Roundtable Discussion (contd.)

0:00 I like the way that you use the foundation conception as something which is not just theoretically 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 aim 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 foundations we were speaking before. In this case, let's say, the infinitesimal calculus provide 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, tensorial calculus, how it was called by the Italians, the absolute differential calculus. This is the foundation of general relativity. Even though it is a development of plain infinitesimal calculus, but it's small, if you stay only with differential calculus, the standard one, you cannot do it. Then you have quantum mechanics, and you have there some other things like Hilbert space and some things like that. And not to speak much later about non-linear dynamics of all kinds and say, no, what you were doing with your linear point of view is not enough. Do you see how 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...

2:30 Group theory and topology, and then you explain me how all of this fits into category theory, but do you at all consider this 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? Well, we spent a few years preparing a textbook. The first year of university, one of the main goals was to get around what you just said, namely the necessity for a prior knowledge of modules, topological spaces, or some mathematical categories, as might be gained in upper classrooms, by instead providing directly a sufficient number of simple and useful, very useful stuff. Reflexive graphs and irreflexive graphs, discrete time dynamical systems, so three or four examples of each kind of abstract which you can learn directly without at the same time are sufficiently different among them to justify having a general theory. So that's the strategy. We carried it out. I get it. I get letters from people saying, oh, this is indeed how they first started to learn, and they found it very useful, often from computer scientists, not necessarily children, but people who are even advanced students of computer science who find that they missed out on modules, topological spaces, and all that, but nonetheless realized that they need category theory, so this is another audience, a distinct audience. This book is now available in Spanish, by the way. In case you read Spanish better than English, you can read it there as translated. But this is certainly intended as an effort in that direction.

5:00 Make available the allegory, theoretic, and even in the last chapter, the sub-object classifier. Because in these concrete examples that we have, the sub-object classifier has a very concrete meaning. I don't know if graph theorists have ever noticed that it has this universal role, but I mean it has all sorts of universal roles, or the sub-object classifier for not necessarily reversible dynamical systems, it's itself a dynamical system, a very non-trivial one, so again the fact that the... Not only the notion of a category or not only the notion of a Cartesian closed function spaces is illustrated very, very concretely by these examples. 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 more combinatorial applied fields to do something with them. Provide some kind of basis for actual children and high school students, some assistance in forming mathematical concepts. Now this is a problem we think about every day because I have a grandson, but I don't have any systematic answer to this, but I have in connection with this. With the obituaries for McLean, I'm talking, did you get one of these? Yeah, yeah, I saw one of these. Yeah, you're right. Oh, this is a kind of naturalization. This is an attempt to, it's really just an outline of a slightly longer article, which the record page is providing a completely, in other words, I'm not pulling any punches here,

7:30 this is the real equation that defines naturality. And moreover, this is a genuine example. The next page after this goes into the question of the collective versus the individual. So there was an orchestra with a conductor and so forth, not just a single performer. Due to natural conditions, it brings out the sort of opposed nature of time and states. In a way, it's applicable in many contexts. It derives from the theorem of Bob Walter, the existence to constrained equations in the field of concurrency. So my idea was that if this thing works at all, then I will expand it a bit more and publish it also in other places. Of course, the Italians, like Renato Vetti, the editor, are very open to this kind of stuff. I tend to get more sympathy there than I get in those places. I want to ask just a minute. Let's assume this. The question is if a person whose world view is based on perceiving different laws of physics, better laws, more general, or there you can just reformulate things as 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, he said, the book of nature is written in the language of mathematics.

10:00 So the first question is, who wrote it? So if you take is, I don't even know the original formulations, but if you take is, it means già fatto. It means, when it was written by somebody, it already, well, who is this somebody? Of course, you know, right? So, therefore, he's comfortable with the church, okay? On the other hand, if you interpret it as the is, like it's is being written, it's a process, then, you know, he's saying, well, actually, I'm doing part of the writing, me, Galileo, but especially, it's mathematics. Now what does mathematics mean? I mean, various people who know Greek, including Greeks, you know, have told me various things about it. But one feature, one feature, which seems to always be there among others, is the fact that it's something that's teachable. In other words, it's the aspect of all of our... All of this can be translated to the next generation. It's the one which can be used in order to establish the unity between people who have to produce in the world. So, for example, this house... All of this was built to have, you know, this width approximately the same as that width. Some workers were working over there, others were working over here, without mathematics even taken in the narrow sense of numbers and geometry. You know, they wouldn't have sufficient unity to carry out any collective production at all. So, you ask about physics. Well, the present situation is... That our world depends on physics and yet nobody understands it. 99% of the people have no understanding whatsoever. Another 1% think they understand it. So the fact of having a collective understanding of physics, classical and modern physics, is something that mathematics must achieve. And that it does by... By clarifying, in particular by clarifying through unification, so that's what, it's not that the theorems of physics, which of course the theorems of physics would change if more people were thinking about it in a coherent way, but the main point is that in fact more people can think about it, can develop their thinking in a way more than just...

12:30 Oh gee whiz, atoms crash together, don't they? They can actually take definitions and results and develop their own thinking about them. 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 the statement in two steps, yes. But not because we start off with, oh, we worship category theory. No, because we see that that's the only available tool, the best available tool for bringing about this mathematics, i.e., the communicability of the maximum amount of knowledge about the real world. Let me take a try at this. The idea, as I see it, is that categorical thinking actually says less than subtheoretic thinking. The set theorist who wants to study, say, groups is going to see all the same relationships that the categorist 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 founded on set theory, they will have then told you to forget the set theory. The point about a free group is that the group homomorphisms from end to another are determined from, you know, where the generators hold. 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's really defined. That was a definition. The definition is that this is an adjoint to the underlying subfactor. That you know a map for a free group as soon as you know what it is. That statement, which the set theorist had to make anyway, defines the free group. The category simply says less about the free group. It's easier to say less than to say more, especially when the war wasn't carrying any content. That's the idea. Category theory is an articulate framework for only saying the parts that matter. Not for saying different things, but for saying less things, i.e. only the ones that matter. Yeah, I mean, you could put it this way. It's talking about the relationship between set theory and mathematics. Well, I mean, it's a common... It is commonplace that each part of mathematics can be interpreted in set theory.

15:00 Topology, etc. Everything can be interpreted in it. In particular, the category of sets can be interpreted in set theory. The point is that all those other interpretations factor through that one. And the category of interpretations... This is a categorist description of the situation, but I think it's very striking. Everything that's mathematically significant about any one of those things is interpreted in the category of sets. In other words, the interpretation in set theory factors through this one interpretation. And the category of sets is a much simpler thing in the universe of ZF sets. Right. Just to fix on that particular point, because every ZF set has this infinitely iterated membership trait. Which plays no role except individuating the set. The set's already individuated up to hasomorphism up here. So the difference between categorical set theory and Zermelo-Frenkel set theory is you don't worry about that iterated membership, which played no role anyway. 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 in the footnote, this is not a good way to try to proceed. Well, he was mistaken. You can. And the elements of the category of sets show that you can. Now, anybody would rather do set theory without worrying about the anatomy of the elements, and every Zagrelo-Fraenkel set theorist, by page 10 of Kuhn and Svein's set, are. The first two or three pages, we have to worry about specific... By page 10, it's all done up to isomorphism. You know how many set elements a set has and which ones. ETCS, that's all there is to a set, it's how many elements it has. Can I ask one question? The point that was made about every structure factoring through calculus has to do, I don't understand, I mean, it doesn't simply, it's simply, you need it. Well, the definition of, so you think, not this individual structure, but the theory of, the theory of the group, all right, so you interpret the theory of the group by saying, well, you have, you have a Cartesian product, you have a math defined on that, and so forth. Well, in the category of sets, you can define that. Therefore, you can define it in the ordinary sets, which are cobblestones, and stigmas, and ellipses.

17:30 And so the usual, without category theory, the usual way of interpreting group theory and set theory is to directly go to these kind of things. Well, at least in principle, although in reality, what you say, you could take that paragraph and throw it into the middle of sets for mathematics, And it would still read well there, because they weren't really using ZF features, but in principle they were. In principle, in principle. In principle, you say. In that principle, you can simply mention that. I'm still not quite clear of what they meant specifically. Composition. Just composition. But see, see, people look at it. Sorry, back to theory was just a word for dividing. Yeah, yeah. Dividing, there are two notions of division in the category, because... Composition of non-community. Yeah, yeah, okay. Sometimes I cross back and it just falls back going through. It means there exists a third map that's inclusive of community. Okay, okay. Now I'm not sure about that. But this is the kind of thing that you've been doing all along. And even now I hear it. I miss some of it. When you say factors through, my first thought is, you mean like by underlying set functors? Because we know. Among other things, underlying set functors. There's lots of other functorial... Sure, if you're really going to make this interpretation explicit, say a group theory, you can look at a lot more than the underlying sets of groups, but all of those will be functors, when appropriately conceived. Yeah, subgroups, quotient groups, all those things. But obviously not necessarily form capable functions into the calculus set, because we know that they've always been proven to be formable, therefore all of them are not. They are formable, they are. That is the source of my confusion as well. So still the conclusion would be the standard tools of mathematics. When you talk about fiber bundles, you want to know what symmetries they have, you want to know what mass they have to others. We don't know what the elements of the set-theoretic mobilization are, but the stuff you want to know always exists, even just to take the zero-level categorical foundation.

20:00 We'll just use, wherever we talk about sets, we'll mean the category of sets. I mean, there are more elegant categorical foundations where you bring it. But let's just, all we're going to do is replace Zermelo-Fraenkel by the elementary theory of the category, 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 those sets had added nothing. Now in principle on ZF foundations you did have to do it. This is a principle that the ZF people themselves ignore all the time. Well, all we're saying is that here we have a foundation where you don't have to ignore it because it doesn't follow the principle. If you've done some concrete constructions on fiber bundles, better interpret it as an adjoint to this function. You can systematize it by using some adjunction. It may or may not be a better way of doing it, but at the sort of zero level, at least, rather interpret it as an adjoint to this function. Well, I think this question is a little different, though. How the physics, the formulation of physics, the results of physics might be different as a result of it. What happens when you do organize in more terms? So, I mean, I think you have this paper there, right? Categorical, algebra. No, no, there was one, there was one. Ah, yeah, yeah, yeah. It's sitting out there. It's one of the ones that I was talking about. So that, for that, that already, that already gives a much... Well, I think deeper meaning to this, I give a second order differential equation, you see, in other words, the first order differential equations are studied commonly in mathematics as dynamical systems, what they call dynamical systems, but they're not really dynamical systems in the sense that Galileo, I can put in the X ingredient that makes them second order in a very general categorical way, which provides for

22:30 State spaces, function spaces, and state spaces, and so forth, and that is intended just as a first step towards a broader, again, I keep emphasizing, a broader understanding by a larger community of the role of continuum mechanics, continuum thermomechanics, continuum electromagnetism, and so forth, which in fact physicists themselves don't appreciate. On that note, I have just remembered the name of the young German. His name is Thomas Knight, so I don't get his details for you, so please remind me to do that. Before we leave this topic, and perhaps go on to probably the last one before we run it for shit, just ask John perhaps to expand a little bit more on his remark a little while ago. about the naturalness standard of taking mappings as the fundamental notion. There's a very strong tradition in modern philosophy of mathematics which regards any field of understandable classification, basically, which is very suspect of abstractionist epistemology, and your mappings to some ears might seem to smack at that, so I'd like you to expand on it a bit No, no, I mean, you could, you can. Well, I think, okay, yes, true, I do think that the notion of which could be taken as a primitive, I mean, why not? I mean, it could be taken as a primitive and axiomatized and much the same as membership was. I mean, it's just because one actually takes the notion of that.

25:00 As a primitive, it doesn't mean that it has to be suspect in any logic. I mean, I know the point you're trying to make, and that's the first observation. It can be taken, of course, as something that is primitive, and also at the same time, it's primitive, but it also reflects value more directly. 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 it should be regarded as suspect. I think the reason, yes, okay, that's the first one. The second is that... Notions involving our understanding of space. Yeah, yeah, exactly. Because we move around. 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 of, actually, as a matter of fact, of functions, it was Fourier analysis. That led to an enormous amount of, a lot of it led to the question of the nature of function, the nature of the domain, right, on which these functions are defined. In other words, you know, one of the notions, you coached Colin Schoenfliess. One of your, it's, I forget exactly where it is, where he says the whole advance really, and 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, as Shertley says.

27:30 The other part of the analysis, I think, was of course, as Shertley pointed, is 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, I, I, that really is the source of set theory, at least it is in the case of real good counting. So the idea of using, if we take the idea of function as another primitive, as a primitive notion, well, I mean as a central notion. In mathematics, of course, it's going to be quite natural to me, 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. And they don't today. I mean, it's just because immediately, as soon as the program was introduced, you had to then talk about many, you know, many value functions. And you then had to sort of expand that notion more or less back to where it was, I mean, before the definition was made. It seems to me that in many, it's true that as far as, if you look at it in the 19th century, the analysis of the continuum, that is the domain, I think, of the X, the event 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 the domain of continuous variation actually was and given its properties, it's a very long struggle to understand that at some useful level. But I don't think that really affected... In a way, that didn't affect directly the idea. 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 the continuum, except that it made it look as if it was, I think, when it made the definition of the function of a set of moving pairs. But I don't think mathematicians, I'm still sort of thinking of functions in something like the... I always, you know, there's a very, very, very, very elementary point here, right, that this view of picturing the function as its graph reflected in this sort of Paris business is only one of the ways the picture of the co-graph is at least as often used in teaching or describing, you know... The picture which represents the co-graph, now that can be, you know, and sometimes equally well describable in category theory, but in terms of set theory, what does it mean, this co-graph?

30:00 You just have to, well, you could express it, but only by, you know, going through the circumlocution of, you know, redefining the category of sets in the usual complicated way and then applying it in that particular case, whereas it's a direct... I remember I used to give my daughter exercises when she was four years old, just translating a co-graph picture into a graph picture and back again, and I think that way you learn what a function is, really. Since this is a very natural point at which to introduce what I think could probably be the last topic of all, this analysis of continuous variation. From that point on, the 19th century was in the orthodox version. It seems to have come to an end with the set theorem, which has all the pairs in this whole. And of course the natural numbers aspect of that observation, the slogans, achieve more explicit understanding that the natural numbers are the source of, in a lot of all evil, in any way, much evil. And I'd just like him to expand on that and say a little bit more about that. They were given by God, weren't they? Well, yeah, exactly. I'd like to say that I'd like to hear a little bit more about that theme of, and was it where the wrong setting was taken? It may be by God. By Piano, and perhaps in this crystallization of the notion of number, which has now come down to us, the sources in. That utility, which has affected our understanding of large parts of analysis and the platforms of mathematics in the last, well, for over a century now, perhaps we could take that as the last thing for discussion?

32:30 Well, maybe you can explain. We began with this. We already discussed this, didn't we? Yeah, we discussed this way back last week sometime. What, in the very first day we discussed the chain topology, but one thing, Leo wasn't here on that occasion. My reflection is that that was much more specific. Nor was I. But all of you. And in fairness, that was, as I recall, a rather more technical... I don't think what I'm talking about is supposed to present empirically, but in fact it means that set theory has very rapidly, very rapidly indeed been programmable and essentially defined in a very nice way.

35:00 Explicit bounds in the rate of growth of functions. There is none of the pathology. That's fine, but that universe of terms, for example, in many cases, there's no hierarchy theory. Typically, all you ever define after the projections are the equation of things. And you get the bonuses from that. And this has been used in personal mathematics traditionally, not mainly in molecular biology, but in hostile biology. Hard core representation of very serious real needs. The fact that one of these, their discussions could in fact, although they were traditionally carried out in the wild universe that came down to us, the general ocean of function, most of their discussions can be carried out in these ten universes, but the fact is that there's a, don't get at the integers, infer discrete indices, you simply can't get there by means of the definitions natural to that subject.

37:30 You get both the subjective effect of it. ...incompleteness and pruning, but also the object of the fact of the space-filling curves messing up the geometry. This is supposed to be, this is not just about music, this is about the fact that pure geometry, we're describing space. The mathematics have come more and more to involve the thinking about space as well as the space itself. As I was trying to say before, I think Galileo was perceiving that when he said it must be written in the language of mathematics. 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 on similar things. By the way, it's a nice interpretation. I'm not sure if it's historically true, but I will take it as what should have been rather than... No, I'm always taking about it. So then the question comes that if in mathematics we have both ideal conceptions which represent idealizations of space and motion and so on. We also have idealizations which represent the thinking itself. So what shall be the relationship between these two? Now when I say idealizations represent the thinking itself, I mean the natural numbers, the free monoid, and all these kind of discrete, infinite discrete structures. Because, you know, 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 one, but precisely the set of natural numbers. In other words, an infinite set which contains nothing except that which you can reach by a purely subjective process, namely, thinking of the next more complicated formula.

40:00 Natural numbers are slightly more abstract, but the idea of a system of formulas where we try to move to the next one. This is what I call the subjective. It's really just the thinking. So now, to objectify that, put it into the same world with the idealizations of space. We invented this idea of a set of natural numbers, not just the individual ones, 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, that the line and the functions of the whole minimum analysis and then all that stuff. There is a conceptualization of the objective, whereas the idealization of the objectification of the subject in the form of a set of formulas, even Goebbels' formulation of the incompleteness theory, you start off with a framework in which you presuppose that there is 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, well it's clearly the case it needs to be investigated at great length, but using topos is we can separate these two aspects a little bit. They interact, but they can be pulled slightly apart so that one can see the proper relationship. Namely by taking a suitable model of O-minimal analysis as a site for Groton-Deke topos. The Groton-Deke topos automatically contain natural number objects because of the Freyd's version of Dedekind's construction by infinite intersections. This construction by infinite intersections of Dedekind actually works in any topos. All you need is the axiomatic properties of the power set and the punter.

42:30 So as soon as you introduce the idealization of truth, the truth value object is an idealization of something that seems simpler than the natural numbers, but in fact its natural properties imply that this construction of data can be done, and hence that you get the objectification of subjective infinity. So the site has to miss the truth value object, but that's nearly always the case anyway, it's a good idea to post-construct. Because we want to miss the static and the chromatic. That's right. So the old minimal geometry, which analysis can be done there, functions in a subcategory, full subcategory, that does not contain the natural number of. Now, you can say, well, why is it that the natural number object is contained in the line? The line is in this. It's true. There's an inclusion map from the natural number object, which lives outside, into this real number object. And, in fact, it's even an equalizer. It's defined by an equation. But the equation does not have its codomain among the geometrical objects. It, for example, is a truth-guiding object. The geometrical portion is closed and can come in from the more idealized world, which mixes up the objective and the subjective to a greater extent. If you take the construction of Piano's curve, this is so clearly a mixing up of subjective and objective, because you have a sequence of approximations. Well, then you think of the next approximation, then you think of the next approximation, and somehow, magically, by passing to the limit, this is transformed into a so-called objective setting that will occur. Well, in the usual formulations, these things are all squashed together so that you can't pull them apart, but in the setting that I just described, it's clear, I mean, it's implicit that I already pulled them apart.

45:00 I'm not, the term bad infinity, I don't really know what Hegel meant by that actually. Sometimes from reading I get the feeling that he did mean the objective part of subjective infinity in that precise sense but maybe he meant something else so I mean the more accurate philosophical will be the objectification of subjective control. I put this as much that might be easier to get It lives inside a topos and it has standard arithmetic in it, but this part of it, you can do an awful lot of constructions 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, a certain vocabulary, the line just looks like it does in Tarski's theory, so the simplifications that you get in this where you can't express arithmetic. It's not a whole topos like this in the topos outside. The gable stuff is waiting pretty freely as if there was no gable phenomenon. Tarski does already in his elementary geometry, but only on the way you get a lot more of a kind of symmetry around it, these good products. And then again the lizard... There's that aspect in a serious way, you see, because the only way they arrived at this was by careful work, by actually analyzing things.

47:30 I'm quite opposed to the usual idea of foundationalism, that somehow you're going to make a sweeping statement as the axiom, and that'll imply everything. You know, if you happen to be a real, real genius like from Neumann, 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 come out of the curriculum of mathematics and that gave the... Sammy Eilenberg, I remember, at the 1990 category theory meeting, the last time I heard him speak, remarked, I think actually in conversation with Neil on this very point, he said, foundations only come into focus working from the inside out, from the inside outwards. And they come into focus piecemeal. It's more like a revolving space station being... And this topos instruction is one way of articulating how you really have, so to speak, it articulates what you could mean by all your math, you really could have done all of your mathematics in this own minimal world. ... phenomena don't arise. This is a technical thing, but it articulates this idea that you could have just thought as if everything was like this. I have a question, I think, I'm sorry, I cut across. You had a question earlier about this. When you were talking about applications of the, I guess, of the O-minimality to lead groups of people, is it a fair first approximation to say that what these lead group people were doing was largely expressible with definable things, and what they've learned is how much that really matters? Yes, yes, exactly that. I mean, it was basically, it's very much, of course, exactly what's in there. They were trying to, I mean, it was in this cachoir. Microfunction, kind of, the exact third of the paper is characteristic cycles for a constructed machine. And they were practiced stratified as various real spaces which occur in the representation theory. And they're about to construct the cohomology, exactly. And that's the reason that the Sumner, Gershon, Gershon and Goretzky, in their book, use that universe.

50:00 And they remark, kind of thing, and it's defined in terms of ordinal polynomial operations under a logarithm. They focused on this problem, they couldn't do it, and they did something very sensible, they put an SOS up on the internet. Gabriele, who knew both, is a marker that constructed a normal universe above the subalternative one. In which all the definitions could be done in which there's fineness, I think. So their problems and their methods were in a tame world, but they hadn't noticed tameness. They didn't notice. I mean, they could write out the definitions, but they didn't know that this particular universe was tame. I mean, that has powerful... It's not looking at various sources. The first paper appeared in the ground in November 60 of the proceedings, but the second author has a paper he proceeds to ICMU. It's a nice table that points out exactly, there's a sort of commuting diagram on one form of physicalism and analogy, and it's founded there, it's that they found they were already doing something. I mean, they were just, they were stratifying and they wanted to construct something out of it, and, in other cases, to use restricted analytic functions and the logarithm, and then they ran out again, they just didn't know what they said to define it in this form. And then we did the right thing, we asked, you know, because in principle they expect this kind of thing to come up very regularly. You don't. You do not do bad things. You don't do wild things.

52:30 The inductions we use are relatively natural. You use the natural things. You never do that kind of thing. You just said you never pick an infinite sum and construct, you know, you construct the function of an infinite sum in some ghastly state. That never happens in that kind of geometry. That's basically what Gruden teased. There was never, there was never something that geometrists either needed or wanted. Analysis, you know, contained these horrible pictures, you know, infinite snowflakes and... You know, they always come to, you know, if you have a reasonable intuition about how extensive and intensive quantum mechanics are, hey, they'll say, oh no, there's a huge number of these things, but they all, I think it's fair to say that, well, they all have this, they're like the piano pair, they're the sort of unwarranted intrusion of the subjective infinity into the object. You'll probably recall that in 1989, I mean, Cambridge was the very first meeting that Colin, you and I were all at, and John of course, that Anders Koch gave a lovely little talk, actually it was an informal talk, he didn't get into proceedings, called, Just When Did Nature Become Fractal? Just When Did Nature Become Fractal? And he made this point very eloquently. All the kind of pathologies that can result from the analysis are relevant over the years. Other things like equivalence between having interior and posterior nature and all these kinds of things. I mean this again, this kind, the pathology there kind of happens. So what happens to number theory? I mean, is there a separate... Well, it's just not there. The point is that you make a wild enough function. The number theory gets brought in by the sense that you're stuck with a girdle thing. The point is, if you set out to do something geometrical, there's no particular reason for you to... No, no, no, I understand. Then in fact, at least in many classical situations, you're not going to get enough. My question only was, yes, but there is still some... Some basic arithmetic that one uses in any context.

55:00 I mean, one of the striking things about, you know, working with calculus, infinitesimal, all the classical things, is that really numbers are not involved. I mean, except you see them as, in partial, you see two, you might see three, you know, partial differential equations. D to x, dy squared. You hardly see, I mean, some very basic facts about numbers, no doubt, are used. And I presume that the K-models actually realized that, that they had this visual about it. They just used the... In the present times, they used the kind of things that... Exactly. Exactly. I mean, that's the whole point. No matter how big they are. I mean, people can say... Oh, we don't use things like space-filling curves. But the point about minimality is you've identified a sense in which you wouldn't be using them if it has dual consequences. If you're avoiding them to this extent, that's got real useful consequences. The point is if you have something, there's always the idea that you can't understand nature without oscillating functions, but there, I mean, that's the passage into the discrete. And the moment you have an oscillating function together with a... The real structures, you will be able to define using that as I've said in the space film, actually define it, and not using this infinitely process, because you'll have the power of the set of quantifiers that will give you that. That's the point. Well that's beautifully connected the theme on which we began this afternoon, the continuous versus the discrete, and very raw philosophical and historical setting to the subject with which we began the discussions a week ago. I think it represents the perfect point at which to break this to the chair. Congratulate everybody for quite incredible stamina, intellectual stamina, and physical stamina that you have displayed over the last seven days. It's a nice shake of all by the way. If you have a postcard, maybe we should send it out to everyone saying, we made it. John, for the quarter, for a quarter, but can I copy those two?

57:30 I mean, were you meaning to? Have these returned to you, right? I'm assuming they were. You want these returned to you, right? I just want to reply. I can't take them, so... Well, may I copy them tonight? Michael, it's already 8.15, shall we... No, I make it just 8 o'clock. Is it? My watch, no, I make it just 8 o'clock. We will be late. You're fast. Oh, I make it, I make it too fast, don't I? Well, we should be there within the next five minutes. Where is it? It's the same place we went to. Oh, no, you were out there. You know where the local pizzeria is? It's the next, it's right next door to that. As you go up the hill on the right, it's the... You know the beer bar? Thank you for your attention.