Conversations on Extensionality Principles & Topological Spaces

0:00 By the fathers of the honeyed philosophy of physics, it is the self-analysis of the world. Which sits the kind of entities I refer to are totally manifested, that is, activities of the body as moving, combining, and all of these are not definitely introspective. Okay, then could we propose, okay, well, to deal with that point, could we say notice in passing how this viewpoint differs radically? Well, this is not because it would be a very weak argument, because the answer could be, well, but cognitive psychology is no longer as that kind of psychology. I agree it's a very weak argument, because at the moment there's no argument at all, so this is sort of a very weak argument. The point is that if you look at the following kind of thinking in psychology, as for instance both in, well, the main trans first through behaviourism and then through cognitivism, External analysis of perceived processes of thought through behavior. According to behaviorism, that's all, I mean, the issue about stimuli and answers to stimuli, and so, given this way, we're okay, but psychology is no longer of these gears.

2:30 Recursive ingredients because of course the patterns compose and compose in a way that can be the way that an exact extensional analysis of functions of involving in description, mathematical description of moving, combining, transforming, the last part of it is the main So the strength of the argument, which I argue elsewhere, about the fact that this is not psychological, actually I think the argument goes back to from country and financialism, is that the idea is that what traditionally were called mental answers cannot be defined through... Processes as computational procedures, as in cognitivism, because it provides you with only a formal sense of the mind. Previous arguments I gave or later, I don't know exactly which point of the paper. This would be, say, the libertian approach to the mind. And one which naturally feeds into the kind of functionalism. These aspects are external content related to activities and the activities of patterns and these patterns are intrinsically connected to the macro world so in this sense here enters the idea of naturalism and these but also for another reason the reason is that psychologist

5:00 It tends to ground the nature of mathematical entities and concepts in aspects of the human mind, which is not my position at all. I think that these patterns of schemes, of combining, of moving, can apply to whatever... There is a complex system of internal representation. There's no particular relation with mind in this sense. So in this sense, this is also the reason why... So there's no reason why you shouldn't have a dynamical systems, theoretic view of the operation of... Right. In this sense, my view is not at all an epistemological foundation of mathematics. Because I ground the nature of epistemology in the structure of, in the macro structure of the world. Yes, that's a very important point, but you've brought it out more clearly in what you've said now than in any place that I can speak for myself, that I have actually found it in your writings. It has always been a worry on my part that your view does, as it were, let in a variant of Psychologism by the back door with all of them. The problems of, you know, the logical problems of the circularity of definition that that entails, that I think, you know, entwined naturalism certainly skirts that danger zone. I agree with you. I think it avoids it. But now, okay, that was an important, very, very helpful and clarifying...

7:30 Now, on the practical side, do we simply leave this sentence as it stands, which is just a bare claim, you're not going to put in another page and a half to encapsulate what you've just said to me because you don't have room, or are we just going to amend it so that we have at least an allusion to the At least to the two or three weakened versions of the argument, rather than no argument at all, so is it worth putting in, say, notice in passing how this viewpoint differs radically from psychologism in either its introspectionist, behaviorist, or cognitivist. I claim, yes, I think it is worth putting a reference here, because at the moment it just simply stands as a bare claim, which sounds a little bit theological in the sense that it seems designed to turn aside an obvious objection without actually answering that objection. You haven't got to answer it, but it's not going to, unless somebody's read all your writings, the deeply unconnected, the, okay, I just thought in passing to mention that. But thank you for telling me a little bit more about your viewpoint because I had always, I had always had that, and I do still have the sense that there is a danger there. I think it comes back to my, the difficulty I have with the epistemological dimension of your, with the view of mathematics as ultimately kind of grounded in a form of phenomenology and given that, given that I, For the foreseeable future, the only available understanding of fundamental physical theory is a structuralist understanding. I don't see how we, as it were, can think the relationship of those theories to the theories that they other than in a purely structuralist term.

10:00 We certainly don't have anything remotely like a positive characterisation of the ontology of those theories. It seems kind of hopelessly premature, and at the least, if we can seek in principle to even ask one, it will clearly involve a recasting in ways which we can only apprehend perhaps in principle, a purely mathematical, a purely structural understanding of the structure of one theory and that of another. So how can we rely on more affordances and the content of one theory? And so on and so forth. And says, yes, I mean, the structuralists were actually right about category theory. It is precisely because it is the most flexible and advanced framework that we have for the... But all of Alberto's... This is just one wing, as it were, of me speaking, not all of me. But Alberto's stress on the reconciliation of content and form, that there is no global separation of form and content. Very promissory dotage and the constructions. I'd love to believe that an account of abstraction would ground the role of universal constructions across the areas of mathematics and the functions and bring it back to the understanding of the relationship between the epistemological and the ontological dimension of the foundation.

12:30 But at the moment I'm not certain, I'm convinced. My doubts, I think, as a program of using category theory as a framework for rigorizing a kind of account of mathematics as formal phenomenology, but it is, but I have this physicalist, well, philosophy of physics The content of our intuitive notions, given the discontinuity, the fundamental paramotor of the discontinuity, the pessimistic meta-induction of physics. The main part that I did not include in this paper consisted in a criticism of structuralism. First, second, the kind of formal phenomenology, you are right, it seems I've closed, it's a formal phenomenology of a very poor kind, nature sort of, because in light of the principle I stated in From Country and Plant to Nature, it is the principle of... Yes, but I have never understood that clearly, the local suspension, coupled also with this other principle of invariance of the referential potential.

15:00 Yes, this is another, but in relation to what you said, the relevant principle to remind is the principle of local suspension. The idea is this, that there is no global cut between form and content, although we always can divide how is it possible. It is possible in the sense that in dealing with any domain or class of means, we can rely on content. Ingredients of contents which are rooted in other domains. So we can suspend any semantic relationships between language and world or any domain of abstraction. Only because what we suspect has a sort of support in meaning that is our talking about forms. In other words, there is no separating the epistemological from the methodological and ontological dimensions of ISA Foundation 4 with respect to mathematical concepts. Everything at the same time. This is the idea. And so this is the main, with respect to Husserl, whose notion of the bouquet was presumed to be global in order to achieve the ontology of pure meanings.

17:30 In different occasions, in different situations, in different theories, you can suspend the relationships you are interested in treating as formal, other sources of schematic nature and nature as contemplative, which are suitable for the description of that domain. And you want to say, I'm sorry to interrupt, but I'd just like a clarification, and you want to say particularly in relation to the interior architecture of mathematics itself, the way that, you know, it's these concepts put together in a manner of kind of systemic easiness rather than in terms of some hierarchy of once and for all directions, that this explains the absence of central precision that you assign to geometrical. And the geometrical intuition in understanding the kind of contention. But the problem with that position is always seems to me that, I mean, I agree it's a tremendously exciting way of trying to close the circle. The gap between our understanding of the relation of dimensions of concepts of the kind of epistemological, logical, semantic, and ontological dimensions of that. The notion of foundation, on the one hand, and the position of mathematics as an entire corpus of techniques, obviously, is by far the most developed of systems that we have, within, overall, the one sees in the role of geometrical.

20:00 We are breathing mathematics, and particularly within the applications of mathematics, to the most developed parts of physical science, because how can one think of the constructions involved, say, in choral metaphors, the signature group or the kind of constructions that one has in quantum field theory or in string theory, as Resting on a contentual notion of geometrical that connects with a, it seems that these are purely, that these are entirely kind of self-standing, self-subsistent notions that, you know, that receive support only from the overall kind of methodological examinag of the whole of mathematics, has actually transformed, increasingly, from any kind of sense of fluctuation. It's just always been my difficulty. Yeah, but there is a position of this argument which I don't agree with. And that is that, it is the idea that we are in front of an atlantic concept, I suppose either we describe, we try to characterize, or we treat it in a totally free way because it is, so to say, so strongly schematic to be able to

22:30 We describe that in an unbounded range of ways, different ways. So for instance, this is the point you made when we were here last year in June about when you were talking about Croton-Deep, about being in this position about the relationship of geometry and algebra, the geometric sources of algebra. I think this alternative does not pay justice. The notion of what we achieve through different characterizations of space is a series of aspects. We place it in this very complex notion. So we describe different aspects. We realize that if we emphasize some relationship between different notions, we obtain mutually inconsistent theories, although each of them can be perfectly consistent. Well, this is perfectly in line with what I say in the sense that the identification of the building blocks of our intuition allows for a lot of bricolage, as the French say. That is, we can combine them exactly as it occurs in biological evolution. Different strains combine in different ways and obtain different organisms. So I don't see... I mean, if I can go on in this analogy, the previous way of looking at foundations applied to biology would mean that either there is only one kind of...

25:00 It's a useful analogy, whereas clearly there are evolutionary constraints on the space of possibility of biological diversity. In fact, a point which, if I may say, I think I tried to bring out very clearly in my rewriting of the footnotes to Dongo, in which I was pleased, yes, all that stuff about the virgin shale, the citational rule, which I think is absolutely right, and I think it's a useful analogy. But if I can play devil's advocate for a second, it is the view that there is some ultimately metaphysically justified notion of the spatial that seems to be operating in the background here, and certainly with the view that you put strongly when we were talking here in June in response to the claims of Elaine Landry that It's precisely the position, the geometrical sources of category theoretic constructions and broadly algebraic constructions in general that is the kind of key theme that runs through much of both Grote and DeCandil's work. A metaphysical come, well a wager on the future space of metaphysical possibilities and understanding of mathematical constructions. It seems to me that that is at bottom why Bill is so hostile towards con precisely because he sees it as getting away from this direction of understanding of algebraic constructions ultimately justifying in practical terms and seeing it as trying to transform geometrical notions in a completely unconstrained way or constrained just as it were by the bare possibility of the form of manipulation that algebra is part of geometry. If, however, the direction that fundamental physics takes is in the direction, then by a variant of the kind of indispensability of mathematics, I will have to say that structuralism has in fact turned out, on many logical grounds, to be, given our understanding, not our understanding of mathematics and physics, the only available...

27:30 And to predict which way it will go, one has to be a physicist, one has to be a mathematical physicist of the caliber of Korn or, you know, perhaps, you know, the kind of caliber of Korn to play this game. Of course, I don't have that, but... It does put me in mind, I have to say, this position of Bill's, these non-conventional jobs, a little bit in mind of your remarks about the history of science being a cemetery of attempts to overwrite or to constrain the development of theories in conformity with a metaphysical agenda. We shall see. I mean, the jury is out. The reason I almost never mention indispensability arguments is that it is complex growth. I mean, what is indispensable for present-day science is not necessarily indispensable to understanding reality. I mean, so, after all, the picture provided by... But then I suppose the structuralist response to that would be to say, you know, talk of reality here is cheap and worthless. The source is, you know, we must agree with Hume, or at least the Kantian variant of Hume, that the... The ultimate sources of springs of knowledge of the real forever shut up to us. We only have the structure of our theories to go on. My point is very down-blurred. I'm interested not in the variations, but in what is used in different, in so many different ways. And this is a constant, so by means of which we understand they are susceptible to be described in different ways, okay, no problem.

30:00 We have to understand the architecture of this bricolage in a very general way, and that's exactly what I'm trying to do here in connection with the Foundations of Mathematics. So, in a sense, it is... Well, perhaps this also is not very generous for me, but it sounds to my mind as a recovery of the layman view of the world. I mean, we understand what we can understand through what we have in our mind and in our resources, but we cannot forget it. I mean, we kind of forget these ingredients by means of which we make all the variations we can make. No global separation of form and content and no tabula rasa. No, I agree on both the methodological and the grounds of an overarching kind of naturalism. Which I feel myself near to Russell is that part of the evidence for what I have said is through the analysis of knowledge. That is, when we look at most algebraic theories of whatever domain, from quantum mechanics to cosmology, from homological to logical, we use a sentence. Which is deeply of space, prepositions have a special meaning.

32:30 So if our theories are understandable, even as syntactic objects, we have to appeal to aspects of our understanding of space. That's all. I would like to have one sentence in the most abstract theory, which is an articulated sentence, deprived of partial ingredients. The point which you made is specifically in the essay on the concept of scheme. In order to realize this, don't look at the nouns. No, look at the verbs. Look at the verb to look at the prepositions. Absolutely agree. Yes, yes, yes. And which brings me in fact to the question I was going to ask about the... Can I ask now? Because I don't know how much time you have. I'll have to go through this later. But the passage on extensionality principles, which of course brings... What you've just said brings me very directly to the questions I wanted to ask about this. I've really regarded it as a triumph of this day, enormously lift my spirits if I could go away fully understanding this, believe me. And this is just ignorance on my part, this is because I haven't had the energy to study the relevant... The relevant material, as carefully as I should have done, and I've tried to pick it up on the wing, or I've given up because I've quailed in the face of Johnston, the 1977 or 19, you know, 2002 versions. I'd just like to get a clearer feel for the way that the global version of extensionality principles, called the kind of, the one that, the meaning of which is naturally thought of in

35:00 In terms of the Kantorian framework, in terms of elements and membership, is to be seen as fitting into this broader, more flexible, geometrical. There's a sentence on page 12 of the paper, in fact it's the whole passage on page 12, which is your exposition of the geometric core of the invariant constructions which express. Versions of originality in the theoretic framework and of their failure. Okay, category theory allows a clear expression for different forms of extensionality. I won't get through my... actually this section I left virtually unchanged. I don't think there was anything in this that I hardly... apart from a few cases of word order that I changed. It's on page 12. Yes, yes, the paragraph that contains the formula for well-pointedness. Oh, incidentally, I think there's a, yeah, there's an index missing. Was it from that expression? Yes, there's an index missing from that expression, I think. Or is it the one over the page? No, no, it's not that one. It's okay, there's one very small typo. It's all right, I've caught it. It's not a, it's just one very small typo. There's an index that... Missing in one, but it's not that one. But to get back to the question, category theory allows clear expression for different forms of extensionality by making use of an idea from algebraic geometry, where the notion of point is suited but generalized any map... Now, I should be able to grasp it quite clearly from what you say here, but it's always good to hear you explain things and, you know, to fill them out. Any map x from t to a can be considered as a generalized element of a, varying over t, and this is noted, this is symbolized. The classical set-theoretic version of extensionality is simply well-pointedness, i.e. for two parallel maps there is a separator, which is the terminal object, and any singleton can be taken as such an object of sets.

37:30 Well-pointedness claims that there are enough points to separate any two parallel maps, and here is the reason why such a principle of global extensionality is also expressed as one is a separator. Classical, in other words, Cantorian sets are totalities variable on a space composed by, slightly better in English, of, but only one point, therefore elements reduced to the constant ones, and any part of a totality is just a subset determined by its global points so that any set is totally disconnected. Now, in the talk that Bill gave at Bolzano back in 1998, in a version of an earlier paper that he prepared, which I've got and which you'll which was his no the the the part I'll talk. So in the Cambridge lectures and also in the lecture notes on synthetic differential geometry which he prepared at the beginning of the same year in 1998. There are these references to the connectivity of the space to the the kind of Cantorian the totally disconnected totality. In which he alludes to this intuitively presented condition, the points being the disconnected on the set and of the one where there is complete cohesion in that one point can become another in an entirely trivial way without it. Now I really want to try and understand this more clearly, particularly the kind of co-discrete case, how it fits into this.

40:00 The underlying idea of the notion of figure being the more general notion of space in general and the notion of the domain of variation with quantities varying in the space in the general case as involving this kind of lattice theoretic relationship between parts of a domain, which in the case where the domain of variation is restricted to be determined by points, is a special condition. I still haven't grasped how the kind of part-whole relationship, which involves this kind of lattice-theoretic relationship between parts of the domain of variation, which obviously also connects with this greater flexibility of the idea that objects are not analyzed purely in terms of static and part-to-form elements, connects with the point about the connectivity. ...in the set and how that in turn connects with the geometrical dimension of this whole construction and indeed the homological, cohomological dimension, I guess because I don't have enough of the necessary technical background. I'm sorry I'm just waving rather than drowning at the moment in what I'm saying. I'm trying, I'm even trying to formulate clearly the question I want to ask... Actually, it's a lovely sound, normally I like the sound of church bells, but not at the moment, not when I'm trying to think. I mean, can you explain to me again specifically how the... The determination of the totality by global points is related to this condition and said it's totally disconnected. What's an example of a domain of variation that's not totally disconnected and how much control does one have over the connectedness in question and how does it relate to the role of the figures? This is a big and wide question, I know. I think, yes, drawing, sketching things will help me a lot. I was going to ask if we could actually look in diving some of this.

42:30 The principle implies classicality. This is the well-pointedness version of the global set-theoretic version of extensionality. Which is much more than disconnectedness. Right, yes, Bill has made this point, yes, but it is much more, it certainly doesn't, the converse doesn't hold, so connectedness is a much weaker condition than, yes. If there is no such pair, such that the space is connected, only at the quantification, quantifications, in order to have a disconnected space, it is enough to have two such that they split. What you have is that whatever subset is a source of a disconnection for the space, since if I take for example x and I find a point, take the same thing, this is open, but also it's complemented somewhere. Right. So, the remaining part of this is that whatever is this x, the result is a disconnection of the space. And the same occurs independently of the point chosen. So, whatever is for any x, x determines disconnection of the object, in this case of space.

45:00 Now, if the space is determined by points, it means that It is not to look at the points in order to have a determination of the open sets, but since there is not any difference... Right, so this is the case where the kind of lattice theoretic relationship between parts and domains of variation just collapses to the case of one point for every... Yeah, so if you have in this case a map from... From x to y, you have two maps. They are different. They of course have to be different on something which is inherently definable in this category. It is a category of spaces. And in this case, they differ on points. They differ on a point, which is what is meant by saying that the space is variable over a single point. But in general, I mean, it is, this point has to be open or closed in order to be defined as a domain of a map from, say, in this case, it could be, you replace it with... With any subset, for any two different maps, there is a subset such that, open subset, it has to be open because the notion you are dealing with in this category is spaces, so the notion has to be topological. But usually, in classical topology, they mix of ingredients which are topological in nature. And ingredients which are not topological in nature, as the notion of membership, the notion of membership is not a topological one, but is used in classical topology as an intrinsic part of it, because spaces are defined as a set of points, defined as a proper subset. In the case of indiscreteness,

47:30 Is the case in which the only proper subse, open subse of space is the space itself. In this case, if you have two different maps, they have two different globally on X, but you can't say exactly where because you have no possibility of identifying a particular subse in topological terms. There are many different ways in which they differ, apart from the whole set, the whole collection acts. Which is where the membership-based version seems to be a very artificial way of dealing with this. And the sort of latched theoretical relationship between the different parts of the domain becomes a much more natural way. Exactly. Right. So I see what he was driving at with his remark about... ...a very artificial way of looking at it, because if I take, say, this structure, take the line, the fabric, the sample room, I have to say, I use it as an universal counter-example to the classical way of looking at topology. Good. You join it with a circle, that's one. You join... Ah, sure. And then, which parts does this space have? Space defined as the union. Well, which are these parts? You can't say this, this. The parts have to be, in a way, connected to one another. So, this, there's no connection. If there is a function defined on the side to another space,

50:00 I'm looking at the one-dimensional, two-dimensional, three-dimensional component, but not uniformly for three-dimensional components. This is only a one-dimensional line. If you look here, it means that there is a path in X, such that this path is a sort of space. It depends, of course, which kind of map is this, because, of course, if it collapses, if it collapses the bidimensional and the three-dimensional into a one-dimensional entity, you obtain, in exit, the image of this is only a sentence. So, in this way, why don't you consider differences between two maps? You always have to look at the faithfulness of the man here, so these are three figures which connected together provide you with the structure in acts which can be a source of differences to another object. This map is usually taken in, for instance, it builds examples of subcategories of figures as always, and not injected in the sense of set theory like a sermonic, because in this way you preserve the structure of the figures and X, but not necessarily, it means that sometimes you could have... You could have a collapse of complexity of the figures and x. And if you collapse this, it might be that this is not enough to separate.

52:30 But of course, if the arrows have no need to preserve these differences, they can be split one from each other in the same way. So, what we ask, actually, as the requisite condition for a category, is that the category of figures, which in the case of set theory reduces to just this one, ... of, say, deforming closed lines, so, say, this sort of thing in space, you know, you take knots in space, and you take bolts, and so you deform bolts, and anyway, and path, which you combine, you cross, double, yes, with the weight, so that, the category of figures... Is sufficient as a whole to distinguish any from that right that is basically this is in fact what the notion of adequacy and care adequacy implies okay i really i was going to come on to ask you about that incidentally this connects presumably with your observation bringing this back to the um with the geometric roots of a fundamental construction uh and indeed of the ingredients of categoretic um formulation This presumably connects with the remark that you made in the schema paper about the notion of set itself having, as it were, this schematic geometric content of resting, in fact, on the, yeah, actually resting in some sense being the kind of the transpose of spatial saturation. In the case where one has closed and bounded components of a path that's connected, would that be a... I mean, is there a connection there?

55:00 I mean, because the connection is this, that if we take a space and you take, let's define on this space... I'm trying to get my head around the way that that insight, the remark about closed and bounded components are a part of this connected space, is connected with this idea of Bill's about figures as the generating notion, as it were, for concepts of space in general. They can be reconstructed from the from the homology that the space can can also be disconnected as a whole so take this sample and take this space the the the unit line the unit the interval What matters in constructions defined on this entity is the behavior of functions defined on a different object, say X, the behavior of these functions on the different components of the space, which are connected components, three different connected components. The adjunctions which field elaborates on in relationship between the notion of the smaller here, various adjunctions here, along the other side.

57:30 Well, the connection is what we call the discrete components. And then we use the term adjunctions because it's all the rest of the components. I really want to understand that, especially in relation to the action of the... I describe this very, very simply in my 93 paper, Conscience on Universals. You look at it, you find a solution of this notion, where the notion of components identified by the rule it plays within this. You have a full range of possibilities. This one is an extreme. On the other side, you have a total code script. In between, you have a range of possibilities. One of the major possibilities is that you look at... You take the most elementary components out of which, by union, by connected union, you obtain any part of this connected part of the space. And the case of points would be an extreme limiting case. Yeah, so, this provides you with, for each of these stages, between the extreme case that was considered in foundations as entitled, what point, what point, the only figures to be considered are points, where on the other side, the figures... I have no particular identity apart from the global figure, which is the space or the object itself. Hence the remark of characterizing the co-discrete case that one point can become any other in a completely arbitrary way without having to pay any attention to the parametrization of the variation of the motion involved in that.

1:00:00 I'm beginning to get more of a feel for the way that this notion of figures operates, particularly in connection with the role of connected components. But say a bit more on how this connects with the underlying ideas in homology. The periods, I mean, have their own topological structure. So this is not reducible to a set of... Because you are interested in continuous maps. So if I say this is the map, the image of this here would be this way, for instance, the map, this is here, this is here. This is here, and this is here, the way, in total it is, for instance, for rational points they go to the extreme, zero, and for irrational ones they go to the sequence, the growing sequence from one middle to one, this is the continuous function, so, the... The point is that you look at the figures in relation to the preservation of their, or at least part of their topological structure. Generally, you consider monomorphic maths. Which in turn connects with the connectedness of the space. Yeah, because in this case the figure is a continuous deformation up to a homeomorphism. But in general you might have... So in this sense the notion of figures which works as a generalization of the notion of... The notion of generalized point is too general to be considered as a particular kind of figures because when we speak about figures, we speak about objects.

1:02:30 Each can be embedded into any object of your category, but of course, I mean, a generalized point can be, say, a generalized point, so if you have, for instance, if you take as category the power set of an x, of a given set x, and you consider as max, say, a, b. Now, take a subset of X, say B, and take X as a figure. So you have a point, you have a point from X to B as a matrix, which is Bill's definition, a generalized point is X. But the occurrences cannot be monomorphic, in this case, objective, because this is the whole set, and this is just the subset. So this again brings out how the conditioners of epi-mono-factorization, where there were captures, the essence of the set theoretic, the set theoretic version of the notion of point, the non-generalized. Say a bit more about this. Again, because this all connects up with the existence of obstructions to inverses and maps, doesn't it? In the general case. Well, it connects with the fact that you can look at data through selecting objects. The way the choice and the exception. Also, also, also, yes, yes. With the argument, I mean, that extensionality was only a particular rational choice, because this is a principle that means that for any method there is a possible choice of an element in X, but if you take X as a collection of these joint sets, it's a principle.

1:05:00 So the fact that when I realized that the form, that MacLean's formulation of the axon choice, it is in the form that for any F, for such that F, G, F, this is the original, this implies my remarking, and then there was the problem that I wrote some notes, I had issues. It is that there is a choice principle than this which is almost coded as the ethics split, that is that instead of having for any f, you take for any f. In this case, you have, you say for any f, f y, and there is a writing such that f g. There is the identity. The identity is a section. It means that there is a section. Why? Because this means that F has to be happy, of course, because... Has to be? Happy, happy, happy. Oh, okay, sorry, sorry. Yeah, yeah, yeah, yeah, yeah. Right. Just remind me again of the definition of... Well, in the case of, in the case of, of its right consumable, which is in the category of setters just a subtractive function. So this is a lot weaker than the condition of the epi-mono-factorization. It's weaker, because then my question was which extensionality principle, and the answer is what is called the weak extensionality.

1:07:30 Sub-objects of the terminal is a separating terminal, which I don't know whether... Yes, which you mention in fact here. And this also, of course, connects up with the last question I wanted to ask you about, about vibrations. So does this condition, does this weaker condition of it, you've got, as it were, an epiz of right cancelable, which is dysregulated. Is that also... You say that's weaker than well-pointed. That's implied by well-pointedness. This is weaker than well-pointedness. So does that fail also in the category of... Well, this is independent. This is independent from well-pointedness because this implies... This is weaker than well-pointedness. Oh, this is actually stronger. No, no, no, these are the strongest. This is the strongest choice, this is the choice, that's the full strength, and that means that there are no abstractions to inverses at that score. This is the extensionality principle, this is the choice principle, and here we have the F, G, F, F. I'm still not sure if I've understood fully. That's this condition here. Right. So for instance, I mean, for example, for example, look, if you have a set... What is the, I'm so sorry, what's the meaning of the dotted line? There's a section for math. There you go. If you compose, you, first you take a section, then you take f, and you obtain the identity, in the sense that if you have a set x, for instance, and you have, say, here... If you want to know the difference between X and Y, suppose that the image of X and Y is not subjective, so in the image of X, then you are claiming that there is a G, and the G is defined on all of that Y.

1:10:00 There is a line here, this point, which is not in the image of F, say this is A, this point A, so G of A goes here somewhere, then you take F, but the image of this, A, can't be here because F brings it necessarily in one of the points in the image. So here choice fails. Yeah, it fails, you see, because FGA is different from A. This choice, of course, would imply that this is not A. This is a choice related to subjective functions. Which is what corresponds exactly to the usual version of the axiom of choice, because you take a collection of non-empty sets, you take a map from this collection. So, let's say I do a set A, and you say there exists a set composed of one element of each depth, okay, and this means that you join all this together, okay, in this set, and you think, what does it mean? It means that... There is obviously a subtraction from this paneling to this set, but it's also sectioned, because from each point, you go exactly from its source, and then you go back and you obtain the identity exactly on that, which is exactly, it's a global section, which obviously one doesn't have here, but okay, I mean, I'm sorry to ask such simple-minded questions. Now, here you have the weak form of extensionality, and here you have, actually, there was, this is, booleanity, and weak extensionality does not imply booleanity, they are totally independent, what you need is that it adds,

1:12:30 If you have this principle hanging on zero as a section on zero, if you have this, you'll get, you'll get the strong... You'll get back to the... Yes, this is the principle implied by this. This is the... And this is implied by this. There is also another... I was going to ask you about the force, but can you explain the meaning of that? It means that since the support of a map is related to any object in a category which is thought to be a unique map. Now, the issue is that even if the point of this does not hold, that is, even if you have two maps, All of these are different from X to Y, and there is no effect such there, at all.

1:15:00 However... Which would be the case, for instance... Yeah, if this is not happy, right? If it's not happy, yeah. It would essentially say that although there is no global X of this sort, there is a U, sub-object of... One, such that, this, such that, that, why, okay, it means that, it's because, because the U is a proper, this is a monic, I wrote it as an inclusion, but it's a monic, it is, you can't find, This is usual, it's true in the case where you have a space defined over another space, which is basically a situation of a vibration, because if you take, for instance, a space, say, this way, and you have a line here, a vertical projection, beginning to get a much better So what's going on, for instance, in the case of the eton do is why they are not sort of set. Okay, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. That's very, very well motivated in topological geometrical terms that we're doing right now, Tom, that's why I'm doing it. Yeah, different continuous maps from X to Y, they have to differ on, and this is the, say, all these spaces are the base space which is our unit, you want to say. Oh, okay, right.

1:17:30 So this is the case where the base space is endowed with a very sort of straightforward topological structure. Right, right, right, right. In this case, they differ, they differ, okay, but they, they, G, T2, is all inside there. It's the image of Y. So if there are two maps which are different, take them here, the point of difference is... Here, because this is not an image of 2 and this is not a codomain of p, you have to search for it here, to take a section here, to find the point x such that fx is different from jx, and of course there is a part of the continuous sections of p, the s. But this means that, of course, this is the base space. So there's no global section. There is no global section. This has to be the part of the base from which you take this point, a particular point, or this interval. It is a proper subspace of the base space. And so you have, in the case of... Topological spaces defined over a base space by a continuous function. These holds, I mean, they substitute... That's this condition that you're talking about. Although, is this the condition of support splits? No, this matches exactly with support splits because support splits means that the support is... It's a sub-object of the third one. This could be not a subjective function because one, not only, because one can be very large, larger than x, so for instance, if you have, if you take a category of segments and the segment x is, say, the open segment, say, open segment,

1:20:00 There are many examples of the real mind. It is zero, and you take space, space, the human hand, of course this is not subjective, it is subjective in the sense that it has to be continuous, it is a mind, mind arrow, which takes the first part of, first half of, so it's not heavy, but it could be, it has a support in the sense that The image which is the subset of the term, the sub-object of the terminal, which by epimonofactualization, you have, since you have epimonofactualization when mapped, you have this map called the internal relationship. Which is this part, which is this part of the possibly much larger space. This epic split, this is not related to expansion mathematics, it's only related to the invertibility of the map which sends any object to a sub-object of the terminal. So this means that this epic in particular, not also, the epic split means that any epic split, for any epimorphic map, there is an inverse. This point is weaker because it says that at least, perhaps also otherwise, at least, and you can't ask for more, the epimorphism from the objects to their supports as sub-objects of the terminal is split. Yes, but you can't ask for less. Yeah, but for less, I don't know, it's engineering, but I don't know. Okay, so that's very helpful indeed. But I'm sorry, but I'm still not quite... Okay, so just the condition of epi, epi mono, epi is a weak condition than supports. I still haven't quite understood what the difference is. Okay, just looking at the two spaces in the case of the topological character of the related model.

1:22:30 Can you just run right past me? I'm sorry to be so slow. The case of vibration is the case where you have the L space, X, and the P is used for the global space and the B space. Right. Helps me to actually draw in the spaces as it were. And you could have here, say, you could take B as a unit circle, you could take E as an elliptic, finitely elliptic, or even projection, or any continuous intervals in the elliptic center, but there is no global section. Sure, but it doesn't have to be an interval, it can just be a double comma. No, it doesn't. There are two conditions for having a vibration. This is a special case for this. The first is that... Two steps. First, this is a vibration. I'm sorry if you'll have to remind me of the definition. What does it mean, Cartesian? It means that to be Cartesian is that it is a triangle. And one by three mathematics, take a triangle, suppose that you have a triangle, suppose that you have here one map, then there exists a map here which closes the triangle and makes this committee.

1:25:00 So for any map which can be decomposed in the Bay, there is a map in the Braavos Bay of which they've given the code. In which one part of the diagram... In the base. Right. So this connects with the limits and so on. Right. And asking that this holds for any math in the base means that you have a vibration. That is, any math is very broadly because... What's the restriction? Sorry to interrupt. The condition that there's no discrete vibrations is a stronger condition, if you can have discrete vibrations, it's a stronger condition. So I've never been able to get my head around the idea of non-discrete vibrations. Can you give me an example of a vibration which is not a discrete vibration? Oh, yeah. I'm sorry, I thought... I don't mind, because too much, too much, too much. Actually, though, this was... remind me... On my definition, no, no, I'm not going to criticize, but you said that my definition of vibration and the one I gave in the 1950 paper in front of Lenin-Retro was not sufficiently general. All of this can be reconstructed out of what is in any fiber of the covering space.

1:27:30 Yeah, right, right. And discrete vibrations is a factor. They presuppose that there is a factor condition. I find a position on the decomposition of maps, that is, we have to this, that if you have any triangle, it's not only that they exist, but this existing map is unique. So you have more than just commutative diagrams reflected in the global space, but that this addition of any diagram, commutative diagram, can be done in one and unique way. So it's more restrictive. So it is clearly more restrictive. Again, I would love to understand more, I know you didn't want to talk about that, but about the constructions in the several categories of Aton do, where of course, if I understand it, that condition of discrete vibration fails because you've got this kind of internal variation in it going on in the domain, which I'd like to understand more about. It means, for instance, that co-equalizers of mathematics fail in general. General under-callback, you don't necessarily preserve or equalize under-callback, and this connects with, as it were, the kind of topological structure of the space, which is interesting because, I mean, it would seem intuitively to connect with the guiding idea that the ultimate topological and geometrical roots of formal notions include even the content of the formal notion of. The apparently unrestricted generality of the notion of object that one has in set theory is actually coming from a topological, geometrical root, which involves a restriction on the kind of variation that the spaces can undergo as captured in these conditions on vibrations, because that's all terribly intuitive and I don't intend to understand enough of it, but I'd be interested to see if there was anything in that intuition.

1:30:00 Can I ask you one other thing about, because I don't want to tire you, but this is the first time I've had a serious conversation about mathematics or philosophy for six months. But let me finish. Oh, no, finish what you were going to say about this condition. I started some pages at work. Sure, no, no, finish, finish, finish the lesson, please. The point, no, no, no. No, no, no, this is very important for us. The point was about the notion of figure. Yes, yes, yes, terribly important. I want to go on. I suspect the question I was going to ask was relevant to what you were going to say. So you have a category of figures, not reduced to one point, but of different kinds, because your objects have, say, dimensionality. That's the example you made right at the beginning. So you have this, and you can have different figures. You have maps to excel. So the condition of adipocyte is that the puncture from F to C, actually from F0 to C, will cancel. I didn't understand that diagram. Can you say that again? I'm sorry. If you have a category of figures... You could also have maps from one figure to the other, so this can resolve in the fact that when you take... Which needs to take account of, for instance, the connectedness of components. Yeah, and you get less information, for instance, if this is not a monomaniac map.

1:32:30 Okay, and so you compose, then you obtain a figure which is defined on the domain, not right on the domain of this map, which turns this map from the category A to the category C as a contravariant map, contravariant factor. Depending on the fact that for any B, from B to X, if there is a map from A to B, you obtain the figure not from A to X, you assign to a map from B to X a map from A to X. And the meaning of a covariant functor is just the congruence condition that you have. You respect the order of composition. Whereas, in contrary to Erich, you, as it were, reverse the order of composition. What is important is, first, this notion of electricity, that is, this man, this function, his fate, means that if you have different figures, they are preserved as different. There is a balance. This provides you with only the topological, in a very general sense, description of the general category. Now enters the algebraic aspect, and the logical aspect, which is implicit, but you have to define which sense it is implicit. There are different ways to put it. The way we could build, putting it clear... That is, you consider, say, it's a specular image on the other side, which is weak.