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

10:00 I'm very excited about the size of what you've got there, or smaller, but you better help me eat it, you know, but I had no idea, I thought it was, I'm really sorry, I had no conception. See if you can get a, can we get a plate for you and Alberto? Well, I can't possibly do it, and I'm not being coy, I really couldn't.

12:30 Yes, and then, where did you, oh yes, it was in Nice, wasn't it, that you gave the paper on quantifiers and sheaves? Yeah. Now, how was that, you see, that was the first time that I'd... You know, reference to those ideas that I knew of in print. Obviously, Alberti had the same impression. When... How was that paper received? The French hadn't yet rejected it either. Even though it was the very occasion that Rodenbeek stopped, Rodenbeek spoke there to give the appeals medal to Hironaka, but at the same time he announced that he was leaving mathematics, but his students, all his students were there and not yet, and so I was invited.

15:00 As a category theory, a year before the Congress, I had predicted that I would find some relations between quantifiers and sheets. Sure enough. But how happened that such a large amount of contextual developments in the categorical spirit that you can find within the seminar? All of this, even from the point of view of the people who contributed to it, independently from Grava, I guess, it is, let's say, even from a nationalistic point of view. We were inclined to develop such an heritage, and they didn't. I don't understand such things. I don't know for sure that Grundig was ever accepted as being French. I think there's probably many reasons, but they probably feel they were forced to do this gigantic exercise. I think it's well, reluctantly, that part of it is the difference of the mental power of Grotendieck.

17:30 It's such a powerful mind, you see, that he can swallow a 2,000-page definition and use it routinely, as soon as he uses it. Whereas us normal mortals, we need very compact definitions in order to be accepted as something simple. For them, what they were doing, these details, was gigantic, something they would rather escape from. Whereas for him, it was just rounding out this tool which he was using habitually. I must say it comes as an unnerving experience to hear Bill all this, the three of us, whereas us for us is ordinary mortals. Sorry Bill, I didn't mean to. Well, I think most of us would be very happy to have one percent of your mathematical insight. Well, it's a different thing, you see. I never pretended or felt to be a powerful mathematician in that sense, but I dressed to hold and actually worked with it. When did you first, when would you say that you first came to think about the axiom of choice in this deeply geometrical way, in terms of constancy and variation? And was that there almost implicitly from the moment that you first, you know, doing set theory without elements, or did that come a little later with the isolation of the topos axiom?

20:00 And obviously as a mathematician you must have used the axiom of force then. How did it come from there? You have the projective and you have the substitute. At least I think it's there. It really... it can't be right. Well anyway, it's somehow there. Put together, what he says there, is to bring it around backwards. I mean, the cohomology is from the theory of the axiom. It measures the theory of the axiom. It's all the instructions. But when did you first relate that, in your own mind, directly to this very deep way of thinking of physical variation, for instance, to the real-world reasons why Banach-Zipinski-Tarski paradox fails?

22:30 So it's the future of the algebra and not the future. In Rome, in March, with Cerny, we faced the actual, the one which gave me the omega object, the power set. And somehow, I don't know, the original handwritten version was dated January 1st. I was attending a meeting, again of logicians, actually, you know, with Waldbach at Spring, when this was distributed still in the handwritten form, and somehow it hit me immediately when I saw it, that this is really just a special case of logistic theory. I've often thought, helplessly naive, that

25:00 Again, in introducing the subject to philosophers, introducing the subject toposthenically to philosophers, introducing your way of thinking in mathematical structure to philosophers, that perhaps the Sierpinski-Banach-Tarski paradox is not a bad place to start, purely pedagogically. It's well known, it's another of these pathologies, but it's a pathology which, because of its directly, very naturally, comes out of only that. It's a very natural kind of result, which can't have anything to do with parameterization of variation in the real world. It's something much more general. It must be, but it shows that obviously mathematical structures have long since escaped our geometrical intuition. That's the ideological claim that you hear in undergraduate lectures. It's probably a good place by making them see and in relating it in ways which are easily accessible to as much mathematical knowledge. Two ideas of constancy and local and global.

27:30 See, there's something else that I've been saying. I'm sorry, it's such a mundane remark. I'm sorry, it's such a mundane remark. There's something else I've been saying. Even my friends go around saying, well, no one knows if V equals L is true or not. Well, I've given a definite line on it. Everybody will disagree with this line, in which case, you know, they can repeat it. But just to pretend that I didn't say it is like, I mean, it's like the same thing. I mean, I was reminded of it because the question is, do we have enough variation? And also just to ask is, even roughly, if there's more variable and less variable sets, then we realize that, of course, Gödel showed that they can be so constant. And the other hand, obviously, Machiavelli's good variable. So that could be elaborated. The theorems could be proved about the logical orientation. Why does everybody ignore this? Well, did you know that Colin had sent a little note to the Journal of Symbolic Logic, entitled The Category Theorists? A category theorist's perspective on V equals L. I think that's the title. A category theorist's response to V equals L. And it was rejected. And I have some neat little notes. Particularly interesting is historical remarks about Euler and the notion of real variables that predated the MWC theory.

30:00 And did you, of course, in fact, yes, John Bell remarked in the meeting, and there was this man, this man, this man called Allenstein, who actually published a paper arguing that the, sorry, arguing the axiom of choice must be true because of these results in elementary particles. And who is stressing about the making of the objective? I mean, it needs to be. Oh, it's in the media, I think that's true. How is your, what is the fact that you're having there? It's good, it's very good. That's right, because it's 1973. Of course. Yes, that was the first place I came across this very striking remark, which is something you have to think deeply about. You know, how wrong, how badly I had been taught to think about foundations of mathematics when you remarked that set abstraction is just an instinct to speak vibrations. I suddenly realized that, you know, there could be this beautiful, this comprehensive geometric setting for thinking about structure, which also related it to the real graph, to graph beings as reflected in things, but in a dialectical way.

32:30 At that point, when I came across your paper, I had almost given up. I almost completely ceased to be a Marxist, because I had become convinced by the way I had been taught, by the way there's mathematics and the necessity to believe in universal mathematical objects in order to have a semantics to do mathematics, in order to have an ultimate... The concepts that are used in mathematical proof. I had come to believe that all this was necessary for science and that one just had to accept it even if it did make the world seem more and more like the medieval view of the world. But since this is the best theory of our modern science, I just have to accept that Marx and all those people of the 19th century were just wrong and they were confused and then I suddenly realized, no, no, no, I haven't thought about it at all. I was saved from that phase. Yes, I was saved from that error entirely by coming across your papers at the right time. That's fine as well. But explicitly put down, it's not hard to see that implicitly,

35:00 Eisenberg's uncertainty principle. That means you can't really tell if rotation is going on or not. In fact, this man, Lucas, who wrote the paper, argued from Gödel's theory to the necessity of embracing a dualistic ontology of mind and matter, possibly to show that mind is reflection of matter. Is a theologian, he's a philosopher at Oxford, he's also a member of the Board of Doctrine of the Church of England, he's one of their leading board of doctors. They actually have such a thing called the Board of Doctrine of the Church of England, and Lucas was at one time, I think the secretary, he was one of the leading members anyway of this Board of Doctrine of the Church of England, who formulate alleged... There's another such man at Cambridge too, Polkinghorne, the former professor of physics, an elementary particle theorist, who became a priest, became an Episcopalian priest, who is now the master of St. Catherine's College, and who publishes articles showing how mathematics is a reflection of the mind of God, who had some influence, I think, on Penrose. He's nothing like as good a... well, he hasn't done such. I mean, for instance, I notice whenever Penrose comes to speak to philosophical audiences about his ideas, Hawkinghorn always seems to be there in the audience, and respectful questions, and then always seems to go off with him afterwards, like a guy on high table, confused.

37:30 He's been exposed to these religious people. Hawkinghorn is, I think, a notable case. Hawking wrote a terrible book. Well, it wasn't a book, it was actually an article, but it was in what is anyway a pretty terrible book, which was a series of papers by mathematicians and philosophers in praise of Wigner's paper, The Reasonable Effectiveness of Mathematics in Science, and there's a paper, I saw Liz McLean in there, I have to say. Yes, please. I'd like to order. Okay, all right, you're right back here. I'd better go first and walk him. I just think he's got better at it. See, mom likes the chicken. One chicken, one beer. Okay. Did you like feta cheese on those? Any idea if Mabry likes feta cheese? I think he does. Yeah, he likes cheesy things because we have, yeah, I think he does like cheesy things. Which one, the beef or the chicken? Beef. Anyway, Polkenkorn wrote in reply to this symposium on Wigner's The Unreasonable Effect, he wrote an article called The Reason Within and The Reason Without, The Reason Within, Reason Without for the Unreasonable Effectiveness of Mathematics.

40:00 The Reason Without is, of course, that it is given to us by God. It is a reflection of God. ...of the divine mind. The reason for that I never quite discovered. But these are the sort of people who certainly had an influence on the project Penrose, I think, in the last few years. Maurice, is that how you say him? No, no, no, when he was here, he was in Korea, I think. Talking about the network mind, right? He actually worked with Hawking on this business, right? Yes, he and... I think that initially they worked independently and then later they collaborated. Penrose, of course, was a mathematician rather than a physicist. I think he discovered the singularity of theorems independently. But it was through being exposed to Penrose's lectures when I was trying to learn mathematics after I had come away down from Cambridge, having done a degree in philosophy, and had finished spending four years trying to learn mathematics, I went to Oxford to listen to Penrose. Later went to a couple of his seminars. Uh, that I had become convinced, this was just before I came across your quantifiers and shoes and then in algebraic geometry with the electric logic, it was at that point that I had become very confused, thought it was impossible to reconcile mathematics and any ideas I had about understanding the world and seeking to change it and political action along those lines. Because of course, you know, you are very easily, um...

42:30 In a climate like the British Universal System, if you're not terribly bright, but just as passionately wanting to understand things, it's very easy to fall into these. My impression is that the way that you have thought about mathematics from the beginning has always been very closely connected. I'm conscious that you can be fast, give you a little trouble, but the actual instance of progress is possible, so let's do progress. And an explicit way to defend that is... And variation in motion are things of experience. There are seven features of... The idea that they have to all be thought away and say, well, it's not something that people fall into without creating an intellectual confusion. But as I say, particularly, I think it's...

45:00 I think it's particularly useful for somebody in my position to try and analyze and spread awareness of where these particular errors, I mean the error involved in the metaphysical abuse of that. But now I understand much more about the developments which led to these pathologies and then to the use that was made, the ideological use that was made in the various pathologies like the so-called, that have led to so many undergraduates being exposed to. It's about the unknowability, where this extreme expansion, this emptying of all material content from the notion of object, it became, could only be seen as either coming from language, so that the world was a phrase of language, or else true source, because of the apparent width, extreme ontological width that was needed to unify mathematics, so that then the mathematics came to be seen as prior to physics, the whole of... The real world is just a tiny little blob of the ordinal kernel. And the other question is extremely strong. Newton, all that Newton was really doing, especially around the three symbols,

47:30 invented the cosmos, invented calculus. But where the hell had I arrived at this, this sort of entirety of the culture? Including the scientific advances. There's deep knowledge to fall into that. Thank you for your attention.