Afternoon Discussions, incl. FW Lawvere, Leo Corry, Angus MacIntyre, John L Bell, Colin McLarty

0:00 But in French, when the boulevard piece says ensembliste, it always means just the contrast between structure and unstructure. But I think it has to be structured in a very informal way. Yeah, that's what I'm saying, in an informal way. It always means that things are less... So that would be inherited from Emmy Netter's school, their use of set theoretic. Set theoretic is a subtle thing. It's certain ways of invoking set theoretic properties are set theoretic. So what you're getting at, I think it's, especially Dedekind, but also Cantor, I think you made a little more conscious this idea that you study the mathematical objects as interpretations of some kind of abstract structure into abstract sets, where the abstract sets are relatively structural. And if you, independently of these historical controversies, if you say, well, why should one do that? Basically, the completeness theorem, the first order logic, it means that you're interpreting your theory into this background. The background doesn't really have any contaminating effect on the theory itself. I think that the early stage is more a practical thing. It's the gradual understanding that starts with Dedecky, and then goes a little bit with Hilbert, and then strongly with Netter. If you start treating things in this way, you discover certain mathematical underlying processes that you were not aware of.

2:30 For example, the question of factorization in fields of numbers, two polynomials. Suddenly, I mean, things were, people did know these things. It's not that it was completely unknown, but then she's able to see, look, there is a structure. If you have this property, which is a property of sense, you have a change. If you have a change, then you have unique factorization. If you have an ascending change, you have this. If you have, so it's, you know, people, it's a great discovery, I think, that people see that people can look at those things that you haven't looked before. It's a tremendous instrument. What I'm inquiring about is, why is it so useful and to what extent is it useful? So I'm saying, making a general statement about that, the completeness theorem is the first order of objects. Anything that's going to be true about the interpretations of the theory in this particular kind of background is going to be provable. Intuitively, this is exactly the sort of thing that's not true about the cohesive world, the continuous world, the continuum. It surely must be having some influence. So, particularly, that has various expressions. I mean, one is the fact that there is no Putin's theorem for higher order modeling in the same way that there is for first order modeling. In other words, it seems that... This, well, it happens always when you try to make universal elements which have no properties, well that's precisely a property, you know, so that the point of trying to make things property-less, in fact, introduces a very strong, is a very strong property, so the higher order things do not interpret faithfully in the same way.

5:00 So in terms of the quest for some kind of completeness for higher order, which is seemingly very difficult. But the other point is that, again, simply from a practical point of view, one wants to interpret these theories not in a featureless background. If you take, for example, the theory of groups and interpret it in a smoother background, it becomes the theory of weak groups automatically. Or you take the theory of vector spaces and you automatically get Bonnock spaces, or at least something like that, so that interpreting the simple algebraic theory in categories that have a lot of cohesion produces a lot of stuff, and this is certainly a practically very useful kind of point of view because On the one hand, you get a lot of stuff for free, and on the other hand, you create a lot of questions that are, so this question of the discrete background has to be taken in a dialectical way, at least in these two aspects, the higher order part and the cohesion on its own further properties of the structures. You know, I think it's a good point, isn't it? There really was a question in the 19th century, and it arose also, I think, in connection with Dedican. What were ideal numbers? What are they? There was always some kind of, mathematicians may not have taken ontology as seriously as philosophers, but nevertheless, these questions do arise, the whole nature of ideal numbers, points of infinity, and look at the debate over imaginary, and then look at the terminology. So I think ontology was an important... I think so. And I think that set theory did, after all, come to answer certain kinds of ontological questions. For example, the use of ideal, you know, a set, if you like, of numbers or whatever, is an ideal number. And all that term ideal comes, of course. And also, what do we mean by this whole thing about using equivalence classes, which is a very important use, early use of set theory. Where did Dedekind's theory of change live?

7:30 Dedekind just talks about these things. And Dedekind, I think it's fair to say that from a certain point of view, Dedekind really does confuse proving there are infinitely many things from proving there's a set of them. Right, right. And it was worth spelling that out. Yeah, so I do think there was a fairly simple... Of course, certain things were lost with this discreet bet, but I think the ontological thing was an important drug. You say, what is a group? Well, it's a set. It's an answer to the question, what is? And I think that question, although maybe not a primary, it's not perhaps the major question in mathematics, but it always played an important role, I think, in the underlying development of mathematics, except that we provided an answer which is now, I think, being transcended. Yeah, but on the other hand, people, once you make, you know, a definition of group and define your various group theoretic notions, your quotients, et cetera, set theoretically, the most unhealthy thing to do in life, people, then it would be to continue to study general set theoretic. No, that's right. No, that was, I mean, that's still going on. It was helped a little bit, again, by Cohen coming along and showing that some of these problems are likely done. I mean. What you've said is unquestionably correct. We wouldn't be where we are now. We probably wouldn't have any of what we're talking about now had one not been able to get it. One just solved it out. I mean, in some cases, one didn't need set theory for it, but one, in general, one needed some language, this set language, to do it. Yeah, if John Mayberry were here, he'd be saying, well, of course, set theory isn't abstract at all. It's actually concrete. This is why I pushed for it. There's a certain, it did solve for the what do we mean by an abstract structure or form. It did give, that's true, that you could question the answer. I've already edited it. A number of people questioned set theory. But nevertheless, it did give that kind of definite answer, counter-grabbing carried it. Perhaps to extremes in his rejection of saying everything really has to fit into this and that's what all the mathematics is. But nevertheless, I think it did answer some of the questions. I'm not sure that's one of the misreadings of Cantor. Well, no, I think... I like your local point that contrarian analysis says this is how I'm going to describe analysis. That's all there is. I'm not even going to justify it. But on the other hand, to say that he's got all of mathematics is this, I think. All right, all right, I may exaggerate, but I do think I, I, yeah, okay.

10:00 It's just that I was so struck at reading the gridlocking recently that, again, that I, you know, he's, and then other things, that he really does have this vision, this incredible vision of discreteness. It's somehow a genre, everything else has to kind of fit in, no, no intuition of continuity. Anyway, maybe I exaggerated, but. Because I started to read Kant from the other side. I was a sec theorist by origin. And so, of course, Kant wrote. Wow, it's amazing. You're fascinated. It's a fantastic achievement. Well, I was looking at basically writing a book on the continuous and the infinitesimal, you see, from the other side. You see, it's very interesting to see what Kant wrote. In other words, trying to understand why he came to have this rather, well, essentially rather narrow view of the continuum as opposed to this Really quite a rebellious review. Anyway, so perhaps I'll exaggerate. But anyway, it's... But only continue. Reading Cantor's article, that's incredibly revealing. Oh yes, yes, it is. It's amazing. It's an amazing book. I mean, I had to point out to Jose Ferreros several points of a very fundamental nature. After studying, you know, Cantor and all these for a long, long time, you still haven't picked these things up. It's very dense, I guess. It's very dense with content. And plus we come with a real strong preconception. There's a whole literature on Cantor explaining the role of the extensionality acts in the Antwerp. Well, he never quite says it, but he meant it here. He never quite says it, but he can't have imagined it wasn't true. Because we know that sets are just collections and extensions. And then you read that Zermelo in 1935... ...and criticized it for having a theory that wouldn't let him do that. Because the whole point is that, yes, it's a Bellos concept. You can't have a set of dots, so you have a P of dot. Exactly. And it always says it's a reasonable P of dot. There are no, they don't have any properties. At least not in that sense. One of the things that most strikes me about this kind of thing is that Nobody can doubt that these are, I think, huge areas of mathematics connected with geometry.

12:30 I mean, of course, they're all in one dimension. And also the good living in this business of connecting the number theory yield case and the geometry yield case is a central theme. This is a central theme leading up to the solution of the Vatican directors. But Cantor, so this is a kind of induction. We started with induction in Dedekind and then we have this notary induction. More geometrical in some sense. But Cantor's induction, which is so much more fantastic, in fact, is rarely used in mathematics at all. This is a big mystery to me. Well, it was, of course, we understand the count, for example, time, and then the attempts to get at the continuum of this, but in, in, for example, does anybody know an example where there's an essential use of any kind of transfinance induction in geometry? Do you know? The Tohoku paper. Thank you for your attention. All of this is completely unnecessary for the topological case. Yes, yes, yes. But still, the whole group refuses this long transponder. But I just want to point an emphatical point that transponder induction, well, recursion really, All of this was used rather a lot, I think, up to the time, I mean, in connection with the use of the axiom of choice, you know, for example, it must have been used in the proof of the Hubble, you know, that they're here continually. Oh, yes, yes, yes, yes. And then all that got swept away by the Maxwell principle. But this, yeah, but even more, I mean, this is the realm of... It's designed to do it. This is the realm of wild analysis, basically. Yeah, I guess that's true. But the acceptable coin system, I'm trying to... I think of many examples where, despite the fantastic power and beauty of the thing, I mean, it just doesn't seem to... I mean, if someone told me that Sayer or Atiyah had ever, in their life, used transformation... Well, what about the construction of an actual idea? Oh, well, it's not using the... No, no, no.

15:00 Fair enough. Logically equivalent, but not necessarily psychologically equivalent. I agree with you on that, too. No, no, they aren't. No, of course, these guys certainly slur over the basic thing. Deligno, on the one hand, will say in way two, I abhor the axiom of choice. He says it explicitly, but Deligno says it. ...explicitly. And on the other hand, he certainly is quite hard at work with his own mathematical consequences early on. ...fundamental or powerful-seeming application of, like Cantor himself, an iterating that derives... Yes, that's what... Yes, of course. No, that is right. Exactly. No, that wouldn't regard that. That wouldn't regard that. You don't need it there. I mean, the Cantor-Bendix theorem, I wouldn't put that quite in the realm of wild analysis. I mean, that certainly is possible. That's a theorem ultimately about, it's a non-trivial thing, about closed sets, which is maybe a tame version of the continuum, I thought, something like that. I think another important, just to give some context, we spoke about Cantor, we spoke about Dedeckin, very nice, and then we have all the developments. What's not a necessary one, historically speaking, I mean... Certain things happened. For example, the role of Hilbert was fundamental in attracting the attention of mathematicians to the continual hypothesis, let's say, and things like that, and at the same time adopting Dedekind's point of view in number theory as opposed to chronicles.

17:30 I want to say with this point that we look at these words very strongly and we of course find very interesting things, etc. Contemporary mathematicians not always did. And if, for example, Hilbert had opted in number theory to go the other way, the chronicle, and not to put, for example, the continuum hypothesis in the list, I mean, at least chronologically things would have been different. I'm not sure at all that people would have paid so much attention at the beginning to these techniques, to this approach, and things may have developed differently. Perhaps in the end, I mean, when you have powerful tools, they tend to appear somewhere and to be used, but at least it was relatively to happen, from the cantor, let's say, papers, to the centrality that it was accorded, and it was because certain historical things could have been otherwise. Very much so. I mean, at least from the fragmentary knowledge I had, it seems that there was a major discontinuity. Things could have gone many different ways. But because Zermelo, in particular, swore a philosophical allegiance to Frege instead of to Cantor, therefore a whole line of development, if people had followed Cantor and Dedekind more directly, it might have been quite different, at least what we know as set theory, because this decision of Zermelo, whatever, why ever why he made it, he never even gave an argument. But it was not just on one point. It was about Frege versus Cantor. I'm for Frege. And so he followed this other method, but then you come to all the later people, as far as the philosophical starting point is concerned, they just refer back to Zermalin. Von Neumann refers back to Zermalin. So this bifurcation, this seizing upon the integrated membership, and making that the central formalism, it could have gone quite a different way, and apparently it was just this quirk of this one man. So I'll ask you something about that. Sarmengo started as a mathematical physicist. His doctoral dissertation was with Planck, and he was involved in questions of kinetics.

20:00 He came to Göttingen that at a certain moment was a very powerful center and Hilbert was telling people what to do and he just started with his axiomatic program and he said to Cermelo, here we have this problem, please go on and take it. Of course Cermelo could have said no, I don't want that. Something happened, you know, there is a lot of historical contingency, I would say at least in what has to do with chronology, because maybe in the end you have no choice and you would have things, but things happen in a certain way, in a rather, in a rather contingent way. This is the very nice thing of this story. But the order in which they happen does affect the sections of the seminar for a long time. But DeRigo wasn't concerned with developing his idea because he's trying to secure his proof of the law of order in theory. Yeah, yeah. I mean, there wasn't a specific problem, I mean, he didn't even have the first paper, the 1904 paper, he doesn't really have a system, no word to say, you know, and it was really criticized, we know from Moore's book describing it, you know, there was a big criticism of it, they thought it was circular, you know, you reweighed choices, there is a book by, what's his name, not Moore, I don't know. What I think is that it's very interesting that in the later development, I mean, yes, he did develop this and he did it, of course, he came up with this system of domains and, you know, structured by membership and the probability of membership. Well, of course, it had to play a central role in his book. Oddly enough, it played, the interrelationship, the interrelationship plays no role whatsoever really in his proof of the action of choice, of what order he's going to have to do it. I mean, you can look at the proof of, you know, there's a beautiful proof of this, of this, it's all called the Zermelo's theorem of Zermene, they call it in Bourbaki in chapter three, that I've analyzed quite a lot, quite a lot about that proof.

22:30 It's nothing to do. Of course, there's not really any set theory as such in the word by key at all, and they don't use it. But in Cervelo, it's somewhere between a trick that he found worked when he needed axioms and a philosophical commitment that he eventually attributes to something he learned. It's not in the mathematics that it requires that. No, no, you're right. No, no, I think he, no, he wanted to get it beyond quick enough to remove it from criticism. Yeah, yeah. And he, it's brilliant, and there are other reasons, but it was rather effective. I mean, it actually, it convinced, I suppose, more mathematicians than before. It also, incidentally, of course, had the effect of, in that system, I mean, we later have Frankl, you know, and all these people who... I'm quite interested in the actual formalism, if you like, the framework, if you like, that Zermelo had introduced. But it's interesting that we've come from a specific problem. Precisely if you believe that mathematics would necessarily have solved the well-ordering problem, that you have to realize it wasn't necessarily going to be through Zermelo. Yeah, that's right. In Gettysburg in 1905. But you know the argument that... All of the elements are not put in it. Because he doesn't do something really not in the 19th century. Sure, but the previous construction of the natural numbers and Hilbert's proof and actually the construction of the von Neumann ordinals are all special cases of the same kind of general result. I mean, and you could formulate it really well, or you do it, it's more easily formulated, you know, you could formulate it, of course, also in terms of maps from power, you know, from exponential objects to objects, but it only needs one, you know, it only needs one, one membership, one, one, the member of freedom between an object and a set, perfectly reasonable notion, membership, it's iterated membership, it's really quite, you need such and such, that's what you do. But you don't see any homogeneity in that. No, you don't. And indeed, you know, Zermelo doesn't use it. It's not really a, it's just... And that's why Zermelo states it clearly, not in 1905, but in 1933 or 35, whenever, when he writes his footnotes. That's right. Well, at that time, he's... That's when he writes his principles. Well, he may not have been also very, I'm not sure, very happy with...

25:00 What had happened in logic, he didn't understand Goebbels. Well, and he felt that Fragnell had betrayed him. Exactly, exactly. The stress on iterated membership, it becomes explicit years later. Yeah, which, I mean, the notation is due to Piano, actually, but Piano and Frege conspired on it. Absolutely, it's SD, you know, in Greek. Yeah, we have the singleton, and then that stuff. Yes, but this was, you know, Frege explicitly wrote to Piano about, oh, right on, that's what we want to do. So, in that sense, it's Frege's influence as well. Although, Frege, no, I just say, although, although... Frank, of course, wasn't doing set theory. Frank actually only has one level. He has objects and concepts. And in a sense, of course, the concepts can, they get stratified in some way, but basically he's only got two levels. And he doesn't use membership. He's got these things called extensions, yes, he admits those. He doesn't want to go into what they are, and in fact it was later assimilated to sets, but actually he doesn't say that. I mean, Frege isn't doing set thinking as such. He's working in extensional logic. That's right. It's a different... I mean, there are other reasons why one might find that objectionable, but it isn't because he's too big set. Well, one of the things that recalls Bill remarking in Florence is that the whole line of development of Frege elaborates a little bit on that point. I think, I keep coming back to this idea that one of the main advances of category theory is simply to note that math has definite domains and definite codomains, and that in order to be able to compose the codomain of one has to be exactly the same as the domain of the other, even, you know, the things that come even before associativity and all that is just this feature because... Well, first of all, the properties of maps depend very much on whether a map is surjective or not, obviously it depends on fixing the co-domain, and surjectivity is dual to injectivity, you see.

27:30 There are these contravariant functors that you apply even in elementary set theory to transform one into the other, so it's very unnatural to restrict oneself to surjective functions. To assume that the codomain of every map is the universe, which is essentially how, you know, if you look at the notion of function that's formalized in ZF, that's essentially what it is. If there is a codomain, it's the whole... You've got a domain, and for every element of the domain you've got a value, but not a set. But where is the value? It's just in the universe, you don't know. You'd have to add an axiom of replacement to ensure that it really does have an axiom, to put a bound on it. So just this simple fact somehow, which I think could have been appreciated. Well, topologists already in the 19th century, a continuous map went from this space to that space because it didn't make sense to worry about its geometrical properties once you knew where they were. Did it divide off a region? Well, I mean, a map from the circle into the plane divides off a region, not into three spaces. You have to know it's into the plane. The only people who knew how to do it were the Netter school after they got interested in topology, the ones who were doing topology, because they needed to talk about maps between cohomology groups and they needed to know whether it was onto or not. For Netter, still, a group holomorphism is onto. We now say it's onto. But at her school, all of a sudden, it is not, oh, there's a group here, a group there, a homomorphism onto some groups in this one. So how do you relate this to Frege? Well, because once you decide that everything in the universe is identifiable in its own terms, you apply a function to an object, you get another object. You don't have to ask what set is it in. No, because Frege makes this unjustified identification. Concepts with properties. So everything is really a property of this one universe, you see.

30:00 But that comes a bit later, because what he does, no, that's true, but he actually starts with the idea of a function, right, which is defined on the universe of all objects. In the domain. Yeah, yeah, the domain. He doesn't, he has no typing, you know, he just, and then he says, well, a concept is such a function. Right? Which only has two values. It takes a value of zero and one. And then of course, he then assimilates that to property. That's right. That's right. That's important. And that's the order. But of course, because he's got this kind of unbounded notion in the first place, where the domain of the thing is already huge. It's the universe of objects. Right. Well, what's the code in it? Well, the universe of objects. I mean, it's still, for Graham or for Seth here, it's the rest of the 20th century, you can talk about what's the square root of 2. It's the square root of 2. You don't have to say, did you mean the algebraic number square root of 2, the real number square root of 2, the complex number square root of 2? It's just the square root of 2. Whereas computer programmers now, these are different functions. You need to know which square root of two it is. And, in fact, now in set theory books, a function, as defined in set theory books today, has a set that it's defined on, takes values in another set. It is normal now that a function does have a domain. But it was not normal up to 1950 or so for programming. Because, for example, if you want to talk about whether a function is onto or not, you've got to have said onto what? Of course. I mean, it doesn't appear before 1950. It doesn't become standard in set theory books. This is part of the basic conception of what a function is. Yeah. Well, for that, you say, here is this function. It's a reflection of Frege's point of view. And the change comes because of the stress given by category theory to that or for other reasons? Well, the category theory summed up and concentrated the features of topology and algebra. And where that notion was next? It's very explicit. Well, it's absolutely explicit because simple functors like Take the set of components of a space that does not preserve the fact of being a monomorphism.

32:30 A monomorphism may become a surjection or vice versa, you see. And these are very basic punctures to this. Well, one sees the same thing. In other words, if you're in a context where the punctures did really preserve monomorphism, then you can sort of get away with this view. There are countless qualitative features of cohesion and algebraic topology without even introducing groups or anything, just as components. Functions, there are no functions in there at all, there's relations, there's a one-sorted structure, it just suppressed almost all the most delicate things you wanted to find in the structure of definitions. There's somehow a misguided idea that this is a simplification. You simplify things because you omitted to mention the code. But then, you see, on the other hand, the type theory did the opposite. Type theory did the opposite thing, because in type theory, the types themselves are co-domains. You don't specify clearly what the domains are. You have to look at the formula and figure out what the few variables are. And even then, you often want to consider the functions which are nominally of certain variables, but don't really depend on some of them. Things are factored through projection maps and so forth. So there's complete ambiguity in the domain. So many, many, many, many foundationists defending set theory, they say, well, okay, we'll look at category theory. Well, that's really type theory. You already know what type theory is. They're equally wrong. Because in both domains, it's the most elementary thing. Well, at the current time, they do have to make that distinction, of course, because they're actually being applied and they have to work. It's important. You can't get it off the ground without doing it. No. Also, obviously, the computer science, yes, exactly. But it has to work in computer science.

35:00 That's what I mean. Computer science has to work. So this has now come into presence that this neglect remains. The idea is that there are no obstructions to the existence of inverses, which again, of course, aren't very special because of the constant. One has to think a little bit less obvious to spot the geometric meaning of these abstracts. The property is an additional given datum rather than something that's just there. They're intrinsically so, you know. If you take a set, maybe you know it's countable, but is it the integers, the naturals? You haven't decided yet. Until you've decided what structures you're going to put on it, you're going to see it with a successor function. Then it must be the naturals. But it only became the naturals because you saw it this way. You've got a countable set, another countable set mapping into it but not onto. What's that? Well, it's only if you say, okay, on my base, I'm going to put this successor, and this is the events. So even becomes this insertion. It's not that anything down here was even. It's the insertion of another set into it is the evens. In other words, a subset is not a set. It's more than a set. It's an exertion. You know, I think the axiom of choice is... Well, if you think of the axiom of choice as... I mean, there were two great ones. One was the question of uncountably many choices, you know, there was that issue, then what does it mean, and so on. It came up from the criticisms of the, you know, some Burrell and some other people, but the other one really was, I think, the idea that some of the choices really aren't, in some ways, definable. I mean, a set was something that had to be thought of, not just as a bunch of

37:30 A combinatorial thing, is it, right? Where, where did combinatorial arguments in some intuitive way tell you, in some, you know, sense, that a choice set exists, you know, you simply shrink, you have a bunch, a bunch of disjoint sets, say, you know, you think of those, you know, bands of dots, whatever you like, you don't know what, and then you have a choice set, it's one that intersects, right, each set. No, no, no, no, no, no, a choice, a general choice, and just either set. So you know they exist, you simply take the union. Now you shrink. You know, you have this idea that they're just what you can throw out. Eventually, you tell me intuitively, in this combinatorial conception, you'll get a set which only has one element in common with each one. Now, Bernays, actually, I think somewhere, he calls, and he has, he's an argument guy, he distinguishes this combinatorial notion of that, which he clearly means something like, like bags of dots, you know, I think that they're, that they're just, yeah, I think that's what, that's what I understood what he means by the other notions, you know, extensions of properties and so on. Now, the combinatorial notion, in other words, I think the notion of an abstract or a cognosome, a cognosome. The actual choice is correct. It's intuitively very natural. You don't have to specify. All of these are examples of how to define the choice. That's not the point. It's something built into the combinatorial structure. And I think that's why I take it. That's one good reason for taking the axiom of choice as being correct, for the category of constant sets conceived in this way. And this is what Pfeiffer has some ambivalence. In 1912, all his career took the axiom of choice as obviously true. Because it only gives us this combinatorial information, we can't get our hands on it, it can't answer any of our questions. It's not that it's not true, it can't answer our questions. Now this is also a mistake, but it's a different criticism. In this 1912 Sir Anthony, he's got like four different answers.

40:00 I didn't know anything about the axiom choice. Yeah, it's just pragmatists say that Zerbello's well-ordering theorem is Pavlov-Bitra-Katz. Because it gives you a well-ordering. It shows you there is a well-ordering. But it doesn't really tell you anything about it. You don't have it. You can't have it. So it's a constructivist... Well, no, because he doesn't... Even his pragmatists don't say the well-ordering doesn't exist. They just say they can't use it. And he doesn't unambiguously endorse the pragmatists. In fact, as I say, he says they can't understand the Cantorians, and he can. So evidently he's not just one of them. Now, orienting people on this question, on this kind of broad foundation which starts that everything could be geometric and topological structures in some sense, in terms of additional structure imposed on some underlying set, was precisely the Baloch-Telsky paradox. The force of the world around us and the congress of precisely this force. I mean, that's what I wanted to do. The real world. Yes, because I mean that is something, it's true the axiom of choice is sort of the culture. You should be able to make explicit what the mathematics is. Well, I mean, in the end it just meant that you have a line, it's true, the Matochski paradox is correct for a color that's just not a paradox, if you can see within the... In this pure combinatorial sense, it's not really a paradox at all. It's when you represent continuous structures that you get this paradoxical aspect, and it is a very stark thing, admittedly. But then, since we should expect our mathematics to be able to make explicit concepts needed to capture the notion of intuitively and constitutively continuous, and there are many forms of intuitive, quantitative, and continual, but certainly... Well, I think there are a lot of notions of set and set theory floating around which got conflated. I mean, one appears quite natural, so to speak, in one manifestation, if you like, in set theory, and paradoxical in another. It did, of course, have a, it was a startling result and it did, it wasn't the first of them, of course, and the Hausdorff two-thirds of the sphere being equivalent to one-third and so on.

42:30 Yeah, Hausdorff really did the main step. Yeah, the main step was Hausdorff. So Pinsky did something on that side as well. But I guess it's true that since set theory was going to be used as the official, it had become the kind of, well, foundation, framework. Or language within which advances in analysis, particularly in this case, you know, and major theory and so on, were being made. I mean, all the work being done in Polish school, for example, at the time. I mean, Bartok-Tarski would consider this problem because, you know, as Polish mathematicians, many of the advances that were being made in the use of set theory in analysis were made in Poland in the 1920s. I'd like to be able, or someone to be able, to say something more about how this, all this idea of using logic and theory, etc. in the Polish school arises, because I have the feeling that it's not the same continuity like, you know, in Germany, something strange happens there. The first issue of Fundamenta. Yeah, yeah. When is that? 1913, something like that? Yeah. I think it's 1920, it's after the war, you will see a Polish officer in a full uniform, and then there is a manifesto saying that this is for Polish nationalism. Yeah, they definitely have their own ideas. Well, I think as far as logic is concerned, there was a strong influence of scholastic, you know, there's all sorts of logicians that can code all those, all the evil forms. Neotomism. Yes, and it's quite a strong tradition, which, certainly as far as logic is concerned. And you can see that in the work of Leshnevsky, the more specifically philosophical logicians, Vyshevich, Leshnevsky, and so forth. I've forgotten it, but the school was in fact founded by a humanist who wasn't really one of the practitioners. I think that's right. And he had, he brought, I've forgotten now, he brought in this scholastic. No, not Kildare-Vitsky. Kildare-Vitsky, yeah. Was it Kildare-Vitsky?

45:00 They could have been Kildare-Vitsky. That was it, yes. But they have a protector who's been him. I mean, they have been him. They have been him. But I'm not sure. The fact that this book does have an adequate account of all of this stuff before it gets sexy. But it's like a chronology, and I really, also with the Hungarian mathematicians, but that's a different story now. It doesn't have so much logic and set theory, but all these topological, set theoretical ideas, it's a strange... I think it was because there were philosophers who were not really very mathematical at all and who had been instrumental in educating people like Tarski and most of the automatists. It was a strange mixture, it's true. And there was a strategy that Brouwer seized on and that I suspect the Poles picked up from him, although Brouwer had it. This topology, this is hot stuff, this is new, it's not being done in the world centers of mathematics. We can be great in this, in Amsterdam, or in, you know. The so-called general department also played a key role. Not a math department, but a general department. And who was there in Duval? Rotated. Rotated. When Philzutsky wanted to step back and not be the absolute dictator, except in reality, he wanted to put up a figurehead, you see, then there'd be somebody like that, and then he'd change his mind, that person would go through the same general department. So this kind of people were completely mixed up and sometimes identical with the ones who were actually doing the mathematics. This is described in the... By Kwiatkowski, I believe. Kwiatkowski, 50 years of Polish mathematics. Mostowski wrote the book. Was that by him? No, Mostowski was 50 years of foundational science. It wasn't specifically Polish. 30 years. 30, 30 years. It wasn't specifically Polish. It probably was Kwiatkowski. Kwiatkowski's book was a huge sport at the time. It was. No such thing has been written since.

47:30 Even like the way Tarski was in Berkeley. I always remember when Gonzalo Reyes, I think, was in the science section. This pre-lecture of Tarskis for a simple-sorted verse, do you think that came from his own exposure to the... I think it was directly with Fregean tradition. I mean, that you sort of definitely, somehow, you know, they didn't want too many... I plead guilty. We're going to re-print... Doe is going to re-print Dello-Slobson and we're not going to get a chance to change that, Dello-Slobson. Oh, too bad. That's the way it is. We don't want to... Dello-Slobson! I hope you take that as a compliment, because almost everybody who's even now who reads it regards it as a delightful book, really, because things were done in such a way that it functions We went down that road initially, and then, of course, we reintroduced them, of course, when they really needed it. But it's still the case. We're doing algebraic theories now. That's the problem, you see. Even now, many of my contemporaries that say they're Newton are insistent on using relational language. And Zilber, who has a very deep understanding of algebra and geometry, frequently insists on using relational language. It's funny because, you know... It's like, I think Church's lectures also, but you see somehow these are basic principles which are not justified at all. That is, there's no argument given why you should accept single-sorted relations and not functions and so forth. No, this is just the way we do it, and you learn that on the first day. Well, this is how to think. And it's incredibly infectious. I remember from one year at Berkeley, from one year at Berkeley, incredible habits. In my early papers, you'll see, you know, the variables are x0, x1, x2. I mean, this is utterly crazy, but they always did it that way. And so I thought, well, I want to make use of this, you know, this logical...

50:00 An outstanding analysis of Robertson's original formulation. It was very much like that. I mean, that was slightly influenced by the Russell tradition, you know, type theory, because he does... But Robinson was like a slobby version of the Terraski. No, I know, I know, I have a story about him... You're going to do that again at the event. I have a story about him that's actually in Dobbins, not very good, but he does reproduce this story. I remember when I, this I'll mention it, when I heard Robinson... I had a lecture in 1965 in Oxford when I was my first year as a graduate student there. And, you know, he came for a term, he was visiting, I can't remember his college name, and he gave me, you know, these lectures on non-standard analysis and Moshe Makarov, who had been his student, you know, that's where I got to know Moshe. Anyway, a little bit of that, but it's sort of funny. He gave these lectures, and he was a very charming person, of course, but he was very nice, and the party at the end was really very, very nice. I have the impression that during these lectures that he had a rather muddled way of lecturing, at least from what I know, I mean, there's a sort of muddle, you know, and I remember thinking... He gave a, I can't remember what the result was, although the result, the actual presentation was fascinating because you were seeing the concept for the first time as marvelous things he did, but his method of proving things puzzled me because I, there was one point at which he went to some proposition, say P, and then he, to prove P, he would seem to assume not P. And if you think he'd be arguing for a contradiction, all right, so we're going to do not P, and what he'd do is then somehow prove P quite independently, and then that contradicted non-P, and that gives you double the answer, so you can't just P. Subtitles by the Amara.org community

52:30 But I, this is the way, this is in Dorbin's, I didn't, I didn't read Dorbin, but he asked for it, and I don't think, but this little thing is in his book. Somebody else noticed that. It is very strange. It gets into writing in Brouwer at one point, when he's proving the degree theorem, that the degree is invariant under homotopy, he gets these four numbers. And he says, I want to show these two are equal. Well, I can show these two are equal, and these two, and I can show these are equal. Now, suppose these two were equal. No, I've heard Dr. Robinson's death. I went to New York and I met William Magnus, who was a great geometrical groupist and a very charming old man. And he almost made me fall off my chair. He said to me, Robinson, I said... Michael Schumacher had been Robertson's student of course in Jerusalem and of course he was really impelled to try to clean up you know the presentation of non-standard and of course he wrote that thing with Hirschfeld you know non-standard about tears and you know and because he said there's got to be a better way. Was it better? Oh yes oh yes oh yeah no because of course it's become if you look at Well, it's interesting to say that Maché has a very nice account of a kind of distilled version of his approach to it, his version of his approach. In that book I wrote with, he wrote with me, he had built a mock-over rule, you know, a mock-and-bell-over rule. But it's slob and welson, you know. I'll leave you to throw your own conclusions. It's a very nice presentation. It's very pretty. He substitutes, of course, for the types. You just do essentially what he calls the Melo structures.

55:00 You do build this type structure by simply iterating the power set up. Anyway, it's much prettier than that. But you see, here I see, I can raise a question related to what we spoke about. You have here this new approach to a foundational problem. People are doing, did a little bit but it of course did not became mainstream in any way and some people, I don't know if still are people searching something about it. So what happens? Does it mean that we have to forget it and it was just an at the end or in a certain situation it could have been different and perhaps it will be different, I don't know, 10 or 20 years from now. We are structurally speaking to what happened with Cantor. The outcome is different so far. But what will happen? How will we judge it? There was no Hilbert to push this view. There were historians. For example, Darwin always speaks about the new revolution in mathematics. I asked myself whether in physics there is any of that. Well, there have been attempts. There have been attempts. I simply cannot judge the value. They have been at least done by people with very serious credentials. He certainly has attempted. That's Snoop Dogg giving me, but I can't really judge it. In the development of these serious attempts on serious unsolved problems, it's very difficult for mathematicians to understand that, and it's certainly true that these practitioners in the UK, the PDEs people in the UK working here, are not totally hostile to what's been done by these people, but they also find it very difficult to judge what... So there might be some development... Yes, I think so.

57:30 These high-level stochastic things are on. Enough has been tried. Nothing sensational has been done. But I think it's more than just honest work. Something might happen. It's very tricky. I mean, the total mass of analysis at the very beginning, not by Robinson, but by others, it was oversold. Including with Goethe. Well, he says that he thinks that it's a surprise that it wasn't discovered earlier and that somehow it's the analysis of the... and so on, and I can't remember the answer to that. There's always a solution to... He says isn't it extraordinary that we still have unsolved these arithmetic problems which can be stated in such an elementary form? Maybe it's non-standard analysis holds the key to their solutions. Well, I think we come down to the end of the chapter on non-standard analysis. I think it's the section which Moshe wrote, of course. I think it's called a cautionary remark, a cautionary tale. Yeah, yeah, I know. I mean, I saw it. I guess we say afterwards, well, you know... That's my memory. Because there's no unique mark, you know, it's not really... It's something relevant. But I think, unless my memory serves me wrong, I saw the... The original of that letter of Gödel to Robinson as Robinson received it. I mean, Robinson, I saw Robinson every day. When, sorry, what year are we speaking? 1974. No, Robinson died in 1974, and he was ill from the end of 73 onwards, and he was, much of the time, he was in the health center, just across the road from the math department. So I went in every day and spoke animation. He was getting letters from Fairfield. But Gödel's had more than Gödel's, of course. I think he reassured Robinson that he was in charge of proteins or something like that. There was no Ben Comfort in my time. High pressure, deep pressure. A blob of protoplasm.

1:00:00 I don't know what happened to that later. There's a Robinson archive somewhere where it is. No, there's something very sad here. Because Gogol was obsessed with... Yeah, that's what he said, wasn't it? Yeah, he was quite obsessed about it. But you see, there is another point here, if we compare that there is not just that by chance there was no Hilbert to push this, also perhaps that it's not strong enough to do it, I mean the theory, but also there is no possible Hilbert to do it, because, you know, just compare the hundred, the list of problems in 1900 now came to 2,000. So we were told to give a list because mathematics has become so diversified, disunified, let's say, that you have to put together some people, right, there is no single person who can have a vision, a general view of the field and say, we should do this, this is unimportant. You have to put together, I don't know how many people were in the client commission who came up with this. It's very, I mean, the disunity of mathematics at this moment makes such things very difficult. Or the diversity, I don't know if disunity is the name. As a social fact, that's undeniable. The profession is far huger than it was then. I'm not happy with people who say that math is conceptually less unified now. I mean, it's difficult, I mean, it is broad and you can't expect somebody in... In the center of the tetrahedron, to understand, they will see into every little corner of algebra and analysis, but even the dark people. Sevis, for example, one of the younger people. Manning, for instance. Manning certainly... Manning came up with a list, I think, and it was... Yeah, but Manning is certainly one of the rare individuals who's done this, I think, I'm assuming. Of course, he writes the book on mathematical logic, which I like very much. I think it's an extraordinary book. He likes it because it's a whole bunch of attacks. George Bull has attacked him for it.

1:02:30 Well, okay, his treatment of the continuum hypothesis was not as nice at times. But, I mean, the thought that one of our community might go in and write on the kind of things that Manny has done is beyond... You know, I think it's a really remarkable book. Concentration on some of the most beautiful things that have been done. And there are also speculations in which no one's ever followed up. He's got various diagrams about recursion theory which look really very much connected to serious issues in geometry and so on. He had pictures there about exactness and things like that. I mean, it's virtually a universal... The institutional situation is different also. For example, even the case, you know, you have so many important centers. Of course, in 1900 you also have, but not that many. Yeah, Hilbert was unquestionably the leader of the, what, three dozen German mathematicians. Exactly. Mathematics was relative to mathematics in the rest of the world, you know. No, but also the range... How many thousands are there in Germany now? I mean, it's just... Yeah, exactly. I think there are two quite distinct issues. We're talking about the disconnectedness or disunity of these two, probably three, quite different senses here. The enormous increase in the total sum of mathematical activity in the senses in which it occurs, in some sense I suppose carries much greater ease of exchange, and the quacks for the conceptual organisation of the subject are such as to provide. I'm not sure if mathematics is in now than they were in 1900, and I would say that actually the situation now was, it wasn't 1900. More helpful. More helpful, yes. In 1900. But because of the development of calculus theory. Well, no, I don't think so. No, no. If I talk about physics, I look at least to the main axis. That's saying, quantum, whatever you want to call it, and relativistic, people can talk through these things. Physicists working in different fields can always find an easy way to communicate.

1:05:00 I don't see that that much in mathematics. I see that in a more difficult way among mathematicians. Which is Atiyah in? Is Atiyah in geometry, analysis, number theory, topology? You can't even say which field Atiyah is in. Of course, if you mean that the average physicist can communicate across this barrier, whereas the average mathematician can't, you're probably right, but there are still a large number of people, conceptualized and abstract, and so on, people like Atiyah, who thinks... No, no, I'm not talking about the exception. I'm talking about what I think was an exception. Yeah, there is again, there is a dialectic here with the exceptional and the other way. Yeah, yeah. It's a very important issue because it raises this whole issue of the relation of individuals. No, it would be nice, for example, if category theory could have provided such an axis to... Well, I think it's... Most likely that that is the category theory. It has to be involved in any kind of... Yeah, but would the average mathematician agree with that? No, the thing is I can speak from better experience within logic that this is not agreed by the average logician. It seems to me close. I mean, given that many of them, at least at one time, were known for forcing and so on, you cannot get them to go just a very small distance and look at it. A fast generalization of the thing. This is quite bizarre, but then logicians are notoriously weird, I mean. Yeah, but algebra, perhaps they know the categorical language and the mainstream, let's say, algebraic, but I don't know about functional anatomy. Is he able to make some, you know, to enter the dialogue? You're right. Functional analysis is a difficult case because functional analysis went through a period of near-degeneration too. I mean, there are independence problems in functional analysis as well, but functional analysis gets revised. In what direction? Well, for example, in the direction of... Gauss? No, no, I think Gauss is a bit... Gauss is a combinatorial kind of thing, but functional analysis is certainly involved in Kohn's work.

1:07:30 Proximal analysis is an instructive case. It's certainly true, it's true. By the 60s it was probably washed out, but there were still a lot of people doing it, but they were doing little. Let's say German. This theory, this theory, weapons deal with problems. Here you have the problem, now bring whatever you can to solve this problem. I think this creates... There are many different kinds of education, of training, etc., and then you get these people who have these burdens, you know, I mean, I can't really follow, but I understand that he takes from everything around us. Exactly. There are so many kinds of people. This is the whole point. By this time, I realize that sometimes it's useful to take categories.

1:10:00 It's part of their... Some guys in Chicago, the representation, they're assuming they are computers. But that's clear. They have fundamental contributions. Hardcore representations. There is. And yet, you don't want to call them categories. No, no, they are. They are one language. Absolutely. That's absolutely clear. That's clear. But that doesn't in fact provide, It could also lead to an instrumentalist, what I call, kind of tool kit. You don't have to think about the five concepts as long as you've mastered how to use this particular piece of the tool kit on that problem.

1:12:30 I mean, a versatility, let's say, might. Unification has, in just a few practical terms, been so powerful. Yeah, in spite of... The question is, what is the aim? I mean, I know this better. Everything is very systematic, very clear. Is this more important than that? I don't know. I mean, I don't want to have an opinion on that. I'm just looking at that. And if you can solve so many problems, that's also an advantage. No, I think the comparison, or a possible comparison, is with the Italians in the late 19th century, beginning of the 20th. You know, you have all these geometrical, algebraic geometry that do all kinds of crazy things that later on kind of Germans or Americans say... You have this problem here, this is not defined, this is not defined. Okay, not defined, not defined, but there are so many problems. They were also unreadable to most people. The paralogue is important, Hilbert's standards for proof are important. If you just read student diaries in 1900, 1920, 1930, mathematics became so much easier to learn as text books appeared. You didn't have to go to the reading room. In 1920, you still studied by going to the reading room because you couldn't own those books. Springer-Berlaug puts out these series of textbooks in a very uniform notation, uniform standard of form. And then comes Burbaki taking a further step. We will say the terminology for all of mathematics. And some people say it also has negative things. It limited, or delimited at least, the kind of things that were allowed. For example, we spoke about geometry. Then it's the complete varying of, at least, on the text, etc. I think, I mean, you know, you have to take from wherever it comes. I think everything has its positive sides. The individual mathematician can only do that much, but if you look at the, you know, from the vantage point of the historian who can look at things and say, this was good, this was good, this was good, there are limitations. I mean, I don't see any problems. Pointing out, I see a difference because I look at the physicists communicate with each other, I think, better than mathematicians.

1:15:00 There's a famous remark, and I'd like Colin to remind me who made it and when, but sometimes you know.