Discussions, incl. M Wright & A Peruzzi on Lawvere's paper

0:00 Yes, I must admit, I found it, apart from the fact it was in Italian, I did actually find it quite difficult to read. I was going through some questions that I wanted to ask you about. It belongs to the, it still belongs to the period in which I made, it seemed that I made everything that was not to be understood. No, I don't think there was ever such a period. Though I must say I did find, the first time I read your papers, I did find, well, for instance, the capitalism logic paper and the internal, local, external, global paper very, very much easier to understand than even a political paper like the one on the theory of descriptions. But then, of course, I didn't know a great deal of the area of sheet theory that I didn't know about at the time. I read the theory of descriptions better. But there were parts of that I certainly wanted to ask you about. I think that John had a strong influence on changing my attitude to write in English. It had a back effect also on my style in Italian. Do you think it's clear enough? Yeah. That's interesting. I think that my attempts to write in English Yes. Interesting. Well, of course, I can't say if you're Italian style, but I think your English style is certainly very distinctive. It's you. But it does convey a tremendous amount of excitement, I have to say, of enthusiasm. Okay, I'm sorry it's not fitting for you. It must get completely, you know, it must get completely well before we go off to London. No, I understand. What are your plans this evening, as far as eating is concerned?

2:30 Well, I think that perhaps we could... Yeah, that's fine. That would be great. I spoke to Francesca and my friend Gary, in fact he's not gone back to LA, he's going to stay until Sunday morning, but he's staying at a, he's going to, he's moving to a little hotel in the station. If, I thought it would be nice to invite him perhaps just to meet, well, Francesca, when we go around the, and if you want to... I said I would bring him at the hotel where he's staying before he goes out tonight to let him know, you know, where we would meet tomorrow. So if I could do that, I can do that now, and then write about this paper, the one you were talking about. I've just managed to get through. It is, in fact, right in my skimming it, that it is very close. I've actually made a copy of the one. I'm not sure if I've, I'll look for it, but I'm not sure whether I left it here. It's not here. Oh, in that case, I'm sorry, in that case it's still in my case, but I'll dig it out so you can have a look at that. That was for you to look at before Sebastian, sorry, before. But there's a good deal in that that is particularly the idea of this layered construction of categories whose right of joint function is category one lower down.

5:00 The sergeant of the lecture gave an analysis note. Yes, I would love to attend this hour. It's almost the same thing as you oversee the category, of course. Yes, well, there's a good deal I think we need to talk about. Particularly how it connects with some of the ideas about the et en deux, which I think, and particularly I need to understand the role that he assigns to the orbits, to the orbits of either of the group action, well, the orbit group. So anyway, can I just make a quick call to Gary first, just to tell him? Because I said before he went out that I would let him know what was happening. I just dialed directly from here, do I? No, nothing at all, no, but I was going to wait until I could, because I didn't know what your plans were, whether you were going to go out to eat or whether we were going to stay in. It's just a matter of courtesy, I just said I'd let him know. But perhaps it would be better for me. This evening, early into the night, the house is no more than I know home. Yeah, well, shall we just have a pizza on the way back? Yeah, that's exactly right. That's why I'm very much in there. I just simply dialed the number, did I? I'm really sorry, I'm not feeling too good. Parsi, how are you doing? Excuse me, camera di agiotto.

7:30 Di agiotto. Yeah, hello. I'm going to just ring from Alberta just right now. He's not actually feeling too good at the moment, he's got rather a bad cold, so he's going to turn in early tonight. We're just going to go out and have a pizza. So I think the best thing is we'll meet up tomorrow, we'll sort out the business at the station. Yeah, looks like it, yeah, I think it must be the same thing, he really isn't in too good shape. And well, I'll let you know at lunchtime tomorrow, but certainly we'll have a drink before dinner tomorrow night. So, okay, have a good time tonight, and I'll see you tomorrow. Okay. One o'clock at the Uffizi. Yeah, that entrance. Yeah, just up by the square. Actually, the entrance is up by the square. It's exactly where I showed you, you know, where we were today. You remember? Right there, where we were today. Almost as, you know, as you're in the square, facing the Paris and the Palazzo. No, because the actual entrance is up by the square, but it's the same gallery that runs down to the river. Yeah, I'll see you there at one. Unless you want to meet at the station first. No, that's pointless, I'll meet you there at one. Okay, and if you need to get in touch with me here, I'll just give you the number of where I am. You don't need it? Okay, well have a good time tonight. Okay, cheers. I gave it yesterday in the film to other people. And the translation is not much different from what I thought it was a good idea to find out. The translation is, the other political cause is he who is able to open paths.

10:00 Who creates a path, right? I said that who invents paths. I think that that is... I just silly pedantic point by some somebody. It's nice also to say that he was able to open paths. I agree I can't see that there's anything wrong with the translation at all. I think it's fine. I'm not even sure I should have bothered to pass it on but since I don't know Greek myself I thought you know that might be worth just checking the translation in case but I also found that actually very helpful. This article, this long historical article of MacLean on conceptual categories and perspectives, he says a lot in here that is very illuminating about the history, the historical development, but particularly, okay, but particularly and obviously in the striking it was that Wendell Bitt set out to act in a paragraph. No, what I found particularly interesting in this... I haven't really got as far as the first two pages. I should have gone around Florence today to study it. It's in my Tasha case, the thing I wanted to get out. It's my notes in my thing. Did I put it in here?

12:30 No, in that case it must be... Do you remember the blue box that I had? I'm sorry, forgive me. It's just that there's... Haven't you left it? I think maybe I left it in... Yeah, I probably left it in here. Oh yes, here is the paper, in fact, that I thought I'd left out. It was here in the box. No, it's okay. Here is the paper that I promised you that I'd left out. Yeah, I'd left it here in the box. It's okay. That's the one that I wanted you to have. But I'm sorry about that. As you've seen, the notes I made following your suggestions, this is all of them, right from the first time I read them. Yes, excellent. Yes, I think that's important to make a reference to. That would be perfect. You've got one typo in deeply, otherwise... Yeah, yeah, yeah, I think that's very clear. And the other one was this, for... Yes, perfect. Yes, I hope you don't mind my making those suggestions, but I think it probably was important to introduce it right, you know, to say something about this peculiar term.

15:00 I think that actually takes to a third point as well. Yeah, the term has a well-understood meaning in group theory, doesn't it? Yes, of course, but I would like to see which kind of meaning. I was going to say, can I have a look at this? This I must make a copy of. I'll bring it with me. Since I have read the remarks, there are points which I would like to discuss with you. A remarkable paper. Questions of Decidable Objects, okay, we are. Well now, this is, whose paper is this that is referred to here, the Questions of Decidable? I haven't. Okay, right, that's one of the first questions. No reference. No, it's Johnston's question. No, that is the paper. Yes, this is the paper by Johnston that I haven't seen. Right, okay, we must get hold of that. It's a 1980s, yes, that's a 1983 claim that was decided on, yes. I said that assignment was between classes of topologists. This assignment is in a way the literature of geometry and logicism in its plot. Oh, so he sees this as an important clarification of his ideas in this paper. You see, this is so important. I think it's absolutely right. This is excellent. This is the paper that I've been wanting him to produce for years to actually provide this kind of orientation for.

17:30 The existence of, yes, now obviously I don't know about the quotations of decidable objects in the paper, but certainly the fact that when he sees set abstraction, I think in the context of the behavior of quotients and equivalence relations ultimately have a local or global topological character, equivalence relations ultimately have a topological character, which is the connection with the, you know, his remarks about Eton, which he expanded in Como. That leads me to try to see how to formulate the right question to him in connection with this paper. Space is generated by and lays the foundations for motion, and hence general space is... Yes, because one can't retain what is said here. You'll see here that there's a great deal in common between these papers, well, in fact, the relation, the third section of this paper is very closely related to the opening section of this paper, although the technical note is actually, I think, rather clear here because he hasn't assumed using diagram notation, as much as I would deeply like to understand, but the First of all, talking of the zeroth form structure, when he speaks of the zeroth form of mathematical structure, he speaks in the same connection here as the, this is the structure which is given, which is already the, what do you call it, the out-paving of the dialectic, the out-paving of dimension minus infinity, in many cases dimension zero.

20:00 The left of joint inclusion, providing the discrete spaces of non-becoming, the opposite code of street spaces forming an identical category in itself, which is however now included as their chaotic pure becoming, in which any point can become any other point in a unique trivial way. Now, is that the, any point can become any other point in a unique trivial way, is that the condition that is already violated? The orbits set functor, in the case of cohesive active sets, where this induces a dynamical action on the space such that two elements of the space-space could be moved into a common element, so that they are therefore already internally cohesive in a pure disorder. Is this in fact? Can I actually read the remarks that he made to Gerri and me? I should really like to, this is me talking, one question, I should really like to make a point about what Etan did back in the 1975 paper on variable quantities and variable structures in tropoid. At that point, he introduced me to say that his student, Kimo Rosenthal, had written in 1980, especially in relation to foliation, Etendue to foliation is the Miebus band, because the Miebus band is a good quote, a genuine Etendue, as opposed to one which is merely an ordinary space.

22:30 Remember, this is conversational, Mark. What you have to have to have a genuine Etendue, as opposed to one which is merely an ordinary space. So to have a genuine etendue, as opposed to one which is merely an ordinary space, is the action of a group, perhaps more generally a monoid. The action does not have to be invertible, but you have to have a... Can you repeat what you just said? Sorry, I'll repeat it very carefully. Well, I'll just read exactly what is written here, but you have to remember that we were actually taking this conversation within the...

25:00 And Jerry was, in other words, an étendu, if it's globalist, what you have to have is, well, he's going to go on and explain, I think, but you see, it comes up to this sort of thinking of, which are not necessarily locally, globally, compatible. The action doesn't have to be invertible. What you have to have in order to have a non-trivial étendu. I'd like to understand whether a non-trivial etendue is simply the same thing as a cohesive active set, or whether it's something rather more specific. Is an action, and the action doesn't have to be invertible, but what you need to have is a bad action in the sense that stuff in the space gets more mixed up as it gets moved around, as in ergodic theory. I'm not sure he would, but anyway, an action which mixes, he actually uses the term stuff, I think he's talking very informally. And we told her that we have to have this kind of thing. Yeah, we have to, in order to have a non-triple A tango, in order to have a non-triple A tango, what you have to have is the action of a monoid, or a group, or more generally a monoid, the action doesn't have to be invertible, it has to mix stuff up as it goes, as it gets moved around. At the end, just to make the side remark, much of the mystery surrounding ergodic theory is due to the fact that it has been forced into the framework not of general topological spaces, but of Borel space, instead of being seen as spread out in an etendue, which is its more natural setting. The idea of ergodic theory is that the dynamics mixes everything up, so to speak. In other words, the good quotient are those which give an action, even when you take the topos-theoretic quotient.

27:30 There are those spaces in which, for every point, there is a neighborhood moved away from itself by each act, group, or monoid. So that for each point there should exist a neighborhood which g moves. Sorry, I should have mentioned that g is not the identity. For any g which is not the identity, and for each point of the space, there should be a neighborhood such that g moves that neighborhood completely to a disjoint neighborhood. So that things are kept nicely separated by the action. In that sense, the étendu associated with the action is an ordinary space. But in the non-trivial case, where you have more mixing up of the original space under the action, there you will have a genuine étendu, where you have a real twisting of the space itself going on. It could be seen as splitting into the way one thought of the domain of variation in topological terms in the case of the set, and particularly how the contrast between Etendu, he already mentions here, for instance, both the petiturpuses of Grotendijk and the growth. This problem arises from the category of all possible ways in which a category can parametrize an act on abstract sets. This is already a proper class in the sense of it. But it is not cardinality but another quality which distinguishes them. He makes an almost exactly parallel remark in the 1975 paper on variable structures and variable quantities, The first section of that paper, that they are too large to be sets. We have domains of variation which are too big to be sets, but not for the reasons that the set theorists distinguish proper classes from sets.

30:00 In other words, the reasons have nothing to do with cardinality, but rather, it is rather through deepening our understanding of the reasons that... The two topological structures of the domain of variation, in the case of Etendu, makes it impossible for that domain to be a set. We thought it was a set. That we shall come to have a correct understanding of cardinals, of cardinal number. He actually makes that remark in the 1975 paper. So I asked him how the set, how the distinction between the non-trivial Etendu and the case of sets considered in the topological structure related to the set class distinction. He said, I wanted to get out, that is what I wanted to get at in asking you this question, the point is, the point about the topological character of the structure in the domain. He said, in order to understand this, you must understand... In order to understand this, you must understand what it is that keeps the arrows between one portion of the space and another separate. The word separator is usually used for a class of small objects, which is to separate any parallel pair of morphisms in a category. But that's why I asked him if there was anything to do with the condition that the identity is attributed to the separator of a pair of arrows, which is actually mentioned in your paper on the topic, and he said, what do you mean the separator?

32:30 I said, well, I come across the use of this term, the separator of a pair of arrows, in the paper of Volusian. The word separator is usually used for a class of small objects which are sufficient to separate any parallel pair of morphisms in a category, just in the sense that there exists a map between the separating objects. But what you are really asking me is what are the generators of an atom doing? This is Bill talking to me. But you're really asking me what are the generators of an atom doing? Well, those are explained in great detail in Rosenthal's thesis. The objects which play this role, in the case of the category of set value sheaves on an ordinary space, are just the open set of the underlying space, viewed as a very special sheaf with just one layer which extends over that open set, whereas a general sheaf has many layers which may cross each other. In the case of the étendue, there is something similar. The open sets are, as it were, and he says, no, I'm sorry, I can't recall the condition very precisely, but basically there are, because the étendue has this covering, it is the quotient of an ordinary space which has these, which has these opens. Which go partially between them, to be regarded as extra information about the structure of the space, of the dynamical motions which are possible, as extra information which, yes, and then I'm afraid I very foolishly interrupted him, I interrupted him at this point, and I then said, I see this extra information, yes, as extra information, which in the case of the question space of the domain of variation, which is because in that case, whereas here in the case of the non-

35:00 You have little pieces with some trace of the variation, which is precisely why one has to think of this as a domain of variation, which is in some sense enlarged, but here we must think of it as internally moving, and not just sitting there in that structure is reflected in the generators of the etendue, as you correctly intuit. That was basically the point that we got to. But I found that very illuminating series of remarks. But it does seem to me to connect up very much with what you were saying the other night, talking about the way in which the more abstract notion of element of a domain, in the case of classical quantification theory, has actually arisen historically. Now, what I would deeply like to understand in the context of this quotient, seeing the quotient What he sees as the significance of that paper of Johnston's hangs together with the important role that he assigns to the quotients of equivalence relations in topology. In other words, the most fundamental notion of equivalence relations in the most general setting is already... In a topological way, and it's the relation between the local and global features of topology that decides whether you have...

37:30 Since the same notion of shift defined... Well, which was the second one of these little germs? The second was shift as a shift... But the notion of germ function is... The other way is this, you make a portion of the cake and then you take two of them and then you take another of them and you, x and y are equivalent, you can go with the intersection of you and x and y. This gives you a construction that is a quotient of the topology on which you arrive. So in this sense you see that in the notion of quotient, the objects you are dealing with have a stronger, stronger possibility of...

40:00 Hence, of course, again coming back to the point about uniform severability conditions as being at the basis of extensionality. But what I'm trying to get at is that it's not simply one more way, one more correlate of extensionality principles. It is really the absolute conceptual core of extensionality. The second one is that the two things that are valid as a specific and an object A can be defined as a T, an A element, a variable over T that is an element of A puncturized by T. It appears almost to me that it is natural to deal with variables, A-variables, as interpreted as any other definition, or as non-definition, of variables,

42:30 which would oblige you to take account of the particular way this A-parametric variable in A has been generalized. So it would not be a key. It would be a variable that is tied to a certain horizon logic, a key variable. You have to give to this notion the most unconstrained interpretation and the only error there. And this is something that actually I am very, very wise. I think it collapses in what you conducted earlier here, that perhaps one could say that we lost the, we gave, we put away the absolute number of elements, but the first phase is to go towards them. It seems to force the variability of the relativization of the notion of element back into the setting of an absolute framework.

45:00 But how to weaken the extensionality condition in the internal language? It's not said that every kind of experiment has to be done. No, no, it's not written in stone. Yes, and there are conditions to inform of one, as we have said in the past. They are ideal, they're not real, they're drawn, they are presented in variations of the same. But still, I prefer to think of the knowledge of the world as an element. I think that this element is a given one. And this is a definition of fine mobility. You think of the possible elements of any given object as nothing but a glass, plus or minus a second, of a generator.

47:30 Which is exactly what you're saying, here it is, yeah. And the fact that we've got a class of generators together, because since, you know, the notion of dynamic purpose is given by these two conditions, that they have to have a set of generators, and they have closure under arbitrary products. But the first one is the condition of width extension. And this means that, in some way, the notion of hooping cannot be captured by the internalization of language, otherwise this condition would be almost trivial, since it is not trivial, why, since the possible, if you think of a variable as a possible element of an object, you see here that it cannot be reduced to just the identity of an object. A strange point here is that it could suggest that the notion of three variables is a kind of byproduct of logic not intrinsically related to the category of ontology. Well, I think that's a very profound suggestion. I've always thought that historically the way in which the ontology of extensionist Platonism arose was obviously very clearly tied to the progressive detachment of the notion of the variable from its actual role in parameterization.

50:00 In fact, a role in parameterization of variation. I think one sees that, I think it's interesting to know how Bill rolled aside the points that Colin makes in his paper, the technical paper that was the companion of the points as points of space, one which appeared in Journal of Symbolic Logic, where he sets out the strength, the weaker version of the decidable sub-object and shows both the geometrical point and also the topological degree. Those seem to actually be quite strong conditions, you know, freezing the possible... Once one thinks of variables, as you say, potentially, I know potential elements of concrete guys is in the process of undergoing variation that is actually parameterized by, you know, demands of parameterization by themselves varying in deeper ways than they are by...

52:30 I mean, for instance, there's this interesting remark about the orbit set. The orbit set already reveals some of the cohesiveness induced by dynamical action. Well, first of all, to how far is the notion of cohesiveness itself connected with ideas of change of state induced? He actually cites specifically the elements of the state space here as an example, those which can be moved into a common element, thereby already sticking together. The condition on the trivial, to have a particular trivial in Canggu, where it's generally globally explained, therefore every trivial in any Canggu is precisely that, the action will always move any, the neighborhood of any. I'm immediately sort of taken by the connectivist for understanding those observations, particularly what you have just said about the understanding of the notion of variable detail. The first of all remarks he makes at the end of, I think, the 1975 book on models in the Topoy collection, and it's a short paper, it's his first paper on that, and he's talking about the delineation here in Topoy, and then he remarks that the, in the case where the, the case where the, of burying over a Well, he actually says it's an organic theory, but I wonder whether he does actually see a connection, and this is something I wanted to ask him, it's very bold, because it's so vaguely posed.

55:00 He actually sees many connections. Do you know the papers of Rosen, the American mathematical biologist? Rosen is a mathematical biologist who has had some exposure to Catholics in it. He makes a couple of quite striking remarks about the relationship between... And the condition that the notion of the state of the system should have an absolutely deterministic sense, either that there is an absolute notion of the state of a system. Well, I've got, I've got, I think I've got, hang on, I'll just pop back into the other room very quickly. I think I've got an opportunity, one of these. Hang on, hang on. Yes. But it just seems that this might connect up with what you're saying about the, as it were, the component aspects of the notion of variable in logic.

57:30 This is something which I have thought about or been trying very hard to think about without seeing clearly how the topic's connected. But whilst being quite certain that there was a deep connection between them. And just let me see if I have the reference here. It's a short paper, and I don't think it's here. It's very annoying. I had a couple of, yes, a couple of, it's in a book called Mathematics and the, it's in a book edited by a man called Mickens, an American philosopher, which was published, it was a symposium which was held, contributions from mathematicians, McLean contributed to it. Biomathematicians, physicists and philosophers, and one or two psychologists and biologists, on the Wigners paper, the unreasonably perfect Wigners paper. Which point is fine? Well, that's precisely what I want to show you. Just let me find the reference, because... It's an interesting point. It's a fairly straightforward one. Yes, here it is. I've got it here. Unfortunately, I'm not sure that I have the whole paper. I think I only have part of it. I mean, it will just save time wasting, you know, my just reconstructing his algorithm if I can just see exactly without any distortion what he says. It's a paper called The Modeling, Relation, and Natural Law. It's in this volume, as I say, about... It's not a deep paper. I mean, well, I think it is actually a deep paper. What I mean is it's not a difficult paper. It's a straightforward, accessible paper. But he remarks, in the case where we have two formalisms, which we suppose that a natural system simultaneously realises, in the case where it is always possible...

1:00:00 To embed either one formalism into the other or both into a larger formalism, which is again a model for naturalism, and from which the, by purely formal means, the existence of such a largest formalism, which itself models a given natural system and from which all others can be formally generated, constitutes a strong postulate about the nature of the material world itself, which I do not believe is valid. It seems that some kinds of natural systems omit such a largest model, that is to say basically a model of which any other model is a quotient or submodel, and others do not. And the distinction between them has many of the properties of the distinction between inanimate and animate, or of simple and complex. This is not the place to pursue this matter, but the interested reader may consult Rosen, 1985b, for a fuller discussion, for having had a chance to consult. It is at any rate apparent, I hope, that, okay, this case, which he sees as very strongly related between semanticism and having a recursive, decidable system. He also remarks, and this is the interesting part, he makes this connection to category theory. I don't know how much category theory he really understands. His other book, he has no idea how good he is in, I believe, mathematics or essay for philosophers like this. He then goes on to say what are the ramifications for understanding of the modeling relation within mathematics itself for this feature, this particular feature of the modeling of the relationship between formalisms thought of as potential models of the behavior of the natural system, coming back to your point about the variable. And he then makes very few introductory remarks about modeling and mathematics.

1:02:30 And then, since 1945, an entirely new branch of mathematics has been developed initially to formalize the developments in methodology of topology initiated by Poincare. This is the theory of categories. For sources, C. Eilenberg and MacLean, 1945, blah, blah, blah. And, in fact, is to be regarded as a general theory of the modeling relation within mathematics. Space precludes my going into category theory in depth here. I must refer the reader to one or other of the excellent texts which now exist, and it gives a list of eccentric sources. That in the category theory, this is very introductible, the active agent of comparison between structure and categories, i.e. between different kinds of formalisms or inferential structures, are functions. Now this is interesting. The formal counterpart of natural law in this purely abstract setting is the existence of non-trivial functions between categories. And then he goes on just to make a few introductory remarks about natural transformations and now this is interesting, let us for example look briefly at the original motivation for the development of category theory, algebraic topology, and just that one side remark leads me to think that he is probably a man who has read deeply about the subject because most Biologists would probably not know that. They'd probably just assume the distortion of the history of category theory is derived from Goldblatt or something like that. One of the most basic problems of algebraic topology is the classification problem to be able to tell whether two given topological spaces are homeomorphic or not. The group theoretic models provide partial answers to this question. Talking to an introductory audience, an introductory talk. They generate group theoretic invariants which are very much like observables of the associated topological space. If they are different for two given spaces, the spaces cannot be homeomorphic. The basic question of algebraic topology is whether there are enough models to discriminate any two topological spaces in some kind of effective way.

1:05:00 Which we also know is directly related to the axiom of choice, because actually toposphere is such an equivalent of Krull's theorem, isn't it? It appears that sometimes there are enough such models and sometimes there are not, and when there are, we can build a largest algebraic model of each of the spaces involved and settle the classification question by comparing these largest models to see which of them is a quotient or some model of the other. But when not, there is in effect no set of invariants whose values settle the questions. It will be seen that this is an abstract reflection of the question that I have raised in the preceding section, and one whose answers should have a most profound effect on our understanding of the relation of biology to physics. You do not have in the case of biological systems this condition which he also remarks elsewhere is clearly connected to the fact that one has a stable notion of the state of the system given in advance on which the dynamics and so that you have a separation of the notion of state and dynamical transformation and that in biology you do not have a separation. ...of the notion of state of the system and dynamical transformation operating on the system. It's a very good idea, but I would like to see how one could... Oh yes, it's a very, very, very vague and weighty idea. But I also would like to see how one could perhaps make it more precise. It's just that it was in connection with this remark of Bill about the... Significance that one already has a departure from the level zero discrete chaotic case in the case where one has cohesiveness induced by dynamical action and it does seem to me that there's a notion of cohesiveness induced by dynamical action particularly in the case where you have what he termed a mixing up of the structure of the domain.

1:07:30 To be viewed as internally varying within itself in a way which is not captured even in the case of the topos of variable sets, is what I understand is the point he made in his conversation with Thoma about Descambure, whether this would have any connection with the behavior of, well, with the definition of state of, well, actually with our ability to treat the system as a state space with a dynamic from which... Which one thinks of as actually, well, our ability to separate out dynamics and initial conditions and stable notion of state of the system. Well, the way I see the... I'm sorry, in that case I wasted your time. No, no, not at all. It's an interesting problem. Because these are the identification of, these are the unique separation, definition of state. I agree, I think it is a profoundly interesting problem. Of course it would change our definition of phase-phase, and of course, probably, which, you know, are very basic ingredients of our actual... What is the view of nature since determinism? Since Newton, essentially. Yes, but precisely his point that determinism, in fact, one of the points that Rosen makes, precisely the way that determinism, the structural classical dynamics, actually hangs together with, well, he suggests there is a direct connection with the action of choice. I agree completely with this kind of, with the relevance.

1:10:00 This is such a long talk, but my way of thinking to the forum was different, as you have seen in the previous one, can't you, in the sense that, first of all, I relativized the problem to the problem of the emergence of the cognitive structures, perceptual structures, and then the problem of the emergence of the... Exceptional structures from levels of, let's say, recursive... Do you want to use that? Because it's probably more... Oh, sorry, we're getting all mixed up. Okay, fine. I haven't, of course, had a chance to copy these, unfortunately. This is your copy. Oh, is it? Oh, that's very kind of you. Thanks. I was thinking I needed to... So, here you have cosmophiles. You have the problem of defining properties of depth, The doctrine of recursion theory, under the hypothesis of determinism, is a very direct reduction of the physical, since if you have a domain of completely recursive actions, this is just a particular way of precising the notion of being deterministic. So, the key was reference to the ontogenesis of the same cognitive ability. And I took this as a correct marked example of this emergence.

1:12:30 I'm not sure that this does not at all really suggest in any way, which is essentially suggesting, even if the passage we read before is that of the observation that it is essential to consider a material function, but you don't require a material function if you have a group, if you have a quotient of this group, whereas definitely this is not a material function, but we are at the same ontological level. What is relevant, in my opinion, to the Americans is the notion of a joint. Since, take, for example, the category of set, and take the notion of group. How can you pass from a set to a group? Of course, it's easy to say how we pass from a group to a set. We just forget the group structure, we remain with a collection of blocks, but to understand this way has been possible only in the case, in this present case with the notion of pregroup, but once we realize that this construction This is just an example of the general construction of a joint pointer to the forgetful pointer. We find here a way of generating from something that it hasn't. What we have is that it takes the active, the constructive, and we say that this is

1:15:00 In the case where this is a category of space, we know that there is also another junction, there is a U that is left adjoined to another, which corresponds to the fact that whereas F is the discrete space, The last find follows that emergence from an ontological level to another one happens when there is possible not only to reduce, and in my opinion, what is this reduction? This reduction is codified, in my opinion, by the second law of time of the animate. It says that you have always given a how, however complicated and rich structure can systematically reduce exactly what the second law was telling us.

1:17:30 But what the second law doesn't say, and what is represented in my opinion, is the fact that we can have at least and at best our joints go in the opposite direction. For example, and this corresponds to the, for each level of complexity corresponding, of course, to algebraic refinement, I was trying to say yesterday that perceptual objects, as functions of perceptual objects, can be reduced to just a collection of points. Which of course brings us back to what we were saying yesterday about the transpositional possibility of arriving at kinds of abstract objects, particularly mathematical objects, being delimited by assumptions about separability of abstraction, which effectively ignore... Really look only at one, the one direction.

1:20:00 Now, I just considered, actually, I just considered the possibility of having a latter joint that gives you the free structure of the next level, of this level. Before we go any further, can I just, very quickly, I'm sorry, just very, very quickly, this is the name of the lecture. So I can concentrate properly.