Conversations

0:00 This is of course the motivation for me, but of course I hope at least that what I say on the horrible part or the graphic part would not be dependent on the epistemologies or anti-epistemologies. It would be horrible. If a cosmologist comes into the room and says that she wants to determine the global topological structure of three-dimensional space, then at that level she is indeed using the continuum in a way that makes it possible. You can do it in a lot of ways, but not as a carrier of physical properties. The problem with using the continuum is that infinite divisibility. It doesn't make much sense. It can't be a continuum. But you can describe it as describing the large chunk behavior of something. Describing, of course. But this is the... I understood. The worst moment I didn't understand what you mean was the beer glass being a continuum. Because I looked, so to say, in a... In the wall of the glass, I said, my goodness, atoms, and in the end it is not a continuum, because it would have more mass than the rest of the universe, but of course what you meant, it has a topological property which can hold the water, the beer, or whatsoever, and this of course exists, but it just exists. Due to the level of cross-training, I only speak about, you go down to the points of the continuum, and then attach some whatever to it. Yes, but the point is being blown. Why does an mathematician want to have a continuum in back of the beer glass? Because there are many, many different ways of writing it as a concatenation of combinatorial elements that have the same global topological... So I want to have a noun behind all of that that refers to the structure itself. What I never wanted to suggest is to forbid the mathematician to use the means of the continuum picture.

2:30 This is of course completely justified. But this was not my intention at all. I couldn't do it, but I also didn't want to do it. I have nothing against speaking of the continuum, just, you know, I didn't lost the reference John Archibald Wheeler said in one paper, I think in Frontier of Time, that the continuum doesn't exist, he didn't say so, and of course he also didn't mean the continuum doesn't exist as a very proper means of describing a lot of physical phenomena and everything else. But he meant, I suppose, there is not a physical entity, and then there's a lot of things, a lot of things are not an entity in this sense, but it is not even an intangible entity if you say I attach anything physical to the point of the continuum, because if I do so, it blows up. In Greek geometry, the continuum was not an infinite collection of points, and it would have been crazy for them to think of taking these infinite points. I should have made this more. It's simply the reads, the power of the reads. Property can have the power of the universe, because suddenly it would blow away everything else, it would blow up actually infinitely. This is what I want to say. It's just the power of the continuum, which I'm against. The size of it is infinite. Sorry? The size of it is infinite? Yeah, yeah, the power. The best part. I think it's the way we use it that's the problem. I mean, the idea that you could actually imagine that space points can be infinitely divided. Yeah, yeah. Space points or, you know, space points, probably not. But it's no problem to think of the real. And to this power, you cannot attach a physical problem.

5:00 The curious thing is that the global properties of this are part of our intuition. We are continuous motion, we are the continuum of space. Many of the properties of the continuum are properties that we attempted to abstract and make a mathematical model. But we find that those mathematical models, if we think of them as collections of individual points, are not. This is one indication. It seems to me that the mistake is in assuming that the continuum is an existing collection of points all there. It's a mathematician's fancy to write such a big number. And it's of his best pride of the world, because it works with good mathematics. There's no problem with that. I understand the continuum. Maybe I, in difference to Wheeler, I was more careful. No, I've got a very definite study on how math continues. Kantorianity. Kantorianity. Kantorianity. Kantorianity. Kantorianity. Kantorianity. Now, then again, one should think twice, once again, because... So let's suppose that I imagine that I could construct an infinitely branching tree. Now it seems that I could begin to approximate it very well, right? I just keep on making this and I make these branch and branch. Now, then in your imagination... In a mathematician's imagination, you take the limit and you get an infinite branch of a tree. But then there's a continuum of paths down through the tree, and the power of the continuum number of paths. So that the continuum as an order of infinity comes into existence if you allow this apparently countable process to come into existence. Yes, but on no stage. I think it's about omega. But of course the countable process can't come into physical existence either. Only some approximation. But that doesn't give you, that gives you... What you've just done there, except the counting process there, is essentially the kind of Aristotelian potential. It's just an unending, it's not a discrete, complete infinity, as it is in the Cantoria.

7:30 Yeah, but if you imagine that you could take the tree to infinity, the number of paths going down through the tree is a continuum. But the question is, do you need for every branching level another tick? It is, you have one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, twenty, twenty, twenty, twenty, twenty, twenty, twenty, twenty, twenty, twenty, twenty, twenty, twenty, twenty, twenty, twenty, twenty, twenty, twenty, twenty, twenty. I'm just not quite sure in this context what the number of... It's always fighting about it. And, you know, I do not have anything against it. Omega. Omega just being an inaccessible natural number. Bigger than you say any number, and then say it's at least double the number, and so forth. Of course, that is never yet the power of the continuum. That's what I want to say. And it's the power of the continuum that makes the guidance. Power makes it horrible. It's the power set. You've got the power set construction. Incidentally, it puts an end to problems. It is not. It's the extension of the continuum as such. But what you have between the nearest two points, you can design on the continuum, and between these two, you have the same power as the rest of it. This is the problem, because if you would attach any physical property to the continuum, it would bring out the smallest, which is not that descriptive. It means you take the two nearest constructable real numbers, what of course is not possible, and then between them you have the same... ...the same magnitude as we saw. Which is what? You mean by saying it blows up? This is... this is... blows it up. That's what you mean by saying it blows up. It blows up. Yeah, and that is the problem with the arithmetic. Of course, and you... Incidentally, your concept you were just making around Iran in the last part of the, you know, exchanges in Egypt would be good. Um, around what time did you want it? About... Well, no, well, about continuum, about continuum. Figures, as opposed to their basic generation, are rather bad. This is a very interesting one, and I'm sure obviously since you obviously know a lot of topics doing it. Do you know that that very, very interesting paper by Orloff here, it's about 1990, it's from the proceedings of the KOMO, 1990 KOMO category theory conference. Is that it, right? No, no, it's just called, no, no, oh no, that's a much earlier paper, that's from Bristol, that's the first one I've read so far.

10:00 No, this is from 1990. I'll tell you where it is, it's in, I can send you a copy. It's something like 1,143. It's the proceedings of the 1990 KOMO conference, which is one of these big, decennial reviews of the category theory, and it's called something like, you know, Thoughts on the Future of Capital Theory, so it's a pretty big sum. But in the middle of that he has this very interesting theme. He thinks that topology essentially starts the wrong term around the time of... Rather, they should have stuck with what the people in the Italian school in the 19th century, like Voltaire and Habermas, looked for morphologically. The idea is that figures are the fundamentals of the notion of space in general. You know, where you've got adequacy and co-adequacy points as a very special limiting case within the way in which the cohomology groups, the spaces, the kind of relationship between space, universe, and common space and its fundamental group, are all put together in some one big category which You can look at the case where space does live in space, where you've got, for instance, the localization of coverings, all coverings, like I said, the smallest element for every cover, which is precisely the case, which gives you a kind of decidability of the cycle of this classical identity that space is a set of parts in that very strong way. He has a kind of extreme limited case, which in some senses is unrealistic, because he thinks of the topos and spin spaces to capture what's really going on, with of course this inherent, quite lax, inherent, you know, Leibnizian idea of the kind of intrinsic dynamism of the plane. So they're really not points, they're more like the kind of notion that people of the 17th, 18th century had of kind of linets or... Linets, yes. Linets, yes. Barrel with linets with the person.

12:30 It's a very interesting stuff, and it gives you what you don't have in the topics of smooth spaces, which is very strong transit figures, very, very stable construction. Once you're inside it you've got all these lovely properties which seem to recapture these very conceptualistic intuitions about the inherent dynamism of mathematical engineering. But the problem is the technical construction. First of all, de Boeck, who was the guy who did a lot of the work on Thomas and Smith's spaces, I mean, he and Lorbeer both did. De Boeck, oh no, de Boeck was very important. No, very important guy. Here's the guy who actually showed Daniel Goodman's script as rigorously over a base topos. You see, the base topos, the only topos, the only base topos over which it has actually been shown rigorously how to describe the topos of smooth spaces is the topos of sets. That isn't where they do it, it's the secret of the synthetic differential geometry. But it does actually, at this point, although there are lots of... There's a lot of people still working on constructing a little bit more generally based topos, but it's very, very difficult because at the moment, the only base topos over which you can completely construct and show that it has all the necessary coherence properties is the topos of sex, so it's only as it were once you've really got the construction up and going and then, so to speak, you've entered inside it, that you can have all these lovely, exciting topos of smoothness. Continuity in the sense that it really gives you this aerobaticsy notion of the dynamism of math. What's that? Why are you signing? You signed with the wrong name. You need to sign. That's what you said. Maybe under pressure. It could be, of course it could be. Why don't you just sign and date this thing and I can bring it up next year. Never want to sign an open check for a man like you. No, no, no, very dodgy. I definitely take legal advice. And it's an object, not an entity. That was the point. Yeah, that's why I was asking him. I wanted to find out how...

15:00 That means it's very concrete, right? August 2002. And so, what does that mean? It means it's an object, which means it's not an entity. To be honest, it shouldn't be, you know, but I do not suppose it really does not fit in there. Of course, anyway, it is, I still haven't quite grasped it. It seems so straightforward, but I've probably been completely crass. Objects are, they're the ones that don't kick, it's entities that kick back, right? Yeah. Okay, so objects, why can't we just call objects deposits or theoretical constructs and entities of a... So objects are less physical than entities? They are completely empty, in a sense. Okay, well then I disagree with you, which is why he's good at this. No, I didn't actually state, I wanted to make sure I understood which side you were on before I started dueling with you. And the reason is because, and some of you people didn't know this before, but if you take a quantum bit, and I mean that may be different than a bit. But if you take a quantum bit and you throw it into a black hole, it increases the surface area of the black hole by the minimum physical amount, which is the box area, which means that a bit is more than a mathematical thing, it's physical. Now I will correct myself. You better get another one of those and sign it the other way. And so the point is that most people don't understand that a bit is more than a mathematical representation. Wait a second. Let me wait a second. You know what, my idea is completely always philosophical, and you can change it every other year. But I didn't. It would be, of course, possible, but I didn't for a long while. But anyway, we heard a lot of physics of information, and somehow physics might transcend itself, why shouldn't it, to reach that sphere, you know, which is not actually underlying, but it is hidden, concealed behind, it might be this tangible reality, it might be such a sub-state.

17:30 And it might be, especially what I'm interested in as a philosopher, the state of the mind of science. And not necessarily the scientific ones. Why don't you all give it a time? May I have a look? My point is that if you would be interested, I have two copies of these. You can actually ask the question, how heavy is a bit? Of course, it does what it does in the black hole. In the black hole, yes. So you actually can say that. Well, it does that in the black hole, otherwise the black hole would violate thermodynamics. So you can kind of believe that. In fact, you can do it in the nonlinear linear case. Doug, if you are really right, I would have to correct a little bit my performance. It's just homology, but you would say and hone my ontology. Okay, so you would owe me something. I would owe you. Okay, well, it's true. So get in your philosophy. Get in your philosophy. The point is that it's important. The radical non-existence, absolutely radical non-existence, sorry, of the ideological object. It's a little bit, I'm annoyed over it. I mean it, of course, but I also would think that if they're in convention, they must be. Well, that was my next comment, is that if a bit is physical and it weighs something, effectively, it has an effective mass by this idea, right, then the question is, how heavy is a thought? Because a thought is an information scope. Well, any kind of pattern that we don't know how to explicitly measure. I don't know, I don't know, I don't know your name. Mike, it's Mike. Very nice to meet you. If you want, I have this... Really appreciate it. And I was wondering, I thought this might involve going to... This is so crazy. Horribly crazy. Well, why the hell not? I'm ashamed by what I have to lose. Thank you for watching.

20:00 No, no, it is, it is, because I want to write you a little bit about it. By the way, I must send you the references for that paper. Please, please. Whereabouts are you based in Europe? Whereabouts in Germany or in Frankfurt? In Frankfurt. I thought I'd have the mic. I have my e-mail. Oh, it's on there, is it? And the other first. Oh, this is from Peter O. Okay, keep that for me. Not just the second, third, fourth. Don't give me anything you don't know. Yes, I think your first impression was right. My first impression, well, I do not know. I'll tell you the truth. I do not know. No, you do not. No, I do not. This is actually true. I do not know. I do not know category theory, and I do not know a couple of theories. I hope that I'm sticking to some points which you may even call ideologic points. It is the first mathematics which really got to do that, with the idea of the matrix. Suddenly, suddenly they had it. They never had it. They only had to simulate it. And suddenly it was... In two ways, what fascinates me most, and this is one of my axioms then, you know, what fascinates me most is it is as well usable in metamathematics, which really has nothing to do, by goodness, with this physical application of mathematics. Horribly! I'll explain. Very, very interesting. I mean, my own work. Then, just let me go on. Then, it has such a trivial, such an absolute trivial... The most crude application, and this is another old Enlightenment idea, that everything is an algorithm, and they are always spit on because everyone thinks they mean machines, and it's what they don't. You were the man who made the remark about Robert Rosen's ideas, weren't you, this morning? Yes, yes, yes. You know Saber, he died. No, he died about three years ago. Who's corresponding with his daughter Judy, who's got a very large strength archive of unpublished papers, who would have really different biology, the foundation which doesn't have it. It's very sad, he died very young, he was only about 53 or 54 when he had diabetes.

22:30 Some of which she has sent to me, which unfortunately I'm not going to be able to send to her, but I'm going through with a few because I'm buzzing her. And I'm very interested to hear about his ideas. He incidentally is a very good on this theory because in my opinion it's a very, very good paper on mathematics. And some of his papers on the reasons he didn't believe in the term clearing model. I'm sure they came out of this. He captures a lot of how much that I think is important. The philosopher is always the last to know, by the way. Yes, the physicists have already known about whether the pen is physical for over ten years or not. And of course, the point is, you know, tell me, come back in 20 years and find out if physics, physicists still believe in physics, but whether you believe, I mean, if obviously there is some fundamental discrete level of, whether it's in terms of information or in terms of quantum, then obviously you would have. You know, your, what is it, will happen in a most macabre way. You know, what I like to see from mathematics is that it simply is not expected by any of these things. You know what, in the sense of philosophy, it's simply a way to do numerical mathematics. Well, then let's go find out. Yes, this is very, very likely. It's empty. That's the whole point of it. This is an array of indignations, which is... You know, I know what I say. Listen to anyone as idiosyncratic. What I say about theories, of course I say the opposite, just to avoid this, because I can say this, I am either inconsistent or no idealist, but no one believes me. No, I'm not sure, I think a lot of people believe me.

25:00 Yes, I know it's inconsistent, but in effect, to overcome it, you must do neutral mathematics. Or something really new to us, sorry for that. And then I'm also inspired by Davidson. I could have a dialectical monism, but then that's it. Davidson gives this idea of anomalous monism, which allows for supervenience and emergence, and emerges anyway, I'm obsessed with. But you know, of course, to remain monist, it must have an invariant. To all the levels of metamorphosis is what I wanted to say in the butterfly. It is like a butterfly. A butterfly looks very different in the different stages. Sure. And why shouldn't the hell, uh, yeah. He is nothing, or whatever, which is the real thing, or nothing, or Peter Holland, and so on, and all these ideas, why shouldn't this be just the same as what is the invariant, through all the levels of appearance, one of the appearance being not appearing, not appearing, the other is appearing as nature, and the other one is appearing as mind. Now you have this idea, but you have no interpretation in the public information of your life. To actually get an idea of the nature of symbolic concepts, of just what it is, what is the content of the symbol one. There's no answer to me, only some of them. There's some that are fundamentally lifting. No one ever gave you a broth of what is the content of it. All these guys are empiricists. They think nature gives us that. It is not true. It is not what they do. No, I think the great attraction of some forms of mathematics is that it does. There is no absolute global separation between forms of mathematics. I mean, even logical form, as in some sense. I think that's what Topper's doing, he's trying to spot the deep limits of geometry on topology. So it all comes together.

27:30 There is no ultimate. I never saw you before. First time here, yeah. From where did you come from? From Berkley. Basel's people, Basel's gang. I won't see that. No comment. Anyway, you're going to be around for the next part of the day. I thought you were speaking again. Yes, very rudely it is. I may not be able to speak at the very last minute. Tomorrow. Tomorrow? Oh, I was sitting around for that. Thanks very much again. That depends on how you might want to use it. You can start to think about different ways in which you might apply, in which you in fact apply quantities to categories, and then see whether it formalizes this way, or how you have twisted it to make it formalize this way. But whenever you are calculating something about something and taking it into another form, it's probably a function. For example, if you calculate amplitudes from climate factors, then you have a category of climate diagrams, and then you have a method by which you associate amplitudes. There is a function called finalizing, and it does follow the class of human. Now, I can give you a toy. If we now see the definition of quantum physics. Quantum physics. Quantum physics is a form of physics. A funker is essentially, without worrying about having formalized it into the context exactly, it is some way in which you went from one kind of mathematics to another, so if you associated a group of symmetries to a physical situation then you would be having a funker which took you from a physical situation to groups. And it might well be a functor. It might be good to know that it's a functor. Knowing that it's a functor means that if you had this situation which was physically, or a physical association with this one, then the symmetry groups would also be associated by this one.

30:00 Okay. And so on. Yeah. Now a favorite of mine, which is like the Feynman diagram, is totally like this, only I'm going to say it in my extract. Let's suppose that to some set of graphs, special set of graphs, I have associated input and output matrices and charges. Now, if I take a combination of processes, then I also have amplitudes. And how I can do that? Well, the answer is, the amplitude associated with a combination of processes is dependent on the preparation and detection. And then I sum them all and give you the possibilities, and I take the products and the variables. So that's a kind of mini final diagram, where I didn't explain how the amplitudes are calculated and how they're produced and fixed. But however they are calculated, then I can calculate them in whatever way. This is what Tom Ehrich was doing. Anyway, this is... If I have a process, which means that in here there are some of these other energy values, If I have two processes that have the same number of inputs, so I have a category, right? And if the composition is intelligent, and it is, by definition, I mean, the end composition is just this graph, and it doesn't matter whether I put it together, this one first, and that one first, and that one first, and that one first, just the graph. And the method by which you calculate the amplitude just depends on the graph, so the amplitude also depends on the graph.

32:30 So what do I have? I have a functor from this graph category to mappings, to what I, I guess I should just call it, vector space of the various dimensions from, let me see, I guess it would be, I have to take cm, so I'm taking the sum over m. This mapping here is a mapping from C to the 4th, to C to the 5th, to the 6th, to the 7th, to the 8th, to the 9th, to the 10th, to the 11th, to the 12th, to the 13th, to the 14th, to the 14th, to the 15th, to the 16th, to the 16th, to the 17th, to the 18th, to the 19th, to the 20th, to the 21st, to the 22nd, to the 22nd, to the 22nd, to the 22nd, to the 22nd, to the 22nd, to the 22nd, to the 22nd, to the 22nd, to the 22nd, to the 22nd, to the 22nd, So I have to get my list going. I'll get my list going. So then you see that now what we have, if you have a mapping from Cn to Cm, if you have a mapping from n points to m points, you're correct. See, now the optics is just n points in order to set up... And the output is an order of seven points. But the graph carries all the information for computing this amplitude, so everything is in the morpher. And this goes under my functor to Cn, and goes by f of g to this linear mapping to find its interior matrices, m. And there are my amplitude calculations, and I have defined a functor from the graph category to the category of amplitude calculations.

35:00 And that is, in fact, a skeleton version of track 1-0, where the base calculations depend on track writing down some integral during the length of whatever sequence. The external structure is the same as it depends on the particular theory. Okay. So, I see it as a systematizing new terminology. But this is useful language. At least I can understand. But in practice, as you said, we have already seen many instances of this. And you give me the general picture. You'll notice that the general pictures definition is so simple that once it's been said, I mean, whether or not, I mean, you know, there's another question of might there be a lot of interesting mathematics that you could develop just using this axiom system. Of course, it's an axiom system which is very simple and has a vast number of examples of that. So you could regard it basically as an organizing definition for a long time about worrying about whether you're interested in theorems. Most mathematicians, in fact, have behaved this way with respect to categories. But if you're curious, then, about what you can really do with categories, there's a book by McLean called Categories for Working Mathematicians, and there are other people who recently have written about how this actually is a very useful way to organize your concepts. And probably eventually there will be a lot of literature available in terms of motivating it and using it, but the thing is it is a systems description of mathematics in general at the level of mathematics where we think about groups, rings, fields, topological spaces, and so that's sort of like the first year graduate school level of mathematics.

37:30 And so, you know, it didn't get to be widely known the way logic is widely known, because it wasn't phrased in a way that looked like you could start there and do a lot of math, which I find a little peculiar because, as you see, in the abstract, it's a directed graph, it's an abstract. And the graph theorists, of course, took their definition, which is nothing more than vertices and digits, are these points connected or not, and then they started matching it with all sorts of different patterns all over the place, and then thinking about funny little combinatorial problems that you could ask about graphs in general, and the whole thing has a texture where you could explain a lot of graph theory problems to 10-year-olds, and anyway, 10-year-olds naturally play games on structures that are questions about graphs. So these guys were a little too intellectual and kind of didn't make the subject popular. That doesn't make it unimportant. Yeah. So what was the first name? Saunders McLean. Saunders McLean. Saunders McLean. This category actually is the one that convinced me that I was interested in category theory. You see, because it turns out that this basic graph category, because of the fact that it's related to some physical ideas and at the same time it's related to math theory, you see, if it was not theory that I was dealing with, then the elementary gifts would be the quantum and also the maxima and minima. So there are things that are in sync with the math diagram as an element of the graph category, but in this case, for the graph, these both come. I lied. I just realized I lied about something here. I got the drift. I mean, the point is that you could think of a knot as a, the thing that was exciting to us and still is that you could look at, you could take a knot, which is a thing, actually, this is a really important idea. Let me just draw a picture of that. A knot is a geometric entity and it doesn't have a direction associated with it, but if you associate it with a direction,

40:00 Then it looks like a morphism, although it's a funny kind of morphism. It gets interpreted as a morphism in the graph tag and the base ring, the hook to the base ring. If I had a triangle, a bit of a grade, something like this, then it goes from some kind of... But in any case... Nots and grades and things become morphisms, and then you can calculate things from them if you knew these individual parts, just as though it was a quantum amplitude. And then if you get the matrices in here adjusted right, you can actually get topological information out of it. You see, so the category idea that you can think about the relationship between things that look like part and parcel. Now, on top of that, is topos theory built? Yes, topos theory is. It's a further development? No, it's not. Topos is a specific category. It's a specific kind of category. Which gives you very nice, which really isolates it. It generalizes the idea of topological space on the one hand, and it generalizes the idea of logic on the other hand, because it turned out that it started in algebraic geometry, but then it turned out to have deep implications for logic. You see, in logic you decide whether something is true or false. Now what does that mean if you're thinking in terms of maths? Or you decide whether you have a subset. For example, if X is a subset of Z, that means that for every point in Z, I either set N or X. So that means that I have a mapping from the containment space into a set consisting of two objects, X and a force. So that's the sub-object classifier in the category. You see, there's some space in the map to it, and what map you have is telling you about sub-optics of this one, X is a sub-optic. It's a set of things that go to N. So I can say that in a scientific way that there's a factorization like this with mapping and injection.

42:30 There are things to be injected with said category, and you see that makes this a sub-optic. Well, you can say everything in terms of thought, but that's the other thing about the category ideas. You're not allowed to refer to the internal structure, so you have to rethink in terms of external relations how things go. And that's another reason why I think category is a good concept to think about in relation to physics, because it's asking you to think everything through relationally. So it asks you to stop thinking about what clever little devices are making up whatever you're doing and describe it in terms of how the entities that we have are related to one another. And maybe you can do that and maybe you can't, but it's challenging to ask yourself whether you can. Oh, well anyway, in a topos you can have different kinds of psychological classifiers and that means you can change the logic. Well, you actually have, in fact, the internal logic is always algebraic. It's not a good old tool. No, no, precisely. It's when you specialise in the Boolean case that you get that kind of logic. That's because you've got, well, there's actually a couple of topos. But the subject classifier has a more general structure, and that may allow us to classify the subjects which are, as it were, neither in a varying, in some varying state, can be in and out of the group. So this exercise of writing things conceptually in terms of maps and relations is actually very illuminating. This book by Laverre and Scheuner, out near the end, they explain how to do Cantor's diagonal arc using methods. And you get Laverre's very beautiful way of reformulating it. It's just something that everybody should know. It's very beautiful. And you don't have to know about categories, it's just confusing to say it in a set. It's just gorgeous. I mean, he's thinking categorically, and he actually just sort of revolutionized everybody's way of, although not everybody knows about it, but I mean, he really did revolutionize ways of thinking about sets and logic and basic mathematics.

45:00 And in fact, he's got a very interesting program, which connects up with the same technology program, to look at. What the effect of, because in a topos you have a power set object, or a topos is a power set object, which is connected with the structure of the classifier, and you can define what's called a natural number object, and the thing is there are different ways of doing this, which actually give you quite fine distinctions, which are not a level, it's between different kinds of real numbers, between a she real and a delicate real, they have different definitions in terms of... The structure that you're actually putting in, in a way, they're very geometrically motivated, the structure you put on the objects and so on. And he's got this lovely theory that you can in fact dispense with... The Cantorian infinity altogether by changing the endo maps in the topos in such a way that you would not get the... You'd get a subversion of the... you'd still have that number of them, but it would be more like the potential infinity of Aristotle. It wouldn't give you this complete infinity that you get in the Delicat case, and it wouldn't therefore have a lot of the pathologies that come out of the Delicat case. Where do you find that? You just need to talk about it internally.