.500)
I think that's, and I think we're agreeing on that point
Jordan Ellenberg (2:15:51.580)
that like, it's good that there's like a lot
Lex Fridman (2:15:53.740)
of different ways in and a lot of different ways
Jordan Ellenberg (2:15:55.540)
to describe these ideas because different people
Lex Fridman (2:15:57.500)
are gonna find different things illuminating.
Lex Fridman (2:15:59.780)
But that said, I think there's a lot to be discovered
Lex Fridman (2:16:04.460)
when you force little like silos of brilliant people
Jordan Ellenberg (2:16:11.060)
to kind of find a middle ground
Lex Fridman (2:16:15.300)
or like aggregate or come together in a way.
Lex Fridman (2:16:20.260)
So there's like people that do love visual things.
Lex Fridman (2:16:23.580)
I mean, there's a lot of disciplines,
Jordan Ellenberg (2:16:25.740)
especially in computer science
Lex Fridman (2:16:27.020)
that they're obsessed with visualizing,
Jordan Ellenberg (2:16:28.900)
visualizing data, visualizing neural networks.
Lex Fridman (2:16:31.500)
I mean, neural networks themselves are fundamentally visual.
Jordan Ellenberg (2:16:34.100)
There's a lot of work in computer vision that's very visual.
Lex Fridman (2:16:36.700)
And then coming together with some folks
Jordan Ellenberg (2:16:39.140)
that were like deeply rigorous
Lex Fridman (2:16:41.020)
and are like totally lost in multi dimensional space
Jordan Ellenberg (2:16:43.620)
where it's hard to even bring them back down to 3D.
Lex Fridman (2:16:48.220)
They're very comfortable in this multi dimensional space.
Lex Fridman (2:16:50.300)
So forcing them to kind of work together to communicate
Lex Fridman (2:16:53.500)
because it's not just about public communication of ideas.
Jordan Ellenberg (2:16:57.300)
It's also, I feel like when you're forced
Lex Fridman (2:16:59.180)
to do that public communication like you did with your book,
Jordan Ellenberg (2:17:02.100)
I think deep profound ideas can be discovered
Lex Fridman (2:17:05.780)
that's like applicable for research and for science.
Jordan Ellenberg (2:17:08.740)
Like there's something about that simplification
Lex Fridman (2:17:10.780)
or not simplification, but distillation or condensation
Jordan Ellenberg (2:17:15.380)
or whatever the hell you call it,
Lex Fridman (2:17:17.020)
compression of ideas that somehow
Jordan Ellenberg (2:17:19.860)
actually stimulates creativity.
Lex Fridman (2:17:22.140)
And I'd be excited to see more of that
Jordan Ellenberg (2:17:25.220)
in the mathematics community.
Lex Fridman (2:17:27.820)
Can you?
Jordan Ellenberg (2:17:28.660)
Let me make a crazy metaphor.
Lex Fridman (2:17:29.500)
Maybe it's a little bit like the relation
Lex Fridman (2:17:31.140)
between prose and poetry, right?
Lex Fridman (2:17:32.620)
I mean, if you, you might say like,
Lex Fridman (2:17:33.740)
why do we need anything more than prose?
Lex Fridman (2:17:35.020)
You're trying to convey some information.
Lex Fridman (2:17:36.460)
So you just like say it.
Lex Fridman (2:17:38.500)
Well, poetry does something, right?
Jordan Ellenberg (2:17:40.500)
It's sort of, you might think of it as a kind of compression.
Lex Fridman (2:17:43.340)
Of course, not all poetry is compressed.
Jordan Ellenberg (2:17:44.940)
Like not all, some of it is quite baggy,
Lex Fridman (2:17:47.660)
but like you are kind of, often it's compressed, right?
Jordan Ellenberg (2:17:53.340)
A lyric poem is often sort of like a compression
Lex Fridman (2:17:55.620)
of what would take a long time
Lex Fridman (2:17:57.740)
and be complicated to explain in prose
Lex Fridman (2:18:00.300)
into sort of a different mode
Jordan Ellenberg (2:18:03.300)
that is gonna hit in a different way.
Lex Fridman (2:18:05.380)
We talked about Poincare conjecture.
Jordan Ellenberg (2:18:10.180)
There's a guy, he's Russian, Grigori Perlman.
Lex Fridman (2:18:14.620)
He proved Poincare's conjecture.
Jordan Ellenberg (2:18:16.620)
If you can comment on the proof itself,
Lex Fridman (2:18:19.220)
if that stands out to you as something interesting
Jordan Ellenberg (2:18:21.580)
or the human story of it,
Lex Fridman (2:18:23.220)
which is he turned down the field's metal for the proof.
Jordan Ellenberg (2:18:28.380)
Is there something you find inspiring or insightful
Lex Fridman (2:18:32.780)
about the proof itself or about the man?
Jordan Ellenberg (2:18:36.180)
Yeah, I mean, one thing I really like about the proof
Lex Fridman (2:18:40.620)
and partly that's because it's sort of a thing
Jordan Ellenberg (2:18:42.940)
that happens again and again in this book.
Lex Fridman (2:18:45.140)
I mean, I'm writing about geometry and the way
Jordan Ellenberg (2:18:46.940)
it sort of appears in all these kind of real world problems.
Lex Fridman (2:18:50.220)
But it happens so often that the geometry
Jordan Ellenberg (2:18:52.780)
you think you're studying is somehow not enough.
Lex Fridman (2:18:56.940)
You have to go one level higher in abstraction
Lex Fridman (2:18:59.260)
and study a higher level of geometry.
Lex Fridman (2:19:01.660)
And the way that plays out is that Poincare asks a question
Jordan Ellenberg (2:19:05.380)
about a certain kind of three dimensional object.
Lex Fridman (2:19:07.900)
Is it the usual three dimensional space that we know
Lex Fridman (2:19:10.340)
or is it some kind of exotic thing?
Lex Fridman (2:19:13.100)
And so, of course, this sounds like it's a question
Jordan Ellenberg (2:19:15.140)
about the geometry of the three dimensional space,
Lex Fridman (2:19:17.660)
but no, Perelman understands.
Lex Fridman (2:19:20.260)
And by the way, in a tradition that involves
Lex Fridman (2:19:21.980)
Richard Hamilton and many other people,
Jordan Ellenberg (2:19:23.580)
like most really important mathematical advances,
Lex Fridman (2:19:26.360)
this doesn't happen alone.
Jordan Ellenberg (2:19:27.460)
It doesn't happen in a vacuum.
Lex Fridman (2:19:28.540)
It happens as the culmination of a program
Jordan Ellenberg (2:19:30.220)
that involves many people.
Lex Fridman (2:19:31.340)
Same with Wiles, by the way.
Jordan Ellenberg (2:19:32.460)
I mean, we talked about Wiles and I wanna emphasize
Lex Fridman (2:19:34.400)
that starting all the way back with Kummer,
Jordan Ellenberg (2:19:36.700)
who I mentioned in the 19th century,
Lex Fridman (2:19:38.220)
but Gerhard Frey and Mazer and Ken Ribbit
Lex Fridman (2:19:42.260)
and like many other people are involved
Lex Fridman (2:19:45.260)
in building the other pieces of the arch
Jordan Ellenberg (2:19:47.260)
before you put the keystone in.
Lex Fridman (2:19:48.340)
We stand on the shoulders of giants.
Jordan Ellenberg (2:19:50.380)
Yes.
Lex Fridman (2:19:53.900)
So, what is this idea?
Jordan Ellenberg (2:19:56.100)
The idea is that, well, of course,
Lex Fridman (2:19:57.460)
the geometry of the three dimensional object itself
Jordan Ellenberg (2:19:59.940)
is relevant, but the real geometry you have to understand
Lex Fridman (2:20:02.500)
is the geometry of the space
Jordan Ellenberg (2:20:04.740)
of all three dimensional geometries.
Lex Fridman (2:20:07.420)
Whoa, you're going up a higher level.
Jordan Ellenberg (2:20:10.540)
Because when you do that, you can say,
Lex Fridman (2:20:12.040)
now let's trace out a path in that space.
Jordan Ellenberg (2:20:18.260)
There's a mechanism called Ricci flow.
Lex Fridman (2:20:19.840)
And again, we're outside my research area.
Lex Fridman (2:20:21.100)
So for all the geometric analysts
Lex Fridman (2:20:23.380)
and differential geometers out there listening to this,
Jordan Ellenberg (2:20:25.820)
if I, please, I'm doing my best and I'm roughly saying it.
Lex Fridman (2:20:29.500)
So the Ricci flow allows you to say like,
Jordan Ellenberg (2:20:32.220)
okay, let's start from some mystery three dimensional space,
Lex Fridman (2:20:35.400)
which Poincare would conjecture is essentially
Jordan Ellenberg (2:20:37.740)
the same thing as our familiar three dimensional space,
Lex Fridman (2:20:39.520)
but we don't know that.
Lex Fridman (2:20:41.260)
And now you let it flow.
Lex Fridman (2:20:44.180)
You sort of like let it move in its natural path
Jordan Ellenberg (2:20:47.500)
according to some almost physical process
Lex Fridman (2:20:50.140)
and ask where it winds up.
Lex Fridman (2:20:51.460)
And what you find is that it always winds up.
Lex Fridman (2:20:54.360)
You've continuously deformed it.
Jordan Ellenberg (2:20:55.740)
There's that word deformation again.
Lex Fridman (2:20:58.340)
And what you can prove is that the process doesn't stop
Jordan Ellenberg (2:21:00.180)
until you get to the usual three dimensional space.
Lex Fridman (2:21:02.100)
And since you can get from the mystery thing
Jordan Ellenberg (2:21:04.660)
to the standard space by this process
Lex Fridman (2:21:06.840)
of continually changing and never kind of
Jordan Ellenberg (2:21:09.900)
having any sharp transitions,
Lex Fridman (2:21:12.980)
then the original shape must've been the same
Jordan Ellenberg (2:21:16.300)
as the standard shape.
Lex Fridman (2:21:17.500)
That's the nature of the proof.
Jordan Ellenberg (2:21:18.780)
Now, of course, it's incredibly technical.
Lex Fridman (2:21:20.460)
I think as I understand it,
Jordan Ellenberg (2:21:21.500)
I think the hard part is proving
Lex Fridman (2:21:23.360)
that the favorite word of AI people,
Jordan Ellenberg (2:21:25.640)
you don't get any singularities along the way.
Lex Fridman (2:21:29.460)
But of course, in this context,
Jordan Ellenberg (2:21:30.500)
singularity just means acquiring a sharp kink.
Lex Fridman (2:21:34.360)
It just means becoming non smooth at some point.
Lex Fridman (2:21:37.020)
So just saying something interesting about formal,
Lex Fridman (2:21:41.020)
about the smooth trajectory
Jordan Ellenberg (2:21:42.740)
through this weird space of geometries.
Lex Fridman (2:21:45.380)
But yeah, so what I like about it
Jordan Ellenberg (2:21:46.740)
is that it's just one of many examples of where
Lex Fridman (2:21:49.620)
it's not about the geometry you think it's about.
Jordan Ellenberg (2:21:51.680)
It's about the geometry of all geometries, so to speak.
Lex Fridman (2:21:55.980)
And it's only by kind of like being jerked out of flatland.
Jordan Ellenberg (2:21:59.600)
Same idea.
Lex Fridman (2:22:00.440)
It's only by sort of seeing the whole thing globally at once
Jordan Ellenberg (2:22:04.160)
that you can really make progress on understanding
Lex Fridman (2:22:05.860)
the one thing you thought you were looking at.
Jordan Ellenberg (2:22:08.440)
It's a romantic question,
Lex Fridman (2:22:09.520)
but what do you think about him
Lex Fridman (2:22:11.140)
turning down the Fields Medal?
Lex Fridman (2:22:12.980)
Is that just, are Nobel Prizes and Fields Medals
Jordan Ellenberg (2:22:17.020)
just the cherry on top of the cake
Lex Fridman (2:22:19.980)
and really math itself, the process of curiosity,
Lex Fridman (2:22:25.220)
of pulling at the string of the mystery before us?
Lex Fridman (2:22:28.500)
That's the cake?
Lex Fridman (2:22:29.580)
And then the awards are just icing
Lex Fridman (2:22:33.780)
and clearly I've been fasting and I'm hungry,
Lex Fridman (2:22:37.220)
but do you think it's tragic or just a little curiosity
Lex Fridman (2:22:44.620)
that he turned down the medal?
Jordan Ellenberg (2:22:46.380)
Well, it's interesting because on the one hand,
Lex Fridman (2:22:48.460)
I think it's absolutely true that right,
Jordan Ellenberg (2:22:50.780)
in some kind of like vast spiritual sense,
Lex Fridman (2:22:55.500)
like awards are not important,
Jordan Ellenberg (2:22:57.380)
like not important the way that sort of like
Lex Fridman (2:22:59.260)
understanding the universe is important.
Jordan Ellenberg (2:23:01.260)
On the other hand, most people who are offered that prize
Lex Fridman (2:23:04.620)
accept it, so there's something unusual
Jordan Ellenberg (2:23:07.740)
about his choice there.
Lex Fridman (2:23:11.740)
I wouldn't say I see it as tragic.
Jordan Ellenberg (2:23:14.420)
I mean, maybe if I don't really feel like
Lex Fridman (2:23:16.220)
I have a clear picture of why he chose not to take it.
Jordan Ellenberg (2:23:19.260)
I mean, he's not alone in doing things like this.
Lex Fridman (2:23:22.060)
People sometimes turn down prizes for ideological reasons,
Lex Fridman (2:23:26.540)
but probably more often in mathematics.
Lex Fridman (2:23:28.020)
I mean, I think I'm right in saying that
Jordan Ellenberg (2:23:30.020)
Peter Schultz turned down sort of some big monetary prize
Lex Fridman (2:23:33.940)
because he just, you know, I mean, I think he,
Jordan Ellenberg (2:23:36.660)
at some point you have plenty of money
Lex Fridman (2:23:39.260)
and maybe you think it sends the wrong message
Jordan Ellenberg (2:23:41.340)
about what the point of doing mathematics is.
Lex Fridman (2:23:45.740)
I do find that there's most people accept.
Jordan Ellenberg (2:23:47.540)
You know, most people give it a prize.
Lex Fridman (2:23:48.820)
Most people take it.
Jordan Ellenberg (2:23:49.660)
I mean, people like to be appreciated,
Lex Fridman (2:23:50.900)
but like I said, we're people.
Jordan Ellenberg (2:23:53.020)
Not that different from most other people.
Lex Fridman (2:23:54.700)
But the important reminder that that turning down
Jordan Ellenberg (2:23:57.900)
a prize serves for me is not that there's anything wrong
Lex Fridman (2:24:01.500)
with the prize and there's something wonderful
Jordan Ellenberg (2:24:03.580)
about the prize, I think.
Lex Fridman (2:24:04.940)
The Nobel prize is trickier
Jordan Ellenberg (2:24:07.660)
because so many Nobel prizes are given.
Lex Fridman (2:24:10.340)
First of all, the Nobel prize often forgets
Jordan Ellenberg (2:24:12.300)
many, many of the important people throughout history.
Lex Fridman (2:24:15.660)
Second of all, there's like these weird rules to it
Jordan Ellenberg (2:24:18.940)
that it's only three people
Lex Fridman (2:24:20.260)
and some projects have a huge number of people.
Lex Fridman (2:24:22.460)
And it's like this, it, I don't know.
Lex Fridman (2:24:26.220)
It doesn't kind of highlight the way science is done
Jordan Ellenberg (2:24:31.180)
on some of these projects in the best possible way.
Lex Fridman (2:24:33.660)
But in general, the prizes are great.
Lex Fridman (2:24:34.980)
But what this kind of teaches me and reminds me
Lex Fridman (2:24:37.380)
is sometimes in your life, there'll be moments
Jordan Ellenberg (2:24:41.180)
when the thing that you would really like to do,
Lex Fridman (2:24:47.580)
society would really like you to do,
Jordan Ellenberg (2:24:50.740)
is the thing that goes against something you believe in,
Lex Fridman (2:24:53.940)
whatever that is, some kind of principle.
Lex Fridman (2:24:56.060)
And standing your ground in the face of that
Lex Fridman (2:24:59.860)
is something I believe most people will have
Jordan Ellenberg (2:25:03.060)
a few moments like that in their life,
Lex Fridman (2:25:05.100)
maybe one moment like that, and you have to do it.
Jordan Ellenberg (2:25:07.460)
That's what integrity is.
Lex Fridman (2:25:09.100)
So like, it doesn't have to make sense
Jordan Ellenberg (2:25:10.460)
to the rest of the world, but to stand on that,
Lex Fridman (2:25:12.340)
like to say no, it's interesting, because I think.
Lex Fridman (2:25:16.060)
But do you know that he turned down the prize
Lex Fridman (2:25:17.740)
in service of some principle?
Jordan Ellenberg (2:25:20.020)
Because I don't know that.
Lex Fridman (2:25:20.980)
Well, yes, that seems to be the inkling,
Lex Fridman (2:25:22.740)
but he has never made it super clear.
Lex Fridman (2:25:24.540)
But the inkling is that he had some problems
Jordan Ellenberg (2:25:26.900)
with the whole process of mathematics that includes awards,
Lex Fridman (2:25:30.220)
like this hierarchies and the reputations
Lex Fridman (2:25:33.500)
and all those kinds of things,
Lex Fridman (2:25:34.500)
and individualism that's fundamental to American culture.
Jordan Ellenberg (2:25:37.660)
He probably, because he visited the United States quite a bit
Lex Fridman (2:25:41.140)
that he probably, it's all about experiences.
Lex Fridman (2:25:47.380)
And he may have had some parts of academia,
Lex Fridman (2:25:51.500)
some pockets of academia can be less than inspiring,
Jordan Ellenberg (2:25:54.740)
perhaps sometimes, because of the individual egos involved,
Lex Fridman (2:25:57.580)
not academia, people in general, smart people with egos.
Lex Fridman (2:26:01.180)
And if you interact with a certain kinds of people,
Lex Fridman (2:26:05.620)
you can become cynical too easily.
Jordan Ellenberg (2:26:07.460)
I'm one of those people that I've been really fortunate
Lex Fridman (2:26:10.700)
to interact with incredible people at MIT
Lex Fridman (2:26:12.820)
and academia in general, but I've met some assholes.
Lex Fridman (2:26:15.500)
And I tend to just kind of,
Jordan Ellenberg (2:26:17.060)
when I run into difficult folks,
Lex Fridman (2:26:19.140)
I just kind of smile and send them all my love
Lex Fridman (2:26:21.340)
and just kind of go around.
Lex Fridman (2:26:23.100)
But for others, those experiences can be sticky.
Jordan Ellenberg (2:26:26.700)
Like they can become cynical about the world
Lex Fridman (2:26:29.820)
when folks like that exist.
Lex Fridman (2:26:31.660)
So he may have become a little bit cynical
Lex Fridman (2:26:35.500)
about the process of science.
Jordan Ellenberg (2:26:37.220)
Well, you know, it's a good opportunity.
Lex Fridman (2:26:38.620)
Let's posit that that's his reasoning
Jordan Ellenberg (2:26:40.220)
because I truly don't know.
Lex Fridman (2:26:42.380)
It's an interesting opportunity to go back
Jordan Ellenberg (2:26:43.820)
to almost the very first thing we talked about,
Lex Fridman (2:26:46.340)
the idea of the Mathematical Olympiad,
Jordan Ellenberg (2:26:48.380)
because of course that is,
Lex Fridman (2:26:50.540)
so the International Mathematical Olympiad
Jordan Ellenberg (2:26:52.100)
is like a competition for high school students
Lex Fridman (2:26:54.620)
solving math problems.
Lex Fridman (2:26:55.860)
And in some sense, it's absolutely false
Lex Fridman (2:26:59.180)
to the reality of mathematics,
Jordan Ellenberg (2:27:00.380)
because just as you say,
Lex Fridman (2:27:02.060)
it is a contest where you win prizes.
Jordan Ellenberg (2:27:07.380)
The aim is to sort of be faster than other people.
Lex Fridman (2:27:11.860)
And you're working on sort of canned problems
Jordan Ellenberg (2:27:13.860)
that someone already knows the answer to,
Lex Fridman (2:27:15.700)
like not problems that are unknown.
Jordan Ellenberg (2:27:18.500)
So, you know, in my own life,
Lex Fridman (2:27:20.580)
I think when I was in high school,
Jordan Ellenberg (2:27:21.900)
I was like very motivated by those competitions.
Lex Fridman (2:27:24.260)
And like, I went to the Math Olympiad and...
Jordan Ellenberg (2:27:26.140)
You won it twice and got, I mean...
Lex Fridman (2:27:28.580)
Well, there's something I have to explain to people
Jordan Ellenberg (2:27:30.180)
because it says, I think it says on Wikipedia
Lex Fridman (2:27:32.220)
that I won a gold medal.
Lex Fridman (2:27:33.420)
And in the real Olympics,
Lex Fridman (2:27:35.220)
they only give one gold medal in each event.
Jordan Ellenberg (2:27:37.420)
I just have to emphasize
Lex Fridman (2:27:38.460)
that the International Math Olympiad is not like that.
Jordan Ellenberg (2:27:40.820)
The gold medals are awarded
Lex Fridman (2:27:42.220)
to the top 112th of all participants.
Lex Fridman (2:27:44.980)
So sorry to bust the legend or anything like that.
Lex Fridman (2:27:47.260)
Well, you're an exceptional performer
Jordan Ellenberg (2:27:48.860)
in terms of achieving high scores on the problems
Lex Fridman (2:27:51.860)
and they're very difficult.
Lex Fridman (2:27:53.220)
So you've achieved a high level of performance on the...
Lex Fridman (2:27:56.300)
In this very specialized skill.
Lex Fridman (2:27:57.900)
And by the way, it was a very Cold War activity.
Lex Fridman (2:28:00.540)
You know, in 1987, the first year I went,
Jordan Ellenberg (2:28:02.900)
it was in Havana.
Lex Fridman (2:28:04.620)
Americans couldn't go to Havana back then.
Jordan Ellenberg (2:28:06.140)
It was a very complicated process to get there.
Lex Fridman (2:28:08.740)
And they took the whole American team on a field trip
Jordan Ellenberg (2:28:10.860)
to the Museum of American Imperialism in Havana
Lex Fridman (2:28:14.140)
so we could see what America was all about.
Lex Fridman (2:28:17.580)
How would you recommend a person learn math?
Lex Fridman (2:28:22.980)
So somebody who's young or somebody my age
Jordan Ellenberg (2:28:26.380)
or somebody older who've taken a bunch of math
Lex Fridman (2:28:29.740)
but wants to rediscover the beauty of math
Lex Fridman (2:28:32.060)
and maybe integrate it into their work
Lex Fridman (2:28:34.260)
more solid in the research space and so on.
Jordan Ellenberg (2:28:38.540)
Is there something you could say about the process of...
Lex Fridman (2:28:44.220)
Incorporating mathematical thinking into your life?
Jordan Ellenberg (2:28:47.620)
I mean, the thing is,
Lex Fridman (2:28:48.900)
it's in part a journey of self knowledge.
Jordan Ellenberg (2:28:50.860)
You have to know what's gonna work for you
Lex Fridman (2:28:53.740)
and that's gonna be different for different people.
Lex Fridman (2:28:55.940)
So there are totally people who at any stage of life
Lex Fridman (2:28:59.260)
just start reading math textbooks.
Jordan Ellenberg (2:29:01.980)
That is a thing that you can do
Lex Fridman (2:29:03.660)
and it works for some people and not for others.
Jordan Ellenberg (2:29:06.580)
For others, a gateway is, I always recommend
Lex Fridman (2:29:09.620)
the books of Martin Gardner,
Jordan Ellenberg (2:29:10.820)
another sort of person we haven't talked about
Lex Fridman (2:29:12.620)
but who also, like Conway, embodies that spirit of play.
Jordan Ellenberg (2:29:16.860)
He wrote a column in Scientific American for decades
Lex Fridman (2:29:19.460)
called Mathematical Recreations
Lex Fridman (2:29:20.980)
and there's such joy in it and such fun.
Lex Fridman (2:29:23.940)
And these books, the columns are collected into books
Lex Fridman (2:29:26.580)
and the books are old now
Lex Fridman (2:29:27.660)
but for each generation of people who discover them,
Jordan Ellenberg (2:29:29.740)
they're completely fresh.
Lex Fridman (2:29:31.420)
And they give a totally different way into the subject
Jordan Ellenberg (2:29:33.860)
than reading a formal textbook,
Lex Fridman (2:29:36.420)
which for some people would be the right thing to do.
Lex Fridman (2:29:40.060)
And working contest style problems too,
Lex Fridman (2:29:42.260)
those are bound to books,
Jordan Ellenberg (2:29:43.740)
especially like Russian and Bulgarian problems.
Lex Fridman (2:29:45.660)
There's book after book problems from those contexts.
Jordan Ellenberg (2:29:47.740)
That's gonna motivate some people.
Lex Fridman (2:29:50.060)
For some people, it's gonna be like watching
Jordan Ellenberg (2:29:51.580)
well produced videos, like a totally different format.
Lex Fridman (2:29:54.300)
Like I feel like I'm not answering your question.
Jordan Ellenberg (2:29:56.060)
I'm sort of saying there's no one answer
Lex Fridman (2:29:57.780)
and it's a journey where you figure out
Lex Fridman (2:30:00.340)
what resonates with you.
Lex Fridman (2:30:01.900)
For some people, it's the self discovery
Lex Fridman (2:30:04.300)
is trying to figure out why is it that I wanna know?
Lex Fridman (2:30:06.780)
Okay, I'll tell you a story.
Jordan Ellenberg (2:30:07.620)
Once when I was in grad school,
Lex Fridman (2:30:09.500)
I was very frustrated with my lack of knowledge
Jordan Ellenberg (2:30:11.540)
of a lot of things as we all are
Lex Fridman (2:30:13.140)
because no matter how much we know,
Jordan Ellenberg (2:30:14.100)
we don't know much more and going to grad school
Lex Fridman (2:30:15.820)
means just coming face to face
Jordan Ellenberg (2:30:17.220)
with the incredible overflowing vault of your ignorance.
Lex Fridman (2:30:20.340)
So I told Joe Harris, who was an algebraic geometer,
Jordan Ellenberg (2:30:23.740)
a professor in my department,
Lex Fridman (2:30:26.140)
I was like, I really feel like I don't know enough
Lex Fridman (2:30:27.620)
and I should just take a year of leave
Lex Fridman (2:30:29.340)
and just read EGA, the holy textbook,
Jordan Ellenberg (2:30:32.620)
Elements de Géométrie Algebraique,
Lex Fridman (2:30:34.420)
the Elements of Algebraic Geometry.
Jordan Ellenberg (2:30:36.900)
I'm just gonna, I feel like I don't know enough
Lex Fridman (2:30:38.660)
so I'm just gonna sit and read this like 1500 page
Jordan Ellenberg (2:30:42.060)
many volume book.
Lex Fridman (2:30:46.900)
And he was like, and Professor Harris was like,
Jordan Ellenberg (2:30:48.300)
that's a really stupid idea.
Lex Fridman (2:30:49.500)
And I was like, why is that a stupid idea?
Jordan Ellenberg (2:30:50.820)
Then I would know more algebraic geometry.
Lex Fridman (2:30:52.660)
He's like, because you're not actually gonna do it.
Jordan Ellenberg (2:30:53.940)
Like you learn.
Lex Fridman (2:30:55.780)
I mean, he knew me well enough to say like,
Jordan Ellenberg (2:30:57.140)
you're gonna learn because you're gonna be working
Lex Fridman (2:30:58.860)
on a problem and then there's gonna be a fact from EGA
Jordan Ellenberg (2:31:01.060)
that you need in order to solve your problem
Lex Fridman (2:31:03.020)
that you wanna solve and that's how you're gonna learn it.
Jordan Ellenberg (2:31:05.300)
You're not gonna learn it without a problem
Lex Fridman (2:31:06.820)
to bring you into it.
Lex Fridman (2:31:08.020)
And so for a lot of people, I think if you're like,
Lex Fridman (2:31:10.660)
I'm trying to understand machine learning
Lex Fridman (2:31:12.420)
and I'm like, I can see that there's sort of
Lex Fridman (2:31:14.460)
some mathematical technology that I don't have,
Jordan Ellenberg (2:31:19.460)
I think you like let that problem
Lex Fridman (2:31:22.580)
that you actually care about drive your learning.
Jordan Ellenberg (2:31:26.060)
I mean, one thing I've learned from advising students,
Lex Fridman (2:31:28.340)
math is really hard.
Jordan Ellenberg (2:31:32.260)
In fact, anything that you do right is hard.
Lex Fridman (2:31:38.180)
And because it's hard, like you might sort of have some idea
Jordan Ellenberg (2:31:41.420)
that somebody else gives you, oh, I should learn X, Y and Z.
Lex Fridman (2:31:44.540)
Well, if you don't actually care, you're not gonna do it.
Jordan Ellenberg (2:31:46.460)
You might feel like you should,
Lex Fridman (2:31:47.500)
maybe somebody told you you should,
Lex Fridman (2:31:48.940)
but I think you have to hook it to something
Lex Fridman (2:31:51.820)
that you actually care about.
Lex Fridman (2:31:52.780)
So for a lot of people, that's the way in.
Lex Fridman (2:31:54.580)
You have an engineering problem you're trying to handle,
Jordan Ellenberg (2:31:57.260)
you have a physics problem you're trying to handle,
Lex Fridman (2:31:59.580)
you have a machine learning problem you're trying to handle.
Jordan Ellenberg (2:32:02.100)
Let that not a kind of abstract idea
Lex Fridman (2:32:05.020)
of what the curriculum is, drive your mathematical learning.
Lex Fridman (2:32:08.420)
And also just as a brief comment that math is hard,
Lex Fridman (2:32:12.260)
there's a sense to which hard is a feature, not a bug,
Jordan Ellenberg (2:32:15.300)
in the sense that, again,
Lex Fridman (2:32:17.060)
maybe this is my own learning preference,
Lex Fridman (2:32:19.820)
but I think it's a value to fall in love with the process
Lex Fridman (2:32:24.500)
of doing something hard, overcoming it,
Lex Fridman (2:32:27.980)
and becoming a better person because of it.
Lex Fridman (2:32:29.740)
Like I hate running, I hate exercise,
Jordan Ellenberg (2:32:32.180)
to bring it down to like the simplest hard.
Lex Fridman (2:32:35.700)
And I enjoy the part once it's done,
Jordan Ellenberg (2:32:39.740)
the person I feel like in the rest of the day
Lex Fridman (2:32:41.980)
once I've accomplished it, the actual process,
Jordan Ellenberg (2:32:44.100)
especially the process of getting started in the initial,
Lex Fridman (2:32:47.540)
like it really, I don't feel like doing it.
Lex Fridman (2:32:49.580)
And I really have, the way I feel about running
Lex Fridman (2:32:51.660)
is the way I feel about really anything difficult
Jordan Ellenberg (2:32:55.060)
in the intellectual space, especially in mathematics,
Lex Fridman (2:32:58.460)
but also just something that requires
Jordan Ellenberg (2:33:01.820)
like holding a bunch of concepts in your mind
Lex Fridman (2:33:04.820)
with some uncertainty, like where the terminology
Jordan Ellenberg (2:33:08.220)
or the notation is not very clear.
Lex Fridman (2:33:10.220)
And so you have to kind of hold all those things together
Lex Fridman (2:33:13.300)
and like keep pushing forward through the frustration
Lex Fridman (2:33:16.020)
of really like obviously not understanding certain like
Jordan Ellenberg (2:33:20.220)
parts of the picture, like your giant missing parts
Lex Fridman (2:33:23.660)
of the picture and still not giving up.
Jordan Ellenberg (2:33:26.580)
It's the same way I feel about running.
Lex Fridman (2:33:28.980)
And there's something about falling in love
Jordan Ellenberg (2:33:32.820)
with the feeling of after you went through the journey
Lex Fridman (2:33:36.140)
of not having a complete picture,
Jordan Ellenberg (2:33:38.180)
at the end having a complete picture,
Lex Fridman (2:33:40.620)
and then you get to appreciate the beauty
Lex Fridman (2:33:42.460)
and just remembering that it sucked for a long time
Lex Fridman (2:33:46.020)
and how great it felt when you figured it out,
Jordan Ellenberg (2:33:48.780)
at least at the basic.
Lex Fridman (2:33:49.940)
That's not sort of research thinking,
Jordan Ellenberg (2:33:52.020)
because with research, you probably also have to
Lex Fridman (2:33:55.220)
enjoy the dead ends with learning math
Jordan Ellenberg (2:34:00.980)
from a textbook or from video.
Lex Fridman (2:34:02.540)
There's a nice.
Jordan Ellenberg (2:34:03.380)
I don't think you have to enjoy the dead ends,
Lex Fridman (2:34:04.580)
but I think you have to accept the dead ends.
Jordan Ellenberg (2:34:06.340)
Let's put it that way.
Lex Fridman (2:34:08.900)
Well, yeah, enjoy the suffering of it.
Lex Fridman (2:34:11.020)
So the way I think about it, I do, there's an.
Lex Fridman (2:34:17.060)
I don't enjoy the suffering.
Jordan Ellenberg (2:34:18.300)
It pisses me off.
Lex Fridman (2:34:19.140)
You have to accept that it's part of the process.
Jordan Ellenberg (2:34:21.220)
It's interesting.
Lex Fridman (2:34:22.060)
There's a lot of ways to kind of deal with that dead end.
Jordan Ellenberg (2:34:24.540)
There's a guy who's the ultra marathon runner,
Lex Fridman (2:34:26.460)
Navy SEAL, David Goggins, who kind of,
Jordan Ellenberg (2:34:30.340)
I mean, there's a certain philosophy of like,
Lex Fridman (2:34:34.020)
most people would quit here.
Lex Fridman (2:34:37.740)
And so if most people would quit here and I don't,
Lex Fridman (2:34:42.420)
I'll have an opportunity to discover something beautiful
Jordan Ellenberg (2:34:45.140)
that others haven't yet.
Lex Fridman (2:34:46.380)
And so like any feeling that really sucks,
Jordan Ellenberg (2:34:52.940)
it's like, okay, most people would just like,
Lex Fridman (2:34:56.980)
go do something smarter.
Lex Fridman (2:34:58.620)
And if I stick with this,
Lex Fridman (2:35:01.220)
I will discover a new garden of fruit trees that I can pick.
Jordan Ellenberg (2:35:06.140)
Okay, you say that, but like,
Lex Fridman (2:35:07.500)
what about the guy who like wins
Lex Fridman (2:35:09.100)
the Nathan's hot dog eating contest every year?
Lex Fridman (2:35:11.300)
Like when he eats his 35th hot dog,
Jordan Ellenberg (2:35:13.020)
he like correctly says like,
Lex Fridman (2:35:14.180)
okay, most people would stop here.
Jordan Ellenberg (2:35:17.020)
Are you like lauding that he's like,
Lex Fridman (2:35:18.460)
no, I'm gonna eat the 35th hot dog.
Jordan Ellenberg (2:35:20.020)
I am, I am.
Lex Fridman (2:35:21.580)
In the long arc of history, that man is onto something.
Jordan Ellenberg (2:35:26.300)
Which brings up this question.
Lex Fridman (2:35:28.420)
What advice would you give to young people today,
Jordan Ellenberg (2:35:30.980)
thinking about their career, about their life,
Lex Fridman (2:35:34.020)
whether it's in mathematics, poetry,
Lex Fridman (2:35:37.300)
or hot dog eating contest?
Lex Fridman (2:35:40.660)
And you know, I have kids,
Lex Fridman (2:35:41.940)
so this is actually a live issue for me, right?
Lex Fridman (2:35:43.900)
I actually, it's not a thought experiment.
Jordan Ellenberg (2:35:45.740)
I actually do have to give advice
Lex Fridman (2:35:47.140)
to two young people all the time.
Jordan Ellenberg (2:35:48.540)
They don't listen, but I still give it.
Lex Fridman (2:35:53.180)
You know, one thing I often say to students,
Jordan Ellenberg (2:35:55.420)
I don't think I've actually said this to my kids yet,
Lex Fridman (2:35:56.820)
but I say it to students a lot is,
Jordan Ellenberg (2:35:59.860)
you know, you come to these decision points
Lex Fridman (2:36:03.100)
and everybody is beset by self doubt, right?
Jordan Ellenberg (2:36:06.620)
It's like, not sure like what they're capable of,
Lex Fridman (2:36:09.780)
like not sure what they really wanna do.
Jordan Ellenberg (2:36:14.780)
I always, I sort of tell people like,
Lex Fridman (2:36:16.540)
often when you have a decision to make,
Jordan Ellenberg (2:36:20.260)
one of the choices is the high self esteem choice.
Lex Fridman (2:36:22.740)
And I always tell them, make the high self esteem choice.
Jordan Ellenberg (2:36:24.620)
Make the choice, sort of take yourself out of it
Lex Fridman (2:36:26.780)
and like, if you didn't have those,
Jordan Ellenberg (2:36:29.620)
you can probably figure out what the version of you
Lex Fridman (2:36:31.980)
that feels completely confident would do.
Lex Fridman (2:36:35.020)
And do that and see what happens.
Lex Fridman (2:36:36.500)
And I think that's often like pretty good advice.
Jordan Ellenberg (2:36:40.100)
That's interesting.
Lex Fridman (2:36:40.940)
Sort of like, you know, like with Sims,
Jordan Ellenberg (2:36:44.060)
you can create characters.
Lex Fridman (2:36:45.420)
Create a character of yourself
Jordan Ellenberg (2:36:47.820)
that lacks all the self doubt.
Lex Fridman (2:36:50.260)
Right, but it doesn't mean,
Jordan Ellenberg (2:36:51.340)
I would never say to somebody,
Lex Fridman (2:36:52.940)
you should just go have high self esteem.
Jordan Ellenberg (2:36:56.220)
You shouldn't have doubts.
Lex Fridman (2:36:57.180)
No, you probably should have doubts.
Jordan Ellenberg (2:36:58.220)
It's okay to have them.
Lex Fridman (2:36:59.340)
But sometimes it's good to act in the way
Jordan Ellenberg (2:37:01.660)
that the person who didn't have them would act.
Lex Fridman (2:37:04.240)
That's a really nice way to put it.
Jordan Ellenberg (2:37:08.420)
Yeah, that's like from a third person perspective,
Lex Fridman (2:37:13.020)
take the part of your brain that wants to do big things.
Lex Fridman (2:37:16.500)
What would they do?
Lex Fridman (2:37:18.140)
That's not afraid to do those things.
Lex Fridman (2:37:20.020)
What would they do?
Lex Fridman (2:37:21.480)
Yeah, that's really nice.
Jordan Ellenberg (2:37:24.420)
That's actually a really nice way to formulate it.
Lex Fridman (2:37:26.300)
That's very practical advice.
Jordan Ellenberg (2:37:27.540)
You should give it to your kids.
Lex Fridman (2:37:31.140)
Do you think there's meaning to any of it
Lex Fridman (2:37:32.660)
from a mathematical perspective, this life?
Lex Fridman (2:37:36.700)
If I were to ask you,
Jordan Ellenberg (2:37:39.180)
we talked about primes, talked about proving stuff.
Lex Fridman (2:37:43.540)
Can we say, and then the book that God has,
Jordan Ellenberg (2:37:47.340)
that mathematics allows us to arrive
Lex Fridman (2:37:49.540)
at something about in that book.
Jordan Ellenberg (2:37:51.800)
There's certainly a chapter
Lex Fridman (2:37:52.820)
on the meaning of life in that book.
Lex Fridman (2:37:54.980)
Do you think we humans can get to it?
Lex Fridman (2:37:57.380)
And maybe if you were to write cliff notes,
Lex Fridman (2:37:59.500)
what do you suspect those cliff notes would say?
Lex Fridman (2:38:01.520)
I mean, look, the way I feel is that mathematics,
Jordan Ellenberg (2:38:04.860)
as we've discussed, it underlies the way we think
Lex Fridman (2:38:07.580)
about constructing learning machines.
Jordan Ellenberg (2:38:09.240)
It underlies physics.
Lex Fridman (2:38:11.780)
It can be used.
Jordan Ellenberg (2:38:12.620)
I mean, it does all this stuff.
Lex Fridman (2:38:15.740)
And also you want the meaning of life?
Jordan Ellenberg (2:38:17.180)
I mean, it's like, we already did a lot for you.
Lex Fridman (2:38:18.820)
Like, ask a rabbi.
Jordan Ellenberg (2:38:22.580)
No, I mean, I wrote a lot in the last book,
Lex Fridman (2:38:25.900)
How Not to Be Wrong.
Jordan Ellenberg (2:38:27.700)
I wrote a lot about Pascal, a fascinating guy who is
Lex Fridman (2:38:32.380)
a sort of very serious religious mystic,
Jordan Ellenberg (2:38:35.180)
as well as being an amazing mathematician.
Lex Fridman (2:38:37.260)
And he's well known for Pascal's wager.
Jordan Ellenberg (2:38:38.900)
I mean, he's probably among all mathematicians.
Lex Fridman (2:38:40.260)
He's the one who's best known for this.
Lex Fridman (2:38:42.340)
Can you actually like apply mathematics
Lex Fridman (2:38:44.140)
to kind of these transcendent questions?
Lex Fridman (2:38:49.900)
But what's interesting when I really read Pascal
Lex Fridman (2:38:53.020)
about what he wrote about this,
Jordan Ellenberg (2:38:54.780)
I started to see that people often think,
Lex Fridman (2:38:56.300)
oh, this is him saying, I'm gonna use mathematics
Jordan Ellenberg (2:39:00.060)
to sort of show you why you should believe in God.
Lex Fridman (2:39:03.340)
You know, mathematics has the answer to this question.
Lex Fridman (2:39:07.220)
But he really doesn't say that.
Lex Fridman (2:39:08.940)
He almost kind of says the opposite.
Lex Fridman (2:39:11.900)
If you ask Blaise Pascal, like, why do you believe in God?
Lex Fridman (2:39:15.140)
He'd be like, oh, cause I met God.
Jordan Ellenberg (2:39:16.580)
You know, he had this kind of like psychedelic experience.
Lex Fridman (2:39:20.140)
It's like a mystical experience where as he tells it,
Jordan Ellenberg (2:39:23.400)
he just like directly encountered God.
Lex Fridman (2:39:24.980)
It's like, okay, I guess there's a God, I met him last night.
Lex Fridman (2:39:26.820)
So that's it.
Lex Fridman (2:39:27.980)
That's why he believed.
Jordan Ellenberg (2:39:29.100)
It didn't have to do with any kind.
Lex Fridman (2:39:30.340)
You know, the mathematical argument was like
Jordan Ellenberg (2:39:33.700)
about certain reasons for behaving in a certain way.
Lex Fridman (2:39:36.780)
But he basically said, like, look,
Jordan Ellenberg (2:39:38.340)
like math doesn't tell you that God's there or not.
Lex Fridman (2:39:41.100)
Like, if God's there, he'll tell you.
Jordan Ellenberg (2:39:43.420)
You know, you don't even.
Lex Fridman (2:39:45.180)
I love this.
Lex Fridman (2:39:46.020)
So you have mathematics, you have, what do you have?
Lex Fridman (2:39:50.500)
Like a way to explore the mind, let's say psychedelics.
Jordan Ellenberg (2:39:53.780)
You have like incredible technology.
Lex Fridman (2:39:56.620)
You also have love and friendship.
Lex Fridman (2:39:59.700)
And like, what the hell do you want to know
Lex Fridman (2:40:01.820)
what the meaning of it all is?
Jordan Ellenberg (2:40:02.920)
Just enjoy it.
Lex Fridman (2:40:03.980)
I don't think there's a better way to end it, Jordan.
Jordan Ellenberg (2:40:07.020)
This was a fascinating conversation.
Lex Fridman (2:40:08.540)
I really love the way you explore math in your writing.
Jordan Ellenberg (2:40:14.140)
The willingness to be specific and clear
Lex Fridman (2:40:18.460)
and actually explore difficult ideas,
Lex Fridman (2:40:21.200)
but at the same time stepping outside
Lex Fridman (2:40:23.100)
and figuring out beautiful stuff.
Lex Fridman (2:40:25.060)
And I love the chart at the opening of your new book
Lex Fridman (2:40:30.380)
that shows the chaos, the mess that is your mind.
Jordan Ellenberg (2:40:33.280)
Yes, this is what I was trying to keep in my head
Lex Fridman (2:40:35.540)
all at once while I was writing.
Lex Fridman (2:40:38.020)
And I probably should have drawn this picture
Lex Fridman (2:40:40.300)
earlier in the process.
Jordan Ellenberg (2:40:41.300)
Maybe it would have made my organization easier.
Lex Fridman (2:40:43.100)
I actually drew it only at the end.
Lex Fridman (2:40:45.420)
And many of the things we talked about are on this map.
Lex Fridman (2:40:48.640)
The connections are yet to be fully dissected, investigated.
Lex Fridman (2:40:52.660)
And yes, God is in the picture.
Lex Fridman (2:40:56.740)
Right on the edge, right on the edge, not in the center.
Jordan Ellenberg (2:41:00.820)
Thank you so much for talking to me.
Lex Fridman (2:41:01.660)
It is a huge honor that you would waste
Jordan Ellenberg (2:41:03.460)
your valuable time with me.
Lex Fridman (2:41:05.820)
Thank you, Lex.
Jordan Ellenberg (2:41:06.660)
We went to some amazing places today.
Lex Fridman (2:41:07.820)
This was really fun.
Jordan Ellenberg (2:41:09.620)
Thanks for listening to this conversation
Lex Fridman (2:41:11.220)
with Jordan Ellenberg.
Lex Fridman (2:41:12.380)
And thank you to Secret Sauce, ExpressVPN, Blinkist,
Lex Fridman (2:41:16.660)
and Indeed.
Jordan Ellenberg (2:41:17.980)
Check them out in the description to support this podcast.
Lex Fridman (2:41:21.380)
And now let me leave you with some words from Jordan
Jordan Ellenberg (2:41:24.140)
in his book, How Not To Be Wrong.
Lex Fridman (2:41:26.720)
Knowing mathematics is like wearing a pair of X ray specs
Jordan Ellenberg (2:41:30.600)
that reveal hidden structures underneath the messy
Lex Fridman (2:41:33.500)
and chaotic surface of the world.
Jordan Ellenberg (2:41:35.780)
Thank you for listening and hope to see you next time.
Lex Fridman (30:00.280)
He's always motivated by physics,
Lex Fridman (30:02.120)
but the physics drove him to need to think about spaces
Lex Fridman (30:06.200)
of higher dimension.
Lex Fridman (30:07.240)
And so he needed a formalism that was rich enough
Lex Fridman (30:09.440)
to enable him to do that.
Lex Fridman (30:10.520)
And once you do that,
Lex Fridman (30:11.560)
that formalism is also gonna include things
Jordan Ellenberg (30:13.600)
that are not physical.
Lex Fridman (30:14.680)
And then you have two choices.
Jordan Ellenberg (30:15.600)
You can be like, oh, well, that stuff's trash.
Lex Fridman (30:17.760)
Or, and this is more of the mathematicians frame of mind,
Jordan Ellenberg (30:21.320)
if you have a formalistic framework
Lex Fridman (30:23.680)
that like seems really good
Lex Fridman (30:24.920)
and sort of seems to be like very elegant and work well,
Lex Fridman (30:27.200)
and it includes all the physical stuff,
Jordan Ellenberg (30:29.040)
maybe we should think about all of it.
Lex Fridman (30:30.560)
Like maybe we should think about it,
Jordan Ellenberg (30:31.400)
thinking maybe there's some gold to be mined there.
Lex Fridman (30:34.520)
And indeed, like, you know, guess what?
Jordan Ellenberg (30:36.640)
Like before long there's relativity and there's space time.
Lex Fridman (30:39.120)
And like all of a sudden it's like,
Jordan Ellenberg (30:40.080)
oh yeah, maybe it's a good idea.
Lex Fridman (30:41.560)
We already had this geometric apparatus like set up
Jordan Ellenberg (30:43.880)
for like how to think about four dimensional spaces,
Lex Fridman (30:47.240)
like turns out they're real after all.
Jordan Ellenberg (30:48.600)
As I said, you know, this is a story much told
Lex Fridman (30:51.680)
right in mathematics, not just in this context,
Lex Fridman (30:53.080)
but in many.
Lex Fridman (30:53.920)
I'd love to dig in a little deeper on that actually,
Jordan Ellenberg (30:55.640)
cause I have some intuitions to work out.
Lex Fridman (31:00.800)
Okay.
Jordan Ellenberg (31:01.800)
My brain.
Lex Fridman (31:02.640)
Well, I'm not a mathematical physicist,
Lex Fridman (31:03.560)
so we can work them out together.
Lex Fridman (31:05.600)
Good.
Jordan Ellenberg (31:06.440)
We'll together walk along the path of curiosity,
Lex Fridman (31:10.000)
but Poincare conjecture.
Lex Fridman (31:13.720)
What is it?
Lex Fridman (31:14.560)
The Poincare conjecture is about curved
Jordan Ellenberg (31:17.320)
three dimensional spaces.
Lex Fridman (31:18.880)
So I was on my way there.
Jordan Ellenberg (31:21.240)
I promise.
Lex Fridman (31:23.360)
The idea is that we perceive ourselves as living in,
Jordan Ellenberg (31:27.580)
we don't say a three dimensional space.
Lex Fridman (31:29.160)
We just say three dimensional space.
Jordan Ellenberg (31:30.280)
You know, you can go up and down,
Lex Fridman (31:31.480)
you can go left and right,
Jordan Ellenberg (31:32.320)
you can go forward and back.
Lex Fridman (31:33.200)
There's three dimensions in which we can move.
Jordan Ellenberg (31:35.480)
In Poincare's theory,
Lex Fridman (31:36.600)
there are many possible three dimensional spaces.
Jordan Ellenberg (31:41.680)
In the same way that going down one dimension
Lex Fridman (31:45.320)
to sort of capture our intuition a little bit more,
Jordan Ellenberg (31:48.420)
we know there are lots of different
Lex Fridman (31:49.700)
two dimensional surfaces, right?
Jordan Ellenberg (31:51.080)
There's a balloon and that looks one way
Lex Fridman (31:54.080)
and a donut looks another way
Lex Fridman (31:55.520)
and a Mobius strip looks a third way.
Lex Fridman (31:57.640)
Those are all like two dimensional surfaces
Jordan Ellenberg (31:59.120)
that we can kind of really get a global view of
Lex Fridman (32:02.360)
because we live in three dimensional space.
Lex Fridman (32:03.900)
So we can see a two dimensional surface
Lex Fridman (32:05.500)
sort of sitting in our three dimensional space.
Jordan Ellenberg (32:07.200)
Well, to see a three dimensional space whole,
Lex Fridman (32:11.260)
we'd have to kind of have four dimensional eyes, right?
Jordan Ellenberg (32:13.220)
Which we don't.
Lex Fridman (32:14.060)
So we have to use our mathematical eyes.
Jordan Ellenberg (32:15.020)
We have to envision.
Lex Fridman (32:17.440)
The Poincare conjecture says that there's a very simple way
Jordan Ellenberg (32:22.080)
to determine whether a three dimensional space
Lex Fridman (32:26.480)
is the standard one, the one that we're used to.
Lex Fridman (32:29.640)
And essentially it's that it's what's called
Lex Fridman (32:31.880)
fundamental group has nothing interesting in it.
Lex Fridman (32:34.640)
And that I can actually say without saying
Lex Fridman (32:36.040)
what the fundamental group is,
Jordan Ellenberg (32:36.960)
I can tell you what the criterion is.
Lex Fridman (32:39.000)
This would be good.
Jordan Ellenberg (32:39.840)
Oh, look, I can even use a visual aid.
Lex Fridman (32:40.880)
So for the people watching this on YouTube,
Jordan Ellenberg (32:42.360)
you will just see this for the people on the podcast,
Lex Fridman (32:45.240)
you'll have to visualize it.
Lex Fridman (32:46.160)
So Lex has been nice enough to like give me a surface
Lex Fridman (32:49.120)
with an interesting topology.
Jordan Ellenberg (32:50.400)
It's a mug right here in front of me.
Lex Fridman (32:52.320)
A mug, yes.
Jordan Ellenberg (32:53.360)
I might say it's a genus one surface,
Lex Fridman (32:55.140)
but we could also say it's a mug, same thing.
Lex Fridman (32:58.580)
So if I were to draw a little circle on this mug,
Lex Fridman (33:03.000)
which way should I draw it so it's visible?
Jordan Ellenberg (33:04.360)
Like here, okay.
Lex Fridman (33:06.280)
If I draw a little circle on this mug,
Jordan Ellenberg (33:07.520)
imagine this to be a loop of string.
Lex Fridman (33:09.380)
I could pull that loop of string closed
Lex Fridman (33:12.080)
on the surface of the mug, right?
Lex Fridman (33:14.640)
That's definitely something I could do.
Jordan Ellenberg (33:15.880)
I could shrink it, shrink it, shrink it until it's a point.
Lex Fridman (33:18.360)
On the other hand,
Jordan Ellenberg (33:19.200)
if I draw a loop that goes around the handle,
Lex Fridman (33:21.840)
I can kind of zhuzh it up here
Lex Fridman (33:23.100)
and I can zhuzh it down there
Lex Fridman (33:24.040)
and I can sort of slide it up and down the handle,
Lex Fridman (33:25.640)
but I can't pull it closed, can I?
Lex Fridman (33:27.300)
It's trapped.
Lex Fridman (33:28.840)
Not without breaking the surface of the mug, right?
Lex Fridman (33:30.680)
Not without like going inside.
Lex Fridman (33:32.380)
So the condition of being what's called simply connected,
Lex Fridman (33:37.160)
this is one of Poincare's inventions,
Jordan Ellenberg (33:39.840)
says that any loop of string can be pulled shut.
Lex Fridman (33:42.640)
So it's a feature that the mug simply does not have.
Jordan Ellenberg (33:45.120)
This is a non simply connected mug
Lex Fridman (33:48.540)
and a simply connected mug would be a cup, right?
Jordan Ellenberg (33:51.120)
You would burn your hand when you drank coffee out of it.
Lex Fridman (33:53.600)
So you're saying the universe is not a mug.
Jordan Ellenberg (33:56.520)
Well, I can't speak to the universe,
Lex Fridman (33:59.360)
but what I can say is that regular old space is not a mug.
Jordan Ellenberg (34:05.320)
Regular old space,
Lex Fridman (34:06.160)
if you like sort of actually physically have
Jordan Ellenberg (34:07.840)
like a loop of string,
Lex Fridman (34:09.600)
you can pull it shut.
Jordan Ellenberg (34:11.000)
You can always pull it shut.
Lex Fridman (34:12.660)
But what if your piece of string
Lex Fridman (34:14.060)
was the size of the universe?
Lex Fridman (34:14.980)
Like what if your piece of string
Lex Fridman (34:16.340)
was like billions of light years long?
Lex Fridman (34:18.120)
Like how do you actually know?
Jordan Ellenberg (34:20.180)
I mean, that's still an open question
Lex Fridman (34:21.480)
of the shape of the universe.
Jordan Ellenberg (34:22.560)
Exactly.
Lex Fridman (34:25.520)
I think there's a lot,
Jordan Ellenberg (34:26.480)
there is ideas of it being a torus.
Lex Fridman (34:28.620)
I mean, there's some trippy ideas
Lex Fridman (34:30.400)
and they're not like weird out there controversial.
Lex Fridman (34:33.440)
There's legitimate at the center of a cosmology debate.
Jordan Ellenberg (34:38.160)
I mean, I think most people think it's flat.
Lex Fridman (34:40.000)
I think there's some kind of dodecahedral symmetry
Jordan Ellenberg (34:42.160)
or I mean, I remember reading something crazy
Lex Fridman (34:43.600)
about somebody saying that they saw the signature of that
Jordan Ellenberg (34:45.920)
in the cosmic noise or what have you.
Lex Fridman (34:48.520)
I mean.
Jordan Ellenberg (34:49.800)
To make the flat earthers happy,
Lex Fridman (34:51.380)
I do believe that the current main belief is it's flat.
Jordan Ellenberg (34:56.380)
It's flat ish or something like that.
Lex Fridman (34:59.820)
The shape of the universe is flat ish.
Jordan Ellenberg (35:01.980)
I don't know what the heck that means.
Lex Fridman (35:03.140)
I think that has like a very,
Lex Fridman (35:06.660)
how are you even supposed to think about the shape
Lex Fridman (35:09.900)
of a thing that doesn't have any thing outside of it?
Jordan Ellenberg (35:14.140)
I mean.
Lex Fridman (35:14.980)
Ah, but that's exactly what topology does.
Jordan Ellenberg (35:16.740)
Topology is what's called an intrinsic theory.
Lex Fridman (35:19.420)
That's what's so great about it.
Jordan Ellenberg (35:20.340)
This question about the mug,
Lex Fridman (35:22.580)
you could answer it without ever leaving the mug, right?
Jordan Ellenberg (35:26.000)
Because it's a question about a loop drawn
Lex Fridman (35:28.980)
on the surface of the mug
Lex Fridman (35:29.980)
and what happens if it never leaves that surface.
Lex Fridman (35:31.820)
So it's like always there.
Jordan Ellenberg (35:33.500)
See, but that's the difference between the topology
Lex Fridman (35:37.820)
and say, if you're like trying to visualize a mug,
Jordan Ellenberg (35:42.480)
that you can't visualize a mug while living inside the mug.
Lex Fridman (35:46.660)
Well, that's true.
Jordan Ellenberg (35:47.500)
The visualization is harder, but in some sense,
Lex Fridman (35:49.180)
no, you're right.
Lex Fridman (35:50.000)
But if the tools of mathematics are there,
Lex Fridman (35:51.980)
I, sorry, I don't want to fight,
Lex Fridman (35:53.700)
but I think the tools of mathematics are exactly there
Lex Fridman (35:55.580)
to enable you to think about
Lex Fridman (35:56.980)
what you cannot visualize in this way.
Lex Fridman (35:58.700)
Let me give, let's go, always to make things easier,
Jordan Ellenberg (36:00.740)
go down to dimension.
Lex Fridman (36:03.000)
Let's think about we live in a circle, okay?
Jordan Ellenberg (36:05.780)
You can tell whether you live on a circle or a line segment,
Lex Fridman (36:11.260)
because if you live in a circle,
Jordan Ellenberg (36:12.340)
if you walk a long way in one direction,
Lex Fridman (36:13.800)
you find yourself back where you started.
Lex Fridman (36:15.220)
And if you live in a line segment,
Lex Fridman (36:17.300)
you walk for a long enough one direction,
Jordan Ellenberg (36:18.740)
you come to the end of the world.
Lex Fridman (36:20.180)
Or if you live on a line, like a whole line,
Jordan Ellenberg (36:22.900)
infinite line, then you walk in one direction
Lex Fridman (36:25.860)
for a long time and like,
Jordan Ellenberg (36:27.080)
well, then there's not a sort of terminating algorithm
Lex Fridman (36:28.720)
to figure out whether you live on a line or a circle,
Lex Fridman (36:30.460)
but at least you sort of,
Lex Fridman (36:33.340)
at least you don't discover that you live on a circle.
Lex Fridman (36:35.680)
So all of those are intrinsic things, right?
Lex Fridman (36:37.360)
All of those are things that you can figure out
Jordan Ellenberg (36:39.700)
about your world without leaving your world.
Lex Fridman (36:42.060)
On the other hand, ready?
Jordan Ellenberg (36:43.300)
Now we're going to go from intrinsic to extrinsic.
Lex Fridman (36:45.220)
Boy, did I not know we were going to talk about this,
Lex Fridman (36:46.920)
but why not?
Lex Fridman (36:48.020)
Why not?
Jordan Ellenberg (36:48.860)
If you can't tell whether you live in a circle
Lex Fridman (36:52.500)
or a knot, like imagine like a knot
Jordan Ellenberg (36:55.580)
floating in three dimensional space.
Lex Fridman (36:56.900)
The person who lives on that knot, to them it's a circle.
Jordan Ellenberg (36:59.660)
They walk a long way, they come back to where they started.
Lex Fridman (37:01.740)
Now we, with our three dimensional eyes can be like,
Jordan Ellenberg (37:04.280)
oh, this one's just a plain circle
Lex Fridman (37:05.580)
and this one's knotted up,
Lex Fridman (37:06.700)
but that has to do with how they sit
Lex Fridman (37:09.700)
in three dimensional space.
Jordan Ellenberg (37:10.620)
It doesn't have to do with intrinsic features
Lex Fridman (37:12.120)
of those people's world.
Jordan Ellenberg (37:13.120)
We can ask you one ape to another.
Lex Fridman (37:14.880)
Does it make you, how does it make you feel
Jordan Ellenberg (37:17.120)
that you don't know if you live in a circle
Lex Fridman (37:19.860)
or on a knot, in a knot,
Lex Fridman (37:24.460)
inside the string that forms the knot?
Lex Fridman (37:28.680)
I don't even know how to say that.
Jordan Ellenberg (37:29.940)
I'm going to be honest with you.
Lex Fridman (37:30.940)
I don't know if, I fear you won't like this answer,
Lex Fridman (37:34.580)
but it does not bother me at all.
Lex Fridman (37:37.140)
I don't lose one minute of sleep over it.
Lex Fridman (37:39.380)
So like, does it bother you that if we look
Lex Fridman (37:41.700)
at like a Mobius strip, that you don't have an obvious way
Jordan Ellenberg (37:46.100)
of knowing whether you are inside of a cylinder,
Lex Fridman (37:49.740)
if you live on a surface of a cylinder
Lex Fridman (37:51.780)
or you live on the surface of a Mobius strip?
Lex Fridman (37:55.700)
No, I think you can tell if you live.
Lex Fridman (37:58.740)
Which one?
Lex Fridman (37:59.580)
Because what you do is you like tell your friend,
Jordan Ellenberg (38:02.500)
hey, stay right here, I'm just going to go for a walk.
Lex Fridman (38:04.140)
And then you like walk for a long time in one direction
Lex Fridman (38:06.700)
and then you come back and you see your friend again.
Lex Fridman (38:08.260)
And if your friend is reversed,
Jordan Ellenberg (38:09.380)
then you know you live on a Mobius strip.
Lex Fridman (38:10.740)
Well, no, because you won't see your friend, right?
Jordan Ellenberg (38:13.860)
Okay, fair point, fair point on that.
Lex Fridman (38:17.060)
But you have to believe the stories about,
Lex Fridman (38:19.820)
no, I don't even know, would you even know?
Lex Fridman (38:24.220)
Would you really?
Jordan Ellenberg (38:25.100)
Oh, no, your point is right.
Lex Fridman (38:26.860)
Let me try to think of a better,
Jordan Ellenberg (38:28.220)
let's see if I can do this on the fly.
Lex Fridman (38:29.420)
It may not be correct to talk about cognitive beings
Jordan Ellenberg (38:33.900)
living on a Mobius strip
Lex Fridman (38:35.380)
because there's a lot of things taken for granted there.
Lex Fridman (38:37.940)
And we're constantly imagining actual
Lex Fridman (38:39.820)
like three dimensional creatures,
Jordan Ellenberg (38:42.300)
like how it actually feels like to live in a Mobius strip
Lex Fridman (38:47.940)
is tricky to internalize.
Jordan Ellenberg (38:50.140)
I think that on what's called the real protective plane,
Lex Fridman (38:52.860)
which is kind of even more sort of like messed up version
Jordan Ellenberg (38:54.900)
of the Mobius strip, but with very similar features,
Lex Fridman (38:57.540)
this feature of kind of like only having one side,
Jordan Ellenberg (39:01.340)
that has the feature that there's a loop of string
Lex Fridman (39:04.500)
which can't be pulled closed.
Lex Fridman (39:06.740)
But if you loop it around twice along the same path,
Lex Fridman (39:09.700)
that you can pull closed.
Jordan Ellenberg (39:11.300)
That's extremely weird.
Lex Fridman (39:12.980)
Yeah.
Lex Fridman (39:14.860)
But that would be a way you could know
Lex Fridman (39:16.260)
without leaving your world
Jordan Ellenberg (39:17.260)
that something very funny is going on.
Lex Fridman (39:20.380)
You know what's extremely weird?
Jordan Ellenberg (39:21.980)
Maybe we can comment on,
Lex Fridman (39:23.260)
hopefully it's not too much of a tangent is,
Jordan Ellenberg (39:26.900)
I remember thinking about this,
Lex Fridman (39:29.020)
this might be right, this might be wrong.
Lex Fridman (39:31.820)
But if we now talk about a sphere
Lex Fridman (39:35.460)
and you're living inside a sphere,
Jordan Ellenberg (39:37.580)
that you're going to see everywhere around you,
Lex Fridman (39:41.180)
the back of your own head.
Jordan Ellenberg (39:44.820)
That I was,
Lex Fridman (39:46.140)
cause like I was,
Jordan Ellenberg (39:47.980)
this is very counterintuitive to me to think about,
Lex Fridman (39:50.820)
maybe it's wrong.
Lex Fridman (39:51.660)
But cause I was thinking of like earth,
Lex Fridman (39:54.260)
your 3D thing sitting on a sphere.
Lex Fridman (39:57.260)
But if you're living inside the sphere,
Lex Fridman (40:00.140)
like you're going to see, if you look straight,
Jordan Ellenberg (40:02.300)
you're always going to see yourself all the way around.
Lex Fridman (40:05.580)
So everywhere you look, there's going to be
Jordan Ellenberg (40:07.620)
the back of your own head.
Lex Fridman (40:09.300)
I think somehow this depends on something
Jordan Ellenberg (40:10.980)
of like how the physics of light works in this scenario,
Lex Fridman (40:13.180)
which I'm sort of finding it hard to bend my.
Jordan Ellenberg (40:14.820)
That's true.
Lex Fridman (40:15.660)
The sea is doing a lot of work.
Jordan Ellenberg (40:16.660)
Like saying you see something is doing a lot of work.
Lex Fridman (40:19.540)
People have thought about this a lot.
Jordan Ellenberg (40:20.740)
I mean, this metaphor of like,
Lex Fridman (40:22.340)
what if we're like little creatures
Lex Fridman (40:24.740)
in some sort of smaller world?
Lex Fridman (40:26.100)
Like how could we apprehend what's outside?
Jordan Ellenberg (40:27.700)
That metaphor just comes back and back.
Lex Fridman (40:29.580)
And actually I didn't even realize like how frequent it is.
Jordan Ellenberg (40:32.140)
It comes up in the book a lot.
Lex Fridman (40:33.540)
I know it from a book called Flatland.
Jordan Ellenberg (40:35.620)
I don't know if you ever read this when you were a kid.
Lex Fridman (40:37.780)
A while ago, yeah.
Jordan Ellenberg (40:38.620)
An adult.
Lex Fridman (40:39.460)
You know, this sort of comic novel from the 19th century
Jordan Ellenberg (40:42.900)
about an entire two dimensional world.
Lex Fridman (40:46.940)
It's narrated by a square.
Jordan Ellenberg (40:48.260)
That's the main character.
Lex Fridman (40:49.820)
And the kind of strangeness that befalls him
Jordan Ellenberg (40:53.580)
when one day he's in his house
Lex Fridman (40:55.220)
and suddenly there's like a little circle there
Lex Fridman (40:57.860)
and they're with him.
Lex Fridman (40:59.220)
But then the circle like starts getting bigger
Lex Fridman (41:02.460)
and bigger and bigger.
Lex Fridman (41:04.340)
And he's like, what the hell is going on?
Jordan Ellenberg (41:06.020)
It's like a horror movie, like for two dimensional people.
Lex Fridman (41:08.500)
And of course what's happening
Jordan Ellenberg (41:09.820)
is that a sphere is entering his world.
Lex Fridman (41:12.140)
And as the sphere kind of like moves farther and farther
Jordan Ellenberg (41:15.060)
into the plane, it's cross section.
Lex Fridman (41:16.700)
The part of it that he can see.
Jordan Ellenberg (41:18.420)
To him, it looks like there's like this kind
Lex Fridman (41:20.140)
of bizarre being that's like getting larger
Lex Fridman (41:22.420)
and larger and larger
Lex Fridman (41:24.700)
until it's exactly sort of halfway through.
Lex Fridman (41:27.300)
And then they have this kind of like philosophical argument
Lex Fridman (41:29.140)
where the sphere is like, I'm a sphere.
Jordan Ellenberg (41:30.180)
I'm from the third dimension.
Lex Fridman (41:31.020)
The square is like, what are you talking about?
Jordan Ellenberg (41:32.220)
There's no such thing.
Lex Fridman (41:33.380)
And they have this kind of like sterile argument
Jordan Ellenberg (41:36.220)
where the square is not able to kind of like
Lex Fridman (41:39.140)
follow the mathematical reasoning of the sphere
Jordan Ellenberg (41:40.980)
until the sphere just kind of grabs him
Lex Fridman (41:42.380)
and like jerks him out of the plane and pulls him up.
Lex Fridman (41:45.820)
And it's like now, like now do you see,
Lex Fridman (41:47.380)
like now do you see your whole world
Lex Fridman (41:50.260)
that you didn't understand before?
Lex Fridman (41:52.100)
So do you think that kind of process is possible
Lex Fridman (41:55.500)
for us humans?
Lex Fridman (41:56.580)
So we live in the three dimensional world,
Jordan Ellenberg (41:58.380)
maybe with the time component four dimensional
Lex Fridman (42:01.520)
and then math allows us to go high,
Jordan Ellenberg (42:06.620)
into high dimensions comfortably
Lex Fridman (42:08.220)
and explore the world from those perspectives.
Jordan Ellenberg (42:13.580)
Like, is it possible that the universe
Lex Fridman (42:19.620)
is many more dimensions than the ones
Lex Fridman (42:23.060)
we experience as human beings?
Lex Fridman (42:25.180)
So if you look at the, you know,
Jordan Ellenberg (42:28.820)
especially in physics theories of everything,
Lex Fridman (42:32.020)
physics theories that try to unify general relativity
Lex Fridman (42:35.380)
and quantum field theory,
Lex Fridman (42:37.400)
they seem to go to high dimensions to work stuff out
Jordan Ellenberg (42:42.660)
through the tools of mathematics.
Lex Fridman (42:44.620)
Is it possible?
Lex Fridman (42:46.140)
So like the two options are,
Lex Fridman (42:47.700)
one is just a nice way to analyze a universe,
Lex Fridman (42:51.500)
but the reality is, is as exactly we perceive it,
Lex Fridman (42:54.740)
it is three dimensional, or are we just seeing,
Jordan Ellenberg (42:58.620)
are we those flatland creatures
Lex Fridman (43:00.740)
that are just seeing a tiny slice of reality
Lex Fridman (43:03.740)
and the actual reality is many, many, many more dimensions
Lex Fridman (43:08.860)
than the three dimensions we perceive?
Jordan Ellenberg (43:10.960)
Oh, I certainly think that's possible.
Lex Fridman (43:14.580)
Now, how would you figure out whether it was true or not
Jordan Ellenberg (43:17.900)
is another question.
Lex Fridman (43:20.440)
And I suppose what you would do
Jordan Ellenberg (43:22.180)
as with anything else that you can't directly perceive
Lex Fridman (43:25.060)
is you would try to understand
Lex Fridman (43:29.380)
what effect the presence of those extra dimensions
Lex Fridman (43:33.220)
out there would have on the things we can perceive.
Lex Fridman (43:36.940)
Like what else can you do, right?
Lex Fridman (43:39.180)
And in some sense, if the answer is
Jordan Ellenberg (43:42.260)
they would have no effect,
Lex Fridman (43:44.720)
then maybe it becomes like a little bit
Jordan Ellenberg (43:46.180)
of a sterile question,
Lex Fridman (43:47.020)
because what question are you even asking, right?
Jordan Ellenberg (43:49.340)
You can kind of posit however many entities that you want.
Lex Fridman (43:53.740)
Is it possible to intuit how to mess
Jordan Ellenberg (43:56.900)
with the other dimensions
Lex Fridman (43:58.240)
while living in a three dimensional world?
Jordan Ellenberg (44:00.320)
I mean, that seems like a very challenging thing to do.
Lex Fridman (44:03.860)
The reason flatland could be written
Jordan Ellenberg (44:06.900)
is because it's coming from a three dimensional writer.
Lex Fridman (44:11.420)
Yes, but what happens in the book,
Jordan Ellenberg (44:13.880)
I didn't even tell you the whole plot.
Lex Fridman (44:15.220)
What happens is the square is so excited
Lex Fridman (44:17.220)
and so filled with intellectual joy.
Lex Fridman (44:19.960)
By the way, maybe to give the story some context,
Jordan Ellenberg (44:22.200)
you asked like, is it possible for us humans
Lex Fridman (44:25.180)
to have this experience of being transcendentally jerked
Lex Fridman (44:28.460)
out of our world so we can sort of truly see it from above?
Lex Fridman (44:30.980)
Well, Edwin Abbott who wrote the book
Jordan Ellenberg (44:32.760)
certainly thought so because Edwin Abbott was a minister.
Lex Fridman (44:35.940)
So the whole Christian subtext of this book,
Jordan Ellenberg (44:37.880)
I had completely not grasped reading this as a kid,
Lex Fridman (44:41.840)
that it means a very different thing, right?
Jordan Ellenberg (44:43.420)
If sort of a theologian is saying like,
Lex Fridman (44:45.700)
oh, what if a higher being could like pull you out
Jordan Ellenberg (44:48.220)
of this earthly world you live in
Lex Fridman (44:50.000)
so that you can sort of see the truth
Lex Fridman (44:51.360)
and like really see it from above as it were.
Lex Fridman (44:54.460)
So that's one of the things that's going on for him.
Lex Fridman (44:56.500)
And it's a testament to his skill as a writer
Lex Fridman (44:58.420)
that his story just works whether that's the framework
Jordan Ellenberg (45:01.760)
you're coming to it from or not.
Lex Fridman (45:05.220)
But what happens in this book and this part,
Jordan Ellenberg (45:07.500)
now looking at it through a Christian lens,
Lex Fridman (45:08.980)
it becomes a bit subversive is the square is so excited
Jordan Ellenberg (45:13.240)
about what he's learned from the sphere
Lex Fridman (45:16.780)
and the sphere explains to him like what a cube would be.
Jordan Ellenberg (45:18.740)
Oh, it's like you but three dimensional
Lex Fridman (45:20.020)
and the square is very excited
Lex Fridman (45:21.020)
and the square is like, okay, I get it now.
Lex Fridman (45:23.180)
So like now that you explained to me how just by reason
Jordan Ellenberg (45:26.060)
I can figure out what a cube would be like,
Lex Fridman (45:27.380)
like a three dimensional version of me,
Jordan Ellenberg (45:29.020)
like let's figure out what a four dimensional version
Lex Fridman (45:31.340)
of me would be like.
Lex Fridman (45:32.740)
And then the sphere is like,
Lex Fridman (45:33.980)
what the hell are you talking about?
Jordan Ellenberg (45:34.860)
There's no fourth dimension, that's ridiculous.
Lex Fridman (45:36.520)
Like there's three dimensions,
Jordan Ellenberg (45:37.740)
like that's how many there are, I can see.
Lex Fridman (45:39.300)
Like, I mean, it's this sort of comic moment
Jordan Ellenberg (45:40.880)
where the sphere is completely unable to conceptualize
Lex Fridman (45:44.660)
that there could actually be yet another dimension.
Lex Fridman (45:47.940)
So yeah, that takes the religious allegory
Lex Fridman (45:49.900)
like a very weird place that I don't really
Jordan Ellenberg (45:51.340)
like understand theologically, but.
Lex Fridman (45:53.140)
That's a nice way to talk about religion and myth in general
Jordan Ellenberg (45:57.880)
as perhaps us trying to struggle,
Lex Fridman (46:00.960)
us meaning human civilization, trying to struggle
Jordan Ellenberg (46:03.860)
with ideas that are beyond our cognitive capabilities.
Lex Fridman (46:08.620)
But it's in fact not beyond our capability.
Jordan Ellenberg (46:10.620)
It may be beyond our cognitive capabilities
Lex Fridman (46:13.300)
to visualize a four dimensional cube,
Jordan Ellenberg (46:16.460)
a tesseract as some like to call it,
Lex Fridman (46:18.320)
or a five dimensional cube, or a six dimensional cube,
Lex Fridman (46:20.800)
but it is not beyond our cognitive capabilities
Lex Fridman (46:23.940)
to figure out how many corners
Jordan Ellenberg (46:26.640)
a six dimensional cube would have.
Lex Fridman (46:28.040)
That's what's so cool about us.
Jordan Ellenberg (46:29.380)
Whether we can visualize it or not,
Lex Fridman (46:31.060)
we can still talk about it, we can still reason about it,
Jordan Ellenberg (46:33.400)
we can still figure things out about it.
Lex Fridman (46:36.160)
That's amazing.
Jordan Ellenberg (46:37.260)
Yeah, if we go back to this, first of all, to the mug,
Lex Fridman (46:41.620)
but to the example you give in the book of the straw,
Lex Fridman (46:44.980)
how many holes does a straw have?
Lex Fridman (46:49.800)
And you, listener, may try to answer that in your own head.
Jordan Ellenberg (46:54.140)
Yeah, I'm gonna take a drink while everybody thinks about it
Lex Fridman (46:56.220)
so we can give you a moment.
Jordan Ellenberg (46:57.060)
A slow sip.
Lex Fridman (46:59.020)
Is it zero, one, or two, or more than that maybe?
Jordan Ellenberg (47:04.900)
Maybe you can get very creative.
Lex Fridman (47:06.660)
But it's kind of interesting to each,
Jordan Ellenberg (47:10.640)
dissecting each answer as you do in the book
Lex Fridman (47:13.100)
is quite brilliant.
Jordan Ellenberg (47:14.140)
People should definitely check it out.
Lex Fridman (47:15.540)
But if you could try to answer it now,
Jordan Ellenberg (47:18.420)
think about all the options
Lex Fridman (47:21.220)
and why they may or may not be right.
Jordan Ellenberg (47:23.420)
Yeah, and it's one of these questions
Lex Fridman (47:25.260)
where people on first hearing it think it's a triviality
Lex Fridman (47:28.300)
and they're like, well, the answer is obvious.
Lex Fridman (47:29.780)
And then what happens if you ever ask a group of people
Jordan Ellenberg (47:31.380)
that something wonderfully comic happens,
Lex Fridman (47:33.960)
which is that everyone's like,
Jordan Ellenberg (47:34.800)
well, it's completely obvious.
Lex Fridman (47:36.500)
And then each person realizes that half the person,
Jordan Ellenberg (47:38.980)
the other people in the room
Lex Fridman (47:39.820)
have a different obvious answer for the way they have.
Lex Fridman (47:42.940)
And then people get really heated.
Lex Fridman (47:44.420)
People are like, I can't believe
Jordan Ellenberg (47:46.100)
that you think it has two holes
Lex Fridman (47:47.440)
or like, I can't believe that you think it has one.
Lex Fridman (47:49.660)
And then, you know, you really,
Lex Fridman (47:50.580)
like people really learn something about each other
Lex Fridman (47:52.420)
and people get heated.
Lex Fridman (47:54.440)
I mean, can we go through the possible options here?
Lex Fridman (47:57.700)
Is it zero, one, two, three, 10?
Lex Fridman (48:01.300)
Sure, so I think, you know, most people,
Jordan Ellenberg (48:04.660)
the zero holders are rare.
Lex Fridman (48:06.100)
They would say like, well, look,
Jordan Ellenberg (48:07.740)
you can make a straw by taking a rectangular piece of plastic
Lex Fridman (48:10.180)
and closing it up.
Jordan Ellenberg (48:11.020)
A rectangular piece of plastic doesn't have a hole in it.
Lex Fridman (48:14.820)
I didn't poke a hole in it when I,
Lex Fridman (48:16.980)
so how can I have a hole?
Lex Fridman (48:18.320)
They'd be like, it's just one thing.
Jordan Ellenberg (48:19.580)
Okay, most people don't see it that way.
Lex Fridman (48:21.940)
That's like a...
Lex Fridman (48:23.820)
Is there any truth to that kind of conception?
Lex Fridman (48:25.900)
Yeah, I think that would be somebody who's account, I mean,
Lex Fridman (48:33.740)
what I would say is you could say the same thing
Lex Fridman (48:39.420)
about a bagel.
Jordan Ellenberg (48:40.460)
You could say, I can make a bagel by taking like a long
Lex Fridman (48:43.100)
cylinder of dough, which doesn't have a hole
Lex Fridman (48:45.060)
and then schmushing the ends together.
Lex Fridman (48:47.720)
Now it's a bagel.
Lex Fridman (48:49.060)
So if you're really committed, you can be like, okay,
Lex Fridman (48:50.580)
a bagel doesn't have a hole either.
Lex Fridman (48:51.700)
But like, who are you if you say a bagel doesn't have a hole?
Lex Fridman (48:54.100)
I mean, I don't know.
Jordan Ellenberg (48:54.940)
Yeah, so that's almost like an engineering definition of it.
Lex Fridman (48:57.980)
Okay, fair enough.
Lex Fridman (48:59.000)
So what about the other options?
Lex Fridman (49:02.240)
So, you know, one whole people would say...
Jordan Ellenberg (49:07.740)
I like how these are like groups of people.
Lex Fridman (49:09.900)
Like we've planted our foot, this is what we stand for.
Jordan Ellenberg (49:12.940)
There's books written about each belief.
Lex Fridman (49:16.260)
You know, I would say, look, there's like a hole
Lex Fridman (49:17.580)
and it goes all the way through the straw, right?
Lex Fridman (49:19.140)
It's one region of space, that's the hole.
Lex Fridman (49:21.900)
And there's one.
Lex Fridman (49:22.740)
And two whole people would say like, well, look,
Jordan Ellenberg (49:24.140)
there's a hole in the top and a hole at the bottom.
Lex Fridman (49:28.420)
I think a common thing you see when people
Jordan Ellenberg (49:34.020)
argue about this, they would take something like this
Lex Fridman (49:35.960)
bottle of water I'm holding and go open it and they say,
Lex Fridman (49:40.260)
well, how many holes are there in this?
Lex Fridman (49:41.540)
And you say like, well, there's one hole at the top.
Jordan Ellenberg (49:44.100)
Okay, what if I like poke a hole here
Lex Fridman (49:46.380)
so that all the water spills out?
Jordan Ellenberg (49:48.940)
Well, now it's a straw.
Lex Fridman (49:50.860)
Yeah.
Lex Fridman (49:51.700)
So if you're a one holder, I say to you like,
Lex Fridman (49:53.140)
well, how many holes are in it now?
Jordan Ellenberg (49:56.020)
There was one hole in it before
Lex Fridman (49:57.340)
and I poked a new hole in it.
Lex Fridman (49:59.280)
And then you think there's still one hole
Lex Fridman (50:01.580)
even though there was one hole and I made one more?
Jordan Ellenberg (50:04.700)
Clearly not, this is two holes.
Lex Fridman (50:06.740)
Yeah.
Lex Fridman (50:08.180)
And yet if you're a two holder, the one holder will say like,
Lex Fridman (50:10.380)
okay, where does one hole begin and the other hole end?
Jordan Ellenberg (50:13.220)
Yeah.
Lex Fridman (50:16.340)
And in the book, I sort of, you know, in math,
Jordan Ellenberg (50:18.660)
there's two things we do when we're faced with a problem
Lex Fridman (50:20.380)
that's confusing us.
Jordan Ellenberg (50:22.940)
We can make the problem simpler.
Lex Fridman (50:24.540)
That's what we were doing a minute ago
Jordan Ellenberg (50:25.740)
when we were talking about high dimensional space.
Lex Fridman (50:27.100)
And I was like, let's talk about like circles
Lex Fridman (50:28.500)
and line segments.
Lex Fridman (50:29.340)
Let's like go down a dimension to make it easier.
Jordan Ellenberg (50:31.740)
The other big move we have is to make the problem harder
Lex Fridman (50:35.100)
and try to sort of really like face up
Jordan Ellenberg (50:36.700)
to what are the complications.
Lex Fridman (50:37.660)
So, you know, what I do in the book is say like,
Jordan Ellenberg (50:39.580)
let's stop talking about straws for a minute
Lex Fridman (50:41.220)
and talk about pants.
Lex Fridman (50:42.860)
How many holes are there in a pair of pants?
Lex Fridman (50:46.220)
So I think most people who say there's two holes in a straw
Jordan Ellenberg (50:48.820)
would say there's three holes in a pair of pants.
Lex Fridman (50:51.980)
I guess, I mean, I guess we're filming only from here.
Jordan Ellenberg (50:54.020)
I could take up, no, I'm not gonna do it.
Lex Fridman (50:56.580)
You'll just have to imagine the pants, sorry.
Jordan Ellenberg (50:58.300)
Yeah.
Lex Fridman (50:59.700)
Lex, if you want to, no, okay, no.
Jordan Ellenberg (51:01.220)
That's gonna be in the director's cut.
Lex Fridman (51:04.580)
That's that Patreon only footage.
Jordan Ellenberg (51:06.380)
There you go.
Lex Fridman (51:07.820)
So many people would say there's three holes
Jordan Ellenberg (51:09.500)
in a pair of pants.
Lex Fridman (51:10.340)
But you know, for instance, my daughter, when I asked,
Jordan Ellenberg (51:11.980)
by the way, talking to kids about this is super fun.
Lex Fridman (51:14.860)
I highly recommend it.
Lex Fridman (51:16.380)
What did she say?
Lex Fridman (51:17.900)
She said, well, yeah, I feel a pair of pants
Jordan Ellenberg (51:21.020)
like just has two holes because yes, there's the waist,
Lex Fridman (51:23.660)
but that's just the two leg holes stuck together.
Jordan Ellenberg (51:26.620)
Whoa, okay.
Lex Fridman (51:28.500)
Two leg holes, yeah, okay.
Jordan Ellenberg (51:29.860)
I mean, that really is a good combination.
Lex Fridman (51:31.420)
So she's a one holder for the straw.
Lex Fridman (51:32.340)
So she's a one holder for the straw too.
Lex Fridman (51:34.460)
And that really does capture something.
Jordan Ellenberg (51:39.580)
It captures this fact, which is central
Lex Fridman (51:42.860)
to the theory of what's called homology,
Jordan Ellenberg (51:44.420)
which is like a central part of modern topology
Lex Fridman (51:46.060)
that holes, whatever we may mean by them,
Jordan Ellenberg (51:49.700)
they're somehow things which have an arithmetic to them.
Lex Fridman (51:51.980)
They're things which can be added.
Jordan Ellenberg (51:53.940)
Like the waist, like waist equals leg plus leg
Lex Fridman (51:57.180)
is kind of an equation,
Lex Fridman (51:58.420)
but it's not an equation about numbers.
Lex Fridman (52:00.100)
It's an equation about some kind of geometric,
Jordan Ellenberg (52:02.900)
some kind of topological thing, which is very strange.
Lex Fridman (52:05.420)
And so, you know, when I come down, you know,
Jordan Ellenberg (52:09.220)
like a rabbi, I like to kind of like come up
Lex Fridman (52:11.500)
with these answers and somehow like dodge
Jordan Ellenberg (52:13.340)
the original question and say like,
Lex Fridman (52:14.740)
you're both right, my children.
Jordan Ellenberg (52:15.860)
Okay, so.
Lex Fridman (52:17.260)
Yeah.
Lex Fridman (52:19.140)
So for the straw, I think what a modern mathematician
Lex Fridman (52:23.460)
would say is like, the first version would be to say like,
Jordan Ellenberg (52:27.980)
well, there are two holes,
Lex Fridman (52:29.260)
but they're really both the same hole.
Jordan Ellenberg (52:31.340)
Well, that's not quite right.
Lex Fridman (52:32.260)
A better way to say it is there's two holes,
Lex Fridman (52:34.740)
but one is the negative of the other.
Lex Fridman (52:37.500)
Now, what can that mean?
Jordan Ellenberg (52:39.860)
One way of thinking about what it means is that
Lex Fridman (52:41.420)
if you sip something like a milkshake through the straw,
Jordan Ellenberg (52:44.620)
no matter what, the amount of milkshake
Lex Fridman (52:48.100)
that's flowing in one end,
Jordan Ellenberg (52:49.940)
that same amount is flowing out the other end.
Lex Fridman (52:53.380)
So they're not independent from each other.
Jordan Ellenberg (52:55.780)
There's some relationship between them.
Lex Fridman (52:57.660)
In the same way that if you somehow
Jordan Ellenberg (53:00.780)
could like suck a milkshake through a pair of pants,
Lex Fridman (53:05.140)
the amount of milkshake,
Jordan Ellenberg (53:06.580)
just go with me on this thought experiment.
Lex Fridman (53:08.860)
I'm right there with you.
Jordan Ellenberg (53:09.700)
The amount of milkshake that's coming in
Lex Fridman (53:11.660)
the left leg of the pants,
Jordan Ellenberg (53:13.300)
plus the amount of milkshake that's coming in
Lex Fridman (53:15.060)
the right leg of the pants,
Jordan Ellenberg (53:16.620)
is the same that's coming out the waist of the pants.
Lex Fridman (53:20.740)
So just so you know, I fasted for 72 hours
Jordan Ellenberg (53:24.140)
the last three days.
Lex Fridman (53:25.540)
So I just broke the fast with a little bit of food yesterday.
Lex Fridman (53:27.740)
So this sounds, food analogies or metaphors
Lex Fridman (53:32.060)
for this podcast work wonderfully
Jordan Ellenberg (53:33.740)
because I can intensely picture it.
Lex Fridman (53:35.740)
Is that your weekly routine or just in preparation
Lex Fridman (53:37.860)
for talking about geometry for three hours?
Lex Fridman (53:39.460)
Exactly, this is just for this.
Jordan Ellenberg (53:41.900)
It's hardship to purify the mind.
Lex Fridman (53:44.100)
No, it's for the first time,
Jordan Ellenberg (53:45.180)
I just wanted to try the experience.
Lex Fridman (53:46.660)
Oh, wow.
Lex Fridman (53:47.500)
And just to pause,
Lex Fridman (53:50.060)
to do things that are out of the ordinary,
Jordan Ellenberg (53:52.060)
to pause and to reflect on how grateful I am
Lex Fridman (53:55.980)
to be just alive and be able to do all the cool shit
Jordan Ellenberg (53:59.220)
that I get to do, so.
Lex Fridman (54:00.220)
Did you drink water?
Jordan Ellenberg (54:01.340)
Yes, yes, yes, yes, yes.
Lex Fridman (54:03.060)
Water and salt, so like electrolytes
Lex Fridman (54:05.740)
and all those kinds of things.
Lex Fridman (54:07.180)
But anyway, so the inflow on the top of the pants
Jordan Ellenberg (54:10.620)
equals to the outflow on the bottom of the pants.
Lex Fridman (54:14.620)
Exactly, so this idea that,
Jordan Ellenberg (54:18.020)
I mean, I think, you know, Poincare really had this idea,
Lex Fridman (54:21.340)
this sort of modern idea.
Jordan Ellenberg (54:22.820)
I mean, building on stuff other people did,
Lex Fridman (54:25.020)
Betty is an important one,
Jordan Ellenberg (54:26.700)
of this kind of modern notion of relations between holes.
Lex Fridman (54:29.900)
But the idea that holes really had an arithmetic,
Jordan Ellenberg (54:32.540)
the really modern view was really Emmy Noether's idea.
Lex Fridman (54:35.540)
So she kind of comes in and sort of truly puts the subject
Jordan Ellenberg (54:40.820)
on its modern footing that we have now.
Lex Fridman (54:43.300)
So, you know, it's always a challenge, you know,
Jordan Ellenberg (54:45.380)
in the book, I'm not gonna say I give like a course
Lex Fridman (54:48.620)
so that you read this chapter and then you're like,
Jordan Ellenberg (54:50.300)
oh, it's just like I took like a semester
Lex Fridman (54:51.940)
of algebraic anthropology.
Jordan Ellenberg (54:53.100)
It's not like this and it's always a challenge
Lex Fridman (54:55.500)
writing about math because there are some things
Jordan Ellenberg (55:00.260)
that you can really do on the page and the math is there.
Lex Fridman (55:03.300)
And there's other things which it's too much
Jordan Ellenberg (55:05.860)
in a book like this to like do them all the page.
Lex Fridman (55:07.380)
You can only say something about them, if that makes sense.
Jordan Ellenberg (55:12.620)
So, you know, in the book, I try to do some of both.
Lex Fridman (55:14.780)
I try to do, I try to, topics that are,
Jordan Ellenberg (55:18.740)
you can't really compress and really truly say
Lex Fridman (55:22.100)
exactly what they are in this amount of space.
Jordan Ellenberg (55:27.420)
I try to say something interesting about them,
Lex Fridman (55:28.980)
something meaningful about them
Lex Fridman (55:30.180)
so that readers can get the flavor.
Lex Fridman (55:31.740)
And then in other places,
Jordan Ellenberg (55:34.380)
I really try to get up close and personal
Lex Fridman (55:36.620)
and really do the math and have it take place on the page.
Jordan Ellenberg (55:40.740)
To some degree be able to give inklings
Lex Fridman (55:44.100)
of the beauty of the subject.
Jordan Ellenberg (55:45.820)
Yeah, I mean, there's a lot of books that are like,
Lex Fridman (55:48.260)
I don't quite know how to express this well.
Jordan Ellenberg (55:49.740)
I'm still laboring to do it,
Lex Fridman (55:51.020)
but there's a lot of books that are about stuff,
Lex Fridman (55:57.380)
but I want my books to not only be about stuff,
Lex Fridman (56:01.020)
but to actually have some stuff there on the page
Jordan Ellenberg (56:03.620)
in the book for people to interact with directly
Lex Fridman (56:05.620)
and not just sort of hear me talk about
Jordan Ellenberg (56:07.060)
distant features of it.
Lex Fridman (56:10.500)
Right, so not be talking just about ideas,
Lex Fridman (56:13.700)
but the actually be expressing the idea.
Lex Fridman (56:16.860)
Is there, you know, somebody in the,
Jordan Ellenberg (56:18.700)
maybe you can comment, there's a guy,
Lex Fridman (56:21.420)
his YouTube channel is 3Blue1Brown, Grant Sanderson.
Jordan Ellenberg (56:25.460)
He does that masterfully well.
Lex Fridman (56:27.860)
Absolutely.
Jordan Ellenberg (56:28.700)
Of visualizing, of expressing a particular idea
Lex Fridman (56:31.620)
and then talking about it as well back and forth.
Lex Fridman (56:34.540)
What do you think about Grant?
Lex Fridman (56:37.020)
It's fantastic.
Jordan Ellenberg (56:37.980)
I mean, the flowering of math YouTube
Lex Fridman (56:40.180)
is like such a wonderful thing
Jordan Ellenberg (56:41.460)
because math teaching, there's so many different venues
Lex Fridman (56:47.020)
through which we can teach people math.
Lex Fridman (56:48.860)
There's the traditional one, right?
Lex Fridman (56:51.500)
Where I'm in a classroom with, depending on the class,
Jordan Ellenberg (56:55.300)
it could be 30 people, it could be a hundred people,
Lex Fridman (56:57.700)
it could, God help me, be a 500 people
Jordan Ellenberg (56:59.460)
if it's like the big calculus lecture or whatever it may be.
Lex Fridman (57:01.860)
And there's sort of some,
Lex Fridman (57:02.700)
but there's some set of people of that order of magnitude
Lex Fridman (57:05.060)
and I'm with them, we have a long time.
Jordan Ellenberg (57:06.460)
I'm with them for a whole semester
Lex Fridman (57:08.340)
and I can ask them to do homework and we talk together.
Jordan Ellenberg (57:10.300)
We have office hours, if they have one on one questions,
Lex Fridman (57:12.140)
a lot of, it's like a very high level of engagement,
Lex Fridman (57:14.500)
but how many people am I actually hitting at a time?
Lex Fridman (57:17.180)
Like not that many, right?
Lex Fridman (57:20.460)
And you can, and there's kind of an inverse relationship
Lex Fridman (57:22.940)
where the more, the fewer people you're talking to,
Jordan Ellenberg (57:27.740)
the more engagement you can ask for.
Lex Fridman (57:29.300)
The ultimate of course is like the mentorship relation
Jordan Ellenberg (57:32.020)
of like a PhD advisor and a graduate student
Lex Fridman (57:35.020)
where you spend a lot of one on one time together
Jordan Ellenberg (57:38.020)
for like three to five years.
Lex Fridman (57:41.260)
And the ultimate high level of engagement to one person.
Jordan Ellenberg (57:46.380)
Books, this can get to a lot more people
Lex Fridman (57:50.420)
that are ever gonna sit in my classroom
Lex Fridman (57:52.700)
and you spend like however many hours it takes
Lex Fridman (57:57.060)
to read a book.
Jordan Ellenberg (57:58.700)
Somebody like Three Blue One Brown or Numberphile
Lex Fridman (58:01.140)
or people like Vi Hart.
Jordan Ellenberg (58:03.140)
I mean, YouTube, let's face it, has bigger reach than a book.
Lex Fridman (58:07.900)
Like there's YouTube videos that have many, many,
Jordan Ellenberg (58:09.660)
many more views than like any hardback book
Lex Fridman (58:13.300)
like not written by a Kardashian or an Obama
Lex Fridman (58:15.820)
is gonna sell, right?
Lex Fridman (58:16.660)
So that's, I mean,
Lex Fridman (58:20.100)
and then those are, some of them are like longer,
Lex Fridman (58:24.860)
20 minutes long, some of them are five minutes long,
Lex Fridman (58:26.580)
but they're shorter.
Lex Fridman (58:27.820)
And then even some of you look like Eugenia Chang
Jordan Ellenberg (58:29.740)
who's a wonderful category theorist in Chicago.
Lex Fridman (58:31.540)
I mean, she was on, I think the Daily Show or is it,
Jordan Ellenberg (58:33.620)
I mean, she was on, she has 30 seconds,
Lex Fridman (58:35.820)
but then there's like 30 seconds
Jordan Ellenberg (58:37.060)
to sort of say something about mathematics
Lex Fridman (58:38.740)
to like untold millions of people.
Lex Fridman (58:41.100)
So everywhere along this curve is important.
Lex Fridman (58:43.980)
And one thing I feel like is great right now
Jordan Ellenberg (58:46.580)
is that people are just broadcasting on all the channels
Lex Fridman (58:49.220)
because we each have our skills, right?
Jordan Ellenberg (58:51.900)
Somehow along the way, like I learned how to write books.
Lex Fridman (58:53.820)
I had this kind of weird life as a writer
Jordan Ellenberg (58:55.700)
where I sort of spent a lot of time
Lex Fridman (58:57.100)
like thinking about how to put English words together
Jordan Ellenberg (58:59.620)
into sentences and sentences together into paragraphs,
Lex Fridman (59:01.880)
like at length,
Jordan Ellenberg (59:03.300)
which is this kind of like weird specialized skill.
Lex Fridman (59:06.940)
And that's one thing, but like sort of being able to make
Jordan Ellenberg (59:09.140)
like winning, good looking, eye catching videos
Lex Fridman (59:13.000)
is like a totally different skill.
Lex Fridman (59:15.060)
And probably somewhere out there,
Lex Fridman (59:16.740)
there's probably sort of some like heavy metal band
Jordan Ellenberg (59:19.540)
that's like teaching math through heavy metal
Lex Fridman (59:21.820)
and like using their skills to do that.
Jordan Ellenberg (59:23.340)
I hope there is at any rate.
Lex Fridman (59:25.060)
Their music and so on, yeah.
Lex Fridman (59:26.580)
But there is something to the process.
Lex Fridman (59:28.820)
I mean, Grant does this especially well,
Jordan Ellenberg (59:31.740)
which is in order to be able to visualize something,
Lex Fridman (59:36.420)
now he writes programs, so it's programmatic visualization.
Lex Fridman (59:39.560)
So like the things he is basically mostly
Lex Fridman (59:42.900)
through his Manum library and Python,
Jordan Ellenberg (59:46.220)
everything is drawn through Python.
Lex Fridman (59:49.340)
You have to truly understand the topic
Jordan Ellenberg (59:54.600)
to be able to visualize it in that way
Lex Fridman (59:58.140)
and not just understand it,
Lex Fridman (59:59.700)
but really kind of think in a very novel way.
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