FW Lawvere / Discussions, incl. M Wright & A Peruzzi (contd.)

0:00 Oh, he is, he is actually an undergraduate mathematician at Poplar, yeah, yeah, he's only just begun there. Hello. Hello. Excuse me. Do you want to come and, do you want to try and get his attention? Thank you for your attention. We already met yesterday. I'm not, that's why. Unattached. Oh yes. Was it Tim Lambeck who wrote that strange paper about Bill called Heraclitus and Modern Mathematics? Or was that somebody, was that a different, no sorry, that was somebody called Lambert. Sorry, that was Lambert. I don't think, I really don't think I'm happy. It seems to boil down to this idea that you need something like a direction of time before the composition from the category exists.

2:30 These tools may just be an artifact of the fact that you're in the wrong category. Look at the distance between you and the people of our time and the people of our time and the people of our time and the people of our time and the people of our time and the people Thank you for your attention. What do they go into? The Basil Miley Beer Fund? Oh, to darling little Fabio. Thank you for your attention and see you in the next lecture. I think there is here. Very, very English. Cambridge man, all people. Out of what? Out of people.

5:00 That's the definition of the logic of physics. Well, it's not difficult to get a definition of projectile out of you. That is easy. Well, thank you. It wouldn't be useful for us having a meeting before the presentation. I'd like a sandwich. I'd like a sandwich. I'd like a sandwich. I'd like a sandwich. I'd like a sandwich. I'd like a sandwich. I'd like a sandwich. I'd like a sandwich. I'd like a sandwich. I'd like a sandwich. I'd like a sandwich. I'd like a sandwich. I'd like a sandwich. I'd like a sandwich. I'd like a sandwich. The problem with the kind of things, topics that interest Alberto, and I think in... What certainly interests me, and I think interests others a bit, is that they're not really the kind of things where you can ask for clear-cut definitions because you're, as it were, upstream from the point at which rigor begins. I mean, certainly in something like that paper on the forms of the axiom of extensionality and topos theory, you can draw carefully. You can very carefully draw distinctions between various versions of extensionality and how they express themselves, but if you're going to say something about how the... ...and what the chap from the... Well, I think he was just a bit shy. No, it's what the... After all, he's only a star. You're a star. He probably feels a bit intimidated. God, I wouldn't blame him if he did. For all he knows, he's talking to the equivalent. For all he knows, he's talking to the equivalent. Well, not quite, but you know what I mean. The equivalent. Right. Yeah, let's... Excuse me, excuse me. Ah, si, si, si. Capisco.

7:30 More progress. More progress. Very high cheating. I think so, too. He puts up with almost any amount of vague hand-waving. No, no, correct waving. Thank you for your attention. I've got to write something first haven't I? But I think I've got enough in the way of ideas just from talking to Alberto in the last three days to write something of my own that in fact would not be entirely derivative and would have something to do with it. So I'm not making heavy weather of this point about the relationship between extensionality and his definition of the functorial version of extensionality and its breakdown in the case of projectile, but I do think I have something to say about the significance for that as a general setting. Thank you for understanding this point that he's making about the way that one thinks of the domain, the very strongly geometrical topological aspect of the way that one thinks about structure in the domain in any logic, and also historically the reason for the strength of the extension or tradition, the determination as a way to cling to extensionality at almost any price, and the refusal by Quine and other logicians to countenance modal logic or any logic that didn't deal with the strict extension. There's a very strong metaphysical commitment there. It's not purely a methodological commitment. That view of truth involves a view of that relationship between the internal logic of the category and the internal and external distinction in categories, which in fact only category-theoretic language allows one to grasp and to see where... And I also think that that determination to hold on to extensionality in that context is in fact one of the chief sources of... Thank you for your attention.

10:00 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, 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 Upwind I should say, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, upwind, up Oh, I'm surprised the Georgians haven't caught on to that, why it's called the categorism. The categorism, yes. In the new church we'll all be made to learn your categorism. Categism. Categism, yes. But, no, I think in fact there is a deep connection between the extensionality and Platonism, the point of view that because of the need to hypothesize abstract objects because of the assumption that In any domain you have to have an inbuilt criterion of identity which is associated with an absolute identity relation and which carries with it this commitment to separability and to a geometrical view of the domain consisting of points and the role that set theory plays in the cognitive representation of space and spatial mapping. All of that classical apparatus I think is deeply connected with the way the completeness is built here. The drive to the closure of the system takes a form that necessarily leads to postulating, as it were, levels of structure in terms of an ontology of already existing objects rather than in terms of looking at the geometrical concept in terms of separability.

12:30 This is why it's so difficult to reconcile quantum theory with the peculiarities of quantum theory and non-separability with an extension view of mathematics, except by assuming that the physical universe is just this blob of structure at the very bottom tip of the orbital cone with lots of loose ends in it and that non-separability is just an artefact of the formalism which is no real ontological. We are discovering something about the world. We know that the fact that you cannot factor on, you have this kind of non-factorizability of product, states of systems, of coupled systems, which after all is experimentally detected and welcomed by the spin experiments, the Bell experiments and spin variables. We know that we do have this kind of... Ontological non-locality builds into the theory. So this does tell us, I really do believe this makes a difference to our notion of variables. I think I'd like to hear him talk to Noura a little bit. Yes, that is fine. Well, I want to meet him. I'd like to know very much what he's going to say to Noura. I'd like to see... It would be very interesting to hear what he has to say. If he has this letter of knowledge, I think you can get him to do it, because you can make him more interested in the world of physics, if there is any money or interest in the world of physics. What you have to do is get someone made of science to do it, or to eat in some other country. And then you make your take. Whoever would want to take it is a physicist. He's going to get it for you because it's a play. You know, science is science. But if you take it on behalf of the city, you're going to appreciate it. And I better know why it's coming in handy because if I'm going to go to Paris, I'm going to go to Paris, yeah.

15:00 I give up the travel business tomorrow if I thought I could get that. Mr. Bradke wouldn't. Well, yes, exactly. Anybody would go after us, exactly. As you gather, the nice thing about Bradke, he bombed highly. And I grant him this. He had complete confidence. And that's what I wanted. There are some of the viruses who treat you as a skibby. Oh yes, they treat you as a skibby. They give you a problem and you report it to a witness, if not a biologist. And I'm not a biologist. I don't want the money to... Some of them, yes. And in fairness, this is the great strength of the British tradition, with all its weaknesses. That is the strength of the British tradition, unlike the German universities, where there is far too much of this kind of thing. I did go and see these other guys. Yeah, I remember you telling me. And I don't think he was fast thinking, fast talking. Very, very fast thinking, good mathematical physicist. Very, very good mathematical physicist. Very good. He was going to push me, so I ran a mile. He would have pushed you very hard indeed. Your feet wouldn't have touched the ground, and you certainly wouldn't have had much time to think. I'm told that Ray Streeter still doesn't know at the age of 50 what an axiom is. John Bell told me that. He literally had to explain to Streeter what an axiom was. But the guy is a very good analyst, I mean, he's there after all the... Curiously, there is, in fact, a whole area of algebraic quantum field theory, constructed algebraic quantum field theory, which is actually captured by a structure called the Streeter-Weitman axioms, but Ray Streeter doesn't know what an axiom is. Weitman talked him into it. Extraordinary.

17:30 This is the person who nearly became Jerry's PhD supervisor, mathematical physicist. I'm going to come one day, let me collect my thoughts, get this thing in sync, and as you say, put it in a good context. Well, I apologize for my friend. But I want to... Oh, he's busy. Because he's busy on the... He has a lot to do with going to Montreal. For him, to speak in English is not easy to hear. No, we should apologize. We're at a conference in Italy. You know, it's disgraceful that neither of us speaks Italian. We don't have to apologize. I accept all credit cards, by the way. I was told by this pa, you know, late at night in London, and he said, do you want some fun? Okay, just let me put this over there. Here we are, that's the sinister. The man with the dark glasses here. The non-separability, the non-separability, and the extension, Alberto, Alberto, they can't do this, there's a glitch, there's a, Sherlock, you've detected her, yes, we're gonna sweat now, no, it's not, it's perhaps it's parapsychology, perhaps it's, perhaps I forget, nonsense. If anyone can, can and can't, in this context. If anyone can, can and can't, in this context. Just let me go and see if this fax has come. Actually, if we can call from here, it would be very convenient.

20:00 I would be very grateful. And, obviously, if you have any more ideas about how to do quantum logic in a topos, I shall be very interested to hear them. But I think these insights... I mean, the thing I'm mainly interested in is the insights that one has into what is going on at the level of the relationship between extensionality and separability and the way one might think about non-separability in the context of... Because it does seem to me that the nature of the variables and the domain is at issue here, at least. Now, if it all are positive mathematics, if the two have been extensional, have extensionality built in, and yet the category theoretic framework promises a way of looking at extensionality in terms of the behavior of separation of arrows Yes, you don't always have the chance to stand back from it. I had that impression very much when we were listening to your talk on representable functions in topology, the last talk that won. Who was this? He wasn't on the list because it was a change of speaker and a change of topic. No, no, this was this morning, this morning. The last, the man who spoke just before Lambeth. Well, I wasn't there. Yes, you were. Yeah, but this is, this is wrong. You know, as I say, it was a change of speaker, a change of speaker.

22:30 But you were there because I was sitting right next to you. It was, it was on representable functions in topology. It was, I thought, actually a very interesting talk. It was the talk after which Kelly made the intervention about some technical point about... But listening to him, again, I thought over and over again the way that pullbacks preserve the unit object, the way in which one has in the local homeomorphic mapping a more general situation than that of the category of sets, and yet still within a topos. Reading that paper of Isham on quantum topology, I began to get very vague heuristic ideas as to how one might further possibly modify the category that one's in in order to try to do quantum topology without, where pullback would not necessarily preserve the unit object, where one might have... Well, after all, the motivation for Locale, which is that of space without points, without intrinsic structural points, already gives you something of an ocean, and connects also with the intuition about the way one thinks of the variable in general, and in connection with the... Conception of the domain. I think this area of quantum topology is one which is now being pursued seriously by good physicists like Isham and where I think there will be a fertile interstate of category theoretic methods once the best people in the, the people who have the good physical intuitions have, like Isham, have mastered the category theory language because... There, the interplay of the physical, the heuristic of the physical notions and of the properties of the mathematics seem to be particularly close, just as they are, I think, also in the context of extensionality and topos. But the geometrical intuition involved seems especially clear. I found I followed that talk without... Knowing very much about topology apart from elementary topology, the Hausdorff axiom, it seems to be very clear. I need to learn much more.

25:00 What is the Islamic prayer of God? Grant me more knowledge. This perhaps connects up with the suggestion as to how we need to generalize the notion of cover, of covering, perhaps by taking projections rather than... That you can present the connection with quantum complexes in which you have no point. I can't tell you. Because you have only to work on functions to a particular privileged object, which I don't know if it would be useful to maintain as true or in other...

27:30 Another privileged object, one has only to try, to have a lot, I think, to have a sufficient motivation about the kind of content you manage to find something, the only thing is to try to find it on the technical side. Yes, I think it's certainly, you are going to have mappings into privileged algebraic structure, but it's not going to be two. It's clearly going to be something more general. And in fact, if it were two, you would recover factorizability in product, which is one of the things which is lacking, possibly into a lattice of some sort. They've done simplicial topology over lattices. In fact, they've actually tried this, again, work of Finkelstein on the quantum simplex, which was an attempt to redo simplicial topology over a kind of quantum domain without, where the vertices were superposed rather than being defined as points, but you still got a combinatorial, you still had a... I think you might recover this at the level of the category theory, at the level of the way one thinks about the mappings of the function, the mappings of the function space in the topology, in the category of topological spaces, but if you were, you know, the category of quantum topological spaces, or the category in which quantum topology lives, would, as you say, certainly not. In the way that you recover the notion of point in the case of the Le Carle, it obviously involves building in a level of the Le Carle itself.

30:00 ...of the frames, the sort of classical assumptions, which offer the choice of two, as the subject gives you, and, you know, I think it's, um, hmm. The voltage is leaving at 430. 430. Do you know where it leaves from? Well, it's silly. It was announced. I just didn't pay attention. It is the number five, but I'm not sure, so I don't want to go into it. It's what time? I think that's the best time. I'm not an error, I know anything at all about the simplicity of topology. But I think that this point that you made earlier in the general context of... The way we think about variables, the way we should try to think about variables in quantum logic and the kind of non-extensionality of the quantum context about the contrast between the local and global setting will show up again in, must show up in the quantum topology, the fact that you don't have the... The breakdown of the covering condition, I think, must be an indication that this global-local distinction is not being held fixed in the way that it is where variation is, where we're arrested in the case of where choice, we've had choice. I mean, it seems to me this is a deeply physical

32:30 An intuitive physical comes metaphysical or protophysical notions, that is to say an element of structure is being put into place there right at the fundamental level of the metaphysical conception of the domain that conditions the way one thinks about the way the elements, the direction of fit of the elements of structure of the level of that is elaborated within technical operations and mathematical operations. And that's the kind of... You can easily become one can just say things that sound so vague that they are of no value at all but at the same time you have to try and strip away right down to the most basic level of which is slightly upstream from the point at which rigor begins because in any you know the definition even of A function of mapping is in question. I mean, whether, I mean, Finkelstein is right to try and tackle it the way he does is a moot point. I don't think he is. I don't think that, but I think he is pursuing the problem right back to the, to the point where the foundations of physics and the foundations of mathematics meet. Holdsworth was trying to find... Holdsworth was trying to find... I think it's pretty... I don't think, you know, he had the aptitude to find the right construction, but he was, I think, pursuing the right kind of questions. Again, you know, pursuing matters right back to the point where... Foundations of physics, foundations of mathematics actually merge. The point he made about trying to understand. For instance, one of the candidates he looked at for category of quantum sets was the category of now. I'm trying to remember which category it was. It was the dual to...

35:00 The analogy was between the dual to the category of complete Boolean algebras and the dual to this category, but it wasn't an obvious one like simply that of all the complemented lattices, it was something a little more specific and there was a technical reason for that. But he made this point that, maybe it's a very naive way of thinking, but he really did... You know, he was saying that we should think of set theory and geometry as united at a very deep level, this is what the topos axioms illuminate, and as both in some sense dealing with the theory of the constitution of objects, or the theory of the constitution relation of systems, complex systems in the world, or... and that... Something very true, something very naive indeed, but simply the contrast between things which retain their identity and interaction on, as it were, and the language that one uses informally, that of mappings of taking an object or taking an element under that mapping into a structure, the informal language that one uses to express the... Preservation or modification of relations of elements of structure in the domain is very strongly influenced by spatial intuitions. Clearly you see that in the category of finite sets, where the notion of mapping has, it seems to me, a very strong degree of physical origin, which does involve the notion of things retaining their identity in interaction. Perhaps this is too naive to be of any value. It probably is too naive to be of any value. Yes, too, too many, too many. I need to concentrate on the, yeah, but in a controlled way. Yes, I certainly shall try to do this, but as soon as I get back.

37:30 But it would be good if one was going to talk, to have something worthwhile to say, rather than just making a lot of vague motivational remarks, to have something quite concrete to say about how we normally say things. So your stuff has all been moved, hasn't it? Oh, yes, that's right. You haven't even got a toilet, have you? It's now only a one-sided mapping because of all the stuff in here. Oh, yes, right next door. Because I like something. It should all be in here. No, your bag is in here, Alberto. And I think these are your things as well. Oh, yes, yes. And I would like to leave it here to cover the microphone. If you have next days the opportunity to read that paper on category and logic, and if you find there some error or some... Well, if you mean simply error of sort of English colloquial expression, yeah. I'm certainly not going to find, I'm not going to find technical errors in what was theory in your papers. You know very well. I'm not going to find technical errors in your papers. And if there's a side condition of the kind that Peter Johnson knows all about, and you know that I'm not going to be the one who...

40:00 Who has mastered it yet? Not yet. Maybe when I give myself a problem with it. But yeah, as a matter of fact, I think that your papers probably could, when they're published in English journals, probably would benefit slightly from being... Just looked at by a colloquial English speaker, but some of the constructions you use are, I mean, they're perfectly good. I mean, they're absolutely correct grammatically, but they are not the way that an English speaker would actually express. No, it's okay. It's not, it's nothing. It doesn't obscure the sense. They're still completely lucid, but I mean to take a trivial example the Hang on, do I have your, yes, do I have your paper, the, well I was thinking of the Husserl one actually, well no, in fact the category, the one on category theory is a good case in point. I mean it would only take half a day to clean up the, English said it was completely colloquial, but it would take literally half a day. I mean it wouldn't alter the sense of anything, but there are one or two. Absolutely, these are purely points of, stylistic points, they don't affect any point of substance at all. This, just to take a second, ever since Jurassic World, mathematics is the best I've ever done. Alberto, is there a time for me to have a little crap before I say goodbye? Do you not have time for me, five, six minutes? No, I will have a little crap. Oh, take it, I'll go. I don't know, because I should be at the station at 3, 3 to 4, 3 minutes to 4. I have to be there. Well, better get going like that. I take, it takes to me to go there 15 minutes. So I can stay here still for 10 minutes. The station is new. The station is very new. So in 15 minutes we can say that we did fine. Yeah, in 10 minutes.