Pm session of discussions

0:00 And we know also that when their geometry is continuous, there is the analysis of the continuance as well. And this led, of course, as we know, again, to the problem that you mentioned, because They have claimed that the quantities in the measurement comparison of whole numbers in their own, and this seems to have been an important aspect of their own claims and arguments. And so, it's often said that the first crisis of mathematics, the first one, is that it is a clash in some way between the continuous on the screen, the idea that continuity, In terms of what one can measure, the ratio can be reduced to ratios of numbers, in other words, to exclusions. This led to a para-crisis. Again, it's not known exactly what the result was for 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 problem, This difficulty on which apparently early geometry, the idea of commensurability, seems to have been the basis on which early geometry was directed. 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 of the various lengths involved.

2:30 And they're fairly simple, as we know. This was a kind of extension of the original five-diagram notion of measurement, the idea of reduction to the discrete, into the continuous again, while admitting, of course, that in that case, continuity, in other words, geometric magnitude, had to be taken as some kind of given, in a sense that it's not reducible directly to the idea of number. The way that we see, I think, the continuous and the discrete as a primary kind of opposition that led to important discoveries in the development of mathematics, and of course later on the Fusino's paradox, which wasn't properly thought of mathematically, but of course the analysis of the continuous itself then leads rather quickly to the other opposition, or one of the other oppositions. There's an infinite invisibility in the continuum, right, to the contradiction between the finite and the infinite, which is, of course, one of the other driving forces, and there are all the, of course, you can see these interconnections among these very compositions. I take that, in my own, it's an oversimplification, but there's also the... The variation of static is also a major problem. The only, as far as variation and status are concerned, the Greeks really only, they had, they did have a theory of the varying, but it was the constant varying. It was in some way the idea that... Because they took all 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 theory, included astronomy as one of the elements, components of mathematics, mainly quantity, what they called varying, this was quantity, varying, in geometry it was quantity, you know, static quantity. Now... The contradiction between the varying and the static assumed, of course, already is an 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 that there were difficulties in trying to analyze that.

5:00 And, of course, this problem ultimately led, of course, to the declaration of calculus. Anyway, I see, on the broad outline, much of mathematics or the development of mathematics has been driven by these occupations in a kind of schematic way. Whether one... 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 as the same. Formalism seems to me another example of that. Mathematical formalism is another example of this contradiction, if you like, or opposition, in this case, I think, between the varying and the static. But I do see the opportunity with the discreteness of what are the two great areas of mathematics, which is geometry on the one hand, which is either spatial or temporal variation on the one hand, and the other is still driving the... I see the physics now as quantum gravity. This is obviously an example. 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, one can see, let's say, general relativity as a very clear and I think rather well-axiomified theory in the continuous in some sense. Quantum theory is a rather impure amalgam of a lot of things, but fundamentally the revolution in quantum theory, the term quantum itself means an isolated quantity. And at least there's some idea of discontinuity of restrictions apparently aligned by summary of it. I think, just as history, most of that about the second one is probably just not... But Aristotle does tell us that most people find nothing more surprising than that the diagonal is not commensurable with the side. Yeah, that's right. On the other hand, he says that the geometry would find nothing more surprising than if it were. So, whatever may or may not be true, at the end, this stood out as a striking thing.

7:30 Well, Aristotle also says that 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 not very clear. Are they atomists in the sense, let's say, of, you know, of Democritus and Eusebius? But anyway, there was, according to him, I'm sure he's right about this, something surprised about the lack of an extreme measure of the relation of dynamics. Yeah, no, it obviously came as a shock from the pipe anyway. Well, I don't think that's obvious. Well, it's not obvious, but... But it's indescribable that Aristotle expects his readers, or at any rate, perhaps he expects mathematicians to expect his readers to be struck by it. Yeah. This might be a good approach to say something about Aristotle's views of mathematics as a science of abstraction. Yeah, yeah, yeah. A viewpoint which tends to be rather disregarded in the philosophical tradition as precisely the conviction that Frege and the Aramaic school have settled all of his philosophical questions. Oh, yeah, yeah, no, no, no, no. Mathematics and physics, and these, you know, monogamy questions as well. Okay, we'll have some more. I don't think I mentioned it because you did actually listen to some of the other topics. Yeah, but I do think the idea that geometry, that 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 conventional case. And it's very clear because, you know, you can cut things up in the appropriate way and rearrange. And when it turned out that these, that these likes turned, I don't know, we don't know the history of it, but there's a rather clear model for the commensurable 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 or Proust, except, of course, one has to introduce the Eudoxian theory in order to be able to... No, I mean, when you try to show that the ratio of areas of triangles with a given altitude is as to their basis, it's you choosing, you know, the simplest way is simply to choose a unit, and then divide, right, of the same area, and then cut up the thing into so many triangles, which is represented by the unit. And that indeed is the way that essentially it's done, even in... In later geometry, except, of course, you can't assume, well, you use inequalities, but essentially the natural proof is this business of dividing it up into smaller triangles according to the unit, with a unit, a common unit on the basis.

10:00 This is a standard thing. You can see it in basic, simple, I mean, conventional geometry, right, I mean, it's very simple. It's just cutting and pasting. It's cutting things up and choosing a unit. Exactly as, I think, people measuring floors and so on. It's exactly that kind of thing. This is where geometry evidently must have come from. You know, one interesting point is that in Euclid geometry there is no unit. No, no, no. Debates about what he was doing there have to do with a theorem that says that whenever you have a polygon, it starts with a triangle, you take a polygon and a length of any given size, then you can always build a parallelogram of the same area. And the standard interpretation, which was given by Hayes, says, what Hayes is saying here is that you can always transform a polygon into a parallelogram of any given area, of any given length, for example, a unit. And when you say, for example, a unit, then you are saying that you can transform areas Thank you for your attention. And this is an interesting question. If that is the case, then the standard account that's given in introductory histories of mathematics to what Eudoxus was doing with his theory of ratio becomes, well, suspect, because you'd expect that if it was written by this Italian... No, no, because if one's ratio is... No, no, the thing was that Eudoxus, at least as presented in Euclid, does seem to be...

12:30 You know, that you can make equalities of ratios, right, of similar things. I mean, the ratio of 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 are. He only does, well, he just says, you introduce an equality. 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 very typical mathematical move, of course. Right? This is what we've been saying all along. But no, no, you're right. Of course, there was this problem. I mean, another interesting thing, the idea that an area, that there would be, well, let me, 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, a curvilinear figure. And a linear, rectilinear one seems to have been taken for granted. Well, hence the whole problem of, you know, it goes back a long way, the term of the quadrature of the circle. On the other hand, apparently, Aristotle didn't think you really could do that with curving up curved lines. In other words, curved lines, it's an interesting point, you know, that there was no, like, really real theory or idea of ratios. Well, of course, in the case of the circumference of a circle, there were arguments. 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 clear what it meant. You know, the idea of saying that the ratio of a curved line... You know, to a particular straight one. 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 which, you know, Archimedean spirals and so on. But there were very few of them. And it's very interesting that the part I know, the circumference of a circle,

15:00 Given as a length, I'm not sure Aristotle would have regarded it as a well-founded question to ask what the ratio, well of course there are formulas that tell you that if you take the circumference of the circle given as a length, 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 can I read it? Aristotle. Well, I 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 thought, 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 systematic account of everything around biology, astronomy, whatever, but the people were doing all kinds of things, you know, mathematicians, even their... At that time, we're not very happy about having to work according to certain constraints. They did any, you know, including the question of the quadrature of the circle and all these. There was no real prescription that you can do it. But rectification occurs. There was a problem in principle about the idea of rectifying curves. In a way, we're trying to... No, that's later, but it was... So Comcrates, if we can trust those fragments, was happy talking about the areas... And I think that 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 of the diameter, I'm less sure. So there's an interesting question about alchimedes. The first sphere and the circle and the sphere have the following meaning. It's very interesting. It is clear that there is a constant proportion between circumferences and diameters. Always. And there is a constant proportion between areas of the circle and square of r. We call this pi and we call this pi.

17:30 There was not at all clear that this is the same pie in both cases. So 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 of a circumference is the same as the area of a triangle, which one side is the radius of the circle and the other is the circumference. 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, of the, how do you say, the circumference. So that it goes, both questions go through the same proportion. No, we don't say that. We say, no, the ratio between circumference is the same as the ratio between, it's not the same. That's what I'm saying. When do people become confident that the circumference has a ratio of the radius? They could express what we express that way in this other way, but when did they become convinced that this way is correct? I'm not convinced of it. No, I think it's later. For example, even in the Jewish sources it disappears. Of course, the number is not correct. But there is a clear idea, there is a constant between both things. And so on. Also in the Greeks. But you don't, in the early times, you don't have the tools to express it very correctly. And then when you have the Dobson's Theory of Proportions... Then you can express it in the way I said. On the one hand you say circumference one, circumference two is the square radius of the square radius, or length one... Sorry, sorry, area, area, r, r, r square, r square. Well, I think Hippocrates was convinced of that. Because you can see. When you look at a picture, you know it doesn't matter what scale it is. But how clearly can you state it? And Archimedes takes it for granted. I mean, Archimedes does this very clever construction that does take for granted there's an answer.

20:00 And he's going to show you how it works. Yes, 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. Oh, yes, absolutely. But I do think that's a good point. Curve lines and straight lines really could have a ratio. I mean, really could be, because they're really two different types of objects. Yes, which brings me to a slightly broader question. I mean, from what you've said, it's clear that, certainly by the time of the doctors, that the theory that's being proposed is an absolutely general theory of proportion. I suppose it's all about possibly being quite a relatively different kind of subject, which you say was problematic earlier. At what point does it become clear that the subject matter of a theory of discrete quantity is the same kind of subject matter, belongs together in one overarching framework with a theory of a figure of geometry and algebra? Well, I'm saying, is it evident that the Greeks thought of the subject matter of arithmetic and geometry as necessarily fitting together within one single... 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 of my question. Yeah, it does. That's the point of my question. I don't think, I mean, it's like to make sure. It's like you don't really need it, but there are certain special things that you say about numbers that you cannot say. I'll give you an example. For example, if you compare area-area with line-line, you cannot cross. It's not an equality. A divided by B equals C. You cannot multiply, you cannot do anything. With numbers you can. If you have A, the ratio from A to B is like the ratio from C to D, that means, in the case of numbers, A to C is the same ratio as B to D. So you have a particular theory. The theory of proportions with numbers is more specialized and therefore has many theories that do not apply to the geometry.

22:30 What's the problem with two equalities, not volumes? You can do two ratios of different kinds of things, but you don't want to form one ratio of two different kinds of things. So you can't rearrange your ratio. The way you would if they were all the same kinds of things. Yeah, you can, I mean... Did you say volume 1? No, you cannot grow. Because it's not an equality. In fact, you know, we even write it today A to point B, 4 point C. Even this thing doesn't appear there. 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 B. 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 type. What does it mean to be of the same type? 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 will never be bigger than the volume. So you only compare things... But it's not comparable. Yes, it's not comparable. Even in a matter of commensurability, it's not comparable. So the theory, it sounds strange, but you have to 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. 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 between force and acceleration, and the third one is action and reaction. Let's leave action and reaction aside. If we formulate inertia and proportionality in modern terms, what do we say? The second one is M times F equals A. Did I say the right thing? No. F equals MA.

25:00 What does inertia say? Inertia says that when no force is applied to the body, then acceleration is zero, right? So take the second one and put zero in the acceleration, or zero in the force, you get acceleration zero. So in these terms, inertia is a particular case of the proportionality between force and acceleration. So why did Newton need to have three principles if he could have only two? You understand? The first one, in modern terms, inertia and proportionality between force and acceleration are equivalent. One is a particular case of the other. And the only way to explain this historically is that the principle of proportionality is formulated in the way I said before. 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, it plays later on because it says why there is a proportionality because this is mass, mass is the opposition to force to create the acceleration. You know, but because the Principia of Newton is written in this language, in the language of proportions, of geometrical proportions. Except that on the face it does talk about the proportion of clear force and acceleration. But yeah, but not in the definition of the 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. Yeah, it's not a quantity at all. Well, what's a quantity then? A quantity for the Greek is clear. For example, it's clear but you cannot formulate it in a sentence. One of the properties of a quantity is the ability to compare with similar quantities. This is true for volumes, this is true for numbers. Later on, it is true also for forces. I mean, this is for a Newton. That was not the case with the Greeks. So the idea of what is a quantity, of course, varies a lot from the Greeks to the 17th century. But the mathematical tool with which you deal with them doesn't change. I mean, if you look at the

27:30 It's much closer in form and in tools, etc., to a book by Euclid or by Apollonius than it is to a book by Lagrange or even by Bernoulli, because they already use the calculus. There is a discussion among historians whether he derived some of the theorems using the calculus. It's hard to say. What you see in the text is just, you know, triangles, radials, How far does he state the second law? As I said. I think he states, he talks about one force and one acceleration. Now you can say this implies a theory of ratios. He says that forces and accelerations are proportional. So meaning two forces are proportional to two accelerations. I even see, it may even be stated in those terms, F1, but I don't know. I really doubt it. No, no, no, I'm sure. What do you think it says? Well, it's not stated in the terms you're talking about. F1 is F2. No, no, but it's F2, so proportionals. The ratio is the same. Ratio between forces is ratio between other ratios. I mean, I'm sure. Well, I think the whole issue of failure... Exegesis of Newton's speech on physics is a very interesting one, but since I'm tentative, I think I'd like us to move on to... No, no, no, that's not in time. No, no, no, it's great. We could easily spend the rest of the time here talking about that. That would be very interesting. We would love to have the free key here. Perhaps we could move on to John. Yes, it is. We'll have to bring it later on. Yeah, it's buried in one of the boxes. There is an opposition between continuous and the continuous in the street, and the varying of the static, and you see it as developing later in mathematics. Let's in fact look now at the 70s, 80s, and going into the 19th century, would you like to say something about how you see this continuing, this opposition continuing, and developing? Well, I'll do my best.

30:00 The continuum was always there, I mean, in the development of geometry. But I think a new phase in this opposition between the variant and the static emerged in the, well, I suppose from the Middle Ages on, when one had to consider... There are many non-constant variations, and you can see it in the idea, for example, of, well, one does genuinely have accelerations, for example, in motions, which the Greeks, well, they didn't really, well, the Greeks, of course, you know, I mean, they analyzed circular motion, which, you know, came from motion in a circle, which comes from motion in some... Well, a circle, really. 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 at any rate, 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 the further kind of... The deepening, if you like, of the opposition between the varying and the static, which was really present, I think, in Big Four. 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. Motions, of course, of actual objects in, you know, in space. I mean, and sublunary. I mean, not astronomical. Galileo specifically talks. There's a lot about motions that occur in the 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, a change, a double variation, so to speak. 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.

32:30 And so on and on, perhaps as linearizing the curve. And then the connection, of course, the great achievement of calculus was to see the connection, if you like, between the direction of the tangent vector and, of course, the area of the curve, which is the fundament of calculus. That effort, as we know, went on to struggle to kind of establish a solid... The basis for working with these things went on for a long time, and of course the history of the calculus is 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 this fundamental connection 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 a kind of, like between two quantities, which, both of which are undergoing simultaneous variation. And one more point. It was the notion of a function, 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 where you saw the connection, the correspondence between them. Without that notion, none of this would have emerged. That's why we see... The notion of a function, similarly, I believe, is also emerging, you know, between this opposition between the varying static. As far as the continuous and discrete are concerned, of course, the calculus... Variation always meant, of course, really continuous variation, and what the term continuity meant, of course, the notion that 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 used 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...

35:00 Right? Continuous variations 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. It's so much of mathematics. It's what even other mathematics is about. Hence, of course, introducing the theme of the relationship between theory and its representation. Yes, it's often, it's all very similar, Tim. Theory and its representation, I mean, the topic which has come up several times in the conversation this week, the nature of mathematical formalism. In fact, I was wondering, at some point, you say quite a little bit about this in your early paper on adjoinments in foundations, I know that you have revised some of the ideas that you have adopted. Not to brag, of course. Well, it would be interesting. I was right. So was I. Thank you very much. You're welcome. Yeah, one thing that comes out of that is to propose the idea that syntax means some other things in terms of mathematics and mathematics and linguistics, and commonly speaking that that was the case, but if you analyze it, you're not really quite right. Well, 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 general. There's this third thing, the abstract general, and the two just merely overlap in the middle, you know. There's no direct way of going from... well, let's say there's no direct way of going from... The concrete to the syntax represent a language which would describe this. It's a two-step process, each of which is an adverb, but if 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. That sort of thing is not really...

37:30 The syntax of group theory, it's an important part of developing a whole theory of syntax. Normally we know it's addition, it's multiplication and division and the association of all of this. That's the typical syntax, the presentation. Appropriate to a given notion of abstract general is determined by the latter, so that it's not an additional choice. I'm not talking about a particular presentation, but I mean what a presentation is. In other words, the presentation and representation, each of which involves abstract general, is one of the poles. It seems to me that one of the fruits of the lack of understanding of this point, the lack of understanding of the relationship between them, is precisely the... The view that the formalists have that syntax isn't sometimes in substance primary, and also on what I call the ontological wing of the era, the extraordinary view that these people who call themselves structuralists have, who don't have a theory of structure but they keep talking about structure, 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 that was doing some useful work for us, either in mathematics or physics, one sees, for instance, precisely this issue of power. I mean, there are several examples of propositional calculus. What I mean is, nobody would ever write down the propositional calculus unless they had an idea of what Boo and Algebras are.

40:00 They may not know the name of Boo and Algebras, they may not have a perfectly clear idea, but the idea that there is something like Boo and Algebras is in the back of their mind by the way he writes down propositional calculus syntaxically. I think that, in other words, the fact that you cannot... Maybe you cannot perfectly know what a Boolean algebra is until you have perfectly worked out the compositional calculus to not misuse 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-specificating of the Boolean algebra? Is it important to you? And the non-colonists, as syntactical things are crystallized, at least the partial consciousness, full consciousness, some more operations on that concept, I think. Why would you ever write down a syntactical theory if you had no idea whatsoever what it was going to be about? You will get the impression, well, somehow people just do that sort of thing. I write down strings and symbols, and I have rules for manipulating them, and this is not the way the knowledge that went into Church's book came into existence. So I'm going to call one topic that Colin mentioned to me, and it comes into the context of the relevant writing section, which is in connection with the Latin conference, which is Bill's Theorem of Current Highlights. You know, I'm still not clear. You know, you've talked about the history of the Korean War, and it's more of a matter of whether you've written it down or not. Well, I've written it down. You know, there I talked about...

42:30 You know, to avoid spending my friend's hours, it's very likely to be an adjunct, so I have two main objections to the term three-hour isomorphism. First, mathematics is not an isomorphism, except possibly at the level of pure syntax, because the thing that you are trying to... In one case, it presents the Cartesian quotas category, but perhaps in other cases it's a hiding algorithm. So the content of the idea is that if you want to present a hiding algorithm, perhaps a hiding algorithm, one way to do it would be to first present the corresponding Cartesian quotas category, work with that, and then collapse it to get to the hiding algorithm. So the fact that you can do that with a collapsing map is sort of not an isomorphic, but essentially you're introducing a new rule, hiding focus, introducing the fact that the Cartesian product of two spaces or whatever is isomorphic, sorry, of a thing with itself, is isomorphic to itself along the 90-minute line, and if you can introduce that axiom you can introduce this proposition here. 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 kind of thing is an objective content.

45:00 Which you led me to believe that there was already existing theory about this at the time in the early 60s, mid 60s, early 60s. So I gave this talk at the Los Angeles Center and sat in the conference thinking that, well, I'm really giving a categorical, hopefully terrifying presentation of something that is already known. This whole tendency of investigation really emerged simultaneously or even after the categorical formulation. Howard, who was a student of McLean, somehow I didn't even meet him when I was in Chicago, he circulated this letter in the late 60s. It wasn't actually printed until he gave it to me on the Curry test trip. So there is this strange reluctance of this. A group of proof-hearers should recognize the categorical origin of their subjects. Yeah, that's what I was interested in then. Howard's not in there at all. By name, no. But you haven't mentioned the reason. Well, yeah, but it's all just a discussion. So, um... The mathematical concept? Yeah, the math is, but the history isn't. Right. And that's what I was interested in. Well, the mathematical script, of course, is false. It also brings up this theme that, you know,

47:30 I mean, not to be fooled about it, but any number of mathematicians told me that, oh, they all, everybody did the act of choice in life, you know, long before the act of choice came up. And, uh, yeah, yeah, yeah, and it must have been in Myhill and Goodman somewhere. Well, Myhill and Goodman did a group of us. Which has, the first sentence refers to Diokonescu, which is the sole reference in the bibliography. They don't claim to. No, no, exactly. They don't claim to. Their first sentence refers to Diokonescu. Diokonescu is the sole entry in their bibliography. But any number of logicians have assured me that Myhill and Goodman did this before Diokonescu. Oh, and I don't believe that's rubbish. It's rubbish. You're saying that they simply don't read it. Yeah, and so it's hard to find a, you know, a forum that... Any number of logicians? They do, they do. I can't believe you did that. I mean, I can't believe it. Well, I'm sure they probably promised to reform it. It came as a great revelation to me when I, well, actually it was the Diakonets group here who came as a revelation. I mean, I thought that was extraordinary, and it's not for the work in my case. Let me end on top of this theory, actually. Well, I don't know. That was an amazing result. I don't doubt a lot of them first understood it in my help. Yeah, yeah, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know. I was astounded to hear that. Well, I was astounded when I looked at the Myho Goodwin article. I quickly, you immediately learned that it's after Diognescu. And then when you look at it, you see it's all about being after Diognescu. That's what it says, I think so. Well, I don't think logicians have the slightest idea. No, I didn't even, well, I know that some logicians claim that. I had no idea they were referring just to Myho. That the action of choice was a complete principle in the sense of a principle omniscience, a principle that would likely solve all questions. So you shouldn't, an intuitionist cannot accept the action of choice because it will answer all questions.

50:00 But this is a much vaguer notion. Yeah, because, because... Well, and also, also... The triad law, I think, maintains that there was... Or do they simply maintain the work they use in proving the intuition of the theory? This is something quite typical of Christ, isn't it, sometimes? It's something working, like noticing something, isn't it? I can always ask him about this. I mean, it comes as a bit of a surprise to me, because I know this is a student, a graduate student, and this... Yeah, I mean, how I didn't want to get into it before that. I certainly don't recall anything. I mean, I was interested in all of these things. I don't recall really anything about Cary Howard until I've actually ever seen him, so it's the first time I've ever heard him say what he just said, but really, it actually, it doesn't, it confirms my experience. I certainly, it wasn't that I was in ignorance of the subject at all, I wasn't. I think, I mean, okay, one big failure was that I didn't have to publish my Los Angeles talk, per se. I mean, it was one of those mix-ups, you know, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, the editor, And the idea of category reality is that if you squash the category into a post-act, that this is the procedure for going from proofs to propositions and so on and so forth. Let's go back in some great detail. As I say, the basic material was split into three other papers and published in the next two years. Certainly when I was lecturing at the University of Chicago, I was in the conference at the same seminar that we had. I wasn't in Los Angeles, but I mean I was around before and after this, but this is awfully fuzzy. Look how this suddenly sprang up later. I don't know, how would, I believe how would give something of that perspective to the Pittsburgh meeting once you were there. Yeah, I saw that. He didn't really, you know, he didn't mention my name or me or, you know, let alone, you know, sort of the, that's where the history started after.

52:30 Well, since the issue of choice has come up, and since time presses, I'd like to actually ask for it as well. This is bearing in mind this session we're having now is intended to focus on broad philosophical themes as connected concepts. To say a little bit about this issue of the opposition between the varying and the static, in the context of the answer to the choice is one of the implications of constancy, and the way that its negation of topology is referred to as an organically varying domain, i.e. bits which take more of the nature of the world's variation of the union. This is something that has always intrigued me, because I think it's pedagogically a very good place to start in, to start into things of category theory, the driving force in general philosophy. I may be quite wrong about that, but I've been prepared to say a little bit about it. Well, first about Diogonescu's result, which was a very good result. He came up with an entire on his own, by the way. However, 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, but it has this same particular consequence that the action of choice implies the logic. So, Diakonescu does it in the context of something or a category that's already known to be a topos, but has a sub-object classifier. And then he, assuming that every match splits, he constructs certain simple quotients overlapping two copies of something and saying that splits in the sense of having a section, but it also implies splitting in the sense of having a complement.

55:00 So it's in one way rather general, because it could be any topo, it just happens to happen. Everything's splitting, but it is a topo, so you assume everything is involved in the general power set of mathematics. Well, in a different level of different generality, actually in my set theory paper at ETCS, which has now been reprinted thanks to Colin, you can read it in TAC also, I have some commentary about this. I'm not going to satisfy some other axioms, but I don't assume it has a sub-object to satisfy it. Rather, I prove that it has, and that it's two at the same time. In other words, I show that two... So I'm simultaneously proving that these very strong axioms are different from topos axioms, which imply that it is a topos, and also that it's a Boolean topos. And the method there is simply that using the Cartesian fluid, you have 2 to the x. What does 2 to the x classify? It classifies, of course, the complement of some object. That's clear. You can classify the complement of some object. And on the other hand, the notion of infinite union is... You know, it's just actually the image of some math on the infinite sum. 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 excitable sub-objects. So that is a map f from the parametrizer into 2 to the x. The sub-object of that, which is intuitively the union of that family, okay? So now, given an arbitrary sub-object, so an arbitrary monomorphism not known to be complemented, 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 need a certain is in fact the union of those two is everything.

57:30 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, 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 the complement. So it's a very different argument. It uses a higher order structure, and so it would. That issue about, no, about the action... I was going to say, specifically, I'm sorry, I didn't get out, obviously you thought, but in connection with the broader discussion, as I say, of variation and adhesion, I'd like to get a little bit into the way that the negation of the answer to the Chauvinism categories, which reflect the real world of adhesion and non-arbitrary. All of this variation is connected with the different ways that choice can be expressed, for instance the failure of the absolute choice, the failure of the absolute choice, the failure of the absolute choice and the fact that the Banach-Tarski paradox is evidently not true of the world in which we live, and the way in which... As I say, it's fairly obvious to me that there is a cohesion and non-authorization of the structures around us, and perhaps which of the specific ways in which one thinks of the accident shots, including the way one thinks of it in the context of topology, is that there are no structures of the existence of inverses of maps. It gives one some insight into what is going on here. It's just that it's so atypical. A typical situation is that you have objects that are not projected, that are not ejected, in other words, there are maps that are epic and don't have sections, and maps that are monics and don't have retractions, and so on, and most of the objects are like that. You're lucky if you can even get enough of them to cover other objects, so you've got enough projectives to serve. Halfway around is the failure of the acting of choice in some actual mathematical context. It seems to me that's the most profound insight, because it's the most profound insight to get across the philosophers, because it suggests a way of seeing how logic, or something like a logic, actually fits into our understanding of the world, rapidly different from the way that people would think of logic as something which has to underpin... Well, Yekoneski himself, actually, not just as a scientist, but he had a theory, the way that homological algebra measures...

1:00:00 So he takes, I don't remember quite exactly, he takes a model of topos, maybe, anyway. You can define cohomology groups for it. And then there's a theorem that these vanish if an employee has a reaction of choice. I actually remember that. I think Andres Blas. Yeah, Blas has done that. Blas had a paper in which he... I think Blas actually published it. Yeah, he published the paper on that. I've written about it before about this. Well, I mean, it's a little... I think actually more than one... It's just fitting, you know, there's this sort of this general pattern, as Cartier, you know, you know, 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, and that's very natural cohomology theory, which... It's true, but otherwise you get homology groups which, I don't know if anybody's ever used this, you see, but in principle, you're getting groups which measure the extent to which it fails. Absolutely. This is a very nice idea. It's not only a very nice idea of mathematics, but it's one that seems to me to connect very deeply to the theme of Collins, that one sees in all this world of school. In fact, in the whole line of development, the program of re-description of structure in the way of different approaches to mathematics in terms of concepts which are in danger of consequences, or glimpses of consequences, at the most profound time, actually for broad philosophical topics, for the subject matters of the understanding of real-world structure.

1:02:30 But to really make any use of this, you'd actually have to compute these groups. That's right. I don't understand. Say you construct a model of that theory, which makes the actual choice, and it has such a group now. What's the relation of that group to the forcing? There are a number of 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. It's an n-indexed sequence of groups, or how do you say it? It just only depends on H1, actually. Okay, that's the first number. As I recall. So did Bosch do this at the outset of his career? Last was a student of Bauck's. Yeah, at Harvard. Do you know when he began his career? Bauck? Was it Harvard? Bauck. Bauck. Bauck. Bauck. Bauck. Bauck. Bauck. Bauck. Bauck. Bauck. Bauck. Bauck. Bauck. Bauck. Bauck. Bauck. Bauck. Bauck. No, I haven't heard from somebody. Bob is very ill now. He's got some very limited survival time. Somebody mentioned, actually, that there's somebody at the university who told me what I hadn't known about the rest of these students. Yeah, you know, I saw him about a couple of weeks ago from across the room and didn't much recognize him. Really? Yeah. Well, I'm not very known. Well, no, no, no, I didn't see him in a very... You know, I'm pretty sure this is that I did. Among your personality, I mean, I mean, I've seen several of you, and you write something that's extremely good. Yeah. Yeah. Well, that, by and large, is quite a sentence. It can't be illuminating, and obviously the technical aspects of this, where one sees sort of a set theory and homology

1:05:00 putting together a huge interest for the mathematician, but I come back to my point. I think that this is an area in which one sees the imprint of. Thank you very much. Well, that's a broad, distinct philosophical discussion. To do with understanding the way in which the expression of cohesion and variation is guided, the understanding of that as a sound element of mathematical structure actually infringes on our understanding of the world. I mean, you yourself mentioned you want to discuss the issue of abstraction mathematics as the book of nature, where this mathematics comes from. You're the guy who wanted the big, broad philosophical discussion, but I'm giving it to you. Right, right. In the case of the accident of choice, I mean, It always was, from the beginning of its formula, it was identified as a kind of, as we know, as a controversial... There's a whole principle that was regarded as natural by some and unreasonable by others. There's a whole history of it. What's known in set 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. It's not 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 okay with. ... about how it can fail in some way, because countable choice is an important principle, which is true even in constructive settings and certain kinds of formulations, but what I mean is that it's very curious to me that, it is interesting that it was taken as a principle of logic, you know, it was taken as a kind of basic... I don't think anybody has made the explicit connection with logic. You see, Hilbert actually says... He likes, he uses, they added the acts of choice for him, it was a classical principle 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 we use, if you add, if you take the intuitionistic epsilon calculus, right, in other words, don't assume the law, he's trying to justify, Hilbert's trying to justify the law a little bit, he can't assume it.

1:07:30 So if you take intuitionistic logic, take the epsilon calculus, you can prove it's classical. Well, I mean, is that a novel? In other words, the justification of the epsilon calculus for Hilbert was essentially because of the axiom of choice. He thought the epsilon terms were two variables, right? It gives you, essentially, a choice function. Hilbert was aware of this. But he didn't know, yeah, but he didn't know, of course, that, I mean, it's Diakonescu 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 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, needed to be justified in some way along with classical logic. So it's sort of interesting that Gilbert had you. 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. When I first saw the Diagonescus there, I was amazed. I thought, well, yeah, now we see what the connexion is. 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. It happens, and if you take Hilbert's epsilon calculus... You just take the pure epsilon calculus, without what's called Ackermann's scheme, which says that the epsilon terms are extensional, that is, if a and b are two predicates, such as for all x, a x is equal to b x, then epsilon a equals epsilon b. If you don't have that, Right? Then you just get, you get a genuine extension of the... 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 law of the student middle. I showed it was independent. And of course, if you have Ackermann's Pencil, you get the fooled Iconescu argument, naturally, because you have extensional functions, etc. And then that gives you the law of the student middle. So it's a sort of, yeah, that's very strong, and of course, yeah, and, but even without it, you get, you get a, you get a, you know, you get a, you get this, this dumb, whatever it's often called, the A, the linear form, the linear, it does sound a little bit like taking a howitzer to an act.

1:10:00 Yeah, I mean, but it's interesting that you're sort of forced into that, where, you know, since the natural framework for Hilbert was a kind of, you know, the logic, was a kind of intuitionistic logic, we don't assume excluded middle, and he wanted just, but he's using absolute terms, but he's already got something like classical logic. It's just another question, but I'm looking at it as just a question of classical, classical logic, classical set, you know, this, you know, imaginative. The result of Chrysler is that the maximum choice can be limited in ZNFC proofs of a number of things, and also in proofs of some things that are on a number of things. Is that result somewhat tied to the possible? Thank you for your time, and I hope to see you again soon. This comes to finding some kind of natural topos theory of constructible, of constructible in the sense of constructible sets. It's not going to be the object, it's going to be Lindsay F, every set is a constructible one. It's going to be arrows or inclusions, because those are not always constructible ones. You need some theory that distinguishes some inclusions as constructible and others not, in a total sense. And then you can talk about, can we relativize these two places? But it just hasn't been really done. It probably doesn't. It probably isn't all theory and maths, really. It's just a project. Well, you know, Bill Mitchell's paper does... He gives a vocabulary for doing it, but all he does is translate the definition. It's just a boo-hoo for the internal language. Yeah. No, I agree. He doesn't show the upshot of it at all. Thank you for your time, and I hope to see you again soon.

1:12:30 And you can interpret some of the electrical sets very that way. He distinguishes constructible trees from non-constructible by translating the definition of constructible sets and non-constructible. Yeah, but when you get done, you found it in the original category, some maps are constructed. 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. I have a partial argument for this, because if you think of a map and you look at the family of sets and that's like a co-domain, and now you, let's just suppose it's not the case that the fibers are finite, and that the co-domain that is in the branch structure is non-countable, then the growth of the size of the fibers as you move Most of the co-domain is a certain number theory of the function. Some number theory of the function is a construction of its own art, but in a particular way, it's unhelpful. Now this isn't a definition that the original math is a construction, but it's showing that it has an invariant function, which is either constructible or not, and so it's a condition on the math. Yeah, but it doesn't show that some are constructible or some aren't. He shows that the relative consistency theorem for a diversity lifts to this context. It doesn't, you can't prove from ETCS that someone has a construction model. Because one of your models in ETCS is supposed to control any construction model. Well, 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, no, there's a predicate. Somehow there's a predicate. All I'm saying is that you can define this predicate that has the significance of that. There are different models then. It might be the whole thing, it might be something much smaller.

1:15:00 But just the fact that there is this predicate is at least this derived thing seems to be derived from it. This is a perspective that Sathya and Narendra Pringle-Sathya 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 runs on the level of relations, eras, inclusions, problems, and interresults should be a category anyway, and there shouldn't be a big difference between interresults and inclusions. Whether it's maths or inclusions, it doesn't really matter, but this is why I'm saying they don't think. They think it's about subsets. Yeah, but yet at the same time, they're well aware of the phenomenon of cardinal collapsing, which is exactly about maths and its whole... I'm sure they know it affects maths. They don't know that it actually lives on the web world. Yeah, but I mean, but the cardinal does live on the web world. They do. They don't know the situation, which is really, they don't know, sorry, can I just ask, can we talk about this one? Well, these are all extremely interesting issues, certainly. But since the state's trying to present some new concepts, we'll try to cover as much as possible of the broader, the subtle topics that John's looking at. Perhaps we could move on to the third of his suggestive topics, which was the whole conception of Galileo's 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 Billy in the account of his role of projectives and injectives vis-a-vis the failure of the action of choice and these instructions which pick up with cohesion. There is a very non-arbitrary variation of calculus that is discussed in the real world, but perhaps you'd like to say a little bit more about how you see it spinning into such a conception of Galileo's conception of black today, and indeed is the conceptual organisation of mathematics today such as to suggest any sort of possibility of really reviving Galileo's conception in a strong form?

1:17:30 No, I mean, the reason I mentioned that was precisely, you know, because, really, because I, because of the development of category theory, I mean, I, well, of course, remember that, as it happens, you remember that Bill 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. Um, 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, I'll call them the book of, uh, the book, look, look to nature to look at mathematics provides a book of, in other words, that in order to understand nature, right? I mean, to, to. To understand it at some conceptual level, it's necessary 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, on 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, 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 that we 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 formulated in its final form, or in its developed form, but I think that the idea there was that the calculus, I mean, one thing that struck me, when you learn calculus, You always learned it, I learned it, in connection with physical problems, and the way you understand the calculus, really, was by, well, you understand that by analyzing motion, it's rather obvious that the, it is language of nature.

1:20:00 Right, right, right, here, here. Yes, no, that's the way you learn it, that's how you understand it. 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. And then one finds, it's rather obvious, Galileo, it's interesting, it's exactly what Galileo meant by, you know, exactly what mathematics Galileo had in mind. I mean, it's obvious that new things are emerging. 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, you know, and I think in particular, I remember when it comes, for example, to Euler, I love the derivation of Euler's equations of hydrodynamics, which I... It can be done beautifully, actually, it's with infinitesimal math. In your book? Yeah, and I've done another one, but second, I've done it in a more general form in this new book. It's very pretty. Oh, okay. It's very, very pretty. And what you see is that, in the case of the, let's say we call it the infinitesimal calculus, that is the language, you know, you see, 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 Bile 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 can be manipulated, and then you have to have this bridge, of course, to the larger world. But the language there is actually the language of nature in the small. A lot of mathematics, mathematics still does often split off into various branches and lots of internal concerns. The idea of mathematics, the lack of a central role of mathematics, in some ways enabling nature in some way to be understood. I mean, you can construe nature in broad sense. It doesn't necessarily have to be... The kind of problems that Galileo was concerned with, but in the broad sense that it also has, well, in the case of category theory, I think it covers both, at least it has the potential. I'm covering both the internal aspects of life. After all, mathematics does develop on 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 a schematic level, but of actual processes and the fundamental way in which they occur at some point in time.

1:22:30 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, 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 has beauties of its own, but set theory is extremely far removed. For 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 big deal. You're preaching to the choir. You are! No, I'm so sorry. It's great what you've said so far. I'm so sorry. Sorry, I certainly do want to intervene, but I must say this now. Firstly, because you are preaching to the choir, and I'm certainly a member of it, and I applaud what you said. There is, of course, a position strongly opposed to this, which I think, were John Mayberry here, he would probably want to put, which is precisely this, that the notion, this view that... The ultimate ingredients of definition of all concepts rest in some sense of deceptively, because that's the only source of definition of structural motions. The geometrical motions, the geometrical and topological motions, as they have been originally suggested by our experience in the real world, have been enormously deeply generalized through being structuralized. And now I see the form where, as they are used in physics, and particularly in physics that is penetrating new levels of structure in the world, it has taken a form in which they're so far removed from anything which geometrical intuition can do.

1:25:00 But it's very misleading to say that... But the fact that category theory deals with maps and geometrics and the awareness of motion... Okay, well I'm just trying to say... I believe it, I'm almost your side. I'm just trying to say that there's probably 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 of the non-combatant geometry program. The algebra is fundamental, the geometric concepts just fall out of that, but they fall out of the new structuralized form which is kind of any kind of... But algebra doesn't fit very well into set theory either. No, no, I'm not speaking here directly of the issue of whether set theory provides a foundation, that's a separate issue. You should probably not have introduced that. I'm sorry, Leo. I like the way that you use the foundation conception as something which is not just theoretically foundational but practically conceptual. So obviously set theory doesn't count as a foundation for, let's say, the language of nature because we are not going to identify or characterize any physical process. Let's talk about physics at the moment. With the aid of set theory. We assume that the concepts can be, you know, numbers and functions and everything can be somehow reduced to that, but that's not the kind of foundation we were speaking before. So the foundation, in this case, let's say, The infinitesimal calculus provides a foundation for physics because it's the language, it's the main tools, etc. But if we look what happened then, some things are added to this. For example... This is the foundation of general relativity, even though it is a development of plain infinitesimal calculus, but if you stay only with differential calculus, the standard one, you cannot do it.

1:27:30 Then you have quantum mechanics. You have there some other things like Hilbert spaces and things like that. And not to speak much later about, how would we call it, chaos theory or nonlinear dynamics of all kinds. We say, no, what you were doing with your linear point of view is not enough. Category theory can fit into this picture, in providing a foundation of this kind for physics. And even I would ask you the following. Let's assume that you are given the opportunity to start educating children, children from a very early age. So that they become, they see the world categorically. It's not that, you know, we know group theory and topology and then you explain me how all of this fits into category theory, but... Do you at all envision such a situation, that people start seeing the world from the categorical point of view and then this provides the foundation or the gate to understand the world as the infinitesimal calculus did for physics for many years, for many centuries? One of the main goals was to get around what you just said, namely the necessity for prior knowledge of some opulence, topological spaces, or some mathematical categories that might be gained in upper class and other levels in the university. By instead of writing directly a sufficient number of simple and useful, very useful structures, I mean reflexive graphs and irreflexive graphs, discrete time dynamical systems, so three or four examples scattered with abstract sets, which you can learn directly without any special preparation in which. At the same time, are sufficiently different among them to justify having a general theory, which applies to all of them. So that's the strategy. We carried it out, I guess.

1:30:00 I get letters from the people saying, you know, this is indeed how they first started to learn and they found that to be very useful. Often these are from computer scientists, not necessarily children, but people who are even advanced students of computer science who find that they've missed out on modules and topological spaces and all that, but nonetheless realize that they need the category theory. So this is another audience, a distinct audience for this. The book is now available in Spanish. This is certainly intended as an effort in that direction, to make available the categories. Even in the last chapter, the sub-object classifier. Because in these concrete examples that we have, the sub-object classifier has very complete meaning. Subobject classifier for graphs is itself a very different classifier, which I don't know if graph theorists have ever noticed. It has its universal role, but I mean, it has all sorts of... Or the subobject classifier for not necessarily reversible dynamical systems. So it's itself a dynamical system. Very non-trivial. So again, the fact that... Not only the notion of a category, but not only the notion of a category, but the use of function spaces for the last chapter is illustrated very, very concretely by these, and I hope that other people are better prepared to learn about modules and topological spaces after they've gone through this, but in any case, they know enough about these to come to the world and do something with them. Now, to push this further... Provide some kind of a basis for, you know, actual children in high school and some assistance in forming mathematical concepts, knowing this is a problem we think about every day because of the grants and whatnot, but I don't have any systematic answer to that. But I have a connection with this...

1:32:30 Obituaries for McLean. Did you get one of these? Yeah, I saw it. The music thing. Yeah, right. Oh, it's a kind of naturality. This is an attempt to... It's really just an outline of a slightly longer article, I think, around three or four pages, providing a completely honest... I mean, in other words, I'm not pulling any punches here. This is the real equation as far as naturality. And moreover, this is a genuine example. The next, I mean, the next page after this goes into the question of the collective versus the individual. In other words, the orchestra, the conductor, and so forth. Not just a single performer, but further, further natural conditions. It brings out the, it brings out the sort of opposed nature of time and space. And so forth. And the way that it's applicable in many contexts. It's a formulation that derives from the theorem of Bob Walters in Australia and Italy about the existence of solutions to certain kinds of constrained equations in the field of concurrency. My idea was that if this thing works at all then I would expand it a bit more and publish it also in other places. Of course the Italians are very open to this kind of thing. I tend to get more sympathy there than I get in other places. Just a minute. Let's assume this... The question is if a person whose world view is based on...

1:35:00 We perceive differently, different laws of physics, better laws of physics, more general, or there you can just reformulate things that existed before. What is the role, the foundational role, of category theory? Or the book of nature? What's the role of mathematics itself? Coming back to this quotation of Galileo, Well, so the first question is, who wrote it? So if you take, if you take is, I don't even know the original Italian for relations, but if you take is, it means, it means, well, it was written by somebody, it already, well, who was a somebody? Of course, you know, right? So, so therefore, you know, he's comfortable with the church. On the other hand, if you interpret it as it is, like it is being written, it's a process, then he's saying, well, actually I'm doing part of the writing, me Galileo, but especially it's mathematics. And what does mathematics mean? Various people who know Greek, including Greeks, have told me various things about Galileo. But one feature, one feature which I think, which seems to always be there among others, is the fact that it's something that's teaching. In other words, it's the aspect of all of our experience and knowledge which in summed up form can be actually transmitted to the next generation. It's the one which can be used. In order to establish the unity between people, we have to produce in the world. For example, this house was built to have this width approximately the same as that width. Some workers were working over there, others were working over here, without mathematics even taking into account the sense of numbers and geometry. They wouldn't have sufficient unity to carry out any collective production. You ask about physics. The present situation is that our world depends on physics and yet nobody understands it.

1:37:30 99% of the people have no understanding whatsoever. Another 1% think they understand it. So the fact of having a collective understanding of physics, both classical and modern physics... It's something mathematics must achieve. And that it does by clarifying, in particular, by clarifying three unification. So that's what... It's not that the theorems of physics will change. Of course the theorems of physics will change if more people were thinking about it. But the main point is, in fact, more people can think about it, can develop their thinking. In a way, more than just reading, oh, gee whiz, atoms crash together, don't they? No. They can actually take the definitions of results. They can develop them in thinking about them. Listen, this is a goal. That is the goal and the best tool we have to achieve it right now is category theory. Let me take a try at this. Taking a few steps, yes. But not because we start off with the idea, oh, we worship category theory. No, because we see that that's the only available tool. That's the only tool for bringing about this mathematics, i.e., the Jewish ability of the maximum amount of work. Let me take a try. The idea, as I see it, is that categorical thinking actually says less than set theory. The set theorist who wants to study state groups is going to see all the same relationships that the categorism is going to see but is also going to be worrying about which the elements are. Now, in fact, they won't. In fact, having said that this is not an upset theory, they will have then told you, forget the set theory. The point about a free group is that the group homomorphisms committed to another are determined when you know when to generate them. That's what you need to know. You don't need to know how the free group is really defined. There has to be a way it was really defined. The categorical approach says forget there has to be a way it was really defined. That was the definition. The definition is that this is an adjoining to the underlying set doctrine, that you know a mathematical pre-group is different from what you know originally. That statement, which the set theorists had to make anyway, defines the pre-group for the categories. The category simply says less about the pre-group. It's easier to say less than to say more, especially when the more wasn't carrying any content.

1:40:00 That's the idea. Category theory is an articulate framework for only saying the part that matters, not for saying different things, but for saying less things, i.e. individual things that matter. You could put it this way, talking about the relationship between set theory and mathematics, it's a commonplace that each part of mathematics can be interpreted in set theory. Theory, topology, etc., everything can be interpreted. In particular, the category of sets can be interpreted in sets. The point is that all those other interpretations factor through that one in the category of interpretations. This is a categorized description of the situation, but I think it's very striking to see that everything that's mathematically significant about any one of those things is interpreted in the category of sets. In other words, the interpretation in sets factors through one of them. And the category of sets is a much simpler thing in the universe of ZF sets, just to fix on that particular point, because every ZF set has this infinitely iterated membership tree, which plays no role except individuating the set. The set's already individuating that set, so why would it be? So the difference between categorical set theory and Zermelo-Frenkel set theory is you don't worry about that iterating membership, which plainly no one will. You worry about what functions there are between sets. Now who would have thought you could describe what functions there were between sets without seeing what their elements are? Not Zermelo, he denies it. He says that foot numbers are not the way to practice speed. Well, he was mistaken. He can't. And I don't want to give you the gags today. I want to categorize sets. I'm sure that you can. Now, anybody would rather be sent here without worrying about the identity of the elements, and every single male or female sent here by page 10 of Kuhnen. We don't care what the elements of any set are. Maybe for the first two or three pages, we have to worry about specifics. By page 10, it's all done up by isomorphism. You know how many set of elements a set of elements is. Well, ETCS, that's all there is to a set of elements. Can I ask one question? The point that you made about every structure... I think that's all I have to say.

1:42:30 I thought I'd just present it so that you can hear me a little bit better. The definition of mathematics is not the same thing that you said. The theory of it, the theory of it is not right. So you interpret the theory of the proof by the sense of our key product, the other math, the final math, and so forth. Well, in the category of sets, you can find that. Therefore, you can find it in some of the stuff in the ordinary sets. And it's unusual, even without categories, it's unusual for a group theory and a category theory to directly relate to each other. Well, at least in principle, although in reality, what you say, you could take that paragraph and throw it into the middle of the steps for mathematics, and it would still read well there. But they weren't really using it yet, because in principle they weren't. In principle. In principle. In principle. And that principle should write it. And you can still read it. Well, I don't know. I don't know. I don't know. Just go. Thank you for your attention. But this is the kind of thing that you've been doing all along, and even now when I hear it, I miss some of it. When you say factors through, my first thought is meaning like by underlying set factors. Among other things underlying set factors. There's lots of other functorial... Sure, if you're really going to make this interpretation explicit, say a group, you're going to look a lot more at the underlying set of truths. But all of those will be functors of what we really perceive. Thank you for your attention.

1:45:00 We need other tools of mathematics. When you talk about fiber bubbles, you want to know what symmetry they have. You want to know what mass they have to others. You don't want to know what the elements of the set theoreticalization are. Well, the stuff you want to know all exists even just to take the sort of the zero-level categorical foundation or just use whatever we talked about sets will mean the category of sets. I mean, they're more elegant categorical foundations. With some kind of categorical foundation. Everything we want to know in physics only exists up here. All the symmetries, all the transformations, all the naturalities, they all exist here. Specifying which elements that's at, that is not there. Now in principle, in ZF foundations, you didn't have to do it. This is a principle that the ZF people have been solving for all their lives. We've done some concrete constructions on fiber bundles that are interpreted as an adjoining dysfunction systematized by using some adjunction. That may or may not be a better way of doing that, but at the sort of zero level at least, it doesn't bother children. You're also going to look back on it. You're not going to have to discipline yourself. Well, I think your question is a little different from the concept that you had today, about how the physics, the formulation of physics, results in physics, right? So that already has a much deeper meaning because I did the second part of the presentation. The first part of the presentation was a study of computing and mathematics and dynamical systems, what they're called, but they're not really dynamical systems.

1:47:30 I can put in the extra ingredients, I think, in a very general categorical way, which provides for all kinds of state spaces, function spaces, and state spaces. And that is intended just as a first step for a broader, again, I keep emphasizing it, a broader understanding by a larger community of the role of a continuum of effects, Pneumo-mechanics continue to reprimand, which, in fact, physicists themselves don't appreciate, but my teachers do, though, and I think it's still true. On that note, I have just remembered the name of the other gentleman. I don't think it's possible to have an email from him after I've said a few words. Not very good. So keep reminded to do that, because that's something else. Before we leave this topic, I think I've gone on to the last one. Just to ask John perhaps to expand a little bit more on this remark about the naturalness standard of taking mapping because there's a very strong tradition which regards any field of geometric and magnetic intuition. I understand that Aldrich was very suspect of old abstractions of cosmology, and you're well-marked about mathematics, but to some ears might seem to have smacked at that, so I'd like to understand a little bit about it. Don't worry, I'll sit in the back.

1:50:00 Well, I think, okay, you know... Yes, true. I do think back to the notion of which could be taken as a primitive. I mean, why not? It could be taken as a primitive and axiomatized in much the same way as membership was. I mean, just because one actually takes the notion of map as a primitive doesn't mean that it has to be suspect in any logic. I mean, I know the point you're trying to make. That's the first observation. It can be taken, of course, as something that is primitive and also that does have the, also at the same time, it's primitive, but it also reflects rather more directly. These are notions with which we are familiar and which we encounter, merely because those actually do arise from intuition, or we are presented with, it hardly doesn't necessarily mean that they should be regarded as suspect. I think the reason, yes, okay, that's the first point. The second is that... The notion is over, let's not expect it. Yeah, exactly. No, exactly. And if you took those as axiomatic, I mean, there's certainly a number of geometers who would have taken those as axiomatic notions. Of course, we know there is this history. It's complicated because it's connected with the development, really, of the continuum, the definition of the continuum in analysis. I think that caused a lot of difficulties, and there are problems there, where indeed the notion of intuition, the advances in knowledge, actually, as a matter of fact, of functions. It was Fourier analysis. All of this led to an enormous amount of, a lot of it led to this questioning of the nature of function, the nature of the, the nature indeed of the domain, right, on which these functions are defined. I mean that was a, that was, you know, not in the notion of, it was, you coached Colin Schoenfleiss. One of your, it's, I forget exactly where it is, where he says the whole advance really in that was in understanding function and argument, you know, they rarely, it's a very interesting quotation because you can see that the, in that case, the notion of function wasn't sort of, didn't disappear, it was still in some way a kind of notion that was in the forefront, it surely says.

1:52:30 The other part of the analysis, I think, was, of course, it's certainly the point of f of x. And it's a question of the x, right? It's a question of the nature of the domain over which the x varies. That was the part that introduced, well, that really is the source of set theory, at least it is in the case of Newton and Campbell. So the idea of using, if we take the idea of function as a sort of primitive, as a primitive notion, well, I mean as a central notion in mathematics, then of course it's going to be quite natural to, which it always was, it never disappeared, set theory gave a definition of it, but a lot of mathematicians didn't take it all that seriously. I mean, they don't today. I mean, it's just because immediately, as soon as a function is introduced, you had to then talk about many, you know, many valued functions and you then had to get to, you know, sort of expand that notion more or less back to where it was, I mean, before the definition was made. So, it seems to me that in many... It's true that as far as, if you look at it in my contemporary, the analysis of the continuum, the domain, I think, of the x, the net of x, sort of forced some kind of consideration. Continuity is a difficult notion. There is an intuitive idea behind it, but actually when it came down to analyzing what a domain of continuous variation actually was and give its properties, it's a very long struggle to understand that. Right? At some useful level. But I don't think that really affected, in a way that didn't affect directly the idea, but I mean the notion of function was evolving in some way, you know, it was, oh, side by side perhaps, but it wasn't reducible, if you like, to that, right, to the definition of a continuum. Sector, he made it look as if it was, I think, when it made the definition of a function instead of organ pairs. But I don't think mathematicians are still sort of thinking of functions in something like these. Very rarely might. There's a very, very, very, very elementary point here, right? That this view of picturing the function as this graph is reflected in those sort of pairs. It's only one of the ways the picture of the co-graph is at least as often used. In teaching, we're describing the picture which represents the Kogrash.

1:55:00 Now that can be, you know, sometimes equally well describable in a category theory, but in terms of set theory, what does it mean, this Kogrash? You just have to, or you could express it, but only by, you know, going through the circumlocution. You know, redefining the category of sets in the usual, complicated way, and then applying it in that particular case, where it's a direct, I remember I used to give my daughter an exercise when she was four years old, just translating a co-graph picture into a graph picture, and that became, you know, that way. You learn what a function is. Since this is a very natural point of which to introduce what I think was probably the last topic of all, this analysis of continuous variation in the orthodox function, it seems to have come to an end. Now Bill is well known for his observation that the natural numbers are the source of I'd like to say a little bit more about that theme and where the wrong term was taken. I dedicate to you in this presentation the sources of the bad infinity, which is our understanding of life and our sense of the masses for the further century. Perhaps we can take a look at the last bit of the lecture.

1:57:30 Oh, maybe we can. We began with this. We began, that's why I thought it might be appropriate. We already discussed it. Yeah, we discussed this way back last week. Well, the very first day we discussed the same topology, but one thing, Leon wasn't here on that occasion. My reflection is that that was much more specific. Nor was I. And in fairness, that was, as I recall, a rather more technical discussion. Whereas here I'm trying to go a little bit out of the way, a little bit broader. I was trying to think of an example, an example of an orientation. You've gone admirably, admirably, admirably. Bill, we prefer not to argue. No, no, no, that's fine. I mean, it would be good if we could. I mean, I can't look at what they're talking about. That's very good. In fact, it means that, in the end, it's very rapid. It's not making it. Very rapid, indeed. Structurally. In the case of arithmetic, if you come up with a term for a moment where you apply a function to such and such a point across a point across a point across a point across a point across a point across a point across a point across a point across a point across a point There are higher references to mathematicians than there are scientists, and that's why I'm trying to introduce more of a quantified sense of life, and essentially only very well if we ever, in any direction, include the geometry of the state of the world. On the other hand, in the way, in the over-extending, more depth and height, an exact, an exact version of that is what we would be interested in.

2:00:00 I suppose that all is connected to one another. The pathology is connected to the space and the planet. What I've said, unfortunately, is that the planet, the matter, is like a thing. One might say, okay, that's fine, but that universe, the task of what it is, in general, it isn't the original. It exists. All these people believe it's the fossil of the cosmos. Consolidated forever. It definitely does. It has an impact on what we believe in. This is the universe of mathematics and mathematics in general, and the universe of mathematics in general is the universe of mathematics in general. In fact, most of their discussions can be carried out on these ten units of the complement. The fact is that those are clear splits. You don't get at the end of the course of anything. They include discrete enemies that you simply can't get there by means of the definitions natural to that subject. What would you count, of course? Well, algebras. What would you include in a final discussion? No, I don't understand. You get both the subjective effects of impugnance and pruning.

2:02:30 But also the objects of the space-building curves. This is supposed to be, this is not just about music, this is about... Yeah, human applications. In fact, the point is that, okay, pure geometry, you're describing space. But mathematics will become more and more involved in thinking about space and knowledge of space itself. As I was trying to say before, I think Galileo in some way was perceiving that when he said it must be written in language like Penrose. Because he knows that it's a collective enterprise, that you get these ideas and proofs and examples and everything from a society of people working to the progress that has to be in it. By the way, it's a nice interpretation. I am not sure if this story is coming true, but I would take it as what should have been rather than... No, I'm taking it. I'm always taking it. So then the question comes, in mathematics we have both ideal conceptions which represent idealization of space and motion and so on. We also have idealizations of representative thinking itself. So what shall be the relationship between them? Now when I say idealizations of representative thinking itself, I mean the natural numbers between all the speech structures. Because, I mean, again, maybe we would like to know who coined this term natural numbers. It's a big swindle. They don't arise in nature. I mean, the individual ones arise in practice, of course, but precisely the set of natural numbers, in other words, an infinite set which contains nothing except that which we can reach by a purely subjective process, in the thinking of the next more complicated formula. This is the essential example, you know, not... The idea of a system of formulas where we kind of move to the next one. This is what I call the subjective. It's really just the thinking.

2:05:00 So now, to objectify that, put it into the same world with the idealization of space. We invented this idea of a set of natural networks, not just an individual network, but a set, this huge idealization. So my basic idea is that we should recognize that idealization as being more ideal than the idealization of space, the line and the functions of the whole minimum analysis and all that stuff. It's an idealization, a conceptualization of an objective, whereas the idealization, the objectification of the subject, the whole idea, the set of formulas, even Gödel's formulation of it, you start off with a framework in which you presuppose that there's such a thing as a set of formulas, a total... Objectification of totally subjective things. And now it goes on from there. So the thing is that I can see, you know, it's clearly the case, it needs to be investigated, but using topos we can separate these two aspects. They interact very much, but they can be pulled slightly apart so that one can see the proper relationship. Mainly by taking a model of, you know, a suitable model of a mammal analysis as a site where both the topos, both the topos are automatically contained in a natural number of objects because of the, you know, pride spurting of Dedekind's construction by infinite intersection. This construction by infinite intersection, this actually works in any topos, but all you need is the maximum properties, the power. So, as soon as you introduce the idealization of truth, the truth value object is an idealization of something that seems secret, but in fact, its natural properties imply that this construction of anything can be done, and hence they should get the objectification of subject.

2:07:30 So, the site has to miss the truth value object. That's nearly always the case. That's right, so there's no minimal geometry in there, but there are functions in a subcategory, a full subcategory, a full subcategory, that does not contain the natural number object. Now you could say, well, why is it that the natural number object is contained in the line? The line is in this. There's an inclusion map from the natural number object, which lives outside, into this real number object. In fact, even an equalizer is defined by an equation, but the equation does not have its co-domain among the geometrical objects, for example, the geospatial objects. So this geometrical portion is closed under equalizers and co-equalizers. Other things can come in from the more idealized world, which mixes up the objective and the subject to a greater extent. If you take the construction of Keanu's curve, this is so clearly a mixing up of subject and object. Because you have a sequence of approximations, you think of the first approximation, well then you think of the next approximation, then you think of the next approximation. And somehow, magically, by passing through the limit, this is transformed into a so-called objective thing. Well, in the usual formulation, these things are all squashed together so that you can't pull it apart, but as you described, it's clear. So that's, at the moment, the way I see it. I'm not, the term bad infinity, I don't really know. Sometimes from reading I get the feeling that he did mean, you know, the identification of subjectivity and that's precisely, but maybe he meant something else. The more accurate philosophical interpretation of subjectivity.

2:10:00 Maybe one way to put this, it might be easier to get started with, but you're saying we take these ideas along the international minimality of Toko's tools, we get a universe that lives inside Toko's and it has standard arithmetic. But this part of it, you can do an awful lot of construction very naturally, as if Tarski's reels without the natural numbers in them, really, that's all there is to know about the reels. From inside this world, this isn't a model with a certain vocabulary. The line just looks like it does in Tarski's theory. So the simplifications that you get in this feudal, free world, where you can't express arithmetic, you can now do a lot of set theory who are fairly freely in apparatus. But none of the gables will ever arise. Even though it's true that even in this, it's not a whole topos like this, it's a topos that's like the gable stuff is waiting, but you can now operate pretty freely as if there was no gables at all, the way that Tarski does already in his elementary geometry, but only elementary geometry. Well, you get a lot more of them, except when you're around a piece of it. If that comes the next week, that is going to be the end of this. And then again, I would just like to thank you for the three minutes that you have, particularly the four of them, for being with us. I'm going to leave it at that. I'm going to get out of this moment with a hearty one. But instead of taking a whole minute of this, I'm just going to say that we are the reals. Subtitles by the Amara.org community Quite opposed to the usual idea of foundationalism, that somehow you're going to make a sweeping statement as the axiom, metal and time, you know, it happens to be a real, real genius, like an anointment, it'll fall into your head, and after that everybody else is just a servant of that. It doesn't work like that. This is a very excellent example to show that it doesn't work that way. Philosophical principles themselves are not going to be.

2:12:30 Sandy Eilenberg, a member of the 1990 Catholic Theory meeting in Köln, at the last time I spoke, remarked in a conversation with you on this very point that foundations only come into focus working from the inside out, from the inside out, and that they come into focus piecemeal, it's more like a Space station. No, no, no. It's a revolving space station being reconstructed in orbit. It's not the best orbit we usually have in the foundation of the building of the rest of the space. And this topos construction is one way of articulating how you really could have done all your math, so to speak. It's hard to articulate what you could mean by all your math. You really could have done all of your mathematics in the sub-minimal world. This is a technical thing, but it articulates. I'm sorry, I cut across. Do you have a question? Oh, was it when you were talking about applications of the, I guess, of the old minimality to the Negro people, is it reasonable, is it a fair first approximation to say that what these Negro people were doing was largely expressive of the mathematical things, and what they've learned is how much that really matters? Yes, yes, exactly. I mean, it was basically, it's very much, of course, they were trying to represent it as you are. There were quite a few stratified various spaces which occurred in those institutions, and thereby they could start from home. And that's the reason that some of the professionals who used that universe, they remarked, and they needed to know the stratification system. But they did what they did when they found out that character analysis was the functions defined in terms of all the conformal languages of the logarithm near the origin of the structure.

2:15:00 It's a very interesting story. They focused on this problem. They couldn't do it and they did something very sensitive. They put, for example, SOS up in the internet. And Gabriele, who was working, he knew both the sub-memory and the logic of this macro and constructive anomaly of the universe about the sub-memory and the logic, which was quite interesting. So their problems and their methods were in a tamed world, but they hadn't noticed teamness. They didn't notice. I mean, they could write out the definitions, but they didn't know that this particular universe was tamed. And that has problem. It's all looking at various sources. The first paper appeared in the Rembrandt 60 proceedings, but the second one hasn't appeared. It's a nice table of words that exactly is a sort of cubic diagram at one point of the physicalism and then the whole thing gets pushed. But it's not that they brought in high-powered machinery and founded them, it's that they found they were already doing something. Yeah, I mean, they were just, they were stratifying and they wanted to can sort something out a bit, but they found that what they needed in this world is to use restricted elements and functions of quantum computing. And then they just didn't know that the sets that are in this form would came with us, basically, and then they did the right thing, they asked, you know, it's quite free. And then we need to look at the after, because in principle they expect this kind of thing to come out of the fragment, but in movies, it's not a mutation procedure. If you don't typically end this kind of thing, you do not do bad things, you don't do any other wild things. The inductions we use, we use the natural things we want to do, and we never do a thing with it. You just said you never pick up an infinite sum of the sort of function of an infinite sum of the sum of gasoline and steroids. That never happens. That kind of junk. That's business for me. It's implying something. This kind of thing we analysts forever think. There's never something the journalists either needed or wanted to do. So they're really not using it.

2:17:30 The books on analysis contain these horrible pictures. You know, they always come... if you have a reasonable intuition about how expensive an instrument is, you'll say, oh no, look, there's a huge number of these things, but they all, I think it's fair to say, they all have this, they're like the piano player. They're the sort of unwarranted intrusion of the subjective. Colin Yeo and I were lax and jovial, but Angus Koch gave a lovely little talk, actually a very informal talk, called Just When Did Nature Become Fractal? Just When Did Nature Become Fractal? And he made this point very eloquently. Obviously all the kind of pathologies account for the analysis of relativity and other things like the equivalence between having inferiority. This again, this can't, the pathology there cannot help. So what happens to number theory? I mean, is that a separate... Well, it's just not there. The point is that if you deal with wild enough functions, then number theory gets brought in by the set of... You're stuck with a girdle thing. The point is, if you set out to do something geometrical, but there's no particular reason for you to do it... No, no, no, I understand that. Then, in fact, at least in many classical situations... My question only was, yes, but there is still some, some basic arithmetic that one uses in any context. You know, I mean, one of the striking things about, about, oh, you know, working the calculator, infinitesimal, all the classical things, is really numbers are not involved. I mean, except you see them as, in partial D you see two, you might see three, you know, partial differential equations. D to, you know, D to X, D to Y squared. You hardly see some very basic facts about numbers, no doubt, are used, and I presume that Tain-Tain models actually realize that, that it has this minimal amount of... They just use the... Sometimes they use the kind of things that... Exactly. Exactly. No, that's the whole point. No matter how big they are. I mean, people can say, and have always been able to say, oh, we don't use things like space-time curves.

2:20:00 But the point about minimalities, you've identified a sense in which you wouldn't be successful. It has no consequences. If you're avoiding them to this extent, that's got to be all useful consequences. What this is, if you have something, there's obviously one of these, which you can't understand nature without, you can't understand nature without oscillating functions, but there, I mean, that's the passage into the screen. The moment you have an oscillating function, you get a real discussion point. And so on and on and on and on and on and on and on and on and on and on and on and on and on and on Subjects of which we began the discussion, and I think representatives of which will draw the conclusion. Breakers and the champagne. Excuse me, I'm the one who doesn't deserve a round of applause. And to congratulate everybody, including Pierre-Claude, who is still not with us. Not least, but a quite incredible stamina, intellectual stamina, physical stamina, that you have displayed for the last seven days, so may I shake you all by a hand? If you're not a post-grad, maybe we should say, we made it. Thank you, John. Yes, as I say, on that note, if you want to take a 10-minute breather, we've reserved the table at the Barleyville Chefs for the quarter-punch. Can I copy those to my computer? I mean, Bill, were you meaning to... Have these returned to you, right? I assume you were. You want these returned to you, right? Could you put them here? I just want to reply. Yeah, sure, sure. Thank you. I've taken some. May I copy them tonight? Sure. No, I make it just eight o'clock. My watch, no, I make it just eight o'clock. We will be late anyway. You're fast. Unless I'm suddenly slow. Oh, I make it too fast, don't you? Well, we should go within the next five minutes. But I'll be very easy.

2:22:30 It's the same place we went to. Oh, no, you went over there. You went over to the pizzeria. It's right next door to that. As you go up the hill on the right, it's the... You know the beer bar? The Breton beer bar. Through the arch. It's beside the pizzeria. It's right next to the pizzeria. It's the Robert Clancy-Schneitzer one, isn't it? Well, it's okay, but they never expect you to be there. No, no, no, for us. As long as we're there, we're in quite hard class. Yes, we're fine. Okay, seriously, I'm in awe of your stamina. Yeah, sure, you want to go online and... I just want to log on to your email and reply to this message. Oh, yeah, sure, sure. Quickly, because I felt I ought to.