back to index## Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries | Lex Fridman Podcast #190

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The following is a conversation with Jordan Ellenberg,

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a mathematician at University of Wisconsin

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and an author who masterfully reveals the beauty

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and power of mathematics in his 2014 book,

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How Not To Be Wrong, and his new book,

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just released recently, called Shape,

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The Hidden Geometry of Information, Biology,

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Strategy, Democracy, and Everything Else.

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Quick mention of our sponsors,

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Secret Sauce, ExpressVPN, Blinkist, and Indeed.

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Check them out in the description to support this podcast.

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As a side note, let me say that geometry

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is what made me fall in love with mathematics

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It first showed me that something definitive

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could be stated about this world

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through intuitive visual proofs.

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Somehow, that convinced me that math

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is not just abstract numbers devoid of life,

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but a part of life, part of this world,

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part of our search for meaning.

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This is the Lex Friedman podcast,

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and here is my conversation with Jordan Ellenberg.

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If the brain is a cake.

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Well, let's just go with me on this, okay?

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Okay, we'll pause it.

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So for Noam Chomsky, language,

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the universal grammar, the framework

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from which language springs is like most of the cake,

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the delicious chocolate center,

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and then the rest of cognition that we think of

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is built on top, extra layers,

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maybe the icing on the cake,

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maybe consciousness is just like a cherry on top.

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Where do you put in this cake mathematical thinking?

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Is it as fundamental as language?

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In the Chomsky view, is it more fundamental than language?

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Is it echoes of the same kind of abstract framework

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that he's thinking about in terms of language

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that they're all really tightly interconnected?

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That's a really interesting question.

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You're getting me to reflect on this question

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of whether the feeling of producing mathematical output,

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if you want, is like the process of uttering language

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or producing linguistic output.

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I think it feels something like that,

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and it's certainly the case.

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Let me put it this way.

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It's hard to imagine doing mathematics

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in a completely nonlinguistic way.

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It's hard to imagine doing mathematics

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without talking about mathematics

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and sort of thinking in propositions.

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But maybe it's just because that's the way I do mathematics,

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and maybe I can't imagine it any other way, right?

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Well, what about visualizing shapes,

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visualizing concepts to which language

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is not obviously attachable?

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Ah, that's a really interesting question.

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And one thing it reminds me of is one thing I talk about

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in the book is dissection proofs,

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these very beautiful proofs of geometric propositions.

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There's a very famous one by Baskara

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of the Pythagorean theorem, proofs which are purely visual,

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proofs where you show that two quantities are the same

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by taking the same pieces and putting them together one way

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and making one shape and putting them together another way

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and making a different shape,

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and then observing that those two shapes

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must have the same area

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because they were built out of the same pieces.

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There's a famous story,

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and it's a little bit disputed about how accurate this is,

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but that in Baskara's manuscript,

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he sort of gives this proof, just gives the diagram,

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and then the entire verbal content of the proof

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is he just writes under it, behold.

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And it's like, there's some dispute

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about exactly how accurate that is.

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But so then there's an interesting question.

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If your proof is a diagram, if your proof is a picture,

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or even if your proof is like a movie of the same pieces

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like coming together in two different formations

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to make two different things, is that language?

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I'm not sure I have a good answer.

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What do you think?

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I think it is. I think the process

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of manipulating the visual elements

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is the same as the process

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of manipulating the elements of language.

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And I think probably the manipulating, the aggregation,

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the stitching stuff together is the important part.

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It's not the actual specific elements.

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It's more like, to me, language is a process

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and math is a process.

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It's not just specific symbols.

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It's ultimately created through action, through change.

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And so you're constantly evolving ideas.

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Of course, we kind of attach,

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there's a certain destination you arrive to

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that you attach to and you call that a proof,

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but that's not, that doesn't need to end there.

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It's just at the end of the chapter

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and then it goes on and on and on in that kind of way.

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But I gotta ask you about geometry

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and it's a prominent topic in your new book, Shape.

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So for me, geometry is the thing,

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just like as you're saying,

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made me fall in love with mathematics when I was young.

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So being able to prove something visually

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just did something to my brain that it had this,

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it planted this hopeful seed

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that you can understand the world, like perfectly.

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Maybe it's an OCD thing,

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but from a mathematics perspective,

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like humans are messy, the world is messy, biology is messy.

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Your parents are yelling or making you do stuff,

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but you can cut through all that BS

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and truly understand the world through mathematics

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and nothing like geometry did that for me.

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For you, you did not immediately fall in love

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with geometry, so how do you think about geometry?

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Why is it a special field in mathematics?

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And how did you fall in love with it if you have?

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Wow, you've given me like a lot to say.

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And certainly the experience that you describe

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is so typical, but there's two versions of it.

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One thing I say in the book

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is that geometry is the cilantro of math.

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People are not neutral about it.

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There's people who like you are like,

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the rest of it I could take or leave,

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but then at this one moment, it made sense.

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This class made sense, why wasn't it all like that?

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There's other people, I can tell you,

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because they come and talk to me all the time,

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who are like, I understood all the stuff

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where you're trying to figure out what X was,

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there's some mystery you're trying to solve it,

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X is a number, I figured it out.

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But then there was this geometry, like what was that?

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What happened that year? Like I didn't get it.

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I was like lost the whole year

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and I didn't understand like why we even

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spent the time doing that.

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So, but what everybody agrees on

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is that it's somehow different, right?

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There's something special about it.

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We're gonna walk around in circles a little bit,

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but we'll get there.

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You asked me how I fell in love with math.

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I have a story about this.

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When I was a small child, I don't know,

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maybe like I was six or seven, I don't know.

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I think you're from a different decade than that.

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But in the 70s, we had a cool wooden box

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around your stereo.

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That was the look, everything was dark wood.

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And the box had a bunch of holes in it

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to let the sound out.

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And the holes were in this rectangular array,

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a six by eight array of holes.

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And I was just kind of like zoning out

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in the living room as kids do,

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looking at this six by eight rectangular array of holes.

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And if you like, just by kind of like focusing in and out,

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just by kind of looking at this box,

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looking at this rectangle, I was like,

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well, there's six rows of eight holes each,

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but there's also eight columns of six holes each.

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So eight sixes and six eights.

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It's just like the dissection proofs

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we were just talking about, but it's the same holes.

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It's the same 48 holes.

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That's how many there are,

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no matter whether you count them as rows

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or count them as columns.

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And this was like unbelievable to me.

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Am I allowed to cuss on your podcast?

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I don't know if that's, are we FCC regulated?

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Okay, it was fucking unbelievable.

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Okay, that's the last time.

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This story merits it.

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So two different perspectives in the same physical reality.

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And it's just as you say.

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I knew that six times eight was the same as eight times six.

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I knew my times tables.

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I knew that that was a fact.

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But did I really know it until that moment?

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That's the question, right?

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I sort of knew that the times table was symmetric,

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but I didn't know why that was the case until that moment.

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And in that moment I could see like,

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oh, I didn't have to have somebody tell me that.

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That's information that you can just directly access.

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That's a really amazing moment.

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And as math teachers, that's something

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that we're really trying to bring to our students.

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And I was one of those who did not love

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the kind of Euclidean geometry ninth grade class

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of like prove that an isosceles triangle

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has equal angles at the base, like this kind of thing.

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It didn't vibe with me the way that algebra and numbers did.

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But if you go back to that moment,

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from my adult perspective,

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looking back at what happened with that rectangle,

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I think that is a very geometric moment.

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In fact, that moment exactly encapsulates

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the intertwining of algebra and geometry.

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This algebraic fact that, well, in the instance,

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eight times six is equal to six times eight.

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But in general, that whatever two numbers you have,

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you multiply them one way.

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And it's the same as if you multiply them

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in the other order.

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It attaches it to this geometric fact about a rectangle,

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which in some sense makes it true.

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So, who knows, maybe I was always fated

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to be an algebraic geometer,

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which is what I am as a researcher.

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So that's the kind of transformation.

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And you talk about symmetry in your book.

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What the heck is symmetry?

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What the heck is these kinds of transformation on objects

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that once you transform them, they seem to be similar?

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What do you make of it?

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What's its use in mathematics

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or maybe broadly in understanding our world?

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Well, it's an absolutely fundamental concept.

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And it starts with the word symmetry

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in the way that we usually use it

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when we're just like talking English

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and not talking mathematics, right?

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Sort of something is, when we say something is symmetrical,

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we usually means it has what's called an axis of symmetry.

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Maybe like the left half of it

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looks the same as the right half.

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That would be like a left, right axis of symmetry.

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Or maybe the top half looks like the bottom half or both.

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Maybe there's sort of a fourfold symmetry

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where the top looks like the bottom

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and the left looks like the right or more.

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And that can take you in a lot of different directions.

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The abstract study of what the possible combinations

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of symmetries there are,

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a subject which is called group theory

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was actually one of my first loves in mathematics

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when I thought about a lot when I was in college.

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But the notion of symmetry is actually much more general

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than the things that we would call symmetry

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if we were looking at like a classical building

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or a painting or something like that.

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we could use a symmetry to refer to

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any kind of transformation of an image

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or a space or an object.

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So what I talk about in the book is

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take a figure and stretch it vertically,

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make it twice as big vertically

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and make it half as wide.

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That I would call a symmetry.

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It's not a symmetry in the classical sense,

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but it's a well defined transformation

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that has an input and an output.

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I give you some shape and it gets kind of,

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I call this in the book a scrunch.

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I just had to make up some sort of funny sounding name

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for it because it doesn't really have a name.

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And just as you can sort of study

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which kinds of objects are symmetrical

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under the operations of switching left and right

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or switching top and bottom

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or rotating 40 degrees or what have you,

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you could study what kinds of things are preserved

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by this kind of scrunch symmetry.

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And this kind of more general idea

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of what a symmetry can be.

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Let me put it this way.

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A fundamental mathematical idea,

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in some sense, I might even say the idea

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that dominates contemporary mathematics.

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Or by contemporary, by the way,

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I mean like the last like 150 years.

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We're on a very long time scale in math.

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I don't mean like yesterday.

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I mean like a century or so up till now.

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Is this idea that it's a fundamental question

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of when do we consider two things to be the same?

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That might seem like a complete triviality.

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For instance, if I have a triangle

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and I have a triangle of the exact same dimensions,

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but it's over here, are those the same or different?

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Well, you might say, well, look,

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there's two different things.

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This one's over here, this one's over there.

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On the other hand, if you prove a theorem about this one,

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it's probably still true about this one

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if it has like all the same side lanes and angles

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and like looks exactly the same.

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The term of art, if you want it,

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you would say they're congruent.

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But one way of saying it is there's a symmetry

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called translation, which just means

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move everything three inches to the left.

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And we want all of our theories

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to be translation invariant.

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What that means is that if you prove a theorem

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about a thing that's over here,

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and then you move it three inches to the left,

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it would be kind of weird if all of your theorems

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like didn't still work.

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So this question of like, what are the symmetries

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and which things that you want to study

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are invariant under those symmetries

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is absolutely fundamental.

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Boy, this is getting a little abstract, right?

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It's not at all abstract.

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I think this is completely central

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to everything I think about

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in terms of artificial intelligence.

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I don't know if you know about the MNIST dataset,

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what's handwritten digits.

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And you know, I don't smoke much weed or any really,

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but it certainly feels like it when I look at MNIST

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and think about this stuff, which is like,

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what's the difference between one and two?

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And why are all the twos similar to each other?

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What kind of transformations are within the category

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of what makes a thing the same?

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And what kind of transformations

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are those that make it different?

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And symmetries core to that.

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In fact, whatever the hell our brain is doing,

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it's really good at constructing these arbitrary

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and sometimes novel, which is really important

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when you look at like the IQ test or they feel novel,

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ideas of symmetry of like playing with objects,

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we're able to see things that are the same and not

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and construct almost like little geometric theories

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of what makes things the same and not

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and how to make programs do that in AI

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is a total open question.

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And so I kind of stared and wonder

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how, what kind of symmetries are enough to solve

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the MNIST handwritten digit recognition problem

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and write that down.

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And exactly, and what's so fascinating

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about the work in that direction

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from the point of view of a mathematician like me

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and a geometer is that the kind of groups of symmetries,

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the types of symmetries that we know of are not sufficient.

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So in other words, like we're just gonna keep on going

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into the weeds on this.

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The deeper, the better.

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A kind of symmetry that we understand very well

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So here's what would be easy.

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If humans, if we recognize the digit as a one,

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if it was like literally a rotation

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by some number of degrees or some fixed one

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in some typeface like Palatino or something,

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that would be very easy to understand.

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It would be very easy to like write a program

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that could detect whether something was a rotation

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of a fixed digit one.

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Whatever we're doing when you recognize the digit one

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and distinguish it from the digit two, it's not that.

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It's not just incorporating one of the types of symmetries

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that we understand.

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Now, I would say that I would be shocked

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if there was some kind of classical symmetry type formulation

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that captured what we're doing

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when we tell the difference between a two and a three.

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To be honest, I think what we're doing

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is actually more complicated than that.

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I feel like it must be.

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They're so simple, these numbers.

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I mean, they're really geometric objects.

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Like we can draw out one, two, three.

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It does seem like it should be formalizable.

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That's why it's so strange.

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Do you think it's formalizable

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when something stops being a two and starts being a three?

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Right, you can imagine something continuously deforming

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from being a two to a three.

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Yeah, but that's, there is a moment.

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Like I have myself written programs

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that literally morph twos and threes and so on.

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And you watch, and there is moments that you notice

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depending on the trajectory of that transformation,

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that morphing, that it is a three and a two.

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There's a hard line.

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Wait, so if you ask people, if you showed them this morph,

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if you ask a bunch of people,

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do they all agree about where the transition happened?

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Because I would be surprised.

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Oh my God, okay, we have an empirical dispute.

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But here's the problem.

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Here's the problem, that if I just showed that moment

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Well, that's not fair.

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No, but say I said,

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so I want to move away from the agreement

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because that's a fascinating actually question

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that I want to backtrack from because I just dogmatically

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said, because I could be very, very wrong.

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But the morphing really helps that like the change,

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because I mean, partially it's because our perception

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systems, see this, it's all probably tied in there.

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Somehow the change from one to the other,

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like seeing the video of it allows you to pinpoint

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the place where a two becomes a three much better.

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If I just showed you one picture,

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I think you might really, really struggle.

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You might call a seven.

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I think there's something also that we don't often

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think about, which is it's not just about the static image,

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it's the transformation of the image,

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or it's not a static shape,

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it's the transformation of the shape.

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There's something in the movement that seems to be

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not just about our perception system,

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but fundamental to our cognition,

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like how we think about stuff.

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Yeah, and that's part of geometry too.

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And in fact, again, another insight of modern geometry

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is this idea that maybe we would naively think

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we're gonna study, I don't know,

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like Poincare, we're gonna study the three body problem.

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We're gonna study sort of like three objects in space

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moving around subject only to the force

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of each other's gravity, which sounds very simple, right?

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And if you don't know about this problem,

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you're probably like, okay, so you just like put it

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in your computer and see what they do.

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That's like a problem that Poincare won a huge prize for

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like making the first real progress on in the 1880s.

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And we still don't know that much about it 150 years later.

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I mean, it's a humongous mystery.

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You just opened the door and we're gonna walk right in

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before we return to symmetry.

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What's the, who's Poincare and what's this conjecture

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that he came up with?

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Why is it such a hard problem?

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Okay, so Poincare, he ends up being a major figure

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in the book and I didn't even really intend for him

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to be such a big figure, but he's first and foremost

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a geometer, right?

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So he's a mathematician who kind of comes up

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in late 19th century France at a time when French math

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is really starting to flower.

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Actually, I learned a lot.

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I mean, in math, we're not really trained

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on our own history.

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We got a PhD in math, learned about math.

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So I learned a lot.

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There's this whole kind of moment where France

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has just been beaten in the Franco Prussian war.

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And they're like, oh my God, what did we do wrong?

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And they were like, we gotta get strong in math

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We have to be like more like the Germans.

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So this never happens to us again.

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So it's very much, it's like the Sputnik moment,

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like what happens in America in the 50s and 60s

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with the Soviet Union.

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This is happening to France and they're trying

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to kind of like instantly like modernize.

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That's fascinating that the humans and mathematics

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are intricately connected to the history of humans.

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The Cold War is I think fundamental to the way people

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saw science and math in the Soviet Union.

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I don't know if that was true in the United States,

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but certainly it was in the Soviet Union.

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It definitely was, and I would love to hear more

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about how it was in the Soviet Union.

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I mean, there was, and we'll talk about the Olympiad.

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I just remember that there was this feeling

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like the world hung in a balance

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and you could save the world with the tools of science.

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And mathematics was like the superpower that fuels science.

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And so like people were seen as, you know,

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people in America often idolize athletes,

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but ultimately the best athletes in the world,

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they just throw a ball into a basket.

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So like there's not, what people really enjoy about sports,

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I love sports, is like excellence at the highest level.

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But when you take that with mathematics and science,

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people also enjoyed excellence in science and mathematics

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in the Soviet Union, but there's an extra sense

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that that excellence would lead to a better world.

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So that created all the usual things you think about

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with the Olympics, which is like extreme competitiveness.

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But it also created this sense that in the modern era

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in America, somebody like Elon Musk, whatever you think

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of him, like Jeff Bezos, those folks,

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they inspire the possibility that one person

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or a group of smart people can change the world.

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Like not just be good at what they do,

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but actually change the world.

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Mathematics was at the core of that.

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I don't know, there's a romanticism around it too.

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Like when you read books about in America,

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people romanticize certain things like baseball, for example.

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There's like these beautiful poetic writing

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about the game of baseball.

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The same was the feeling with mathematics and science

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in the Soviet Union, and it was in the air.

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Everybody was forced to take high level mathematics courses.

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Like you took a lot of math, you took a lot of science

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and a lot of like really rigorous literature.

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Like the level of education in Russia,

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this could be true in China, I'm not sure,

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in a lot of countries is in whatever that's called,

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it's K to 12 in America, but like young people education.

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The level they were challenged to learn at is incredible.

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It's like America falls far behind, I would say.

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America then quickly catches up

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and then exceeds everybody else as you start approaching

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the end of high school to college.

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Like the university system in the United States

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arguably is the best in the world.

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But like what we challenge everybody,

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it's not just like the good, the A students,

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but everybody to learn in the Soviet Union was fascinating.

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I think I'm gonna pick up on something you said.

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I think you would love a book called

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Dual at Dawn by Amir Alexander,

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which I think some of the things you're responding to

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and what I wrote, I think I first got turned on to

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by Amir's work, he's a historian of math.

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And he writes about the story of Everest to Galois,

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which is a story that's well known to all mathematicians,

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this kind of like very, very romantic figure

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who he really sort of like begins the development of this

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or this theory of groups that I mentioned earlier,

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this general theory of symmetries

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and then dies in a duel in his early 20s,

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like all this stuff, mostly unpublished.

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It's a very, very romantic story that we all learn.

link |

And much of it is true,

link |

but Alexander really lays out just how much

link |

the way people thought about math in those times

link |

in the early 19th century was wound up with,

link |

as you say, romanticism.

link |

I mean, that's when the romantic movement takes place

link |

and he really outlines how people were predisposed

link |

to think about mathematics in that way

link |

because they thought about poetry that way

link |

and they thought about music that way.

link |

It was the mood of the era to think about

link |

we're reaching for the transcendent,

link |

we're sort of reaching for sort of direct contact

link |

And part of the reason that we think of Gawa that way

link |

was because Gawa himself was a creature of that era

link |

and he romanticized himself.

link |

I mean, now we know he wrote lots of letters

link |

and he was kind of like, I mean, in modern terms,

link |

we would say he was extremely emo.

link |

Like we wrote all these letters

link |

about his like florid feelings

link |

and like the fire within him about the mathematics.

link |

And so it's just as you say

link |

that the math history touches human history.

link |

They're never separate because math is made of people.

link |

I mean, that's what, it's people who do it

link |

and we're human beings doing it

link |

and we do it within whatever community we're in

link |

and we do it affected by the mores

link |

of the society around us.

link |

So the French, the Germans and Poincare.

link |

Yes, okay, so back to Poincare.

link |

So he's, you know, it's funny.

link |

This book is filled with kind of mathematical characters

link |

who often are kind of peevish or get into feuds

link |

or sort of have like weird enthusiasms

link |

because those people are fun to write about

link |

and they sort of like say very salty things.

link |

Poincare is actually none of this.

link |

As far as I can tell, he was an extremely normal dude

link |

who didn't get into fights with people

link |

and everybody liked him

link |

and he was like pretty personally modest

link |

and he had very regular habits.

link |

You know what I mean?

link |

He did math for like four hours in the morning

link |

and four hours in the evening and that was it.

link |

Like he had his schedule.

link |

I actually, it was like, I still am feeling like

link |

somebody's gonna tell me now that the book is out,

link |

like, oh, didn't you know about this

link |

like incredibly sordid episode?

link |

As far as I could tell, a completely normal guy.

link |

But he just kind of, in many ways,

link |

creates the geometric world in which we live

link |

and his first really big success is this prize paper

link |

he writes for this prize offered by the King of Sweden

link |

for the study of the three body problem.

link |

The study of what we can say about, yeah,

link |

three astronomical objects moving

link |

in what you might think would be this very simple way.

link |

Nothing's going on except gravity.

link |

So what's the three body problem?

link |

Why is it a problem?

link |

So the problem is to understand

link |

when this motion is stable and when it's not.

link |

So stable meaning they would sort of like end up

link |

in some kind of periodic orbit.

link |

Or I guess it would mean, sorry,

link |

stable would mean they never sort of fly off

link |

far apart from each other.

link |

And unstable would mean like eventually they fly apart.

link |

So understanding two bodies is much easier.

link |

When you have the third wheel is always a problem.

link |

This is what Newton knew.

link |

Two bodies, they sort of orbit each other

link |

in some kind of either in an ellipse,

link |

which is the stable case.

link |

You know, that's what the planets do that we know.

link |

Or one travels on a hyperbola around the other.

link |

That's the unstable case.

link |

It sort of like zooms in from far away,

link |

sort of like whips around the heavier thing

link |

and like zooms out.

link |

Those are basically the two options.

link |

So it's a very simple and easy to classify story.

link |

With three bodies, just the small switch from two to three,

link |

it's a complete zoo.

link |

It's the first, what we would say now

link |

is it's the first example of what's called chaotic dynamics,

link |

where the stable solutions and the unstable solutions,

link |

they're kind of like wound in among each other.

link |

And a very, very, very tiny change in the initial conditions

link |

can make the longterm behavior of the system

link |

completely different.

link |

So Poincare was the first to recognize

link |

that that phenomenon even existed.

link |

What about the conjecture that carries his name?

link |

Right, so he also was one of the pioneers

link |

of taking geometry, which until that point

link |

had been largely the study of two

link |

and three dimensional objects,

link |

because that's like what we see, right?

link |

That's those are the objects we interact with.

link |

He developed the subject we now called topology.

link |

He called it analysis situs.

link |

He was a very well spoken guy with a lot of slogans,

link |

but that name did not,

link |

you can see why that name did not catch on.

link |

So now it's called topology now.

link |

Sorry, what was it called before?

link |

Analysis situs, which I guess sort of roughly means

link |

like the analysis of location or something like that.

link |

Like it's a Latin phrase.

link |

Partly because he understood that even to understand

link |

stuff that's going on in our physical world,

link |

you have to study higher dimensional spaces.

link |

How does this work?

link |

And this is kind of like where my brain went to it

link |

because you were talking about not just where things are,

link |

but what their path is, how they're moving

link |

when we were talking about the path from two to three.

link |

He understood that if you wanna study

link |

three bodies moving in space,

link |

well, each body, it has a location where it is.

link |

So it has an X coordinate, a Y coordinate,

link |

a Z coordinate, right?

link |

I can specify a point in space by giving you three numbers,

link |

but it also at each moment has a velocity.

link |

So it turns out that really to understand what's going on,

link |

you can't think of it as a point or you could,

link |

but it's better not to think of it as a point

link |

in three dimensional space that's moving.

link |

It's better to think of it as a point

link |

in six dimensional space where the coordinates

link |

are where is it and what's its velocity right now.

link |

That's a higher dimensional space called phase space.

link |

And if you haven't thought about this before,

link |

I admit that it's a little bit mind bending,

link |

but what he needed then was a geometry

link |

that was flexible enough,

link |

not just to talk about two dimensional spaces

link |

or three dimensional spaces, but any dimensional space.

link |

So the sort of famous first line of this paper

link |

where he introduces analysis of Cetus

link |

is no one doubts nowadays that the geometry

link |

of n dimensional space is an actually existing thing, right?

link |

I think that maybe that had been controversial.

link |

And he's saying like, look, let's face it,

link |

just because it's not physical doesn't mean it's not there.

link |

It doesn't mean we shouldn't study it.

link |

He wasn't jumping to the physical interpretation.

link |

Like it can be real,

link |

even if it's not perceivable to the human cognition.

link |

I think that's right.

link |

I think, don't get me wrong,

link |

Poincare never strays far from physics.

link |

He's always motivated by physics,

link |

but the physics drove him to need to think about spaces

link |

of higher dimension.

link |

And so he needed a formalism that was rich enough

link |

to enable him to do that.

link |

And once you do that,

link |

that formalism is also gonna include things

link |

that are not physical.

link |

And then you have two choices.

link |

You can be like, oh, well, that stuff's trash.

link |

Or, and this is more of the mathematicians frame of mind,

link |

if you have a formalistic framework

link |

that like seems really good

link |

and sort of seems to be like very elegant and work well,

link |

and it includes all the physical stuff,

link |

maybe we should think about all of it.

link |

Like maybe we should think about it,

link |

thinking maybe there's some gold to be mined there.

link |

And indeed, like, you know, guess what?

link |

Like before long there's relativity and there's space time.

link |

And like all of a sudden it's like,

link |

oh yeah, maybe it's a good idea.

link |

We already had this geometric apparatus like set up

link |

for like how to think about four dimensional spaces,

link |

like turns out they're real after all.

link |

As I said, you know, this is a story much told

link |

right in mathematics, not just in this context,

link |

I'd love to dig in a little deeper on that actually,

link |

cause I have some intuitions to work out.

link |

Well, I'm not a mathematical physicist,

link |

so we can work them out together.

link |

We'll together walk along the path of curiosity,

link |

but Poincare conjecture.

link |

The Poincare conjecture is about curved

link |

three dimensional spaces.

link |

So I was on my way there.

link |

The idea is that we perceive ourselves as living in,

link |

we don't say a three dimensional space.

link |

We just say three dimensional space.

link |

You know, you can go up and down,

link |

you can go left and right,

link |

you can go forward and back.

link |

There's three dimensions in which we can move.

link |

In Poincare's theory,

link |

there are many possible three dimensional spaces.

link |

In the same way that going down one dimension

link |

to sort of capture our intuition a little bit more,

link |

we know there are lots of different

link |

two dimensional surfaces, right?

link |

There's a balloon and that looks one way

link |

and a donut looks another way

link |

and a Mobius strip looks a third way.

link |

Those are all like two dimensional surfaces

link |

that we can kind of really get a global view of

link |

because we live in three dimensional space.

link |

So we can see a two dimensional surface

link |

sort of sitting in our three dimensional space.

link |

Well, to see a three dimensional space whole,

link |

we'd have to kind of have four dimensional eyes, right?

link |

So we have to use our mathematical eyes.

link |

We have to envision.

link |

The Poincare conjecture says that there's a very simple way

link |

to determine whether a three dimensional space

link |

is the standard one, the one that we're used to.

link |

And essentially it's that it's what's called

link |

fundamental group has nothing interesting in it.

link |

And that I can actually say without saying

link |

what the fundamental group is,

link |

I can tell you what the criterion is.

link |

This would be good.

link |

Oh, look, I can even use a visual aid.

link |

So for the people watching this on YouTube,

link |

you will just see this for the people on the podcast,

link |

you'll have to visualize it.

link |

So Lex has been nice enough to like give me a surface

link |

with an interesting topology.

link |

It's a mug right here in front of me.

link |

I might say it's a genus one surface,

link |

but we could also say it's a mug, same thing.

link |

So if I were to draw a little circle on this mug,

link |

which way should I draw it so it's visible?

link |

If I draw a little circle on this mug,

link |

imagine this to be a loop of string.

link |

I could pull that loop of string closed

link |

on the surface of the mug, right?

link |

That's definitely something I could do.

link |

I could shrink it, shrink it, shrink it until it's a point.

link |

On the other hand,

link |

if I draw a loop that goes around the handle,

link |

I can kind of zhuzh it up here

link |

and I can zhuzh it down there

link |

and I can sort of slide it up and down the handle,

link |

but I can't pull it closed, can I?

link |

Not without breaking the surface of the mug, right?

link |

Not without like going inside.

link |

So the condition of being what's called simply connected,

link |

this is one of Poincare's inventions,

link |

says that any loop of string can be pulled shut.

link |

So it's a feature that the mug simply does not have.

link |

This is a non simply connected mug

link |

and a simply connected mug would be a cup, right?

link |

You would burn your hand when you drank coffee out of it.

link |

So you're saying the universe is not a mug.

link |

Well, I can't speak to the universe,

link |

but what I can say is that regular old space is not a mug.

link |

Regular old space,

link |

if you like sort of actually physically have

link |

like a loop of string,

link |

you can pull it shut.

link |

You can always pull it shut.

link |

But what if your piece of string

link |

was the size of the universe?

link |

Like what if your piece of string

link |

was like billions of light years long?

link |

Like how do you actually know?

link |

I mean, that's still an open question

link |

of the shape of the universe.

link |

I think there's a lot,

link |

there is ideas of it being a torus.

link |

I mean, there's some trippy ideas

link |

and they're not like weird out there controversial.

link |

There's legitimate at the center of a cosmology debate.

link |

I mean, I think most people think it's flat.

link |

I think there's some kind of dodecahedral symmetry

link |

or I mean, I remember reading something crazy

link |

about somebody saying that they saw the signature of that

link |

in the cosmic noise or what have you.

link |

To make the flat earthers happy,

link |

I do believe that the current main belief is it's flat.

link |

It's flat ish or something like that.

link |

The shape of the universe is flat ish.

link |

I don't know what the heck that means.

link |

I think that has like a very,

link |

how are you even supposed to think about the shape

link |

of a thing that doesn't have any thing outside of it?

link |

Ah, but that's exactly what topology does.

link |

Topology is what's called an intrinsic theory.

link |

That's what's so great about it.

link |

This question about the mug,

link |

you could answer it without ever leaving the mug, right?

link |

Because it's a question about a loop drawn

link |

on the surface of the mug

link |

and what happens if it never leaves that surface.

link |

So it's like always there.

link |

See, but that's the difference between the topology

link |

and say, if you're like trying to visualize a mug,

link |

that you can't visualize a mug while living inside the mug.

link |

Well, that's true.

link |

The visualization is harder, but in some sense,

link |

But if the tools of mathematics are there,

link |

I, sorry, I don't want to fight,

link |

but I think the tools of mathematics are exactly there

link |

to enable you to think about

link |

what you cannot visualize in this way.

link |

Let me give, let's go, always to make things easier,

link |

go down to dimension.

link |

Let's think about we live in a circle, okay?

link |

You can tell whether you live on a circle or a line segment,

link |

because if you live in a circle,

link |

if you walk a long way in one direction,

link |

you find yourself back where you started.

link |

And if you live in a line segment,

link |

you walk for a long enough one direction,

link |

you come to the end of the world.

link |

Or if you live on a line, like a whole line,

link |

infinite line, then you walk in one direction

link |

for a long time and like,

link |

well, then there's not a sort of terminating algorithm

link |

to figure out whether you live on a line or a circle,

link |

but at least you sort of,

link |

at least you don't discover that you live on a circle.

link |

So all of those are intrinsic things, right?

link |

All of those are things that you can figure out

link |

about your world without leaving your world.

link |

On the other hand, ready?

link |

Now we're going to go from intrinsic to extrinsic.

link |

Boy, did I not know we were going to talk about this,

link |

If you can't tell whether you live in a circle

link |

or a knot, like imagine like a knot

link |

floating in three dimensional space.

link |

The person who lives on that knot, to them it's a circle.

link |

They walk a long way, they come back to where they started.

link |

Now we, with our three dimensional eyes can be like,

link |

oh, this one's just a plain circle

link |

and this one's knotted up,

link |

but that has to do with how they sit

link |

in three dimensional space.

link |

It doesn't have to do with intrinsic features

link |

of those people's world.

link |

We can ask you one ape to another.

link |

Does it make you, how does it make you feel

link |

that you don't know if you live in a circle

link |

or on a knot, in a knot,

link |

inside the string that forms the knot?

link |

I don't even know how to say that.

link |

I'm going to be honest with you.

link |

I don't know if, I fear you won't like this answer,

link |

but it does not bother me at all.

link |

I don't lose one minute of sleep over it.

link |

So like, does it bother you that if we look

link |

at like a Mobius strip, that you don't have an obvious way

link |

of knowing whether you are inside of a cylinder,

link |

if you live on a surface of a cylinder

link |

or you live on the surface of a Mobius strip?

link |

No, I think you can tell if you live.

link |

Because what you do is you like tell your friend,

link |

hey, stay right here, I'm just going to go for a walk.

link |

And then you like walk for a long time in one direction

link |

and then you come back and you see your friend again.

link |

And if your friend is reversed,

link |

then you know you live on a Mobius strip.

link |

Well, no, because you won't see your friend, right?

link |

Okay, fair point, fair point on that.

link |

But you have to believe the stories about,

link |

no, I don't even know, would you even know?

link |

Oh, no, your point is right.

link |

Let me try to think of a better,

link |

let's see if I can do this on the fly.

link |

It may not be correct to talk about cognitive beings

link |

living on a Mobius strip

link |

because there's a lot of things taken for granted there.

link |

And we're constantly imagining actual

link |

like three dimensional creatures,

link |

like how it actually feels like to live in a Mobius strip

link |

is tricky to internalize.

link |

I think that on what's called the real protective plane,

link |

which is kind of even more sort of like messed up version

link |

of the Mobius strip, but with very similar features,

link |

this feature of kind of like only having one side,

link |

that has the feature that there's a loop of string

link |

which can't be pulled closed.

link |

But if you loop it around twice along the same path,

link |

that you can pull closed.

link |

That's extremely weird.

link |

But that would be a way you could know

link |

without leaving your world

link |

that something very funny is going on.

link |

You know what's extremely weird?

link |

Maybe we can comment on,

link |

hopefully it's not too much of a tangent is,

link |

I remember thinking about this,

link |

this might be right, this might be wrong.

link |

But if we now talk about a sphere

link |

and you're living inside a sphere,

link |

that you're going to see everywhere around you,

link |

the back of your own head.

link |

this is very counterintuitive to me to think about,

link |

But cause I was thinking of like earth,

link |

your 3D thing sitting on a sphere.

link |

But if you're living inside the sphere,

link |

like you're going to see, if you look straight,

link |

you're always going to see yourself all the way around.

link |

So everywhere you look, there's going to be

link |

the back of your own head.

link |

I think somehow this depends on something

link |

of like how the physics of light works in this scenario,

link |

which I'm sort of finding it hard to bend my.

link |

The sea is doing a lot of work.

link |

Like saying you see something is doing a lot of work.

link |

People have thought about this a lot.

link |

I mean, this metaphor of like,

link |

what if we're like little creatures

link |

in some sort of smaller world?

link |

Like how could we apprehend what's outside?

link |

That metaphor just comes back and back.

link |

And actually I didn't even realize like how frequent it is.

link |

It comes up in the book a lot.

link |

I know it from a book called Flatland.

link |

I don't know if you ever read this when you were a kid.

link |

A while ago, yeah.

link |

You know, this sort of comic novel from the 19th century

link |

about an entire two dimensional world.

link |

It's narrated by a square.

link |

That's the main character.

link |

And the kind of strangeness that befalls him

link |

when one day he's in his house

link |

and suddenly there's like a little circle there

link |

and they're with him.

link |

But then the circle like starts getting bigger

link |

and bigger and bigger.

link |

And he's like, what the hell is going on?

link |

It's like a horror movie, like for two dimensional people.

link |

And of course what's happening

link |

is that a sphere is entering his world.

link |

And as the sphere kind of like moves farther and farther

link |

into the plane, it's cross section.

link |

The part of it that he can see.

link |

To him, it looks like there's like this kind

link |

of bizarre being that's like getting larger

link |

and larger and larger

link |

until it's exactly sort of halfway through.

link |

And then they have this kind of like philosophical argument

link |

where the sphere is like, I'm a sphere.

link |

I'm from the third dimension.

link |

The square is like, what are you talking about?

link |

There's no such thing.

link |

And they have this kind of like sterile argument

link |

where the square is not able to kind of like

link |

follow the mathematical reasoning of the sphere

link |

until the sphere just kind of grabs him

link |

and like jerks him out of the plane and pulls him up.

link |

And it's like now, like now do you see,

link |

like now do you see your whole world

link |

that you didn't understand before?

link |

So do you think that kind of process is possible

link |

So we live in the three dimensional world,

link |

maybe with the time component four dimensional

link |

and then math allows us to go high,

link |

into high dimensions comfortably

link |

and explore the world from those perspectives.

link |

Like, is it possible that the universe

link |

is many more dimensions than the ones

link |

we experience as human beings?

link |

So if you look at the, you know,

link |

especially in physics theories of everything,

link |

physics theories that try to unify general relativity

link |

and quantum field theory,

link |

they seem to go to high dimensions to work stuff out

link |

through the tools of mathematics.

link |

So like the two options are,

link |

one is just a nice way to analyze a universe,

link |

but the reality is, is as exactly we perceive it,

link |

it is three dimensional, or are we just seeing,

link |

are we those flatland creatures

link |

that are just seeing a tiny slice of reality

link |

and the actual reality is many, many, many more dimensions

link |

than the three dimensions we perceive?

link |

Oh, I certainly think that's possible.

link |

Now, how would you figure out whether it was true or not

link |

is another question.

link |

And I suppose what you would do

link |

as with anything else that you can't directly perceive

link |

is you would try to understand

link |

what effect the presence of those extra dimensions

link |

out there would have on the things we can perceive.

link |

Like what else can you do, right?

link |

And in some sense, if the answer is

link |

they would have no effect,

link |

then maybe it becomes like a little bit

link |

of a sterile question,

link |

because what question are you even asking, right?

link |

You can kind of posit however many entities that you want.

link |

Is it possible to intuit how to mess

link |

with the other dimensions

link |

while living in a three dimensional world?

link |

I mean, that seems like a very challenging thing to do.

link |

The reason flatland could be written

link |

is because it's coming from a three dimensional writer.

link |

Yes, but what happens in the book,

link |

I didn't even tell you the whole plot.

link |

What happens is the square is so excited

link |

and so filled with intellectual joy.

link |

By the way, maybe to give the story some context,

link |

you asked like, is it possible for us humans

link |

to have this experience of being transcendentally jerked

link |

out of our world so we can sort of truly see it from above?

link |

Well, Edwin Abbott who wrote the book

link |

certainly thought so because Edwin Abbott was a minister.

link |

So the whole Christian subtext of this book,

link |

I had completely not grasped reading this as a kid,

link |

that it means a very different thing, right?

link |

If sort of a theologian is saying like,

link |

oh, what if a higher being could like pull you out

link |

of this earthly world you live in

link |

so that you can sort of see the truth

link |

and like really see it from above as it were.

link |

So that's one of the things that's going on for him.

link |

And it's a testament to his skill as a writer

link |

that his story just works whether that's the framework

link |

you're coming to it from or not.

link |

But what happens in this book and this part,

link |

now looking at it through a Christian lens,

link |

it becomes a bit subversive is the square is so excited

link |

about what he's learned from the sphere

link |

and the sphere explains to him like what a cube would be.

link |

Oh, it's like you but three dimensional

link |

and the square is very excited

link |

and the square is like, okay, I get it now.

link |

So like now that you explained to me how just by reason

link |

I can figure out what a cube would be like,

link |

like a three dimensional version of me,

link |

like let's figure out what a four dimensional version

link |

of me would be like.

link |

And then the sphere is like,

link |

what the hell are you talking about?

link |

There's no fourth dimension, that's ridiculous.

link |

Like there's three dimensions,

link |

like that's how many there are, I can see.

link |

Like, I mean, it's this sort of comic moment

link |

where the sphere is completely unable to conceptualize

link |

that there could actually be yet another dimension.

link |

So yeah, that takes the religious allegory

link |

like a very weird place that I don't really

link |

like understand theologically, but.

link |

That's a nice way to talk about religion and myth in general

link |

as perhaps us trying to struggle,

link |

us meaning human civilization, trying to struggle

link |

with ideas that are beyond our cognitive capabilities.

link |

But it's in fact not beyond our capability.

link |

It may be beyond our cognitive capabilities

link |

to visualize a four dimensional cube,

link |

a tesseract as some like to call it,

link |

or a five dimensional cube, or a six dimensional cube,

link |

but it is not beyond our cognitive capabilities

link |

to figure out how many corners

link |

a six dimensional cube would have.

link |

That's what's so cool about us.

link |

Whether we can visualize it or not,

link |

we can still talk about it, we can still reason about it,

link |

we can still figure things out about it.

link |

Yeah, if we go back to this, first of all, to the mug,

link |

but to the example you give in the book of the straw,

link |

how many holes does a straw have?

link |

And you, listener, may try to answer that in your own head.

link |

Yeah, I'm gonna take a drink while everybody thinks about it

link |

so we can give you a moment.

link |

Is it zero, one, or two, or more than that maybe?

link |

Maybe you can get very creative.

link |

But it's kind of interesting to each,

link |

dissecting each answer as you do in the book

link |

is quite brilliant.

link |

People should definitely check it out.

link |

But if you could try to answer it now,

link |

think about all the options

link |

and why they may or may not be right.

link |

Yeah, and it's one of these questions

link |

where people on first hearing it think it's a triviality

link |

and they're like, well, the answer is obvious.

link |

And then what happens if you ever ask a group of people

link |

that something wonderfully comic happens,

link |

which is that everyone's like,

link |

well, it's completely obvious.

link |

And then each person realizes that half the person,

link |

the other people in the room

link |

have a different obvious answer for the way they have.

link |

And then people get really heated.

link |

People are like, I can't believe

link |

that you think it has two holes

link |

or like, I can't believe that you think it has one.

link |

And then, you know, you really,

link |

like people really learn something about each other

link |

and people get heated.

link |

I mean, can we go through the possible options here?

link |

Is it zero, one, two, three, 10?

link |

Sure, so I think, you know, most people,

link |

the zero holders are rare.

link |

They would say like, well, look,

link |

you can make a straw by taking a rectangular piece of plastic

link |

and closing it up.

link |

A rectangular piece of plastic doesn't have a hole in it.

link |

I didn't poke a hole in it when I,

link |

so how can I have a hole?

link |

They'd be like, it's just one thing.

link |

Okay, most people don't see it that way.

link |

Is there any truth to that kind of conception?

link |

Yeah, I think that would be somebody who's account, I mean,

link |

what I would say is you could say the same thing

link |

You could say, I can make a bagel by taking like a long

link |

cylinder of dough, which doesn't have a hole

link |

and then schmushing the ends together.

link |

So if you're really committed, you can be like, okay,

link |

a bagel doesn't have a hole either.

link |

But like, who are you if you say a bagel doesn't have a hole?

link |

I mean, I don't know.

link |

Yeah, so that's almost like an engineering definition of it.

link |

Okay, fair enough.

link |

So what about the other options?

link |

So, you know, one whole people would say...

link |

I like how these are like groups of people.

link |

Like we've planted our foot, this is what we stand for.

link |

There's books written about each belief.

link |

You know, I would say, look, there's like a hole

link |

and it goes all the way through the straw, right?

link |

It's one region of space, that's the hole.

link |

And two whole people would say like, well, look,

link |

there's a hole in the top and a hole at the bottom.

link |

I think a common thing you see when people

link |

argue about this, they would take something like this

link |

bottle of water I'm holding and go open it and they say,

link |

well, how many holes are there in this?

link |

And you say like, well, there's one hole at the top.

link |

Okay, what if I like poke a hole here

link |

so that all the water spills out?

link |

Well, now it's a straw.

link |

So if you're a one holder, I say to you like,

link |

well, how many holes are in it now?

link |

There was one hole in it before

link |

and I poked a new hole in it.

link |

And then you think there's still one hole

link |

even though there was one hole and I made one more?

link |

Clearly not, this is two holes.

link |

And yet if you're a two holder, the one holder will say like,

link |

okay, where does one hole begin and the other hole end?

link |

And in the book, I sort of, you know, in math,

link |

there's two things we do when we're faced with a problem

link |

that's confusing us.

link |

We can make the problem simpler.

link |

That's what we were doing a minute ago

link |

when we were talking about high dimensional space.

link |

And I was like, let's talk about like circles

link |

and line segments.

link |

Let's like go down a dimension to make it easier.

link |

The other big move we have is to make the problem harder

link |

and try to sort of really like face up

link |

to what are the complications.

link |

So, you know, what I do in the book is say like,

link |

let's stop talking about straws for a minute

link |

and talk about pants.

link |

How many holes are there in a pair of pants?

link |

So I think most people who say there's two holes in a straw

link |

would say there's three holes in a pair of pants.

link |

I guess, I mean, I guess we're filming only from here.

link |

I could take up, no, I'm not gonna do it.

link |

You'll just have to imagine the pants, sorry.

link |

Lex, if you want to, no, okay, no.

link |

That's gonna be in the director's cut.

link |

That's that Patreon only footage.

link |

So many people would say there's three holes

link |

in a pair of pants.

link |

But you know, for instance, my daughter, when I asked,

link |

by the way, talking to kids about this is super fun.

link |

I highly recommend it.

link |

She said, well, yeah, I feel a pair of pants

link |

like just has two holes because yes, there's the waist,

link |

but that's just the two leg holes stuck together.

link |

Two leg holes, yeah, okay.

link |

I mean, that really is a good combination.

link |

So she's a one holder for the straw.

link |

So she's a one holder for the straw too.

link |

And that really does capture something.

link |

It captures this fact, which is central

link |

to the theory of what's called homology,

link |

which is like a central part of modern topology

link |

that holes, whatever we may mean by them,

link |

they're somehow things which have an arithmetic to them.

link |

They're things which can be added.

link |

Like the waist, like waist equals leg plus leg

link |

is kind of an equation,

link |

but it's not an equation about numbers.

link |

It's an equation about some kind of geometric,

link |

some kind of topological thing, which is very strange.

link |

And so, you know, when I come down, you know,

link |

like a rabbi, I like to kind of like come up

link |

with these answers and somehow like dodge

link |

the original question and say like,

link |

you're both right, my children.

link |

So for the straw, I think what a modern mathematician

link |

would say is like, the first version would be to say like,

link |

well, there are two holes,

link |

but they're really both the same hole.

link |

Well, that's not quite right.

link |

A better way to say it is there's two holes,

link |

but one is the negative of the other.

link |

Now, what can that mean?

link |

One way of thinking about what it means is that

link |

if you sip something like a milkshake through the straw,

link |

no matter what, the amount of milkshake

link |

that's flowing in one end,

link |

that same amount is flowing out the other end.

link |

So they're not independent from each other.

link |

There's some relationship between them.

link |

In the same way that if you somehow

link |

could like suck a milkshake through a pair of pants,

link |

the amount of milkshake,

link |

just go with me on this thought experiment.

link |

I'm right there with you.

link |

The amount of milkshake that's coming in

link |

the left leg of the pants,

link |

plus the amount of milkshake that's coming in

link |

the right leg of the pants,

link |

is the same that's coming out the waist of the pants.

link |

So just so you know, I fasted for 72 hours

link |

the last three days.

link |

So I just broke the fast with a little bit of food yesterday.

link |

So this sounds, food analogies or metaphors

link |

for this podcast work wonderfully

link |

because I can intensely picture it.

link |

Is that your weekly routine or just in preparation

link |

for talking about geometry for three hours?

link |

Exactly, this is just for this.

link |

It's hardship to purify the mind.

link |

No, it's for the first time,

link |

I just wanted to try the experience.

link |

And just to pause,

link |

to do things that are out of the ordinary,

link |

to pause and to reflect on how grateful I am

link |

to be just alive and be able to do all the cool shit

link |

that I get to do, so.

link |

Did you drink water?

link |

Yes, yes, yes, yes, yes.

link |

Water and salt, so like electrolytes

link |

and all those kinds of things.

link |

But anyway, so the inflow on the top of the pants

link |

equals to the outflow on the bottom of the pants.

link |

Exactly, so this idea that,

link |

I mean, I think, you know, Poincare really had this idea,

link |

this sort of modern idea.

link |

I mean, building on stuff other people did,

link |

Betty is an important one,

link |

of this kind of modern notion of relations between holes.

link |

But the idea that holes really had an arithmetic,

link |

the really modern view was really Emmy Noether's idea.

link |

So she kind of comes in and sort of truly puts the subject

link |

on its modern footing that we have now.

link |

So, you know, it's always a challenge, you know,

link |

in the book, I'm not gonna say I give like a course

link |

so that you read this chapter and then you're like,

link |

oh, it's just like I took like a semester

link |

of algebraic anthropology.

link |

It's not like this and it's always a challenge

link |

writing about math because there are some things

link |

that you can really do on the page and the math is there.

link |

And there's other things which it's too much

link |

in a book like this to like do them all the page.

link |

You can only say something about them, if that makes sense.

link |

So, you know, in the book, I try to do some of both.

link |

I try to do, I try to, topics that are,

link |

you can't really compress and really truly say

link |

exactly what they are in this amount of space.

link |

I try to say something interesting about them,

link |

something meaningful about them

link |

so that readers can get the flavor.

link |

And then in other places,

link |

I really try to get up close and personal

link |

and really do the math and have it take place on the page.

link |

To some degree be able to give inklings

link |

of the beauty of the subject.

link |

Yeah, I mean, there's a lot of books that are like,

link |

I don't quite know how to express this well.

link |

I'm still laboring to do it,

link |

but there's a lot of books that are about stuff,

link |

but I want my books to not only be about stuff,

link |

but to actually have some stuff there on the page

link |

in the book for people to interact with directly

link |

and not just sort of hear me talk about

link |

distant features of it.

link |

Right, so not be talking just about ideas,

link |

but the actually be expressing the idea.

link |

Is there, you know, somebody in the,

link |

maybe you can comment, there's a guy,

link |

his YouTube channel is 3Blue1Brown, Grant Sanderson.

link |

He does that masterfully well.

link |

Of visualizing, of expressing a particular idea

link |

and then talking about it as well back and forth.

link |

What do you think about Grant?

link |

I mean, the flowering of math YouTube

link |

is like such a wonderful thing

link |

because math teaching, there's so many different venues

link |

through which we can teach people math.

link |

There's the traditional one, right?

link |

Where I'm in a classroom with, depending on the class,

link |

it could be 30 people, it could be a hundred people,

link |

it could, God help me, be a 500 people

link |

if it's like the big calculus lecture or whatever it may be.

link |

And there's sort of some,

link |

but there's some set of people of that order of magnitude

link |

and I'm with them, we have a long time.

link |

I'm with them for a whole semester

link |

and I can ask them to do homework and we talk together.

link |

We have office hours, if they have one on one questions,

link |

a lot of, it's like a very high level of engagement,

link |

but how many people am I actually hitting at a time?

link |

Like not that many, right?

link |

And you can, and there's kind of an inverse relationship

link |

where the more, the fewer people you're talking to,

link |

the more engagement you can ask for.

link |

The ultimate of course is like the mentorship relation

link |

of like a PhD advisor and a graduate student

link |

where you spend a lot of one on one time together

link |

for like three to five years.

link |

And the ultimate high level of engagement to one person.

link |

Books, this can get to a lot more people

link |

that are ever gonna sit in my classroom

link |

and you spend like however many hours it takes

link |

Somebody like Three Blue One Brown or Numberphile

link |

or people like Vi Hart.

link |

I mean, YouTube, let's face it, has bigger reach than a book.

link |

Like there's YouTube videos that have many, many,

link |

many more views than like any hardback book

link |

like not written by a Kardashian or an Obama

link |

is gonna sell, right?

link |

So that's, I mean,

link |

and then those are, some of them are like longer,

link |

20 minutes long, some of them are five minutes long,

link |

but they're shorter.

link |

And then even some of you look like Eugenia Chang

link |

who's a wonderful category theorist in Chicago.

link |

I mean, she was on, I think the Daily Show or is it,

link |

I mean, she was on, she has 30 seconds,

link |

but then there's like 30 seconds

link |

to sort of say something about mathematics

link |

to like untold millions of people.

link |

So everywhere along this curve is important.

link |

And one thing I feel like is great right now

link |

is that people are just broadcasting on all the channels

link |

because we each have our skills, right?

link |

Somehow along the way, like I learned how to write books.

link |

I had this kind of weird life as a writer

link |

where I sort of spent a lot of time

link |

like thinking about how to put English words together

link |

into sentences and sentences together into paragraphs,

link |

which is this kind of like weird specialized skill.

link |

And that's one thing, but like sort of being able to make

link |

like winning, good looking, eye catching videos

link |

is like a totally different skill.

link |

And probably somewhere out there,

link |

there's probably sort of some like heavy metal band

link |

that's like teaching math through heavy metal

link |

and like using their skills to do that.

link |

I hope there is at any rate.

link |

Their music and so on, yeah.

link |

But there is something to the process.

link |

I mean, Grant does this especially well,

link |

which is in order to be able to visualize something,

link |

now he writes programs, so it's programmatic visualization.

link |

So like the things he is basically mostly

link |

through his Manum library and Python,

link |

everything is drawn through Python.

link |

You have to truly understand the topic

link |

to be able to visualize it in that way

link |

and not just understand it,

link |

but really kind of think in a very novel way.

link |

It's funny because I've spoken with him a couple of times,

link |

spoken to him a lot offline as well.

link |

He really doesn't think he's doing anything new,

link |

meaning like he sees himself as very different

link |

from maybe like a researcher,

link |

but it feels to me like he's creating something totally new.

link |

Like that act of understanding and visualizing

link |

is as powerful or has the same kind of inkling of power

link |

as does the process of proving something.

link |

It doesn't have that clear destination,

link |

but it's pulling out an insight

link |

and creating multiple sets of perspective

link |

that arrive at that insight.

link |

And to be honest, it's something that I think

link |

we haven't quite figured out how to value

link |

inside academic mathematics in the same way,

link |

and this is a bit older,

link |

that I think we haven't quite figured out

link |

how to value the development

link |

of computational infrastructure.

link |

We all have computers as our partners now

link |

and people build computers that sort of assist

link |

and participate in our mathematics.

link |

They build those systems

link |

and that's a kind of mathematics too,

link |

but not in the traditional form

link |

of proving theorems and writing papers.

link |

But I think it's coming.

link |

Look, I mean, I think, for example,

link |

the Institute for Computational Experimental Mathematics

link |

at Brown, which is like, it's a NSF funded math institute,

link |

very much part of sort of traditional math academia.

link |

They did an entire theme semester

link |

about visualizing mathematics,

link |

looking at the same kind of thing that they would do

link |

for like an up and coming research topic.

link |

Like that's pretty cool.

link |

So I think there really is buy in

link |

from the mathematics community

link |

to recognize that this kind of stuff is important

link |

and counts as part of mathematics,

link |

like part of what we're actually here to do.

link |

Yeah, I'm hoping to see more and more of that

link |

from like MIT faculty, from faculty,

link |

from all the top universities in the world.

link |

Let me ask you this weird question about the Fields Medal,

link |

which is the Nobel Prize in Mathematics.

link |

Do you think, since we're talking about computers,

link |

there will one day come a time when a computer,

link |

an AI system will win a Fields Medal?

link |

Of course, that's what a human would say.

link |

Is that like, that's like my captcha?

link |

That's like the proof that I'm a human?

link |

Is that like the lie that I know?

link |

What is, how does he want me to answer?

link |

Is there something interesting to be said about that?

link |

Yeah, I mean, I am tremendously interested

link |

in what AI can do in pure mathematics.

link |

I mean, it's, of course, it's a parochial interest, right?

link |

You're like, why am I interested in like,

link |

how it can like help feed the world

link |

or help solve like real social problems?

link |

I'm like, can it do more math?

link |

Like, what can I do?

link |

We all have our interests, right?

link |

But I think it is a really interesting conceptual question.

link |

And here too, I think it's important to be kind of historical

link |

because it's certainly true that there's lots of things

link |

that we used to call research in mathematics

link |

that we would now call computation.

link |

Tasks that we've now offloaded to machines.

link |

Like, you know, in 1890, somebody could be like,

link |

here's my PhD thesis.

link |

I computed all the invariants of this polynomial ring

link |

under the action of some finite group.

link |

Doesn't matter what those words mean,

link |

just it's like some thing that in 1890

link |

would take a person a year to do

link |

and would be a valuable thing that you might wanna know.

link |

And it's still a valuable thing that you might wanna know,

link |

but now you type a few lines of code

link |

in Macaulay or Sage or Magma and you just have it.

link |

So we don't think of that as math anymore,

link |

even though it's the same thing.

link |

What's Macaulay, Sage and Magma?

link |

Oh, those are computer algebra programs.

link |

So those are like sort of bespoke systems

link |

that lots of mathematicians use.

link |

That's similar to Maple and...

link |

Yeah, oh yeah, so it's similar to Maple and Mathematica,

link |

yeah, but a little more specialized, but yeah.

link |

It's programs that work with symbols

link |

and allow you to do, can you do proofs?

link |

Can you do kind of little leaps and proofs?

link |

They're not really built for that.

link |

And that's a whole other story.

link |

But these tools are part of the process of mathematics now.

link |

Right, they are now for most mathematicians, I would say,

link |

part of the process of mathematics.

link |

And so, you know, there's a story I tell in the book,

link |

which I'm fascinated by, which is, you know,

link |

so far, attempts to get AIs

link |

to prove interesting theorems have not done so well.

link |

It doesn't mean they can.

link |

There's actually a paper I just saw,

link |

which has a very nice use of a neural net

link |

to find counter examples to conjecture.

link |

Somebody said like, well, maybe this is always that.

link |

And you can be like, well, let me sort of train an AI

link |

to sort of try to find things where that's not true.

link |

And it actually succeeded.

link |

Now, in this case, if you look at the things that it found,

link |

you say like, okay, I mean, these are not famous conjectures.

link |

Okay, so like somebody wrote this down, maybe this is so.

link |

Looking at what the AI came up with, you're like,

link |

you know, I bet if like five grad students

link |

had thought about that problem,

link |

they wouldn't have come up with that.

link |

I mean, when you see it, you're like,

link |

okay, that is one of the things you might try

link |

if you sort of like put some work into it.

link |

Still, it's pretty awesome.

link |

But the story I tell in the book, which I'm fascinated by

link |

is there is, okay, we're gonna go back to knots.

link |

There's a knot called the Conway knot.

link |

After John Conway, maybe we'll talk about

link |

a very interesting character also.

link |

Yeah, it's a small tangent.

link |

Somebody I was supposed to talk to

link |

and unfortunately he passed away

link |

and he's somebody I find as an incredible mathematician,

link |

incredible human being.

link |

Oh, and I am sorry that you didn't get a chance

link |

because having had the chance to talk to him a lot

link |

when I was a postdoc, yeah, you missed out.

link |

There's no way to sugarcoat it.

link |

I'm sorry that you didn't get that chance.

link |

Yeah, it is what it is.

link |

Yeah, so there was a question and again,

link |

it doesn't matter the technicalities of the question,

link |

but it's a question of whether the knot is slice.

link |

It has to do with something about what kinds

link |

of three dimensional surfaces and four dimensions

link |

can be bounded by this knot.

link |

But nevermind what it means, it's some question.

link |

And it's actually very hard to compute

link |

whether a knot is slice or not.

link |

And in particular, the question of the Conway knot,

link |

whether it was slice or not, was particularly vexed

link |

until it was solved just a few years ago

link |

by Lisa Piccarillo, who actually,

link |

now that I think of it, was here in Austin.

link |

I believe she was a grad student at UT Austin at the time.

link |

I didn't even realize there was an Austin connection

link |

to this story until I started telling it.

link |

In fact, I think she's now at MIT,

link |

so she's basically following you around.

link |

If I remember correctly.

link |

There's a lot of really interesting richness to this story.

link |

One thing about it is her paper was rather,

link |

was very short, it was very short and simple.

link |

Nine pages of which two were pictures.

link |

Very short for like a paper solving a major conjecture.

link |

And it really makes you think about what we mean

link |

by difficulty in mathematics.

link |

Like, do you say, oh, actually the problem wasn't difficult

link |

because you could solve it so simply?

link |

Or do you say like, well, no, evidently it was difficult

link |

because like the world's top topologists,

link |

many, you know, worked on it for 20 years

link |

and nobody could solve it, so therefore it is difficult.

link |

Or is it that we need sort of some new category

link |

of things that about which it's difficult

link |

to figure out that they're not difficult?

link |

I mean, this is the computer science formulation,

link |

but the sort of the journey to arrive

link |

at the simple answer may be difficult,

link |

but once you have the answer, it will then appear simple.

link |

And I mean, there might be a large category.

link |

I hope there's a large set of such solutions,

link |

because, you know, once we stand

link |

at the end of the scientific process

link |

that we're at the very beginning of,

link |

or at least it feels like,

link |

I hope there's just simple answers to everything

link |

that we'll look and it'll be simple laws

link |

that govern the universe,

link |

simple explanation of what is consciousness,

link |

what is love, is mortality fundamental to life,

link |

what's the meaning of life, are humans special

link |

or we're just another sort of reflection

link |

of all that is beautiful in the universe

link |

in terms of like life forms, all of it is life

link |

and just has different,

link |

when taken from a different perspective

link |

is all life can seem more valuable or not,

link |

but really it's all part of the same thing.

link |

All those will have a nice, like two equations,

link |

maybe one equation, but.

link |

Why do you think you want those questions

link |

to have simple answers?

link |

I think just like symmetry

link |

and the breaking of symmetry is beautiful somehow.

link |

There's something beautiful about simplicity.

link |

I think it, what is that?

link |

So it's aesthetic.

link |

It's aesthetic, yeah.

link |

Or, but it's aesthetic in the way

link |

that happiness is an aesthetic.

link |

Like, why is that so joyful

link |

that a simple explanation that governs

link |

a large number of cases is really appealing?

link |

Even when it's not, like obviously we get

link |

a huge amount of trouble with that

link |

because oftentimes it doesn't have to be connected

link |

with reality or even that explanation

link |

could be exceptionally harmful.

link |

Most of like the world's history that has,

link |

that was governed by hate and violence

link |

had a very simple explanation at the core

link |

that was used to cause the violence and the hatred.

link |

So like we get into trouble with that,

link |

but why is that so appealing?

link |

And in this nice forms in mathematics,

link |

like you look at the Einstein papers,

link |

why are those so beautiful?

link |

And why is the Andrew Wiles proof

link |

of the Fermat's last theorem not quite so beautiful?

link |

Like what's beautiful about that story

link |

is the human struggle of like the human story

link |

of perseverance, of the drama,

link |

of not knowing if the proof is correct

link |

and ups and downs and all of those kinds of things.

link |

That's the interesting part.

link |

But the fact that the proof is huge

link |

and nobody understands, well,

link |

from my outsider's perspective,

link |

nobody understands what the heck it is,

link |

is not as beautiful as it could have been.

link |

I wish it was what Fermat originally said,

link |

which is, you know, it's not,

link |

it's not small enough to fit in the margins of this page,

link |

but maybe if he had like a full page

link |

or maybe a couple of post it notes,

link |

he would have enough to do the proof.

link |

What do you make of,

link |

if we could take another of a multitude of tangents,

link |

what do you make of Fermat's last theorem?

link |

Because the statement, there's a few theorems,

link |

there's a few problems that are deemed by the world

link |

throughout its history to be exceptionally difficult.

link |

And that one in particular is really simple to formulate

link |

and really hard to come up with a proof for.

link |

And it was like taunted as simple by Fermat himself.

link |

Is there something interesting to be said about

link |

that X to the N plus Y to the N equals Z to the N

link |

for N of three or greater, is there a solution to this?

link |

And then how do you go about proving that?

link |

Like, how would you try to prove that?

link |

And what do you learn from the proof

link |

that eventually emerged by Andrew Wiles?

link |

Yeah, so right, so to give,

link |

let me just say the background,

link |

because I don't know if everybody listening knows the story.

link |

So, you know, Fermat was an early number theorist,

link |

at least sort of an early mathematician,

link |

those special adjacent didn't really exist back then.

link |

He comes up in the book actually,

link |

in the context of a different theorem of his

link |

that has to do with testing,

link |

whether a number is prime or not.

link |

So I write about, he was one of the ones who was salty

link |

and like, he would exchange these letters

link |

where he and his correspondents would like

link |

try to top each other and vex each other with questions

link |

and stuff like this.

link |

But this particular thing,

link |

it's called Fermat's Last Theorem because it's a note

link |

he wrote in his copy of the Disquisitiones Arithmetic I.

link |

Like he wrote, here's an equation, it has no solutions.

link |

I can prove it, but the proof's like a little too long

link |

to fit in the margin of this book.

link |

He was just like writing a note to himself.

link |

Now, let me just say historically,

link |

we know that Fermat did not have a proof of this theorem.

link |

For a long time, people were like this mysterious proof

link |

that was lost, a very romantic story, right?

link |

But a fair amount later,

link |

he did prove special cases of this theorem

link |

and wrote about it, talked to people about the problem.

link |

It's very clear from the way that he wrote

link |

where he can solve certain examples

link |

of this type of equation

link |

that he did not know how to do the whole thing.

link |

He may have had a deep, simple intuition

link |

about how to solve the whole thing

link |

that he had at that moment

link |

without ever being able to come up with a complete proof.

link |

And that intuition maybe lost the time.

link |

Maybe, but you're right, that is unknowable.

link |

But I think what we can know is that later,

link |

he certainly did not think that he had a proof

link |

that he was concealing from people.

link |

He thought he didn't know how to prove it,

link |

and I also think he didn't know how to prove it.

link |

Now, I understand the appeal of saying like,

link |

wouldn't it be cool if this very simple equation

link |

there was like a very simple, clever, wonderful proof

link |

that you could do in a page or two.

link |

And that would be great, but you know what?

link |

There's lots of equations like that

link |

that are solved by very clever methods like that,

link |

including the special cases that Fermat wrote about,

link |

the method of descent,

link |

which is like very wonderful and important.

link |

But in the end, those are nice things

link |

that like you teach in an undergraduate class,

link |

and it is what it is,

link |

but they're not big.

link |

On the other hand, work on the Fermat problem,

link |

that's what we like to call it

link |

because it's not really his theorem

link |

because we don't think he proved it.

link |

So, I mean, work on the Fermat problem

link |

developed this like incredible richness of number theory

link |

that we now live in today.

link |

Like, and not, by the way,

link |

just Wiles, Andrew Wiles being the person

link |

who, together with Richard Taylor,

link |

finally proved this theorem.

link |

But you know how you have this whole moment

link |

that people try to prove this theorem

link |

and there's a famous false proof by LeMay

link |

from the 19th century,

link |

where Kummer, in understanding what mistake LeMay had made

link |

in this incorrect proof,

link |

basically understands something incredible,

link |

which is that a thing we know about numbers

link |

is that you can factor them

link |

and you can factor them uniquely.

link |

There's only one way to break a number up into primes.

link |

Like if we think of a number like 12,

link |

12 is two times three times two.

link |

I had to think about it.

link |

Or it's two times two times three,

link |

of course you can reorder them.

link |

But there's no other way to do it.

link |

There's no universe in which 12 is something times five,

link |

or in which there's like four threes in it.

link |

Nope, 12 is like two twos and a three.

link |

Like that is what it is.

link |

And that's such a fundamental feature of arithmetic

link |

that we almost think of it like God's law.

link |

You know what I mean?

link |

It has to be that way.

link |

That's a really powerful idea.

link |

It's so cool that every number

link |

is uniquely made up of other numbers.

link |

And like made up meaning like there's these like basic atoms

link |

that form molecules that get built on top of each other.

link |

I mean, when I teach undergraduate number theory,

link |

it's like, it's the first really deep theorem

link |

What's amazing is the fact

link |

that you can factor a number into primes is much easier.

link |

Essentially Euclid knew it,

link |

although he didn't quite put it in that way.

link |

The fact that you can do it at all.

link |

What's deep is the fact that there's only one way to do it

link |

or however you sort of chop the number up,

link |

you end up with the same set of prime factors.

link |

And indeed what people finally understood

link |

at the end of the 19th century is that

link |

if you work in number systems slightly more general

link |

than the ones we're used to,

link |

which it turns out are relevant to Fermat,

link |

all of a sudden this stops being true.

link |

Things get, I mean, things get more complicated

link |

and now because you were praising simplicity before

link |

you were like, it's so beautiful, unique factorization.

link |

Like, so when I tell you

link |

that in more general number systems,

link |

there is no unique factorization.

link |

Maybe you're like, that's bad.

link |

I'm like, no, that's good

link |

because there's like a whole new world of phenomena

link |

to study that you just can't see

link |

through the lens of the numbers that we're used to.

link |

So I'm for complication.

link |

I'm highly in favor of complication

link |

because every complication is like an opportunity

link |

for new things to study.

link |

And is that the big kind of one of the big insights

link |

for you from Andrew Wiles's proof?

link |

Is there interesting insights about the process

link |

that you used to prove that sort of resonates

link |

with you as a mathematician?

link |

Is there an interesting concept that emerged from it?

link |

Is there interesting human aspects to the proof?

link |

Whether there's interesting human aspects

link |

to the proof itself is an interesting question.

link |

Certainly it has a huge amount of richness.

link |

Sort of at its heart is an argument

link |

of what's called deformation theory,

link |

which was in part created by my PhD advisor, Barry Mazer.

link |

Can you speak to what deformation theory is?

link |

I can speak to what it's like.

link |

What does it rhyme with?

link |

Right, well, the reason that Barry called it

link |

deformation theory, I think he's the one

link |

who gave it the name.

link |

I hope I'm not wrong in saying it's a name.

link |

In your book, you have calling different things

link |

by the same name as one of the things

link |

in the beautiful map that opens the book.

link |

Yes, and this is a perfect example.

link |

So this is another phrase of Poincare,

link |

this like incredible generator of slogans and aphorisms.

link |

He said, mathematics is the art

link |

of calling different things by the same name.

link |

That very thing we do, right?

link |

When we're like this triangle and this triangle,

link |

come on, they're the same triangle,

link |

they're just in a different place, right?

link |

So in the same way, it came to be understood

link |

that the kinds of objects that you study

link |

when you study Fermat's Last Theorem,

link |

and let's not even be too careful

link |

about what these objects are.

link |

I can tell you there are gaol representations

link |

in modular forms, but saying those words

link |

is not gonna mean so much.

link |

But whatever they are, they're things that can be deformed,

link |

moved around a little bit.

link |

And I think the insight of what Andrew

link |

and then Andrew and Richard were able to do

link |

was to say something like this.

link |

A deformation means moving something just a tiny bit,

link |

like an infinitesimal amount.

link |

If you really are good at understanding

link |

which ways a thing can move in a tiny, tiny, tiny,

link |

infinitesimal amount in certain directions,

link |

maybe you can piece that information together

link |

to understand the whole global space in which it can move.

link |

And essentially, their argument comes down

link |

to showing that two of those big global spaces

link |

are actually the same, the fabled R equals T,

link |

part of their proof, which is at the heart of it.

link |

And it involves this very careful principle like that.

link |

But that being said, what I just said,

link |

it's probably not what you're thinking,

link |

because what you're thinking when you think,

link |

oh, I have a point in space and I move it around

link |

like a little tiny bit,

link |

you're using your notion of distance

link |

that's from calculus.

link |

We know what it means for like two points

link |

on the real line to be close together.

link |

So yet another thing that comes up in the book a lot

link |

is this fact that the notion of distance

link |

is not given to us by God.

link |

We could mean a lot of different things by distance.

link |

And just in the English language, we do that all the time.

link |

We talk about somebody being a close relative.

link |

It doesn't mean they live next door to you, right?

link |

It means something else.

link |

There's a different notion of distance we have in mind.

link |

And there are lots of notions of distances

link |

that you could use.

link |

In the natural language processing community and AI,

link |

there might be some notion of semantic distance

link |

or lexical distance between two words.

link |

How much do they tend to arise in the same context?

link |

That's incredibly important for doing autocomplete

link |

and like machine translation and stuff like that.

link |

And it doesn't have anything to do with

link |

are they next to each other in the dictionary, right?

link |

It's a different kind of distance.

link |

In this kind of number theory,

link |

there was a crazy distance called the peatic distance.

link |

I didn't write about this that much in the book

link |

because even though I love it

link |

and it's a big part of my research life,

link |

it gets a little bit into the weeds,

link |

but your listeners are gonna hear about it now.

link |

What a normal person says

link |

when they say two numbers are close,

link |

they say like their difference is like a small number,

link |

like seven and eight are close

link |

because their difference is one and one's pretty small.

link |

If we were to be what's called a two attic number theorist,

link |

we'd say, oh, two numbers are close

link |

if their difference is a multiple of a large power of two.

link |

So like one and 49 are close

link |

because their difference is 48

link |

and 48 is a multiple of 16,

link |

which is a pretty large power of two.

link |

Whereas one and two are pretty far away

link |

because the difference between them is one,

link |

which is not even a multiple of a power of two at all.

link |

You wanna know what's really far from one?

link |

Like one and 1 64th

link |

because their difference is a negative power of two,

link |

two to the minus six.

link |

So those points are quite, quite far away.

link |

Two to the power of a large N would be two,

link |

if that's the difference between two numbers

link |

then they're close.

link |

Yeah, so two to a large power is in this metric

link |

a very small number

link |

and two to a negative power is a very big number.

link |

Okay, I can't even visualize that.

link |

It takes practice.

link |

It takes practice.

link |

If you've ever heard of the Cantor set,

link |

it looks kind of like that.

link |

So it is crazy that this is good for anything, right?

link |

I mean, this just sounds like a definition

link |

that someone would make up to torment you.

link |

But what's amazing is there's a general theory of distance

link |

where you say any definition you make

link |

to satisfy certain axioms deserves to be called a distance

link |

See, I'm sorry to interrupt.

link |

My brain, you broke my brain.

link |

Cause I'm also starting to map for the two attic case

link |

to binary numbers.

link |

And you know, cause we romanticize those.

link |

So I was trying to.

link |

Oh, that's exactly the right way to think of it.

link |

I was trying to mess with number,

link |

I was trying to see, okay, which ones are close.

link |

And then I'm starting to visualize

link |

different binary numbers and how they,

link |

which ones are close to each other.

link |

Well, I think there's a.

link |

No, no, it's very similar.

link |

That's exactly the right way to think of it.

link |

It's almost like binary numbers written in reverse.

link |

Because in a binary expansion, two numbers are close.

link |

A number that's small is like 0.0000 something.

link |

Something that's the decimal

link |

and it starts with a lot of zeros.

link |

In the two attic metric, a binary number is very small

link |

if it ends with a lot of zeros and then the decimal point.

link |

So it is kind of like binary numbers written backwards

link |

is actually, I should have said,

link |

that's what I should have said, Lex.

link |

That's a very good metaphor.

link |

Okay, but so why is that interesting

link |

except for the fact that it's a beautiful kind of framework,

link |

different kind of framework

link |

of which to think about distances.

link |

And you're talking about not just the two attic,

link |

but the generalization of that.

link |

Why is that interesting?

link |

And so that, because that's the kind of deformation

link |

that comes up in Wiles's proof,

link |

that deformation where moving something a little bit

link |

means a little bit in this two attic sense.

link |

No, I mean, it's such a,

link |

I mean, I just get excited talking about it

link |

and I just taught this like in the fall semester that.

link |

But it like reformulating, why is,

link |

so you pick a different measure of distance

link |

over which you can talk about very tiny changes

link |

and then use that to then prove things

link |

about the entire thing.

link |

Yes, although, honestly, what I would say,

link |

I mean, it's true that we use it to prove things,

link |

but I would say we use it to understand things.

link |

And then because we understand things better,

link |

then we can prove things.

link |

But the goal is always the understanding.

link |

The goal is not so much to prove things.

link |

The goal is not to know what's true or false.

link |

I mean, this is something I write about

link |

in the book, Near the End.

link |

And it's something that,

link |

it's a wonderful, wonderful essay by Bill Thurston,

link |

kind of one of the great geometers of our time,

link |

who unfortunately passed away a few years ago,

link |

called on proof and progress in mathematics.

link |

And he writes very wonderfully about how,

link |

we're not, it's not a theorem factory

link |

where you have a production quota.

link |

I mean, the point of mathematics

link |

is to help humans understand things.

link |

And the way we test that

link |

is that we're proving new theorems along the way.

link |

That's the benchmark, but that's not the goal.

link |

Yeah, but just as a kind of, absolutely,

link |

but as a tool, it's kind of interesting

link |

to approach a problem by saying,

link |

how can I change the distance function?

link |

Like what, the nature of distance,

link |

because that might start to lead to insights

link |

for deeper understanding.

link |

Like if I were to try to describe human society

link |

by a distance, two people are close

link |

if they love each other.

link |

And then start to do a full analysis

link |

on the everybody that lives on earth currently,

link |

the 7 billion people.

link |

And from that perspective,

link |

as opposed to the geographic perspective of distance.

link |

And then maybe there could be a bunch of insights

link |

about the source of violence,

link |

the source of maybe entrepreneurial success

link |

or invention or economic success or different systems,

link |

communism, capitalism start to,

link |

I mean, that's, I guess what economics tries to do,

link |

but really saying, okay, let's think outside the box

link |

about totally new distance functions

link |

that could unlock something profound about the space.

link |

Yeah, because think about it.

link |

Okay, here's, I mean, now we're gonna talk about AI,

link |

which you know a lot more about than I do.

link |

So just start laughing uproariously

link |

if I say something that's completely wrong.

link |

We both know very little relative

link |

to what we will know centuries from now.

link |

That is a really good humble way to think about it.

link |

Okay, so let's just go for it.

link |

Okay, so I think you'll agree with this,

link |

that in some sense, what's good about AI

link |

is that we can't test any case in advance,

link |

the whole point of AI is to make,

link |

or one point of it, I guess, is to make good predictions

link |

about cases we haven't yet seen.

link |

And in some sense, that's always gonna involve

link |

some notion of distance,

link |

because it's always gonna involve

link |

somehow taking a case we haven't seen

link |

and saying what cases that we have seen is it close to,

link |

is it like, is it somehow an interpolation between.

link |

Now, when we do that,

link |

in order to talk about things being like other things,

link |

implicitly or explicitly,

link |

we're invoking some notion of distance,

link |

and boy, we better get it right.

link |

If you try to do natural language processing

link |

and your idea of distance between words

link |

is how close they are in the dictionary,

link |

when you write them in alphabetical order,

link |

you are gonna get pretty bad translations, right?

link |

No, the notion of distance has to come from somewhere else.

link |

Yeah, that's essentially what neural networks are doing,

link |

that's what word embeddings are doing is coming up with.

link |

In the case of word embeddings, literally,

link |

literally what they are doing is learning a distance.

link |

But those are super complicated distance functions,

link |

and it's almost nice to think

link |

maybe there's a nice transformation that's simple.

link |

Sorry, there's a nice formulation of the distance.

link |

Again with the simple.

link |

So you don't, let me ask you about this.

link |

From an understanding perspective,

link |

there's the Richard Feynman, maybe attributed to him,

link |

but maybe many others,

link |

is this idea that if you can't explain something simply

link |

that you don't understand it.

link |

In how many cases, how often is that true?

link |

Do you find there's some profound truth in that?

link |

Oh, okay, so you were about to ask, is it true?

link |

To which I would say flatly, no.

link |

But then you said, you followed that up with,

link |

is there some profound truth in it?

link |

And I'm like, okay, sure.

link |

So there's some truth in it.

link |

It's not true. But it's not true.

link |

That's such a mathematician answer.

link |

The truth that is in it is that learning

link |

to explain something helps you understand it.

link |

But real things are not simple.

link |

A few things are, most are not.

link |

And to be honest, we don't really know

link |

whether Feynman really said that right

link |

or something like that is sort of disputed.

link |

But I don't think Feynman could have literally believed that

link |

whether or not he said it.

link |

And he was the kind of guy, I didn't know him,

link |

but I've been reading his writing,

link |

he liked to sort of say stuff, like stuff that sounded good.

link |

You know what I mean?

link |

So it's totally strikes me as the kind of thing

link |

he could have said because he liked the way saying it

link |

made him feel, but also knowing

link |

that he didn't like literally mean it.

link |

Well, I definitely have a lot of friends

link |

and I've talked to a lot of physicists

link |

and they do derive joy from believing

link |

that they can explain stuff simply

link |

or believing it's possible to explain stuff simply,

link |

even when the explanation is not actually that simple.

link |

Like I've heard people think that the explanation is simple

link |

and they do the explanation.

link |

And I think it is simple,

link |

but it's not capturing the phenomena that we're discussing.

link |

It's capturing, it's somehow maps in their mind,

link |

but it's taking as a starting point,

link |

as an assumption that there's a deep knowledge

link |

and a deep understanding that's actually very complicated.

link |

And the simplicity is almost like a poem

link |

about the more complicated thing

link |

as opposed to a distillation.

link |

And I love poems, but a poem is not an explanation.

link |

Well, some people might disagree with that,

link |

but certainly from a mathematical perspective.

link |

No poet would disagree with it.

link |

No poet would disagree.

link |

You don't think there's some things

link |

that can only be described imprecisely?

link |

As an explanation.

link |

I don't think any poet would say their poem

link |

is an explanation.

link |

They might say it's a description.

link |

They might say it's sort of capturing sort of.

link |

Well, some people might say the only truth is like music.

link |

Not the only truth,

link |

but some truths can only be expressed through art.

link |

And I mean, that's the whole thing

link |

we're talking about religion and myth.

link |

And there's some things

link |

that are limited cognitive capabilities

link |

and the tools of mathematics or the tools of physics

link |

are just not going to allow us to capture.

link |

Like it's possible consciousness is one of those things.

link |

Yes, that is definitely possible.

link |

But I would even say,

link |

look, I mean, consciousness is a thing about

link |

which we're still in the dark

link |

as to whether there's an explanation

link |

we would understand it as an explanation at all.

link |

I got to give yet one more amazing Poincare quote

link |

because this guy just never stopped coming up

link |

with great quotes that,

link |

Paul ErdÅ‘s, another fellow who appears in the book.

link |

he thinks about this notion of distance

link |

of like personal affinity,

link |

kind of like what you're talking about,

link |

the kind of social network and that notion of distance

link |

that comes from that.

link |

So that's something that Paul ErdÅ‘s.

link |

Well, he thought about distances and networks.

link |

I guess he didn't probably,

link |

he didn't think about the social network.

link |

Oh, that's fascinating.

link |

And that's how it started that story of ErdÅ‘s number.

link |

It's hard to distract.

link |

But you know, ErdÅ‘s was sort of famous for saying,

link |

and this is sort of long lines we're saying,

link |

he talked about the book,

link |

capital T, capital B, the book.

link |

And that's the book where God keeps the right proof

link |

So when he saw a proof he really liked,

link |

it was like really elegant, really simple.

link |

Like that's from the book.

link |

That's like you found one of the ones that's in the book.

link |

He wasn't a religious guy, by the way.

link |

He referred to God as the supreme fascist.

link |

He was like, but somehow he was like,

link |

I don't really believe in God,

link |

but I believe in God's book.

link |

but Poincare on the other hand,

link |

and by the way, there were other managers.

link |

Hilda Hudson is one who comes up in this book.

link |

She also kind of saw math.

link |

She's one of the people who sort of develops

link |

the disease model that we now use,

link |

that we use to sort of track pandemics,

link |

this SIR model that sort of originally comes

link |

from her work with Ronald Ross.

link |

But she was also super, super, super devout.

link |

And she also sort of on the other side

link |

of the religious coin was like,

link |

yeah, math is how we communicate with God.

link |

all these people are incredibly quotable.

link |

She says, you know, math is,

link |

the truth, the things about mathematics,

link |

she's like, they're not the most important of God thoughts,

link |

but they're the only ones that we can know precisely.

link |

So she's like, this is the one place

link |

where we get to sort of see what God's thinking

link |

when we do mathematics.

link |

Again, not a fan of poetry or music.

link |

Some people will say Hendrix is like,

link |

some people say chapter one of that book is mathematics,

link |

and then chapter two is like classic rock.

link |

So like, it's not clear that the...

link |

I'm sorry, you just sent me off on a tangent,

link |

just imagining like Erdos at a Hendrix concert,

link |

like trying to figure out if it was from the book or not.

link |

What I was coming to was just to say,

link |

but what PoincarÃ© said about this is he's like,

link |

you know, if like, this is all worked out

link |

in the language of the divine,

link |

and if a divine being like came down and told it to us,

link |

we wouldn't be able to understand it, so it doesn't matter.

link |

So PoincarÃ© was of the view that there were things

link |

that were sort of like inhumanly complex,

link |

and that was how they really were.

link |

Our job is to figure out the things that are not like that.

link |

That are not like that.

link |

All this talk of primes got me hungry for primes.

link |

You wrote a blog post, The Beauty of Bounding Gaps,

link |

a huge discovery about prime numbers

link |

and what it means for the future of math.

link |

Can you tell me about prime numbers?

link |

What the heck are those?

link |

What are twin primes?

link |

What are prime gaps?

link |

What are bounding gaps and primes?

link |

What are all these things?

link |

And what, if anything,

link |

or what exactly is beautiful about them?

link |

Yeah, so, you know, prime numbers are one of the things

link |

that number theorists study the most and have for millennia.

link |

They are numbers which can't be factored.

link |

And then you say, like, five.

link |

And then you're like, wait, I can factor five.

link |

Five is five times one.

link |

Okay, not like that.

link |

That is a factorization.

link |

It absolutely is a way of expressing five

link |

as a product of two things.

link |

But don't you agree there's like something trivial about it?

link |

It's something you could do to any number.

link |

It doesn't have content the way that if I say

link |

that 12 is six times two or 35 is seven times five,

link |

I've really done something to it.

link |

So those are the kind of factorizations that count.

link |

And a number that doesn't have a factorization like that

link |

is called prime, except, historical side note,

link |

one, which at some times in mathematical history

link |

has been deemed to be a prime, but currently is not.

link |

And I think that's for the best.

link |

But I bring it up only because sometimes people think that,

link |

you know, these definitions are kind of,

link |

if we think about them hard enough,

link |

we can figure out which definition is true.

link |

There's just an artifact of mathematics.

link |

So it's a question of which definition is best for us,

link |

Well, those edge cases are weird, right?

link |

So it can't be, it doesn't count when you use yourself

link |

as a number or one as part of the factorization

link |

or as the entirety of the factorization.

link |

So you somehow get to the meat of the number

link |

by factorizing it.

link |

And that seems to get to the core of all of mathematics.

link |

Yeah, you take any number and you factorize it

link |

until you can factorize no more.

link |

And what you have left is some big pile of primes.

link |

I mean, by definition, when you can't factor anymore,

link |

when you're done, when you can't break the numbers up

link |

anymore, what's left must be prime.

link |

You know, 12 breaks into two and two and three.

link |

So these numbers are the atoms, the building blocks

link |

And there's a lot we know about them,

link |

or there's much more that we don't know about them.

link |

I'll tell you the first few.

link |

There's two, three, five, seven, 11.

link |

By the way, they're all gonna be odd from then on

link |

because if they were even, I could factor out

link |

But it's not all the odd numbers.

link |

Nine isn't prime because it's three times three.

link |

15 isn't prime because it's three times five,

link |

Two, three, five, seven, 11, 13, 17, 19.

link |

Not 21, but 23 is, et cetera, et cetera.

link |

Okay, so you could go on.

link |

How high could you go if we were just sitting here?

link |

By the way, your own brain.

link |

If continuous, without interruption,

link |

would you be able to go over 100?

link |

There's always those ones that trip people up.

link |

There's a famous one, the Grotendeek prime 57,

link |

like sort of Alexander Grotendeek,

link |

the great algebraic geometer was sort of giving

link |

some lecture involving a choice of a prime in general.

link |

And somebody said, can't you just choose a prime?

link |

And he said, okay, 57, which is in fact not prime.

link |

It's three times 19.

link |

But it was like, I promise you in some circles

link |

it's a funny story.

link |

But there's a humor in it.

link |

Yes, I would say over 100, I definitely don't remember.

link |

Like 107, I think, I'm not sure.

link |

Okay, like, I mean.

link |

So is there a category of like fake primes

link |

that are easily mistaken to be prime?

link |

Like 57, I wonder.

link |

Yeah, so I would say 57 and 51 are definitely

link |

like prime offenders.

link |

Oh, I didn't do that on purpose.

link |

Didn't do it on purpose.

link |

Anyway, they're definitely ones that people,

link |

or 91 is another classic, seven times 13.

link |

It really feels kind of prime, doesn't it?

link |

But there's also, by the way,

link |

but there's also an actual notion of pseudo prime,

link |

which is a thing with a formal definition,

link |

which is not a psychological thing.

link |

It is a prime which passes a primality test

link |

devised by Fermat, which is a very good test,

link |

which if a number fails this test,

link |

it's definitely not prime.

link |

And so there was some hope that,

link |

oh, maybe if a number passes the test,

link |

then it definitely is prime.

link |

That would give a very simple criterion for primality.

link |

Unfortunately, it's only perfect in one direction.

link |

So there are numbers, I want to say 341 is the smallest,

link |

which pass the test but are not prime, 341.

link |

Is this test easily explainable or no?

link |

Ready, let me give you the simplest version of it.

link |

You can dress it up a little bit, but here's the basic idea.

link |

I take the number, the mystery number,

link |

I raise two to that power.

link |

So let's say your mystery number is six.

link |

Are you sorry you asked me?

link |

No, you're breaking my brain again, but yes.

link |

We're going to do a live demonstration.

link |

Let's say your number is six.

link |

So I'm going to raise two to the sixth power.

link |

Okay, so if I were working on it,

link |

I'd be like that's two cubes squared,

link |

so that's eight times eight, so that's 64.

link |

Now we're going to divide by six,

link |

but I don't actually care what the quotient is,

link |

only the remainder.

link |

So let's see, 64 divided by six is,

link |

well, there's a quotient of 10, but the remainder is four.

link |

So you failed because the answer has to be two.

link |

For any prime, let's do it with five, which is prime.

link |

Two to the fifth is 32.

link |

Divide 32 by five, and you get six with a remainder of two.

link |

With a remainder of two, yeah.

link |

For seven, two to the seventh is 128.

link |

Divide that by seven, and let's see,

link |

I think that's seven times 14, is that right?

link |

Seven times 18 is 126 with a remainder of two, right?

link |

128 is a multiple of seven plus two.

link |

So if that remainder is not two,

link |

then it's definitely not prime.

link |

And then if it is, it's likely a prime, but not for sure.

link |

It's likely a prime, but not for sure.

link |

And there's actually a beautiful geometric proof

link |

which is in the book, actually.

link |

That's like one of the most granular parts of the book

link |

because it's such a beautiful proof, I couldn't not give it.

link |

So you draw a lot of like opal and pearl necklaces

link |

That's kind of the geometric nature

link |

of this proof of Fermat's Little Theorem.

link |

So yeah, so with pseudo primes,

link |

there are primes that are kind of faking it.

link |

They pass that test, but there are numbers

link |

that are faking it that pass that test,

link |

but are not actually prime.

link |

But the point is, there are many, many,

link |

many theorems about prime numbers.

link |

There's a bunch of questions to ask.

link |

Is there an infinite number of primes?

link |

Can we say something about the gap between primes

link |

as the numbers grow larger and larger and larger and so on?

link |

Yeah, it's a perfect example of your desire

link |

for simplicity in all things.

link |

You know what would be really simple?

link |

If there was only finitely many primes

link |

and then there would be this finite set of atoms

link |

that all numbers would be built up.

link |

That would be very simple and good in certain ways,

link |

but it's completely false.

link |

And number theory would be totally different

link |

if that were the case.

link |

It's just not true.

link |

In fact, this is something else that Euclid knew.

link |

So this is a very, very old fact,

link |

like much before, long before we've had anything

link |

like modern number theory.

link |

The primes are infinite.

link |

The primes that there are, right.

link |

There's an infinite number of primes.

link |

So what about the gaps between the primes?

link |

Right, so one thing that people recognized

link |

and really thought about a lot is that the primes,

link |

on average, seem to get farther and farther apart

link |

as they get bigger and bigger.

link |

In other words, it's less and less common.

link |

Like I already told you of the first 10 numbers,

link |

two, three, five, seven, four of them are prime.

link |

That's a lot, 40%.

link |

If I looked at 10 digit numbers,

link |

no way would 40% of those be prime.

link |

Being prime would be a lot rarer.

link |

In some sense, because there's a lot more things

link |

for them to be divisible by.

link |

That's one way of thinking of it.

link |

It's a lot more possible for there to be a factorization

link |

because there's a lot of things

link |

you can try to factor out of it.

link |

As the numbers get bigger and bigger,

link |

primality gets rarer and rarer, and the extent

link |

to which that's the case, that's pretty well understood.

link |

But then you can ask more fine grained questions,

link |

A twin prime is a pair of primes that are two apart,

link |

like three and five, or like 11 and 13, or like 17 and 19.

link |

And one thing we still don't know

link |

is are there infinitely many of those?

link |

We know on average, they get farther and farther apart,

link |

but that doesn't mean there couldn't be occasional folks

link |

that come close together.

link |

And indeed, we think that there are.

link |

And one interesting question, I mean, this is,

link |

because I think you might say,

link |

well, how could one possibly have a right

link |

to have an opinion about something like that?

link |

We don't have any way of describing a process

link |

that makes primes.

link |

Sure, you can look at your computer

link |

and see a lot of them, but the fact that there's a lot,

link |

why is that evidence that there's infinitely many, right?

link |

Maybe I can go on the computer and find 10 million.

link |

Well, 10 million is pretty far from infinity, right?

link |

So how is that evidence?

link |

There's a lot of things.

link |

There's like a lot more than 10 million atoms.

link |

That doesn't mean there's infinitely many atoms

link |

in the universe, right?

link |

I mean, on most people's physical theories,

link |

there's probably not, as I understand it.

link |

Okay, so why would we think this?

link |

The answer is that it turns out to be like incredibly

link |

productive and enlightening to think about primes

link |

as if they were random numbers,

link |

as if they were randomly distributed

link |

according to a certain law.

link |

Now they're not, they're not random.

link |

There's no chance involved.

link |

There it's completely deterministic

link |

whether a number is prime or not.

link |

And yet it just turns out to be phenomenally useful

link |

in mathematics to say,

link |

even if something is governed by a deterministic law,

link |

let's just pretend it wasn't.

link |

Let's just pretend that they were produced

link |

by some random process and see if the behavior

link |

is roughly the same.

link |

And if it's not, maybe change the random process,

link |

maybe make the randomness a little bit different

link |

and tweak it and see if you can find a random process

link |

that matches the behavior we see.

link |

And then maybe you predict that other behaviors

link |

of the system are like that of the random process.

link |

And so that's kind of like, it's funny

link |

because I think when you talk to people

link |

at the twin prime conjecture,

link |

people think you're saying,

link |

wow, there's like some deep structure there

link |

that like makes those primes be like close together

link |

And no, it's the opposite of deep structure.

link |

What we say when we say we believe the twin prime conjecture

link |

is that we believe the primes are like sort of

link |

strewn around pretty randomly.

link |

And if they were, then by chance,

link |

you would expect there to be infinitely many twin primes.

link |

And we're saying, yeah, we expect them to behave

link |

just like they would if they were random dirt.

link |

The fascinating parallel here is,

link |

I just got a chance to talk to Sam Harris

link |

and he uses the prime numbers as an example.

link |

Often, I don't know if you're familiar with who Sam is.

link |

He uses that as an example of there being no free will.

link |

Wait, where does he get this?

link |

Well, he just uses as an example of,

link |

it might seem like this is a random number generator,

link |

but it's all like formally defined.

link |

So if we keep getting more and more primes,

link |

then like that might feel like a new discovery

link |

and that might feel like a new experience, but it's not.

link |

It was always written in the cards.

link |

But it's funny that you say that

link |

because a lot of people think of like randomness,

link |

the fundamental randomness within the nature of reality

link |

might be the source of something

link |

that we experience as free will.

link |

And you're saying it's like useful to look at prime numbers

link |

as a random process in order to prove stuff about them.

link |

But fundamentally, of course, it's not a random process.

link |

Well, not in order to prove some stuff about them

link |

so much as to figure out what we expect to be true

link |

and then try to prove that.

link |

Because here's what you don't want to do.

link |

Try really hard to prove something that's false.

link |

That makes it really hard to prove the thing if it's false.

link |

So you certainly want to have some heuristic ways

link |

of guessing, making good guesses about what's true.

link |

So yeah, here's what I would say.

link |

You're going to be imaginary Sam Harris now.

link |

Like you are talking about prime numbers

link |

but prime numbers are completely deterministic.

link |

And I'm saying like,

link |

well, but let's treat them like a random process.

link |

but you're just saying something that's not true.

link |

They're not a random process, they're deterministic.

link |

And I'm like, okay, great.

link |

You hold to your insistence that it's not a random process.

link |

Meanwhile, I'm generating insight about the primes

link |

that you're not because I'm willing to sort of pretend

link |

that there's something that they're not

link |

in order to understand what's going on.

link |

Yeah, so it doesn't matter what the reality is.

link |

What matters is what framework of thought

link |

results in the maximum number of insights.

link |

Yeah, because I feel, look, I'm sorry,

link |

but I feel like you have more insights about people.

link |

If you think of them as like beings that have wants

link |

and needs and desires and do stuff on purpose,

link |

even if that's not true,

link |

you still understand better what's going on

link |

by treating them in that way.

link |

Don't you find, look, when you work on machine learning,

link |

don't you find yourself sort of talking

link |

about what the machine is trying to do

link |

in a certain instance?

link |

Do you not find yourself drawn to that language?

link |

Well, it knows this, it's trying to do that,

link |

it's learning that.

link |

I'm certainly drawn to that language

link |

to the point where I receive quite a bit of criticisms

link |

for it because I, you know, like.

link |

Oh, I'm on your side, man.

link |

So especially in robotics, I don't know why,

link |

but robotics people don't like to name their robots.

link |

They certainly don't like to gender their robots

link |

because the moment you gender a robot,

link |

you start to anthropomorphize.

link |

If you say he or she, you start to,

link |

in your mind, construct like a life story.

link |

In your mind, you can't help it.

link |

There's like, you create like a humorous story

link |

You start to, this person, this robot,

link |

you start to project your own.

link |

But I think that's what we do to each other.

link |

And I think that's actually really useful

link |

for the engineering process,

link |

especially for human robot interaction.

link |

And yes, for machine learning systems,

link |

for helping you build an intuition

link |

about a particular problem.

link |

It's almost like asking this question,

link |

you know, when a machine learning system fails

link |

in a particular edge case, asking like,

link |

what were you thinking about?

link |

Like, like asking, like almost like

link |

when you're talking about to a child

link |

who just did something bad, you want to understand

link |

like what was, how did they see the world?

link |

Maybe there's a totally new, maybe you're the one

link |

that's thinking about the world incorrectly.

link |

And yeah, that anthropomorphization process,

link |

I think is ultimately good for insight.

link |

And the same is, I agree with you.

link |

I tend to believe about free will as well.

link |

Let me ask you a ridiculous question, if it's okay.

link |

I've just recently, most people go on like rabbit hole,

link |

like YouTube things.

link |

And I went on a rabbit hole often do of Wikipedia.

link |

And I found a page on

link |

finiteism, ultra finiteism and intuitionism

link |

or into, I forget what it's called.

link |

Yeah, intuitionism.

link |

That seemed pretty, pretty interesting.

link |

I have it on my to do list actually like look into

link |

like, is there people who like formally attract,

link |

like real mathematicians are trying to argue for this.

link |

But the belief there, I think, let's say finiteism

link |

that infinity is fake.

link |

Meaning, infinity might be like a useful hack

link |

for certain, like a useful tool in mathematics,

link |

but it really gets us into trouble

link |

because there's no infinity in the real world.

link |

Maybe I'm sort of not expressing that fully correctly,

link |

but basically saying like there's things

link |

that once you add into mathematics,

link |

things that are not provably within the physical world,

link |

you're starting to inject to corrupt your framework

link |

What do you think about that?

link |

I mean, I think, okay, so first of all, I'm not an expert

link |

and I couldn't even tell you what the difference is

link |

between those three terms, finiteism, ultra finiteism

link |

and intuitionism, although I know they're related

link |

and I tend to associate them with the Netherlands

link |

Okay, I'll tell you, can I just quickly comment

link |

because I read the Wikipedia page.

link |

The difference in ultra.

link |

That's like the ultimate sentence of the modern age.

link |

Can I just comment because I read the Wikipedia page.

link |

That sums up our moment.

link |

Bro, I'm basically an expert.

link |

So, finiteism says that the only infinity

link |

you're allowed to have is that the natural numbers

link |

So, like those numbers are infinite.

link |

So, like one, two, three, four, five,

link |

the integers are infinite.

link |

The ultra finiteism says, nope, even that infinity is fake.

link |

I'll bet ultra finiteism came second.

link |

I'll bet it's like when there's like a hardcore scene

link |

and then one guy's like, oh, now there's a lot of people

link |

I have to find a way to be more hardcore

link |

than the hardcore people.

link |

It's all back to the emo, Doc.

link |

Okay, so is there any, are you ever,

link |

because I'm often uncomfortable with infinity,

link |

like psychologically.

link |

I have trouble when that sneaks in there.

link |

It's because it works so damn well,

link |

I get a little suspicious,

link |

because it could be almost like a crutch

link |

or an oversimplification that's missing something profound

link |

Well, so first of all, okay, if you say like,

link |

is there like a serious way of doing mathematics

link |

that doesn't really treat infinity as a real thing

link |

or maybe it's kind of agnostic

link |

and it's like, I'm not really gonna make a firm statement

link |

about whether it's a real thing or not.

link |

Yeah, that's called most of the history of mathematics.

link |

So it's only after Cantor that we really are sort of,

link |

okay, we're gonna like have a notion

link |

of like the cardinality of an infinite set

link |

and like do something that you might call

link |

like the modern theory of infinity.

link |

That said, obviously everybody was drawn to this notion

link |

and no, not everybody was comfortable with it.

link |

Look, I mean, this is what happens with Newton.

link |

I mean, so Newton understands that to talk about tangents

link |

and to talk about instantaneous velocity,

link |

he has to do something that we would now call

link |

taking a limit, right?

link |

The fabled dy over dx, if you sort of go back

link |

to your calculus class, for those who have taken calculus

link |

and remember this mysterious thing.

link |

And you know, what is it?

link |

Well, he'd say like, well, it's like,

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you sort of divide the length of this line segment

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by the length of this other line segment.

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And then you make them a little shorter

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and you divide again.

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And then you make them a little shorter

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and you divide again.

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And then you just keep on doing that

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until they're like infinitely short

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and then you divide them again.

link |

These quantities that are like, they're not zero,

link |

but they're also smaller than any actual number,

link |

these infinitesimals.

link |

Well, people were queasy about it

link |

and they weren't wrong to be queasy about it, right?

link |

From a modern perspective, it was not really well formed.

link |

There's this very famous critique of Newton

link |

by Bishop Berkeley, where he says like,

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what these things you define, like, you know,

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they're not zero, but they're smaller than any number.

link |

Are they the ghosts of departed quantities?

link |

That was this like ultra burn of Newton.

link |

And on the one hand, he was right.

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It wasn't really rigorous by modern standards.

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On the other hand, like Newton was out there doing calculus

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and other people were not, right?

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It works, it works.

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I think a sort of intuitionist view, for instance,

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I would say would express serious doubt.

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And by the way, it's not just infinity.

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It's like saying, I think we would express serious doubt

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that like the real numbers exist.

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Now, most people are comfortable with the real numbers.

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Well, computer scientists with floating point number,

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I mean, floating point arithmetic.

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That's a great point, actually.

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I think in some sense, this flavor of doing math,

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saying we shouldn't talk about things

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that we cannot specify in a finite amount of time,

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there's something very computational in flavor about that.

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And it's probably not a coincidence

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that it becomes popular in the 30s and 40s,

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which is also like kind of like the dawn of ideas

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about formal computation, right?

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You probably know the timeline better than I do.

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Sorry, what becomes popular?

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These ideas that maybe we should be doing math

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in this more restrictive way where even a thing that,

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because look, the origin of all this is like,

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number represents a magnitude, like the length of a line.

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So I mean, the idea that there's a continuum,

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there's sort of like, it's pretty old,

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but just because something is old

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doesn't mean we can't reject it if we want to.

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Well, a lot of the fundamental ideas in computer science,

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when you talk about the complexity of problems,

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to Turing himself, they rely on an infinity as well.

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The ideas that kind of challenge that,

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the whole space of machine learning,

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I would say, challenges that.

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It's almost like the engineering approach to things,

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like the floating point arithmetic.

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The other one that, back to John Conway,

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that challenges this idea,

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I mean, maybe to tie in the ideas of deformation theory

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and limits to infinity is this idea of cellular automata

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with John Conway looking at the game of life,

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Stephen Wolfram's work,

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that I've been a big fan of for a while, cellular automata.

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I was wondering if you have,

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if you have ever encountered these kinds of objects,

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you ever looked at them as a mathematician,

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where you have very simple rules of tiny little objects

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that when taken as a whole create incredible complexities,

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but are very difficult to analyze,

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very difficult to make sense of,

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even though the one individual object, one part,

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it's like what we were saying about Andrew Wiles,

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you can look at the deformation of a small piece

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to tell you about the whole.

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It feels like with cellular automata

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or any kind of complex systems,

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it's often very difficult to say something

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about the whole thing,

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even when you can precisely describe the operation

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of the local neighborhoods.

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Yeah, I mean, I love that subject.

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I haven't really done research on it myself.

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I've played around with it.

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I'll send you a fun blog post I wrote

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where I made some cool texture patterns

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from cellular automata that I, but.

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And those are really always compelling

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is like you create simple rules

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and they create some beautiful textures.

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It doesn't make any sense.

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Actually, did you see, there was a great paper.

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I don't know if you saw this,

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like a machine learning paper.

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I don't know if you saw the one I'm talking about

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where they were like learning the texture

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as like let's try to like reverse engineer

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and like learn a cellular automaton

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that can reduce texture that looks like this

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And as you say, the thing you said is I feel the same way

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when I read machine learning paper

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is that what's especially interesting

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is the cases where it doesn't work.

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Like what does it do when it doesn't do the thing

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that you tried to train it to do?

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That's extremely interesting.

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Yeah, yeah, that was a cool paper.

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So yeah, so let's start with the game of life.

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Let's start with, or let's start with John Conway.

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So yeah, so let's start with John Conway again.

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Just, I don't know, from my outsider's perspective,

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there's not many mathematicians that stand out

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throughout the history of the 20th century.

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And he's one of them.

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I feel like he's not sufficiently recognized.

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I think he's pretty recognized.

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I mean, he was a full professor at Princeton

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for most of his life.

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He was sort of certainly at the pinnacle of.

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Yeah, but I found myself every time I talk about Conway

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and how excited I am about him,

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I have to constantly explain to people who he is.

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And that's always a sad sign to me.

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But that's probably true for a lot of mathematicians.

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I was about to say,

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I feel like you have a very elevated idea of how famous.

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This is what happens when you grow up in the Soviet Union

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or you think the mathematicians are like very, very famous.

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Yeah, but I'm not actually so convinced at a tiny tangent

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that that shouldn't be so.

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I mean, there's, it's not obvious to me

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that that's one of the,

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like if I were to analyze American society,

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that perhaps elevating mathematical and scientific thinking

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to a little bit higher level would benefit the society.

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Well, both in discovering the beauty of what it is

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to be human and for actually creating cool technology,

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But anyway, John Conway.

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Yeah, and Conway is such a perfect example

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of somebody whose humanity was,

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and his personality was like wound up

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with his mathematics, right?

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And so it's not, sometimes I think people

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who are outside the field think of mathematics

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as this kind of like cold thing that you do

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separate from your existence as a human being.

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No way, your personality is in there,

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just as it would be in like a novel you wrote

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or a painting you painted

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or just like the way you walk down the street.

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Like it's in there, it's you doing it.

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And Conway was certainly a singular personality.

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I think anybody would say that he was playful,

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like everything was a game to him.

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Now, what you might think I'm gonna say,

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and it's true is that he sort of was very playful

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in his way of doing mathematics,

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but it's also true, it went both ways.

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He also sort of made mathematics out of games.

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He like looked at, he was a constant inventor of games

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or like crazy names.

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And then he would sort of analyze those games mathematically

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to the point that he,

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and then later collaborating with Knuth like,

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created this number system, the serial numbers

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in which actually each number is a game.

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There's a wonderful book about this called,

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I mean, there are his own books.

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And then there's like a book that he wrote

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with Berlekamp and Guy called Winning Ways,

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which is such a rich source of ideas.

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And he too kind of has his own crazy number system

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in which by the way, there are these infinitesimals,

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the ghosts of departed quantities.

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They're in there now, not as ghosts,

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but as like certain kind of two player games.

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So, he was a guy, so I knew him when I was a postdoc

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and I knew him at Princeton

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and our research overlapped in some ways.

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Now it was on stuff that he had worked on many years before.

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The stuff I was working on kind of connected

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with stuff in group theory,

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which somehow seems to keep coming up.

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And so I often would like sort of ask him a question.

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I would sort of come upon him in the common room

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and I would ask him a question about something.

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And just anytime you turned him on, you know what I mean?

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You sort of asked the question,

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it was just like turning a knob and winding him up

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and he would just go and you would get a response

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that was like so rich and went so many places

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and taught you so much.

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And usually had nothing to do with your question.

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Usually your question was just a prompt to him.

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You couldn't count on actually getting the question answered.

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Yeah, those brilliant, curious minds even at that age.

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Yeah, it was definitely a huge loss.

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But on his game of life,

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which was I think he developed in the 70s

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as almost like a side thing, a fun little experiment.

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His game of life is this, it's a very simple algorithm.

link |

It's not really a game per se

link |

in the sense of the kinds of games that he liked

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where people played against each other.

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But essentially it's a game that you play

link |

with marking little squares on the sheet of graph paper.

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And in the 70s, I think he was like literally doing it

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with like a pen on graph paper.

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You have some configuration of squares.

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Some of the squares in the graph paper are filled in,

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And there's a rule, a single rule that tells you

link |

at the next stage, which squares are filled in

link |

and which squares are not.

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Sometimes an empty square gets filled in,

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that's called birth.

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Sometimes a square that's filled in gets erased,

link |

that's called death.

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And there's rules for which squares are born

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and which squares die.

link |

The rule is very simple.

link |

You can write it on one line.

link |

And then the great miracle is that you can start

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from some very innocent looking little small set of boxes

link |

and get these results of incredible richness.

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And of course, nowadays you don't do it on paper.

link |

Nowadays you do it in a computer.

link |

There's actually a great iPad app called Golly,

link |

which I really like that has like Conway's original rule

link |

and like, gosh, like hundreds of other variants

link |

and it's a lightning fast.

link |

So you can just be like,

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I wanna see 10,000 generations of this rule play out

link |

like faster than your eye can even follow.

link |

And it's like amazing.

link |

So I highly recommend it if this is at all intriguing to you

link |

getting Golly on your iOS device.

link |

And you can do this kind of process,

link |

which I really enjoy doing,

link |

which is almost from like putting a Darwin hat on

link |

or a biologist hat on and doing analysis

link |

of a higher level of abstraction,

link |

like the organisms that spring up.

link |

Cause there's different kinds of organisms.

link |

Like you can think of them as species

link |

and they interact with each other.

link |

They can, there's gliders, they shoot different,

link |

there's like things that can travel around.

link |

There's things that can,

link |

glider guns that can generate those gliders.

link |

You can use the same kind of language

link |

as you would about describing a biological system.

link |

So it's a wonderful laboratory

link |

and it's kind of a rebuke to someone

link |

who doesn't think that like very, very rich,

link |

complex structure can come from very simple underlying laws.

link |

Like it definitely can.

link |

Now, here's what's interesting.

link |

If you just pick like some random rule,

link |

you wouldn't get interesting complexity.

link |

I think that's one of the most interesting things

link |

of these, one of these most interesting features

link |

of this whole subject,

link |

that the rules have to be tuned just right.

link |

Like a sort of typical rule set

link |

doesn't generate any kind of interesting behavior.

link |

And I don't think we have a clear way of understanding

link |

which do and which don't.

link |

Maybe Steven thinks he does, I don't know.

link |

No, no, it's a giant mystery where Steven Wolfram did is,

link |

now there's a whole interesting aspect to the fact

link |

that he's a little bit of an outcast

link |

in the mathematics and physics community

link |

because he's so focused on a particular,

link |

his particular work.

link |

I think if you put ego aside,

link |

which I think unfairly some people

link |

are not able to look beyond,

link |

I think his work is actually quite brilliant.

link |

But what he did is exactly this process

link |

of Darwin like exploration.

link |

He's taking these very simple ideas

link |

and writing a thousand page book on them,

link |

meaning like, let's play around with this thing.

link |

And can we figure anything out?

link |

Spoiler alert, no, we can't.

link |

In fact, he does a challenge.

link |

I think it's like rule 30 challenge,

link |

which is quite interesting,

link |

just simply for machine learning people,

link |

for mathematics people,

link |

is can you predict the middle column?

link |

For his, it's a 1D cellular automata.

link |

Can you, generally speaking,

link |

can you predict anything about

link |

how a particular rule will evolve just in the future?

link |

Just looking at one particular part of the world,

link |

just zooming in on that part,

link |

100 steps ahead, can you predict something?

link |

And the challenge is to do that kind of prediction

link |

so far as nobody's come up with an answer.

link |

But the point is like, we can't.

link |

We don't have tools or maybe it's impossible or,

link |

I mean, he has these kind of laws of irreducibility

link |

that he refers to, but it's poetry.

link |

It's like, we can't prove these things.

link |

It seems like we can't.

link |

It almost sounds like ancient mathematics

link |

or something like that, where you're like,

link |

the gods will not allow us to predict the cellular automata.

link |

But that's fascinating that we can't.

link |

I'm not sure what to make of it.

link |

And there's power to calling this particular set of rules

link |

game of life as Conway did, because not exactly sure,

link |

but I think he had a sense that there's some core ideas here

link |

that are fundamental to life, to complex systems,

link |

to the way life emerge on earth.

link |

I'm not sure I think Conway thought that.

link |

It's something that, I mean, Conway always had

link |

a rather ambivalent relationship with the game of life

link |

because I think he saw it as,

link |

it was certainly the thing he was most famous for

link |

in the outside world.

link |

And I think that he, his view, which is correct,

link |

is that he had done things

link |

that were much deeper mathematically than that.

link |

And I think it always aggrieved him a bit

link |

that he was the game of life guy

link |

when he proved all these wonderful theorems

link |

and created all these wonderful games,

link |

created the serial numbers.

link |

I mean, he was a very tireless guy

link |

who just did an incredibly variegated array of stuff.

link |

So he was exactly the kind of person

link |

who you would never want to reduce to one achievement.

link |

You know what I mean?

link |

Let me ask you about group theory.

link |

You mentioned it a few times.

link |

What is group theory?

link |

What is an idea from group theory that you find beautiful?

link |

Well, so I would say group theory sort of starts

link |

as the general theory of symmetries,

link |

that people looked at different kinds of things

link |

and said, as we said, oh, it could have,

link |

maybe all there is is symmetry from left to right,

link |

like a human being, right?

link |

That's roughly bilaterally symmetric, as we say.

link |

So there's two symmetries.

link |

And then you're like, well, wait, didn't I say

link |

there's just one, there's just left to right?

link |

Well, we always count the symmetry of doing nothing.

link |

We always count the symmetry

link |

that's like there's flip and don't flip.

link |

Those are the two configurations that you can be in.

link |

You know, something like a rectangle