Morning Discussions, incl. FW Lawvere, Pierre Cartier, Angus MacIntyre, John L Bell (contd.)

0:00 The Cantorian points to the components is an epimorphism. Points to components should be an epimorphism. That natural transformation is induced because of the adjoinments and because of the fact that the occlusion is full, basically goes through the original space, right? So now we can... So each component has a point. Yes, that's right. If it were actually surjective, that's what it would mean, but I just required that. But you say epic in the given category. Epic in the given category, right. And I don't really need that, but it makes the construction easier. So now comes the construction of the intermediate category. So that's a new segment, that's your formulation of the new segment. That's one of these which is suitable for this new context. So that the intermediate category is simply those objects for which that map is an isomorphism, for which every component has exactly one point, which doesn't mean that it looks like a point, because it has maybe a little infinitesimal, the point is a little infinitesimal cloud around it, it's quite different from an actual point. For any given space, the number of these is the same. The map is a bijection. So now it turns out that this is now a bigger subcategory, because it includes the original discrete support. It's another subcategory which plays the role of something like local or infinitesimal. It also enjoys three adjoins. Which I have invented now the colorful name of hot and cold adjoins. Left and right, you get tired of the left and right. This is quite precise in the case of reflexive graphs. Everything would be exemplified there. So you have this space, a non-trivial object in some way.

2:30 You push it down on the left to this category with points and coordinates of the same. And on the right, and there's a map between those even, and it has sort of this flavor to it that you, when you take a portion of an interesting space with that property, it's sort of like catastrophes, you've got something that's defined by a curve and you just look at the singularities by themselves and you throw away the global cohesion. For the physical, you heated up this material to such an extent, you superheated it to the extent that there are no more interactions between atoms, but the atoms themselves retain their electrons and they have their own... So high temperature. Yeah, this is the high temperature adjoint. H-O-A-G, I think. When you throw away the connections. On the other hand, the other one is the supercooled version. Because you, you collect together atoms that have any interaction at all into one atom, it's like one of these Einstein and... I suppose Einstein. You collect all the atoms that have any reasonable connection at all into one, and it has a huge number of, huge atomic number because all the original bonds have become particles. 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 and on And you can't predict that from looking only at the gas because you've thrown away the connection. You said that the interactions are negligible. But from the original object, yes. That's why it's a natural transformation, it's parameterized by the original. I like this. And it applies in every situation.

5:00 In every situation, there is exactly this kind of thing. It's astonishing. It's a Buddha record. It's a Buddha record. But it exists in every such situation. Whenever you have these four functions, that's when Milstein exists, which means, for example, whenever you have this infinitesimal theory, so it has many, many, in principle, in every field of mathematics there are things like that. So what does superheating, supercooling, what does the cooling math mean? So now you need Van der Waals equations before you actually make the transition. Currently we have only the qualitative extremes, but a map between them. So the Van der Waals equation would have to be compatible with that, whatever that means. But then there's a strong, I haven't proved this, I don't know what generality it's true, but there are sort of two classical ideas about the Noostrandsatz. That you have enough field value points. But the other one is really essentially the Birkhoff so directly irreducible thing, which in algebraic geometry is really just about these Gorenstein rings, or they have three different names. Gorenstein, Birkhoff, and different groups of ring theorists study the same set of rings under these different names. From a functional and analytic point of view, or a stochastic point of view, it means that there is a distribution of global support. It's a local around nilpotent maximal ideal.

7:30 So the stronger null stone sets, first of all for the rings, it says that the finitely presented algebra has enough maps to those things, not just that there exist maps, but that they're jointly monic. You can separate quantities by evaluating them at points and all the derivatives. Two different elements of the community of ring A, they finally present an algebra over k, will be distinguished by some map into an algebra over k which is finite dimensional as an algebra over k. And hence has one of these stochastic sections and all that. So this is sometimes called the no-so-and-so. It's stronger because you're getting not just the existing, you're getting it mono. Of course, it will never be an epi because that's where the Zariski topology comes in. To glue these sections back together, they're really quite exploded. If you just look at them, they're yet less exploded than if you just looked at them together, actually. So there's an intermediate picture there, which is a definition of a nearby point. Yeah, these nearby points. So there are enough nearby points to actually distinguish all you can say in dimension is one way to describe it, which is basically a problem. No, but you just say that the intersection of the maximal ideal is a set of new portions. So I conjecture that the analysis thing can be applied as this intermediate level should play the thing for the...

10:00 Because those finite dimensional local rings, you see, they have an interesting property in that they have an idempotent endomap. If you divide out by the maximal idea, you get a field extension. But that has a section. Yes. It's undecivated. Yes. I'm always assuming acceptability because that's what came out from the other definition. In the case of the left and right adjoints, it's the same thing. In fact, in any situation where you have left and right adjoints isomorphic, really it's all coming just from the central adjoint, from the category. How does this look in the case of... I call this quality. This is intensive quality. This feature having a central input, or criminally having left and right adjoints that are equal. It's true also for homotopy. Completely different. Extensively. You know homotopy? The homotopy category is over sets. There is an adjoint. You include the discrete spaces that have the same homotopy as themselves, and that's coming back, because both left and right adjoin. If you look at maps from one, rational points of the homotopy type, others are the same as components.

12:30 They are the same as components, but that's just the beginning. In fact, that carries on to every level. The front is represented by one, which secretly is really components. It's also morally the points, but not only morally, in the precise sense of which adjoint it is, the sort of points and discrete inclusion you see. So you take the projection from homotopic category to set, and the left and the right are adjoint points. This you have to, yeah, I mean, it's of course, but it's almost like a trivial statement, but then you realize, well... Other adjuvants are not like that at all. This is really a very... No, no, I understand geometrically why you do so much of... ...put it into a single adjuvant. Another example, by the way, is graded commutative code alphabets. Yeah, right. And then ungraded ones below. And the underlying is just to pick off the zero component. And so then, well, you can... You can include, you have a single thing, you just put zeros in there, so that's the inclusion, but then you can, given the general one, you can extract the zero part, and you can also mod out all the higher degree stuff, so this modding out gives the same result as plucking out, so again, left and right is your degree. Which is one variant of Anselm-Lemaire in the finite output. Anselm-Lemaire wrote that the main point is twilight.

15:00 According to my weird philosophical terminology, you see that homology is also equality. Homotopy is equality. Homology is a graded co-object. The left and right agilites agree. I mean that's sort of floating equality. Concretely, if you have some sort of interesting category, you may have a functor two quality, and that measures, you see, given the object, its value is the quality of that object, which can again be that kind of quality of that object, two levels, that can then be measured by any number of those. So there's both intensive and extensive quality, you see, this business about the hot and the cold, or the local rains and all that, that's one. There are seemingly different sort of homotopy and homology, but they both share this feature of being quality when you attach them. One's coming in on the left, one's coming in on the right. And could you repeat your objection? Well, there's no objection because it's true in the case of complex numbers. Okay. It's just misleading that people then think, oh, the way to define points is to take some kind of ideals. But it is immediately wrong as soon as you go away from complex numbers or from logic where you have two. But in any other case, you have really a whole category of things.

17:30 And since there is group action on the points, the maps will be determined by the ideals, only if you have some work in them. So by taking the ideals, you've formed that. No, so, you know, Galtham was perfectly, nothing wrong with what he said, it's just that... No, I don't know what... It's a special case, isn't it, I mean... Because, you know, analysts, you know, they go around saying maximal ideal, maximal ideal here, maximal ideal there, when in fact, that may be a bad... It's assumed, for example, that's right, people try to imagine, well, what could be non-community of algebraic geometry? Well, this is really just a kind of speculation, for the most part, and so they say, well, we should take some kind of ideals. You see, they've already been possibly misled because one should have taken home a work of it into, it's just kind of a ritual which may or may not be justified, I don't know. I was in school that called the substance still retaining the kind of stuff it is, and so therefore its behavior going from gaseous to supercooled and all of it, but that's just part of the, but then if you build a house out of it, it has a form, and even the plan for the house is still the same form, but neglecting the substance, but that's still a quality, namely the homotopy. The homotopy is just, you know, it's a new category.

20:00 The new hams are the components of the old hams. That's a definite construction also. That's the form neglecting the substance. So in a way, I think these things about derived categories are very, very interesting, but really it's just the form aspect that neglects the substance of the space.

22:30 So the notion of reconstructing an object, a given object, from its points. What do you know its form and its substance, i.e., say it's homotopy type and it's hot and cold manifestation, can we still reconstruct it? Well, I don't know, maybe in the case of reflexive graph, we're just thinking, maybe you can reconstruct in that case, I don't know, but it seems like it's... There'd have to be some condition. There's an additional feature which is actually telling you how to assemble the substance into the form that you're throwing away when you just extract the measurement of those two qualities. So the next thing we're going to do is try to calculate it in the case of reflexive graphs, at least, and see if there's a clear example where you... So, you can reconstruct essentially some variety of needs and you can reconstruct the space from its points. I mean, that's a very simple, you know, in that case you have a very... Well, only up to the amount of work it is. Yeah. In other words, that's a stronger emotional and such type of thing, but still a long way from reconstructing the actual object. Just having enough points doesn't even really allow you to construct a resolution of the object. I mean, having enough projectives... ...different situation, but it means you can cover an object by a projective and then you get an induced equivalence relation that you can again cover by a projective and the idea is that that resolution really does determine the object.

25:00 Points behave differently. They're partial information which might turn out to be, if you have enough of them, to be faithful, but sort of only in very, very extreme cases to you. It's only if the space consists of nothing but a sort of infinitesimal collision. Daniel is not always sober. Yeah, that's right. Scott is not always sober. Yeah, Scott is not always sober. That's right. You know what I mean. That's very much because, of course, the points have some kind of structure as points. They're not as bare. And so if you take that into account, then you can kind of recover the space. And you would think, right, that Scott's topology... There's a set, filterably complete set. Since it's really sort of defined in terms of that order structure, you really think, well, come on, if we take the points with their structure, we should be able to reconstruct. And again, there's an adenoid function in the comparison there, but as Johnstone says in that little, it's not always an isomorphism. I still find that mysterious. I still find that very mysterious. What about the case of synthetic differential geometry, where you have these grad joints, you know, from the, say, from smooth topos to, that's an interesting case to look at, to discuss. From topos to? Yeah, to the category of sets. I mean, you have, right, and you can get sets in two different ways, you know, by taking double negation sheaves or... Well, I don't know. I don't know. I was thinking about that last night, actually. I was always hung up on this idea that sometimes because there's construction of really petite toposes and it always turns out to be a subtopose and often turns out to be a quotient topose as well with the rejection.

27:30 On the other hand, I had a construction of a very simple case of something like points and it turned out to be a quotient and not a sub. So if you have a subcategory, a full subcategory of the topos, which is closed under finite limits even, so all the simple algebra looks the same, it could be that it's a subtopos, it could be that it's a quotient topos, it could be that it's even both, but it's sort of confusing from a naive point of view. You can say, well, these objects ought to be playing that role. No, and you suddenly find they're on the other side. So this idea of trying to extract, or general topos in an external way, some approximation to the street myths there too. I mean, there's taking the Bevel Vigation Sheet, which is part of the sub-topos. Then there's Johnstone's Netelgash Functor that gives you a quotient topo. The inverse image being the inclusion of the so-called direct image being the extraction of... The locally decidable portion of any object. And maybe it's like hot and cold, I was thinking. Maybe I have to accept both of these things. And, of course, there's an induced map between them, which goes through the original, more interesting topos. But what you have then is this Boolean one and this QD one. Boolean's a special case of QD. And a map between them. So maybe that is what, in a very general sense, could be playing the role. In the case of differential geometry, there's really just enough non-classicality, so to speak, to allow for the presence of these. You really haven't produced a kind of non-standard. It's not like non-standard. I mean, you could do non-standard analysis and such things, but that's not fundamentally what it is.

30:00 It's been standard for 300 years. Yes, that's exactly right. Recognizing it. No, sorry, I was just... No, no, I think that's true. So you're saying that the logic is... Well, it's pretty close. I mean, my point when I wrote that little book was... In fact, I didn't want to say anything about, well, I would have said something about the logic, but, you know, just parenthetically, that you don't really notice any of this. Of course, one of the reasons why smooth infinitesimal analysis, or analysis, right, an infinitesimal calculus, is it doesn't use non-constructive reason, I mean, even though it's necessary, you know, to introduce... Oh, well, maybe that's your confusion, because hiding logic is one thing. Constructive philosophy. Oh, sure, sure. But you do need to introduce, you know, if you, when you look at the, there were puzzles, of course, in the use of infinitesimals that were actually also logical puzzles. You can see the history of it that there were difficulties, but actually it only required a very minimal, you know, it's something, of course, it's difficult to construct the models and so on, but it only required in some way a minimal sort of recognition that, that... Well, actually in this case of the algebraic, you know, the fact you could use, that you used, you know, quantum quantities, that really doesn't change the practice. What I mean is it doesn't change the actual practice of these computations with, you know, derivatives and so forth. It just really simplifies them. And yet, on the other hand, the, you know, after hundreds of years of trying to get this, you know, get this straight, It's interesting that with full classical logic, you had to do quite drastic, you know, very drastic, perform really drastic operations, right? You had to throw out infinitesimals, you had to turn everything into discrete sets, eliminate all the, as you would call it, cohesion that was already there, and so on. And yet all the, it looked, and then later on, right, synthetic differential geometry smoothed infinitesimals out.

32:30 You realize that in some sense only rather minimal, so to speak, needed to be made at a level that's not even noticed, actually, at the level of practice. Whereas, of course, we know that the practice, people introduced it, it's true it was an effort of precision and so on. But, you know, there were lots of arguments that went on in the beginning of the 19th century that people didn't really want to use limits and all the rest of it. Lagrange, for example, didn't, although, well, this is in the 18th century, but a lot of people were quite reluctant, you know, to jump on that particular bandwagon, even though, of course, in the end, it's quite interesting that, you know, there were rather small changes in a way, at least for a certain purpose, you can't say. That it would have sufficed, smooth infinitesimal amounts, it would have sufficed for doing everything, but never less for the, you know, the purposes of infinitesimal calculus, for example. Including the infinite dimensional, calculus and variations. Exactly. They also use the same methods. Yes, and it only meant in a certain sense rather minimal, rather minimal adjustments, I mean, which doesn't actually affect the practice. I mean, you're still working with more or less the thing. There was some definite intuition, if you like, a feeling of cohesion or smoothness or continuity, which got thrown out with the discretization of it. If you have a better word, I'd be happy to entertain it. Cohesion? I think cohesion is a very good word. I couldn't think of any other. No, I think it's good. You don't want to say connected. It doesn't mean anything. Oh, no, no, it's not connected. It means hanging together. It's continuous, but it's... How? It's translated. How rather than how? I don't know. So how and how? No, cohesion means hanging together. Adhesion is putting two things together and sticking them. So cohesion is definitely the word. I've always thought so. And it hasn't been appropriated by the other branch of mathematics. Exactly. No, that's right. Because all the other terms have. I'm in good shape. Yeah. It's interesting. No, it's a good point. I mean, nobody... It's true, that word, that term hasn't been appropriated or co-opted by. I've never seen it used in mathematics. You know, there's connectedness and connectedness in the small. There's all these terms they try to use, but funny enough, cohesion doesn't seem to have been one of them.

35:00 These two qualities could also be called large and small. The substance is in the small, the form is in the large. So that distinction was, that dialectic was noticed long ago by our grandfathers. Yeah. Anyway, now I'm just trying to, can I make this more precise, your point about how this is Camilla Miles? A mild extension of classical logic. Well, it's just that you don't notice. The point is that computation, I mean, just looking at it from a level, just say a basic calculus. The principle that using infinitesimal, the infinitesimal never appears at the end of a calculation. No basic idea. I mean, it's a sort of intermediate step, which is discharge. And this idea, of course, of infinitesimal, the use of infinitesimal, is very old. Yeah. You never do the, of course, it was mysterious just how they entered, right? But the point is, and this was, of course, there the difficulty was precisely that, in fact, it violated the laws of long non-contradiction, and this had been pointed out long before, you know, that you assume at some point that it's zero, and not zero, and then you divide, right, and then you divide by, and so on. I always meant to follow up on that, because it was actually Nick Goodman. I think Aristotle first brought this out. I always meant to study that. Actually Aristotle considered hiding logic and co-hiding logic and said, well, let's start with Boolean as simple, you know, rather than the law of excluded middle was really a conscious oversimplification as a starting point in the whole enterprise of formalized logic. Rather than, you know, an internal law laid down by Plato. It's a different thing, right? Yeah. But I think in this case, it's in that the reason why... In fact, the fact that we have both hiding and co-hiding was sort of viewed in as the inner circle. So maybe we can take the push out. Yeah, perhaps, perhaps, perhaps, perhaps, eventually, right? But I got the idea from what Nick was saying that Aristotle already thought about that and then realized that for a practical program we have to study it.

37:30 Yeah, but he does talk about it with these future contingents, you know, the idea that you may not have the law of excluded middle. You know, he does recognize this for tense statements, Aristotle. For tense statements. For tense statements it's quite natural that you don't have the law of excluded middle. Yeah. Of course. Right. But I think one interesting fact is that I think it was felt that, well, you know, of course there was the analysis also of variation by Hegel and the idea that somehow you would, that this would, that you would need some of the logic of contradiction, a dialectical logic in the sense that you would have a law of contradiction, the law of non-contradiction would be violated or would be violated in some sense. The interesting thing was that the, it was argument about the law of non-contradiction rather than the law of excluded middle that dominated, you know, all of the discussion, I mean, certainly on the basis of dialectical philosophy, you know, the whole idea of the law of non-contradiction is a thing that's really questioned, you know, not the law of excluded middle. I was very, I looked into it. I was wanting to see where, where can we find examples? You find it actually curiously in Plato's Parmenides. Which is a very difficult dialogue, you know, it's very hard to understand, and where there does seem to be something, there's some kind of recognition that there's a connection between the law of excluded middle and the law of non-contradiction. To violate one is something like violating the other. And then you find it in Aristotle, the future contingents. And then, well, you hardly find it at all. The law of non-contradiction is the one that's... You know, Leibniz, of course, talks a lot about the law of non-contradiction. Not very much about the law, excuse me. And then, of course, later on, the basis of dialectical philosophy is somehow the unity of opposites and, you know, the idea in some way of violating the law of non-contradiction. Whereas in the case of smooth infinitesimal analysis, it's very interesting that if you actually look at some of the earlier arguments, well, you know, these quantities, these infinitesimals are really... Easy or more easily understood as being things which are neither zero nor not zero rather than being both zero and zero. Yes, hiding and not co-hiding. That's right. It's definitely hiding and not co-hiding, at least in many of the arguments you can see that if they recognize that in a way, you know, the arguments of Fermat, the objections that Barclay and others made, and Descartes too, I mean, sometimes. He was unhappy with it.

40:00 And so on and so forth. Well, of course, you know, he realizes that there's something somewhat fishy about this, but it's interesting that all that can be resolved simply by assuming this is a quantity that doesn't satisfy the law of excluded middle. And yet that point was not recognized until, well, as far as I know, who was the first to recognize it? Peirce sees it. And to continue, and before Brauer, he says, he's saying this at the end of the 19th century, but it's quite interesting that that, that it was the law of, I think partly because the dialectical philosophers hardly mentioned the law of exclusion. It just doesn't play, of course, a central role as far as I know. Certainly it doesn't in Hegel or in his successes or in, Marx doesn't mention, you know, in his work in the calculus. It's very interesting. I'm sure you've seen it. His notebook's on the counter. Well, I mean, I gave a specific model for it in my paper. Right, right. I mean, he doesn't also really consider this... I don't know. It's an interesting point. Identity and identity of opposites. Yeah. Oh, I see the way out. What you're saying is that it's a... Somehow the infinitesimal calculus and all that is a mild extension. Yes, yes, I think it is. I mean, they made this drastic, of course, for a little reason. I mean, one way in which it's mild is this, you know, you have this first-order infinitesimal object in the topos. Yeah. But the topos... The topology is the smallest topos containing them all. Yes, that's right. That's a very good point. Because if you take t to the power of t, that already contains the line as a retract. Because multiplication by scalars is faithfully represented just by multiplying.

42:30 I mean, I call this Euler reals. So just having exponentiation, which is any sheet subcategory is closed under exponentiation. Then products equalize you so you have all the algebraic varieties. And so on. And that's normally taken as a site for the gross topos. So there is no topos in between. There are lots of sub-topos, but none that contains this kind of object, infinitesimally generated. Yes, yes. Well, I make this point also, acknowledging in my book that's going to be published soon. But I don't think that really makes the logic. No, no, no, I don't think that has much to do with... No, no, no, this really has nothing much to do directly with the question of the... I mean, most of these extensions turn out to already have the maximum complication, like as Peter Fry points out, the sheaves on the unit interval, in the usual sense, the unit interval of ordinary sheaves, faithfully interprets the full intuitionistic proposition of high-order calculus. In other words, it seems like a small extension, but it's already maximally complicated from the logical point of view. Yes, but I think this is an important point for the methodology, well, philosophically for the methodology of science. Because I remember, Hermann Weill was a great advocate of the idea of infinitesimal physics. I mean, he says it in FaceTime Matter, he says it all over the place. You gain an understanding, right, of the physical world by analyzing its infinitesimal parts, and he, of course, he says this goes back to Riemann, and of course it may go back further than that. On the other hand, Weil does say, also, that you have, he admits, You have to have limits. He says this, and I can't remember the term because I've got it all in my chapter on why I'm doing it. You made it in space time, not in... It's actually in the Philosophy of Mathematics and Natural Science. He says that the purpose of limits... There's a bunch you do remember. Yeah, the Philosophy of Mathematics and Natural Science. He says really the purpose of limits is to link, you know, for him in physics, the purpose is to link this micro-world with the macro-world. Because in some way you don't have any, this very simple infinitesimal rule, the laws are essentially linear, you know, and you have great simplicity, just isn't linked, it's just not connected with the nonlinear rule, so to speak, the macro rule, except in limits.

45:00 Well, I think the idea that it's generated, that everything's actually generated by the infinitesimal object establishes a complete link, I mean a complete linkage, without the use of limits. Limits, exactly. I wish Weill, I often wish Weill had lived long enough to, you know, to see these developments, you know, he died in 1955, so. And he was very concerned, you know, that there should be some very natural way of doing, of, well, really of founding a continuum, of course, that was what he was looking at intuitively, and I think it's a pity, you see, that he didn't see this because, well, I mean, there's, this realizes, you know, the idea of the connection, that the macro world, you know, the world, the physical world of observation somehow emerges. All of this is generated by this simple infinitesimal world, and of course that's what was something that would have pleased him greatly, because he thought limits had to be the linkage. He couldn't see, of course, the way that you could get it by this means. I think it's a very interesting, a very important observation. This link, what I said, you know, is infinitesimally generated. This could not be... We assume this being is here, this category is here, and then we investigate it, but it gives its own notion of natural transformation, you know, the morphisms are natural transformations, and that's going to impact everything, right? But on the other hand, the first order of infinitism, but the whole hierarchy of infinitism is more complicated, but it is a second order. These are just exterior powers, those are symmetric powers. But it's very pretty because it really is generated, you know, generated essentially by the first order infinitesimals. It's the simplest case. And then the functions of infinitesimals.

47:30 I mean, like Euler is, Euler is 0 over 0, you see. The attempt to invert multiplication, yes, there's a whole story about that. Factorial, additional sections that were stuck in there before. That's a sort of relatively generalized morphism of the gene and I have to know H to isomorphism,

52:30 space over G times H, and the direction commutes. So it's really a space only, G times A to the eighth. But then the composition is just that you have G, H, and K, and then you have the inauguration of the tensor problem.

55:00 There is a system involving algebra of any kind of algebra. If you apply that to L1, it runs in the algebraic sense, in the algebraic sense, and covers most of it within its thesis. So you change the terminology on the space of Lewis. And now, of course...

1:00:00 We'll see the crucial point, I think, is the linearization, really.

1:02:30 Yeah, exactly. Because you're looking at these odd modules, not just... So in fact there is this kind of course equivalence. But you could say for topos, for example, to put in every Grotendieck topo complex number object, you could talk about range topos as in general, but let's say in each one there is the sort of the Dennecon version of the complex number, which is probably the appropriate one. Now, if you look at the category of all modules that is internal to the topos, it's of course the Grotendieck AB5 category. To have given two toposes, look at these internal complex vector spaces, as it were, and ask for equivalences between those categories, this is precisely, in other words, this coarse equivalence makes linearized coarse equivalence by a bimodule, because, of course, these functors are going to be implemented by bimodules. I mean, the point of view of Houghton, he taught you, with his, I think he's a great, mathematical, great son of Erasmus, he was a student of Paddington who was a son, a mathematical son of Erasmus, I mean, but he went into very complicated, I mean, fresh, and I'm not so happy about his solution. There is something there, but I'm not so happy because...

1:05:00 Again, again, I mean, we discussed it already, I mean, this point of view of infinite dimensional manifolds and using this approach via Banach algebra or generalization, fresh account, doesn't seem to be the right approach, and so we should have to take it, we should have to take it, let's say, maybe taking into account the idea about calculus of variation, calculus of variation, that means that you approximate and... An infinite dimensional space by finite order, they say. Instead of working with the space of all functions, you work with the space of the jet of finite order and the tower of jet spaces. So, instead of working with the functional space, you work with the tower of jet spaces. As I said, I mean, the work of Chen. KT chain. KT chain. KT chain. KT chain. KT chain provides a model of the loop space in the beginning until people like what linear theory will algebra of continuous factor, smooth factor, new work with represented or something like that, something like that. And most of the integration theory. But without knowing the real number.

1:07:30 It's slightly different. I observe that from a distance. I think it's more or less the same basic idea, but something like a transference result. Ambitious mathematics means that you are careful enough about deciding the computation rules and being as explicit as possible about what is permitted to do in the calculation. It's not a complete axiomatic. I would say it's

1:10:00 It's calculus of the 19th century. In the 18th century, people knew perfectly well how to do integration by part and so on, manipulate according to certain rules, but they did not draw the bound line, they did not draw the bound line, they did not define the extent, to me, means that, I mean, axiomatization, so you have a first step, which is so, you have an algebraic analysis, and which is a certain rule, and you manipulate. There is some infinity, I mean, when you say sum of one over n squared. That's infinite. Bakoshi and so on, which is to delineate. And at the moment, so I consider the first phase axiomatic.

1:12:30 And at the moment in physics, we are about the same situation. We are in the half axiomatic phase. We can't delineate exactly the computation rules and also explain the pitfalls and the dangerous curve. We cannot say. I mean, it's like, you know exactly what you are doing and if you have lost your key, you can find them easily. It's the essence of life. It's the essence of life. It's the essence of life. It's the essence of life. It's the essence of life. It's the essence of life. It's the essence of life. It's the essence of life. It's the essence of life. It's the essence of life. It's the essence of life. It's the essence of life.

1:15:00 It's the essence of life. It's the essence of life. It's the essence of life. It's the essence of life. It's the essence of life. It's the essence of life. It's the essence of life. It's the essence of life. It's the essence of life. She said she did not have to go to a fight and so on. She would say, oh Dominique, I know she visited you last week.