Unity of mathematical work & outlook of FE Lawvere

0:00 And yet at the end, a highly non-trivial theorem is used. But I say to you, the fact is, that for me, there are 880 words that are easier to read than any two pages of the theorem. You can have two pages of Rietveld, two pages of Rietveld. All right, yeah, yeah, yeah. I gave the money to André for the... So then, usually it doesn't even prove their results. It's something that's extremely general and much more generalistic. Sometimes you just realize that they have anything to do with the results. And if we go again and again and again... It seems to talk about something else, but then in the end it didn't work. And this is, to me, one of the interesting things. It's a good place to drop that, and then take it particularly seriously. Sarah, this is exactly what I'm talking about. Every time you make a good argument, it doesn't go all the way to hell. But it works. Yeah, it really works. I think we're going to have to go through the first two pages of the book, because I think we're going to have to go through the first two pages of the book, because I think we're going to have to go through the first two pages. Let's look at that sort of thing, let's look at the future of that, because that is other. No, no, no, they tell you the only thing they want to make sure of now is that it will be seen as an item of value for a wide range of people like that, especially if you're a therapist and you want to help them. And I say to them, that's going to be my answer, it's the other way around. But, to get it back, it would be not going to be possible to do it with a lot of people.

2:30 Thank you very much. I think I'm a good kid. 10% of the time, I'm going to go to Levis. Oh, no, no, it's not at all. It's, you know, it's just, you know, it's a note. Yeah. And I... Is that, is that better? Anyway, there he is. Thanks. I'll get back to you in a minute. I'll leave. I should just bring you a note. Yeah. Yeah. I'll... So, where are you from? So... New York. I'm thinking... Well, I'll be... Denmark. I'm thinking... I'm thinking... I'm thinking... I'm thinking... I'm thinking... I'm thinking... I'm thinking... I'm thinking... What, why, why did I make that up? I don't know. Now, I'll be at the hotel at 9 p.m. before you check out. It will help you bring me in the room. If there is any, I'm getting ready to come out tonight. Then you can come in before I tell you how to do it. If not, you can have a seat. I'm sorry if I've done it. No, no, it's fine. I want to publicly give my gratitude to you, but let me express it once more privately. I really appreciate all the work you've put into this. I think it's very successful. Well, actually, I think it is very successful. I think it's picking up. Thank you very much.

5:00 Oh, what's it now? What is it now? I'm sorry, I just have 500 cards to put out there and you don't realise what's involved in being a world master of one of those elitism stuff. I'm sorry. I'm just saying we're in time, Graham. Oh, you're in time? I thought you said you're in trouble. Sorry. Yes, we are! I'm always in trouble. I never knew you were. So don't change that. Now these talks are being recorded, aren't they? Yeah, and they're going to be put on the service of Diffusion de Savoie, which is the website of the ENS, so you can listen to them all, and indeed watch them. I think it's almost unnatural because in some way his trying to articulate that unity. I think two themes in particular. One of them is that Laverre has a principle, which he doesn't put in these words, whatever is important to say can be said clearly. And more than that, what's important to say at some time can be said clearly by the conceptual means available or becoming available at that time. He takes this lately from Hegel and Marx, although he got similar ideas from Clifford Truesdell and Samuel Eilenberg earlier in his career.

7:30 Along with that is his interest in dialectics, which he's lately explained by talking about Grassman's view, not exactly his, that dialectics seeks the unity in all things and mathematics is the art and practice of taking each thought and pursuing it to its end. So dialectics starts with the general and tries to find this in the particular, mathematics takes the particular and tries to make it general, but this is not exactly Laver's conception. In Laver's conception, the road up and the road down are the same, that they are both, as they've been articulated the best we have so far, they're both best understood as category theory. Finally, the way to say something clearly is, today, the best we've got, is to put it in categorical terms. Laverre, as an undergraduate, was studying with Klipper-Trusdell. Klipper-Trusdell was doing his rational mechanics. The idea there was to produce rigorous foundations for mechanics that would solve specific problems in thermodynamics and continual mechanics. That the philosophy will come from the math and not the other way around. This was philosophy from the start for Bill, but the philosophy will come from the math. You won't have to take it to the mathematics. Truesdell, of course, being interested in thermodynamics, he's doing a kind of mathematics where there is a lot of need for the conceptual. Truesdell is looking for non-equilibrium thermodynamics. He wants to know what is entropy outside of equilibrium. For example, this is Hart and Trudell had issued a challenge to theoretical mechanics to cleave the stinging fog of pseudo-philosophical mysticism around ideas like entropy and information by coming up with a properly mathematical theory of it. Now this was part of a widespread project in the 1950s. A lot of people were trying to bring formal foundations closer to practice. Bourbaki, of course, they want a foundation along with their encyclopedic approach to mathematics.

10:00 John Kelly's General Topology, extremely influential book. An awful lot of people will tell you that the first math book that really excited them was Kelly's own set theory that will be adequate to the needs of his topology. And Kelly was originally going to title the book, What Every Young Analyst Should Know. This was meant to be an acculturation in mathematics. LaVere took this from him. LaVere was particularly moved by a quotation from Kelly. Kelly, through, he does local properties, he does a few global properties, and towards the end he starts talking about this other approach to topology associated with Eilenberg and MacLean. I think he says Eilenberg and MacLean rather than Eilenberg and Steenrod. I mean, he knows all the people. And he says that their approach The study of objects and maps might be called the galactic theory, continuing the analogy whereby the study of a topological space is called global. This, I think, we don't see this now the way that Bill did in the 1950s, let alone the way that Kelly did when he wrote it. We take galaxies for granted now. Of course, we've all sailed through them at warp speed. We know what that's like. Astronomers had only for about 25 years had clear evidence that there was anything outside the Milky Way. They've been inclined to believe it for a long time. But there was no clear evidence that there was anything outside the Milky Way until around 1920, I forget the exact. This idea that there could be lots of galaxies was a very new one for Kelly. Less new for Bill, Bill grew up on science fiction, so Bill can sail through these things in his mind, even though you couldn't do it at the movie theater yet, but he was very taken by this image, pointed out to him that what Kelly was talking about was called categories, and so he went off to try to learn some category theory, and we're talking now, what, about 1958, it's no easy thing to go off and learn some category theory. Sort of catch as catch can in various research articles.

12:30 But in particular, Kelly has, this remark he makes is in a series of exercises around functional analysis, LaVier presented these in functorial terms in Truesdell's seminar, as an undergraduate, but he's in a graduate seminar, and he then became very interested in putting topological dynamics in categorical form, I mean there was this idea that some number of dynamical questions could be understood As invariant under continuous deformation, right, so we're not really going to have energies or velocities, but just equilibria, and there were, Gottschalk and Hedlund in particular had a book on this, Bill was interested in it, but he wanted to make it properly topological, and he wanted to do that by making it categorical, he discovered Godemont, and as he says, Godemont's book became my bible. This was the right way to do topology. Actually, when you look at the particulars, it's not easy to connect Goethe's chief theory with topological dynamics. This was a project very much in formation, but it was Bill's project. So, Truesdell wants to encourage Bill. I mean, he's very impressed with his work. But he does say to him, look, you're not really a physicist. What you need to do is go work on the mathematics, and you need to do this at Columbia with Eilenberg. Eilenberg is the one who can take you farther in this direction, though got there, and he heard a confirmation of Trudell's philosophy in one of Eilenberg's slogans. Eilenberg was a good one for putting things in a few words. He never explained anything. If a theorem needs an explanation, then it wasn't a good theorem, or it wasn't well proved, or it's in the bad or wrong concept. In a way, to me, this echoes the remark someone gave of Eilenberg saying, the theory and the example should be the same. The theory should be transparently motivated by the example. The theory should utterly penetrate the example. You shouldn't need an explanation.

15:00 So we have Alex Heller explaining about Sammy Eilenberg. Sammy considered that the highest value in mathematics was to be found not in specious depth nor in overcoming overwhelming difficulty, but in providing the definitive clarity that would illuminate its underlying order. This was to be accomplished by elucidating the true structure of the objects of mathematics. And he points out this was not to be understood in any philosophical or metaphysical sense. Sammy was not a Platonist. Equally, he was not a non-Platonist. He was interested in making everything as clear as possible. In categories at this time we're expanding across mathematics, because we're in, what, 1961, I believe, now. Eilenberg and Steenrod had made them the standard framework in topology, and not even just in the 1952 book, but already in a note in 1946 in the Proceedings of National Cabinet of Science. It just gives the axioms and asserts the unicity for simplicial spaces of called axioms. So they made it central there. Cartan and Eilenberg had made category theory central to all of the subjects in Cartan, Eilenberg, abstract algebra, Liebhoud theory. By this point, of course, Grotendieck's ideas, while his functional analysis was established, the value of categorical methods. I did want to mention that around the same time, Frank Adams used Steenraas and Grotendieck's ideas to solve the vector field problems for spheres. Everywhere independent tangent vector fields exist on an n-dimensional sphere. Well, on S n-1. You take S1, the circle. Well, there is a non-zero vector field on that. Obviously, there are no two, because it's a one-dimensional manifold. How many in general? What is the maximum integer Km for which there exists Km, oh, I forgot to say independent, everywhere independent, everywhere pointwise independent, continuous vector fields on S n-1? And he had solved this using Grundy's and Steenrod's ideas. At Columbia, this was just obvious that category theory would solve classical geometric problems. It tells me that when this was presented at Princeton, everybody was,

17:30 wow, you use category theory and you get geometry. But to Eilenberg, in this climate, his special contribution was to say he saw more than anyone else that simple ideas also have clear categorical versions. He found a categorical Peano postulate early on to define the natural numbers. There are set M. It's got a successor function on it. It's got a selected element zero. But these are not sets. Well, that's a thoroughbred ambiguity in Bill's work. Bill will call the objects of any category sets. He does not mean what set theorists mean by sets. They're things. Things are sets. Things aren't necessarily like what the logicians think sets are. We'll get more into that. But at any rate, they're a thing with a successor function and a zero. And here's their key property. If you take anything with an endomorphism and a selected element, there's exactly one function from the natural numbers to that thing that tracks. It takes zero to x, and it takes the successor of zero to f of x. It takes each natural number, and to the nth. And this, this entirely characterizes the natural numbers. Now in what way? In some sense, what I've told you so far doesn't characterize them at all, because I haven't told you where it lives. But once you decide where it lives, it defines the natural numbers up to isomorphism in that category. It defines them up to isomorphism. If you have any two things like this, well, there's a function here from this one to that one because this is a natural number object. There's a function here because this is a natural number object. But this composite has the natural number property and so does the identity.

20:00 And so that composite is the identity. And so these things are inverses to each other. Determines not a unique natural number object, but determines it uniquely up to isomorphism, up to a unique isomorphism. So, the invariant statements about the natural numbers follow from this definition in a given category. If you do it in a category of sets, all the isomorphism invariant properties of the natural numbers, of the usual natural numbers follow, because the usual natural numbers obviously satisfy the definition. And so everything that does is equal to that up to isomorphism. Later Bill saw how you could, in fact, have much of familiar arithmetic with, of course, in steps, a large part of primitive recursive arithmetic follows just from the category axioms, plus this. More follows from the Cartesian closed category axioms, plus this. More follows from the axioms for a category of steps, plus this. He saw at around the same time that functor categories are the key to characterizing the category of categorism, and Pierre has said some things about this that you might have thought that since so much of category theory involves constructing functors or even the functor category from one category to another, you'd say, oh, well, that's a higher order idea. And from some point of view, yes. The functors are in some way maybe higher order than the objects, but what Laguerre finds is precisely that there is a first order characterization of it. This really isn't leaving the first order, as Pierre has been saying. We can describe, we know what the functors to it are from any category C. What is a functor from a category C to this functor category? Well, it's just a functor small g from the product of C with A to B.

22:30 This is already familiar as lambda conversion. In thinking of these as sets, if each element of C is associated to a function from A to B, well, you might as well have said, look at functions of each element of C and an element of A and assign an element of B. You can define a binary function either of those ways. But La Vieux finds that this works, not only does it work formally in the category of categories, but he can derive all important facts about functor categories from this. Plus a few simple other facts about categories. And this, I think, is an example with this idea that the theory should not mean the theory should only have one example. I suspect that when Eilert saw this, he says, right, this explains what we mean by exponentiation in categories. This example of exponentiation has all the key features, uses them all in the key ways. This is in some sense fully exemplified by functor category formation in the category of categories. You see the whole theory there. The theory changes your view of this because until LaMere showed this to people, functor categories looked complicated because they've got all these commutative diagram squares in them and you have to make sure that naturality means what you wanted it to mean and you have to worry about composition in one direction and also composition in another. And Bill says, no, that all follows here. He's not throwing that out. Of course the naturality squares are still here. But they follow. What's a naturality square in a category B? Well, that's just, let's see, the arrow category. So we take our category A, we form the category of squares, and then map them into B. Well, that's just a map from the arrow category into the puncture category. It's an arrow of the puncture category. So the whole nuts and bolts explanation, which it seemed obscure to people, follows from this when you want it, but the definition is a simple adjunction. Well, obviously in the length of this talk I'm not going to prove it's simple, but it is. This, I should say, functions if you're in the category of categories, but it gets all their properties, again because it determines the metahisomorphism.

25:00 This is one of them, and that's one of them. They have these arrows between each other. These arrows have to be inverse. This is a complete characterization of functor categories for the same reason that the natural number axiom was a complete characterization of the natural numbers. It's the same proof. So he's got this idea that simple things have categorical definitions, but we're still around 61, 62. Even his project when he went to Columbia, he didn't plan to do rational mechanics at Columbia. He planned to learn category theory there so that he could do various kinds of mathematics categorically and eventually do rational mechanics this way. This was a long-range plan already at the time. And what he ended up doing, he took a logic course from Elliot Mendelson. In fact, it was one of the two courses where Mendelssohn created the notes that became Mendelssohn's textbook on logic. And as he was taking it, he's got a notion that morphisms of category of models look like natural transformations. I mean, it's not a precise theorem yet, but it looks that way. So he goes to California and he just goes and he sits in on a lot of logic seminars at Berkeley without ever registering at Berkeley. It sits on Tarski's seminar, Michael Raven, Dana Scott. In particular, his attention was called to universal algebra. Universal algebra is going to study theories like, say, group theory that are axiomatized just by operators and equations. So the group axioms would just say, there's this unit E, multiplying anything by it gives you that thing. There's this operation of taking inverse, and the point is you multiply something by its inverse, you get the unit.

27:30 This multiplication is associative. I suppose that with these axioms, I need to say the unit... Does it follow that that's a two-sided inverse? Yeah. Okay, so these are already, these are already, yeah. He gets interested in theories that can be equations, and he says, well, look, an operator is an arrow, and an equation is an equation. The theory of groups is a category. The theory of groups just is a category. This is what becomes his dissertation. He lays this out. Okay, so what category is it going to be? The key is that an operator is an arrow. An operator has an arity. These should be arrows from 2 nests to something that's a 1 nest. The inverse operator, that should be an arrow from 1 to 1. So if you like, the types are tuple lengths. There's a type of 0 tuple, a type of 1 tuple, 2 tuple, 3 tuple. The operator's element is value. So these operators conceived that way, go from two poles to one, satisfy various equations. So he does present an algebraic, so a theory, looking at it categorically, or even just looking at it a priori, I just said, you know, a theory is a category, and I wave my hands about that, but was it doing anything? The point is that a model of that, you don't just define something so that you claim it's there. Type 1 goes to the line set of the group. It's Cartesian squared. The operators devout to elements, to elements, elements to elements.

30:00 This operator satisfies all the equations. The equations were commutativities. They're all satisfied. And this generalizes in the category, I call it top, of topological spaces with continuous maps as arrows. We simply define a group to be a functor from the, a product-preserving functor from the theory of... In the category of topological spaces, these turn out to be the ordinary topological groups with their topological group morphisms, and that a morphism is just a natural transformation between these markers. If we define a group in a category of differentiable manifolds, we get the Lie groups. These are examples of structured sets. This is a discussion that philosophers of math will be familiar with. Are the structures of mathematics structured sets with structure-preserving functions between them? These are. These things have underlying sets. The morphisms between them are functions between those sets. And this was also not a new idea. Mathematicians have long said a topological group is a topological space with continuous group operations, and Lie groups are manifold group operations. That's for the examples I've given so far, but it's not everywhere. In precisely an algebraic geometry, as Delenia says, to construct a scheme, one generally does not begin by constructing the set of points. We want to know what is a group scheme. Well, it's nonsense to find a group scheme as a scheme with group operations on its points. But what a group scheme is, is a functor from Lavier's theory of groups to the category of schemes. Okay, schemes over some base, whatever. I mean, whatever category of schemes you're interested in. These are commutative diagrams. That's what functoriality from this theory is, is that these diagrams should commute. A group scheme is a scheme equipped with a multiplication, a binary operation of multiplication, and a unit, and a selection of a unit that satisfies these laws.

32:30 Now, again, this is not new with LaVier either. This idea is already in the air. What's new with LaPierre's dissertation, though, is formalizing those as logic, as the model theory. He brings it into logic that way. And bringing it into logic, as he says, from this point of view, many other algebraic constructions are viewed in a unified way as functors adjoint to algebraic functors. Now that theories are categories, we can talk about functors between theories. And functors between theories turn out to be exactly what logicians have classically called interpretations of one theory and another in the algebraic. Now, pre-categorical logicians were a little more concerned about what an interpretation even is for rich or first-order theories. But this is quite standard as the definition of interpretation for an algebraic theory. But an interpretation of, say, the theory of groups in the theory of rings gives you a way of looking at each ring as a group. Say we interpret the ring as the group of units of some ring. That takes each ring to its group of units. An algebraic function is always the one theory that's induced by an interpretation. These LaPierre proofs always have adjoins. These are typically things like forming free algebras, forming, say, group rings, which is the free ring over a group, so that LaPierre's viewpoint is unifying all of this. If you have a semantics functor, this one is reasonably familiar to logicians before Loewe here, you take an algebraic theory and you look at the category S of A, that's a script, S of A is the category of models of the theory.

35:00 Precisely because we're categorical, we notice that each interpretation, that is, functor from an algebraic theory A to an algebraic theory B, gives in the routine way a functor from models of B to models of A, since after all these were just functors from B and those were functors from A, so we just composed, nobody could look for before, an algebraic structure functor that goes the other direction. We take each category, the algebraic theory of it, in general this is not the category, but we find the algebraic theory that's the best possible match to it. And this is revealed in exactly the way you would like it to, as an adjunction, so that it oversets, there's a detail, I'm skipping there, a functor from any category of a theory A is just the same as an interpretation of the theory A in the algebraic theory of that category, so that in some sense we know all about these, but of course the particulars remain to be found. So here's another theme in Lavier's work, levels of organization. You can organize a theory as a category. Good. Okay, now that organizes its models as the functors. More good. Now that organizes interpretation as simply composition of functors. Also nice. Now, however, we've got a functor from the category of all theories to the category of all models of theories. In fact, an adjunction between the two. And this can be taken up to higher levels, no doubt, which I don't have on my mind right now. But this idea that things will keep going on and on and on up. Essentially, every theme in Laguerre's later work is presaged in this dissertation, including that he says in the dissertation that this suggests a possible principle of philosophy. We have the structure bunker that gives us their algebraic theories. It takes a concrete to something more abstract.

37:30 We have semantics that takes a theory to its category of models, and these are adjoint. This diagram actually occurs in the article that's been mentioned here, Adjointness and Foundations in Dialeptica, except that it occurs sideways. It's not a road up and a road down in that article. This without mentioning that lots of other kinds of categorical logic were then formulated on the model of Bill's dissertation. This was for theories presented by equations. Okay, what about equations, um, details. There's too many of them. All theories we might look at, and many of them have been studied this way. The result I most like to talk about is on first order theories, and that's Michael Mackay's stone theorem. It's often remarked that the category of models of a first order theory There is no way to determine that theory at all. It's radically deficient. But a first order theory doesn't just have a category of models. It has an ultraproduct to form the ultraproduct of any family of models of that and get another one. So suppose we take abstract categories and equip them with an infinitary product operator that satisfies a few of the formal rules of ultraproducts. Is this good? Can we look at this and tell whether or not this is the theory of models of some first-order theory with its ultraproduct? And Mackay's Stone theory says, yes, exactly. You can take any category with a candidate product on it. You can find the best fit to it by a first-order theory. You can take any first-order theory, get a category of models with an ultraproduct. Not just the category of them with the elementary embeddings, but that category plus the ultraproducts. To do this with all these levels of organization that Bill first wrote up, his ideas that he'd had for a long time, on the elementary theory of the category of sets.

40:00 He starts telling people that he's going to do set theory in categorical terms, and this produces the story, which Saunders liked to repeat a lot, and which we've heard here, of Saunders saying, no, Bill, you just can't do that. Set theory is about elements, and you have to have a set theory about elements. He points out his set theory always had a set theory. But I think a more important thing to know about McLean, McLean hears this project, says, no, Billy, you can't do that. What did it take to convince him that you could do it? It was not some long exposure. He read, yeah. Being told there were such axioms, he doubted it. Seeing there were such axioms, it's a fairly general lesson. If you want to evaluate a theory, one way to do that is to... This work stays on Bill's mind for a while, and when we come to the Dialeptica article, he's interested in foundations, and foundations from two points of view, which we see already. Elementary theory of the category of steps. There was a slide about that off this morning where he explains what the theory does. I almost wish that I had a slide of the same quotation. But he says very clearly from the start that his elementary theory of the category of steps accomplishes two things. We take the category of sets as given to us, and we look at these axioms and say, here we have concentrated what we need to know about them. The things we need to know about this theory follow up from these axioms. And this is foundations in terms of what he calls the study of what is universal in mathematics. We have a very tight description of the category of sets, which is in some way... Universally present in mathematics, foundations in this sense cannot be identified with any starting point or justification for mathematics, although it might be interesting, because on this conception, we're assuming we know all about sets.

42:30 What we looked for in these axioms was description. We knew about them, and we asked whether it's description. But he already says in his publication in the category sets, another way to look at it, you could start from these axioms. And from them deduce the major theorems of number theory, analysis, geometry. That's foundations in the sense of a starting point. It's very clear from the start that his axioms could be used as a foundation in that way, or might only be taken, but that these are different senses of foundation. Again, I think in terms of philosophers' discussions of foundations in mathematics sometimes get confused on this very point. We'll say, is this a good foundation? The actual mathematics reflects the universal. It's not foundations in this sense, because if you think you've got a mathematics on hand, then evidently this is worth distinguishing the two. He notes that being itself part of mathematics, foundations also partakes of this formal conceptual duality that he's interested in. In its formal aspect, foundations is often concentrated on the formal side of mathematics, giving rise to logic. But more recently, the search for universalists has also taken a conceptual turn in the form of category theory. So here's, again, this duality. He's got formal foundations has been about theories, but it could be about the categories that they characterize. It doesn't have to be about the categories. If you want to approach it in any articulate way, you're going to need category theory, because the only thing that's got a good universally oriented description of those realities is both the theories. It is true that LaVere's approach to algebraic theories was always about their categories of models.

45:00 Logicians before Bill would talk about the class of models of a theory. Now they knew what the elementary embeddings between those models were, and particularly if these are models of an algebraic theory, they knew what the homomorphisms between them are. But they thought of the homomorphisms as derivative from the things themselves. You've got the real models, and then subsequently you ask for the maps between them. Bill's orientation was to take those maps between them seriously in the first place, and that's what gives him this adjunction when he says here, if we deal with categories of models in the first place, these will determine their own full sense of natural relation variables. If you take the class of models of an algebraic theory, I mean, take the class of groups, the class of rings, as classes, they're both just proper. There's nothing more to know about them. There's no hope of characterizing the group theory. By knowing that it has a class of models, it determines the operations of that theory. It does not determine which will be taken as primitive. It does not determine whether you will regard the inverse operator on groups as an operator or as just a property that every element has an inverse. It does determine there will be an inverse operator, whether you choose to take it as a theory or not. It is the category of models that determines its theory, not at all the class of models. He works on logic for a long time and I've just been aware, listening to various talks In my crazy of his career, I've neglected how much time he spent in the 60s working on different doctrines of logic and their model theories. The idea of hyperdoctrine, the word hyperdoctrine does not occur here except handwritten as a result of other people mentioning it. Where he goes on to next from that logical interest, He then pulls all of his interests together, all of his interests in logic and dynamics and foundations, he pulls them all together into a huge project based on Grotendieck's algebraic geometry, which he'd been hearing about in Switzerland.

47:30 He learned the latest on that at the ETH in Zurich, and this gave him a project. Later described in three steps, which he listed in reverse order, when he was right to do that. In the spirit of Walter Noll, a Truesdell student and an extremely successful one, and we were going to do this on the basis of a direct axiomatization of the essence of differential geometry. We're going to get differential geometry a lot more simply using results and methods of the French work on algebraic geometry, some of which I learned from Gabrielle. This is a plural. This is Grotendieck's French plural of topo. Well, if people pronounce the S one, I... See, I get too much by reading. But I take a peak. This is topo, so it's not the singular topos. It's the toposes of Grotendieck. Since the most natural form of two is incompatible with usual set theory, he's going to come up with a wholly different approach to sets. These are going to be smooth sets. There are going to be sets, well, as he says here, category smooth sets. These are going to be similar to algebraically structured sets from Grotendieck. What he's looking for here, he wants to axiomatize the idea of a topos. Now, work was already going on. This wasn't a prerequisite to doing continual mechanics in Noll's way. This was meant to extend it and make it more unified, more powerful. This axiomatized differential geometry, hands down the least plausible step, was the first one. Grotendieck's toposes knew something about them. They knew they were orderly, but they were very large and they seemed rather elaborate. Some people found them simple in conception and some people not, Grotendieck did, and powerful in geometry. They had a lot of, a very complicated nature in terms of their set theoretic nuts and bolts, but whatever it's important to say can be said clearly.

50:00 Laver was convinced this was important to say, and it could be said clearly. And it could be said clearly by the same tools that had created the problem. The methods of category theory, in particular adjunction, were going to give the answer. He had no idea just how complicated the answer might be. He told me that he and Tierney at one point thought there might be 20-some axioms for a topos. But they eventually arrived at its terminal. Every object has a unique function to it. That's all you need to know about it. If you want a picture, think of a singleton set. It's got one element. Every set has exactly one function to it. Take everything to that. But we don't say this is a set with elements. We just say every object has exactly one. And for any object A and B, there's a product. I'm going to put the definition of product up in a minute. For any object A and B, there's a function object. Representing all the arrows or all the functions, if you like, from A to B. It's the same diagram that I use for functor categories. And there's a truth value object with a selected element true. Every sub-object has a characteristic. Okay, that one, if you haven't seen it before, you probably don't like it much right now, but we'll get back to it. These are elementary axioms. Their first order, they're just about objects and arrows between them. They don't, we commonly say they don't use set theory, and this is true in a straightforward technical sense. They're first order axioms. On the other hand, from LaBierre's point of view, they are set theory. These are all sets. Okay, just using, he's using set more generally, and he's got historical antecedents for that too.

52:30 The point is that these let you then do a huge amount of mathematics in fairly ordinary terms. You take the description of, oh, I left out natural number objects, you're not going to get real numbers without it. Take any textbook on analysis that gives it a construction in set theoretic terms, you can pretty much reread that in these terms, get the Cauchy real numbers in your topos. Now, there are some technical differences in how it works out that you need to be aware of, and Grotendieck was already aware of this. Grotendieck went to a logic conference in Denmark just around the time that Lavier and Tierney had finished off these axioms and said this very thing. He said, a topos, this should interest logicians because a topos is like a universe of sets, and you really should think about it as a universe of sets. He thought this without forming the first order axioms. Grobnik had the axioms, but commented that he never understood what Grotendieck knew, because he knew the Laguerre-Tierney axioms. That's what persuaded him. Grotendieck didn't know those axioms. How could he think this without pointing to Laguerre-Tierney theory? Grotendieck had his other way of knowing about it. But in particular, he can now talk about a topos that's got other kinds of structure. The second step of his, he's now achieved the first step of his program. He now has axioms for Groton-Deke toposes. In short, how close are these to Groton-Deke toposes? It's, to me, the nutshell description is a category satisfying these axioms need not be a Groton-Deke topos. For example, the one object category that has only the identity arrow satisfies all of these axioms. Gee, that probably is a Groton-Deke topos on an empty site. Okay, but there's other ones. Things that are trivially not Groton-Diktopos. Any category that satisfies these axioms and is the right size is a Groton-Diktopos. There's actually two dimensions to being the right size. The collection of arrows between any two objects has to be small enough, and you have to have large enough coprograms.

55:00 But then it is a Groton-Diktopos. Yes, yes, and it also has to have a small enough collection. And then maybe it's not even fair to call that just a size. Now let's consider a topos. It's got a ring R in it, and that's the definition here. This is a ring R, so it's got addition, multiplication, zero in a unit. And it's got a subset D of the elements of square zero. D is just the set of elements of this ring whose square is zero. You might think of it as a little infinitesimal interval around zero. We haven't defined any ordering, so where can we get off calling this an interval? It has this interesting property that every function from it to... Here I've got the arc that extends the embedding, but it's linear. If you give it the point at center dot A, give it a slope, you've uniquely determined it. Every function from D into this ring has a constant part, a linear part, and is uniquely... So you take any function from the ring to itself. I'm going to draw that as a graph. I'm going to think of this ring as the real line. I'm going to draw its graph. What's its slope at a point? Well, take an infinitesimal neighborhood around that point. On the infinitesimal neighborhood, it's zero. It's linear. It has a well-defined slope. That slope is the derivative at that point. It has to have a slope because that's the nature of infinitesimal intervals, is that maps from them to the ring are linear. The function ring to itself has a uniquely defined derivative at every point, including that function has a derivative. So functions here are all smooth. What's not obvious from this description becomes obvious on a little bit of thought is this cannot possibly...

57:30 There is no ring in the category of sets that's all like this. As a matter of fact, in this topos, the law of excluded middle can't even succeed. It cannot be the case, it simply cannot be the case, that this is true is either equal to zero or unequal to zero. If that case of excluded middle holds, there have to be elements of d that are not affirmatively zero, but also you cannot deny that they're zero. So we need a lot of excludimental to fail in this topos, toposes that have this kind of a ring in them. Now this lesson, then from here he goes back into any continuum mechanics. Continuum mechanics is a mathematical theory of a placement of a body in a space S. We're going to map that from B to S. They're both spaces. This body is a space. The set of all possible placements of this in the room should form a function space. Now, what is the set of all possible placements of this in the room? You have to pick what you're interested in. Am I allowed to bend this? Right? I mean, physically you could bend this. That would give one theory of placements. It would be more polite of me not to bend it. That's going to give another theory of placements. Both of these theories exist. You decide what your parameters are. But the set of all possible placements should form a function space. And everybody thinks that it does. Let me not get into those yet. About the physical properties of different placements. Let me do this. I'm not really bending it, but I'm stressing it some. Now, this thing has a total, a different total energy, when I've stressed it, than when I haven't. So knowing the details of the placement should give me, so each placement should have a total internal energy for this, for this body.

1:00:00 It's got this heat radiating over here, it's going to cool off. So placements have associated with them, obviously each placement has a different, they're not all different from each other, but each placement has a gravitational potential. So we want to study placements of this object in space. That means we want to study this function set from the body into space. And we might like to look at parameterized families of placements. Of course, we can look at these as just differentiable manifolds and then look at the function manifolds between them. But there's lots of different definitions of a function space for differentiable manifolds, depending, again, on what kind of model of it you want to use. What differentiable structure do I place on this function set? What continuous structure do I place on it? Simply the adjunction. That's all I'm going to say about it. It's simply an adjunction. Now, so for example, this should have been r. Think of r as, think of this as the real number line. Think of that as times. So a math like this takes each moment of time and gives you a placement of b. That's a history of this thing as it moves through time. By definition, we can regard that as a function that takes each moment of time and each point in this thing and tells you where in the room it was at that moment. Or another way we can look at it, we can take each point of this thing and map it into the set of functions from R to S. Take any point of this thing, look at the trajectory it followed. Now, I think, rather than try to explain that in any more detail, let me move to the philosophic with this.

1:02:30 This is, from a set theoretic point of view, we're at mathematics. From LaVier's point of view, it's as simple as three different ways of looking at placements varying through time. LaVere does not want to give a philosophical motivation for this. LaVere wants to get the philosophy from the mathematics. He wants to conclude from this that a lot of, say, fairly robust, like 19th century mathematicians talked about movement through space in these ways. Euler, well, of course, you look at Euler's calculus of variations. Euler does calculus of variations in a very robust, freely manipulative way. Historians of math are apt to look at it and say, well, that wasn't really rigorous at the time, but it's really true that it wasn't. But Bill is saying, but it's just this. He's just freely using either one. What he's saying is that you can give these foundations that are this close to mathematical practice. What Vaubir is, is that he doesn't find, well, I'm sorry, I'm getting a little out of order. Loeweer concludes from this that it's a good thing to bring rigorous foundations closer to practice. He doesn't really argue for the claim. It's just a transparent good to him. He gives a few arguments to show that we want foundations that are closer to practice. What he doesn't do in his foundations is find new things to say about sets. He does not find new things to say about sets. He finds less things. What I mean by this is About the product, for example, what do we, I put up those axioms and said in a topos any two objects have a product. What do I mean by a product? The product of sets S and T, which now could be kinds of things other than classical sets, is simply another set which we'll call cross T with a pair of projection arrows.

1:05:00 This is the product. Well, it's not any set with projection arrows, it's the universal one, by which I mean, given any set A and a pair of arrows to S and T, There's a unique arrow down here that projects this way onto F, projects this way onto G. He simply said less about the product. It is a theorem of Zeraylo-Fraenkel set theory that every two sets have such a product. So he hasn't said anything new. His topos axioms are all, if you interpret them in Zeraylo-Fraenkel set theory, they're all theorems. What he has said is much less. The Zeraylo-Fraenkel set theorist is going to say, but you haven't really told me what set this is. Typically, in my experience, they'll say, you haven't told me what set this is until you've told me it's a set of ordered pairs of elements, one element of S and one element of T. You have to tell me what its elements are, then you've told me what set it is. It's a set of ordered pairs. Now, what's an ordered pair? Well, we don't have to say that. And so then I say, yeah, but you haven't told me what an ordered pair is until you've told me what its elements are. Now, every Zermatt and Frankel set theorist knows lots of ways to do that. Precisely, lots of ways. Do they pick one? No, they'll tell you there's no need to pick one. Just say it's a set of ordered pairs. But don't just say it has projection arrows, that's too little. They do. We define, in categorical foundations, we define this product... By the same proof as I showed you for natural numbers, for function sets, the same proof, we've only defined this thing up to isomorphism. As far as I know, most Zermelo-Frenkel set theorists in most conversation really only define it up to isomorphism themselves, except they declare that the elements will be called ordered pairs. Now, they have the option of also defining ordered pair. Now they've defined it uniquely in Zermelo-Frenkel. They usually forget to take that step. But what LaVere said is, again, he's not going to say new things about sets. He's going to say less things. You didn't have to talk about ordered pairs at all. You had to talk about the projection functions, no matter what. You're not going to be able to do math without them. You didn't have to talk about the elements of the set. And in fact, you don't talk about them in any serious way.

1:07:30 But what LaVere finds is, what the set theorists will say is, for rigorous foundations, you would have had to. One of your sound was, no, you can get perfectly rigorous foundations without going into that. He does say, he gives some reason to think that, he articulates some reasons. Let me put it that way. As I say, for a Bill, it's obviously a good thing to bring foundations closer to practice. He doesn't articulate a lot of reasons for why that should be true, but he does. As he says, to set up the problem. From the ongoing investigation of ideas and sets and mappings, you can derive a few statements called axioms. Experience has shown that these statements are sufficient for deriving most other true statements. Just the categorical definition of product, for example, together with a few other things, lets you derive the classically true statements about them. I think I'm going to let that draw. During this time, he's seen this galactic theory in topology. Just a reference in Kelly. It's led him into Eilenberg, McLean, Steenrod, Godemont, Grotendieck topology. It's worked. He's seen Grotendieck's derived puncture cohomology. He's seen Grotendieck's abelian categories. We've seen some references to those schemes, which I don't know if we've really discussed. And in each case, he's found that whatever's important to say can be said clearly. These things can be very tightly summarized in axioms. For Bill, this confirms what Marx says, it can solve, since closer examination will always show the very problem arises only when the material conditions for its solution are ready at hand, or at least in the course of formation.

1:10:00 This is actually a link between its commonly remarked how optimistic Grotendieck is in the 50s and 60s. It's clear that if the bank conjectures look like they have a cohomological solution, they do have one. And that cohomological solution can be put in extremely elegant terms, which of course for Grotendieck may be 500 pages long. But it's going to be a clear 500 pages about that same optimism. Whatever it's important to say can be said clearly. One conclusion from this is that foundations and applications depend ultimately for the existence on each other. Foundations and applications are the same things. Foundations is derived from applications by unification and concentration, in other words, by the axiomatic method. The philosophical interplay between Bill and Saunders, they affect each other a lot. As Steve remarked, Saunders' return to logic was largely because of his encounter with Bill. The two of them have had a lot of influences on each other, but they do disagree about the nature of truth in mathematics. Saunders has said repeatedly in his book, Mathematics, Form, and Function, it's the theme of the whole last chapter. We should think of mathematics as correct. Statements of mathematics falsifiable. He cites Popper in this connection. We have all these geometries, lovely geometries, just like the classical non-Euclidean geometries, just hyperbolic Euclidean elliptic geometry. And we can make any one of them work in space. Now, of course, this is a debated proposition, but Saunders takes it as clearly true. We just select the right metric, space will turn out to be Euclidean. We select the right metric, space will turn out to be hyperbolic.

1:12:30 Which of them is more convenient is a question of physics. Saunders is disjoint from a lot of the debate over this. Saunders doesn't care whether each of them is merely a convention from the physicist's point of view, because he's not taking the physicist's point of view. He does state they're merely conventional from the mathematicians' point of view because space can be interpreted in any one of them. So, none of them is falsifiable, so none of them can be true. Euclidean geometry can't be true, hyperbolic geometry can't be true. Maybe a physical theory of the Euclidean nature of space, but that's not Saunders' job. Or of a non-Euclidean theory. Per Lavier, mathematics is true. And as he had up there, past experience directs us to it. And this ought to be reflected in the philosophy of mathematics and in teaching mathematics. Because foundations is derived from applications by unification and concentration, the two are in constant interaction. La Vieira certainly gives the idea of foundations as an actual starting point for mathematics. Saunders has said over and over again, foundations for mathematics are proposals for the organization of mathematics. None of them can be the starting point. Obviously, Zermelo-Frankel-Seth theory is not the starting point in mathematics because there was mathematics before it. But it's a proposal for organizing mathematics. For LaVier, this should also be true, but he wants to take this, he takes a more aggressive matter of applying this. For example, Bill has his ideas on the axiom of constructability. Penelope Matty has talked about this. We've got Zermelo-Fraenkel axioms for set theory. As commonly remarked, they're incomplete in many ways. The axiom of constructability claims that all sets are constructable in a certain way. Now that's constructable by transfinite induction. It has nothing to do with constructability in any kind of Brouwer philosophical sense. But it has set theoretic consequences. Are we going to take this axiom or not?

1:15:00 The continuum hypothesis follows from that. We know what the power set of the natural numbers is. The generalized continuum hypothesis follows from that. The axiom of choice follows from it. Set theorists are interested in the continuum hypothesis. Should we accept the continuum hypothesis or not? For the most part, set theorists have no interest in the axiom of constructability. They're just going to throw it out. They give some philosophical reasons for this. Some people have mentioned that if you didn't throw it out, a lot of questions would be settled. But as set theorist Keith Devlin has remarked, the action of constructability relates to other questions in mathematics. You can relate it to serious questions in mathematics that arose outside of set theory. You've got the Susslin problem. What does it take to describe the continuum? Well, it's not hard to give definitions of the continuum, but our basic definition, we say, we're going to take a countable set, order it like the reals. They're dense in the continuum. The continuum is an order closure of that. Did we have to talk about a countable dense subset? Could we have done this in terms of patterns of overlap of open intervals instead? And this is the Sousolin hypothesis. You've got a description of the continuum in terms of just overlap, not talking about a countable dense subset. It's not, does it describe the usual continuum? Independent of the axioms of set theory. The axiom of constructability decides, it says no, and as Devlin says, okay, if you think that that countable dense subset should be crucial, then you should like the axiom of constructability for this reason because it says it was crucial. It says you didn't have a description without. For Bill, that's a decisive kind of an argument. That's the kind of argument we should be taking. Candidate axioms should be evaluated in relation to all of mathematics. This is not a popular position in set theory, or most philosophers of set theory, who want to evaluate candidate axioms just in terms of technical problems in set theory, though has a very different approach, because he wants foundations and applications to depend on each other. Foundations shouldn't be studied just in terms of foundations. They should be related to all of the applications.

1:17:30 Let me just come to my concluding line on teaching. Bill does believe that mathematics teaching should reflect foundations with the new math fiasco where we had this idea we're going to take a sort of a Borbachian conception of mathematics we're going to teach it in elementary school and this way kids will really understand math. From Bill's point of view, the problem there is not with the idea. The idea is fine. It's not even that it's Bourbaki's foundations instead of categorical foundations. It's that it wasn't understood. The teachers did not learn the Bourbaki's foundations. They were not taught the foundations. Math teachers don't get a lot of preparation, and they didn't get any preparation in math. And that a less speculative philosophy based on the actual practice of mathematical theorizing should ultimately become one of the important guides to mathematics. So, my question is, actually, when I read Laguerre's thesis and done this 2003 paper, it seemed that actually there is some change of Laguerre's view on the foundation. Of course, the thesis looks, say, more classical in the sense that it still uses the first order logic, and then he makes this kind of conceptual clover, I mean, categories, categories. Which, of course, this 2003 paper, but also it seemed that somehow he came too much. Do you think he really changed? Yeah, let me just outline. No, it develops with time. In my thinking, I wouldn't see that so much as a change. It becomes more flexible, and it becomes more flexible largely because of the discovery of the elementary topos axioms and their range.

1:20:00 In 1963, he just hadn't conceived of a lot of the phenomena that he was later going to see rigorously explained, and so, so he comes to have a more flexible vision just because he's seen the possibilities. At the same time, it's... When you say it has a classical look, that dissertation spends a lot more time, say, with the category of sets than a lot of his later work does. But he's also returned to that, and he's looking at focus into the category of sets. Already in 63, it's not that he thinks the category of sets is the foundation for mathematics. He does think that models in sets are the simplest kind of models in the group axioms. He's doing some model theory here, and the simplest model theory is in the category of sets. Now, he mentions that there's a model theory in the category of smooth spaces and a model theory in the category of topological spaces. For that matter, a model theory in the category of rings, right? You have groups in the category of rings. But it's still true that the category of sets has a central place for him. And as he says, and I believe it's a 2003 paper, one of the central places the category of sets has is precisely in achieving this goal of classical logic. There's this idea that's been around a valid argument for a long time. A valid argument is one where there is no information in the conclusion that was not in the premises. First order arguments in the category of sets. That's a completeness theorem. We know that if the axioms don't entail a contradiction, then they have a model. Anything that's true in all models of them, just reversing that, anything that's true in all models is actually already in the axioms in the sense that it can be deduced from them. Now this is not true, for example, of let's take the category of connected differential manifolds.

1:22:30 In fact, let's not even take the group in the category of differentiable manifolds. Now divisible groups are then all connected. Somehow, connectedness wasn't in the axiom about divisibility, and yet it's in all the models. Because when we looked at models in the category of smooth manifolds, there was some information in our domain category, so But when we look at this extremely structuralist category of sets, the only conclusion, the only information that's in any of our models is the information in the axioms. So for Lavier, this is a fine goal of classical logic, is to come up with a framework in which there will be no information in the conclusion of an argument that wasn't already in the premises. And that's realized by taking models in the category of sets. So the category of sets has still a very deep classical motivation. And it has constant technical roles. But he would have said in 63, as he will say now, this is not the only category we could have looked at models in, we just wouldn't have achieved this classical goal than all the others. We don't need that classical goal, but we have it. It's not the only way to think, but it's a valuable way to think and to be able to articulate it. There's another point, if I understand this correctly, that he has this kind of double talk about models. He writes down the axioms that call about them as models. Well, it's a double point of view that he describes in his paper on the category of sets also. In his thesis, he gives axioms for the category of categories. This is before he's published axioms for the category of sets, because he was interested in category of categories. He gives axioms for the category of categories, and he says two things. We should be able to deduce all the theorems of this dissertation from these axioms. He says, I did this in a hurry, I'm not sure that's right. But if there's anything further on that doesn't follow, take it as one of the axioms.

1:25:00 That's the foundation to a starting point. He's saying these should serve as a starting point. But he also says, now let's look at them from another point of view. Let's take it that the universe of sets exists and ask, what are these axioms describing in those terms? And he says, now let's look at the category of all finite categories as defined set theoretically. It's a model of all these axioms except this. Let's look at the category, let's take a strong inaccessible. Let's take a category of all categories cardinality below this strong and accessible. That's a model of all the axioms. So he looks at these axioms both ways. He says we should be able to use these as a starting point and if we can't patch them. But we should also be able to compare these to what we think about independently of the universe of steps. So it's double talk because there's two different questions you could ask. And those two questions are still both there. The definition you gave, you can give it in any any kind of a race. No, no, it's not even a monoid. You cannot prove that it is a monoid. If it has that kind of an immeasurable property, how do you define addition? Exponential, et cetera. So, in a little bit of time, something pre-monoid with one generator exists, but they are not the same.

1:27:30 If you try to take it with no other axioms, you haven't said anything yet because it talks about composition. If you take it with just the category axioms, you can do trivialities. You can show that there's a doubling function, but you can't show that it relates to addition because you don't have addition function spaces. Yeah, yeah, whereas if you take it, yes, if you take it just, just take the Cartesian closed category axioms, which just says it's a category axioms plus, well, there's products and function spaces. Now, all of a sudden, you can prove basically all of primitive recursive arithmetic. In fact, essentially, all you can do is you can show that there are some other, other functions. Well, you can't even show they aren't the identity because they could all be the identity. Well, you just get trivialities. You get that there is, it has a mono-independent morphisms. Well, everything has a mono-independent morphisms. You mentioned that Laubir talks about dialectics. It's a unity in particular. I was going to give a multiplicity. This is so vague in general. I mean, even Parmenides would say, being is one. And I hardly, anyone would classify Parmenides among them. For example, I think I was able to show that Marx used the concept of polar opposition and puts it to work in the transition from commodities to money in a very precise way. Is there any way in which Ludwig has used dialectical concepts in this precise, useful way rather than just vague generalities?

1:30:00 Yeah. Now, actually, the generality I put up there is not so entirely vague as my slide here because it's... Some hundred pages by Grassman. Remember, this is LaPierre's exposition of Grassman's theory of dialectics. And I think part of the reason LaPierre has spent that time expounding Grassman's theory is that, as far as I know, Bill would not claim to have a theory of dialectics. He's interested in theories of dialectics. But there's passages in Hegel that Bill has looked at and said, you know, I can show you some things that can really mean. He's been interested in expounding unity identity of opposites, and the example of that he gives that I think is most accessible, consider topological spaces. On one hand, you've got discrete spaces, so that every singleton is open, and you've got indiscrete spaces, where only the whole thing is open. From one point of view, these are as different from each other as topological spaces can be. From another point of view, so they're opposites, but they're in a tight unity because they are the two adjoins to the underlying set functor. The left and the right adjoint to the underlying set functor. So the case of zero variation, there's no discrete movement. And the case of chaotic variation, where every movement is continuous, there's no continuous motion here, every movement is continuous here, and yet they are both realizations of the category of sets as topological spaces. But he doesn't view this as explaining what Hegel meant by unity identity of opposites. He says, look, here's one thing that fits this pattern. Hegel's idea is at least as rich as this. We see cases of it. This is a case. Yeah, and here's Bill with me. Bill is interested in reflecting on what Hegel meant in one passage or another, what Grasping means in one passage or another, but certainly in his own work, he's not going to try to take the idea of dialectic and make something of it. He's going to take math and see if he can recognize it.

1:32:30 Yeah, and if I can just add to... Colin's response to John, the opposition, the dialectical opposition in question for Bill, I think, certainly in the mathematical context, almost invariably does involve an adjointness. Exactly. And he's actually explored this issue of unity and identity of adjoint opposites in considerable detail. In several cases, including actually in some very interesting papers on foundations of calculus. But the question I wanted to put to you, I really wanted to put two, but I just have one bite of a cherry now. I was wondering if I could draw you out a little bit more on Bill's attitude towards the natural numbers. One of his more familiar and repeated sayings is that the natural numbers can be regarded as the source of all evil in mathematics. I obviously know that's a soundbite. What I understand and have in mind in that remark is that the view of the natural numbers as forming a completed discrete totality, which obviously goes with the piano construction, is the source of the pathological functions of 19th century analysis, like the space-filling curve and other examples, that these things are profoundly ungeometrical. and conflict with his sense of how the structures of mathematics stick together, the kind of synthesis of mathematical knowledge that he feels that he has in view. And that the solution of this is to find ways of constructing the natural numbers which avoid what he sees as the source of the difficulties in the piano case, which I take to do with what he regards as the wrong choice of endomath in the piano category. And I'm just wondering if you could say a little bit more about that, also how it connects with his views of the Groten-Deet termed topology program. Yeah, I'll say a little bit about that. Let me just, I was just also affected on what you were thinking. I don't know that Bill's inclined to criticize Engels this way because it may be constructive, but Engels doesn't really get it from the math either. It just doesn't come from...

1:35:00 Yeah, with the natural numbers as the root of all evil. I'm not sure I've ever really understood all he means by that. Actually, some of what I've come to understand by it is contained in the program of O-minimality, in particular, tamed topology, where it takes a concrete form. You say, the interpretability of arithmetic in a theory is what makes it undecidable. And we learn from experience. It's what produces pathological objects. And the paradigm case here is Tarsi's description of what he calls elementary algebra. You give first order axioms for the real numbers in which you cannot interpret arithmetic. They are too weak to interpret arithmetic. You even have a unit and you can talk about any number of sums of it with itself, but you cannot define a subset of the natural or integral. And this has two advantages. The advantage that occupied logicians for a long time, this theory is decidable. You can explicitly describe a decision routine for it. I understand it's not very feasible to really use, but it's there. The side of that that interests Angus McIntyre more, and Bill is very keen on this, is that now if you start taking, of course, this is a theory of the real line, but it's got pairs and triples, so it's also a theory of the plane and three-space and four-space, and the figures you can define in the plane or three-space or four-space in this decidable theory are distinctly unpathological. But you don't get space-filling curves, for example. In some clear sense, all the spaces you can describe this way are triangulable, because you can't describe very many. You can only describe them if they're pretty nice, and if they're pretty nice, they're triangulable. So in some sense, this lets you avoid all these pathologies. And Bill is certainly interested in that. He's been concerned about the natural numbers for longer than he's been very interested in them.

1:37:30 One of his concerns there was in synthetic differential geometry. We've got this topos, we've got a line, and every function on that line is differentiable. And from that line we can define a plane, three space, four space, definable subsets of them, the torus, the sphere. Basically all the examples you're ever going to think of of differentiable manifolds, but with luck not the ones you wouldn't have thought of. Only get there if you assume there's a classical natural number object. But let's try to do it without the classical natural number object. And we'll have certain ersatzes for that. Look at, we can define the sine function. Now look at the roots of the sine function. They're a lot like natural numbers. They have a translation. They have addition. If you pick one, they have addition and multiplication. If you start using elementary algebra, you don't have a set of all of them that you can do induction on. And he's felt that, yes, you get... So we'll get... If you allow the classical natural numbers into SDG, you've got this nice non-classical theory where everything is differentiable. There's certain kinds of constructions you can't do. If you throw in a natural number object, you can start imitating arithmetized constructions. And so his feeling is, yeah, we want to avoid the natural number object, and the only technical form I know that's taken is in O-minimality. Absolutely. Well, thanks for a magnificently clear introduction. Is it in order for me to ask the second question? I can abuse the sponsor's privilege. Another thing I'd really like to draw you out on is your remark about the extremely structuralist nature of the character it sets, and that particularly in connection with Bill's attitude to the axiom of choice. He remarks in, I think in the Bristol Colloquium paper, Quite early, about 30 years ago, 1973, in the introductory remarks that every, and it's a very dialectical remark, I might say to John, that every notion of constancy is relative and arises as a limiting case of some underlying notion of variation, and its scope and utility is limited by that origin.

1:40:00 And he connects that in a number of places with the way in which Conditions like the axiom of constructability, which you cited, but also the axiom of choice, enforce a certain degree of constancy in the sets that we're dealing with, and that the axiom of choice in particular is naturally violated by sufficiently, perhaps regarded as naturally varying sets. He actually uses the expression of organically varying sets, and also by what he terms cohesion. And he connects this, of course, with the reasons that the Banach-Tarski paradox fails in the material world. And I was wondering, as I say, if I could draw you out on some of those connections as well. Yeah, he's got this idea that what's crucial about the category of sets is that sets have no internal structure. And this is certainly a classical motivation for a long time. So he was talking about Cantor taking the notion of cardinality of sets, but he takes the word from Steiner. Exactly, the geometry. Yeah, as you're doing projective geometry, Cantor says, okay, Steiner talked about some kind of isomorphism between geometric spaces. I'm going to take a structuralist kind. I'm going to take one where we just, element to element, that's all I'm going to require. So there's certainly this idea that sets don't have any algebraic structure, they don't have any topological structure, they don't have any order structure. These are things you can articulate. But Bill was interested in spelling out deeper ways they're structuralist. For example, the topos of sets satisfies the law of excluded middle. Every subset has a complement, an exact complement, not just the largest disjoint subset, but the two of them exhaust the domain. Now, this rules out a kind of variation. When people want intuitive examples of how the law of excluded middle could fail, they say, it's not the case that I'm either in the room or out of the room. Because at one moment, I'm passing out of the room. At one moment, this variation is so realized that I'm not in the room, I'm not out of the room, I'm passing out of the room.

1:42:30 There's no point on the continuum as passing through zero. Some of them are on one side, some are at it, some are on the other side, but there's no point passing through zero in classical set theory. There are points passing through zero in synthetic differential geometry. You can't say that they are zero. You can't say they're not. You have to not be able to say either of those for the axioms to work. Category of sets will insist on the law of excluded middle. This is a kind of constancy. You don't capture these variations. This, if I can just intervene, of course, knocks the at-at ontology of motion as promoted by the Aristotelians on the head and substitutes a more richer dialectical notion of the kind that, let's say, Hegel was writing about. And as I mentioned, this becomes, and I think Phil's got a persuasive argument here, this becomes the completeness theorem. The completeness theorem is a further witness. It's much stronger than just excluded middle. But it says, yeah, the axioms, if they're not contradictory, they have a model. Because there were no restrictions. There's no structure that has to be preserved. It's just if your axioms are contradictory, there's a model for them. This is a further evidence of constancy. We have not completely described the category of sets, and we're never going to completely describe the category of sets. These are all relative constancy. We're enforcing more and more constancy. And Bill takes the axiom of constructability as an example here, or you can take the axiom of choice. Okay, the axiom of choice, if you've got a family of non-empty sets, you can select one from each. Well, this would not be true in a world where selections all have to be smooth. Take the real line, take a cubic curve over that real line. Every point has curves over it, or under it. But there's no smooth selection of one point on the curve over each point of the axis. And you can theoretically say, right, no smooth selection, no continuous selection. Well, in a world where everything is smooth or continuous, there's no selection function there. So the active choice enforces this, but he also looks at, take it in the Zerbello-Frenkel context, Why should we regard the axiom of choice as a kind of constancy? Well, look at how you refute it. Look at Cohen's construction.

1:45:00 How did we get a model of ZF that didn't satisfy choice? Well, we set up this realm of variable sets. They vary over a post set of conditions. And then we froze the variation along some generic path. And that gave us a universe of sets that's not the one we started with. I.e. a universe that retains some trace of the variation that we put in this construction. That's why it's not the one we started with. So we get a universe that satisfies the Melo-Frenkel axioms, but not choice, by setting up explicit variation and keeping just a little bit of it. So little that we satisfy the other ZF axioms. The ZF axioms are a defense against variation, but we manage to satisfy all those defenses and yet still preserve some. And the action of constructability also, you falsify it by forcing. So we've got these. So this becomes an argument again for constructability. Basically it's an argument, anything whose independence is shown by forcing, so the models that make it true are forcing models, that's prima facie evidence that it contradicts constancy. That it somehow expresses some variation. So we should reject it as a description of constant sets. Now that doesn't mean we should Just don't regard it as a description of constant sets. Of course, Bill is not saying we shouldn't ever talk about non-constructible sets. Nobody is. Zermelo-Fraenkel set theory with the axiom of constructability itself implies that there are universes with non-constructible sets. They just aren't the universe of sets. And so Bill is not against there being such a universe either. But he's saying when you're describing constancy, you should regard constructability as an element of that. I only have five minutes. And there is two questions. Five minutes is for the answers. I have a quick one related to this. Mine is also very small. Just a small question. You mentioned that after describing the axiomatization of topology and before entering on axiomatization of continuum mechanics, You point out that Navier chooses a specific category, this new potent infinitesimal.

1:47:30 A specific kind, yeah, I mean, I'm sorry. Well, is there any argument he gives why he makes this choice? Well, yeah, I mean, we're going to need something like smooth functions. Now, he's not going to try to argue this is the only way we could have gotten something like smooth functions, but it's the way that he sort of picked up from Gabriel, extending somewhat from Gronkic, it's the way he was interested in pursuing. So no, he has no argument that there couldn't be any other way to do it, but it's not just accidental that he's taken that. He wants some kind of smoothness, and when he was looking for an improved theory of differentiable geometry, these were the kinds of toposes that suggested themselves, and they've been valuable. I just want to mention, in connection with your discussion of the completeness of algebraic theories or even first-order logic with respect to the category of sets. Of course, once you have the law of excluded middle, that completeness of first-order logic with respect to the category of sets is equivalent to the axiom of choice of sets. So that fits right into your frame there, right? You use the axiom of choice to prove completeness, and conversely, given completeness, you get the Boolean prime ideal theorem, which is essentially... Yeah, I said the completeness theorem was stronger than excluded middle. We even have a measure of that. It's the axiom of choice. So the other thing that I wanted to ask is, I got a sense from your discussion of the development, The elementary concept of the topos that was actually, you're suggesting, was driven by Bill's program related to synthetic differential geometry. Is that the idea that you were suggesting, that the conception of synthetic differential geometry as a kind of differentiable analog to the growth needle? And this is something that's already stressed in SGA3, is the importance of these no-potent infinitesimals and the theory of deformations that they give. So if Bill looks at this and says infinitesimals, I have a good reason for that.

1:50:00 Can I make one very brief administrative announcement before people go into the buffet? I'm sorry I wasn't here earlier this afternoon. I do need to collect from the people who are coming to the dinner tomorrow night. So perhaps if I could just set up, as it were, here in this chair, and if, as you go into the buffet, you could just let me have a cheque or the space, and I can just tick your names off the list. OK? Thanks. If you're going to the movies, can we leave our things in here? Overnight, I know, but we're not in here tomorrow. Oh, it's a movie. Oh yes, good question, sorry, is anybody there? Andre, sorry, we've just got a question. Can people leave their things in here while they're watching the movie and get them back later? No, but the movie's in here, isn't it? Actually, yeah, probably better we take them. Okay, okay, I'll answer it's no, it's not a good idea. Thank you very much. I know it's a nice one, but it's the only one who's deaf now. Thank you very much. Okay, thanks. I'm so sorry, can you remind me of your name? Of, uh, Senor Merlin. Yeah, that's right. How much is this? Thirty, thirty. Hang on a second, I'll give you a chance. Where are you on the list? Oh, there you are. Thank you very much, and I can give you a drink, thanks very much, we'll see you in the... Where is the microphone? That's a good question, straight to it. Thanks very much, thanks ever so much for the happy challenge. Jean-Yves, I told you, I know the one. Yeah, you couldn't give it to me now, so I'm just picking off the list. I think you're French, so I'm sorry. I can't accept these Euros.

1:52:30 I don't know, I'm just going to be... Sorry, I don't think I can speak to you. Is Johnny on here? No, you're probably not. Oh no, because he wasn't even there yesterday. Get back to the slide and figure it out. Okay. And... Sorry, I'm just going to be your theory, okay? Is it 30 years old? Yeah, absolutely it is 30 years old. Are you on the list? I'm coming, I don't mind. Oh, you're on the list, okay, no problem. But towards the end of this, so the next day... Thanks very much David. We're all going to squeeze in there, aren't we? Oh yes, it's easy, we're only 20 now. So we don't take that over the whole... Well, we won't actually take it all over now, if there's only 20 left. It wasn't so big, was it? It's always a mystery, and you can tell that you can take it up to 50, and you can discipline it with me. Well, there's only the six of us at that table now. No, no, I know, it's very deceptive, believe me. Oh, it is. It is, believe me. I think, actually, it's probably not 50, but it's probably 45. I've been in there for two or three years. Anyway, 20 is easy, that's the right time. I should not forget to give you what I promised you. Oh yes, I really want to. But it's nothing interesting. It will be in English in the book. I should be the judge of that, Ralph. I'm sure it had lots of interesting things in it. Seriously, if your paper was any... Anyway, we... Well, my German is very basic, but it will give me a good incentive to struggle with it. When I... I will publish my thesis in English. Yes, you said. So you will have everything... No, but thank you for giving me that. I really wanted to look at it. And thank you for a really excellent talk. I really was. Thank you for the invitation. It was great. You're doing a great job and it's great to be here. Well, I'm very much at all about today. I think it's been a very good day. I had yours yesterday, and I'm very pleased.

1:55:00 I'm only sorry you couldn't stay for about five minutes. We'll be in touch. But of course I want to at some stage, when you're settled and hopefully got a bit of security. I'd like to come and talk to you about this, you know, this subject, this archive, because it is, you know, it's a pretty amazing archive. We've got stuff there going back to the 70s, interviews with Dirac, Wheeler, Wigner, on the physics side, these 12 days of interviews with Lorvier, which we just did this summer with Kohn and Cartwright, and I think it's a very valuable piece. I'm just hoping that we can get some... It's a matter of fact, Leo Corrie, who, well, of course you know Leo, Leo suggested to me that I should get in touch, he gave me the name of somebody in the Max Clark Institute, who he thinks might be, but the problem is it would have to come from some academic, I mean Leo himself or possibly you and others, if we could... I think in terms of putting together a kind of consortium, a panel of scholars who would be responsible and to make a funding proposal for the task of... I'm very worried about the rate at which the old audio tapes from the 70s and 80s will now deteriorate. And we're looking at something, almost 9,000 items, so the task, it's a full-time job. I mean, obviously I would be looking to get a salary out of it, in no two ways, and that's the point. I've built it up and I've administered it and I've recorded it, but, you know, so I would hope I would be regarded as a fairly strong candidate to the guy who had the job of, you know, of administering the task. But it wouldn't be simply my intention, it would be to try to ensure that the material itself is conserved, not just in... You know, in bookcases in my house in a form where it can be permanently secured and hopefully put on the web. And I would have no problem at all about surrendering the copyright or even physically surrendering the actual tapes themselves as long as I knew that they were going to be digitally, you know, remastered and put in a permanent secure form.

1:57:30 Obviously, I hope to have a role in it, I mean, you know, because you're looking at at least a three or four year job, but, you know, it's something we should think about in the longer term, putting together funding for both. As I say, Leo mentioned, I can't remember his name, he mentioned a guy in Munich. Munich? Well, maybe I've mentioned him in Munich. The History Institute is in Berlin. Oh, in that case it was in Berlin. He thought that they might be interested in the project if we, you know... Well, if they came and actually examined the recordings and saw what there was in them, not to listen to them, I think they might be persuaded that it's a sufficiently malleable art and that they were committing some resources to preserving, or even taking over. I wouldn't have a problem with them taking the whole thing over, but physically, you know, something's got to do the job of remastering it and putting it in a form where you can put it on the web. And I think I'm the best qualified person to do that, because I've gone through the whole thing. I'm slightly, I have a slightly subjective view. But I would want to try and put together a funding proposal anyway, so maybe we could talk about that. I know it's not a good time in Germany for a funding proposal, but, you know, it's always worth trying. They can only say no. Okay, Stefan, thanks. Oh, you're not coming at all. What is the purpose of this buffet? Well, there's a buffet which... But when then? Thank you very much for your time, and I hope to see you again in the next lecture. This is Catherine, it's Karine. Oh, okay. Karine. Ah, oui, j'ai vu, elle avait un t-shirt, elle venait pas et... C'est son petit cadeau pour moi. Ah, c'est bien. Thank you for your attention. But there are a few words that French people don't know anyway, for example, because there are some words, the gender is changing, the sort, for example.

2:00:00 There are also a number of other fields of study, such as mathematics, geometry, physics, and mathematics. Yes, I was aware that there was some way you changed the case, but that's fine now. Okay, who's this? Where am I? Okay. Well, I finally... Wait, hey, don't speak. No, well, you weren't. That's just brilliant. Well, I couldn't see, I couldn't see where. No, all that happened was that you would simply, it's basically, search me, search me, Scott. Subtitles by the Amara.org community The only reason I put that as a rather lengthy question to you was to... Well, if we're going to both, we're in the parameters. No, it's just in every slice. It ain't all in every slice. Mike, just one small point. Yeah, sure. I asked at my hotel today whether they would have a room. No, no, I spoke to them. I spoke to them today. I spoke to them this afternoon. Yeah, Andre, we're coming, we're coming. And also, I have the pro formas from the two hotels. So we can, you know, we can go to Giselle. No, no, no, I'm explaining to you to solve some problems. Well, no, no, but I'm talking to André and saying we're going to Giselle, no? No, no, it's solved. I've solved the problem. I'm explaining to you what to do.

2:02:30 Oh, okay, okay, okay, okay. Well, okay. Well, if he has, it will be the biggest fucking miracle of all time, because, to put it quite bluntly, I have known teenage heroin whores who were better organized and had more of a sense of obligation than Charles O'Dooney. I'm sorry you didn't hear me say that. Well, it's if they think they'll get some heroin for beating themselves, they will. Yeah, but, you know, what can I say? You know, bloody hell. Sorry. You didn't, you know, that last remark was to be, is to be scripted from your memory, members of the jury. Yes, it is. But... What did you say? I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager and I said I'm a teenager. So you were saying... Well, no, I was just saying that the reason that I had you with those two other worthy questions, which obviously I keep them well what you're going to say, the answer was to try and drive home to John Stagestuckle the point that, yeah, Bill is certainly a serious dialectician. I mean, just because he doesn't... Try to formalize Marx? Try to formalize Marx, it's just because he doesn't try to read the map out of the philosophy. Of course he doesn't. I think there are...