Sheaves and Topoi Workshop — SDG (contd.)

0:00 around it. They say the following. Another danger in curved spacetime is the temptation to regard the tangent space as lying in spacetime itself. This practice can be useful for heuristic purposes, but it's incompatible with complete mathematical precision, they say. While the consistency of synthetic differential geometry shows that, on the contrary, yielding to this temptation is compatible with complete mathematical precision. Their tangent spaces may indeed be regarded as lying in space-time itself. If, as Hilbert said, set theory is Cantor's paradise, then I would submit that SDG is nothing less than Riemann's paradise. Thank you very much. a question to you. First of all, please review your bold speculation, Planck leg, but I would think there's actually bolder speculation, is that in fact, there's not a question that you can't order these events in Planck regimes, there aren't any events at all there to order, so that there's nothing to quantify over. That was my own actual guess, but I was suddenly ah you mean in the you mean in the in the in the in the physical yeah yeah yeah yeah because you use the phrase very carefully uh epsilon e are variables ranging over delta yes yes that's exactly no no i think they i only use that as a set theoretical no it's it's it's merely that they have the property you simply use the set theoretical language that if you have a property X is a P. That's why the term epsilon was introduced by piano in 1894 or whatever. We're only using it in this sense. In other words, when I write that, it's just an abbreviation for that. I agree, but you're right, of course. One has to be a little careful about what you know of variables because it's set three to be is to be the value of a variable you see but

2:30 it's not true here variables are different from objects they're true variables you're right but that doesn't mean you can't still use it's still convenient i agree misleading until you you know sort of get used to it but we still use the light you know the light as if in other words you sort of reify the object here might not work no maybe not but nevertheless i what about the principle of comprehension I mean what is the is it going to I mean you you actually work in pre well we're working I'm not sure what you'll explain what you're doing now but pre-sheave tapasas or sheave tapasas do you have the principle of comprehension there well we often work on things that are wrong I mean that's what it is to be a physicist absolutely can I ask you a serious question What I first read about this, DG, one of the things that intrigued me was the point that Leveo often makes about experientiality. On the face of it, it is a very attractive idea that if you've got two manifolds, then the set of maps between them is also a manifold. Now, in many cases in theoretical physics actually these days, where you inordinate language, you do have to think about maps between manifolds. The best example is dipheomorphism group of the manifold. Now, it's very desirable indeed, if one can say, that different and inter-dimensionally grew. But in fact, that involved really quite hairy mathematics. As you know, all inter-dimensional geometry does, because you have different models for a tangent space, that you have other places, or threshold spaces, and so on. And that really is a very important complexity, which has physical means, not just mathematics. Now, I never quite see, well, I don't know actually, where this has got lost in this. Because on the face of it, there's nothing left to ask anymore. I don't know that example but you don't see what I'm talking about yes I do, because the fact is in order to prove properties this opens up no more general question the object you describe which is provably a manifold in the usual class, difficult by some difficult yes it's going to appear there it will show up because it embeds the whole of the category of manifolds now would it be and it will be the presumably the exponential but proving showing that the that those particular that actually that the object you'd have

5:00 to prove that the that the that the could sort of the exponential the classical thing corresponds to these you know to the construction of the exponential within the category of smooth but it is unique in the category of manifolds up to oh yes because the exponential is defined i mean you know if you've identified what the maps are maps, the exponential is to take a space of functions, right? What more than one, say, topological vector structure on it? Yes, but if it's a manifold, right, if the thing is a manifold, then... Well, but if it's a baron manifold, it's one thing, it's the fresh air manifold. Yeah, okay, but I mean... No, but that's important in physics. Where does that get coded into this? That's really what I'm asking, I think. Is that...? I mean, I would presume that it would be the problem of that if the... That if you take two spaces and you have a definition of what it means to be a smooth path between them, then the exponential, if it is there, is uniquely determined. Somehow the topological problem seems to disappear. Yeah, it does remove it. Except in order to verify, no, in order to verify the, I guess in the case of, in the case of how would it work you have to show in those cases you'd have to show that the embedding still you know still acts properly right on the construction that you've made in the classical case I don't know how that would work out in detail right I mean in other words you make you know you identify that and then you have to show that the that the well if you can verify that it really is an exponential right classic then it will automatically be the exponential in the category of S. Of course, but you would have already had to have done the work in the classical case in order to verify the thing that there is an exponential thing. I suppose the problem is that in practice we do this with spaces of functions. Very often you're not taking all functions of the two-round-fold. All those are satisfied, say, certain long-distance behavior in certain ways. And then that's physically relevant, and that affects the differential structure that you use. And that's a difficult mathematics problem.

7:30 I mean, more generally, there's the question of to what extent the properties of this, you know, of S really depend on, essentially, on man. I mean, on the category, on things that are classically established in the category of manifolds. Many classical theorems are actually used and transferred or translated, right, into man in order to show that it has these properties. For example, I mean, even in the case of the micro-refiners or the coclavir axiom, one has to use various features of classical analysis in order to show that the axiom is actually satisfied in the final model. Now, one question, I mean, it's related to, is really, to what extent is this autonomous? I mean, for example, one would like to have, that's not known. like to have some way to say, well, we only need very minimal constructive type of mathematics in order to be able to build models of smoothness in this way. But in fact, you really do need to lean on a lot of stuff from classical analysis, as far as I know. And it's still a sort of open question as to whether these sort of axioms somehow stand by themselves. This is the difference between, I think, between synthetic differential geometry and non-standard analysis, which, of course, non-standard analysis is just equivalent to, you know, it's elementary equivalent to standard analysis. So it's natural that everything, you know, you have this automatic transfer, whereas you don't have it here. It's much more difficult. And it's the same, this strikes me as a similar kind of problem to the one you mentioned, although I don't know the answer. Yes, going back to that distinction, can you envisage the theorem? of in s right a theorem in man which is not provable within man but by embedding it in s it's provable within s and then well apart from this sort of thing i i i that's that that is an open question i mean i mean of course maybe there are as i said the only example i know of is that you'll call it a is this this fact that that Delta that the that is that well that that well I wrote it I can find it now that that that differential forms

10:00 are representable in this way that's something it just isn't true in classical analysis at all it's just not the inverse of Delta simply isn't there yes yes but that's not provable come back to a statement in man that you've obtained that is what they can do you know who wouldn't have that i suppose that's possible i i think that to work but really wanted a better way of proving classical so without that's why i don't want to think that that's just a little bit i want to know is it more than just a language it actually something which gets you to prove that you couldn't reach working what was the formulation of the theorems is different There are things, there are things, there are, there are, I mean, if there's any name system, there will be statements you can make, which you can't necessarily, which you can't do within that system. The, the, I, I, there are formulations of the theorems, of course, you know, that can't be given in classical case, naturally. For example, all this stuff using the direct delta function and distributions and so on. I mean, this is similar to the situation in non-standard analysis. but i mean people don't think that non-standard analysis is valuable because you can every principle every theorem that's expressible the classical language provable in non-standard analysis is provable in the classical framework that's the thing i don't i don't really know any what do you mean by not provable in man well can you make a point you about in algebraic topology, so if you can prove some homotopic classifications by actually going through some of these function spaces, that is assuming you're working in a convenient category of spaces, whereas the final statement actually does not, it is completely in the ordinary so-called authoritatively. So you're actually, I mean, maybe I can't quite pin it down here, but but certainly you use the exponential law. In a sense, that was why I introduced the term convenient category, that you can actually work in there, and when you come back to your ordinary category, you've got something which is just a statement in the ordinary category. I think there are a lot of problems. But to the extent that you construct the convenient category out of the inconvenient one, you are really still proving it in the original category.

12:30 Yeah, I mean, I don't understand how all this is related to this big book by Kriegel and Mikor on a convenience setting for analysis. I don't know the relation between what you said and that book. I don't know, well, I've heard it, but I guess there is some connection, but I don't know what it is. Because they give a very concrete description of this extended category. Well, the thing is that the construction of the category models here is quite difficult. If you actually want, if it uses a lot of fine-tuning and chiefification over several broken deep topologies, it's really quite an effort. Of course, you can then restate the, you know, the fundamental reason why it works in some fairly simple terms. Because you use, you know, you use sort of some intuitive idea of the coverings and so on in order to force the properties that you want. but I don't think the view of the point the topless view point really isn't that we should go on back thinking in the classical case all the time anyway there's a I mean the classical case is already an idealization that is somewhat questionable perhaps I mean one might question but it's true it's been the most convenient place to work in for a long time but I agree there is this question about to what extent If we take, for example, there is an underlying, they call it constructive, you know, the constructive, the sort of constructive meta-theory behind all this. And there is this question about what is really needed in order to be able to, you know, to be able to carry out these constructions. And if it's really dependent finally on the classical idealization, then of course it opens up the question as to what extent this is really in some way autonomous. It's still sort of parasitic on the classical case. But I don't think anybody really knows the answer to it. This is a question that concerns logicians, admittedly, more than, let's say, people actually working in differential geometry. But it's quite the important foundational question, I think. Is there a minimal realization of ethics, just somehow generated by man and one object, Delta?

15:00 That's a good question. I'm not sure. You'd have to look at the construction. It might be the case that if you went through it, you'd find that it was a classifying tapas of some kind. I mean, that there would be some invariant way of describing it. As far as I know, there are in many cases, right, for tapas that are familiar. They're called classifying tapas, and they're the kind of minimal, they have the minimal model, if you like, of the axiom embedded in it, the principle of question embedded in it. I don't know, it's a very good question I don't know whether it's true in this case I don't know whether that's really been investigated you might the construction of these models is really quite complicated and I don't think that question has really been sorted out but it's possible at least in this sense I described of a classifying topis, yes I mean it's a natural question to ask because you know, I mean set theory, all that work of Cohen, which was later understood much better for the work of Mulvere and work in tapas theory, does have that sort of property. These models are, I mean, Cohen already thought in some sort of intuitive sense that he was trying to add, if you like, a generic set to a model of set theory and then, you know, generate the minimal model that had this object in it over the original model. And so it's very, you're right, it's very natural to try to think of that in this way, of this in that way. But I don't, I don't, I don't see myself I don't know whether that certainly hasn't been carried out as far as I know. Can I just quickly ask about the plank, then? Because you said it was 10 to the minus 33. Oh, my speculation, I... Well, okay, you said it was 10 to the minus 33. Well, that's what Reece says. 10 to the minus 33. I don't know what this is. 42 seconds. But, you know, you change your units, right, and it's going to be one. So if the reals are smooth in the way we used to think of the reals being smooth, then that, you know, to think of one to the infinite decimal is... Well, that's where the plot goes. I agree. I wouldn't want to push the analogy to it. You can't, of course, make delta. You know, delta remains infinite decimal no matter what you add. No matter what you multiply by, because, of course, everything is no problem. What I'm saying is you can't just make this work by going through some differential geometry. You're actually going to be working with a real line.

17:30 And I assume it's a big difference that we already have. Maybe. I did wonder whether, is it true that when you take a Planck length and you, is there a sense in which one does have infinitesimal, you know, micro-refineness for that? and what I mean is that if you if you take things that apply and then you apply operations maps to it does the thing remain I mean yes you could change the units I mean totally one but is there a sense in which in which in which the the plank length is preserved well for study does have units so in the ordinary context we go to a moving frame of reference I mean it's it's different I mean that's another thing you have to be careful I mean that mean actually but recently people trying to suggest the plane that it's not like Planck length really is the, to be honest, it's simply the edge of the world. It's just that's the signal of danger as far as problem grammage is concerned. Always something interesting happens. You can't make definitive statements about it. If you could, it would have a written context of a particular approach to problem grammage. So you could ask a string theorist, if actually he can't give you one answer, or you could ask someone working in loop on the graph to give you a different answer. You can't give a universal answer, because it's not very like that. But it seems, John, it seems to me that your your way of thinking of plank length say identifying plank length with Delta it you and you say you treat it as being something fuzzy okay although still yes events are behind it they we do not know how to order them I mean we have lost information about ordering yes yes it is perhaps this building on the market first remark by Chris it is there are no event there at all in the first place. Well, there's only one... But there is a Planck length around a, so to speak, if you have an event. Presumably, there's a fuzzy, you know, thing around it. I mean, look, the only point in delta is zero. It's just the place that you attach the arrow. I mean, it's the base point. Then there's a fuzziness around it. Well, presumably, there's a Planck length around even any... I mean, it's unsense. Is that right? I mean, if you take a definite event, I don't know, and I try to, the analogy, I... It's the word definite event, which I think is very... Well, it's true that one does have the already, in SDG, it's true that there is a sense, you

20:00 have a thing, well, you have a notion of a point. There are many ways of defining points, right? But it's true that this is a notion that has a particular definite, it's an arrow from one, you know, to the object, whatever. And, yes, I guess, yes, and I suppose to assimilate that idea to a definite event, right, in physics, whatever that is, if you don't have to really know, if one doesn't know exactly what a definite event is, then it would seem, I agree, rather stretching the point to assimilate that to the idea of a point, let us say, a global point or whatever, in a category. But I'm flattered that you even, you know, take the, initially, well you don't take it seriously, but that it's even, even considerable. The analogy that you drew with ordering, that you lose information about ordering events, it is something like a black box. Although, the ontology, I mean, you seem to think that it is a, there are points behind that black box, although we cannot, we cannot order. I've gotten over that. I'm great. But this is... Sorry, I don't... A discrete world, I don't really... I think about it, I don't really understand it. It's an idealization. If the world really liked that, I don't see how we'd understand anything. But anyway, that's my own prejudice. You will get problems of this kind coming up when you examine a world of that kind that actually is atomically based. Oh, yes, indeed. It's based on elements. Oh, yes. We're starting from the wrong thing mathematically, philosophically speaking. Yeah. So we're going to get these problems turned on. Yeah, that's quite true. Perhaps we should think again, John. Yeah. I would suggest, okay, we'll run late in the schedule. So we can break for lunch till half past two, please. An hour, and more an hour, and I would suggest that we go for lunch to the senior...