Roundtable Discussion

0:00 The copyright out of Bell and Bokover so that we can try marching. It's ridiculous what they're charging us for that for. But in principle you're right, Michael. Yeah, sorry, I'm listening. I just wanted to say something that I wanted to use. We should not give them for what looks to follow. We should not buy them. Well, what I disagree with, though, is everything else John Roberts says, and what he says about N categories. I do approve of his campaign for people to... You know, to stop reviewing for sitting on the billboards of journals that are published by these outrageous price-ganging corporations. But I realise it's a difficult system. It's now the afternoon of the 16th, a chance to get a few things done. Can I thank everybody for a superlative explanation, and particularly thank Bill for the quite wonderful scene from the categorical standpoint. I'd like to say something about in this session, which I would like to see devoted, since we might be able to fit in an hour or half tomorrow morning, depending on what people feel like, but it's the last effective full session, to be seen in its developments, driven by fundamental oppositions, to ask Bill to identify the discernibles, which is specifying the notion of, one might see that notion from the capital. I'd like to lead off and say a little bit about the fundamental oppositions that you see as driving the development of mathematics.

2:30 Well, I think if one looks back at the mathematics of ancient Greece, in particular, of course, I suppose identifies as one of the sorts that actually took, since many of the basic categories of mathematical thinking were named, varying and static, one of the many, and the continuous and the discrete is something that they, they, they, of course, identified in terms of geometry and the continuous number of discrete. And, of course, they, although... We know that, for one reason or another, the Pythagoreans came to identify number, the discrete, broadly speaking, although where it's really used comes, of course, to Aristotle, as a kind of basic principle.

5:00 I'm thinking of understanding the world. And we know also that when their geometry, the continuous, in other words, the analysis of the continuous developed, they proved at some point the Pythagorean theorem. And this led, of course, as we know, again, to the problem of commensurability. The quantities, and their own, and this seems to have been an important aspect of their philosophy. It's claimed because of the discovery of the music. It's often said that the first crisis, the notion that there were three crises, but the first one is often claimed is that it is a clash in some way between the continuous and the discrete, the idea that continuity is the ratios that emerge from the Pythagoreans. It's claimed that they threw one of their disciples overboard, as we know, when he revealed this problem. All of this is probably just gossip. What we do know is that in order to resolve... This difficulty on which, apparently, early geometry, the idea of commensurability seems to have been the basis on which early geometry was erected. As far as we know, the usual formulas for determining areas of triangles and so on really did. They're very simple proofs, of course, based on the idea of commensurability and the various lengths involved.

7:30 And they're fairly simple, as we know. To develop the theory of ratios, of proportions, this was a kind of extension of measurement of the idea of reductions of the discrete into the continuous again, while admitting, of course, that in that case, continuity, in other words, geometric magnitude, had to be taken at some kind of given. In a sense, they're not reducible directly to the idea of number. Anyway, already we see, I think, the continuous and the discrete as a primary kind of opposition that led to important discoveries and development of mathematics, and of course, later on, the Zeno's paradox, which wasn't properly found in mathematics, it's true. Of course, the analysis of the continuous itself then leads rather quickly to the other opposition, or one of the other oppositions, because of the infinite divisibility of the continuum, to the contradiction between the finite and the infinite, which is of course one of the other driving forces. Of course, you could see these interconnections among these various oppositions. I think that, in my own discipline, is an oversimplification. In the case of the varying of static, it is also a major problem. As far as variation and stages are concerned, the Greeks really only did have a theory of the varying, but it was the constant varying, it was in some way the idea that Because they took a lot of this from astronomy, where there is a certain, you know, there is a form of constancy in the variation, and of course this is why they put astronomy in, as the Pythagoreans in the Greek included astronomy as one of the elements, components of mathematics, namely quantity, what they called varying, this was quantity, varying, and geometry was quantity, you know, static quantity. The contradiction between the varying and the static assumed, of course, already had assumed, when you introduce the idea of genuine variation, in other words, whether variation is actually itself undergoing change, this sort of double variation, which, of course, the Greeks must have recognized, but there were difficulties in trying to analyze that.

10:00 And, of course, this problem ultimately led, of course, to the differential calculus. Anyway, I see the broad outline of mathematics or the development of mathematics as being driven by these oppositions in a kind of schematic way. And I still see it. I mean, it's rather clear. There's still the opposition, indeed, between the notions that are actually being described and the way in which they're presented, for instance, in terms of symbols. Again, we see this same—formalism seems to me another example, right? Not only formalism, but another instance of this contradiction, if you like, or opposition, right, between the—in this case, I think, between the varying and the static. Geometry, on the one hand, which is either, you know, spatial or temporal variation, on the one hand, and the arithmetic on the other, is still driving the... I see it in physics now. It's quantum gravity. This is obviously an example, right? Although one has to say that in the case of quantum theory, it's a funny kind of discreteness. It's very impure. But broadly speaking, I think... You know, one can see, let's say, general relativity is a very clear and, I think, rather well-axiomatized theory of the continuous in some sense. Quantum theory is a rather impure, an algebra of a lot of things, but fundamentally the revolution in quantum theory, the term quantum itself means, you know, an isolated quantity. And it's, at least there's some idea of discontinuity or discreteness apparently underlying it, on some or the other. ...that most people find nothing more surprising than the diagonal. On the other hand, he says, gee, I would find nothing more surprising than if it were.

12:30 Well, Aristotle also says, and he's not quite sure, in what sense the Pythagoreans really meant all these numbers. That's the Pythagorean slogan, but are they atomists? Well, it's only clear. Are they atomists in the sense, let's say, of, you know, of Democritus and Eusebius? Who knows? But at any rate, there was, according to him, I'm sure he's right about this, something surprising about the lack of a discrete measure of the relation of the diagonal. Yeah, no, it obviously came as a shock to the pipe anyway. Well, I don't think that's obvious. Well, it's not obvious, but... But it's indisputably, indisputable that Aristotle expects his readers of the original mathematics to be struck at a viewpoint which, of which vision is precisely the conviction of. I do think the idea of the geometry, the early geometry, we don't know exactly, but the proofs of the area theory and so on are in some ways, the model for them is a commensurable case. And it's very quick, because, you know, you can cut things up in the appropriate way and rearrange, and when it turns out that these legs turn, I don't know, we don't know the history of it, but there's a rather clear model for the conventional case, which is quite natural, and you, you know, you choose a unit and you do these things, and indeed, that is still the model of Euclidean, you know, except, of course, one has to introduce the Eudoxian theory in order to be able to... What is their formula? What do you mean by their formula? No, I mean where you try to show that the ratio of areas of triangles with a given, you know, with a given altitude is as to their basis. It's, you choose a unit, the simplest way is to choose a unit. And then divide, right, of the same, and then cut up the thing into so many triangles, which is represented by the unit.

15:00 And that indeed is the way that, essentially, it's done, even in the... In later geometry, except, of course, you can't assume why you use inequalities, but essentially the natural proof is this business of dividing it up into smaller triangles according to the unit, with a common unit on the basis. And this is a standard thing. You can see it in simple, with conventional geometry, right? I mean, it's very simple. It's just cutting and pasting. It's just cutting things up and choosing a unit, exactly as I think that people, you know, measuring, you know, people measuring floors and so on, you know, I mean, it's exactly that kind of thing, when you're trying to build, and this is where geometry evidently must have come from, so. You know, what's interesting is that in Euclid geometry there is no units. No, no, no. With a theorem that says that whenever you have a polygon, it starts with a, with a, wait, you take a polygon and a length, then you can always, which was given by him, by his, is saying here is that you can always, for example, for example, a unit, you are saying that you can transform, talking about areas, really areas, it's a ratio, right, multiplication, yeah, yeah.

17:30 ...to what Eudoxus was expected if he had lost rhythm. No, no, because it was a ratio. The thing was that Eudoxus, at least as presented in Euclid, does seem to be, you know, that you can make equalities of ratios, right, of similar things. I mean, the ratios are all similar things. Nevertheless, you can introduce equalities of the ratios. So, a length to a length, ratio of a length to a length, is equal, for example, to the ratio of an area to an area. Because he doesn't say what these things, you know, are. All he does, well, he just says, you introduce an equality of length. You don't say what ratios actually are, well, at least not in the, you just call them proportions, and you're able to do certain things with them. It's a pretty typical mathematical move, of course. Practice of what we've been saying all along. Right, of course, there was this problem, I mean, another interesting thing, the idea that an area, that there would be, well, let me, it's interesting, as far as I understand it, the idea that there would be a ratio, let's say, between an area of a circle, between a curved area, a curvilinear area of a curvilinear, and a linear, rectilinear, for granted. Well, hence the old problem of, you know, it goes back a long way, the term of the quadrature of the circle. Now, on the other hand, apparently, Aristotle didn't think you really could do that with clipping up for lines. In other words, curved lines, it's an interesting point, you know, that there was no, I could read it, real theory of idea of ratios. Well, of course, in the case of the circumference of a circle, there were arguments. You're saying there was no theory of a regular... Yes, yes, actually for curved lines. I'm saying there was... Are you saying that there was no theory of ratios of continuously varying quantities? Well, you may say that, but it just comes to the thing that when you say a curved line has a length, it wasn't clearly read.

20:00 You know, the idea of saying that the ratio of a curved line, you know, to a particular straight one, given is... This was apparently not systematically worked out at all. I mean, partly because I suppose the idea of an arbitrary curve was... Well, you know, there are only a very few, you know, what I call transcendental curves, you know, in Greek, you know, Archimedean spirals and so on, but there were very few of them. And it's very interesting that the, as far as I know, the circumference of a circle given as a length, I'm not sure Aristotle would have regarded it as a well-founded question to ask what the ratio... If you take the circumference of the circle given as an angle, then the area, of course, is connected with that. But on the other hand, there's no general theory. Yeah, yeah. It's an old... How confident are you that Aristotle compared the circumference to the area? I don't know. The truth is that Aristotle is not very important in this context. Because he, I mean, usually, until not long ago, I think... What Aristotle said was taken as a good indicator of what mathematicians are. But today, historians of Greek mathematics, they say, well, you know, he was talking, he was saying many things, and he needed to have this very, out of everything around, biology, astronomy, whatever, but the people were doing all kinds of things. Mathematicians, even there, didn't, at that time, were not very happy about the... If we can trust, you know, to work according to certain constraints, they did any, you know, including the question of the quadrature of the circle and all these. But there was no way of making sure that you can do it. But rectification occurs. There was a problem in principle about the idea of rectifying curves, in a way which happened later. Hippocrates, if we can trust those fragments, was happy talking about the areas of women. Hippocrates would happily have said that the circumferences of two circles were in the ratio of their diameters. Whether he thought the circumference itself had a ratio to the diameter, I'm less sure.

22:30 Yeah, yeah. It's an interesting question about Archimedes. The first theorem of the circle and the sphere has the following meaning. It's very interesting. It is clear that there is a constant proportion between circumferences and diameters. Always. There is a constant area. There was not at all clear... The first theorem of Archimedes is to show implicitly that it is the same pie, because what he does is to say that the area is the same as the area of a triangle, which one side is the surface of the circle and the other is the surface of the circle. So, of course, this says nothing because you, how do you build this line? This is a problem, but at least says that there is a connection between the question of the area and the question of the, of the... The question goes through the same proportion, but it doesn't calculate that way. It's just problem of cross ratios. Well, when we say the circumference is proportional... No, we don't say that. We say... Well, we don't. It's the same as the ratio between... It's the same as the ratio between... Right, okay. Look, that's what I'm saying. When do people become confident that this is a conference that they could express what we express that way in this other way? But when do they become confident that this is a conference that they could express what we express that way in this other way? But when do they become confident that this is a conference that they could express what we express that way in this other way? But when do they become confident that this is a conference that they could express what we express that way in this other way? But when do they become confident that this is a conference that they could express what we express that way in this other way? But when do they become confident that this is a conference that they could express what we express that way in this other way? But when do they become confident that this is a conference that they could express what we express that way in this other way?

25:00 But when do they become confident that this is a conference that they could express what we express that way in this other way? Yes. And he's going to show you some of the rest. Yes, yes. But this is the first place where they show, as far as I know, that it's the same ratio in both cases. Yeah, yeah. Oh, yeah. Oh, yes, absolutely. But I do think that's a good point, the idea that curved lines and straight lines really could have a ratio. I mean, really could be compared, because they're really two different types of objects. Yes. Which brings me to a slightly broader question. ...that the theory that's being proposed is an absolutely general theory of proportion. There's no question at all about there possibly being quite radically different kinds of subject matter and theories of different sorts, which you say was problematic. ...come clear that the subject matter of a theory of discrete quantity is the same kind of subject matter that belongs together in one overarching framework with the theory of... Is it evident that the Greeks thought... It's a matter of arithmetic and geometry as necessarily fitting together within one single... Well, if you look at Euclid, there are... I mean, that exposes it as a... You know, the theory he gives of general proportions seems to apply to numbers. But he does it all separately for numbers. That's my question. That's the point. That's the point of my question. That's the point of my question. I don't think... I mean, it's like this. To make sure... You don't really need it, but there are certain special things that you say about numbers that you cannot say.

27:30 I'll give you an example. For example, if you compare area, area, it's not an equality, right? You cannot multiply, you cannot do this with numbers. A, C, so you have a particular theory. The theory of numbers with numbers is more special and therefore has many theories that do not apply. Sorry, what's wrong with the equality of our volumes? It's not equality. So you can't rearrange your ratio? Yeah, you can't. I mean... Did you say volume one? No, you cannot... In fact, you know, we even corrected today A to point B before point C. Even this thing doesn't appear there, is it? It's just a phrase. The relation between A and C is the same, or the ratio between A and B is the same as the ratio between C and D. Now, if A and B are volumes and C and D are lines, You cannot even say the ratio between A and C. There is no such thing. Why? Because they say it very clearly. They have to be the same time. What does it mean to be of the same time? It's an Archimedean condition that if you take the smaller of the two, you can add to itself enough times to be bigger than the bigger. You cannot do this with a line and a volume. No matter how many times you add a line to itself, it would never be bigger than the volume. So you only compare things... It's not comparable. It's not comparable. Even in a matter of incommensurability, it's not comparable. So the theory, it sounds strange, but you have to really feel that it's so. You compare only things of the same... A point that I wanted to make is the following, I think it's an interesting one.

30:00 This thing was so strong until the 17th century that I think it explains an interesting phenomenon which is the following. Look at the three principles of Newton. What do they say? One is inertia, the second one is proportionality, and the third one is action and reaction. Let's leave action and reaction aside. And the proportionality. In modern terms, what do we say? The second one is M times F equals A, right? Did I say the right thing? Now, what does inertia say? Inertia says that when no force is applied to the body, then acceleration is zero, right? Or say zero in the force, you get acceleration zero. So in these terms, inertia is a particular case of the proportionality between force and acceleration. It needs to have three principles, it could have only two. You understand? The first one, in modern terms, inertia and proportionality between force and acceleration, one is a particular case of the other. All of this is formulated in the way I said before. So, force 1, the proportion between force 1 and force 2 is the same proportion like acceleration 1 and acceleration 2. You cannot start, there is no mass. Mass plays no role in the second, in place later on because it says why there is a mass is this. The opposition to force to create the acceleration. It's written in this language, in the language of geometrical proportion. It's not written on the face, it does talk about the proportion.

32:30 Yeah, but not in the definition principle. Not in the definition principle. You're somehow presuming that a ratio must come out to be a pure quantity, is that what you're saying? It's not a quantity, it's a ratio. It's just a ratio. Yeah, it's not a quantity at all. Well, what's a quantity then? It's not complete, it's clear, but it's the ability to compare with similar... It's true also for forces. In Newton, that was not the case. The idea of the century, but the mathematics of which you deal with then, doesn't change. I mean, if you look at the, at the, at the Poiketians, it's much closer in form and to a book by Euclid, or than it is to a book by Lagrange, or by, even by Bernoulli, because they already used the calculus. I think he states, he talks about one force and one acceleration. Now you can say this implies, even, it may even be stated in those terms, F1, but I really doubt it. I doubt it. No, no, no, I'm sure. In the terms you're talking about, F2, F1, F2. No, the ratio is the same. Ratio between forces.

35:00 This is a very interesting one, but since our time is limited, I think I'd like us to move on to... No, no, no, it's true. We could easily spend the rest of the evening here talking about mathematical mathematics. Perhaps we need to move on to John. Yes, it is. Let's move forward, again, looking at this between development and selection of mathematics. Let's in fact look now at the 17th, 18th and... I'm sure you'd like to say something about how you see this continuing, this opposition continuing to, and developing and evolving. Well, I'll do my best. I don't know. In the early 20th century, the idea emerged in the, well, I suppose from the Middle Ages on, when one had to consider non-constant variation. And you can see it in the idea, for example, of, well, one does genuinely have accelerations, for example, in motion. Were the Greeks the Greeks? Well, the Greeks, of course, you know, they analyzed... Circular motion, which, you know, came from, you said motion in a circle, which got into a motion in some, well, a circle, really, because they didn't really see, they had a theory of conic sections, of course, but they didn't, as far as I know, didn't see natural motions, any natural motions in terms of ellipses or any of the conic sections. I don't know whether there was a connection ever made. I don't think so. But anyway, this began to, and of course, when one has the idea of acceleration, of a kind of double variation, a non-constant variation, then, well, then you have to introduce the idea of, introduce a further kind of the opposition between the varying and the static, which wasn't really present, I think, in Greek thought.

37:30 Now, if you actually look at the emergence, and of course, the result of all this, this new analysis, particularly the one we see in Galileo, you know, the idea that there had to be an analysis of actual motion, which was not constant variation, I mean, it wasn't just the classical motions, trajectories, motions, of course, of actual objects in space. I mean, it's sublunary. I mean, not astronomical. Galileo specifically talks about motions that occur in a kind of sublunar history of actual objects rolling down inclined planes and things that you can actually see directly in front of you. And that is much more complicated because you actually do have genuine change, you know, a double variation. Well, I think, of course, the emergence of the calculus from that, the idea that one then had to reduce this idea of non-constant variation to a kind of constant variation, in other words, linearizing, well, we would see it as perhaps as linearizing, right, the curve, and then the connection, of course, the great achievement of calculus was to see the connection, if you like, between the... You know, between the direction of the tangent vector and, of course, the area under the curve, which was a fundamental thing to calculate. That effort, as we know, went on to struggle to kind of establish a solid basis for working with these things. It went on for a long time. And, of course, the history of the calculus, of course, is simply simplified by saying, quite wrongly, that Newton and Leibniz just invented it, which is, of course, ridiculous. But nevertheless, they were probably the first to see explicitly, I guess, and formulate, you know, this fundamental connection, you know, between quadrature... On the one hand, and the idea of the direction of the, well, if you take a moving body, then there's a connection between the area, this quantity, one quantity, and this, well, between the derivative and the integral, but actually between what you might call the kind of, you might, between two quantities, which, both of which are undergoing, you know, simultaneous variation. And one more point, and it was the notion of a function.

40:00 Right, which was struggling to appear without the notion of a general correspondence between two varying, right, between two pretty well, maybe arbitrarily varying quantities, but when you saw the connection, the correspondence between, without that notion none of this would have emerged. That's why we see, The notion of a function, in some way, I believe, is also emerging between this opposition between the varying static. As far as the contiguous and the discrete are concerned, of course, the calculus... Variation always meant, of course, really continuous variation, and what the term continuity meant, of course, the meaning of the term continuity was intimately bound up with the development of the notion of function, and for a very long time, of course, the term function, which was introduced, I think, by Leibniz, I think it's Leibniz who first uses it. Really means, in one way or another, continuous function, I mean, even though that had, you know, something you could draw, it meant some kind of trace of emotion. And this idea of correlating continuous variations... Mathematics was an important, I mean, a major element in the analysis. And the other calculus, of course, was to make it calculable. In other words, to introduce numbers in one way. In other words, to, again, to take the continuous and not reduce it, but to correlate it with discreteness, as so much of mathematics, as I think a lot of mathematics is about. Hence, of course, introducing the thing of the relationship between a pair and its representation. Yes, it's often, it's all very similar. A theory and its representation, I mean, the topic which has come up so many times, in fact, I was wondering whether at some point we were in a mental opposition.

42:30 One thing that comes out of that is the syntax and semantics are often linguists tend to only speak as if that were the case, but if you analyze it, it's not really quite right. I mean, semantics is about the relation between the abstract general and the concrete general on the one hand. And on the other hand, syntax is about the presentations of abstract generals. So there's this third thing, the abstract general, that sits between. The two just merely overlap in the middle, they don't. There's no direct way of going from the concrete to the syntax that would present a language which would describe this. It's a two-step process, each of which is an adjoint. But as you compose them, you just get stupid things, I mean, not stupid, but useful things, like the, you know, the free group generated by all the elements of some given group. You see, and that sort of thing is not really the syntax of group theory. It's an important part of developing the whole theory of the syntax of group theory. But it's interesting that a very close approximation to the specific notion of presentation that's appropriate to a given notion of abstract dental is determined, by the way, so that it's not an additional choice. I'm not talking about the particular presentation, but the meaning of what a presentation is.

45:00 It seems to me that one of the fruits of the lack of understanding of this point is precisely the abstraction and also the extraordinary as if there were some ontologically useful general notion of structure which was somehow prior to any notion of what it is the structure of. A simple example is the propositional calculus. Nobody would ever write down the propositional calculus unless they had an idea of what boolean algebras are. They may not know the name of a boolean algebra. They may not have a perfectly clear idea. But the idea that there is something like a boolean algebra is in the back of the mind of anybody who writes down the syntax of a boolean algebra. In other words, the fact that you cannot perfectly know what a Boolean algebra is until you have perfectly worked out the compositional calculus does not refute the fact that there is a dialectical relation between the two and the growth of the precision of your knowledge about this relationship. What's the general non-specificity of Boolean algebraic hydropotency? I don't know. It depends on, you know, who's making it up. I'm just saying that you notice that there is this kind of relationship. And, and not, and all this are as syntactical things are crystallized of at least a partial consciousness, but a full consciousness of similar operations on the...

47:30 I think it just seems to me, why would you ever write down a syntactical theory if you had no idea whatsoever what it was going to be about? But in other words, I think if you take Church's book, for example, you will get the impression, well, somehow people just do this sort of thing. They write down strings and symbols, and they have rules for manipulating them, and this is not the way the knowledge that went into Church's book came into existence. I'm just meeting that perhaps it would be better to raise him, which isn't in connection with the Bill's view of the Curry-Howard isomorphism. That's something I'd like to raise him. I think that seems to... You've talked about the history of the Curry-Howard isomorphism, but I don't think you've written it down in any place. Well, I've written it down in this Siena meeting. And there's a book of Marsha Deckard, probably recording it. Yeah, here it is. Yeah, there's something, yeah. You know, to avoid being my friend Howard, it's very likely. Yeah. So... Yes, yes. I have two main objections just to the term Perry-Howard isomorphism. First, the mathematical one is one isomorphism. It's just a... except possibly at the level of pure syntax. Because the thing that you are trying to present, in one case, is a Cartesian closed category,

50:00 The content of the idea is if you want to present a hiding algebra with operators, one way to do it would be to first present the corresponding Cartesian closed category, work with that, and then collapse it to get the hiding algebra. So the fact that you can do that, or whatever, the collapsing map is certainly not an isomorphism, you're essentially, you're introducing a new rule, high impotence, you're introducing the fact that the Cartesian product of two spaces, or whatever, is isomorphic, sorry, a thing with itself, is isomorphic to itself along the diagonal map, so if you introduce that axiom into a category that's operating, then you reduce it to... Propositional theory. And moreover, this picture represents, in many ways, a good model of the objective aspect of proof theory. In other words, proof theory is often viewed as a purely presentational thing, a purely syntactical thing, but it has an objective content. This kinder thing, which is objective content, had many conversations with Kreisel. Which led me to believe that there was an already existing theory about this at the time, in the early 60s, mid 60s. So I gave this talk at the Los Angeles Theory Museum 7, and I wrote these papers, thinking that, well, I'm really giving a categorical, hopefully clarifying presentation of something which is already known. Martin, Lerf, Howard, and so that this whole tendency simultaneously or even after the categorical formulations so this diagonal argument state can be printed on the tack very soon which was from it it's explicitly outlined this fact that I just described it was a student of McLean by the way

52:30 Somehow I didn't meet him when I was in Chicago. He must have overlapped. But then he circulated this letter in the late 60s. It wasn't actually printed until 1980, the time when the... So there is this strange reluctance of this group of proof-bearers to recognize the categorical origin of their subject. Yeah, that's what I was interested in, and then Howard's not in there at all. By name, no. No, but he would mention the reasons for that. Yes, the math is, but the history isn't. And that's what I was interested in. Not to be petty about it, but any number of logicians have told me that, oh, they all, everybody knew that the axiom of choice implied excluded mental, you know, but long before. The first sentence refers to Diakonescu, which is the sole reference in the bibliography.

55:00 Theoretical theory is the sole entry in the bibliography, but any number of logicians have assured me that my Helen Goodman did this before Diokonescu. I hope that was extraordinary, and it's not for the work, in my case, that I mean the topos theory, actually. Well, I don't know. That was my basic result. I don't know how a lot of them first understood it in my hell of a good manner. Yeah, I don't know. I don't know why they tried. The purpose in writing it was to cut this result, which was already established, into a language which they thought to understand. Absolutely. And they did a very nice, very pretty formulation. I was astounded when I looked at the Michael Goodman article. You immediately learn that it's after Diokonescu's, and then when you look at it, you see it's all about being after Diokonescu. That's what it says. I don't think logicians have the slightest idea. I know that some logicians claim that. I had no idea they were referring just to my... Well, see, Brouwer had already suggested that the axiom of choice was a complete principle in the sense of a principle omniscience, a principle that would let you solve all questions. So an intuitionist cannot accept the axiom of choice because it will answer all questions. But this is a much bigger notion here than implying it's simple. Also, the cryosol, I think, maintained that this was, in its essence, known. Or did he simply maintain that one could easily prove it with intuition? Sure, but this is quite typical of cryosol. Sometimes, for me, there's something or the other. You know, I've seen something, and then something survived. I can almost ask him about this. I mean, it comes as a bit of a surprise to me. Yeah, I mean, I certainly don't be able to see any of them.

57:30 It's the first time I've ever heard anyone say them. It actually, it doesn't, it sort of confirms my experience. I certainly wasn't, I was in ignorance of this subject at all. I wasn't. Well, I think, I mean, my, okay, one big failure was that I didn't actually publish my Los Angeles Step Theory talk, per se. It's one of those mix-ups, you know, the editor was Dana. Come on, yeah, Dana was one of those notatorians. At least I'm listed in the second volume, given this talk. Namely, it's called Category-Valued Higher Ordinance. That's right. And the idea of category-valued is that if you squash the category into a poset, that this is the procedure for going from proofs to propositions and so on and so forth. As I say, the basic material was split into three other papers and published in the next two or three years, but certainly when I was lecturing at the University of Chicago in the summer of 60, so the same summer where we had a meeting in Los Angeles, I mentioned this. I wasn't in Los Angeles, I came back to Britain, but I mean, I was around before. This is often puzzled when you look at how this suddenly takes the perspective of the Pittsburgh meeting last year. No, I didn't get much of it. I didn't mention McLean or Lee or, you know, sort of the history started after, yeah, 1970 or something. This was already in existence. Well, since the issue of choice has come up, and since time presses, and we've got quite a few other topics I was hoping we could cover, I'd like to actually ask Bill at this point, since bear in mind this session we're having now is intended to focus on broad philosophical... Themes, as connected with the concerns of philosophers and mathematicians, say a little bit about this issue of the opposition between the varying and the static in the context of the excellent choice is one of the strongest indications of constancy and the way that its negation of the 4-4-5 model theory of topology is referred to as organically varying domains.

1:00:00 Demands which are in some sense organically varying, i.e. things which take more into the nature of real world variation and cohesion. And this is something which has always intrigued me because I think it's pedagogically a very good place to start, to start in significance of category theory as a driving force in general philosophy. I may be quite wrong about that. Well, first, I mean, about Iagonescu's result, which was a very good result. He came up with it entirely on his own, by the way. I later realized that actually I had already proved it myself. No, in a different way. I mean, it's actually a different theorem in its actual generality and statement, though. It has this same particular consequence that acting with choice implies moving logic. And then he, you know, assuming that every match splits, he constructs certain simple quotients overlapping two copies of something and saying that's split and that splits in the sense of having a section, but it also implies splitting in the sense of having a complement, something related to having a complement. So it's in one way rather general because, you know, it could be any topo, it just happens to F. Everything's splitting, but it is a topo, so you assume everything that's involved in the general power set. Well, in a different level, actually in my set theory paper at UTCS, which has now been reprinted, thanks to Colin, you can read it in the TAC also, I have some commentary about this.

1:02:30 I don't assume it has a sub-object, but rather I prove that it has, and that it's two at the same time. So I'm simultaneously proving that these very strong axioms, which are different from topos axioms, imply that it is a topos and also that it's a Boolean topos. And the method there is simply that using the Cartesian flow, you have 2 to the x. What does 2 to the x classify? It classifies, of course, the complemented sub-objects. That's clear. You can classify the complemented sub-objects. And on the other hand, the notion of infinite union is... You know, it's actually the image of some map on an infinite sum, and if the infinite sum is uniformly parametrized, it's really a product. And so, you know, it's basically planetary from this point of view to take, say, a parametrized family of a decidable sub-object. So that is a map f from parametrizer into 2 to the h. You can then construct the sub-object. So now, given an arbitrary sub-object, an arbitrary monomorphism not known to be, you can define the object which represents all the complemented objects which don't intersect it, and then you can take the union of all those. So now the object is to prove that in fact this union of all the complemented things that don't meet a certain piece of the model is in fact the union of those two is everything. So now you proceed by contradiction. If it were not the case, then there would be some point that, you know, that it missed and this point, the map that expresses this point would be split by the action of choice. So at this point the action of choice intervenes to show that this union really is a complement of it.

1:05:00 It's a rather very different argument. It uses a higher order structure. That issue about, no, about the axiom... I was going to say, specifically, I'm sorry, obviously, I'll give you the floor, but in connection with the, as I say, variation and cohesion, I'd like to get a feel for the way that the negation of the axiom, reflect, is connected with the different ways that choice can be expressed. What do you mean by he does not agree with that? Yeah, the fact of the axiom. The failure of the axiom. The failure of the axiom. The fact that the Banach-Tarski paradox is obviously not true. I'm sorry, I'm using it. Yeah, the fact that the Banach-Tarski paradox is evidently not true of the world in which we live. And the way in which its failure obviously occurs. And perhaps which of the specific ways in which one thinks of the axiom gives one some insight into what is going on here. It's very typical. The typical situation is that you have objects that are not projective or not injective, so in other words, maps that are epic and don't have sections, maps that are monics and don't have retractions. Most of the objects, you're lucky if you can even get enough of them floating around enough projectives to serve context. It seems to me that's the most profound insight, because it's the most profound insight to get across to philosophy. Because it suggests a way of conceiving of how logic, the subject matter of logic, actually fits into our understanding of the world radically different from the way that the people who think that logic is something which has to underpin... Well, Zia Kaneshi himself, actually, not that this is published, but he had a theorem, the way that homological algebra measures the failure of being projected in cohomology.

1:07:30 So he takes, I don't remember maybe quite exactly, he takes a model, a topos maybe, anyway, you can define cohomology. And then there's a theorem that these vanishes in the whole of the action of choices. I actually remember that. I think Andreas Blas. Blas has done that. Blas had a paper in which he... I think Blas actually published the paper. Yeah, he published the paper. I think there's actually more than one that's taken up a little bit of this idea. But it's sort of this general pattern, as Cartier repeated many, many, many times. Take cohomology in order to measure these kind of things, and so it turned out that this set theoretic question can be inserted into that pattern. The very natural cohomology theory, which vanishes if and only at the accident of choice, is true, but otherwise you get a long regroup. I don't know if anybody's ever used this, you see, but in principle, they're getting roots which measure the extent to which it fails. Right, Mike? Yeah, absolutely. It's not only a very nice idea of mathematics, it's one which seems to me to connect very deeply to the theme that one sees. In fact, this whole line of development in terms of concepts came from the most profound time of the understanding of real-world sculpture.

1:10:00 In the use of this, you'd actually have to compute these groups. That's right. Say you construct a model set theory with the HCI kind of choice, and it has such a group now. What's the relation of that group to the forcing theory? These are the permutations that you use in order to do this. But also, these are... It should be more computational before we say that it really helps at all. But also, these are... It's an n-indexed sequence of groups, or how is it? It just only depends on phase one, actually. Oh, that's the question I was wondering about. As I recall. Well, Blas was a student at Harvard. Yeah, at Harvard, yeah. And who did you know when you began this career? Boach, was it harder? Boach, Raoul Boach. Oh, Raoul Boach, yes. Yes, yes, who in turn was a student of Ernst Strecker. Yes, yes, yes. So, you see, there have been close connections for a long time. That's right, Raoul Boach, right. I didn't know that Blas was... Now, I'd love to hear your thoughts to me from somebody, and Bob is very ill now, very limited, but to somebody of the new, you won't recognize him. Clearly, you know. Well, not that I know him. I'm trying to look at this in several ways, but you're right, it's something that's extremely good. I think there's a lot of sense in this. Profoundly illuminating, and obviously the technoplasts, what I want to see is sort of a set theory, homology fitting together with huge interest to the mathematician, but I come back to my thought, I think that this is an area in which I want to see speed.

1:12:30 Well, I actually want to... Well, I actually want to... Well, I actually want to... Well, I actually want to... Yeah, yeah. From the beginning, it was formulated, it was identified as a kind of, as we know, as a controversial... What's known in such theory about ways choice can fail? I mean, I know nothing. I know it can fail. It can be true or it can not be true. What's known about the extents to which it can be true? Oh, there's a lot of work done on countable choice. There's a whole jungle of results, I think, which I confess I'm not all crawling. About how it can fail in some way. This countable choice is an important principle which is true even in constructive settings in certain kinds of formulations. But what I mean is that it's very curious to me. It is interesting. It was taken as a principle of logic. It was taken as a kind of basic... All of this is a logically evident principle by some, but it's real justification on the very nice point I think I said the other day. I mean, really, as far as he was concerned, it was only really justified on what you might call combinatorial grounds, or so he says. And I don't think anybody had made the explicit connection with logic. You see, Hilbert actually says... He likes it. He uses it. They added the actual choice for him, who was a classical principal that he wanted to justify. And I always find it rather interesting when I, well, I don't know how long ago it was now, 12 years ago, so, I showed that in his epsilon calculus, what he used, if you add, if you take the intuitionistic epsilon calculus, right? In other words, don't assume the law. He tried to justify, Hilbert's trying to justify the law of excludability. He can't assume it.

1:15:00 So we take the, if you take the intuitionistic logic. Take the epsilon calculus. You can prove it's classical. In other words, the justification of the epsilon calculus for Hilbert was essentially because of the axiom of choice. He thought the epsilon terms with two variables gives you essentially a choice function. Hilbert was aware of this. But he didn't know, yeah, yeah, but he didn't know, of course, that, I mean, his Diakonescu, I mean, who actually shows, well, I formulated in terms of the epsilon calculus, but it's an application of Diakonescu's theorem, but it shows really is that this principle kind of forces the logic, you know, Hilbert's minimal sort of logic to be classical anyway. Although he wasn't aware of that, but nevertheless he did think that the Axiom of Choice was essentially a classical principle that justified, that had to be employed, or needed to be justified in some way along with classical logic. So it's sort of interesting that Hilbert had, there was some connection between classical logic and classical reasoning in the Axiom of Choice. He didn't, you know, he made no explicit connection. I was amazed. I thought, well, yeah, now we see what we can do. I mean, I thought that was just so impressive. And I tried to find other ways of formulating it. For example, you know, with the epsilon calculus, which has always interested me. And it's true. I just mentioned it parenthetically, that it happens that if you take Hilbert's epsilon calculus and you just take the pure epsilon calculus without what's called acronyms. Which says that the epsilon terms are extensional. That is, if A and B are two predicates, such as you can form like this, AX equaling BX, then epsilon A equals epsilon B. If you don't have that, Right? Then you just get, you get a genuine extension of what you mean. Well, that's very strong. Well, it is. And that, well, if you don't have that, what you get is the, is A implies, you know, that A implies B or B implies A. Yeah, that's good. But not the log-exclusive middle. I showed it was independent. And, of course, if you have Ackermann's principle, you get the full Diakonescu argument, naturally, because you have extensional functions and so on. And then that gives you the log-exclusive middle. Well, yeah.

1:17:30 So it's a sort of... That's what it is. Yeah, that's very strong, and of course, but even without it, you get this dumb, whatever it's often called, linear form, linear implications. It does sound a little bit like taking a howitzer to a nap. Yeah, I mean, but it's interesting that you're sort of forced into that, where, since the natural framework for Hilbert was a kind of, you know, the logic. Quasar kind of intuitionistic logic, we don't assume excluded middle, and you want to just, when he's using epsilon terms, when he's already lucky, you know, something like classical logic, you know. Just another question, I couldn't really, I alluded to this yesterday, this is a result of Chrysler's, right, I've seen proofs of numbers, and also proofs of some things that are homotopic to you here, so, can... You know, it sort of collapses the classical logic very quickly, I mean, I don't know, there might be forms of it which, you know, I don't know. No, my guess is that, I just wondered if I ever did it. This comes to finding some kind of natural topos theory of constructible, in the sense of constructible sets. It's not going to be the objects, because even in ZF, every constructible, every set is isomorphic. It's going to be arrows or inclusions, because those are not always isomorphic. You need some theory that distinguishes some inclusions as constructible and others not, in a dogma. And then you can talk about, can we relevantize it? But it just hasn't been really done. It probably doesn't. Bill Mitchell's paper does. He gives a vocabulary for doing it, to translate the definition into the internal language. No, I agree. He doesn't show the upshot of it at all. He doesn't really pursue it or show the upshot on the hydrant, but he did at least, I think, show that it can be done.

1:20:00 No, but to me, that's the most surprising thing. It's totally surprising. But he does it in terms of trees. I mean, that just says that in a two-value complex, you can use the theory of trees, and you can interpret it in a mathematical set very that way. All of these are non-constructible by translating the definition of constructible sets and non-constructible. Yeah, but when you get done, you found that in the original category, some maps are constructible and others aren't. That's the amazing thing, because the whole idea is that the sets have no structure whatsoever, but it turns out that maps between them do. You know, so you think of a map and it grows equally as a family of the co-domain, the fibers are finite, and that the co-domain that is the parameterizer is countable. Then, you know, sort of the growth of this amongst the, there's a certain number of theoretic functions. The number of theoretic functions are so large, very deferable. Now, this isn't the definition that the original map was constructable, but it's showing that it has an invariant right, which is either constructable or not, so it's a position on the map. Yeah, but he doesn't show that some are constructable and some aren't. He shows that the relative consistency theorem for diverging lifts to this context. You can't prove from ETCS that some maps are constructable and some aren't. Because one of your models of ETCS is a constructible, any construct is a model of ETCS. Well, no, when I say some or not, I don't mean... You show it's consistent to suppose that some of them are. No, no, that's not the point. No, there's a predicate. Somehow there's a predicate. Constructible. Yeah, and you can't prove that any arrow doesn't have it.

1:22:30 I didn't say that. You can show it's consistent if you don't have it. All I'm saying is that you can define this fact that it has the significance of an arrow. Of course, there are different models then. It might be the whole thing, it might be much, much smaller, but of course I didn't say that. But just the fact that there is this predicate is, at least this derived thing, the thing that you would derive from it. Well, yeah, and this is a perspective that the set theorists I know don't have. They'll say some subsets are constructible, others aren't. But, of course, they think of a subset as some of the things in a set. They don't think of it as an inclusion. They don't realize that constructability really arises on the level of relations, arrows, inclusions... Whether it's maths or inclusions, it doesn't really matter. But this is what I'm saying, they don't. They think it's about subsets. Yeah, but yet at the same time, they're a little aware of the phenomenon of cardinal collapsing, which is exactly about maths and this whole... How should they know it affects maths? They don't know that it actually lives on the level of maths. Yeah, but I mean, but cardinal does live on the level of maths. They do know situations... They do live on the level of maths. They don't know it lives only on the level of maths. Extremely interesting issues, certainly. But since, as I say, time passes and we had said we'd try to cover as much as possible of the broad and subtle topics that John submitted, perhaps we could move on to at least the third of his topics, which was a discussion of the whole conception of Galileo. The whole conception of mathematics as the book of nature and the specific role of category theory within such a conception which indeed I think has just been touched on very clearly by Bill in the account of just the role of projectives and injectives vis-a-vis the failure of the axiom of choice and structures which pick up with cohesion, very non-arbitrary variation. It's characteristic, obviously, of sculptures in the real world. But perhaps you'd like to say a little bit more about how you see it spinning into such a conceptual, Galileo's conceptualist, latter-day, and indeed, is the conceptual organization of mathematics today such as to suggest any sort of possibility of really Galileo's conception in a strong form?

1:25:00 The reason I mention that was precisely because, really, because of the development of category theory. I mean, like, well, of course, remember that as it happened, do you remember the bill that actually uses the example of Galileo in several places where, you know, Galileo's analysis in a simple case where one talks about velocities and trajectories and so on. No, I think, I, I, um, That's right, I mean, it's all, the, but, but, but the idea that, that, I mean, Gallo, I do have the idea that, uh, in some way, the mathematical notions are, uh, well, it's called the book of, you know, the book, Look to Nature, to look, the mathematics provides a book of nature, in other words, that in order to understand nature, I mean, to, to... To understand it at some conceptual level, it's necessary to express it, to formulate it in mathematical terms. And of course, in that case, the issue is to identify what it is about mathematics or what kind of mathematics, if you like, or the core of mathematics that enables that conceptualization of nature to take place. Now, I think, of course, I mean, one thing that struck me is the fact that, of course, in geometry, Well, surely we have to see geometry as some kind of reflection, at least of some abstraction from space or measurement and so on. That's the very term, you know, measurement, if you like. I think the calculus, and I think this is why, although Galileo is really, you might say, pre-calculus, or at least it's sort of emerging in the 17th century, but it isn't. I think that the idea there was that the calculus, I mean, one thing that struck me, when you learn calculus,

1:27:30 You always learned it, I learned it, in connection with physical problems. The way you understand the calculus, really, was by, well, you understand that by analyzing motion. It's rather obvious that the language, it is the language of nature. Right, right, right, right, here, here. Yes, no, that's the way you learn it. That's how you understand. There are, of course, then... Of course, conceptual difficulties, well, after all, conceptual difficulties had emerged and they were side by side, of course, with the actual use of the calculus, and this debate went on for centuries, actually, of course. No, and then one finds, it's rather odd that the Galileo, it's interesting, it's exactly what Galileo meant by, you know, exactly what mathematics Galileo had in mind. The calculus, clearly he meant geometry, he meant a lot of things, but the calculus I think then assumed, you know, it assumed the form of this language for a very long time, even when it was, despite the difficulties, and I think in particular, I remember when one comes, for example, to Euler, I love the derivation of Euler's equations of hyper-dynamics, which could be done beautifully. I've done another one, but I've done it in a more general form in this new book. It's very, very pretty. And what you see is that in the case of the, let's call it the infinitesimal calculus. That is the language, you know, you see that you learn it. It is a kind of, it's a language of nature in the small. It's this idea of infinitesimal knowledge of the, as Vial said, you know, the knowledge of the world in the infinitesimal world with its simplification. The whole point is with the infinitesimal world, that language is linear, it's very simple. It could be manipulated, and then you have to have the bridge, of course, to the larger world, but the language there is actually the language of nature in the small. Now, I think a lot of mathematics still does offer, although it's split off into various branches and there are lots of internal concerns, the idea of mathematics is that you like the central world.

1:30:00 In the case of category theory, I think it covers both, at least it has the potential. I'm covering both the internal aspects. I'm like, I thought mathematics does develop of its own. It isn't physics. It's mathematics, but it also enables the description of, you know, the kind of thing that Galileo had in mind where it enables, it does give a description at a broad, you know, rather schematic level, but of actual processes and the fundamental way in which they occur at some, how, I don't know, let's look at it. Well, I think in the case of maps, I mean, already, I think just the basic vocabulary of category theory already suggests that. I mean, the idea of map, you know, this is something you should be saying, but I'll say it anyway. Just the fundamental idea of map and map composition, you know, rather than set. Set theory, I mean, set theory has beauties of itself, but set theory is extremely far removed. There are many considerations in physics. I mean, physics is just not a natural language for physics at all. Whereas category theory is, because of the idea of map and map composition, the idea of motion, the idea of change. And the idea of composition of motions, which is a, you know, you're presenting to the choir. Yeah, well, exactly. You are, so, but he hasn't heard this. No, so, sorry. I didn't want to hate what you said so far. I'm so, so sorry, but I do, but I do want to say, sorry, I certainly do want to, but I must say this now. Precisely because you are operating the choir, and I'm certainly a member of it, and I follow what you said. I'll see. I'm very much on this point, but there is, of course, There is, of course, a position strongly opposed to this, which I think, were John Mabry here, he would probably want to put, which is precisely this view, that the notion, this view, that the ultimate degree, the geometrical experiences have, through being, has taken a form in which they're so far removed from the...

1:32:30 Any, anything on which geometric or kinetic intuition can get across is that it's very misleading to say that the fact that category theory deals with maths and metric and the awareness of motion is on your side. I'm just trying to say that there's clearly an opposition to this. I'm trying to see where that opposition stems from. For instance, that opposition seems to me to be at the bottom.