Discussions (contd.)

0:00 I'd like to flag up for discussion at some point in the workshop, if you're agreeable, if other people think it would be a good idea, if you could just give us Calais at the category theory meeting last year. Which is your view of set theories, naturally, falling into place as a fragment of algebraic geometry. A set theory? Yeah, that's it. Not now, but I mean at some point in the workshop, I think this will be a very useful topic to explore. It's a sub-grain of a, not systematically explored at all. As you say, it's hard to find an algebra book that actually has an explanation by finite dimensional ones. I always think of it as finite dimensional. Similar to the arts and algebras, it's probably the same idea that there's this interesting class of things which are qualitatively smaller and has to vary to be proposed in their terms.

2:30 That's one. Newton. Newtonian rings. Newtonian rings. Newtonian rings are things like polynomials and finitely variables over the field or any quotient thereof. But also polynomials overseas, so actual finitely presented rings. On the other hand, the ones that are really finite, but among the rings, they're actually finite. Subdirectly, there's Berns, Kornstein, and what's the other name? Those, again, are precise attempts at the coarse idea of the infinitesimal range. They're really punctual ones. So the thought was that if you're over an algebraically closed base, every non-trivial thing has at least a point to cut over that. So an algebraically closed base, then the points are not necessarily defined over the base, but they're defined over some finite extension. That's one way of putting it, without the algebraic disallow for these punctual things that aren't necessarily the same. Rather getting the existence of these things. In large of this class, instead of just points, you have the infinitesimality of Gorenstein rings. These are cheap. Gorenstein rings are sort of everywhere. You can't bring into them. They're not just punctual.

5:00 They're not even fat-type-icratered if they have the infinitesimality of Leibnizian. More ample, indeed, in the sense that every function is generated out of the existence of that. Which is very good. See, again, this is the sort of thing I learned from Chandler, that if you want to consider presentation, you know, like the linear frames, the finite number of variables and the finite number of polynomial equations that you can pose and what are the consequences of the presentation, Chandler's remark is that, well, if you want to see whether some equation follows some sort of other equation, you can always assume that the algebra is not an advantage. Because the ring that you're trying to study has enough maps into finite-dimensional ones. And so if some alleged consequence equation is false, you'll already be false in one of those. The finite-dimensional linear algebra you see has all sorts of tools available that you don't have even for the polynomials. Like, inclusion maps always have retractions, so there's an ubiquity of these distributions. One of the things we learn about in field theory is the trace, which is the larger field retracts back onto the smaller one. Why? Because each element is really acting as a linear transformation on the smaller one. It's a finite dimensional one, and therefore it has a trace, and that trace belongs to the sum of them.

7:30 It's only a retraction if you have the characteristic zero and divide by the dimension, the trace divided by the dimension. It's very useful, this, what I call, I call this the strong millstone science, the fact that you get this monomorphic embedding into the family of the Internet Testament. In my talk in Ogarve, I was trying to show that there actually is a large class of algebras that automatically have this strong millstone property. Think of the things that arise the way R does, as D to the D, you know, that you start with. If you take the infinitesimal objects and take the function spaces, which are now sort of the macroscopic objects, they should have enough maps into the finite ones again, into the infinitesimal. You could use that sort of O'Larian construction of the computations to the class of, there are many, it feeds back on itself very tremendously. As I say, there are these different, so within the class of Newtonian rings, traditionally we always talk about chain conditions in terms of the video. The Artinian rings have changed in one direction and the Artinian in the opposite direction. And the opposite direction implies the other one because there are theorems about how you can compose or embed an Artinian ring. And in turn, the Artinian ones have a sort of concrete structure, themselves being products of local rings.

10:00 The infinitesimal things are basically sums of local ones. The thing is that I'm sure that Hilbert and other people who invented the community of algebra, they thought of it that way, but then when you look at the formal textbooks, you get the neat purely algebraic proofs where this whole intuition of infinitesimal, except the part about the function algebra, they don't seem to know that, that the tangent bundle of an affine scheme is still affine. Although they constantly used the functions on phase space to bring in variables that depended on both p and q's, they didn't quite realize that that was the explanation. What needs to be developed as a double basic is about the community of algebra expressed in this point of view of the geometric intuition. Again, the algebraic geometers, you see, they had this kind of intuition deep down somewhere. They've developed this whole other intuition, which is not bad. It's funny, I talked to Gabriel, for example. He understood things in that sort of a way. And Grotendieb, Grotendieb talks about the category which expressed the, embedded into the, that's also formal mathematics. You've heard my talk, haven't you?

12:30 My talk starts out by saying, first of all, philosophers in mathematics, for example, Grossman, they don't know any sense of that. He thought that the purpose of philosophy of mathematics was to help mathematics along, rather than to criticize it as being maybe misconsistent, maybe not constructive enough, or maybe not consistent enough. But he really tried to explain things, and that spirit remains, although it was thought that somehow brought out explicitly as being considered non-physician. So that's just our thought school. So who are these mathematicians you're right in front of? Well... See, I'm telling you, Grossman definitely had that flavor. Sure. There was one guy around 1910, I saw the paper, once I forgot his name, and then sort of one little trace of it. I mean, it just doesn't go back 200 years. Is there anybody who seriously... There's a guy called Kai Hauser, but he's tried to give philosophical arguments that sort of advance that Wooden's Amiga.

15:00 There's quite a lot of, and this is sort of, in fact, semi-geometrical arguments of subsets of the real mind, so this is very far from the thing like it in that school, but that's quite recent. Hugh Wooden, I think, wasn't part of it. Yeah, he still is. Wooding, yeah, he's a very virtual set theorist, a large cardinal, so at least he's proved a restricted version of the Pupinic hypothesis. But he's particularly interested in fixing...

17:30 To unify the universe. Yes, but anyway, so it's building, and then sort of take those things. Well, intuitive, in your life, of course, it's an enlarged amount of math. It's not the sort of mathematics, and philosophers are mathematicians who are doing philosophy. There's a lot of, you know, criticizing these various approaches that might have been thought and have, um...

20:00 This is the basis for the attitude that led to the paradoxes taking place. But now, philosophy of mathematics has become in such a way that you can make it consistent with mathematics. It is a very un-mathematical subject now. It's essentially a branch of philosophy of language. Take what mathematicians say is just another bit of language that you want to understand. So this is a very, very low level of philosophy of mathematics in comparison to the traditions of the events. Yeah, well, the philosophy of mathematics is hard. I sort of see myself as continuing that traditional grasp in that general sense that, you know, giving a definition of what is quantum, I suppose. All of those ingredients, I'm sure I can find it also in Witten's work, that there are things that correspond to the domains of the quantities, so it may have some relation to that, although I don't know of any contribution to it in large part of other areas. I know, I know, I'm curious. Continuous and discreet, well, it's either a different spirit, but when I say continuous versus discreet, I understand that there's going to be a vast number of examples of it related, not one definite example, but the, the, the, there is true or false, unlikely, if you take it to this point, which is actually my interpretation, we work, we can't force the idea of rigid things that aren't rigid.

22:30 And so you record some of the information about the things that make an extreme idealization about the fix, fix, fix, fix, fix thing. That's the basis of the language. It's just that it is something, and then you just consider these set theoretical words. Your point seems to be that there is the universe of mathematical objects, and then whether they are reducible to one sort or another is not... Something to investigate, but not... The splits are the same to the contiguous and the discrete, or to the quantity and the quality, or the categorization of buying, exploring, and using, but that's something that's unique and impossible in category theory, because as soon as you take the set of theories, you've essentially committed a sort of reductionist view. And of the continuum, well the very fact that they use the definite article when they speak about the continuum, which is itself to introduce this kind of rigidification, that there is only one determination of the notion. Continuous, which is crazy because there are obviously multiple determinations of the notion of different kinds of continuity.

25:00 These train out those points in the intuitive sense in which it should be. It's just, you know, it's small because that big thing is there. Well that's somehow what they would say too, maybe, that big thing in there is small, but why? It's because it's what you would never think of if you thought it would be over the reels, for example. That's the end of the era, something like it. You don't even have that interpretation. There's a structure of one or algebraic theory or something like this. But certainly Grasman's idea of structuring the intuitions about fundamental oppositions, such as that between quantity and space, and regarding that as the core of philosophically informed mathematics and mathematically informed philosophy, which seems to me to be extraordinarily powerful, largely of course because it is essentially exploiting dialectics, has I think been

27:30 I think it's very strongly implicit in some places, it's implicit in Grotendieck, and there have certainly been other people in between Grassmann and articulating philosophical intuitions in the way that they did their mathematics, Herman Violet, so it hasn't been entirely lost sight of, but it certainly has been far more in the background than it was in the 17th and 18th. Yes, which was, and people of course said the same thing about vials. Exactly the same thing about vials. Yeah, absolutely. He gets away with it precisely because... All the people who have popped the pedestal and at the same time, this is beyond your understanding, which is why you must bow down and worship it.

30:00 Yeah, precisely. It's a type of dialectic. Yeah, it's a second to what you were saying about graphs. The astrologers have been sold. It's run by people who will never get it. Let's go back to a second to Grassman. I think Grassman also illustrates your motivation very clearly. That's why these things that were giving a precise explanation of equality versus equality, etc., etc., he serves to, fully at the end, I mean, you could probably say I'm a secure lecturer, to get things so clear that somebody, some day, will write textbooks. And I think that was Grassman's motivation as well. Whereas a contrast would, I must say, something very, very more eloquent. I'm not going to explain to you what they are. The rest of the article, most of the second article, goes on and talks about them, their statements, pseudo- theorems, and so on and so forth, yet he has consciously and deliberately not told you what they are. So this is not him personally, this is the culture that he's embedded in, which is the culture of obscurantism.

32:30 Again, a total contrast to the draftsman. And finally, their success in suppressing precise definition. It's an incredible, incredible thing. And if you see, the funny thing is that Shannon Earl and I and my other friends, Don Shack, we've been reading these articles for years because they ended up as a niche cult. Exposing your article. These articles are useless. There's nothing you can do. By reading this article, you do not come epsilon forward in your knowledge. You don't have any further tool which you can use to develop any...

35:00 You've just been, maybe you've been amazed and dazed at some of it. You don't know what it was, really. We were saying these articles were useless, and then we found out that it was a conscious conspiracy. To be useless. It's unbelievable to say that. You see exactly the same thing in the case of the Scientific American. And Wooden's, well, of course, the Scientific American levels, there's this process of some type called dumbing down. It's not only scientifically illusory, or the New Scientist, or Nature, or any of those. Within mathematics itself, demand was, from mathematicians, that I would like to understand some other mathematicians, some neighboring fields within mathematics that might be relevant to mine, therefore I would need, so this is definitely, the demand is being slapped down, just like the treasurer's secretary is slapped down if you're wanting to know how the economy works. Or even more straightforwardly, what the banks have done with all the trillions of money that had just been given to them. Yeah, well, that's just information that's beyond you. In that period, since they adopted that, yeah, completely, non-democratically, of course, they didn't even reveal that there was a policy for ten years after they... They were so, they were so, they were so succeeded that they... ...especially in large card, large cardinals, turns on one or two clauses in a multi-clause.

37:30 My guess is that there are, you know, hints that could be together. It seems that the main person who wrote these articles about... Ooh, yes. That's too long part of that Summon from the Void thing that she wrote about. Yes, yes, that, yes, that. Alan Jackson. That's right, yes, yes, yes. I went over... I've just been reading that. I was shocked at how little... I mean, anybody reading that is really not going to have any understanding of what Grotendieck did in... ...to make some other... No, no problem. Quite recently, and I, yes, I was very, very surprised indeed. She's not a mathematician, okay, that explains a little bit.

40:00 What is she by profession? I'm formulating it now. Can I just go back for a second? They have this thing, you see, also, that the notices, I don't know. That's quite a lot of endowment anyway, I imagine. Sure. Not to the public free of charge, but to universities. Who are made to pay a great deal of money for it. Well, plus even before the MathSciNet, they already had pretty generous endowments from us, did they not? The AMS, did they? Did they not already have pretty substantial endowments from before? But I'm sure MathSciNet has greatly increased it. It did serve as a dependence for us. The one which Maclean answered so very effectively.

42:30 As an example of somebody whose bleed pedagogy was absolutely crucial to any correct understanding of foundations and obviously of their exposition, I think Grassman again absolutely stands out. I think it's very important. It's always seemed to me that his reputation for wanton obscurity is extremely undeserved because he isolates these absolutely crucial key oppositions between systems and then explores them in a very systematic way, obviously develops the theory of quantity in a way which certainly should be taken up again. It seems to me that is the guiding, probably the best. Even now, after 150 years, probably the best guiding model for overall strategy, for exposition, for foundations of pedagogy that we have. I think, for instance, in your little account of his, Grassman's views of the continuous and discrete, using the example of the, literally, of the wood and the trees, is a beautiful example of that. I hope you'll repeat that at some point in the course of this workshop, if anybody doesn't know it. But that's a perfect, the wood in the trees rather than the wood in the trees is the kind of thing which can get the crucial point about continuity and discreteness across to a philosophically inclined, serious person, even if they're an absolute beginner. Yes, the asking Deng's layer has been translated now, certainly, into a talk about this earlier, not particularly wonderful.

45:00 Which is actually a much better translation. I think it's better. Anyway, I comment, even though it's not the official title. Try it. Try it as best as you can. Very good. It seems that, for example, regarding the philosophical content, someone noticed they had philosophical content. Philosophers have realized the difference between... It couldn't possibly be geometry, geology like that, from Kant, that you fit into. Therefore, it couldn't be geometry. It clearly wasn't philosophy. It was just a very superficial one in that sense. That wouldn't happen now, it's the opposite. You could disagree with Kant that they were making on that. There were a lot more of them in 1865.

47:30 Well this was the original edition, it wasn't the 1844 edition. Of course it would have been even more likely to have been. Of course you misguided the attempt to do it over. I was asking because you, in the Grassman by the Elections, you mentioned the collected works by Edwards, Tully, of Grassman, but it wasn't very... The collected works of Grassman. I'm surprised that Stuartie was still in print. Is he? I have it. Really? It's a black book. Maybe it was reprinted. That came out in the 19th century. It was in the 19th century. It was not long after Grove Grassman. The Grassman died in 1877. If the quagmire works, what it probably saves you is the graph of the dynamics. Yeah, that's absolutely real. That's your paper. It's in the volume. Yes, of my urging you, you very kindly recorded it all. In fact, this is a transcript. This is actually in a proceeding, a very fine proceeding, actually, for the Grassman-Sespa Centennial Conference, which came out in 1994.

50:00 Yes, that's a very, yes, but that's one of the, that's the proceeding, were you talking about the proceedings of the Congress? No, I'm talking about Rasmussen. Yes, and the Alfvénum Flera. Did he in fact publish anything else in his lifetime? I don't think he published, yeah, did he? No, he published. He published some papers? He published a few papers, yeah. Oh, okay, so, okay. Yeah, yeah, yeah. His attorney, as you saw, was a special case of the Alfvénum Flera. Exactly, a special case of the Alfvénum Flera. He may not have a fully grasped, unique importance. But certainly, anyway, he started his first work of the tides describing the tides on the earth. Yes, of course, yes. So is that mathematics? I'll say he was there with some abstract and general people. There are some very interesting things about rotation. Of course. I think it was popular 50 or 60 years later in England. In Maxwell. In Maxwell, of course. Good people worked on this. ...was a subject of widespread investigation. I was reading about this guy, Estenis.

52:30 Estenis, yes. He wrote about this... Yes, he has quite a long paper in that. In Darwin, of course. The door is just on a latch. Shouldn't we lock up afterwards? Okay, but that guy asked us to turn things off, didn't he? Yeah, well... Do you want us to meet up with you and Anders later on? Are there any plans for this evening? I don't know. Yes, sure. Say 10 o'clock again. Yeah, do you want us to exist? Yeah, just come here at 10. Yes, that makes a lot of sense. And, um... Sorry, tomorrow... So it's tomorrow, and then of course you're speaking on the Wednesday, aren't you? I'm not going to do anything on Wednesday morning.

55:00 I think so. 2.30. There's stuff I absolutely must do. If you don't mind me, you were saying at lunch that someone over-introduced you. Yeah, you'll be much more concise and much more informative. I'll try not to say too much. And then we'll go through. So, are you stuck in the room for too long? No, I wouldn't. No, exactly. I think you might not be such a good friend. Oh, so that's yours. Very interesting. I should write it for you sometime. That's all of these.