Naturalness, mathematics, relativised ontology

0:00 But you have to give a foundation for this. And when the axiomatic point of view developed gradually in the 19th century, and when various efforts were made to give an axiomatic foundation, not only for the main or, sorry, metrical geometry, but for the projective geometry, including the point of view, then Newton and Philips were soon realized that the basic notion for the plane projective geometry were points and straight lines and they were... There are postulates of a certain number of properties, to two points you can draw a line, etc. And to check that even axioms are completely similar, you have a set of axioms, and if you replace a point by a line and a line by a point in one axiom, you get another axiom, or at least a consequence of the axiom. And then, by a pure delusion, that's a mathematical, not mathematical, but mathematical theory. And then it's developed. And I think it gave a strong incentive to developing the axiomatic point of view, because you had to make a real axiomatic analysis, logical analysis of your vision. You could not be satisfied by building your vision, you had to think about your vision. Now, similar things occur in the discovery of non-mathematical geometry. People, more and more broad, decided that, of course, You're starting from different set of types, from two lines to one. There are, in some situations, there's more than one pilot or no pilot at all. And then they develop and they consider that by analytical analysis that they want to go into contradiction. The next step is what you could call the beginning of model.

2:30 The beginning of model theory. The next step was a discovery of construction in the same point in the line. No, I mean polar co-ordination. I can quite catch the French. By polar reciprocity. Polar reciprocity, yes. Okay, by polar reciprocity. Then? It's polar reciprocity. Polar reciprocity, yes. Now, no more reasoning about reasoning. It's part of the reflexivity of mathematics. As I said, an important feature of mathematics is that... Mathematical reasoning applies to all kinds of abstracting. The mathematical laws and the language of logical formula or mathematical formula have that function. But then people are immensely happy when one discovers polarity producing a certain geometrical transformation by which, if you have a certain picture with a certain pattern of intersection of marks, you produce explicitly by the geometrical transformation. And therefore, if you have some reasoning about this picture... The transformation, you have the transformation, you have the transformation of the geometry. Each step is a reason. The objectives are there, you transform them, and you get the other one. So, now, the symmetry principle, it doesn't apply to statements, but to mathematical objectives. And in non-Euclidean geometry, a similar development occurred. I heard people like Lopatchevsky were following, I mean... They are following the path of development in ways, assuming that there are no parallel or too many parallels, I'm never drawn into a translation, I'm happy. But then, we discover the many models, the last one was an Ontario model, it was modeled like crazy, like precarious, and so on. And it's a different setting, that you have non-euclidean geometry, it's what you call line decomposition. But now, again, that's a mixture between tactical development and a non-nuclear geometry theory.

5:00 Okay, now, I will finish by a very similar kind of development, and I will draw my conclusion. Something similar occurred in another line of development with the British school of action. I don't know in this country, but in France, I mean, when I speak of... By that I mean Cayley, I mean Bohm, I mean Abisag, I mean Dirac. We could add Bachmann. There is a continuous line of error you can call symbolic. And often in modern physics, Dirac stays isolated. It's really part of this tradition. I mean, the invention of the delta function and all the tricks of Dirac were not born in a dark room. They were born in the good, this time. Of course. Why the first time? My master, George Schwartz, invented it and said, ah, at last, I have a justification. Dirac function and the justificatively heavy side, symbolic calculus. And again, the tool is every other. Gelfand mentioned this in his own textbook about this reduction.

7:30 Provided you discard the irrelevant meaning of functional analysis. And what is the meaning of this line of development? The notion of OPEI. I looked carefully at Boole. When we were studying the book by Boole, I was three years ago in the 48th position. I spent a month in Nagoya University in Japan. There I was on time. And I have three acts, I have a faculty member, I have three acts in the library every time I want it. And I was alone without talking, so of course I think days and nights in the library. I looked to the, all the books by Boole were there. And, well, Boole is, the story of Boole is fantastic. For instance, you know about Mount Everest. Everest was a British geographer. It was sent in India, I mean, to France, India, and in Mount Everest, which I've already named before, in Debatee, Boole married his, and when Boole died, the names that were there, he distinguished Paris as a radical writer, 18th century, and even his thought was in it. But that is the mathematics of it. What you consider as Boolean algebra, in logic, is an application of the principle of operator theory. For Bohr, it's presented in Bohr. It's not usually stated this way. What you have? You have a certain universe. Why is logic a tradition for the universe? And then, what is a predicate? It's an operator. It selects a part of the universe that means the part of the object which satisfies this predicate. Now, what can you manipulate predicate as you want to? It's very complicated.

10:00 Two predicates. What is x, y, u? It means that, by an operator form, it means x, y, u. That's the very definition of an operator. So it means that if you have two predicates, you first select the element in the universe u which satisfies the criteria of the equation, and among them, satisfy x. So, for Boole, the predicate was a logarithm. Boole had written a very interesting book about calculus. Not very, it was quite fairly. I found, in conjunction with what I said, that was a design. At some point, Boole wants to solve the differential equation. And the trick is that he had a logarithm calculation where independent variables, x is interpreted as a multiplication by x, and there is a differential operator. That is fine. Delta of x. And then now something comes through. So we have a non-committal calculus. In another instance, so that's for differential data. In his book on, and the other book on differential equations, in his book on differential equations, he makes a situation where there are two operators to this, and that is this. Now, I know the solution. Since my book on differential equations, that's only on the formal properties of this book, we are actually going to buy algebra.

12:30 And he said, at least now, Laplace has invented a transformation which transforms one operator into another. So, to have the solution I know from my previous chapter, I could use Laplace's transformation to get my new solution. But why? Why? It's not necessary. So, it is the same bool, the same bool, produced the algebra of logic and produced the very bool thing. While I'm in logic, as I said, after a bool we have, after a bool we have Lewis Cowell. Everyone knows about Lewis Cowell, books in logic and... And also, and also a book on determinants, which is one reason that still today people use a certain identity among determinants, which is also an identity theory. And then Eliezer did a similar thing, and the analysis of differential equations by Eliezer is strongly, I don't know, trustworthy. And as I said, Girard also was strongly inspired. And of course it's interesting to say that the invention, the statement of Boole about The Laplace transform reminds me of what happened with the invention of distribution. Again, in a sense, distributions are integral transforms. They simply are integral, a kind of integral transform. And Fourier transform, Laplace transform, integral transform, convolution, and so on. So distribution theory is a very systematic view of integral transform power, or transform that you pretend to be. And that was used to substantiate reasoning by Elisade and others. But 20 years later, and even more now, I mean, with the invention by Sato of D-module, as there was Sato at the Japanese school of D-module, we are about to learn from this kind of development that it's totally acceptable in mathematical reasoning as mathematical. And that's the reflexivity principle of mathematics.

15:00 Mathematics can reason about its own modus operandi in mathematical terms. You mean that you should get metamathematics, and in my opinion, metamathematics is not some kind of superstructure of mathematics. It's a genuine part of mathematics, dealing with certain kinds of problems. Now about the ontology which is behind this. To an ontology of what we see, what do we see. And strings of symbols. And of course, now the basic mathematical object, the basic foundations, philosophical foundations of mathematics rest on the manipulation of logic and metamathematic to reduce every mathematical calculation. It's not enough to say so. It's not enough to say so. And, but the benefit... Well, I don't have to repeat the benefits of this approach, all modern developments in literature and programming memory are based on that. In terms of academic discussion and academic polemics, I remember there was a time in our academic system where number theory was considered as pure proof of mathematics, and then probability as the most applied part of mathematics.

17:30 Logic was considered a few of them, but now, over the years, we have seen the further checks, especially the French school of probability became more and more abstract, coming back, that's more and more abstract. On the other hand, number of theory applied, number of applications, and formal logic became hard. Now, in terms of what justifies a mathematical activity, I would say that In part, if you want something complete, you need symbols. But on the other hand, mathematicians, being what they are, need them. So, in a sense, it's a potential infinity of formulas. I mean, you give words to generate new formulas, but a mathematician cannot. The actual infinity of all these formulas, whether he is able to write them or not, and to make very general reasoning about formulas... I would say that, as a conclusion, I would say that modern mathematics evolved towards a very gigantic, and that would give me a transition to the category of mathematics. It seems to me, in line with my generality, that mathematics is part of my heritage. And, of course, when I said, when I said we manipulate string of symbols, that is, string of symbols are usually, they're safe, and what is important is to, and the same kind of problem occurs in biology today.

20:00 With molecular biology, well, there are the list of the genes, I mean, there are the long sequences of these four. The problem is that, the problem is that these long sequences of symbols in biology have a meaning that we don't know yet the meaning. And in mathematics, if you would reduce mathematical activity to the production of long and long and long and long lists of symbols, and in terms of metaphysics, I have to admit that I cannot give you a clear answer. I say the main purpose of mathematics and the main pride of mathematics are beautiful tactics. From a metaphysical point of view, it's quite difficult to understand. We can discuss it. I don't think they are dealing with a few. Because these patterns are created and developed. They don't exist per se, somewhere. They are developed and created. Now, where, once they have been developed and created, where do they stay? I don't really know. Maybe it's a memory of mankind. Maybe it's a seed of mankind. That's the thing I just wanted to mention. Thank you very much indeed. Since we started a little late and I took up some separate minutes with the introduction, we do still have some time for questions, so perhaps we could take a short break. Thank you, very nice talk. I'm trying to come to terms with some of the things you said. Walking in here today, if you asked me what is mathematics, I probably would have said maybe something like all possible abstractions of numbers and properties of numbers and things that could be created from numbers, but of course you destroyed that whole idea with that. But let me ask you a question. If you say very simplistically, geology is about rocks and biology is about living things.

22:30 But the tools that the geologists would use to analyze rocks are not part of geology, and the tools that the biologists would use for analyzing animals and living things are not part of biology. We sort of have, in a sense, artificially sort of imputed to mathematics this self-reflexivity which maybe leads to category theory and things like that. And two, talking about the methods and using the methods about the methods and drawing some. In a sense, mathematics being self-reflective in a way that biology cannot. My deeper question is something else, maybe you can address that too, but my deeper question is obviously a philosophical one, which I respect, but I'm not sure I understand. You said all along that mathematics is part of our culture, and I agree that perhaps in adding tools and using tools on the tools of mathematics is perhaps... And maybe Zermelo-Frankel is, in a sense, cultural. It's not something that would be true on another planet. But if you take something like Fermat's Last Theorem, could you argue that Fermat's Last Theorem is not true on another planet, another universe, where it's totally divorced and has no relation at all to our culture? And of course, throughout your talk, you used that by saying that Schwarz invented distributions, but I would say he discovered them. I knew that I was on a very steep edge. I said civilization, I did not say society. That's why I'm against all these sociological approaches, which I certainly do not support, and if you believe I was supporting this kind of thing, I don't, I don't, I don't. Maybe a different solution, but the problem is different, I think. So it's not a question of sociology. Civilization is a thing. You say culture, and you say culture, I mean, I understand all, well, I know all this discussion, present discussion about multiculturalism and so on, and I'm not interjecting it. So I can reassure you that I'm not interjecting it. I say civilization on the bottom. That means civilization for me is a totality of all cultures, not a special thing.

25:00 Of course, it's a world culture, you have mankind culture, but as far as we are not conformed to the pitfalls of mankind, we can be happy. No, I'm certainly not indulging in it. We have quite a few people who want to ask questions. First Professor Hindsicke, then Professor Lorbeer, then the gentleman there, I'm sorry, then your name, and then this gentleman. Thank you. I like your talk very much. I think this is the real philosophy of mathematics, and I'm like, what goes on with this thing? But however, I think on a little bit more technical level, I don't think the picture, I think the picture that you've painted in some ways is changing. And the main point that I'd like to suggest is that axiomatic symmetry in its present form is a lost cause. Very good. No, I agree. And in a bad way. It's limited in many possible ways. Not just by the paradoxes, but in a much deeper way. By the incompleteness. No, that's not the problem. That is not the problem. They have many of the competencies. And also, in my mind, I don't think there's absolutely no reason to say that the continuum hypothesis is competing with problems and so on and so forth. I can't give the full technical reasons, but... But I know that you had a lot of ideas about that. No, well, look at the situation. What has to be proved is that the continuum... But that would be shown that continuum hypothesis cannot be proved on the basis of zergomorphic recollection, but that's all that follows, and if the zergomorphic set theory is as bad and the open mark as it is, this doesn't mean anything about the probability of the zergomorphic, or the continuum hypothesis.

27:30 We first consider the theory of the natural and the effect of the natural at the order of the network of the system. Then, the interdependence, the parallel system, the scope. Then, the applied, the good and the bad, what is the mitigation, and so on and so forth. And then the cause is that this system is logically contained according to... Well, I agree that the Central African centuries are much worse off than that, and that's not just incomplete, it's wrong. Give me an example. Everybody says that the axiom is the axiom of choice, but if you look at the intuitive content, the structural content of the axiom of choice is without any limitations, that's incompatible. That's one form of the axiom, which is incompatible with the seminar on mathematics, although everybody pretends that they're telling the truth. I'm not going to try to give you an approval, but this is something that is well known, that we seek. And also, you cannot use axiomatic set theory to theorize about itself. I think I like your point about atmospheric spinning. Also about the capital itself. But in that regard, precisely in that regard, Mr. Lavaterka's set theory is hopeless. You can't define set theory as a whole in set theory. Well, there are a few questions I haven't answered before. Okay. I think, obviously, a very full discussion of Professor Hinton's points would just take us too far, and obviously we can continue. Professor Law there. Yes. Well... You said you started from a monist position. Well, the monist position historically, which we've started to tell, always ended up with the subjective idealist conclusion, like your conclusion that what you see is just a symbol, a string of symbols. But fortunately, you then said, well... I don't quite believe this objective idealism. But then you raise the spectrum. Well, the alternative is objective idealism.

30:00 I think philosophy has been much too preoccupied for a long time in pretending that this is the real dichotomy of objective idealism and not objective idealism. One leads to the other always, anyway. And this is just to avoid talking about dialectical materialism, namely the fact that in the case of mathematics, mathematical concepts come from the material world. They are, what the symbols stand for, they are both invented and discovered, mainly invented collectively over a period of time, and if they become explicit, usually discovered by one or two individuals after they have become developed. So the fact that there is conceptual content, and it's not in heaven, it's in the history of science. It's ignored by the state, and I quite understand why. I think that's what I'm trying to say. It's part of my philosophy. I think it's very important. I think that's what I'm trying to say. And I think that's what I'm trying to say. I think it's very important. Well, whether it's in the genes or in the library, I think it's both, and it isn't really relevant to the main... No, no, I think that's what I'm talking about. But you have to keep in mind that when I sketch, I mean, what is the meaning of communication? Our very function is more to now than we have changed over the years. So, about communication, that of course, I mean, what is fundamental is two, I think. Mathematics is very important, fundamental is two. I would agree. I wanted to make one other point about why truly the strings and cymbals are not the primary mathematics.

32:30 These are really tools for communicating the concepts, and this is illustrated, can be illustrated by the fact that many times we define a particular concept in a way that is not independent of presentation, that is not the presentation that compers the concept that is presented to us, typically. Now my conclusion was very good. Thank you very much. Thank you very much. It doesn't satisfy me, though. I mean, you know, there are people, other people, who say it's quite irritating, because there are all these five and big systems, and you have to deal with one, and you have to deal with another, and try to be cautious, and to be cautious means that I'm taking into account this whole process, and truth, or if mathematics is, or not, as part of the activity of mathematics, can be survived by intuition. We have time just for a few more questions. If I could ask the questioner if he would agree with the answer. Wonderful and wonderfully accessible time. I have four ideas, so I'll only stick to two, which you can answer. So in a recent Mathematical Autobiography by Conrad McClain, he said, one thing about Robachi was they built the foundations on a set, but then sets got outmoded, and they never went back into the world of foundations. So they were using atmospheric theory. So I guess I have to wait till tomorrow for the second question is, try to keep it brief, Alain Kahn. So I think once you get to 19 years of different geometry, the whole connection back to the future seems to be done on the scale of the thing. It seems to break this, so you have this thing of number versus geometry, one follows the other. So what's happening now? Operator algebras rule the world, and whatever happens to pictures? But we need pictures. So what happens then?

35:00 Well, I gave you the diagram, you know, you can... And you can calculate the diagram now. So, it's coming back to the talk I gave some months ago, but we've got two more to go. Well, since you were so good, I'm going to give out your first point. One more. Duality, from projective geometry to reversing arrows and pictures. So how big is this? Or do you see a coherency in this version of duality? Oh yeah, of course, of course. That's it. I just cut. I just wanted to give one example. I don't want to say it's an important thing, but I wanted to make a part of this quite relevant in my discussion. Well, actually, this gentleman right here will answer the next question in line. If we have time for one more, we'll get it in. First of all, it's a great honor to share with this town a vision for mathematics and mathematics. Well, the number of concepts that we throw into this very large number, and basically I have many questions that you don't regret, but it's so rather fascinating that your talk itself sort of answered some of the questions which were coming up in the course of the talk. For instance... There was a lot of talk going on about duality and then you come up with a whole chapter called duality. So it was a very interesting progression from the talk. And since there is such an overlap in your thinking and mine, I would perhaps suggest reviewing myself to a very practical question.

37:30 But even before this, all notion of coming to me is sort of prehistory. And there was no... And when the notion of set came within a counter, there was a certain history leading towards it. It did not come out of nowhere. But I didn't understand it, so I will not say too much about it. But if we take it from there, and then the whole Wilberti group comes into existence, and wishes to base, say, mathematics on it. For the purpose of unity of mathematics. However, my practical question will be this. Since the Bourbaki group was made up, was comprised of such great mathematicians, who obviously knew mathematics so well, the history of math, in particular, I consider the Bourbaki world of the business class, I think, the world health. With the exception of the Lee theory. Okay. If those mathematicians, such as Andre Wey, or Cartan, or you name it, could perceive the unity of math in such methods as duality, reciprocity, introduction of ideal elements, which are basically the implications of light in this world, continuity in this house of knowledge, rather than regularity, so if all those mathematicians could perceive that, why would not the base... Their approach towards the unity of math on those unifying concepts in thinking, rather than the unifying concepts of set, which really gives you nothing, doesn't give you any ideas, just gives you a platform to work with, but not a... You see the question, right? Well, I think it's a historical debate, I think, between Ray and Grossman and Errol. Okay. What you actually need are the fields of Rotman. But Rotman was a philosopher who was a practitioner of mathematics. He knew a lot of mathematics. But he was not a practitioner. He was a physicist and he was a philosopher and a mathematician of multiple analysis. And I remember this question about Leveille and other conflicts.

40:00 And I would say that Leveille would consider that the approach would be too allegorical. Really? And they did not function in practice. I mean, he wanted to do more practice, but that's it. Well, and they were good. I mean, they were good. Philosophically, they were perfect. But what he wanted, he didn't do it, but he wanted students to do it. And... But you mentioned that in the discussions of the students... ...over the notion of algebraic manifolds. Already the point of view was coming out of the discussions that a manifold is not a collection of points, rather it's a collection of nested mathematical structures, and then of course Gratendi takes it wholly towards the functional point of view. So if that was available, was it considered too esoteric for the genuine mathematician? To take it at the time? The point is that the 3D has been developed already to measure 25 bodies, to measure 150 bodies, to measure 150 bodies, about 40 bodies, and then you can read in the account given by Armand Borel And you will see that at some point you have to make a practical distinction. We all knew the physics of people at that time, but the physics was well known and they were head if God only wanted us to start from the beginning of the world.

42:30 It was surprising to me that Andrei Beg himself was such a practitioner of duality. Even his exclusive formulas in the Riemann hypothesis, where he replaces zeros by some summation by numbers of the primes, I mean, his whole mind seems to be driven, and his approach towards Riemann hypothesis on curves, it's all about duality. It's all driven by duality. Why not put it in the foundation? Maybe he was jealous of this concept and wanted to consume it from the general public. My question is, how do you write a treaty, a global treaty of mathematics? If I can just plug in here, this is obviously a fascinating dialogue and I would love to have you to privilege the chair by putting in my own point of view, but I think we have to draw to a conclusion and we have just time for one more very short question and I'm going to give it to you. It's not a question. I just want to say that I like to tell you that there are many things in mathematics to restore the harmony, inventing things to restore some harmony and make the world a different future. That's what I'm trying to say. I was just going to say, Jean-Michel, did you want to... We just have one time to speak. This has to be the last question, I'm afraid. I was not expecting the conclusion you gave. When you started with duality and poncelet, it was the introduction of the individual practice into mathematics, and I thought that talking about Boole, you would also mention laws of thought. And I think that there was another possible conclusion of your speech, your speech of course, I like, that it's a return, there has been some kind of return of the individual practice of mathematicians into mathematics.

45:00 I'm not thinking about meta-mathematics, for which we will all agree, but I think that to ask a question simply and to try to refer to what I mean, maybe I'm influenced by my current work on Russian... You said mathematics is the creation of patterns. So the way, if we don't go into detail, one can imagine a very big computer producing patterns for us. So what will be your philosophical answer to this? Well, of course, the problem is... It has been known since infinity. In invariant theory, people claim that now we have the method of invariant in the 19th century. We can just put together a number of formulae and it would produce automatically a number of general equations. But which one is really interesting? And I did not answer this. This is the basic question. I mean, if you have a way to generate more or less... Automatically I mean the set of patterns. How do you distinguish the really interesting patterns? Why, also in a sense, what is the role of natural selection in concepts? And I hate the question, I don't have the answer. So it should be supplemented by a notion of natural selection of concepts. I think, like all the best discussions, this has raised far more questions than it's answered, but I'd just like to thank everyone to thank Professor Westcott again for an absolutely magnificent talk. And I urge everybody to come to the two days of our colloquium tomorrow to pick up the program and jump outside and you'll hear a great deal more about these matters. Thanks very much for coming.