The continuous & the discrete & set-theoretical thinking

0:00 The emergence of ensemblistic thinking is part of the theory of ensembles by looking at the elements that emerge by looking at other theories than those that are most likely to be studied when we study the emergence of the theory of ensembles. So to see if the emergence of ensemblistic thinking has not taken place elsewhere than the emergence of the theory of ensembles is to study the link between the two. Thank you very much for your attention and I hope to see you again in the next lecture. Recently, there is a conflict with the direction of the seminar. So, as you say, as you stand here today, we decided to do some reflections on connections between the emergence of theoretical thinking, which is Not exactly a centralized set theory, with issues related with geometry and also some other connections, of course.

2:30 The general question of relations between geometric thinking and theoretical thinking, I find... Thank you for your attention and see you in the next lecture. I'll try to decide from there what should be understood as theoretical thinking, what should be the contrast between that kind of way of thinking and geometric thinking. Of course, this would be a critical way of understanding the problem, and I think it might be misleading, There is a very complex and rich background, and so this kind of work has lots of difficulties and lots of rejections. So, having said that, my... Let me begin with some general reflections, and then I will go back to the history and discuss a couple of particular examples in a little detail, not much detail. So, some general reflections to begin with. One important thing is that this theory is about sets, but also about mathematics. So it's too easy to forget, I guess, and maybe also given the way set theory is discussed, the way it is presented in lectures, it's too easy to forget that the concept of set is quite a sophisticated concept and quite on the contrary there is the usual tendency to say, well, you know, this is a very familiar concept. It's just, take the idea of collection.

5:00 And then that is basically what sets are. From a philosophical point of view, there are too many difficulties for that approach. The differences between collections and the normal understanding of the world. And sets are too many to forget. So this is, of course, good policy if you want to teach students mathematics in the early years. A better way of just keeping things in mind is that, of course, mappings can be defined on the basis of sets, so it shows that there is some interesting relations there, but also in the other way, as you may know from Neumann with his axiomatic... The way he presented it was by axiomatizing a notion of function, and on the basis of the axiomatic properties of those functions, he developed set theory. That was published 1925, 1928, in 1920. One can say that the concepts of sets and mapping are inter-defined, but just to give some background to what I said before, I mean, the difference between collections and sets. Just to give a very simplistic, a very simple example that I feel it's enough. Yeah. Have you got something darker? Yeah. It's impossible to read that. As far as I can think, the question with sets is that we want to be able to have sets and sets of sets and sets of sets in all kinds of ways.

7:30 These are things you simply cannot do with collections, if I had any collection of other clients and I tried to, I think it makes no sense at all to start doing this kind of further constructions in the basis of it. And of course this kind of further construction is essential to be able to define any function concept on the basis of physics. So this belongs, of course, it's just a very simple example that one can give to first-year students, but to make them reflect that the notion of set is far more complicated than any familiar concept we use in normal life. Okay, now a very simplistic idea that one could use to begin with, just to throw it on the table and use it. I would say that the concept of Z has an arithmetic origin and has been particularly connected in its development with issues in life. And one could say, too, that the concept of math, mathematics, has geometric origins, or at least has been strongly connected with issues in geometry and analysis. So that is still simplistic, as I say, but it contains a grain of truth, and at least it helps avoid being even more simplistic than that. That is not a kind of point of view that I can actually defend, but what comes now is perhaps something that I can defend. I would say that the origins of the theory are quite intimately related to a hypothesis, something that I cannot call anything but a hypothesis, which is the idea of the line

10:00 The real line can be regarded as a policy. Of course, we know this is a very contentious idea that has been discussed since antiquity, and Aristotle is famous for having rejected this. And again, in the 20th century, Raoult is famous for having rejected this and developed another way of understanding the continuum. So this is perhaps, to a good extent, the ground hypothesis behind the whole development of theoretic thinking in the style of Cantor and also in the style of Delecq and so on. So this, of course, connects both with the emergence of the problem of the continuum problem, the problem of theory, the continuum of hypothesis and control, and the difficulties connected with it, but also with the older problem, the no less basic problem of how to define the continuum, how to define... Asymmetrically, the line and how to define the real numbers. When this problem was solved in 1872, I kept thinking that that was a particularly important time in the development of set theory and set theoretic thinking. Rational numbers plus theoretic constructions. This is then 1872, you know, the well-known definitions given by Macquarie and Byron, and this at the time led to hopes for finding some common measure This is the reason behind the title I have chosen, you know, the title reads continuous and discontinuous measured by a common measure, but actually this comes directly from Cantor in a paper of 1882, he wrote

12:30 With the new ideas that he was presenting there, the discontinuous and the continuous can be regarded from the very same viewpoint and can be measured in a common measure. It's a very beautiful idea, of course, because it's the hope that the problem of the incommensurables will finally disappear with these new developments. This is the first position in the whole book, so can... Yeah, that's in the first place. I will come to that location later. Sorry that I... I will come to that... I will come to that... I will come to that... There were only 15 copies, so maybe there are... This project, the whole project of just making river as a number system and then being able to have some kind of homogeneous treatment of the whole system from the basic steps of the integers and so on all the way to the real numbers. This course, I believe... The project and the idea of developing it by means of using steps forced a dual line of attack from a methodological point of view because it was necessary, of course, to develop some general set theoretic ideas which ended up leading to abstract ideas in set theory like those of the theory of cardinals and ordinals and all that. which we typically associate with the names of theory, but it also of course you have analysis of the topological properties of the real numbers and this is something so we have both a more geometric line of attack meaning not geometric, geologic would be because it is not metric but it is a topological kind of approach.

15:00 And the other is the more abstract kind of line of attack. And this dual nature of early investigations of the theory can be seen, of course, obviously, in Cantor. This is no new, especially during the first ten years of his studies. He had a very little, by way of abstract sources for general theory. There are a lot of developments related with theory of concepts and what we will later recall as twistor theory and so on, and he just made no distinction. There is no distinction at all between the use of concepts of one kind and concepts of the other. There is no distinction at all with Cantor. With Cantor, in the first, say, ten years of the development of his work, he has a lot of work on cardinality, of course, but it's always mixed with the study of policies on the basis of the political properties of the policies. And it's only much later, with Cantor, the real change comes in 1883, when he introduces the trans-finite ordinals, this new idea, this is how this makes it possible to develop a full theory of... It's only at that point, 1883, 1884, 1895, around the middle of the 1980s, that he sees for the first time the possibility of distinguishing between topological issues and theory of concepts and all of that, which now he will start seeing as more particular applications of the more general ideas, and the general This comes very late, because he has been working on it for more than 10 years, I think. Now, with Dedeckin, interestingly, and this is perhaps much less known, we have also the same kind of dual nature of investigations. I will discuss that a little bit later, but he, in fact, did some work on what he...

17:30 It could reasonably present as issues in topology. And of course, it's more famous for the other part of this work, the work that has more a general kind of associated with generalist theories. Well, just to finish a little bit with this general part and comment a little bit on the idea of the common measure. I guess from today's point of view, one could say that hope for a common measure along that kind of line has disappeared to some extent. And this is simply the whole problem of the continuum. It's enough, I think, to present the ideas about it. It's enough to... We can defend, at least, that this was not the way towards a common measure between the continuous and the discontinuous. I mean, well, this could be developed much further, maybe, if there are questions or something. But one can say, for instance, that the... The insistence on the possibility recalls what I call the ground hypothesis of understanding the line as a set of points, and that we just take for granted that this is reasonable to do. And actually, there's only once in his papers that Cantor raises the issue, but he doesn't come into discussing the issue. I take it that because he can't. It's just simply his ground assumption. I mean, there is no way. You try to discuss that, you're lost in all kinds of difficult philosophical speculations, and nothing really becomes rational. So, well, recalling that assumption, once you start thinking that way, of course you are forced to accept infinite sets, that is a very simple idea, and you're even, you will eventually be forced to adopt an axiom of power sets, because the kind of definitions of the real numbers you can do...

20:00 In the end, they will depend on the strength of principles like that. On the other hand, well, you have the other kinds of inputs into the theory coming more from ideas about the natural numbers and these developments and so on. And this, when you combine with this assumption of infinity, then you start thinking about order sets and well-ordering and things like that. And so on. But in the end, since even such a simple problem as what is the cardinality of the totinium, what is the position of the totinium among the scale of ordinals, It is presently regarded, at least by most people, to be unsolvable, or take anyone beyond, except these few ones that cannot be like Phoenix. And so on, take any one you want. Most people just assume that this is simply beyond the problems that the axiom system has to study. In a way, I find that extremely unsatisfactory because, as you can prove, that the cardinality of real numbers is exactly the same as the cardinality of the power set of n, right? In a way, it just emphasizes, again, that we don't really understand what we are talking about when we are talking about all the subsets of the natural numbers. This first part has been more, as I say, a general kind of reflection and perhaps a bit more philosophical. Now I would like to go back to history, then, and discuss a couple of examples, basically, that are relevant. If we start from this general idea suggested by the title of finding a common measure for continuous and discontinuous, well, it would be easy to offer quotations from Riemann about this famous lecture on geometry, showing that he has this vision, if one can only call it a vision, for a future theory that ought to be possible to develop.

22:30 And a theory of what he calls manifolds that would unify the discrete and the continuous as you have it here. No, I haven't because I just wrote here that it would be easy to have that kind of quotation and you can... I think you won't find it yourself, so just look for it in my book, but he has clearly this vision. It's clear that his concept of the manifolds is not only potential manifolds by no means. He wants to talk about discrete manifolds and continuous machines. But I decided when I was preparing this, and maybe because I Well, I didn't know exactly if you wanted me to discuss the issue of Weymouth, but I just was not thinking along that line when I was preparing the talk. And so I decided to concentrate more directly on Cantor and Deleon, but perhaps the way I will discuss it makes it clearer for me. So let's begin with Cantor and let's begin with Deleon. Oh, by the way, I have another... Take this text. If one wants to see connections between Penrose's ideas and geometry, well, these are not the only texts, but it's quite interesting to look at papers he published in 1982 and 1883. I'm just saying that this is exactly the thick of this production. There is an amuse-murality for Cantor that is 1882, for the richness of the papers he published and for the richness of the new ideas he had, because it was exactly in that year that he came to this wonderful idea of the trans-linite terminals and the connections between the trans-linite terminals and the powers of cadmium.

25:00 Well, in one of the papers published that year, he writes as follows, the theory of manifolds, he gives there a very general discussion of what just general ideas about what sets are and so on. You will see there the word manifold because I was just literal in translating, but of course it means set. The theory of manifolds, or sets, as it is conceived here, Embraces, if we restrict our attention to mathematical fields and provisionally disregard the other conceptual spheres, the domains of arithmetic, function theory, and geometry. It reunites them into a higher unity on the basis of the concept of power. I should have updated it there. Power is a metric type in German, and it is the quite characteristic word he was using for what we call the cardinality. Which for him was the most general concept in connection with quantum theory. I included there a literal sentence that I like which comes from the other books. It is the most general genuine aspect or moment in a manifold. By that he means that it is an invariant property that will remain. Whatever you do with the set. It's only by changing the set, by adding new elements or leaving elements out, that you can change the cardinality. So he has this vision that, by now, a few things are here. The theory of sets is quite a general one. It goes beyond mathematics. And actually in some other places he said that there are connections with epistemology and logic. So probably at this point in his life he was still close to the ideas of the logicists somehow. I think that even by Estras he was somewhat close to this kind of view. Later on Kantor will separate himself quite clearly from logicism.

27:30 But he proposes to restrict our attention to mathematical fields, and then we have basically the domains of arithmetic, function theory, and geometry, which are going to be those in which the theory applies, and that will come under a higher unity. And this is where he says that discontinuous and discontinuous can thus be regarded from the very same view, or can be measured with a common measure. And he goes on, and a bit later remarking that arithmetic and algebra offer many examples of the enumerable sets. The enumerable sets, as you know, are those that are countable and can be correlated one by one with the natural numbers. But he says, no less fertile is geometry. And he gives here an example. Well, this is, to some extent, this may have been a... A little bit of a rhetoric way of introducing this particular theorem, which was to be very important for the further development of these studies of point sets. And the thing is that the paper that I am quoting belongs to a series of important papers on infinite linear point sets, because I remember well six parts, and this is the third part. And at the beginning of the series of papers, he had decided to concentrate only on subsets of the line. This is what linear points mean, the subset of points in the line. But now, in order to prove this particular parent, he needs to go up in dimensionality. He needs to work on the n-dimensional spaces. Part of what has been presented is a rhetoric discussion of how this theory of sets is not only meant to apply to points in the line, but also generally to all kinds of geometric issues. In particular, we find among the interesting topics there, things like this theory. Which, by the way, I could say much more about the theorem, but anyway, let us really, given a non-dimensional continuous space that is infinite in all directions,

30:00 infinite in many continuous subdomains, so it's A there, the idea is simply that we have here, well, think about closed subsets there, And they have to be continuous of course, closed because at this point his life is basically, by the way he analyzes point sets, has the effect that he's not considering open sets. And so the condition is that these subdomains are to be disjoint from one another and touching each other at most in their boundaries. Under these conditions, he can prove that the manifold A of such subdomains is always enumerable. This is a very interesting result, especially because it will lead to many important... I mean, it's like a domino effect. And the proof he is going to give will rely on... You take this set of n-dimensional subdomains of Rn and what you will do is to project the subdomains onto the unit hypersphere in our procedure for uses because then he can, it's interesting, I haven't really tried to make all the little details exactly why he needs to, that way he can have a good definition of the volume of the images of the subdomains and can As you see there, and perhaps just to make things clear for everybody, whenever you see the word manifold, simply understand safe.

32:30 There is no special complexity there. It's simply that in German there was not a particularly good word. For example, for example, people were saying a little bit too long, so after a few years you are tired and you want a short word, and you end up saying Menger, because that's short, but it was initially a very bad meaning for Seth. Menger means, so it's a very bad word to begin with, but in the end quite natural. So as you see here, it's just an example in which he will apply the kind of thinking that for some reason he regards as particularly connected with geometry. This idea of having continuous subdomains and predictions from one space to an embedding space and things like that, he's presenting as particularly geometrical. This theorem is very interesting because along a different line I believe that there are good reasons to think that the way he came to this theorem is strongly related to his speculations about the way in which the theory would promote the understanding of the natural world. So I discussed this already. I was mentioning this particular theorem a year and a half ago, but in connection with a completely different line, which is this natural philosophy of Cantor and his works. Because actually, in a letter to Gould, he presents this particular result as follows. I can prove that the number of cells in the universe is the number of them. Excuse me, I didn't quite catch it. The number of what? Cells. Cells. Cells in the universe. Cells, as in? Okay, I understood. He means it specifically in the biological context. He's not using it in a metaphorical sense at all.

35:00 No, no. Absolutely. Really? Interesting. Okay. So this is quite interesting. And if you want applied mathematics, you have here an interesting, a beautiful example of applied physics theory. No, no. It's not a metaphorical sense because it is. Okay. So, but let's go back to the topic. How do you say cell in German? Selling. It's absolutely specific and clear. And actually, you can find the letter in this edition of Letters. This is one of these little cases in which Cantor may offer more than he really can give. The editor said that he cannot really prove it, but it's only because they haven't seen the connection to this paper. He really cannot prove anything in the 1880s. There are also relevant passages about connections between set theory and geometry taught in one of the next papers of this series, the one which Cantor also published independently, the famous moon lagging, foundations for a general theory of manifolds or sets, which is where he introduces the transfinite orbitals. So, for instance, the very last sentence in the paper, this is just the last sentence in the paper, says, with these concepts that I have placed at the top of the theory of manifolds, I render myself answerable to the investigation of all the configurations of algebraic and transcendental, all the configurations of algebraic and transcendental geometry, according to all of their possibilities. In doing so, the generality and precision of the results should not be overcome by any other method. Well, this is simply a case in which he seems to be promising further developments which even come. But anyway, it's interesting. And the particular place at which he is making these promises is an old...

37:30 To his general definition of what a continuum is. So the paper is quite a rich one, it's a very interesting one and a rich one, and it not only contains things related to the transfinite ordinates, it also contains, presents again, for instance, the theories of the real numbers. And then he comes to what was an important part of his, to have a general definition of what is a continuum. And so this is, of course, important in the development of topology. And he has an end note adding new ideas about how to deal with continuous sets and those that may not be semi-continuous and so on. And by the end of this note he introduces this last sentence, which will be the sentence that closes the loop of the paper. Okay, but in this case, I don't know more precise aspects of what we have in mind, so we just have to leave it as just a problem that was not really that we were given. Now let me mention a few things, abandon Cantor for just a few moments, and mention a few things about Heideken. Hawking, with some frequency, is intentioned to establish theoretical concepts as a basis for rigorous development of geometrical ideas, especially if we consider topological concepts. An example, if you read carefully, you already have indications in the famous paper on multirational mathematics. The definition of the continuum by means of Katz, when he gives the definition of this as an analysis of what we mean by saying that the line is continuous, he adds that this is a rigorous basis for the investigation of all continuous problems.

40:00 Maybe one has to know the kind of man that Dilipin was. If he underlines the word all, he's really meaning something. It's typically his careful way of expressing. It would not allow him at all to use the word all and much less to underline it if he were not meaning that you can apply this to n-dimensional spaces. Of course, the way in which you can apply this to n-dimensional spaces would have to be an indirect one by the introduction of coordinates, because that's the only way it can make sense of it. But it's clear that he had in mind this kind of connection, and it's interesting because he had previous experience in trying to analyze n-dimensional spaces. This is something we know only a minute ago. This is something you can read about in one of the letters to Hunter, but actually Emmy Netter found the manuscript and it is published in the collective papers. And a paper that I give you the name there, this Aliamanis et se judo ordine, General Theorems and Spaces, it's quite interestingly a manuscript that can be dated to the middle of the 1860s. This is my own dating. On the basis of all the evidence we have about what is going on, but there is very, I believe, very good reason to, one can be sure that this is before 1870. It's quite interesting because at the time when it arrives, it's introducing the very first idea about some kind of topological issues concerning the real numbers, like a limit point. We have working on this manuscript, which is quite interesting for the generality and the very systematic way in which it's presented. It's only a few pages, very few, but he gives the concept of an open set and of interior point, exterior point, and boundary.

42:30 Very systematic, almost axiomatic, if you say, in fashion, in the context of n-dimensional metric spaces. Why was he working on this? We know that because he tells Cantor that there exists a manuscript like this. He will never send it to Cantor, and Cantor didn't ask to see it. But he just mentioned the existence of this. Dirichlet's coalescence, or after 1863, he started to work on publishing the lectures of Dirichlet on potential theory. So he changed topics completely, he was potentially, and he wanted at that point, because he was aware of the problem already from his time at Göttingen, that the Dirichlet principle needed a better grounding, he wanted to give a proof of the Dirichlet principle, and for that reason of course he needed some kind of basic concepts about continuous domains. And this is the reason why he will start doing quite systematic and what I find more interesting is that he's talking about this as an autonomous study of political issues. There is also the background of Riemann, I believe, of course, because Riemann, I agree with him. He was working on that project from 1863 to a few years later, so we know, we can be quite certain that he He has written that within more or less five years I will be there, because later on, after three months, he will start work actually on differential geometry. So it's interesting that we see Delekingne as a number of theorists, but this is a retrospective thought in many ways.

45:00 I mean, for somebody who may have met him in 1869, it's quite unclear that he's a number of theorists. Sorry, just purely on the point of information, can you remind me how old Dedekind was in 1863? Must have been very young. I was thinking just how old he was at his death. I was thinking he was much younger than that. So that's the context for this kind of idea. So in 1872, Deligant writes that he's proposing his analysis of the continuum as a basis for the investigation of all continuous domains. There is something behind it. One thing that I find interesting is that both Dierken and Cantor have had some contact with the project of geometry. Let me motivate this. Why is this important? Well, I believe that for the development of the kind of theoretical analysis of problems in topology and problems in, well, even in connection with just the real numbers, the understanding of the real numbers, that these people were developing, it's quite essential to have gained distance to older ideas about geometry. The reason why I think this way, what made me think of this for the first time was especially Bolzano.

47:30 If you compare the ideas of Bolzano, you all know that Bolzano wrote things somehow related with set theory. He was using several different concepts in different ways. But he also considers whether one-to-one correspondences can be a measure for the size of the set that he basically creates. My belief is that this is mainly because he's still too close to eukaryotism in a way, or to the metric conceptions of geometry. So he's not yet prepared for a more abstract kind of approach, which means a more topological mode summary. So that was just a short remark in brackets, but this is the reason why I first came to think of the importance of distance from geometric thinking for the possibility of developing not only topology, which is quite clear, but also all these kind of more abstract ideas that Cantor would be developing. So, from that point of view, it's interesting that both Debekin and Cantor had contact with geometric geometry because already there, of course, you are gaining a lot of distance to metric. And in the case of Debekin, his very first lecture course, which he became a professor, was on geometric geometry. He was trying to see whether it's possible to make a combination. And we know that he was reading Möbius especially carefully. Anyway, one of the things that I find interesting in this is that the concept of bijective mapping is emerging very clearly in the work of the physical geometry. Of course, in the particular setting of predictive correlations that we have the concept of mapping, and this is very clear in Möbius, it's very clear in other people too.

50:00 This sentence from Kantor, again from the same paper, where he, as you have in the other piece of paper, explains why he has been using the word power, which is a funny word for the cardinality of the second. I have taken the expression power from Steiner to use it in the very special meaning that however is clearly related of expressing the two configurations. All of these are related to each other by a predictive correlation so that every element in one of the configurations corresponds to one and only one element in the other. This is actually the case with the... The reference he gives for this is the correlation game that we have there. But this is already here in Steiner's most important work, which is the one... Do you mean that when there is such a projected relation, Steiner says that they are the same in this way? Is that what... Yeah. Actually, there are many interesting things about this perspective. This kind of idea, as I was saying, is already found in the systematician, and even though there are a few different types of mathematics that are coming under, this is the most, I believe, I know little about it, I know it's the geometry of all of this kind of, but it's usually taken to be this way. The book is quite interesting because Steiner, he's not a careful mathematician that rejects the infinity, on the contrary, he seems to be accepting actual infinity without much difficulty. This is a talk that is quite similar to later talk that will be used by Cantor and Delecq, like this idea of the totality of all these lines, and he has emphasized previously that there are infinitely many lines. He is the planar bundle of lines, and similarly he defines the bundle of planes, and then of course he establishes this idea of predictive coordination between bundles of lines and points in a line at different times.

52:30 So the kinds of examples are very simple, but the interesting thing is that we have there predictive mathematics. Excuse me, 1999 something? My book. There is a 1999 book. Oh, yeah, maybe I was thinking that I was, I don't know, maybe I took it from somewhere. It shouldn't be there. It's not an important reference for the history of this theory. But for the history of the history. Yeah, it is. And did he have in mind some kind of arithmetic analogy, talking about this, by just magic? No, not at all, as far as I can see. As far as I know, no. But it's funny that he uses this kind of heavy-loaded talk, which seems to be related to the more or less idealistic philosophical talk that he's using at other places in the book. And well, interestingly, there is also a correspondence between Steiner's emphasis on the idea that by analyzing things this way, by, among other things, Considering this kind of projective coordinations, which belong to the heart of the matter, he says, he will be able to show the road followed by nature, this is literal, the road followed by nature in producing the geometrical configurations. And all this again makes me think immediately of Franco's idea that his transfinite numbers are essential for an understanding of nature. We really want to do that. He also has these remarks about ponderomotive matter in this connection. Yes, he also has ideas about, well, quite, maybe too simplistic, I don't know, but he has concrete ideas about how to propose physics. Yes, exactly.

55:00 Well, I should say, or at least just to give Marco a reason to criticize me. Already a lot. So to make you even more happy, I should say something about connections to Riemann. Today I would only like to say one thing that in fact I didn't say in my book, but I think it's important. I just didn't realize it at the time. Later on I have been doing a bit more work on Riemann. And I'm quite serious when I believe that it's too easy to underestimate the importance of his work as... Paving the way for the development of set theoretical thinking in mathematics. So for the set theoretical reformulation of mathematics. Actually, if I were to write on it now, I would start not in a geometry lecture, but I would start in a dissertation on functional theory. So, the famous dissertation of Riemann, The Green Lagoon, The General Theory of Functions, 1851. Just, let me just mention one thing, and maybe connections, well, I will mention two things. One of them that you probably know, or at least many of you know, the Riemann surfaces that he introduces there. The Riemann surfaces he will be understanding as point manifolds. And it's quite explicit in the sense of the idea of a manifold in the geometry lecture. There is this connection to human surfaces. So, to keep it short, I just mention it at the end. I believe this actually belongs to the origin of Riemann's idea of a manifold, his attempt to understand better what kind of humans are these Riemann surfaces, if we come up with a clear graph.

57:30 Now, the other point that I would like to mention from the dissertation on the contraction theory is that Rubin is emphasizing very strongly the idea that analytic functions are conformal mathematics. As they were talking at the time, what was the talk? Let me see if I can hear it. Yeah, but the corresponding minimal parts in the image are similar, so that the angles are the same, and so that's really the problem with this. Minimal parts are in transforming elements or artisans. This is already a terminology that Gauss was using if he was working on mathematics. And so, he's emphasizing that idea of conformal mapping. And actually, he calls them a bit up-building. This is where the term up-building, as far as I can see, comes from. Directly from Riemann's thesis application. And it's quite interesting also... Well, you can say, of course, here we have only a concrete particular example of mathematics, it's not the general notion. The general notion will come only with direction, actually, in this natural. But still, it's a very important background for it, also because it suggests this in a quite a concrete, and you could say, almost geometric setting. This idea of image and original, right? So the very motivation for using the word abbildung. Abbildung is different. It's difficult to translate. I don't like at all the French translation. I assume we say in Spanish the abdication. But it loses all of the substance, I mean all of the simplistic properties of the original. So, a good translation, if it were not because the term is used for all of these in mathematics, so it's not so good, but I would translate it into a representation, I think that...

1:00:00 I think that captures much more, or of course you could say correspondence, but in a way, representation, I like it a lot because it allows one to understand not just the mathematics and to give it a more vivid expression, but also it allows you to understand the kind of generalization, the very abstract term. In the case of Riemann, this idea of a building, he takes up again, but in a much more general context, in his philosophical magnitudes, and he has reflections on a building there, and meaning representation. And this is, at any rate, this is... It is also important for understanding how dedekind can present this notion of a building mapping, and not just as a mathematical notion, but which is very serious about the fact that this is an extremely general notion that applies, that is indispensable to human thought, I think. You can see theoretical reformulations of previous mathematics already at work in the Riemann's dissertation on quantum theory. From there you can see very nice lines of development towards geometry, towards philosophical fragments, and towards basic foundational ideas. Ideas that will later on be taken up and developed in a concrete way. Now, I have been talking a lot about possible connections. As I say, this has not been about any kind of attempt to say a priori what is set theoretical thinking, and so it comes out of that.

1:02:30 But just the contrary, trying to see how some important aspects of set theoretical thinking come together. They may have emerged in connection with problems suggested by geometry and by topology especially and so on. Now, of course, having said all of that, I don't want to give the impression that I wanted to say that the theory of quantum physics emerges not at all from analysis as people have wrongly been saying, but I'm not claiming that at all. I'm not trying to over-synthesize. I think that the problem in the past has been that some people have over-synthesized the emergence of secretic thinking and tried to see it only as a line coming from real analysis towards this and that problem and only from there. But it's very important anyway to avoid the wrong impressions. Let me at least make the following point. One has to emphasize the importance of arithmetization, the general program of arithmetization at the time. As you know, this is the idea within rigor's analysis. Basically, this idea that was promoted by Baestras, but also by Bilgeau. It's important to name the two here because Lius Leibold is the most important figure for dedicating his development, and Weierstorff is of course Cantor's professor at the Rhin. In this idea of arithmetization, in essence you could say that the thing is to elaborate the basic concepts of analysis not by any reference to geometry. But we formulate them in terms of relations between numbers, arithmetic properties and relations. That has to be the key for defining the basic concepts of analysis. So this is the way in which we lose contact with previous or geometric backgrounds. And from there, it will be the way to obtain a rigorous... This is completely rigorous development of analysis. This is Weierstrass by theory, which is already consumed. Now, it's important to emphasize, I believe, just to make the counter-balance to what I have been saying.

1:05:00 It's important to emphasize the concrete proofs that counter-analysis will be looking for in many of the traits. Perhaps most of the traits of the proofs that Derrick is looking for come from the standards introduced by this way of thinking, this idea of promoting mathematicalization. And actually both Cantor and Derrick have quite a clear background in number theory and arithmetic. In the case of Dedekind, we see it very clear. As I say, if you met Dedekind on the street when he was 38, and you asked him what he's doing, then you will not come out with the idea that he's a member of it. But, retrospectively, from today's viewpoint, we think that he was. In the case of Cantor, at the same time, it was much more clear, because Cantor's dissertation and publication were both in one book. And I just gave you a couple of examples there that I find quite interesting. One of them is that, as an example, this kind of talk comes often in the letters between Cantor and Ledecky. To give you a concrete example, in this letter of 1877, when Cantor is proposing this proof, That you can coordinate one by one the line and the dimensional space, right, so that dimensionality is not the key there, you can coordinate r and rn. He offers a first attempt to prove it, and he's asking Dedeckin to do it carefully to see whether this is a strictly arithmetical proof. And that's, of course, quite interesting to see because it's not just a... Thank you very much for your time, and I look forward to working with you in the future.

1:07:30 Sentiment and perhaps for understanding something of how Cantor thought of his own work in the first ten years or so is this text, this little sentence taken from a footnote from again the same paper of 1872 that they have been working on almost all the time, this third part of the On Infinite Linear .6. And he writes there that the great majority of difficulties of principle arising in mathematics seem to me to have their origins in a lack of understanding of the possibility of a purely arithmetical theory of magnitudes and manifolds. To make it possible for you to understand it better what kind of words he's using, this idea of an arithmetical theory of magnitudes, this is basically what Bayer-Strauss, this is the way Bayer-Strauss himself was taught, he was all the time talking about magnitudes, so he's defining the real numbers, he's defining them as a question. And now, this suggests, since Cantor is here playing with this idea of having a purely arithmetical development not only of magnitudes, but also of manifolds, this is a very nice way of understanding what he was doing with his papers on manifolds. Some aspects in the problems related with the point sets of topology, let's leave it at point sets, and introduce into the study of that kind of problem the kind, the style, the line, the typical style of working work on analysis of physics.

1:10:00 I saw the mathematics and ideas about mathematics. You see the map? It's incredible, eh? Sure? I will teach you later on. And I saw the set, but I didn't see when and how the map was connected. Actually, I forgot to mention at least a little bit of this. Would you know this one? It's perfect, Karim, because it's the subject of my lecture. Well, listen... Perfect! I'm glad you had time to ask questions. So from this kind of axiomatic perspective, this comes very well. Also, starting in the 1910s, and the other way, getting sets from functions, this is from Neumann, his dissertation, and then he starts publishing the two papers in this. If you go back, what is the situation with Cantor? With Cantor, the situation is quite interesting because, for one thing, he doesn't thematize the idea of... It doesn't really formalize it in the sense of developing a mathematical analysis of this idea of one-to-one correspondence. It just uses it.

1:12:30 So we are close to having a thematic analysis of this in the case of Dedekind in his book on the book of the natural universe, which is the first piece of schematics in which you have the general idea of function in the setting, the general setting function. So, you might say, I mean, to have a very explicit way of connection, you might have to look there. But in the case of Derekin, his, I believe his way of presenting set theory could be axiomatic, but you need, one thing is clear, you have two basic concepts, which are different things, something related to sets, maybe. Elements, windows, whatever. You have a basic concept in connection with sets. And you have mapping as a basic concept. So he's quite far from 20th century ideas in the sense that he thinks he's needing two things in order to develop what he will be calling a systemia. But of course connections are emerging earlier. The idea of mapping in... In the case of Deukin, he has a long prehistory. One can trace the idea back all the way to the 1850s. He was also doing work on sets, of course. Was there a special internet connection? Yes, because he was... Simply because, I mean, it depends on how you think about what kind of connection you want to have. Early work on sets, nobody is thinking about general sets as something which has no structure and that we can play with it in different ways and give it different structures. Nobody is thinking that way until at least the 1880s. Very late in Kantor's own development. So in the early days, people are thinking about sets with some structure. And this is the way in which, in the case of Dirichlet, the connection comes, because this has mortices, because it has mattings, it has this connection with the structure. But still, there are some connections from there, and of course, in the case of Kantor, there is the connection of the way of analyzing cardinality of sets, which is probably related to that.

1:15:00 Just a reminder to the point that Cantor didn't analyze the notion of one-to-one correspondence, there is a very interesting proof in the Jetson Bikini of Grundlagen where he actually gives a proof that two sets, which stands for one-to-one correspondence, have the same cardinal number, where cardinal number is defined by kind of abstractions, so you take a set and abstract words. It's kind of an informal argument, of course, but he develops an argument, remember, just saying that if you take two things in one long correspondence, it would give the same kind of Montreal fact. And my question actually is about this Arab Humanization project. The question is how much... I completely agree that was crucial and important for the whole story. Just my question, how much that project of arithmetization depended on that idea of Cartesian coordinates? And the proof you mentioned, is it, I understand that correctly, that basically the proof is that how can you get from one-dimensional line to whatever n-dimensional space, just taking n types of, you know, coordinates. And I'm actually wondering if it really was the real reason behind this kind of argument. Of course you can say it's wrong, wrong argument from the point of view of Riemann's perspective, right, you don't have anything like one coordinate system for your whole geometric universe, and then you can go the other way around defining something like manifolds, so we have here just another probably way to look at geometry sets theoretically.

1:17:30 Yes, of course the idea of Cartesian coordinates is very important. This is, I guess, what you meant was my remark on the way in which Lirikin may have made sense for himself, because it is written anywhere, as far as I know, of this idea that his analysis of the continuity of the real line, Yeah, of course the idea there is to have locally, I suppose, and he was doing actual work in differential geometry, meaning trying to consider examples of spaces of variable curvature. He may have been one of the very few people in the 19th century. So I take it that he was understanding the kind of problems that we have in the setting of that kind of setting, so it would be locally. In the neighborhood of a point, you can give a sense to the idea that this is a point of an n-dimensional space. And yes, I think that in that connection, at least I have been trying in the past to think what kind of reasons he may have given for saying that he has a sufficient analysis. And that's the only way I can see it, to use Cartesian measurements. In the case of Cantor, he will not be satisfied with this kind of approach, but he will want to have a direct approach to the continuity of a continual domain in a functional space. So this is the new definition we will give ten years later. It wants to be this kind of a more abstract definition and in this sense closer to general topology. I think you have already answered this partially but just again about the terminology when you translate Manifaldic guides and you say well we find Manifaldic guides and what would you have to mean is set.

1:20:00 And, I don't know, for us, I think there is a problem in reading this from today's point of view, as you said earlier, because we are used to this idea of a set that is, at the beginning, without any structure, and that, for instance, when you find, firstly, there is Cantor, and he talked about manifold disguise, but then when we talk about Riemann, for instance, we are used to translating that manifold, or even surface, and the idea is close to the one we use today in physics, for instance, when we talk about manifolds and stuff like that. And we don't want to call that set. And again, the last quotation by Cantor, for instance, when you talk about the purely mathematical theory of magnitude and manifolds, and you say, well, me and my manifold point sets, or something like that, again, it seems that you can't really say set in that case, just set, because if you say point set, you are already going towards the idea of the geometrical surface. So just my question is, how do we have to, do we have to think that the idea of, in German, the idea of manifold, of manifold, it's almost unpronounceable, it was almost, it was the same as mangel, there was no distinction, and then later on manifold became rather what we mean today by surface or manifold or parieté, And Menger became what we mean by sex. It's just, for me, it's interesting to see. Yeah, for instance, Menger was a very bad candidate to be the name of sex at the beginning. So it's only very late that it's used by more than one people. It was the last of them. Yes, one of the words that Bolzano is using. This is true. This is true. Only Bolzano and Cantor. Cantor doesn't like it very much because he only uses it for concrete mathematical examples. He talks about, actually, it's the contrary. For the first ten years of his life, when he wants to have a good word that you can apply anywhere, you say manifolds.

1:22:30 But then you say salem maybe. It's short and concise. Or you say punk maybe. He only uses it mainly in connection with particular mathematical examples, points or numbers. When he wants to have something more general, he uses mathematical articles. In this room, Lagrange is also the very first time that he's used a general definition of set. He starts saying, I don't know, I don't know what it is, but we understand so on and so forth. So an enormous part of the difficulties have only to do with the question of words. But in that sense, maybe German is a good language. Because if you were reading the ensemble, you wouldn't think. It wouldn't force you to think at all. You would just introduce into it what you want. In German, it forces you to be careful, and to distinguish, and to look more carefully at the way things develop, and it's just the case, I mean, in the case of Riemann, it's clear that he in no way means manifolds to be this differential of manifolds, because he talks about discrediting manifolds with manifolds, and it just makes no sense. Well, if he were an Englishman, he would be saying script sets and continuous sets, and then there would have been reason to use the word set in English for manifolds. Yeah, that's a good way of putting it. It's something very general and it employs, I forgot the term I didn't put in my head, but it employs something like to put something in relation to another, to put something in relation. And I'm a little embarrassed to take your presentation. I can see that application, to produce by application, it's a little bit retroflexive.

1:25:00 But at the same time... I would like to find something that is related to images, because the use of bild for what is related to the element source, with my representations in French, it seems to me to be a little bit connoting a set of significations that may be a little overloaded, it's a little bit… I don't know. So, putting them in correspondence would be an exam, but it's a bit hard because when you say that you have to send, quote unquote, a set in itself, well, you have to put them in correspondence. But representations, it seems to me, are almost too precise in terms of... in French, I speak in French. There is a question of information. I don't really know. Is there an idea of direction? I have the same problem because we are in the same boat because I have both translated into our languages and in Spanish I just decided to write the preposition again. That was, of course, surprised many people, because they were... No, it's just one of the things I did that surprised people. But, for instance, it makes... In that text that you mentioned, there are three different formulations, if I remember well. It says this is the faculty of making one thing correspond to another, make one thing relate to another, make one thing at least a third one. I don't know. But there are three. Perhaps one could try to work to... We can work on the basis of one of those and see. But I believe that it's just impossible to have in our language something as clean as we have in German.

1:27:30 We have the image and something on that root for the general connection. And so, at least when you write a representation, that's very nice, for instance, to make sense of the distinctions that... Some representations are clear, he says. He distinguishes between clear and unclear representations, which is of course the case. If I am a good painter and a really fine painter, I will paint you with a very clear shot and so on. If I am an impressionist, then it would be having an unclear, because at this point... In your case, this relates to the point in the image. So it's very clearly... What is the German word for this? English. English. And he also says, he also uses the Cartesian word at the center. Distant. Distant. Clear and distinct. So, briefly, I don't want to go back to this discussion about Riemann that has been on the web, essentially, but I think that making the distinction between arithmetics and geometry and the influences of... There is a danger of distortion. There is a danger. And I want to make the slight critique the following. You say 1882 is the annus mirabilis, but you should not forget that the first discovery is not of cardinal numbers but of ordinal numbers. And it is really, of course one can... Go and see all the work and later on, but the first infinite numbers to discover were really connected with derivative sets and zero sets of trigonometric series and that was ten years before the Annus Mirabilis. So I'm sure you know that. So I think that the mixture of arithmetics and geometry asks more deeper questions about the cutting separation about the two subjects in mathematics at that time and later on.

1:30:00 And so it goes back to, of course, Zeno's prime view of arithmetics and geometry are different from Cantor's. And also, in terms of distortion of history, when you connect axiomatic construction of von Neumann, this is completely out of Kant's view to have axiomatic construction set and function in connection. But not so much of that, I was connecting developing axiomatic things. I agree here with Sieg, who has been metastasizingly important. Okay, so essentially what I wanted to say is that the fact that he discovers Ordinus V4 goes against his thesis, but I'm sure there's a mixture between the two that's closer to reality. The situation is very complicated because he doesn't discover the Ordinus. He introduces symbols of some operations on polysex, and he needs these nice symbols with an infinity and infinity plus one for operations on polysex. But what is the difference between symbols and numbers? He specifically refers to them as symbolizing the second act of abstraction, doesn't he, in the context in which he first introduces them in, definitely what later comes to be seen as the definition of the ordinals. But of course, I mean, I was saying, if there is an animal journalist, he has many important work working in many different years.

1:32:30 So you don't agree that he discovered Odin before Cardinal? No. No, I don't agree. Let me put it plain. I don't agree. He first discovers Cardinal, and he starts using the notion of Cardinal in connection with his work on poinsettias. And it's only later that he discovers Odin. The first question is about the word measure in the title Common Measure. What is the relationship between continuous and discontinuous and the different kinds of discontinuities? I don't know what to say because one would have to ask Anto what he really means. I guess he didn't mean it very seriously, but it was a very nice way of emphasizing this idea. Well, the very old problem of incommensurability is now superseded because of the development of these new ideas which make it possible to... But just to try to press it into more concrete questions of how much it... I don't know what they really meant... I'm not sure, but it's a question that I ask myself, that we can't take things differently, because then it's against, it's a remorse, that is to say that the notion of the whole would be to establish a kind of rupture, synthetic, but a rupture in relation to the other senses of the word variety, that is to say that...

1:35:00 Let's say that, if the concept of an ensemble is a concept that poses an object devoid of topological structures, isn't that the opposite of what we should do? This is a very nice way of thinking about the role of mathematics. And I basically can agree with that. The problem is that it is exactly the kind of orientation I said at the beginning. I am not going to take this kind of orientation. Starting from a more present understanding of what sets are and deciding from there what is the essence of it. And then, no, I was just trying to do it the opposite way because that is a nice way of analyzing the situation. To some extent, it doesn't allow you to understand why was it really constructed the way it was, the motivation behind having things like the power set axiom, or so on and so forth. There are very complete motivations which we cannot... You cannot set them apart from the program of a particular kind. If you are trying to understand the development of mathematical stuff, you cannot set them apart from concrete problems of understanding what is in the structure. So of course you can set them apart, and you can just consider them as one axiom that you might take among others, but then this is hard to do within the field. It's true that there is an important, a very important point when people start considering sets in an abstract way and start differentiating and being very concrete and precise about introducing structure. This is something that you... I don't know exactly when you can say that he's working that way, but in the case of Cantor, I know it quite precisely, and this is the only, well actually the first letter of him that I know that suggests this kind of thing is taken again the same year.

1:37:30 You know, I talk very much with you, and I like very much what you do, what you say, so I wanted to, I would like to say that there is a misunderstanding about some of our relations, so thank you very much for your talk, very very rich, but let me put quite a general question in a sense from outside of your talk. And I have not a lot of time, we have not a lot of time, so I will be very broad. Sorry for that. The question is, in a sense, is this one. When I wrote this Yang, I was telling in Italy, in the Department of Philosophy, with quite an analytical orientation, the story that I told. This one, the notion of set theory, frame, problem of foundation of arithmetic. Then there was a paradox of that, so there was a very difficult question about that. And then, fortunately, German scientists arrived, and they gave an axiomatization of theory and legal. Of course, when I arrived in Paris to study, they told another story to me that is not so sophisticated and complicated as yours, but was essentially Cavalier's story, completely different. So my question is this one. I think that the first story is wrong, not only. But are you sure that the second story is right? And the studies of the origin are not of Hume, but of the brain. In English it's very complicated because it's a set-theoric thinking. In set-theoric thinking, in French, passant-sant-liste is much more clear the point.

1:40:00 Because you say, you should say, set-thinking. Are you sure? As historians, we have very rough solutions. I mean, I would even say this. Eliminate the illusions of Frege and Russell from essentially the same history. I mean, if you are interested in the place, to your question, the place that the concept of set was playing in 1950, in the middle of the 20th century, you can eliminate Freire and Rossi. I'm being quite sharp, but this is a way of... I'm not sure. Well, I am quite sure, in fact, because... The role of the Gettysburg people, which was completely independent from Russell or Frege, in establishing this taxidermist in the early 20th century was enormous and was essentially independent from them. If you take out Frege and Russell, you still have them at this old school, which is extremely important that people are paying political attention to it. An enormous amount of the input we need towards the logical side of things, and so on and so forth. And one still has Zermelo, one still has the axiom of extensionality, one still has Miramanov, who is a very neglected figure in this.

1:42:30 But you still have Hermann Weil proposing ideas about essentially what is first-order understanding of esoteric axioms and the pain in getting there, without knowing much, I guess. And of course, by bringing in value, you also bring in the link to the algebraic, especially, which is a thing to do. In Brazil, I'm sure you can. But in Australia, it's much more risky. I don't know.