Was sind und was sollen die Kontinua? / discussion, Q&A

0:00 Everyone is invited tonight for the Peretrois restaurant or the Place Colonel and Fabien. And the second information, Professor Courtauld is missing his coat. Is there some exchange? Please tell the teacher who was on the head and co-track in the evening chair yesterday. So, thank you. Good morning. It says that the chair this morning is St. Fischer. The appearance is notwithstanding, I'm not St. Fischer, I'm Dr. Hildegard, and I'm chairing this morning's session. So, welcome. It's my first and beautiful interview, of course, over there. And usually one says that the speaker needs some introduction. I have proof that he doesn't need any introduction. Everybody here, of course, that's the first axiom, knows about category theory, right? And everybody who knows category theory knows about Prof. Solveig. So, conclusion, he doesn't need any introduction.

2:30 So, with great pleasure, I give the ball to Prof. Solveig. Thank you for managing here. You can see so much praise inside the non-introduction. I thank you very much for it. So, the theme, I think, could be described as, what is space? Under space, we will understand ordinary space, time, motion, etc., etc., etc. And more besides, the first answer to what is space becomes simply, it's an object in a space category, but somewhat dramatically analogous to the fact that a vector is an element of a vector space. There are many vector spaces connected by suitable functors that one is forced into that sort of locution, just as there are many vector spaces connected by many transformations and so one does not want to invent a different word for each vector space, one uses the word vector, but on the other hand one has to recognize that there are infinitely many different species of vectors all connected. It was the same, I claim, with spaces. So, to put it in terms of a personal story, when I first met my co-author, Steve Shenwell, we were discussing various things, and I said, let X be a space. And he said, what does that mean? Does that mean a topological space, a differential space?

5:00 I said, well, I don't want to tell you that, because what I'm going to say will apply to all those cases and many more. So, in other words, what I'm saying is, he's gotten used to this now. He does it himself even better than I do. But the point is, one cannot insist on a particular once and once and for all definition. So, space is an object in the space category. Which of course raises the question, what is a space capital? And this is the sort of question to which I don't yet have a complete answer, but for which a great deal, toward which a great deal is known. Now of course we know in principle the kind of actual work that a space is supposed to do for us. We need spaces as domains for variable quantities. Mathematics is full of variable quantities, including not only real and complex, but vector, tensor, but also truth values, and so on, including quantities. But the thing about these quantities is that they are almost always variable. And the question arises, how variable are they? And the first answer is, well, the domain space for this particular variable quantity is such-and-such. So the spaces serve as domains of variation for the fourth variable quantity to both the intensive and extensive sort. I guess I won't have time to analyze this question of intensive and extensive here, It's very, very interesting, as we were reminded yesterday by the talk of Dominique Tramont about Grassland. So beyond these, one use of space is as a domain of variation for variable quantities. Another use is as the arena in which motion takes place, or more generally, in which becoming takes place.

7:30 So, for both these purposes, both for accounting in a spatial arena and for variation of quantity, there immediately arises the kind of qualitative feature that these objects should have, which is loosely called continuity. Up to and including at least Einstein, physicists used the word continuous in a very good way, namely in a way which was changed according to the context. They didn't realize they were changing from one category to another, but we would probably analyze it that way today. Unfortunately, unfortunately the word continuity has become totally rigidified. Now every person who has studied mathematics reacts to say, oh, continuous means that's a morphism in the category of topological spaces, meaning topological space has again been rigidified into a particular determination, a very useful determination, naturally, but still an unnecessarily rigid one. It might mislead you in that way. So, again, as a crude approximation, we want to say that a space category is a category which contains some continuum. So, again, the question of explaining what is a space category will involve, among other things, I'm trying to explain how some of its objects, at least, or all of its objects, participate in this beneficial continuum. The notes are somewhat legible. James Clark Maxwell has an excellent expression. Unfortunately, I remember this, but I don't remember where the reference was.

10:00 In any case, he said we can screw up or down. That has nothing to do with screw up. You can screw down or up. The resolution needed to study a particular situation, a particular kind of motion. Of course, by resolution he doesn't just mean bits per inch or something like that. For example, phenomenological thermodynamics is fine for many purposes, such as heating rooms and so on, but sometimes we need the kinetic theory of gases to deal more accurately with that. And sometimes we have to introduce not just the kinetic theory of gases, but still a qualitatively deeper resolution of statistical mechanics against statistical mechanics of the classical or quantum sort. So, on the other hand, we wouldn't... We wouldn't want to commit ourselves to making every discussion in terms of the most refined resolution that we know because for practical purposes, a course of resolution may in fact even be the best one. So this idea of Maxwell's scoring up or down the qualitative resolution will be very much a feature of my analysis of the continuum. Oh yeah, so here are 1, 2, 3, 4, 5 levels, roughly speaking, combinatorial levels. Well, if we plan this room, an architectural plan or blueprint is essentially a finite instrument as any subjectively produced thing probably must be in any case.

12:30 This may be very complicated, but there are points, there are lines, and so forth, and basically a line means that the architect is simply saying, he's simply declaring, pure thought declares, so he's simply declaring, as you can get from here to there, and that you can't get from here to there, because there's a door, or it's not a door, and all these things. Supposedly the engineers and the construction workers can then develop the real world and actually construct a continuous building. This is what's known in mathematics as the process of geometric realization of a combinatoric scheme by some kind of continuous space. Again, for nearly every kind of space, nearly every space category. There is a much more geometric visualization starting from these finite descriptions. Now, the point is that these finite descriptions themselves do constitute a space category. We can prove that some of these objects are connected, for example. The number of connected components that a space has. These things are, in some sense, the first level of analysis of continuum. And already for the finite plans, those and many, many more supposedly continuous predicates, if you wish, make sense and are in constant use. So I claim that although these objects are finite, one should still consider that to be a very genuine base category because of its, if for no other reason, because of its connection with the other ones. But in fact, as I just said, there's even an internal reason for that. You just look at the structure of the category, you can detect homotopy groups and so on, as I'll explain more in a moment.

15:00 So, the next level, I think, is the idea of approximation. This is, again, pure thought, but pure thoughts that we can never actually do, idealized infinite thought, whereby we say that the continuum somehow consists of these points and that we can approximate them by understandable means, for example, by rational numbers. Whether by Convergence, Cauchy Convergence Sequences or Getty-Concussed, these are significantly different actually in many cases, but both contain this idea of some kind of idealized thoughts which are schemes of approximation whereby we imagine that through infinite iteration of finite schemes we can approximate. But the more typical aspect of space is cohesion. Cohesion of a space has many features, one of which is the passive possibility of motion. I can't get from France to England by walking. So at a certain level, one of Maxwell's below, at a certain level, England and France are disconnected from one another, and so motion between them, by walking for example, is not possible, whereas within France or within England it is possible. So there is that passive possibility of certain motion correspondingly to impossibility according to the level of analysis that has been made. Now, going beyond that, not only the passive possibilities of motion, but the parameterizations of actual motion themselves constitute a deeper kind of continuum.

17:30 The classical definition of motion, which is to be in one place and someplace else at the same time, is exactly captured, as you'll see, by the representor for the tangent model of the functor, which has been around for a long time. Someplace else, at the same time, is brought out by affine isomorphism with the space of pairs of collisions, depending on choice of affine connection. So without the choice, one has the direct identification or notion of this dialectical, this ancient dialectical definition, which turns out to be literally true. Well, beyond really parameterizing motion, there is the question of actually describing the cause of motion. And again, the continuum which contain within themselves the law of motion, description of the cause, will be seen to be a level or a depth of analysis of the continuum. I guess most people have thought about things, at least in a non-platonic way, and realized that there must be some kind of connection between continuum and motion. I just want to point out here that already dedicated construction, let me just say the approximation model, can be seen as a mere logical reflection of

20:00 The category of actual laws of motion, literally using, instead of 2, that's supposed to be a bold 2 there, classifying the truth value of space, instead of using that space, that construction uses it in a definite way, instead we could use the category of all sets, actually any base category, base value of the category of discrete sets, as the co-truth values unfold. Namely, We start with the additive monoid, non-negative rationals. Of course, in that monoid we can define the ordering by the possibility of adding one quantity to another to get another one. So, we consider the topos of state spaces acted on by the rational numbers. In other words, this is a huge category that consists of actual laws of motion, laws in the following simple sense. And given a state and a non-negative rational number, there is a specification of what state the first state would move to if you weighed that long, as the time measured by the rational number, the unit of time for this description. And of course the usual transitivity condition, that if you add together two rationals, Then the composite state change is equal to the resultant state change as such. Then between these, there are the equilibrium maps, that is, maps between the state spaces which are precisely compatible with the law of evolution, law of motion. Now, this being a topos, it has a truth value space. In particular, one of these things, it plays the role of... The role of truth in the sense that, actually the role of truth is largely overrated, as some of you may have already pointed out, but what it means here is that the truth-telling object plays the following role.

22:30 That's what, in its own category, given any category, given the object x in the category, and any subspace of it, problem spaces instead of objects, I hope, but given the subspace of the space, there is a unique characteristic function with all the values of the same truth value space. Phi applied to it gives the constantly true figure in omega. So omega is uniquely determined by the category it's in because of its universal relation to all possible inclusion factors. Now, Dedekind noted the following necessary refinement on the idea of cuts. It's part of his definition. Now, what do I mean by future? A future is an infinite family of states parameterized by the strictly positive rational, with the property that if you take any rational and add it to a strictly positive one, you get another strictly positive one for which the actual dynamics takes one state into the other, so it's a whole... It's a whole future except it hasn't specified anything at time zero. It's compatible with the dynamics and declines to a positive time. Well, a semi-continuous dynamical system of this sort is one such that every future of that sort indeed comes from a unique state at time zero. So if we restrict our terms, we call script R as the sub-category of these semi-continuous systems, it turns out that that's also a tokos, as its own truth value object, which is smaller, and that in fact the truth values are precisely the non-negative reals, so we started with rationals and produced the reals by a method of

25:00 Arbitrary actions, cuts are very special. And the meaning of the truth value is how long do we wait. So in other words, the truth value of being in Y, Y as a sub-object, has to be specified for every figure in X. So a figure in X belongs to Y to the extent that if we wait a certain amount of time, we'll be in it. This is the first type by which a given state winds up in this subsystem of states. That is the truth value of the statement that that state belongs to, that subsystem. And these three values, with the same continuous rational dynamical systems, are exactly the reals. More exactly, in the semi-continuous fields, which we did this in the general Tocos S, other than the Boolean one, it's the appropriate reals to serve as values of metrics. This is, if you take a slight step, this is exactly an extension or a... Dedekind's cuts are a very, very special case of these dynamical systems. At most one state possible at any given time. Now an important feature of space categories was in effect pointed out by Volterra 180 years, 140 years ago, namely that space categories need to have or at least be naturally embeddable in categories that have exponentiation.

27:30 Well, something like what logicians call a lambda converter. It's a right adjoint to Cartesian dot phi and Cartesian dot r, gives the precise view of inference for determining what the exponential of two spaces is, so that in particular the points of an exponential space correspond uniquely to the actual maps from one space to the other. There's more to it because basically the point is that the spaces of intensive and extensive variable boundaries and also the spaces of motions and so forth and so on have their own cohesion, they must have their own cohesion, that's the fundamental axiom or philosophical idea behind functional analysis. Which actually goes back well before Voltaire to Bernoulli, the calculus of variations is based on minimizing or maximizing functions whose domain was infinite dimension. Now, something which has been quite obscured really by the fixation on a particular determination of topological space The notion of cohesion which is appropriate to an exponential space is in fact completely determined. There is no real choice for it. The choices that you're taught in functional analysis are really about different spaces, if you want. Essentially Voltaire and his followers knew this, but their point of view was obscured by the mainstream. But now, what I want to really emphasize here is not functional analysis, but the fact that the exponential operation, which is applied to things like the line, or the point, or the circle, or ordinary finite dimensional spaces, tends to produce these infinite dimensional spaces. In fact, also produce one dimensional infinitesimal spaces.

30:00 If you apply them to infinitesimal spaces. So this is one of the specified ways of approaching the continuum, which I am advocating, actually, is that the continuum is or is closely related to an exponential space, but of some infinitesimal spaces, a little bit more. Because of this concept of motion, which exists in reality, we can concentrate that and postulate something that everyone's going to ask me if all these things really exist. Well, what really exists usually means is that we can construct models based on the category of discrete sets. And yes, we can always do that in all the examples that I mentioned, greater or lesser labor, so that, in fact, these categories do exist even from the point of view of the sector. That's not always the most efficient way to get at them, so I'm saying here I think that fundamentally we have to import from reality the postulation that there is Something called the continuum. That is to say, models that are based on infinite subjectivity may approximate any given finite number of properties that we might attribute to this continuous model. The physiological, I mean, to be, not to be facetious, Newton, Leibniz, Grassmann, and so forth, dealt with real numbers, but they never thought that they were coaching sequences, or that it would be best, I mean, to deal directly with those, you try to write down explicit properties.

32:30 So, a concept which in effect nobody widely has talked about is a space T which has got a specified point, zero, and has in effect no other points, which is spaced with only one point, and yet in itself is not equal to the bear point as a kind of... If you have a joke or something in Italian, I'm going to call it in French, you can do something similar because the word point has two genders. So if you think of the bear point as the masculine one and the naso bear point as the female one, then we have an inclusion here. Casual discussion might well attribute the word point to the latter as opposed to the former. Somehow I don't know the linguistic explanation of why these languages have maintained this beautiful distinction that's very convenient for me. Okay, so, okay, and as a blatant demonstration that this t is not a single, is not equal to one, even though it has only one point, And that's the space which, among other things, whose points are corresponding exactly to the maps from t to t, although it has more to it, but it has its own cohesion, et cetera. That's what it clarifies as the level of points. So you see, there are two definite points of t to t, in any case. There's the constant, the name of the constant map, zero, and on the other hand, like with any space, the identity map also has this. So the constant zero and the identical both points, pair points if you like, of the exponential space, well, you might say, well, they're equal aren't they? No, they're so different that they don't even have any intersection, nothing in common.

35:00 And more blatantly, all of the real numbers are included among the endomorphisms of this infinitesimal. Or, more exactly, with appropriate assumptions about T, it's reasonable to define the reals as this sub-object of T. And by the way, the sub-object, this is something that all those mathematics that preserve zero. Now, you might say, well, any actual... Has to be 0 because it's a little bit more than any other math, applied to 0 gives a point, but there is only one point, and so it must be the same one. That's all true, but simply because of the fact that T has its own cohesion, if you look at the higher types of figures in T, other than just points, you find that some of them... This is just a part of t to the t that satisfies the equation that says I'm a map that takes zero into zero. Why is that intuitive with real numbers? So the composition of the interlapse of T, that is the multiplication of real numbers. Multiplication comes about immediately just because of the fact that it is or is closely related to an exponential object. Now, this object T that I'm talking about has a very, very classical significance because in 300 years of multidimensional calculus, there constantly occurred constructions Radiance and divergences and blah, blah, blah, blah, blah, and one in this century, one has, we now recognize that the, in some sense, the key to all of that that makes that category different from others is the fact that we have a tangent bundle of functions, where every space, there is associated a space of tangents, of tangent vectors. Now, because of the Yoneda Lemma,

37:30 Any function which is representable is representable by only one space, otherwise the representing space is unique. So, it seems reasonable that this fundamental function, the tangent fundamental function, should be made representable. We find that the representing object is nothing like Non-standard analysis with its subjective goal of getting away with making as many statements as you can. So rather the objective content is simply that a tangent vector is an infinitesimal path. So the exponential space, x to the t, it has, among other things, its points for actual maths from t to x. Action maps from T to X. So those maps are at a certain point, because the point zero of T will be mapped to a certain point, but then there's the elsewhere aspect of T, because La Punta is bigger than La Punta, which also, which is precisely a tangent vector, equivalent to a tangent vector at the point where this infinite measurement map is. So this space, T, is actually, from the point of view of analytic algebraic geometry or differential geometry, it's a very well-known space and it's quite simple. It's not something complex. It's sort of like, as Leibniz said about it, it's rather like the complex numbers, i.e. it's a quadratic. It's just that, instead of having x squared equals minus one, we go x squared equal to zero, and it's still, the function space is two-dimensional. Or more generally, as I said, there are a great many basic spaces, E, which can be equipped with an affine connection, the tangent of the force is isomorphic to E cross E.

40:00 So even though these are the infinitesimal paths, with help of the affine connection, we set up this equivalence so that they do come out to be a question of here and somewhere else, a specification of here and somewhere else. Yeah, so the meaning, why we should think of the endomatch of T as deals is because they're acting as speedups, you see. Even a decimal path or a tangent vector to a space x at a point x sub zero, then one thing you can do is you can apply any random vector p could be composed with it, and you get x lambda. So what's x lambda? Well, it's the same path except doing lambda times the speed, which presides over all possible speedups. And, of course, if you compose two speedups, you get a composite speedup. That's the multiplication of the reals. So all the properties of the reals can be developed by combining suitable assumptions about T itself with the general machinery of exponentiation. As being here and elsewhere in a reasonable way, I want to give one of the most important historically interpretations of calls in motion, of spaces which incorporate within themselves actual laws in motion. The kind of things that I referred to earlier as dynamical systems, whether they're acted on by rationals or whatever, very much the sort of thing that our colleagues may study under the name of dynamical system, but actually they lack a crucial ingredient which already was in principle known to Galileo, namely...

42:30 The concept of state as such, of course, is due to Aristotle, but there's a key refinement that was made 400 years ago or so, I guess, maybe you can find it if you're still older, medieval history, I'm not sure, but it's the fact that a state is not a configuration of being, but a state of becoming, so that the forces which are the cause of motion Act on states of becoming, to give other states of becoming, not just on configurations. Now, this is a very general philosophical type of distinction. You can have greater or lesser, again, Maxwell's screwing up which we have, analyses of what you mean by becoming relative to some notion of configuration and so forth. In quantum mechanics, for example, the becoming is taken care of by the imaginary, what do you call it, the angular part of the wave function. The gradient of the angular part of the wave function carries this aspect of becoming together with the amplitude of the wave function, which, of course, describes configurations probabilistically, not in the book. But the traditional one, oh, and my friends in continuum mechanics, Walter Knoll and Cole and so forth, often talked about materials with memory because of the fact that certain plastics and other materials, turns out that their response to stimuli now depends on their whole history. Kind of like children or something, maybe. But even non-living matter often has this property. Well, you see there, you'd have to have a more sophisticated notion of state of becoming.

45:00 State of becoming might actually be the whole history, but then the forces that act on that would be more history. But the traditional model works for many purposes. It's been exactly the second order of differential equations. So that the states, if we consider x to be a configuration space, then states are defined to be x to the t, tangent, the tangent number. So that a state of becoming, motion, is identified just with being in a certain configuration but with a certain tangent vector at that configuration. So, now, here I'm giving a specific definition in terms of this T, postulated infinitesimal space with one point and one point only. I'm giving a specific definition of the second order, but I interpret the first one as first-order infinitesimal, and then second-order infinitesimal. Quite a general thing, but the specific one that works in most cases corresponds to the classical choice. It looks sort of like Taylor's theorem, except this is something you do to the space itself. T squared is the space of pairs. Two factorial is the group that permutes the two factors in any product, and you take the quotient space of what's known as the symmetric power of T. The power, key statement, made symmetric by dividing out by influence the relation. So it turns out that that's exactly the second order of infinitesimals. Somehow the idea is this, that in all of these classical theories, the order doesn't matter. If you have a little infinitesimal show and another infinitesimal show, you could have done it in a diverse order and you'd get the same result. It's only in the infinitesimal level. As soon as you go beyond that, it's assumed that something, this is what makes probably calculus possible, that you have something relatively simple going on at the infinitesimal level in this community.

47:30 But again, as I say, that wouldn't be necessary for anything that I'm saying. The map from T into that has gotten the following way. Remember, T has a point. And so any space with a point, if you look at the square, you have a coordinate axis. In other words, you can map T in to the space of pairs by letting always the second coordinate be zero, or by letting always the first coordinate be zero. So you have these two coordinate axes, let's say, mapping T in to T squared, which are certainly different. Identification, log 2 factorial, we, among other things, have become equal, this becomes equal, when there's really only one map from G into T sub 2. So that's a concrete example of T sub 2. Now what I'm going to say would actually apply to any map from G into some other fixed object. Namely, a law of motion, but a law of motion in the classical sense. They can be analyzed as follows. It's a law of prolongation of these states of becoming. Now, what we mean by prolongation is pretty clear. If we're given a path, an intangible path, which is only first order, the law of motion will extend it in a definite way to Now, actually, it's a neat coincidence that this Brook-Taylor-like formula for the space also applies to the elements. The elements of t sub t can typically be denoted also by t squared mod two factorial, where not t is a temporal quantity. But anyway, you see, so the point is, prolongation, you just diagram this community. So, this next bar restricts back to the original realm, but now it's going a little bit further.

50:00 Under the, under the, what I was just saying about how T sub 2 is essentially, beyond T, it's essentially just quadratic terms in the Taylor series, this is exactly a secondary differential equation. The second derivative given the zero of the first derivative. Now, of course, we want an actual law, not just a point-wise saying, so there's a single map on these spaces, from x to the t to x to the n squared, with the property that if you follow it by the map that's trivially induced by the inclusion, so-called restriction map, In this case, you will get the identity, so the composite here should be the identity on x. You can say, well, of course, on the other end you're going to get an inner potent, so the points of that inner potent are all the slightly longer infinitesimal paths, which in fact follow the law, the specific law of mu and everything else. So it's specified such as a second derivative. Now we get a category, of course, in the usual sort of way. We're given another space U equipped with such a prolongation operator that given a map from U to X will induce something on U to T by raising the power T and induce something else by raising the power T squared. So we can demand that this diagram be commuted. The two diagrams have been induced by a map from U to X. But we demand that it's communicated with the time-bearers as well. In other words, that it precisely preserves the law of motion. Now, so we get a category I call X to the T. So this again is a space category. That is to say, it contains objects which answer to more or less properties that we would demand of the tenure.

52:30 More or less, depending on what x itself was to start with, so in a sense it answers the last thing that I said. A motion itself is actually a morphism of laws where the domain is time. Now, the last one is Scientific American. I like to call it the anti-scientific, anti-nerd, but the whole issue is about time, and they're constantly making this big mystery, do we really know if time flows or not, blah, blah, blah. Well, if we take any reasonable mathematical definition of flows, well, yes, there is a definite flow law. In the second order case, it's surprisingly more difficult than you might think. Because it turns out that in this kind of category, second-order laws, laws that depend on states of becoming, time is not one-dimensional, but it is generated in a certain sense by one-dimensional space in the lower category. Now lower category is when you look at first-order laws on bigger spaces. And the first order of law, of course, on the open interval of the ordinary real line is simply to add an infinitesimal to it. So the addition of infinitesimals to arbitrary times. So I take it that I should stop soon. Basically, there is a kind of category into which all these that I've mentioned fit. And that's the so-called extensive categories. A long echo of Grassland, I suppose, that we chose the name extensive. It's a definite property which, in effect, the general definition is simply to provide within your category a reasonable notion of finite discrete space. Now, again, remember the world was not electrical, so the point is that having that notion of finite discrete space precisely gives you the room or the possibility to analyze the connectivity of the other spaces.

55:00 We actually get the whole Umatobi analysis, the so-called higher connectivities. Again, that's an old word. It probably should be revived. Higher connectivities are connectivities of higher exponential spaces. Research is continuing on making that approximation more precise. Now that we know what time is and motion is, we have some time today to put the discussion into motion. I think most of all I can respond to questions and comments. What's the definition of extensive category? This is not a plan. It's simply a category with coproducts. You say, what's a coproduct? Well, it's the left adjoint to the obvious diagonal one here. So a category with coproducts which are disjoint and exhausted in a definite sense. There are a lot of categories that have coproducts like groups and so on. Which are not at all disjoint and exhaustive, but here they should be, and that could be expressed this way, that if we look at the category of objects over an arbitrary sum,

57:30 and the category of objects over each one separately, then the summing function gives rise to the function there, and that should be the equivalence of categories. That's the shortest way to state it. In other words, the idea is that if we consider things, if we have a space which is broken into parts, non-interacting parts, the idea is that the sum should be sort of a space with some non-interacting parts, then if we have any sort of space that maps into that, it's an arbitrary object here, it should necessarily be neatly split. These are two parts which are again unrelated. So all this part goes here, all this part goes there. What the splitting is, it comes with the math you started with. But this one I've just described should be an inverse to this sort of topological. We've given the addition that this one is topological. So anything over the sum comes from a unique pair, something over each summand. So, in other words, these total spaces, if you wanted to quantify the volume and the mass or whatever that these total spaces had, you would be able to measure the sum of two things in it to get the sum of the measures. That's why the name makes sense. Thank you. Other questions? Comments? Yes. This is probably a measure of my ignorance, but you are looking very hard to real spaces and I wondered through your construction whether you really use real numbers. I mean I wondered for instance what about metra-metric spaces? What about what? Neutrometric spaces, spaces on the other hand, I mean, not real, I mean... No, I was almost always using the word real in the sense of reality, not... It was not, I mean... Neutrometry is also a construction category of...

1:00:00 One example would be real algebraic geometry. Real analytic geometry, real smooth geometry. But there's no reason why nothing that I've said really restricts me to using real numbers as coefficients. But when I said that the endomorphisms of T should be thought of as real numbers, it really means it's the appropriate analog for real numbers in a certain category, 20th century mathematics viewed as a C-infinity manifold or whatever, in the appropriate kinds of categories, which can be characterized by internal properties. Thank you. Other questions? If not, are they? Oh, yes. Bill, you objected in the beginning to the rigidification of the notion of continuity. As given by his ontological space as regarded as sets equipped with a distinguished collection of subsets, maybe one reason, correct me if I'm wrong, to object to that is if that's the only notion of topology and continuity you have at your disposal, then you can't find a space T that will represent a tangent bundle in this way. Right. Right. You need to have more. So we need to have a more general... In order to have the space T, which has this beautiful property of being a way of building the tangent problem, it's simply by explanation. I should say it was implicit in my research, probably, but it was explicitly introduced in the last century by Keeler, I believe. There were some Italians in Sicily who tried to do mechanics a hundred years ago. I think none of these attempts really caught on until Kahler's idea was taken up by Grotendieck, and then of course everyone working in algebraic geometry knows about it.

1:02:30 Not even category theory generally, but just the very idea that domains and pilgrimages are explicitly given. That enables the calculations to be kept track of in a way that, in other words, if you start describing motion and so on without specific domains and pilgrimages, you quickly get quite a mess of things that are infinitesimal and things that aren't insensible. are taken care of by a sort of pre-categorical notion of a graph that is given by a source and target. Thank you. Yes, you refer to physics of the past a number of times. I wonder if you envision this analysis as shedding any light on new possibilities or hitherto unknown. Unthought of possibilities for physics of the future. Well, I hate to encourage this sort of thing, because people then go off and speculate about that instead of learning the stuff that you've learned first. But, nonetheless, as Curtis would go ahead and say, first of all, there is no viable theory of quantum mechanics, relativity, or anything else that doesn't base itself On the continuum of one form or the other. Of course, as I already point out, we discover discrete aspects popping out of the continuum, but to say that it itself, that the space itself is actually discrete has never been seriously worked out. It's speculated about, but it isn't very... On the other hand, you see, something I mentioned earlier should have... Clocked the eye of the people who want to speculate, namely that all of these classical theories, including the underlying differential calculus that quantum mechanics uses, have this feature that the second-order infinitesimals are the symmetric power of the third one. Now, you could certainly think, well, okay, everything should be non-commutative. It's a kind of mania that's going around these days. So that's a possible possible extension well within the general framework for that outline, but at the moment of no value.

1:05:00 Thank you.