back to indexJordan 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 Erdős, 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 Erdős.
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 Erdős number.
link |
It's hard to distract.
link |
But you know, Erdős 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 Poincaré 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 Poincaré 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,
link |
you sort of divide the length of this line segment
link |
by the length of this other line segment.
link |
And then you make them a little shorter
link |
and you divide again.
link |
And then you make them a little shorter
link |
and you divide again.
link |
And then you just keep on doing that
link |
until they're like infinitely short
link |
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,
link |
what these things you define, like, you know,
link |
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.
link |
It wasn't really rigorous by modern standards.
link |
On the other hand, like Newton was out there doing calculus
link |
and other people were not, right?
link |
It works, it works.
link |
I think a sort of intuitionist view, for instance,
link |
I would say would express serious doubt.
link |
And by the way, it's not just infinity.
link |
It's like saying, I think we would express serious doubt
link |
that like the real numbers exist.
link |
Now, most people are comfortable with the real numbers.
link |
Well, computer scientists with floating point number,
link |
I mean, floating point arithmetic.
link |
That's a great point, actually.
link |
I think in some sense, this flavor of doing math,
link |
saying we shouldn't talk about things
link |
that we cannot specify in a finite amount of time,
link |
there's something very computational in flavor about that.
link |
And it's probably not a coincidence
link |
that it becomes popular in the 30s and 40s,
link |
which is also like kind of like the dawn of ideas
link |
about formal computation, right?
link |
You probably know the timeline better than I do.
link |
Sorry, what becomes popular?
link |
These ideas that maybe we should be doing math
link |
in this more restrictive way where even a thing that,
link |
because look, the origin of all this is like,
link |
number represents a magnitude, like the length of a line.
link |
So I mean, the idea that there's a continuum,
link |
there's sort of like, it's pretty old,
link |
but just because something is old
link |
doesn't mean we can't reject it if we want to.
link |
Well, a lot of the fundamental ideas in computer science,
link |
when you talk about the complexity of problems,
link |
to Turing himself, they rely on an infinity as well.
link |
The ideas that kind of challenge that,
link |
the whole space of machine learning,
link |
I would say, challenges that.
link |
It's almost like the engineering approach to things,
link |
like the floating point arithmetic.
link |
The other one that, back to John Conway,
link |
that challenges this idea,
link |
I mean, maybe to tie in the ideas of deformation theory
link |
and limits to infinity is this idea of cellular automata
link |
with John Conway looking at the game of life,
link |
Stephen Wolfram's work,
link |
that I've been a big fan of for a while, cellular automata.
link |
I was wondering if you have,
link |
if you have ever encountered these kinds of objects,
link |
you ever looked at them as a mathematician,
link |
where you have very simple rules of tiny little objects
link |
that when taken as a whole create incredible complexities,
link |
but are very difficult to analyze,
link |
very difficult to make sense of,
link |
even though the one individual object, one part,
link |
it's like what we were saying about Andrew Wiles,
link |
you can look at the deformation of a small piece
link |
to tell you about the whole.
link |
It feels like with cellular automata
link |
or any kind of complex systems,
link |
it's often very difficult to say something
link |
about the whole thing,
link |
even when you can precisely describe the operation
link |
of the local neighborhoods.
link |
Yeah, I mean, I love that subject.
link |
I haven't really done research on it myself.
link |
I've played around with it.
link |
I'll send you a fun blog post I wrote
link |
where I made some cool texture patterns
link |
from cellular automata that I, but.
link |
And those are really always compelling
link |
is like you create simple rules
link |
and they create some beautiful textures.
link |
It doesn't make any sense.
link |
Actually, did you see, there was a great paper.
link |
I don't know if you saw this,
link |
like a machine learning paper.
link |
I don't know if you saw the one I'm talking about
link |
where they were like learning the texture
link |
as like let's try to like reverse engineer
link |
and like learn a cellular automaton
link |
that can reduce texture that looks like this
link |
And as you say, the thing you said is I feel the same way
link |
when I read machine learning paper
link |
is that what's especially interesting
link |
is the cases where it doesn't work.
link |
Like what does it do when it doesn't do the thing
link |
that you tried to train it to do?
link |
That's extremely interesting.
link |
Yeah, yeah, that was a cool paper.
link |
So yeah, so let's start with the game of life.
link |
Let's start with, or let's start with John Conway.
link |
So yeah, so let's start with John Conway again.
link |
Just, I don't know, from my outsider's perspective,
link |
there's not many mathematicians that stand out
link |
throughout the history of the 20th century.
link |
And he's one of them.
link |
I feel like he's not sufficiently recognized.
link |
I think he's pretty recognized.
link |
I mean, he was a full professor at Princeton
link |
for most of his life.
link |
He was sort of certainly at the pinnacle of.
link |
Yeah, but I found myself every time I talk about Conway
link |
and how excited I am about him,
link |
I have to constantly explain to people who he is.
link |
And that's always a sad sign to me.
link |
But that's probably true for a lot of mathematicians.
link |
I was about to say,
link |
I feel like you have a very elevated idea of how famous.
link |
This is what happens when you grow up in the Soviet Union
link |
or you think the mathematicians are like very, very famous.
link |
Yeah, but I'm not actually so convinced at a tiny tangent
link |
that that shouldn't be so.
link |
I mean, there's, it's not obvious to me
link |
that that's one of the,
link |
like if I were to analyze American society,
link |
that perhaps elevating mathematical and scientific thinking
link |
to a little bit higher level would benefit the society.
link |
Well, both in discovering the beauty of what it is
link |
to be human and for actually creating cool technology,
link |
But anyway, John Conway.
link |
Yeah, and Conway is such a perfect example
link |
of somebody whose humanity was,
link |
and his personality was like wound up
link |
with his mathematics, right?
link |
And so it's not, sometimes I think people
link |
who are outside the field think of mathematics
link |
as this kind of like cold thing that you do
link |
separate from your existence as a human being.
link |
No way, your personality is in there,
link |
just as it would be in like a novel you wrote
link |
or a painting you painted
link |
or just like the way you walk down the street.
link |
Like it's in there, it's you doing it.
link |
And Conway was certainly a singular personality.
link |
I think anybody would say that he was playful,
link |
like everything was a game to him.
link |
Now, what you might think I'm gonna say,
link |
and it's true is that he sort of was very playful
link |
in his way of doing mathematics,
link |
but it's also true, it went both ways.
link |
He also sort of made mathematics out of games.
link |
He like looked at, he was a constant inventor of games
link |
or like crazy names.
link |
And then he would sort of analyze those games mathematically
link |
to the point that he,
link |
and then later collaborating with Knuth like,
link |
created this number system, the serial numbers
link |
in which actually each number is a game.
link |
There's a wonderful book about this called,
link |
I mean, there are his own books.
link |
And then there's like a book that he wrote
link |
with Berlekamp and Guy called Winning Ways,
link |
which is such a rich source of ideas.
link |
And he too kind of has his own crazy number system
link |
in which by the way, there are these infinitesimals,
link |
the ghosts of departed quantities.
link |
They're in there now, not as ghosts,
link |
but as like certain kind of two player games.
link |
So, he was a guy, so I knew him when I was a postdoc
link |
and I knew him at Princeton
link |
and our research overlapped in some ways.
link |
Now it was on stuff that he had worked on many years before.
link |
The stuff I was working on kind of connected
link |
with stuff in group theory,
link |
which somehow seems to keep coming up.
link |
And so I often would like sort of ask him a question.
link |
I would sort of come upon him in the common room
link |
and I would ask him a question about something.
link |
And just anytime you turned him on, you know what I mean?
link |
You sort of asked the question,
link |
it was just like turning a knob and winding him up
link |
and he would just go and you would get a response
link |
that was like so rich and went so many places
link |
and taught you so much.
link |
And usually had nothing to do with your question.
link |
Usually your question was just a prompt to him.
link |
You couldn't count on actually getting the question answered.
link |
Yeah, those brilliant, curious minds even at that age.
link |
Yeah, it was definitely a huge loss.
link |
But on his game of life,
link |
which was I think he developed in the 70s
link |
as almost like a side thing, a fun little experiment.
link |
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
link |
where people played against each other.
link |
But essentially it's a game that you play
link |
with marking little squares on the sheet of graph paper.
link |
And in the 70s, I think he was like literally doing it
link |
with like a pen on graph paper.
link |
You have some configuration of squares.
link |
Some of the squares in the graph paper are filled in,
link |
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.
link |
Sometimes an empty square gets filled in,
link |
that's called birth.
link |
Sometimes a square that's filled in gets erased,
link |
that's called death.
link |
And there's rules for which squares are born
link |
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
link |
from some very innocent looking little small set of boxes
link |
and get these results of incredible richness.
link |
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,
link |
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
link |
is bilaterally symmetric.
link |
You can flip it left to right,
link |
but you can also flip it top to bottom.
link |
So there's actually four symmetries.
link |
There's do nothing, flip it left to right
link |
and flip it top to bottom or do both of those things.
link |
And then a square, there's even more,
link |
because now you can rotate it.
link |
You can rotate it by 90 degrees.
link |
So you can't do that.
link |
That's not a symmetry of the rectangle.
link |
If you try to rotate it 90 degrees,
link |
you get a rectangle oriented in a different way.
link |
So a person has two symmetries,
link |
a rectangle four, a square eight,
link |
different kinds of shapes
link |
have different numbers of symmetries.
link |
And the real observation is that
link |
that's just not like a set of things, they can be combined.
link |
You do one symmetry, then you do another.
link |
The result of that is some third symmetry.
link |
So a group really abstracts away this notion of saying,
link |
it's just some collection of transformations
link |
you can do to a thing
link |
where you combine any two of them to get a third.
link |
So, you know, a place where this comes up
link |
in computer science is in sorting,
link |
because the ways of permuting a set,
link |
the ways of taking sort of some set of things
link |
you have on the table
link |
and putting them in a different order,
link |
shuffling a deck of cards, for instance,
link |
those are the symmetries of the deck.
link |
And there's a lot of them.
link |
There's not two, there's not four, there's not eight.
link |
Think about how many different orders
link |
the deck of card can be in.
link |
Each one of those is the result of applying a symmetry
link |
to the original deck.
link |
So a shuffle is a symmetry, right?
link |
You're reordering the cards.
link |
If I shuffle and then you shuffle,
link |
the result is some other kind of thing.
link |
You might call it a double shuffle,
link |
which is a more complicated symmetry.
link |
So group theory is kind of the study
link |
of the general abstract world
link |
that encompasses all these kinds of things.
link |
But then of course, like lots of things
link |
that are way more complicated than that.
link |
Like infinite groups of symmetries, for instance.
link |
So they can be infinite, huh?
link |
Well, okay, ready?
link |
Think about the symmetries of the line.
link |
You're like, okay, I can reflect it left to right,
link |
you know, around the origin.
link |
Okay, but I could also reflect it left to right,
link |
grabbing somewhere else, like at one or two
link |
or pi or anywhere.
link |
Or I could just slide it some distance.
link |
That's a symmetry.
link |
Slide it five units over.
link |
So there's clearly infinitely many symmetries of the line.
link |
That's an example of an infinite group of symmetries.
link |
Is it possible to say something that kind of captivates,
link |
keeps being brought up by physicists,
link |
which is gauge theory, gauge symmetry,
link |
as one of the more complicated type of symmetries?
link |
Is there an easy explanation of what the heck it is?
link |
Is that something that comes up on your mind at all?
link |
Well, I'm not a mathematical physicist,
link |
but I can say this.
link |
It is certainly true that it has been a very useful notion
link |
in physics to try to say like,
link |
what are the symmetry groups of the world?
link |
Like what are the symmetries
link |
under which things don't change, right?
link |
So we just, I think we talked a little bit earlier
link |
about it should be a basic principle
link |
that a theorem that's true here is also true over there.
link |
And same for a physical law, right?
link |
I mean, if gravity is like this over here,
link |
it should also be like this over there.
link |
Okay, what that's saying is we think translation in space
link |
should be a symmetry.
link |
All the laws of physics should be unchanged
link |
if the symmetry we have in mind
link |
is a very simple one like translation.
link |
And so then there becomes a question,
link |
like what are the symmetries of the actual world
link |
with its physical laws?
link |
And one way of thinking, this isn't oversimplification,
link |
but like one way of thinking of this big shift
link |
from before Einstein to after
link |
is that we just changed our idea
link |
about what the fundamental group of symmetries were.
link |
So that things like the Lorenz contraction,
link |
things like these bizarre relativistic phenomenon
link |
or Lorenz would have said, oh, to make this work,
link |
we need a thing to change its shape
link |
if it's moving nearly the speed of light.
link |
Well, under the new framework, it's much better.
link |
You say, oh, no, it wasn't changing its shape.
link |
You were just wrong about what counted as a symmetry.
link |
Now that we have this new group,
link |
the so called Lorenz group,
link |
now that we understand what the symmetries really are,
link |
we see it was just an illusion
link |
that the thing was changing its shape.
link |
Yeah, so you can then describe the sameness of things
link |
under this weirdness that is general relativity,
link |
Yeah, yeah, still, I wish there was a simpler explanation
link |
of like exact, I mean, gauge symmetries,
link |
pretty simple general concept about rulers being deformed.
link |
I've actually just personally been on a search,
link |
not a very rigorous or aggressive search,
link |
but for something I personally enjoy,
link |
which is taking complicated concepts
link |
and finding the sort of minimal example
link |
that I can play around with, especially programmatically.
link |
That's great, I mean,
link |
this is what we try to train our students to do, right?
link |
I mean, in class, this is exactly what,
link |
this is like best pedagogical practice.
link |
I do hope there's simple explanation,
link |
especially like I've in my sort of drunk random walk,
link |
drunk walk, whatever that's called,
link |
sometimes stumble into the world of topology
link |
and like quickly, like, you know when you go into a party
link |
and you realize this is not the right party for me?
link |
It's, so whenever I go into topology,
link |
it's like so much math everywhere.
link |
I don't even know what, it feels like this is me
link |
like being a hater, I think there's way too much math.
link |
Like there are two, the cool kids who just want to have,
link |
like everything is expressed through math.
link |
Because they're actually afraid to express stuff
link |
simply through language.
link |
That's my hater formulation of topology.
link |
But at the same time, I'm sure that's very necessary
link |
to do sort of rigorous discussion.
link |
But don't you think that's what gauge symmetry is like?
link |
I mean, it's not a field I know well,
link |
but it certainly seems like.
link |
Yes, it is like that.
link |
But my problem with topology, okay,
link |
and even like differential geometry is like,
link |
you're talking about beautiful things.
link |
Like if they could be visualized, it's open question
link |
if everything could be visualized,
link |
but you're talking about things
link |
that can be visually stunning, I think.
link |
But they are hidden underneath all of that math.
link |
Like if you look at the papers that are written
link |
in topology, if you look at all the discussions
link |
on Stack Exchange, they're all math dense, math heavy.
link |
And the only kind of visual things
link |
that emerge every once in a while,
link |
is like something like a Mobius strip.
link |
Every once in a while, some kind of simple visualizations.
link |
Every once in a while, some kind of simple visualizations.
link |
Every once in a while, some kind of simple visualizations.
link |
Well, there's the vibration, there's the hop vibration
link |
or all those kinds of things that somebody,
link |
some grad student from like 20 years ago
link |
wrote a program in Fortran to visualize it, and that's it.
link |
And it's just, you know, it's makes me sad
link |
because those are visual disciplines.
link |
Just like computer vision is a visual discipline.
link |
So you can provide a lot of visual examples.
link |
I wish topology was more excited
link |
and in love with visualizing some of the ideas.
link |
I mean, you could say that, but I would say for me,
link |
a picture of the hop vibration does nothing for me.
link |
Whereas like when you're like, oh,
link |
it's like about the quaternions.
link |
It's like a subgroup of the quaternions.
link |
And I'm like, oh, so now I see what's going on.
link |
Like, why didn't you just say that?
link |
Why were you like showing me this stupid picture
link |
instead of telling me what you were talking about?
link |
I'm just saying, no, but it goes back
link |
to what you were saying about teaching
link |
that like people are different in what they'll respond to.
link |
So I think there's no, I mean, I'm very opposed
link |
to the idea that there's a one right way to explain things.
link |
I think there's like a huge variation in like, you know,
link |
our brains like have all these like weird like hooks
link |
and loops and it's like very hard to know
link |
like what's gonna latch on
link |
and it's not gonna be the same thing for everybody.
link |
So I think monoculture is bad, right?
link |
I think that's, and I think we're agreeing on that point
link |
that like, it's good that there's like a lot
link |
of different ways in and a lot of different ways
link |
to describe these ideas because different people
link |
are gonna find different things illuminating.
link |
But that said, I think there's a lot to be discovered
link |
when you force little like silos of brilliant people
link |
to kind of find a middle ground
link |
or like aggregate or come together in a way.
link |
So there's like people that do love visual things.
link |
I mean, there's a lot of disciplines,
link |
especially in computer science
link |
that they're obsessed with visualizing,
link |
visualizing data, visualizing neural networks.
link |
I mean, neural networks themselves are fundamentally visual.
link |
There's a lot of work in computer vision that's very visual.
link |
And then coming together with some folks
link |
that were like deeply rigorous
link |
and are like totally lost in multi dimensional space
link |
where it's hard to even bring them back down to 3D.
link |
They're very comfortable in this multi dimensional space.
link |
So forcing them to kind of work together to communicate
link |
because it's not just about public communication of ideas.
link |
It's also, I feel like when you're forced
link |
to do that public communication like you did with your book,
link |
I think deep profound ideas can be discovered
link |
that's like applicable for research and for science.
link |
Like there's something about that simplification
link |
or not simplification, but distillation or condensation
link |
or whatever the hell you call it,
link |
compression of ideas that somehow
link |
actually stimulates creativity.
link |
And I'd be excited to see more of that
link |
in the mathematics community.
link |
Let me make a crazy metaphor.
link |
Maybe it's a little bit like the relation
link |
between prose and poetry, right?
link |
I mean, if you, you might say like,
link |
why do we need anything more than prose?
link |
You're trying to convey some information.
link |
So you just like say it.
link |
Well, poetry does something, right?
link |
It's sort of, you might think of it as a kind of compression.
link |
Of course, not all poetry is compressed.
link |
Like not all, some of it is quite baggy,
link |
but like you are kind of, often it's compressed, right?
link |
A lyric poem is often sort of like a compression
link |
of what would take a long time
link |
and be complicated to explain in prose
link |
into sort of a different mode
link |
that is gonna hit in a different way.
link |
We talked about Poincare conjecture.
link |
There's a guy, he's Russian, Grigori Perlman.
link |
He proved Poincare's conjecture.
link |
If you can comment on the proof itself,
link |
if that stands out to you as something interesting
link |
or the human story of it,
link |
which is he turned down the field's metal for the proof.
link |
Is there something you find inspiring or insightful
link |
about the proof itself or about the man?
link |
Yeah, I mean, one thing I really like about the proof
link |
and partly that's because it's sort of a thing
link |
that happens again and again in this book.
link |
I mean, I'm writing about geometry and the way
link |
it sort of appears in all these kind of real world problems.
link |
But it happens so often that the geometry
link |
you think you're studying is somehow not enough.
link |
You have to go one level higher in abstraction
link |
and study a higher level of geometry.
link |
And the way that plays out is that Poincare asks a question
link |
about a certain kind of three dimensional object.
link |
Is it the usual three dimensional space that we know
link |
or is it some kind of exotic thing?
link |
And so, of course, this sounds like it's a question
link |
about the geometry of the three dimensional space,
link |
but no, Perelman understands.
link |
And by the way, in a tradition that involves
link |
Richard Hamilton and many other people,
link |
like most really important mathematical advances,
link |
this doesn't happen alone.
link |
It doesn't happen in a vacuum.
link |
It happens as the culmination of a program
link |
that involves many people.
link |
Same with Wiles, by the way.
link |
I mean, we talked about Wiles and I wanna emphasize
link |
that starting all the way back with Kummer,
link |
who I mentioned in the 19th century,
link |
but Gerhard Frey and Mazer and Ken Ribbit
link |
and like many other people are involved
link |
in building the other pieces of the arch
link |
before you put the keystone in.
link |
We stand on the shoulders of giants.
link |
So, what is this idea?
link |
The idea is that, well, of course,
link |
the geometry of the three dimensional object itself
link |
is relevant, but the real geometry you have to understand
link |
is the geometry of the space
link |
of all three dimensional geometries.
link |
Whoa, you're going up a higher level.
link |
Because when you do that, you can say,
link |
now let's trace out a path in that space.
link |
There's a mechanism called Ricci flow.
link |
And again, we're outside my research area.
link |
So for all the geometric analysts
link |
and differential geometers out there listening to this,
link |
if I, please, I'm doing my best and I'm roughly saying it.
link |
So the Ricci flow allows you to say like,
link |
okay, let's start from some mystery three dimensional space,
link |
which Poincare would conjecture is essentially
link |
the same thing as our familiar three dimensional space,
link |
but we don't know that.
link |
And now you let it flow.
link |
You sort of like let it move in its natural path
link |
according to some almost physical process
link |
and ask where it winds up.
link |
And what you find is that it always winds up.
link |
You've continuously deformed it.
link |
There's that word deformation again.
link |
And what you can prove is that the process doesn't stop
link |
until you get to the usual three dimensional space.
link |
And since you can get from the mystery thing
link |
to the standard space by this process
link |
of continually changing and never kind of
link |
having any sharp transitions,
link |
then the original shape must've been the same
link |
as the standard shape.
link |
That's the nature of the proof.
link |
Now, of course, it's incredibly technical.
link |
I think as I understand it,
link |
I think the hard part is proving
link |
that the favorite word of AI people,
link |
you don't get any singularities along the way.
link |
But of course, in this context,
link |
singularity just means acquiring a sharp kink.
link |
It just means becoming non smooth at some point.
link |
So just saying something interesting about formal,
link |
about the smooth trajectory
link |
through this weird space of geometries.
link |
But yeah, so what I like about it
link |
is that it's just one of many examples of where
link |
it's not about the geometry you think it's about.
link |
It's about the geometry of all geometries, so to speak.
link |
And it's only by kind of like being jerked out of flatland.
link |
It's only by sort of seeing the whole thing globally at once
link |
that you can really make progress on understanding
link |
the one thing you thought you were looking at.
link |
It's a romantic question,
link |
but what do you think about him
link |
turning down the Fields Medal?
link |
Is that just, are Nobel Prizes and Fields Medals
link |
just the cherry on top of the cake
link |
and really math itself, the process of curiosity,
link |
of pulling at the string of the mystery before us?
link |
And then the awards are just icing
link |
and clearly I've been fasting and I'm hungry,
link |
but do you think it's tragic or just a little curiosity
link |
that he turned down the medal?
link |
Well, it's interesting because on the one hand,
link |
I think it's absolutely true that right,
link |
in some kind of like vast spiritual sense,
link |
like awards are not important,
link |
like not important the way that sort of like
link |
understanding the universe is important.
link |
On the other hand, most people who are offered that prize
link |
accept it, so there's something unusual
link |
about his choice there.
link |
I wouldn't say I see it as tragic.
link |
I mean, maybe if I don't really feel like
link |
I have a clear picture of why he chose not to take it.
link |
I mean, he's not alone in doing things like this.
link |
People sometimes turn down prizes for ideological reasons,
link |
but probably more often in mathematics.
link |
I mean, I think I'm right in saying that
link |
Peter Schultz turned down sort of some big monetary prize
link |
because he just, you know, I mean, I think he,
link |
at some point you have plenty of money
link |
and maybe you think it sends the wrong message
link |
about what the point of doing mathematics is.
link |
I do find that there's most people accept.
link |
You know, most people give it a prize.
link |
Most people take it.
link |
I mean, people like to be appreciated,
link |
but like I said, we're people.
link |
Not that different from most other people.
link |
But the important reminder that that turning down
link |
a prize serves for me is not that there's anything wrong
link |
with the prize and there's something wonderful
link |
about the prize, I think.
link |
The Nobel prize is trickier
link |
because so many Nobel prizes are given.
link |
First of all, the Nobel prize often forgets
link |
many, many of the important people throughout history.
link |
Second of all, there's like these weird rules to it
link |
that it's only three people
link |
and some projects have a huge number of people.
link |
And it's like this, it, I don't know.
link |
It doesn't kind of highlight the way science is done
link |
on some of these projects in the best possible way.
link |
But in general, the prizes are great.
link |
But what this kind of teaches me and reminds me
link |
is sometimes in your life, there'll be moments
link |
when the thing that you would really like to do,
link |
society would really like you to do,
link |
is the thing that goes against something you believe in,
link |
whatever that is, some kind of principle.
link |
And standing your ground in the face of that
link |
is something I believe most people will have
link |
a few moments like that in their life,
link |
maybe one moment like that, and you have to do it.
link |
That's what integrity is.
link |
So like, it doesn't have to make sense
link |
to the rest of the world, but to stand on that,
link |
like to say no, it's interesting, because I think.
link |
But do you know that he turned down the prize
link |
in service of some principle?
link |
Because I don't know that.
link |
Well, yes, that seems to be the inkling,
link |
but he has never made it super clear.
link |
But the inkling is that he had some problems
link |
with the whole process of mathematics that includes awards,
link |
like this hierarchies and the reputations
link |
and all those kinds of things,
link |
and individualism that's fundamental to American culture.
link |
He probably, because he visited the United States quite a bit
link |
that he probably, it's all about experiences.
link |
And he may have had some parts of academia,
link |
some pockets of academia can be less than inspiring,
link |
perhaps sometimes, because of the individual egos involved,
link |
not academia, people in general, smart people with egos.
link |
And if you interact with a certain kinds of people,
link |
you can become cynical too easily.
link |
I'm one of those people that I've been really fortunate
link |
to interact with incredible people at MIT
link |
and academia in general, but I've met some assholes.
link |
And I tend to just kind of,
link |
when I run into difficult folks,
link |
I just kind of smile and send them all my love
link |
and just kind of go around.
link |
But for others, those experiences can be sticky.
link |
Like they can become cynical about the world
link |
when folks like that exist.
link |
So he may have become a little bit cynical
link |
about the process of science.
link |
Well, you know, it's a good opportunity.
link |
Let's posit that that's his reasoning
link |
because I truly don't know.
link |
It's an interesting opportunity to go back
link |
to almost the very first thing we talked about,
link |
the idea of the Mathematical Olympiad,
link |
because of course that is,
link |
so the International Mathematical Olympiad
link |
is like a competition for high school students
link |
solving math problems.
link |
And in some sense, it's absolutely false
link |
to the reality of mathematics,
link |
because just as you say,
link |
it is a contest where you win prizes.
link |
The aim is to sort of be faster than other people.
link |
And you're working on sort of canned problems
link |
that someone already knows the answer to,
link |
like not problems that are unknown.
link |
So, you know, in my own life,
link |
I think when I was in high school,
link |
I was like very motivated by those competitions.
link |
And like, I went to the Math Olympiad and...
link |
You won it twice and got, I mean...
link |
Well, there's something I have to explain to people
link |
because it says, I think it says on Wikipedia
link |
that I won a gold medal.
link |
And in the real Olympics,
link |
they only give one gold medal in each event.
link |
I just have to emphasize
link |
that the International Math Olympiad is not like that.
link |
The gold medals are awarded
link |
to the top 112th of all participants.
link |
So sorry to bust the legend or anything like that.
link |
Well, you're an exceptional performer
link |
in terms of achieving high scores on the problems
link |
and they're very difficult.
link |
So you've achieved a high level of performance on the...
link |
In this very specialized skill.
link |
And by the way, it was a very Cold War activity.
link |
You know, in 1987, the first year I went,
link |
Americans couldn't go to Havana back then.
link |
It was a very complicated process to get there.
link |
And they took the whole American team on a field trip
link |
to the Museum of American Imperialism in Havana
link |
so we could see what America was all about.
link |
How would you recommend a person learn math?
link |
So somebody who's young or somebody my age
link |
or somebody older who've taken a bunch of math
link |
but wants to rediscover the beauty of math
link |
and maybe integrate it into their work
link |
more solid in the research space and so on.
link |
Is there something you could say about the process of...
link |
Incorporating mathematical thinking into your life?
link |
I mean, the thing is,
link |
it's in part a journey of self knowledge.
link |
You have to know what's gonna work for you
link |
and that's gonna be different for different people.
link |
So there are totally people who at any stage of life
link |
just start reading math textbooks.
link |
That is a thing that you can do
link |
and it works for some people and not for others.
link |
For others, a gateway is, I always recommend
link |
the books of Martin Gardner,
link |
another sort of person we haven't talked about
link |
but who also, like Conway, embodies that spirit of play.
link |
He wrote a column in Scientific American for decades
link |
called Mathematical Recreations
link |
and there's such joy in it and such fun.
link |
And these books, the columns are collected into books
link |
and the books are old now
link |
but for each generation of people who discover them,
link |
they're completely fresh.
link |
And they give a totally different way into the subject
link |
than reading a formal textbook,
link |
which for some people would be the right thing to do.
link |
And working contest style problems too,
link |
those are bound to books,
link |
especially like Russian and Bulgarian problems.
link |
There's book after book problems from those contexts.
link |
That's gonna motivate some people.
link |
For some people, it's gonna be like watching
link |
well produced videos, like a totally different format.
link |
Like I feel like I'm not answering your question.
link |
I'm sort of saying there's no one answer
link |
and it's a journey where you figure out
link |
what resonates with you.
link |
For some people, it's the self discovery
link |
is trying to figure out why is it that I wanna know?
link |
Okay, I'll tell you a story.
link |
Once when I was in grad school,
link |
I was very frustrated with my lack of knowledge
link |
of a lot of things as we all are
link |
because no matter how much we know,
link |
we don't know much more and going to grad school
link |
means just coming face to face
link |
with the incredible overflowing vault of your ignorance.
link |
So I told Joe Harris, who was an algebraic geometer,
link |
a professor in my department,
link |
I was like, I really feel like I don't know enough
link |
and I should just take a year of leave
link |
and just read EGA, the holy textbook,
link |
Elements de Géométrie Algebraique,
link |
the Elements of Algebraic Geometry.
link |
I'm just gonna, I feel like I don't know enough
link |
so I'm just gonna sit and read this like 1500 page
link |
And he was like, and Professor Harris was like,
link |
that's a really stupid idea.
link |
And I was like, why is that a stupid idea?
link |
Then I would know more algebraic geometry.
link |
He's like, because you're not actually gonna do it.
link |
I mean, he knew me well enough to say like,
link |
you're gonna learn because you're gonna be working
link |
on a problem and then there's gonna be a fact from EGA
link |
that you need in order to solve your problem
link |
that you wanna solve and that's how you're gonna learn it.
link |
You're not gonna learn it without a problem
link |
to bring you into it.
link |
And so for a lot of people, I think if you're like,
link |
I'm trying to understand machine learning
link |
and I'm like, I can see that there's sort of
link |
some mathematical technology that I don't have,
link |
I think you like let that problem
link |
that you actually care about drive your learning.
link |
I mean, one thing I've learned from advising students,
link |
math is really hard.
link |
In fact, anything that you do right is hard.
link |
And because it's hard, like you might sort of have some idea
link |
that somebody else gives you, oh, I should learn X, Y and Z.
link |
Well, if you don't actually care, you're not gonna do it.
link |
You might feel like you should,
link |
maybe somebody told you you should,
link |
but I think you have to hook it to something
link |
that you actually care about.
link |
So for a lot of people, that's the way in.
link |
You have an engineering problem you're trying to handle,
link |
you have a physics problem you're trying to handle,
link |
you have a machine learning problem you're trying to handle.
link |
Let that not a kind of abstract idea
link |
of what the curriculum is, drive your mathematical learning.
link |
And also just as a brief comment that math is hard,
link |
there's a sense to which hard is a feature, not a bug,
link |
in the sense that, again,
link |
maybe this is my own learning preference,
link |
but I think it's a value to fall in love with the process
link |
of doing something hard, overcoming it,
link |
and becoming a better person because of it.
link |
Like I hate running, I hate exercise,
link |
to bring it down to like the simplest hard.
link |
And I enjoy the part once it's done,
link |
the person I feel like in the rest of the day
link |
once I've accomplished it, the actual process,
link |
especially the process of getting started in the initial,
link |
like it really, I don't feel like doing it.
link |
And I really have, the way I feel about running
link |
is the way I feel about really anything difficult
link |
in the intellectual space, especially in mathematics,
link |
but also just something that requires
link |
like holding a bunch of concepts in your mind
link |
with some uncertainty, like where the terminology
link |
or the notation is not very clear.
link |
And so you have to kind of hold all those things together
link |
and like keep pushing forward through the frustration
link |
of really like obviously not understanding certain like
link |
parts of the picture, like your giant missing parts
link |
of the picture and still not giving up.
link |
It's the same way I feel about running.
link |
And there's something about falling in love
link |
with the feeling of after you went through the journey
link |
of not having a complete picture,
link |
at the end having a complete picture,
link |
and then you get to appreciate the beauty
link |
and just remembering that it sucked for a long time
link |
and how great it felt when you figured it out,
link |
at least at the basic.
link |
That's not sort of research thinking,
link |
because with research, you probably also have to
link |
enjoy the dead ends with learning math
link |
from a textbook or from video.
link |
I don't think you have to enjoy the dead ends,
link |
but I think you have to accept the dead ends.
link |
Let's put it that way.
link |
Well, yeah, enjoy the suffering of it.
link |
So the way I think about it, I do, there's an.
link |
I don't enjoy the suffering.
link |
You have to accept that it's part of the process.
link |
There's a lot of ways to kind of deal with that dead end.
link |
There's a guy who's the ultra marathon runner,
link |
Navy SEAL, David Goggins, who kind of,
link |
I mean, there's a certain philosophy of like,
link |
most people would quit here.
link |
And so if most people would quit here and I don't,
link |
I'll have an opportunity to discover something beautiful
link |
that others haven't yet.
link |
And so like any feeling that really sucks,
link |
it's like, okay, most people would just like,
link |
go do something smarter.
link |
And if I stick with this,
link |
I will discover a new garden of fruit trees that I can pick.
link |
Okay, you say that, but like,
link |
what about the guy who like wins
link |
the Nathan's hot dog eating contest every year?
link |
Like when he eats his 35th hot dog,
link |
he like correctly says like,
link |
okay, most people would stop here.
link |
Are you like lauding that he's like,
link |
no, I'm gonna eat the 35th hot dog.
link |
In the long arc of history, that man is onto something.
link |
Which brings up this question.
link |
What advice would you give to young people today,
link |
thinking about their career, about their life,
link |
whether it's in mathematics, poetry,
link |
or hot dog eating contest?
link |
And you know, I have kids,
link |
so this is actually a live issue for me, right?
link |
I actually, it's not a thought experiment.
link |
I actually do have to give advice
link |
to two young people all the time.
link |
They don't listen, but I still give it.
link |
You know, one thing I often say to students,
link |
I don't think I've actually said this to my kids yet,
link |
but I say it to students a lot is,
link |
you know, you come to these decision points
link |
and everybody is beset by self doubt, right?
link |
It's like, not sure like what they're capable of,
link |
like not sure what they really wanna do.
link |
I always, I sort of tell people like,
link |
often when you have a decision to make,
link |
one of the choices is the high self esteem choice.
link |
And I always tell them, make the high self esteem choice.
link |
Make the choice, sort of take yourself out of it
link |
and like, if you didn't have those,
link |
you can probably figure out what the version of you
link |
that feels completely confident would do.
link |
And do that and see what happens.
link |
And I think that's often like pretty good advice.
link |
That's interesting.
link |
Sort of like, you know, like with Sims,
link |
you can create characters.
link |
Create a character of yourself
link |
that lacks all the self doubt.
link |
Right, but it doesn't mean,
link |
I would never say to somebody,
link |
you should just go have high self esteem.
link |
You shouldn't have doubts.
link |
No, you probably should have doubts.
link |
It's okay to have them.
link |
But sometimes it's good to act in the way
link |
that the person who didn't have them would act.
link |
That's a really nice way to put it.
link |
Yeah, that's like from a third person perspective,
link |
take the part of your brain that wants to do big things.
link |
What would they do?
link |
That's not afraid to do those things.
link |
What would they do?
link |
Yeah, that's really nice.
link |
That's actually a really nice way to formulate it.
link |
That's very practical advice.
link |
You should give it to your kids.
link |
Do you think there's meaning to any of it
link |
from a mathematical perspective, this life?
link |
If I were to ask you,
link |
we talked about primes, talked about proving stuff.
link |
Can we say, and then the book that God has,
link |
that mathematics allows us to arrive
link |
at something about in that book.
link |
There's certainly a chapter
link |
on the meaning of life in that book.
link |
Do you think we humans can get to it?
link |
And maybe if you were to write cliff notes,
link |
what do you suspect those cliff notes would say?
link |
I mean, look, the way I feel is that mathematics,
link |
as we've discussed, it underlies the way we think
link |
about constructing learning machines.
link |
It underlies physics.
link |
I mean, it does all this stuff.
link |
And also you want the meaning of life?
link |
I mean, it's like, we already did a lot for you.
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Like, ask a rabbi.
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No, I mean, I wrote a lot in the last book,
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How Not to Be Wrong.
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I wrote a lot about Pascal, a fascinating guy who is
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a sort of very serious religious mystic,
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as well as being an amazing mathematician.
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And he's well known for Pascal's wager.
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I mean, he's probably among all mathematicians.
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He's the one who's best known for this.
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Can you actually like apply mathematics
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to kind of these transcendent questions?
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But what's interesting when I really read Pascal
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about what he wrote about this,
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I started to see that people often think,
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oh, this is him saying, I'm gonna use mathematics
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to sort of show you why you should believe in God.
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You know, mathematics has the answer to this question.
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But he really doesn't say that.
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He almost kind of says the opposite.
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If you ask Blaise Pascal, like, why do you believe in God?
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He'd be like, oh, cause I met God.
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You know, he had this kind of like psychedelic experience.
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It's like a mystical experience where as he tells it,
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he just like directly encountered God.
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It's like, okay, I guess there's a God, I met him last night.
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That's why he believed.
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It didn't have to do with any kind.
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You know, the mathematical argument was like
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about certain reasons for behaving in a certain way.
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But he basically said, like, look,
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like math doesn't tell you that God's there or not.
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Like, if God's there, he'll tell you.
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You know, you don't even.
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So you have mathematics, you have, what do you have?
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Like a way to explore the mind, let's say psychedelics.
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You have like incredible technology.
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You also have love and friendship.
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And like, what the hell do you want to know
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what the meaning of it all is?
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I don't think there's a better way to end it, Jordan.
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This was a fascinating conversation.
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I really love the way you explore math in your writing.
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The willingness to be specific and clear
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and actually explore difficult ideas,
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but at the same time stepping outside
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and figuring out beautiful stuff.
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And I love the chart at the opening of your new book
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that shows the chaos, the mess that is your mind.
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Yes, this is what I was trying to keep in my head
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all at once while I was writing.
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And I probably should have drawn this picture
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earlier in the process.
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Maybe it would have made my organization easier.
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I actually drew it only at the end.
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And many of the things we talked about are on this map.
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The connections are yet to be fully dissected, investigated.
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And yes, God is in the picture.
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Right on the edge, right on the edge, not in the center.
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Thank you so much for talking to me.
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It is a huge honor that you would waste
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your valuable time with me.
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We went to some amazing places today.
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This was really fun.
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Thanks for listening to this conversation
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with Jordan Ellenberg.
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And thank you to Secret Sauce, ExpressVPN, Blinkist,
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Check them out in the description to support this podcast.
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And now let me leave you with some words from Jordan
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in his book, How Not To Be Wrong.
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Knowing mathematics is like wearing a pair of X ray specs
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that reveal hidden structures underneath the messy
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and chaotic surface of the world.
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Thank you for listening and hope to see you next time.