Theories of Everything with Curt Jaimungal - The Life of Alexander Grothendieck & Topos Theory | Colin McLarty

Episode Date: June 11, 2024

Colin McLarty is an American philosopher and mathematician renowned for his contributions to category theory, particularly in the philosophy of mathematics, and for his role as a leading scholar on the works of mathematician Saunders Mac Lane. Links Mentioned: - Richard Borcherds Youtube: https://www.youtube.com/watch?v=xu15ZbxxnUQ - Fermat's Last Theorum Paper: https://philpapers.org/rec/MCLWDI-2 Please consider signing up for TOEmail at https://www.curtjaimungal.org  Support TOE: - Patreon: https://patreon.com/curtjaimungal (early access to ad-free audio episodes!) - Crypto: https://tinyurl.com/cryptoTOE - PayPal: https://tinyurl.com/paypalTOE - TOE Merch: https://tinyurl.com/TOEmerch  Follow TOE: - *NEW* Get my 'Top 10 TOEs' PDF + Weekly Personal Updates: https://www.curtjaimungal.org - Instagram: https://www.instagram.com/theoriesofeverythingpod - TikTok: https://www.tiktok.com/@theoriesofeverything_ - Twitter: https://twitter.com/TOEwithCurt - Discord Invite: https://discord.com/invite/kBcnfNVwqs - iTunes: https://podcasts.apple.com/ca/podcast/better-left-unsaid-with-curt-jaimungal/id1521758802 - Pandora: https://pdora.co/33b9lfP - Spotify: https://open.spotify.com/show/4gL14b92xAErofYQA7bU4e - Subreddit r/TheoriesOfEverything: https://reddit.com/r/theoriesofeverything  

Professor McLarty, please tell myself and the audience about your journey into math, how you arrived at where you are now, especially how you became interested in Alexander Grothendieck. Dr. McLarty Well, I mean, I guess a journey into math starts when I was like a 13-year-old. My grandmother gave me the universe and Dr. Einstein by Lincoln Barnett. He was a Time Magazine writer, wrote about relativity. He was terrific. So I went off to college to become a physicist because that was the cool thing to do. But then I got this quantum mechanics course

and we made all these graphical solutions. We didn't really do it mathematically. And I took a course from a logician philosopher Howard Stein on philosophy of math and one on Gödel's theorem and I just thought that's what I want to do. So that's what I did. Yes. At what point was it that you took quantum mechanics? As an undergrad, junior. I see. And then you learned logic at what point?

Well, that was, I was starting that same year and I thought, you know what, I'm not going to switch to a philosophy major because I won't get math in grad school and philosophy. I'll switch to a math major, majored in math, to go into philosophy so that then I would know some math when I was in philosophy. So I liked that. You'll see a lot of spring of air log yellow behind me. Yes. Yes.

Then how did your interest in Alexander Grothendieck develop? Oh, well, I had good fortune to know a great teacher of mine who did category theory and he spent a lot of time with the category theorists up in Montreal, Mike Barr, Marta Bunga, Gonzalo Reyes, Andres Royale. And so because I had this personal connection, I learned some category theories, some topos theory. And at that time, I didn't know anything about Grotendieck. I mean, I knew he'd created the idea, but I didn't pursue Grotendieck at all. But I kept thinking somebody should. Somebody should find out what Grotendieck actually, but I kept thinking somebody should. Somebody should

find out what Grotendeek actually had to do with what I'm learning as category theory, because I knew he didn't do that exactly. And one thing and another I ended up, yeah, I got a three-year NSF grant to go to Harvard Mathematics and find out what Grotendeek did. Did you ever get to meet him? No, no. At the time I got interested in him, mathematics and find out what Grotendieck did. Did you ever get to meet him? No, no. At the time I got interested in him, there were people who met him. If, you know, he wasn't like a mafia boss with, you know, plastic surgery on his face.

If you chased him, you could meet him. But it didn't really tend to work very nicely. He didn't... At first he'd be angry, then he'd decide you were okay, and then he'd worry about why you were really there. Why do you think that is? Well, you know, he was born in 1928. He was born in Weimar, Germany. He went through World War II as the son of anarchist leftists, his father is a Russian Jew. He went through a lot and he was a very passionate person, but he was just always kind of a tormented

person. And he felt very strongly that his students had not pursued his work the way they should have. And in the context of this life, of this very conflictual life, he took that pretty hard. How did he envision his students taking his work and then how did they ultimately end up taking it? Well, he really wanted his students to pursue everything from the ground up, this sort of the Grotendieck way, you completely reconceive the theory. And his students, they did good work and Delinia, for example, sort of, you know, he learned that stuff, but as students, of course, they're trying to solve a problem, a particular problem, and so they would often sort of, they would take what he did for granted, which he intended to do it so it could be taken for granted, but he didn't want his students doing that,

and they would solve the problem not as systematically as he had hoped for. Can you give an example for people who, well, the audience is split. The audience comprises researchers in math and physics, but then there's also a general audience. So you can come up with two examples if you like. It's also my understanding that Alexander Grothendieck didn't like examples, so this is ironic. But please give examples about what Grothendieck would advise, how his systematic way of thinking would manifest. Well, the famous problem that he worked on with his largest group of students for a period of 10, 15 years is called the vague conjectures. You take some algebraic equations, not just on

standard numbers, they're unusual algebraic equations, but you have predictions about how many solutions they should have. And these were clearly famous when from the moment they were proposed, they were important. And Grotendieck got recruited to this by his friend Jean-Pierre Serres, who's important here because Serres does math a completely opposite way, and yet they work together perfectly. And Grotendieck looks at these problems and says, this looks like very insightful geometry and arithmetic, but I'm going to just let it sit in my brain until the solution becomes easy. And he starts everything from the ground up.

And his friend Jean-Pierre Serre keeps saying, Alex, that's going to take too long. Why? The way Serre approaches a problem is you learn everything that's been written about it for the past 150 years and then go to work. Grotendieck, his approach is, well, no, let's rethink it from the beginning. And Grotendieck could never have done what he did without Serre. This is not an opposition. It's a difference between the two. And so Grotendieck would say, well, let's just rethink this whole thing. And so he ends up writing these books that are thousands of pages long. They're not all single authored, but doing everything from the ground up and yet much easier to read than say 10 pages by Serres because Serres assumes

that you've learned everything about the subject for the past 150 years. Grotendeig, you just start from what he's writing. You just start from there and go. Matthew 18 So that's interesting because when I spoke to Richard Borchards, he said reading Serres is lucid. He loves reading Serres because it's extremely clear and he gives several examples and hammers at home. Richard Borchards likes when there's a, when a math textbook doesn't tell you, okay, this is left as an exercise. He actually likes the point to be hammered home. Yeah. And I believe Borchards is a field semetalist, right? Correct. Yeah, well, I'm not, you know.

You know, Sarah has a huge following, and if you can do what he does, then do it. That's great. But Grotendieck has this other approach, and in fact, despite Sarah's misgivings, that's the approach that worked on the vague conjectures. Hmm. I mean, Sarah's constantly saying, Alex, that's too long. Nobody's going to pursue it. Yeah, but why don't you just use such and so?

And yet it's Grotendieck's approach that worked. But for the last step of it, Pierre Dilligne achieves the last step, and he does not do it by the system. He uses Grotendieck's basic tools, but he does not prove all the basic theorems Grotendieck wanted to get to. He uses a very clever geometric insight that goes back to Solomon Leftschatz. Yes. And he solves the problem. And it's correct, but Grotendieck didn't want to go back to

Leftschatz. Grotendieck wanted to do it from the ground up. So why do you call that a systematic way of thinking? Because it sounds more loosey goosey in the sense that, look, Sarah sounds systematic. I'm going to read everything. There's a process. Whereas with Groten Dieck, he's, okay, I'm just going to take it in, let it impress upon my intuition. Let me walk with it and then something will come to me. At what point I don't know.

But what will come to you will be a really vast vision from the foundations up. Tate, Mumford and Tate wrote a very nice article about this. When Delinia solved the problem his way, they said, Grotendieck's approach to crossing a mountain valley is to fill in the valley. Delinia's approach is to build a suspension bridge across the valley. Grotendieck, he's going to do the whole thing from the ground up so that he doesn't need to know any prior math. Matthew 16 Interesting, right.

Matthew 17 Well, he couldn't have done this without the benefit of Seir, who did know any prior math. Interesting, right. He couldn't have done this without the benefit of Seir who did know the prior math, but his approach isn't going to rely on that. So Grotendieck is going to fill in everything. It will be systematic. The system will be explicit in what he writes. Whereas Seir, it's all references, and if only you were a math genius who grew up in the library of the Ecole Normale, you would already know all that stuff. But I don't. Yes, and Alexander Grothendieck himself was a math genius. He was, but not one who read lots and lots. Interesting.

He would invent things. Can you give another example? When he was in high school, they're teaching him calculus. He learns differentiation, integration, and he thinks this way of doing integrals, this isn't right. You should do it this other way. And he reinvents the idea of integrals. And his teacher looks at it and says, this is very smart, but you need to talk to people in Paris. He goes to people in Paris and they say, oh, you've reinvented the Le Beg integral. Ah.

He did reinvent the Le Beg integral. Now, I don't know that he invented it as thoroughly as Le Beg did, but he's just in high school. Well, Kalash, you know, but… So Grotendieck, yeah, he was a math genius, but he didn't have a stable high school education. Most of his high school was in the mountains in Le Chambon, where they're sheltering 5,000 Jewish and leftist children from the Nazis. It's a wonderful story, but also horrifying and long. So he didn't come with a great elite education, but he came with fantastic skill.

Yes, you had mentioned at one point that Grothendieck's upbringing had a profound impact on his mathematical thinking. Can you elaborate on this? Well, on his way of thinking about other people altogether. He's born in Berlin. His parents leave to go to Spain for the civil war to support the left in the Spanish civil war. They send for him when he's 10, they bring him to Paris where they're soon, very soon rounded up the Raft in France. They round up leftist Jews, suspicious foreigners, they take them to internment camps in the south of France, where it's important to say these camps did not kill people, but people died. As people say, they didn't kill people, but they let them die. He gets sent there. This is before 1942. This is when Vichy France controls the South, and the Germans control the North of France.

And then in 1942, the Allies invade North Africa. And the Germans say, no, we're not going to have two France's anymore. And they take the whole South. And now, there's a huge rescue effort to pull children out of these camps before they're sent to Germany. Grotendieck's father is sent to Auschwitz, where he is later killed. His mother is not, she's kept in France. So, he's now with this group of 5,000 Jewish and leftist children being rescued by Protestants in Le Chambon, because the Protestants have their own history of religious persecution in France. They know about this. So there's all these kids, where they went in the mountains of France, Grotendieck lived in a house in a small town, but some of them were out in stone huts. And these were kids who'd grown up in Berlin and Munich and Paris, and now they're with these

families of cowherds. And kids are not grateful. I mean, they understood their lives were being saved, but they're kids. And so this is a very alienating experience for them, not to complain about the people who did it at all, but the kids didn't get it. You know, they… Yes, I understand, yeah. Yeah. And so, Grotendieck, he goes through this.

His father, like I say, does disappear and we now know Kildred Nauschwitz, well, Kildred Birkenau, people like to make the distinction. He finishes high school basically, I mean, it's a different school system, but basically high school in Montpellier, they send him up to Paris, they say, Alex, you're good at math and the only people that good are in Paris, so you have to go there. And they have connections in Paris and the relevant people in Paris say, okay, you say, is that good? We will take him on. And in Paris in 1950, most of the buildings

don't have heat. Um, you don't know when the wartime economy will recover. Um, they had largely run out of gasoline. Cars and trucks had been converted to burning wood, some of them. And this is still going on in his time. So you don't know. And he's in the elite. He's with the people. They all want to get the best job in French mathematics when they finish their degree. Everyone of them wants the best job because there are not that many good jobs. Now today, a lot of good jobs in math in France.

In 1950, it was very reasonable not to know that, to think you're either going to get a top job or you're going to be a tutor in the provinces without television, without high-speed trains. So it was a very weird upbringing. And so how did that affect his ground-up way of thinking? Well, the way he describes it is he has the capacity to be alone. He doesn't need to be with anybody to do his thinking. He doesn't need to have people working with him. Now, eventually, he does have lots of people working with him at the Institute for Higher Studies, but he can innovate. He doesn't need to have a precedent.

He doesn't need to have a comrade. He can be alone. It's how he sees it. Now, of course, he's not quite alone, but his image of it is that he's very much alone. Matthew 16 Did he like to collaborate? Matthew 17 Well, he and Ser spoke every night for hours in Paris, on the phone. They lived in different neighborhoods. They would call and they would talk for hours. When Grotendieck went to Kansas, there's this correspondence that still survives. They wrote back and forth a lot. And it's clear that Grotendieck really did like having this guy

to collaborate with, even though they did completely different approaches to the problem. with, even though they did completely different approaches to the problem. Grothendieck says in his memoir, every important step I made in algebraic geometry began with an idea by Serre. Matthew 14.15 Interesting. Without Serre, Grothendieck would have been one of these maddening grad students who keep saying, I know how to do it, I just can't do it. Well, with Sarah, he was Alexander Grotendieck. Interesting. Okay, so then what was Grotendieck working on in Kansas?

He was beginning to work on this, the VEI conjecture. Sarah had told him, well, when VEI gave these conjectures, he's gonna describe solutions to equations, but he was gonna do it by comparing them to topology in a way that it was very pretty, it was very persuasive, but it made technically no sense at all. Nobody could see how you were actually going to do this. It just looked like a charming metaphor. And they apparently took that attitude himself, but Saer did not. Saer said, no, you can really do this. And he convinced Grotendijk that you can really do this.

Now to do this, you're going to have to utterly redefine this whole subject of topology because the current tools, like Ve thinks, will never do it. But Saer says, we can dress them up. If we, like I say, if we just know everything that's been done since Riemann, we can make them work. And Grothendieck says, well, maybe we don't need to know everything. Maybe we can just rethink them from the ground up and they'll work. And that happened, but only with crucial inputs from Seir. It would never have happened without Seir. Did he coin the Grothendieck topology or did someone else coin it? Like, I mean, the name, the moniker.

Oh, the term, uh, that might very well have been Mike Arton that made up the term. Um, Sarah took a decisive step on this problem by a, a clever technical construction in a talk in Paris, I believe it was in March on March 22nd, 1958, we've got detailed records. I don't remember the exact date, I think it was that, but we've got the records. And he did this clever construction and Grotinet is sitting in the audience and Grotinet comes

up afterwards and says, you know what, this is going to solve the whole problem. You've solved a tiny little part by doing this, but what you did will solve the whole problem. You've solved a tiny little part by doing this, but what you did will solve the whole problem. And Saar, he tells us, he thought this was crazy optimistic, that there's no way this would solve the whole problem. But Grotendieck had, because he had this systematic approach in mind, as soon as he saw Saar's one trick, he said, oh, I know how to generalize that. I know what really happened there. You don't generalize it by adding new cases,

you generalize it by getting rid of special assumptions. Okay. He says, I know which ones to get rid of, and then this will do everything. Can you explain the difference between those two types of generalization? So one is you have several axioms and you just remove one or two of them, but then the other was what? Well, the other is it's like you take coordinate geometry like you learned in high school and you say oh I'm not going to assume that I have one coordinate grid I'm going to

assume I have different coordinate grids and they overlap in different ways and I'm going to invent differential geometry by overlapping coordinate grids. And so I take something that's more complicated, but it has more cases and it's wonderful stuff. But that's generalizing by adding new cases. The other kind of generalization is you say, I'm going to forget coordinate geometry. I'm just going to take Euclid's axioms and then I'm going to stop talking about distances and parallels. And I'm going to call this projective geometry. I'm just going to forget some of the assumptions. And a lot of theorems will go away. You won't have them anymore when you forget about parallels, when you forget about distance.

But they won't all. Some will stay and those are now called theorems of projective geometry. Interesting. You simply forget some of the subject and it still works. Would you say that was his special power or he had multiple powers? Well, that was his really special power was to say all that really used was such and so and I can say what such and so is and I can just start from there. Yes. I can stop ever saying that other stuff. Well, okay,

110 pages down the line I'll bring in ever saying that other stuff. Well, okay, 110 pages down the line, I'll bring in some of that other stuff. But for the first 110 pages, I'll just forget it. In other words, he cares more about what's necessary than what's sufficient. Yeah, yeah, he's got a genuine, but it's not in any mechanical sense. It's in an insightful, genius sense. He finds the right things to drop. He finds the right description of what was necessary here. Did he attempt to teach that to his students?

Yeah, well, that's what he wanted everybody to learn. And like a lot of geniuses, he's just beside himself that people don't all do it as soon as he shows it to them. You know, how could you not do it this way? Now, what would be an example of showing it to them? You've outlined some examples, but I mean, how did he say, how did he convey it? How was he trying to teach this? He would say, yeah, it's hard, hard to think. He would say, look, in topology, you learned about what are called injective sheaves, and they had this long definition, but all you really used about them was if you take two, the combination of them is still one of them.

And if you would just rely on that, you won't get all the same theorems, but we don't want most of those theorems because they won't apply to our problem. But you will get some of the right theorems, and this will turn out to be the right thing to have paid attention to, and this will give us the theorems that will work. And meanwhile, Jean-Pierre Serre, who is the youngest Fields Medalist ever, I mean, everybody knows he's good, is saying, Alex, I don't think you want to forget all that stuff. And Alex is saying, oh yeah, I do really just forget that for now. And then it works. And Sarah recognized that it worked, right? I mean, Sarah was never at a loss to follow Grothendieck, but he was constantly at a

loss to know why anybody would want to do it like that. Yes. So now we're at Kansas, what occurs to growth in Deke afterward? What is his thinking? What is he working on? Well, in Kansas, he says, we want to have these what are called cohomology theories for algebraic varieties. And we don't know what a cohomology theory is.

Up till now, only topological spaces had a cohomology, abstract groups had a cohomology, and these were all built in very complicated ways. But you know what? This pattern is all that mattered. There's this pattern of mappings that's all that mattered. So forget that other stuff. Anything that has this pattern of mappings has a unique cohomology, and I can show you how to define that in just 140 pages, which he did. And Sarah is like, 140 pages, Alex? Why not? I did a similar thing in 10, but it's exactly like Borchard said. You have to be Borchard to read those 10. Now, Borchard can read them. Yay! Barry Mazer can read them.

I could read them if I... but it hasn't been my priority. I can't read them easily now. Grotendieck, he writes 140 pages and everybody says, yeah, but what does this give you? Does this give you new theorems? And he says, no, not yet, but what it will give us is the vague cohomology. We'll solve the vague conjectures by thinking of cohomology this way. And Sarah's like, maybe, I don't know, maybe. But Grothendieck's yes, absolutely we will. And then when Sarah comes up with this one construction in 1958, Grothendieck says, oh, take that, plug it into my axioms,

boom, you'll get the whole theory. And all it took was like 10 years to do it. So would you say that reading Grothendieck is relatively easier than reading Sehr? Yeah, absolutely. I mean, because to read Sehr, you can't just read Sehr. You have to read Wey, you have to read Riemann, you have to read Gauss, you have to read, I mean, Grotendieck. You just read Grotendieck.

He doesn't much cite other theorems. And of course, to Sehr, this is a terrible loss doesn't much cite other theorems. And of course, to say this is a terrible loss because it was the other theorems that are why we ever thought of this. But Grotendieck is saying, yeah, sort of, but I never learned them. I got them from you. You know? At the University of Toronto, there's the infamous MAT 157, which is real analysis. And in it, you use Spivak's book, his first book on calculus. It's a great read, although it's lengthy.

And then for real analysis 257, so the second year version, you read his, his book on manifolds, I believe, which is a short book. Yeah. It's actually a super short. It was a decisive book for me as an undergraduate. Yeah. That calculus on manifolds. Aha.

Meaning it did what for you? I understood it. It's like, I had never understood a whole math book before. That one's so short. You can understand the whole thing. It's dense and it requires work to go through. So would you say that it would be akin to growth and deep would be like his Spivak's first large calculus book,

where you can read it, but it flows much more nicely. It takes a bit more time because it's a lengthy book, versus the Calculus on Manifolds, which is the Serre book. Is that a correct analogy? Well, except that to me, Calculus on Manifolds really pretty much is from the ground up. It's abstract. You don't necessarily see where it's coming from, but it is from the ground up. It's abstract, you don't necessarily see where it's coming from, but it is from the ground up. Whereas, Seir will cite classical results that... Okay, so the differences in references, which one can you read without having to

read the prior literature? That's an important part of the difference, yes. I see, I see. Okay, so now what is Grothendieck up to at this point in the story? So he comes up with this, and then he goes to the Ser talk and he says, you know what, we take my description of cohomology theories from Kansas, but that depends on having picked a category to start with. We'll pick that category based on the construction that Ser used in this talk and that will give us a cohomology that will have all the properties they talked about and that they didn't really even believe you could have,

but Sarah and I believed you could have them and you will have them. And then direct from general properties of that theory, you will prove all the Ve conjectures. And indeed, very directly from general properties, you do prove the first three of the Ve conjectures, but not the fourth one that Delina finally solved. That fourth one, they never got from those very general properties. You had to go through this. What experts tell me is a beautiful, elegant construction, but it's technical and I, you know, I'm not trying to be an algebraic geometer. So, yeah, and then Delina solves it by that. He does use Grothendieck's cohomology theory,

but he doesn't use its most general properties to prove the last Vey conjecture. Matthew 14 So why don't you rattle off, for the mathematicians watching, or the aspiring mathematicians even, some of the accomplishments of Grothendieck. Well, okay, for his doctoral dissertation he's assigned a problem by Laurent Schwartz and, due to Ney I believe, he's told try to explain these results that Laurent Schwartz is getting in functional analysis. And he goes into this and he takes this very complicated field that's doing a lot with infinite dimensional vector spaces, very general kinds of infinite dimensional vector spaces, and he finds a certain categorical property that one of these spaces might have or might not, and

he says all the ones that have that property give a theorem analogous to the Schwarz's famous kernel theorem. That categorical property explains when a category does or does not have a kernel theorem. And this takes a huge impression on people, and it's one of the first uses of category theory in functional analysis, probably the first serious use of category theory. And you can say, well, it's not category theory, it's just categorical language. I don't care what you call it. I mean, he talks about categories and functors, and he solves this problem. He doesn't have a lot of faith in that topic. He later says, I didn't pick that topic, they gave me that topic. And so, I never really felt it. He doesn't

feel like he finished that work. He made decisive progress. It became famous. It's the idea of a nuclear space. But people, analysts will argue about how important that idea really is and Grotendieck says, well, that's because I didn't finish it. Did he ever say where he would have taken it to had he finished it? Well, yeah, he left some open problems that if you would solve them, then the theory would fall together even more nicely. Are they still open to this day? At least the most important of them is not that they've been solved. Some of them only in the last 20 years, and he did this in 1953, so what, 70 years ago. But then he gets sold on the

Vey conjectures by Seir, and again, he and Seir believe we can really do what Vey said you might do, even though Vey doesn't really believe you could do it. He thinks that's just a motivation, but we're going to actually do it that way." And he goes and he… cohomology was this technical tool that you could use it on topological spaces, you could use it on groups, you could use it on what are called Lie algebras, you could use it on number of isolated cases. But Grotendieck says, oh no, it's much more general than that. Anytime you have these categorical properties on a structure, there will be a cohomology theory. If these structures were sheaves on topological spaces,

it'll give you the classical topological case. If the structures are Galois actions by a group, they'll give you the group case. But you don't need to know what they really were. If they just have these abstract properties, they'll give you a cohomology theory. And that's what he develops in Kansas, what are called Abelian categories and cohomology of Abelian categories. And then when he's back in Paris, the great thing about his time in Kansas was he didn't call Saar on the phone, he wrote letters, and that

correspondence has been published. When they're both in Paris, they're on the phone and we can't reconstruct what they said. But there, the Grotendieck-Saar correspondence is published, Saar went to a lot of trouble to get it published well, to make good comments on it. So then he comes back to Paris and Saar gives this talk and Grotendijk says, ooh, Serre gives what's called the one-dimensional cohomology of any algebraic variety. So it's like the n-dimensional cohomology, you take a space and the n-dimensional cohomology tells you how many n-dimensional surfaces you can put in there that are essentially different from each other. I mean, you can put infinitely many in there, but lots of them can be turned into each other,

trivially. How many different ones, how many ways can you put them in that are not equivalent to any of the ones you've done so far? So the one-dimensional cohomology tells you how many curves you can, how many essentially different ways there are to put curves in there. And this is not, it's not great progress on the vague conjectures, except that it was the first progress ever on this aspect of them. And Sarah is saying, well, you know, I had to use so much apparatus to make this work in the one dimensional case. This is not a promising approach to the two

dimensional case. It's just too hard. It was too hard already in one dimension, but then Grotendieck says, no, no, really, no, the parts you needed to do were really pretty simple and they'll work in all dimensions. And by six months later, Sarah agrees. He says, yup, this is going to work in all

dimensions to my astonishment, but it will. And it still took five years to get it to, but it did. It took five years by a bunch of people, but it did. And then it took another five years after that before Delina says, oh, but you know that last vague conjecture, we're not going to do it straight from this. We will use all of this apparatus, but we're not going to do it straight from that approach because I can't, I can't, you know, I still can't make that work. And Grotendieck is very disappointed because Grotendieck doesn't

care where the result is true. He cares whether the method works. Yes. But Delinia, well, he says, but we also care where the result is true and it is true, he cares whether the method works. Yes. But Delinia, well, he says, but we also care whether the result is true, and it is true, and I can prove it, you know, which to me, that's also good, but it's not what Grotendieck wanted. And Grotendieck, again, because of this tortured past and the way he is, he's kind of, he sees this as a betrayal sometimes, and other times not. He goes back and forth. Peter Bregman Betrayal by Pierre? Paul Jay By Delinia, by Pierre Delinia, yeah.

Peter Bregman Uh-huh. So what happens next? Paul Jay Well, okay, a lot of what happens next is May 68 in Paris, you know, there were, there are riots, there are factories that are occupied, universities are shut down. Paris in 1968 was a, well, all of France was very conflictual with these things going on. The Vietnam War is going on in the United States and the civil rights movement is exploding in all kinds of directions. And then, so by 1973, before he knows that the delinia has solved the Las Vegas Conjecture, Grotendieck discovers that the math center he works at has military funding.

Pete Right. Pete Now, a lot of the people involved have said, this was never a secret. How could he not have known it from the start? Did he not come to our receptions? I mean, it's on the letterhead, you know. I mean, how did he not know this before? But he feels like he didn't know it before. And he quits the public practice of mathematics. He says, my students are refusing to publish the seminars we worked on together. They're not using my methods. They're trying to deny that I ever did anything, which is completely false. They were not trying to deny that he ever did anything, but he feels like they are. Are they undervaluing his tools? Well, that's

not completely false. It's a value judgment. And even somebody who values the tools might say nonetheless, I can solve a famous problem not using them, which I can't solve using them. And Grottendyke is like, well, why would you want to? Well, because I can, you know. It would seem to me that Grotendieck would have sympathy for that approach because there were other tools that were available to Grotendieck that he also said, I'm not going to use those. I'm going to use my own methods. So when people did that to him, he didn't, he wasn't receptive of it.

Well, one thing that happened in his life and a lot of people's lives is that French mathematics going into World War II was extremely elitist. There were a tiny number of good jobs, and the people who held them all had world reputations. And this is what, like, he and Serre in 1950, when they go to Paris, they're thinking, I need to get the best job available the year that I finished my degree in fact the decade I need the best job of the decade that I finished my my career so it's extremely competitive what's it more competitive than it is today oh yes much more

viciously, personally competitive. Andre Vey, he gives these conjectures. I wrote him a letter a long time ago now, I don't remember, 1990, 1980s or sometime, I said, Dear Professor Vey, he was at Princeton at that time, when you had this conjecture, were you thinking of a sort of a similar claim in this 1919 book by Veblen and Young? Two weeks later, I get back this envelope, it's slightly puffy. He has taken my letter, he has crumpled it up, he has flattened it back out again, he has written NO across it, and then signed, Entrez-Veille, folded it up and sent it back to me. Pete It's interesting. across it and then signed Andre Vey, folded it up and sent it back to me. Interesting.

It's nice he gave me an autograph. But it was like, when Andre Vey, when he would meet someone, his job was to prove they didn't belong in math. By growing this generation, when, you know, if Ser meets you, he's going to want to know whether you're very good. Of course he's going to want to, but it's not his job to prove you aren't. It's just his problem to tell whether you are. For they, it was his job to prove you're not, you know, and if you survive that great, good for you. But he's not trying to tell whether you're good.

He's going to try to prove you're not. And if you can't do it, yay, you're good. He's going to try to prove you're not, and if you can't do it, yay, you've passed. So, for context, Grothendieck was a pacifist or what? Well, growing up, he was a refugee. He probably could have taken French citizenship a lot sooner than he did, he probably avoided it to not go in the military. Now whether that was because he was a pacifist or because he didn't want to go in the military, I don't really know. 1968 made him a pacifist.

Mm-hmm. That's when the Center for Math that he was working for had revealed or he had found out that they had military funding? Well, that was a little bit before he found it out. He found it out in 71-72, quit by 73. Ah, yes, right, right. Yeah. But by 1970, he's living in a commune. He's the one that has the income, so he's the star of this commune, you know. He becomes interested in the radical ecology movement, they're poisoning our planet. When they're not actually bombing people, they're poisoning the rest of us. And he gets very involved

with this. So then he withdraws from academia in 1971 or so? Yeah. In 1972 and 73, he goes around, he goes on this large tour giving lectures, where he will talk about his math at your university, if you will also let him talk about his ecology movement, because he's trying to save the world, because we're all going to die of poison if we don't do this. So he goes to raise money for his commune, for his ecology movement. So he gives those talks, but he's not doing math research at that time. At the end of one of these talks, somebody from the audience says, what are you working on now? And he says, working? Well, I'm not, because he's protesting. He's not going to be part of mathematics at that time.

Do you see similarities between him and Perlman or only superficially? I think only superficially. Perlman's from a completely different context. Also very competitive math world, but. Please go into detail. Well, because for one thing, in Paris, you had the factory occupations, you had the universities occupied. This has not happened in Russia. Also Perlman, so far as I know, does he even talk about public issues?

I have not specialized in him. He talks about how badly people treat him, but it's not. Yes. I'm not even sure he talks about that. His friends talk about it. Yeah, he, as far as I know, he, he references how the math system abuses students. It uses them and abuses them. Yeah, but he doesn't talk about how the whole military industrial complex is poisoning all the people.

It's not blowing up. And for Grotendieck, of course, Grotendieck didn't invent that, right? He's got a whole movement he can join that's about that. Right. So what did he work on? You mentioned it wasn't math. What was he doing with his time? Well, he did write articles for a radical ecology newsletter, and later on he does start generating

some math notes. He just doesn't tell people about them. What he does between 1973 and 1985, I've never really looked into it. You can probably recover a lot of it. But one resource that people didn't used to know about is that in 1973, there are roughly 100 hours of tape-recorded lectures by Grototendeek in English at the University of Buffalo where he's talking about his mathematics. 100 hours, tape recorded hours. And that's available to the public? Yeah, yeah, you can probably find them online.

Just search Grotendeek Buffalo 1973. You can find them at the Poincare archive in Nancy. That archive has a copy. Where else can you find out about some of the journals of Alexander Grotendieck? Are they published or do you have to go to some specific place to learn more about him? Well, on the Grotendieck Circle website, the Grotendieck Circle website has these, Grotendieck Circle website, the Grotendieck Circle website has these, but also Olivia Caramello's Topos group is collecting those and I might not be quite up to date of what they have online. The ecological journal is called Survivre et Vivre.

It's in French, survive and live. It was just called survivre, it was just called survive. And people said, Alex, that's terrible. You're going to make it sound like we're all going to die. And he said, okay, I'll make it more positive. I'll make a survivre et vivre. Not that it makes it a lot more positive for most of us, but you search survivre et vivre, you can find that. But you search S'River Vivre, you can find that. If you search Grotendieck Buffalo 1973, you can find those. Like say the archive at Nalsee has copies you can listen to there, and you would probably find them on there, but you have to go there to do that. I have published an article describing them in this book, Lecture Grosindiqueienne.

My articles in English, all the other articles in French, that book is being translated, but I don't know how quickly into English. You've done an article on the what, the 100 hours of Grosindique? Well, no, I wrote about 33 hours on topos theory. There's other lectures I didn't do. Okay, sorry. So there's several distinct themes in this 100 hour talk? Topos theory, algebraic geometry and algebraic groups.

I see. Okay. Not ecology. There's no recording of ecology. Can you explain what topos theory is? Yeah topos theory They're taking they've got the vague conjectures they want to apply methods of topology to these Arithmetic equations. Well, so there's got to be groups these equations have to describe spaces These equations have to describe spaces and then you're going to study those spaces.

But there was not known how they described spaces. And Serres gives this step. He gets the one-dimensional cohomology and Grotinik says, oh, that's how to describe these spaces. Serres has told us how you can cover these spaces. He doesn't say what the spaces are, but he says how you can cover them with other things. So, we'll just look at all the ways of covering a given one and say that that is the space. We're going to look at all the ways to cover it. We're not going to ask exactly what it is, but we're going to say we know all about it

because we know all the ways to cover it. Explain me what it is that Sarah did so he counted curves He was throwing one dimensional strings like like what the fundamental group or he extended it to something that's higher dimensional So the homotopy is it analogous to that it's like it's like he showed how to Look at all the different coordinate patches on a space He we don't really have these spaces, we don't really have the coordinate patches, but we say how all these coordinate patches would relate to each other. That's what he can describe, how all the coordinate patches on this thing would relate to each other if it existed. And Grotendieck says, but that's good enough. We know everything

we need by knowing how it's covered by these coordinate patches. And the collection of all the ways to cover it by coordinate patches, it turns out you can look at that as if, well, on the one hand, it's a generalized space. You can give it a cohomology. You can describe it in spatial terms. On the other hand, you can work at it as if each of these ways of covering is a set. This is like a generalized universe of sets. A topos is on one hand a generalized topological space, on the other hand it's a generalized universe of sets, in each case specially adapted to a given equation in algebraic geometry. You get a whole world of sets for this equation, a whole other world for

that equation. You know that these worlds relate the way the solutions here should relate to solutions there. You just don't know what the space of solutions is. But you know how the solutions here should relate to solutions there, and these worlds do relate to each other that way. And so that's what a topos is. It's a world in which you can do mathematics and that mathematics will be specially adapted to some one generalized space. Maybe a topological space, maybe not. Maybe a classical group, maybe not. But it will be something you can think of as like a space.

And to each one of these spaces, is there assigned a unique equation? Well, in algebraic geometry, these spaces come from equations, but not unique equations, because different equations can solve this, can have the same solutions. Right. So we've cancelled out the difference between the different ones. This is just a space of solutions. In order to specify a topos, do you have to specify at least one equation? No, no, you don't have, no, no.

There's a lot of ways to specify a topos. You can paste it together out of other toposes. You could just give axioms that you wanted to satisfy. And would it then be the case that if you were to construct a topos without an equation that there would be at least one equation that corresponds to that topos or has it now moved to such a generality? It might not be algebraic. It's too general to, it might not be, it might not have an algebraic description like an

algebraic geometry. It might turn out to be a topological space or it might not. It might turn out to be a topological space or it might not. It might turn out to be a group or it might not. It might turn out to be a set of equations to some solution, some equation, or it might not. What it will have is a cohomology theory. You can describe it by cohomology, and if it corresponds to a topological space, this will be the classical cohomology of that topological space. If it corresponds to a group, this will be the classical cohomology of that topological space. If it

corresponds to a group, this will be the classical cohomology of that group. But even if it doesn't correspond to anything you know about, it's a cohomology theory, you just can't tell of what. What's the significance of Topos theory in modern mathematics? Well, it is in fact the way that cohomology theory is organized now. A lot of authors on cohomology theory avoid the word topos, but they don't avoid the idea. It is the way cohomology theory is, modern cohomology theory is organized. Is there some stigma against that word or concept? Is there some, why would they have to avoid it? Well, that's a sociological question. For one thing, people have this idea that there's a set theoretic problem about topos theory,

and they don't want set theoretic problems, they don't want to hear about set theory, so let's not talk about topos because there's a set theoretic problem. Well, skipping the word doesn't solve any problems, but it solves your feeling that you needed to worry about them. So there are very nice books on, on Italian cohomology that just never use the word topos, but in fact they're organized in topos terms. And because modern cohomology is organized in topos terms, this becomes the background to a lot of related but different modern ideas, especially

around the concept of homotopy, as you see in homotopy type theory, as you see in infinity category theory. So this topos organization over here became a takeoff point for other more advanced concepts. What's the relationship between topos theory and classical logic? Well you can interpret all of classical logic in topos terms it just won't always work out the same. If you take this the category of standard sets that's a topos you can interpret logic in it, and you'll get perfectly classical logic. But if you take a topos that corresponds to some topological space, even just like the real line, nice simple topological space, the real line,

you get a set theory where statements are not simply true or false, they're sort of true in some parts of the line false and others and indeterminate In the places where they cross between So you now get you get you interpret logic, but you get what's often called intuitionistic logic It's not exactly what people before called intuitionistic logic, but very close very close So that okay people call it that why is it different? Is it like some fuzzy logic plus intuitionist logic or what well for one thing? Intuitionistic logic originally had these epistemological motives. It should be entirely finitary

You should have no infinite airy commitments. Well topos here is don't care if they have infinite airy commitments It's just so that epistemological aspect is just not of interest to them. What they want is it mathematically correct, you know. We know lots of infinitary facts of math, so we'll not worry about them. Jared Sissling help me with the grammar of this. If you take the topos theory of set theory, you then recover classical logic. Is that the right phrase? Pete Slauson If you take the universe of sets that maybe you described in Zermelo-Frenkel in terms of membership and everything and you ignore some of that and you just look at the category of sets and maps so that now you can't distinguish between any two one element sets. They're isomorphic,

you can't distinguish between them. You get a topos and that topos has classical logic. So if you were to take the topos of the real line now, so the first one you just outlined was the universe of sets that would come from ZFC. Yeah. Okay. Now, if you take the topos of the real line, that's something different. Cause that's, but that would be a subset of the universe of Zermila Franco, no? Well, from a set theoretic viewpoint, sure, and it's a very peculiar, completely artificial and ugly sub-universe.

But from Toppo's point of view, it's just not. And it's actually not a sub-universe, it's a covering universe. But you can now, what is a singleton set? Well, in classical set theory, a singleton set is two things. Every set has a unique map to it because it's a singleton. You can only map onto that one value. Also, it has only two subsets, the whole thing and nothing. Cause it was a singleton. You've either, the whole thing and nothing, because it was a singleton. You either got the whole thing or you got nothing. Over here, if you look

at the real line, everything has a unique map to it. So it's like a singleton in that everything has a, but it does not only have two subsets, it has infinitely many subsets, what we would have called the subintervals of the real line. From outside, they are the subintervals of the real line. So the singleton, it doesn't just have two subsets, you can be partly empty. There's infinitely many, there's partly empty things that are disjoint from each other, partly empty subsets that overlap with each other, a lot of partly empty subsets. And this will give you a non-classical logic

in which real analysis is sort of built into this topos. See, that's super interesting because intuitionist logic, maybe I should be saying intuitionistic logic. So that form of logic is the people who like it tend to be constructivists. And in my experience, constructivists aren't fans of the real number line. No, no. And that's why I say the constructivist motivations are gone here.

We still have, what we have is the truth value of a claim in this toe post will be some open subset of the reals. The negation of that claim will be the biggest open subset disjoint from that. That'll always exist and they'll always be disjoint. But their union won't always be the whole thing. Because you got one open subset, another, you missed the boundary point. You only took the two open subsets. You don't regate, you don't recover that boundary point in between. So in this topos, we say every x is either greater, every real number is either greater than zero or, or less that are equal to zero.

Say, no, no, that's not, that's not going to work anymore. That's not going to work anymore. Um, because a variable real might be passing through zero. It isn't greater. It isn't less. It's passing through.

So we're going to miss the law of excluded middle, just like the classical intuitionists didn't get the law of excluded middle. We won't have it, they won't have it. They rejected for epistemological reasons. We say, well, look, a variable function doesn't have to all either, a continuous function doesn't have to, it can cross zero it's not greater it's not less it's not equal it's crossing through but that's completely different from the intuitionist what is the term for you say the topos of the universal sorry the topos of the universe of ZFC is it of or

the topos on the universe the topos of ZFC sets the topos of ZFC sets this is in fact the Grotendieck topos on a single point space this is the topos of a single point topological space there's no there's no question of continuity there's no variability it's these are all they're variable sets that vary over a single point, so they have zero variation. They're classical sets. Now would it be the case that, so there are various logics like pair consistent or fuzzy

or multivalued, modal, etc. Would it be the case that for every logic that we have, there's a corresponding space such that the topos of that space gives rise to that logic? Well, not fuzzy logics, not paraconsistent. No, those are not really topos logics. There's relations. Some people regard paraconsistent logic as what's called a co-topos logic. Instead of taking negation, you take what's called co-negation. So the negation of a subset in topos logic is the

largest subset of the same thing that's disjoint from it. Well, what if we took the smallest one whose union is the whole space? That's a different thing so that so then They won't be disjoint a subset a closed subset and the smallest closed subset whose union is the whole real line Those aren't disjoint. They all overlap at boundary points Yes, so you can you can get a kind of paraconsistent logic that way, but it's not the paraconsistent logic that paraconsistent logicians mostly want. Is there a relation between topos theory and homotopy type theory? Well, a relation of inheritance.

Topos theory, one of the main goals when Grotendieck invented it was to give a homotopy theory of toposes. By the way, is toposes the technical term or is it topoi? Well, see when Grotendieck coined the term in French, and in French the plural is just the singular is topo, plural is topo. When he, when he talked in Buffalo, he did use topoi as the plural. I don't like topoi as a plural. I think Peter Johnston says, you know, you've got a thermos, you've got a thermos full of hot tea when you've got two of them, you don't say you've got

two thermoi, you say you've got two thermoses. Yes. Well, see, I don't want to say too topoi. I want to say, because it's not a Greek word. It's a. Yes. Well, see, I don't want to say to topoi. I want to because it's not a Greek word. It's a French word. Ah, okay. So the logic that corresponds to the topos of a certain space is that called the topos is internal logic? Like is there a term? Yes. Yes. And that reflects the collection of open subsets of that space. Yes. Now, of course, that won't be literally true unless it was a topological space. If it's an

algebraic variety, it's more subtle, but it's still a good motivation. You have a paper, it's a part of a book on what does it take to prove Fermat's Last Theorem. part of a book on what does it take to prove Fermat's Last Theorem. And I believe on page 374 you wrote, much of the large apparatus of Wiles' 1995 proof will one day be bypassed in favor of a more direct use of piano arithmetic. At the same time, progress will continue making the functorial apparatus swifter and more accessible. In growth index terms, more naive. Okay, so what does that mean, more naive? Is that a pejorative? Is that meant as a favorable compliment?

No, no, no, no. It's more natural, more the way you would ordinarily have thought of this. Wiles had no interest in the large structure apparatus. Some of his results use it, but everybody uses them. He doesn't care. He just used them. Wiles is not interested in these questions. And what I'm saying there is that as these techniques get better and better absorbed,

and we see Peter Schultz working on this, it'll just smooth out. People will lose a lot of these concerns. Peter Schultz presses more on the nature of a Grotendieck topos than Wiles ever did, so Schultz is more concerned to deal with these set theoretic questions. And so in his paper on condensed spaces, he comes up with an alternative to Grotendieck toposies. They aren't quite Grotendieck toposies, and yet because they don't require any stronger set theory. Now in his latest work, he's even gotten simpler than that. So these are all technicalities. Grotendieck saw they were technicalities, and you look at just the history of math in general, this isn't special to this topic. Technicalities, well, two things happen to technicalities. Either they come to seem natural and people stop calling them technicalities, or people find a way to do things rigorously a little easier when you don't have them.

You learn to rigorously eliminate them. And that's something I did in a later paper. I showed how to get all the Grotendieck large structure apparatus and all the theorems of SGA-4 without Grotendieck universes. You can call it a topos if you want to, and I kind of do want to, but you just use a weaker set theory to begin with. And so you don't need these large sets. And SGA4 is? That's seminar, Seminar de Geomaterialis J'Abric 4. It's Grotendijk ran this seminar at the Institut des Instituts Scientifiques, Institute of

Higher Studies, Scientific Studies. Your accent is great. Oh, yeah. Your accent is wonderful. But he runs this seminar for several years, and they develop all the tools that he thinks might bear on the vaguejectures in tremendous detail. And it's very much his strategy, let's not worry whether they do, let's just find everything that might, and then the answer will fall out for us.

If we try to go find a minimal path, we don't know what's the minimal path, so let's just do everything from the ground up, everything that might work, and eventually we'll have so much that it will work." And yeah, and this seminar ran for years, and he would assign topics to students, and they would give the talk, or maybe he would give lectures, and then he would assign them to write up the notes. And this stuff was eventually all published by Springer Verlag. Grotten Dieck, he felt like his students, they weren't publishing it fast enough. They weren't working up the notes. They weren't getting it published. His students felt like, I'm trying to make

a living. I have other things to do besides write up this seminar, you know. But Grotten Dieck feels like, no, they're not writing this up fast enough and it's no good. And the decisive one on Topos theory is SGA 4. It's not the only one that bears on it, but it's called a Tao-cohomology and theory of Topos. For him, a Tao-cohomology practically was Topos theory. To someone like Saer, they're polar opposites. They're not the same thing at all. Just Grotendieck used them together.

But to Grotendieck, no, I used them together because they're really the same thing. So why wouldn't Saer see that etal cohomology is the same as topos theory if Alexander Grothendieck had demonstrated it? Well, this topos theory is a vast generalization of italic cohomology. We don't need most of it for the italic cohomology. And Grothendieck is saying, yeah, but it's pretty, and it's the way I thought of this stuff, you know. Did he ever use that word pretty? of this stuff, you know. Did he ever use that word pretty? No, he doesn't say pretty. Oh no, he tends to say things like of infantile simplicity. It's of infantile simplicity, so why wouldn't you? And that's said eulogistically, like it's a compliment. Yeah, yeah, yeah. Yeah, the infant, right? The child in all of us. You don't want to lose the infant.

Interesting. Maybe growth and deep didn't think like this, but did he have a philosophical commitment to one of the forms of logic? Like how I mentioned, computationalists tend to be, or people who work with computer science tend to be constructivists. No, he had no interest in that. How about yourself? I have no sympathy for it. I have an interest, but it's kind of a negative interest.

By all means, if you've got to calculate answers to a problem, calculate those answers. I'm not against calculational methods, but I am against favoring them as sort of the real content of math. They're not the real content. What is not the real content? The logic? The calculational aspect? I see so what I want to know is do you favor? Personally speaking classical logic or intuitionist logic or something else. No to me I spent a lot of time on this question if you're talking foundations of math That's it's got to be axioms in first-order logic classical first-order logic

That's the only thing that can deserve the name foundation of math because that's the only logic that you can really just say this Is what I believe I don't have to define it in any set theory I just believe these principles you're going to get classical first-order logic It took 50 years of progress by logicians to clarify these issues But that's the upshot of it. And I'm talking 1880 to 1930 now. I'm not talking anything while I was alive. But the progress of logic from Perse Schroeder Boole through Gödel said, okay, basic logic is classical first-order logic. That's your basic logic.

And that's the lesson I take from logic. That's your basic logic. That's the lesson I take from it. I believe that's correct. Jared Suellentrop In what ways do you think Gödel's incompleteness theorem is overhyped? Pete Slauson Oh, it's not. It's not. It's misunderstood by people who say, well, Gödel showed that we can't perfectly trust mathematics. It's the end of mathematical certainty. Mathematics can't prove it's correct. Well, that's just silly. I mean, all the people I know who are best at proving they're correct are delusional, right? They're really good at proving they're right because they're not right, because they're delusional. If math could prove its own consistency,

that wouldn't be your grounds for believing it. Your grounds for believing it would be the grounds we have now, right? Just because it could give a proof, that wouldn't persuade you of anything. What persuades you that, say for example, standard set theory is consistent is that you understand it clearly and you can see it's consistent. Oh, okay. I'm referring to the first Gertels incompleteness theorem, not the second one. Okay, yeah.

Okay, the first incompleteness, well, but that's not, that doesn't get sold at all. Popularization is all of the second one. In my experience, many philosophers will say, well, look, given that there's a statement that we can't prove, but we believe it to be true, or we can show that it's a statement that we can't prove but we believe it to be true or we can show that it's true That means that Well, that means math is incomplete But it doesn't it just means power if you prove it for power arithmetic power arithmetic is incomplete you say well Okay, let's go up to any any of the standard set theories They all prove consistency of PA just PA doesn't prove it and they all use the good old theorem that

of PA. Just PA doesn't prove it. They all prove the Gittl theorem. You can give a complete definition of the natural numbers in set theory. Now your set theory will also be incomplete. There will be a Gittl's first theorem for that set theory. So yes, it does show that our grasp of mathematics will never be complete. But that cannot be news. That just can't be news. That just can't be news. Apologies if this is ill formed, but it's like, look, only within the system can you not prove your consistency, but you can jump outside it and then from there prove it, but then you can't in that system prove its own consistency

to have to jump outside it again. Well, yeah, yeah, but we already know this. We know that, okay, there are fragments of pano-arithmetic that are easier to understand and that can't prove some things that full pano-arithmetic can prove. Set theory can prove some things that full pano-arithmetic can't prove. Yeah, so it is news that take, for any consistent axiomatization of mathematics, there will be a stronger one. That was not obvious before Gale proved his theorem. But it's not news that I don't know a complete description of mathematics.

It's really cryingly obvious that I don't know a complete description of mathematics. Now giving that formal character to it that every consistent every consistent theory of of mathematics will be incomplete, that's a very clever fact and that's I think not oversold that's really a great idea that's a great fact. So where I was going is that let's imagine that we know it's the case that you can't prove your consistency within the system but you can go to a so-called higher system and improve but then there's the problem of well you can never ultimately prove because you can't prove from within a system you just have to keep jumping outward. Now in the with natural numbers you also have something called ordinal numbers which is

greater than any natural number so So it serves as akin to infinity, this omega. Is there some omega logic, which is somehow consistent and can prove its own consistency, like second order, then we go to third order, then we dot to dot to omega order? Well, it doesn't prove its own consistency. It's consistent and it proves consistency of PA, but it doesn't prove its own consistency. If you mean axiomatic second-order arithmetic, now if you mean what's sometimes been called full second-order arithmetic, we don't know axioms for that. And it's complete, but you can't say it proves its own consistency because there's no proof system for it. Hmm.

Because it's not a first order theory. So it has no... I see. Yes, yes. Okay. So yeah, some people say that second order or higher order logics aren't as well defined. What does that mean that they're not as well defined? They require you to know what's

really meant by the power set of the natural numbers. You talk as if you know what is the power set of the natural numbers. We don't know what is the power set of the natural numbers. Now in any standard set theory, you've proved there is a unique power set, but if it's an axiomatic set theory, it won't answer all the questions about that power set. It if it's an axiomatic set theory, it won't answer all the questions about that power set. It proves it's unique, but will not answer all your questions about it because you'll have a girdle sentence, but it will prove it's unique. Does girdles and completeness theorem, the first one, have a correspondence work, sorry,

does it work in pair consistent logic or other forms of logic that aren't classical like intuitionist and fuzzy? Yeah, this is by now really well understood. Anytime your logic has what's called an effective proof relation, Gittles' theorem will apply to it. What that means is I've got this logic, I can make statements, I can give proofs in this logic, and I can define what a proof is, and from this definition, if you give me any candidate proof, I can by a routine process tell whether it's a proof. Now, if

you give me a candidate theorem, I can't tell whether it's a theorem. Maybe I'll know a proof of it, maybe I won't know a proof. If I don't know a proof, that's it. I just don't know a proof. But if you give me a candidate proof, I should be able to check whether it's right or not. And if you have an effective relation in that sense, then Gödel's theorem will apply. It's only for systems that don't have an effective proof relation

that Gödel's theorem doesn't apply. So I've heard computationalists say that Gödel's theorem doesn't apply. So I've heard computationalists say that Gödel's incompleteness theorem doesn't apply in intuitionist logic because it's just constructive. Something is only the case if you were able to construct it and by the way that the Gödel theorem or the Gödel statement is defined it's a

non-constructive way of defining it. Is that correct? Oh it is constructive. No that's just incorrect. It is constructive. No, that's just incorrect. It is constructive. Now, it's not finite test. It does involve talking about strings of symbols so long that you could never write them. But it's constructive.

Yes, yes. Okay. I misspoke. Some people would say that the concept of infinity in math leads to contradictions. There's the word that they say, even though I disagree with that, but they say contradictions like Gertl's incompleteness theorem, which is why we need to be finite tests and be more intuitionist. That's not avoiding any contradictions.

If you want to be a finite test by all means, except that I noticed that a lot of people who say they want to be finitists have trouble being finitists. They'll say, okay, I'm gonna be a finitist. I'm just gonna accept the set of all well-formed formulas of panoarithmetic. That's already not finitary. I mean, it's just not. It's recursive. In principle, if I give you a candidate and I claim this might be a well-formed formula of PA, you can check whether it is or not in a finite number of steps, but it might take longer than the age of the universe, so you're never going to get done with it. But if the sense of constructive that lets you take longer than the age of the universe, Gödel's theorem is

completely constructive. What other theorems are there in logic that should have, that more people should know about, that have philosophical import? So maybe Loewenheim-Skolen to some people, or Loeb's theorem, if I'm pronouncing that correctly, Lubs, I believe. Yeah, I'm not sure either. And I'm not sure it is even waltz, but you know, because there's that barred L in Polish, I don't know. Pete Yeah. Pete Well, there really is no other theorem in logic comparable to Gödel's theorem.

But I think the underappreciated aspect of Gödel's theorem, there is a desperately underappreciated aspect. It's been known since Solomon Fefferman in 1960. Gödel's theorem is not really properly understood. It's not about consistency of theories. It's about interpretability of one theory in another. What does that mean? The fact is, it's not just that pionoarithmetic can't prove its own consistency, pionoarithmetic can't interpret itself plus its consistency.

Pionoarithmetic can't even define a non-standard model in which it would turn out to be consistent. Pionoarithmetic can define a non-standard model in The things that are true in it aren't really arithmetic truths, but in that model, P-A is inconsistent. Piano arithmetic can interpret its own inconsistency, but it can't even interpret its own consistency. Piano arithmetic can imagine a way the world

could be where it would be inconsistent, but it cannot even imagine a way the world could be where it would be inconsistent, but it cannot even imagine a way the world could be where it would be consistent. And in this word imagine is the you're using that as a synonym for interpret. Interpret, yeah, yeah, yeah, that's the precise version. And what's the technical definition of interpret? Actually the one definition that works for that there, it's a, it's a protean word. There's lots of things that could mean,

but one definition that works works there is I'm simply going to redefine what I mean by an axiom of PA. I'm going to give a new definition where everything that should be provable is provable, but it will be a redefinition and there will be some things provable that aren't really provable. I'm not following. Yeah, yeah, yeah. Let's assume we know what's meant by an axiom of panorithmetic, I mean something which is an axiom and such that the whole string of enumerated axioms up to there is consistent.

Okay. This is just a new definition. Extensionally, it's the same thing because we really know they're all consistent. We really know this didn't make a difference, but formally it did. We threw consistency into that definition. And from this, it will follow that the lovely axioms of arithmetic, in a sense, will prove their own consistency. What they're not is effective. You can't tell what's a lovely axiom of arithmetic by this definition, because you can't tell whether an arbitrary theory is consistent or not.

But we just said, by lovely axiom, we mean it's an axiom such that the whole string of the axioms up to there is consistent. Now you can prove that that axiom system can prove its own consistency, except it's not a recursively defined axiom system. We don't know what the axioms are. Gödel's incompleteness theorem has a couple axioms. So one is that theory needs to be so-called strong enough to have the piano arithmetic in it, and then it also needs to be recursively axiomatizable. It needs to be a formal theory, and it also needs to be

sound, which is just implicit because it's a formal language, and it needs to be consistent. Yeah. So now we give this new definition of panoramic arithmetic, a new interpretation where we lose that recursiveness. Yes. So non-standard interpretation of piano. Yeah. Where we lose that recursiveness. And now the Gödel theorem is not correct for that theory.

But that theory is not a recursive theory. It's not a usable theory. We can't tell what its axioms are. But it does escape the Goodell theorem. It just also escapes our ability to comprehend it. Super interesting. Yeah, yeah, yeah, yeah. And so this is, and the correct way to say it, we don't, we don't, some textbooks will say, paranormal arithmetic and all its consistent extensions are inconsistent. Inconsistent or incomplete? I mean incomplete. I mean incomplete. Incomplete. But they also know that it applies to set theory, which is not an extension of PA,

it's in a whole different vocabulary. The reason we know it applies is you can interpret PA inset theory. The theory has always been about interpretability. The right way to say it is any formal theory which can interpret, say, Robinson arithmetic and is recursively axiomatized is incomplete. Right. But the right statement is already about interpreting, not just extending, anything that can interpret Robinson arithmetic.

Why are you switching to Robinson right now? Robinson arithmetic is where you forget the axiom of induction, but you only keep a tiny ghost of it, namely the axiom that every number is either zero or a successor. OK. We have no induction. We can prove nearly nothing in this theory, but

Goodell's theorem applies to it. Robinson discovered this by working through the proofs of Goodell's theorem. He was in, Tarkey says, you know, Raphael, why don't you go figure this out? And he did figure it out. So professor, what's something that you think is true, but your colleagues don't. So you Professor, what's something that you think is true, but your colleagues don't,

so you disagree with your colleagues about? Now, colleagues is quite a general term, so it could mean all mathematicians, or it could just be the ones in your department. I try not to believe things that nobody else I know believes. I think that would be a bad idea. Okay. So what do you disagree with your colleagues about? Well, I disagree with some colleagues about whether or not it's worth working hard or

to find out precisely how to prove Fermat's Last Theorem in Peron Arithmetic. I think Angus MacIntyre says, we all know you can do that, and I don't doubt you could do it, but I have to say we don't know how to do it, and I think it would be better if we did know how to do it. We've not proved you can't do it. It would be better if we could prove you can't do it. Angus feels like, no, that would be wasted effort. So there's something, I disagree with him about it. I don't disagree with that. Like I say, I try not to believe things that nobody else I know agrees with.

I mean, Harvey Friedman, you will never hear Harvey Friedman say, that's obviously provable, don't prove it. As you say, well, if it's really obviously provable, don't prove it. As you say, well, if it's really obviously provable, it should be easy to prove it, go spend the afternoon proving it. And if you can't prove it in an afternoon, then I guess it wasn't obviously provable. So, prove it. Angus will say, look, we know how it's going to turn out, so don't spend your time on it. And I do agree we know how it'll turn out, I just don't agree don't spend your time on it. I think it would be better to articulate that proof, I think we would learn things from articulating that proof. Let's see is it really provable, because in particular, if it's provable

in PA, then it's provable in some weaker theory. Every theorem of PA is provable in a weaker theory. Can we tell what that weaker theory is for Fermat's Last Theorem? Well, not if we don't know how you prove it's a theorem of PA. Even if we absolutely believe it is a theorem, so we absolutely believe it follows from some weaker theory, we can't begin to guess what weaker theory until we see how you would prove it's a theorem of P.A. So I think there's stuff to be learned from a more thorough proof-theoretic analysis of Fermat's Last Theorem. What we don't need is a proof-theoretic analysis to tell whether the theorem is correct or the theorem is correct. But we do need a more thorough analysis to tell, okay, what part of arithmetic does it really use? And Angus will agree we needed to tell what part of arithmetic suffices. He just considers

that not an important project and I disagree with him. I think it's a pretty cool project. In terms of philosophers of math, I think it would be great if philosophers of math knew more functional analysis. In some way, friends say, no, that's too much. We don't want to do it. Well, I can't disagree with them. They don't want to do it. If they don't, they don't.

What's specifically about analysis or functional analysis? Oh, well, the thing that most interests me in functional analysis is the realization that lots of things that functional analysts call functions are not function set theoretically. They're called generalized functions. They're not against the set theoretic definition of function. Many of them take that definition as the official definition of function. And then they'll say in their book, but we're not going to use the word like that. We're going to call things functions that aren't technically functions. Such as? Oh, well, say measurable functions.

A measurable function is, it's an equivalence class of set theoretic functions, where any two count as equivalent if they agree on a set of measure one. So an equivalence class is not a function. But when you read their books, they'll keep saying the function f, the function f, the function f. And say of course f has no specific value at any at any argument. Well, because the set theoretic functions in the equivalence class don't all agree at that argument. I don't understand what the problem is. Is it just terminological or are they misusing the object by calling it function?

No, no, no, they're using it perfectly correctly and I just think if you insist that function means what it means in Zerule-Frenkel set theory, you are closing off your understanding of what's going on in measure theory. You can say, oh, I can correctly interpret their textbooks, they just don't mean what they're saying. Say, but they do mean what they're saying. You are the one that don't get what they're saying because you insist on this set-theoric definition of function. If you would use the definition that they're actually using, then you would understand what they're doing better. Uh-huh. You would understand the insights there and then generalized functions are another thing beyond that.

Well, a lot of my friends, they say, no, Colin, we're not going to go read that book. Which book? Do you have one in mind on functional analysis? Oh yeah, yeah, yeah, yeah. Well, unfortunately it's four books. Um, yeah, Stein and C, yeah. Well, unfortunately, it's four books. Okay. Yeah, Stein and Cicacci, Lectures on Analysis, the Princeton Lectures on Analysis by I. Stein, I don't remember what the I stands for, and Cicacci was a student of his. Elias Stein,

it's E, Elias Stein, Elias Stein. Yeah, the Princeton lectures on analysis, lovely thing. I would compare them to the Feynman lectures on physics. Pete Wow. Pete It's a longish read, but it's beautifully selected, expert exposition. Every time I start to get tired of a topic, I think maybe I've learned enough of this now, they change the subject, yay, and I'm encouraged again. It's a beautiful full volume series on analysis. And you can see when you read it that for Elias Stein and his student, Shikarchi, this is what math really is. Set theory is a lovely logical foundation for this, but this is what math really is. Now some of my set theorists friends say, no, no, no, they don't even have, they can't even talk about wouldn't work on the continuum hypothesis. Well, okay, they can't. They can't. Elias Stein happens

not to want to do that. He's not against doing it. Well, he's passed away now, but it's just very much as the category theorists in the 1950s reconceived the foundations of mathematics around category theory. Elias Stein shapes them around the history of analysis. It's not a logical foundation, right? And now the category theory gives a new logical foundation for math. The analysis doesn't give a new logical foundation, but it gives a new synoptic vision of what math really is. And these are visions worth having. As a philosopher of math, if you can't get Elias Stein's vision of analysis, you're missing

something because it's a wonderful vision of math as a whole. It's not the only vision, but philosophers of math ought to have access to that vision. And if you're going to try to translate it all into ZF, these lovely short books are going to blow up. And this is not to say that this is not a critique of ZF, it's just saying it's a different view of math. So it doesn't of ZF, it's just saying it's a different view of math. So it doesn't use ZF to define itself? Well, officially, well, okay, they don't say ZF because they don't care the technicalities. Officially, they use the set theoretic definition

of a function, but they say in this passage, we have used a convention that we have already adopted before of identifying any two functions that agree on a set of measure one. They officially, that's an abusive notation. They shouldn't call them functions, but they do. Unofficially, they do. Constantly. And it's, you're not going to get this book if you keep trying to translate away that abusive notation What's another beautiful book that you've read in math that you recommend well calculus on manifolds? right Yeah, I'll leave the link to that in the description as well Okay, yeah

Well on category theory McLean's categories for the working mathematician. I have gone through and worked all the exercises and when you do that, what you discover, every page of that book, Saunders was aware of every other page when he wrote it. He saw, he had the whole book before his mind when he wrote it. He saw, he saw, he had the whole book before his mind when he wrote each individual page. Every page there is designed to advance the whole thing. Jared Yeah. Please explain, how does that feeling come to you just from doing an exercise? Pete Well, because, you know, he'll say something in chapter two, you think, okay, well, that's

correct, I don't know why he said it that way. Then you do an exercise in chapter seven, you say, oh, that's why he said it that way, so that this chapter would work out, so that this exercise would work the way it does. Pete I see. Pete He used that expression in chapter two because it was going to help you understand how to solve this exercise in chapter seven. And when you look at the history of it, that book is a write-up of a lecture series he gave repeatedly over five or six years. So he just went over it and over it and over it, and then he writes that vision. Now, the problem with reading categories for the

working mathematician is it draws examples from all over mathematics. There's nearly no chance you're going to be familiar with all those examples. So you need to be at the upper undergraduate or graduate level. You also need to be able to say of half the examples, I don't quite get that, I could get it if I looked into it. I see. And many of the proofs in category theory are ones where you would just expand the definition and you move around in symbol chasing. Now are those the types of exercises

that he gives or does he leave those to the wayside because those are more trivial? Oh, he gives lots of exercises of that kind. Yeah. Yeah. So that you come to realize that this chasing is just, it's just true. It's just. Well, what I mean by that is that when you're pushing around the symbols, there's not much insight there. In some sense, you get a bit aggravating. You're like, okay, it works out great.

Thank you. It's like multiplying two matrices together. But this is, this is the insight. There's less going on here than you thought. You can confuse yourself about a math problem by saying, well, I'm going to have to take account of this and that and the other thing when you don't. You're just distracting yourself by taking account of this, that, and the other thing. You didn't need to do any of that. It really just all depended on this little symbol manipulation, boom, which tells you

this wasn't really the problem. Peter Fried has sometimes said, the point of category theory is to take statements which are apparently trivial and show they really are. Oh, that's interesting. And it's good to know when something's really trivial so that you won't focus on it. What would you say or what would you ask Alexander Grothendieck were you to be able to have a dinner with them? I'm afraid what I would ask is for more particular detail about that afternoon, March 22, what exactly what he was thinking before he walked up to Sarah and says this will work in

all dimensions. It's a smallish question, but I don't have large questions about him. Laurent Laforgue has done a beautiful essay about major questions that are raised in Grotendijk's memoir. And if I understood them better, I might ask him what progress he could make on solving it. But I guess I would just want to, I would ask him something about his conversations with Sarah to just try to get more. Because I'm not trying to be an algebraic geometer. You may not know this, but my background is in math and physics and filmmaking. So something I aspire to do at some point, not sure when, is to do a film on the life and times of Alexander Grothendieck as for what form

that would take if it's a documentary or if it's a narrative, like a fiction, sorry, I'm not entirely sure. But I think he's a terribly interesting person. a lot of it is very hard to face, a lot of the unhappiness. I mean, what would you do with his, what he was, he was only in Le Chambon for maybe a year, I don't know, more, a year, give or take. How would you, how would you do that? How would you convey? I'll tell you, when I drove there, Le Chambon is south, I guess it's called the southwest of France, but not very far west. The train got me as far as, I've forgotten the name of the town. I rent a car and I drive 40 miles to this little village of Le Chambon. And it was a Renault Clio. It was a beautiful little car. I love it. It's solidly

built. I like small cars, just fine. Solidly built. And in my beautiful, solidly built Renault Clio, I feel like I'm driving off the face of the earth. I'm going up and up these tiny roads into the woods, and I feel like, mm-hmm, imagine doing this in 1942 as a 16-, a 15-year-old. Yes. You know you're leaving your parents to who knows what. You know there's a war on. There's no war on here. You're apparently not on planet Earth anymore. And the other kids feel the same way. And like I say, you can make it sound horribly ungrateful, and since it's the nature of kids to be ungrateful, but these are very sophisticated kids,

and they're being put up in a shack with cowherds where no one in the family can read, but the Father and He only reads the Bible. For one thing, it's hard not to be offensive trying to convey that, hard not to take cheap shots at these people, the country people. Jared Yes. people. And the kids did, you know, they're stupid, they're kids. Five thousand French peasants sheltered five thousand children and in the whole episode, they lost eighteen people. They lost only eighteen, sheltering five thousand children from the Nazis. Well, because there's only one road through this town. There's only one town down the road, five miles down the road, there's only one town

five miles up the road. If the Germans would come through, the pharmacists had phones. If they come through Thons, the pharmacist in Thons would call the pharmacist in Chambon and say, they're coming, and the kids would all go out into the woods and stay there for three days. That happened? Yeah. The kids, if there weren't Germans coming, they would go out in the woods and gather mushrooms, and now their meal will not only be nutritious, it will have wild mushrooms on it. mushrooms and now their meal will not only be nutritious, it will have wild mushrooms on it.

That you could make a nice scene of a movie, but how do you make the hiding in the woods a scene of a movie? I mean, it can be done. So a movie about someday there has to be a movie about Grotendieck. Um, of course there's this very popular figure of Grotendeek as lunatic genius, and you can decide how many women to involve in it. I hope you won't make that one, but you'll do it. Okay, I'm not going to tell you what to do or not do, but…

Now, speaking of telling people what to do, or at least giving advice, what advice do you have for people entering the field of math and philosophy slash logic? Do not be afraid of current mathematics. Current mathematics is not a shell game. Current mathematics, sadly, does include category theory, so you have to learn that beautiful stuff, even though you might have wished not to. Do not be afraid of the Grotendieck Revolution. There's no future in theory in the Grotendieck revolution. There's no future in fearing the Grotendieck revolution. Among philosophers, there are still people who think, well, that's too new. We don't, come on, new, 60 years old now. So, that would be my only advice. Don't fail to learn all

the classic stuff. Do not fail to learn Gödel's theorem, but don't fear to learn the new stuff either. And what's meant specifically by the growth and deep revolution? Scheme theory and presenting everything as a problem of cohomology. Cohomology isn't just one tool. Cohomology is the central tool. Interesting. Thank you for spending so much time with me, professor. Oh, thank you. It's been a blast. You're a great speaker, a great storyteller. No, it's only because I got good stories. Lucky me, I started studying Grotnick soon enough. Okay, well thank you so much. Firstly, thank you for watching, thank

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