Unity of mathematical work & outlook of FE Lawvere

0:00 For the research topics, I will present to you Colin McLarty, the philosophy of logic, the philosophy of mathematics, the philosophy of science, and the French philosophy. By American standards, yes. I could not teach it here. Trying to look at Lauvière's work as a whole, I think it's almost a natural, because in some way, all of his work is already in his dissertation. There is a tight unity through the whole thing, but trying to articulate that unity. I think two themes in particular are going to be important. 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 it's important to say at some time can be said clearly by the conceptual means available or becoming available at that time. So he takes this lately from Hegel and Marx, although he got similar ideas from Clifford Truesdell and Samuel Eilenberg earlier in his career. Along with that is his interest in dialectics, which he's lately explained by talking about Grassman's view, not exactly his. The 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 theories. Finally, the way to say something clearly is, today, the best we've got is to put it in categorical terms. LaVier, as an undergraduate, was studying with Clifford Truesdell.

2:30 Clifford Truesdell was doing his rational mechanics. The idea there was to produce rigorous foundations for mechanics that would solve specific problems in thermodynamics and continuum mechanics. Loveyer tells us Truesdell taught 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 hard, and Truesdell had issued a challenge already 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. 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 General Topology, and it includes inventing its 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, 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.

5:00 We take galaxies for granted now. Of course, we've all sailed through them at warp speed. We know what that's like. At the time that Kelly wrote this, 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 date. 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 had sailed through these things in his mind, even though you couldn't do it at the movie theater yet. And so he went off to try to learn some category theory. And we're talking now, what, about 1958? It was no easy thing to go off and learn some category theory in 1958. A sort of catch-as-catch-can in various research articles. But in particular, Kelly has this remark he makes 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, so we're not really going to have energies or velocities, but just equilibria. And 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 Godemal. And as he says, Godemal'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 Godemal's chief theory with topological dynamics. This was a project very much in formation. But it was Bill's project.

7:30 He 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. Bill got there, and he... 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, never explain 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 wrong concepts. 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. 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. Categories at this time were 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 the National Academy of Science that just gives the axioms and asserts the unicity of cohomology, the unicity for simplicial spaces of cohomologies and theories that meet these axioms. Sorry. So they made it central there. Cartan and Eilenberg had made category theory central to all of the subjects in Cartan-Eilenberg, abstract algebra, Lie group theory. By this point, of course, Grotendieck's ideas, well, his functional analysis was established, the value of categorical methods in studying tensor products and organizing topology around that. 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.

10:00 How many everywhere independent tangent vector fields exist on an n-dimensional sphere? Well, on S n minus 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 kn for which there exists kn, oh I forgot to say independent, everywhere independent, everywhere pointwise independent, continuous vector fields on Sn-1? And he had solved this using Grotendieck's and Steenwoud's ideas. At Columbia, this was just obvious that category theory would solve classical geometric problems. Freud tells me that when this was presented at Princeton, everybody was, wow, you use category theory and you get geometry. But to Eilenberg, of course you use category theory to get geometry. That's what category theory is for. But Bill, 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 categorical piano postulate early on to define the natural numbers. What are the natural numbers? Okay now I could just position that more nicely. Well they're a set n. It's got a successor function on it. It's got a selected element zero. But don't, these are not sets. These are, that's a thorough going ambiguity in Bill's work. Bill will call the objects of any category sets. He does not mean what set theorists mean by sets. He's saying 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 0 to x, and it takes the successor of 0 to f of x. It takes each natural number to the n-fold iterative f on x.

12:30 And 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. You say this in a particular category. Say the category of sets. 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. And so that composite is the identity. And so these things are inverses to each other. So this definition determines not a unique natural number object, but determines it uniquely up to isomorphism, up to a unique isomorphism. All isomorphism 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. Further, Bill saw how you could, in fact, derive 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 sets, plus this. He saw around the same time that functor categories are the key to characterizing the category of categories, 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 LaVier finds is precisely that there is a first order characterization of it.

15:00 This really isn't leaving the first order, as Pierre's been saying, the functor category from a category A to a category B, because 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 of small g from the product of C with A to B. This is already familiar as lambda conversion. 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 that take 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 Lavier finds that this works, not only does it work formally in the category of categories, but he can derive all the 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 for Eilenberg, the theory should be the example, not meaning the theory should only have one example. But I suspect that when Eilenberg 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. The theory of Cartesian closed categories is, in some sense, fully exemplified by functor category formation in the category of categories. Until LaVere showed this to people, functor categories look complicated, you know, 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. I'm not, 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 a, just let C be the arrow category. So we take our category A, we form the category of squares in it, map them into B.

17:30 Well that's just a map from the arrow category into the functor category. It's an arrow of the functor category. So the whole nuts and bolts explanation, which had seemed obscure to people, follows from this when you want it, but the definition is a simple adjunction. Obviously, in the length of this talk, I'm not going to prove it's simple, but it is. And again, this uniquely characterizes, say, functor categories if you're in the category of categories, but it describes other things in other categories. It gives all their properties, again, because it determines them of isomorphism. If this is one of them and that's one of them, they have these arrows between each other. These arrows have to be inverse. 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... He took a logic course from Elliot Mendelssohn. 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 kept saying, that's really functorial. He's got a notion that morphisms 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, sits on Tarski's seminar, Michael Raven, Dana Scott, and 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.

20:00 So the group axioms, they 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 that you multiply something by its inverse, you get the unit. This multiplication is associative. I suppose that with these axioms, I would need to say the unit, the inverse. Does it follow that that's a two-sided inverse? Okay, so these already, these already, yeah, yeah. He gets interested in... In theory, it can be axiomatized in this way in terms of just operators and 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. And 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. So an operator has an arity. You've got binary operators. These should be arrows from 2 to 1, from something that's a 2-ness to something that's a 1-ness. 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 operators go between...we usually think of operators as taking a single element as value. So these operators conceived that way go from various tuples to one, and they satisfy various equations. So he does present an algebraic theory as a category. Let me catch myself up here. One effect this immediately has... Okay, 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 waved my hands about that, but was it doing anything? The point is that a model of that theory is a functor from that category. Yeah, this does do work. You don't just define something so that you claim it's there.

22:30 A model is so that an ordinary group is a functor from the theory of groups into sets. Type 1 goes to what we call the underlying set of the group. Type 2 goes to its Cartesian square. Type 3 goes to its Cartesian triple. The operators are now arrows between, say, pairs to elements, triples to elements, elements to elements. And, functoriality said that commutativity is preserved, that is, this operator preserves all the, 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, a product-preserving functor, from the theory of groups to the category of topological spaces. These turn out to be the ordinary topological groups with their topological group morphisms. A morphism is just a natural transformation between these functors. If we define a group in the 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. This was also not a new idea. Mathematicians had long said a topological group is a topological space with continuous group operations, and Lie groups are manifolds with group operations. That's for the examples I've given so far, but it's not everywhere. Precisely in 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 define a group scheme as a scheme with group operations on its points, because the points are the wrong place to look. What a group scheme is, is a functor from LaVere'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.

25:00 The equations 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 binary operation of multiplication and a selection of a unit that satisfies these laws. Now, again, this is not new with Love-Year either. This idea is already in the air. What's new with Love-Year's dissertation, though, is formalizing this. He brings it into logic that way, and bringing it into logic is the logic, as he says, from this point of view, free algebras, tensor algebras, many other algebraic constructions are viewed in a unified way as functors adjoint to algebraic functors. 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 case. Now, precategorical logicians were a little more concerned about what an interpretation even is for richer first-order theories, but this is quite standard as the definition of interpretation for an algebraic theory. This gives you a way of looking at each ring as a group. Say we interpret the group as the group of units of some ring. That takes each ring to its group of units. An algebraic functor is a functor from the category of models of one theory to the category of models of another that's induced by an interpretation. Not an arbitrary one, but one that's induced by an interpretation. These LaVier proofs always have adjoints. And these adjoints are typically things like forming free algebras, or more than that, forming, say, group rings, which is the free ring over a group, not just the free ring over some set. So, Love, your viewpoint is unifying all of this. And it sets up a whole lot of structure. We get a semantics functor. This one is reasonably familiar to logicians before Lavier. You take an algebraic theory and you look at the category S of A. That's a script S on my latex. S of A is the category of models of the theory.

27:30 But 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. And, love your guts, nobody could look for an algebraic structure functor that goes the other direction. We take each category and we find an algebraic theory of it. Now, that algebraic theory is not an exact match. In general, this is not the category of models of any algebraic theory, 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 a functor, well, this is a functor over set, so there's a detail I'm skipping there, a functor from any category, the models 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 LaVere'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 up. Essentially, every theme in Laver's later work is presaged in this dissertation, including that he says in the dissertation that this suggests a possible principle of philosophy. And that is captured as an adjunction. We have another road up and road down. From categories, we have the structure function that gives us their algebraic theories, takes a concrete to something more abstract. We have semantics that takes a theory to its category of models, and these are adjoint.

30:00 This diagram actually occurs in the article that's been mentioned here, Adjointness in Foundations in Dialectica, except that it occurs sideways. It's not a road up and a road down in that article. Now, I can't pass on from 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. Operators and equations, I won't go into the details, there's just too many, all the way up to any first order axioms, for example. There's all different kinds of 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 doesn't determine that theory at all. The first order theory doesn't just have a category of models. It has an ultraproduct on that category. You know how 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. Can we look at this and tell whether or not this is the theory of models of some first-order theory with its ultra-product? And Mackay-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 ultra-product. And this is also an adjunction. So now we know exactly what about the models it takes to determine the theory, not just the category of them within elementary embeddings, but that category plus the ultra-products. It was to do this with all these levels of organization that Bill first wrote up his ideas that he had for a long time on the elementary theory of the category of sets. 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, which we've heard here, of Saunders saying, no, Bill, you just can't do that.

32:30 Set theory is about elements, and you're not going to have a set theory without elements. As Bill quite fairly points out, his set theory always had elements. They're just not exactly like the elements in Zeray-Lafrankel set theory. But I think a more important thing to know about the story of McLean, McLean hears this project, says, no, Bill, you can't do that. What did it take to convince him that you could do it? There was not some long exposure. He read the axioms. Being told there were such axioms, he doubted it. Seeing there were such axioms, he read them and saw that they worked. Bill takes this as a fairly general lesson. If you want to evaluate a theory, one way to do that is to read it. Now, all of this work stays on Bill's mind for a while, and when we come to the Dialectica article, he's interested in foundations, and Bill has always worked in foundations from two points of view, which we see already in his first work on the elementary theory of the category of sets, there was a slide about that up this morning where he explains what the theory does. He has the same quotation, but he says very clearly from the start that his elementary theory of the category of sets accomplishes two things. From one point of view, 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 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. What we looked for in these axioms was description. We knew about them, and we asked whether this description was a good one. But he already says in his first publication of the category, he says, another way to look at these is you could start from these axioms and from them deduce the major theorems of number theory, analysis, geometry.

35:00 That's foundations in the sense of a starting point. He's very clear from the start that his axioms could be used as a foundation in that way. Or might only be taken as foundational in this way, but that these are different senses of foundation. Again, I think in terms of philosophers' discussions of foundations of mathematics, sometimes get confused on this very point. Is this a good foundation? Does it describe the actual mathematics? Okay, that's foundations in this sense. Is it an apt description of what's universal? It's not foundations in this sense because if you think you've got a mathematics on hand to ask whether this is describing it, then evidently this is not your starting point. Worth distinguishing the two. But while Abram 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 universals 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's going to have to be about the categories. If you want to approach it in any articulate way, you're going to need the category theory, because that's the only thing that's got a good universally oriented description of those realities as opposed to theories. Actually, I'm afraid this topic is a little out of order. I mean, it is true that, okay, I will make a point. LaVere's approach to algebraic theories was always about their categories of models. 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. 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,

37:30 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 sets 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 classes. 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. Every theory has a class of models, if it doesn't do something weird to make it only a finite set of models. But when you include the morphisms between them, then the category of models of an algebraic theory 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 in your theory or not. So it's 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 here, that my praise to you 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 whole idea of hyperdoctrine, the word hyperdoctrine does not occur here except handwritten as a result of other people mentioning it, but what I go on to is that as he comes, what takes, where he goes on to next from that logical interest, He then pulls all of his interest together, all of his interest 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. He learned the latest on that at the ETH in Zurich, and this gave him a project, which he later described in three steps, which he listed in reverse order when he was right to do that. The last step of the project would be to axiomatize the foundations of continuum mechanics in the spirit of Walter Noll, a Trisdell 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.

40:00 We're going to get at differential geometry a lot more simply using results and methods of the French work on algebraic geometry, some of which I had learned from Gabriel. And to do this was going to require axiomatic study of categories of smooth sets, similar to the topos of Grotendieck. This is a plural. This is Grotendieck's French plural. People pronounce the S one. I get too much by reading. This is his plural of topos, 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. They're going to be sets, well, as he says here, a category of smooth sets. These are going to be similar to algebraically structured sets from Grotenbeek. What he's looking for here, he wants to axiomatize the idea of a topos. Now, obviously, Knoll's work was already going on. This wasn't meant to be a prerequisite to doing continuum mechanics in Knoll's way. This was meant to extend it and make it more unified, more powerful. There have been various attempts to axiomatize differential geometry. Hands down, the least plausible step was the first one. Grote and Dieck's toposes were people who knew something about them, knew that they were orderly, but they were very large and they seemed rather elaborate. Some people found them simple in conception and some people not, Grote and Dieck 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. 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 injunction, were going to give the answer. Now going into it, 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. A natural presentation of axiom, right? You can put and between them all and call it one axiom. But they eventually arrived at an astonishingly simple description, which I'll give because I can give it.

42:30 We've got a category. There's an object 1. It's 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, but we've seen it before. For any objects 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. They're first order. They're just about objects and arrows between them. 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 LaVier's point of view, they are set theory. These are all sets. He's using set more generally, and he's got historical antecedents for that, too. 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 object, 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 up 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 that Lavier and Tierney had the axioms.

45:00 Mike Barr has commented that he's never understood what Grotendieck could have meant by that. Because he knew the Lavier-Tierney axioms. That's what persuaded him. Grotendieck didn't know those axioms. How could he think this without knowing the Lavier-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. He goes back to the second step of his, he's now achieved the first step of his program. He now has axioms for Grotendieck toposes. In short, how close are these to Groton-Dick-Topos? To me, the nutshell description is a category satisfying these axioms need not be a Groton-Dick-Topos. For example, the one object category that has only the identity arrow satisfies all of these axioms. Gee, that probably is a Groton-Dick-Topos on an empty site. Okay, but there's other ones. There's other things that are trivially not Groton-Dick-Topos. Any category that satisfies these axioms and is the right size is a Groton-Dictopos. 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 coproducts, but then it is a Groton-Dictopos. Oh, yes, yes, yes, and it also has to have a small enough collection of objects that cover everything in it. And then maybe it's not even fair to call that just a size restriction. And he can do mathematics in these, but he can consider, now let's consider a topos. It's got a ring R in it, enough of 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. And that subset obviously has an inclusion into the ring, which you might think of as a little infinitesimal interval around zero. We haven't defined any ordering, so where do we get off calling this an interval? But we are going to call it an interval.

47:30 And it has this interesting property that every function from it to the ring... Here I've got the arbitrary case of a function from it to the ring R that extends the embedding, but actually it didn't matter whether it extended the embedding. Well, it's linear. If you give the point it's centered at A and you give it slope, you've uniquely determined it. Every function from D into this ring has a constant part, a linear part, and is uniquely determined by that. And now what we're going to do is we're going to regard this as an infinitesimal interval around zero. So 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 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. So now in this topos, every function from the ring to itself has a uniquely defined derivative at every point, including the derivative of that function has a derivative. What's not obvious from this description but becomes obvious on a little bit of thought is this cannot possibly be the category of sets. 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 if every element of D is either equal to zero or an equal to zero. You don't have one of these. There have to be elements of D that are not affirmatively zero, but also you cannot deny that they're zero. So we need the law of included middle to fail in this topos, but there are toposes that have this kind of a ring in them. From here he goes back into the project of explaining continuum mechanics. A placement of a body B in a space S. We're going to think of that as just a mapping from B to S. They're both spaces.

50:00 This body is a space. The room is a space. This body is placed in the room. That's a function from each point of this marker into a point of the room. Well, I mean, you see it before your eyes. So 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. We can ask 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. Placing it over here gives it a higher internal energy because it's got this heat radiating onto it. Placing it over here, it's going to cool off. So placements have associated with them. Obviously, each placement has a different... Well, they're not all different from each other, but each placement has a gravitational potential. Well, we'd have to pick a zero if we want it to be affine. So we want to study placements of this object in space. That means we want to study this function set. And we might like to look at parameterized families of placements. Of course, we could 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. LaVere's going to say, what... 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.

52:30 It's simply an adjunction. 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 map 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, which I've got here as my arrow m. We've got this history of this thing moving through space through time, but 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 this set of functions from R to S. Take any point of this thing, look at the trajectory it followed. These three all give the same information because they're just adjuncts to each other. Well, I think rather than try to explain that in any more detail, let me move to the philosophic connection with this. This is, from a set theoretic point of view, pretty weird 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 19th century mathematicians talked about 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 that historians of math are apt to look at it and say, well, that wasn't really rigorous at the time, and in some sense this is importantly true that it wasn't, but Bill is saying, but it's just this. He's just freely using these adjunctions. What he's saying is that you can give Rigorous foundations that are this close to mathematical practice. What LaVere is, he doesn't find, well, I'm sorry, I'm getting a little out of order.

55:00 LaVere 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. In his foundations, he finds 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, product for example. I owe you a definition of the product. I put up those axioms that said in a topos any two objects have a product. What do I mean by a product? A product of sets S and T, which now could be all kinds of things other than classical sets. It's simply another set at which we'll call S cross T with a pair of projection arrows. This is the product. It's S cross T with the projection arrows. 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 Zermelo-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 Zermelo-Fraenkel set theory, they're all theorems. What he has said is much less. The Zermelo-Fraenkel set theorist is going to say, but you haven't really told me what set this is. 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. Do they pick one? No. They'll tell you there's no need to pick one. It's good enough. 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 just up to isomorphism 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.

57:30 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. 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. What Laver finds is, what the set theorists will say is, yeah, but for rigorous foundations, you would have had to. And what Laver found was, no, you can give perfectly rigorous foundations without going into that at all. 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. What LaVere has found through his career, look at what LaVere has seen happen during this time. He's seen this galactic theory in topology, just a reference in Kelly. It's led him into Eilenberg, MacLean, Steenroth, Godemont, Grotendieck topology.

1:00:00 It's worked. He's seen Grotendieck's derived functor cohomology, seen Grotendieck's abelian categories. We've seen some references to those schemes, which I don't know if we've really discussed. These things can be very tightly summarized in axioms. For Bill, this confirms what Marx says, that humanity always sets itself only tasks 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. I could have quoted Feuerbach instead. We recognize a problem when we've got the means to handle what that problem is about, i.e. when we've got the means to solve that problem. This is actually a link between LaVier and Grotendieck. It's commonly remarked how optimistic Grotendieck was in the 50s and 60s. They don't look like they have a cohomological solution. They do have a cohomological solution. 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. And Bill's got that same optimism. Whatever it's important to say can be said clearly. 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. One philosophical, actually, how am I doing for time? It's worth mentioning the philosophical interplay between Bill and Saunders. They've affected 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.

1:02:30 That we should think of mathematics as correct but not true. Statements of mathematics are not falsifiable. He cites Popper in this connection. We have all these geometries, lovely geometries, just take the classical non-Euclidean geometries, just hyperbolic Euclidean elliptic geometry, lovely things. 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. 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 say they're merely conventional from the mathematician's point of view because space can be interpreted in any one of them. 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 nature. For LaVere, mathematics is true. It's true, 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. LaVier is certainly against 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-Fraenkel set theory, not the starting point of 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 Matti has talked about this.

1:05:00 We've got Zermelo-Frenkel 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 Brouwers, in any kind of Brouwerian or philosophical sense. But it has set theoretic consequences. Are we going to take this axiom or not? If we take the axiom of constructability, as Gödel noticed, the continuum hypothesis follows from that. 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 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 axiom constructability relates to other questions in mathematics. You can relate 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? Um, and this is, this is the Suis-Werner hypothesis. You've got a description of the, 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 Zermatt-Lofrankel is compatible with saying yes, compatible with saying no. 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 it. 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 among philosophers of set theory who want to evaluate candidate axioms just in terms of technical problems in set theory.

1:07:30 Bill 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. I'm sorry to get a little tangled, but I have gotten a little tangled. Let me just come to my concluding line on teaching. Bill does believe that mathematics teaching should reflect foundations. Everybody's familiar with the new math fiasco where we had this idea we're going to take a sort of a Borbach-Keys 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 Borbach-Keys foundations instead of categorical foundations. It's that it wasn't understood. The teachers did not learn the Borbach-Keys 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 education. When I read Laguerre's thesis and then this 2003 paper, it seemed that actually there is some change in Laguerre's view of foundation. Of course, the thesis looks, you say, more classical in the sense that it still uses first order logic and then he makes this kind of conceptual closure, I mean, categories. Yeah, of course, this 2003 paper is just a different character, it's not a technical paper, but also it seemed that somehow he came to much more flexible understanding of what foundations is and what is soft scores, I don't know, he speaks about concentration, he speaks about education, so do you think he really changed his view or you presume it just has a different aspect of the same, basically the same view?

1:10:00 Let me just jot an outline on that. Of course, his view 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. In 1963, he just hadn't conceived of a lot of the phenomena that he was later going to see rigorously explained. And so he comes to have a more flexible vision just because he's seen the possibilities. 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 F. Bunker's into the category of sets. Already in 1963, 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 of 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. I mean, you can 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 of 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 the 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.

1:12:30 The category of connected differentiable manifolds. We can talk about models of the group axioms in the category of connected differentiable manifolds. No, in fact, let's not even take connected. Let's take the group axioms 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 we're not just getting the information in the axioms. But when we look at this extremely structureless 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 in 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. Another point, if I understand it correctly, is that he has this kind of double talk about models in the thesis. I mean, he writes down first order axioms, then introduces categories as they are, and calls about them as models, just before doing the functorial semantics, and then he does the functorial semantics. And in his later works it seems that this first part is not as redundant, probably. Well, it's the 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 the category of categories before the category of sets. 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 farther on that doesn't follow, take it as one of the axioms.

1:15:00 That's foundations as 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... What a weakening, a strong inaccessible. He doesn't, I mean, this is before he's seen Grotendieck universes. Let's take the category of all categories, cardinality below this strong inaccessible. That's a model of all the axioms. So he looks at these axioms both ways. He says, let's, 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 sets. So it's double talk, because there's two different questions you could ask. And those two questions are still both there. Very closely related to what you said about some kind of universal property of the category of sets, which is everything that's true, etc. If it is okay, you can give it in any category. However, from it, you cannot derive any kind of arithmetic. No, no, it's not even a monoid. You cannot prove that it is a monoid. If it has that kind of universal property, how do we define addition? You have to have exponential, etc. Then, this Lo-Weir definition enables you to permit a recursion, etc., etc., but otherwise, I can give you an example of a category where something which satisfies Lo-Weir axiom exists, the free monoid with one generator exists, but they are not the same. In an arbitrary category, if you try to take it with no other axioms, you haven't said anything yet, because it talks about composition of arrows.

1:17:30 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. Functions defined by recursion a la Louvre, but in function spaces, and then you get everything you want. Yeah, whereas if you take it, 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 as I recall. No, I'm not sure, I'm not sure. Make it Cartesian. Yeah, yeah, yeah. Make it as you presented it. No, no, no, no. In fact, essentially all you can do is you can show that there are some 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 monoid of endomorphisms. Well, everything has a monoid of endomorphisms. You mentioned that Laubir talks about dialectics as a unity, in particular, as a unity of multiplicity. This is so vague in general, I mean, even Parmenides would say being is one, and hardly anyone would classify Parmenides among the dialecticians. Are there any more precise subcategories that he uses in this concept of dialectics, and does he put them to work anywhere? For example, I think I was able to show that Marx uses the concept of polar opposition and puts it to work in the transition of commodities to money in a very precise way. Is there any way in which Loeb has used dialectical concepts in this precise, useful way, rather than just vague generalities? It's not so entirely vague as my slide here, because it's some hundred pages by Grassman. Remember, this is LaBierre's exposition of Grassman's theory of dialectics.

1:20:00 And I think part of the reason that LaBierre has spent that time expounding Grassman's theory is that, as far as I know, Bill would not claim to have a theory of dialectic. He's interested in theories of dialectic. There's various 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 the one hand, you've got discrete spaces so that every singleton is open. And you've got indiscreet 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 adjoints to the underlying set functor. The left and the right adjoint to the underlying set puncture. 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 motion is continuous here, and yet there 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. 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. If I can respond to that, I think that this is what's wrong with Engels dialectics of nature. It's in a bunch of, for instance, a series of examples. It's not a category, it's not a set of categories that he puts to work and does anything useful with. In contrast, what Marx does in capital A. I think if dialectics is going to get anywhere, it has to be made into a set of precise categories which can be put to work. Not just for examples of things that have already been done. Yeah, and here's Bill with Truesdell that the philosophy will come from the math and not the other way around. I mean, Bill is interested in reflecting on what Hegel meant in one passage or another, what Grossman 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 dialectic in that. Yeah, and if I can just add to Colin's response to John. Opposition, the dialect of opposition in question for Bill, I think, certainly in the mathematical context, almost invariably does involve an adjointness.

1:22:30 Exactly. And he's actually explored this issue of the 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'll just have one bite of the 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. But what I understand him to 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 the Cantor set and other examples, but these things are profoundly ungeometrical and conflict with his sense of how the structures of mathematics fit together, the kind of synthesis of mathematical knowledge that he feels and that he has in view. And 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 endomap 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 Grobendieck tamed 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 not seem constructive, but Engels doesn't really get it from the math either, you know, I mean, it just doesn't come from much, yeah, yeah, yeah, yeah, Bill has been real concerned with, yeah, with the natural numbers as the root of all, 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.

1:25:00 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. 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 3-space and 4-space, and the figures you can define in the plane or 3-space or 4-space in this decidable theory are distinctly unpathological. They're fairly rich. You can do a fair amount of what seems like elementary geometry, 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 that. 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. 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. We can define the sine function. Now look at the roots of the sine function.

1:27:30 They're a lot like natural numbers. They have a translation. If you pick one, they have addition and multiplication. But like Tarski's elementary algebra, you don't have a set of all of them that you can do induction on. 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. But the whole impulse for this subject was to not do arithmetized constructions, to just do elementary geometrical 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. Well, the question, if I get this to be clear, 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 category of 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. And he connects that in a number of places in his writings 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, what he would perhaps regard 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.

1:30:00 And this is certainly a classical motivation for a long time. Somebody was talking about, Sir Fatih was talking about Kantor taking the notion of cardinality from equipolence of sets, but he takes the word from Steiner. In doing projective geometry, and Kantor 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're 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 is 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. And so the idea with sets is we're going to throw that out. 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 is, I can just admit, of course, not the at-at ontology of motion as promoted by Anastasios on head and substitutes a more richly dialectical notion of the kind that... 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.

1:32:30 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 aren't contradictory, there's a model for them. This is a further evidence of constancy. But we also learn from Gödel's other theorem that 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 even 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, I mean over in the mathematical sense, 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. 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. 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 Zermatt-Lafranco 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. In the action of constructability also you falsify it by forcing. So this becomes an argument again for constructability. Basically it's an argument anything whose independence is shown by forcing,

1:35:00 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 throw it out of the world, just don't regard it as a description of constant sets. Bill is not saying we shouldn't ever talk about non-constructible sets. Nobody is. Zermelo-Fraenkel set theory, with the action 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 constancy. But it's all going to be relative, there's no complete characterization. Excuse me, for the part of the discussion before the buffet, we only have five minutes. And there is two questions, and five minutes is for the answers too. You mentioned that after describing the axiomatization of Topoi and before entering on axiomatization of continuum mechanics, you point out that Lavir chooses a specific category, this nilpotent infinitesimal. A specific kind, yeah, I mean the sum. 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 Grotendieck, 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 These were the kinds of telepathies 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 for sets. So that fits right into your claim 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...

1:37:30 Yeah, so even... I said the completeness theorem was stronger than the 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 of the elementary concept of a topos that it 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 Grotendieck And this is something that's already stressed in SGA III, is the importance of these no-potent infinitesimals and the theory of deformations that they give. So Bill looks at this and says, infinitesimals? I have another reason for that.