Mathematical knowledge (contd.)

0:00 He says in one important paper, relatively at the core of the development of his career, his new ideas, he's a peer, so this is 1882, he has almost done everything by then. he says I mean, Sir Mello says that by 1884 we have everything so two years later we have everything the quintessence of Cantor's work is already there so this is only two years before he says most of the principal difficulties that are found in mathematics seem to me to have their source in the fact that the possibility of a purely arithmetical theory of magnitudes and of manifolds, which means I'm thinking these sets, but I prefer to keep the literal translation, is not acknowledged. So people don't realize that it is possible to have a purely arithmetical theory of magnitudes, and also of sets. And this is why they get me to trouble. What is he thinking about? As I said, on Monday, now we have an agent that has a lot of peculiar ways of thinking, actually. Some ways of thinking is of the ones that create difficulties for historians and the subtleties of history that you were mentioning. So now we have to try to read here and there and try to come up with some reconstruction of what may be in his mind. Actually, I have to say that I have struggled with this sentence for years, actually. Well, not all the time. That's in a way, I think about it a little bit more. And actually, I think by now, I think I understand a lot of what he had in mind. But I might be wrong. Maybe somebody else comes and shows that I was not interpreting it the right way. The possibility of a purely arithmetical theory of magnitudes, this is biostores. basically. Because actually biographers talk about magnitudes all the time. Now you have to read biographers and then you see that when he introduces, when he introduces the real numbers, actually, so the number system is always writing or saying,

2:30 number of magnitudes, where the main, in German, so the main substantive there is magnitudes. We are talking about magnitudes. What he is doing, of course, is to, he has freed himself from the idea of magnitude as a natural thing that we were mentioning. And he's coming to actually a modernized way of thinking about magnitude, but this is becoming abstract. Some other people thought it is much better to stop talking about magnitude. This was delicate. He was totally against using any more the idea of magnitude here, because he simply thought it was mixing up things too much. and he was making everything neat, he would stop using it. But Baristrad didn't do that, and Cantor didn't do that. If you read the paper in which he introduces the real numbers, he talks about himself, Baristrad, just like Baristrad. So this whole development of the introduction of the number system, starting from the integers or the natural numbers and going all the way up to the complex numbers, actually, it's a matter of having an arithmetical, a purely arithmetical theory of magnitudes so this is the way I understand the first part, so now interestingly he says you can also have a purely arithmetical theory of many folds Manifaltigkeit's Lerbe, which is normally called Mängen Lerbe he himself was talking about Mängen already and later he decided you know, it is a bit shorter so let's talk about Mängen Lerbe So, set theory. So, you can also have a purely arithmetical theory of sets. What is that? What does it mean? Was it really relevant in Cantor's actual work? It was. And the other thing is that the declaration of principles is clearly realized in his previous papers, in his contemporary papers. So, there is a paper, I believe it is the most important paper by Cantor the Rumblagen of a general theory of manifolds you have there he proposes there

5:00 let me see, I think I have the quotation here he proposes there a purely arithmetical concept of a continuum of points very strange way of saying what he's doing but anyway So his abstract definition of what a continuum is, he's calling it purely arithmetical. And that's not, maybe a little bit, maybe to some extent he keeps using phrase theology that he has been using for many years, when he's already in some context where he should start stopping to use it. But anyway, I'll give you another example where everything is much more clear, which is another famous move of Cantor, which, try to do it in, yeah, can I do it in three to five minutes, I hope. Now, in a famous paper that I didn't mention on Monday, but I should, Cantor proves that there is a function from the set of real numbers to Euclidean space of n dimensions. So from a line to the plane, from a line to three space, from a line to n space. He gives you a function doing that, a very arbitrary function, by the way. He doesn't call it a function. That's interesting. So let me tell you a few things. I will not discuss the proof. That's not very relevant. There is an easy proof, and there is a more complicated and very interesting proof. I simply want to give you some ideas about what is going on and what he's thinking about it and the way he's presenting it. So how it fits into this picture of a purely arithmetical theory of sets. The sets he's talking about are mainly, and still by the time that he gives you that quotation, are mainly point sets. Sets of real numbers or sets of points in dimensional Euclidean space. So this is what he was thinking at the time. And he wants to have a purely theoretical theory of roles, sets of real numbers, sets of points in space. Interestingly, one of the things that is interesting here is that in this paper, even though he's given one of the most arbitrary functions

7:30 that people came to know at the time, the topic of arbitrary functions and the easiest, and all of that is not present. Nothing. That doesn't seem to be an important thing, or at least he doesn't emphasize it at all in the context. Actually, the context he's emphasizing is n-dimensional manifolds. He's emphasizing Riemann and Helmholtz and the idea of what n-dimensionality of space is. So this is the context for him. So he's doing theory of manipulance himself. He thinks about it that way. Instead of doing theory of functions. The function is important for what you get out of it in the theory of manipulance. Not the other way around. And actually, he's very worried about arithmetical rigor. In the... There is, of course, the famous correspondence with Dedekin. And in the correspondence with Dedekin, he begins by asking I would like to know if you think this kind of proof this proof procedure that I have been using is arithmetically rigorous so it's actually there I won't give you a lot of if you look at it to make it very short the interesting thing is to read the paper and to realize that he is presenting things actually in a very peculiar way magnitudes, actually. Everything is in the language of real magnitudes, variable magnitudes, and one-to-one correspondences between magnitudes. And that is a very very structural language. And he's very worried about the proof procedure being arithmetically rigorous. So, all of this places out, and it's still, it enables us now in a very tightly historical way, a very concrete case study, how some new framework is being elaborated by an agent in connection with a previous well-established framework, which is the point that I would like to emphasize. The idea is that it is by using carefully and purely ideas from a previously well-established framework that he is able to establish this one-to-one correspondence. Of course, then this becomes an extremely important set theoretical fact.

10:00 And you can look at it from a higher perspective. He will do it himself a few years later. So you can rethink it from a higher perspective. But at the time of the development of the framework, the connection with the previously established framework was essential. So that's the key point. And that's it. Yeah. I guess one, just to probe the keeper, I'm wondering why, I'm just trying to get a sense of what you think of the problems in the direction of the center of the function. The problem, yeah. The problem in the direction, because, I mean, the one thing about the counter function, even if it's a little bizarre. That's what people said about it. It's so bizarre that he would rather keep it for himself. I can't believe it or something like that, but he still would specify it. I mean, he still would define it. I was thinking, I mean, the sort of thing that would push you more in the direction of considering arbitrary functions would be cases where you were forced to, you know, positive existence functions could be specified. Yeah, of course, this is not arbitrary, not very arbitrary, insofar as you can specify it. Yeah, so I'm not sure, you know, what you were saying is that it has become a common place to present it as one of the examples of highly pathological functions, because of course it is so absolutely discontinuous and, you know and it creates assailment with a notion that seems to be so clear in dimensionality and different dimensionality of different spaces that seem to be so intuitive and so intuitive to explain for people. So it became a commonplace at the time in the context of this whole discussion we were talking about the other day between people with different function concepts as one of the quintessential examples of highly arbitrary, pathological strange function. Anyway, this is why what I find interesting is that Cantor is not emphasizing

12:30 that side of it at all. It was not the important side of it. I guess what I'm trying to approach for, though, is there is a picture that sometimes comes up in emailing in histories or read instructions because it has a lot of What you find is that people produce a bunch of things that look really strange, and then because of the strangeness people try to understand it or something like that. But I think more characteristic of the things that drive the theory forward are things which you sort of say not merely seem strange but which you're compelled to address by the need to solve problems. So, you know, the fact that there are these open questions, you know, you're putting a series of time for us. Yes, I think I agree with you. We're much more, so to say, powerful agents. Yeah, definitely. Powerful forces. Yeah, this is not, I don't see it at all as one of the powerful forces. So I agree totally with you. It's much more important in order to understand what drove people towards that. That was one of the reasons why I didn't emphasize this. I don't see that as important the other day. It's a much more important issue of the previous series. It's much more important Riemann's contribution to this integral concept. Because that really opens up the possibility of studying these continuous functions in ways that are interesting for analysis. On the other hand, something that I haven't said today, and maybe relevant, is that actually, in a sense, you can see Cantor here, as late as 1878, doing the difficult Berlin thing with Riemann. So what do Berlin people do with Riemann? They create difficulties, for whatever he's going to say. So Riemann is telling you, you know, n-dimensionality, this is very simple. Because in order to specify a point in an n-manifold, you need n-coordinates. No, no, no, no, it's the Berlin line, you don't need it. So this is Kantor doing that. In a sense, he's praying just like, you know, continuity, well, you know, this is differentiability. No! No, no, no, no. Weistruss comes nowhere differentiable continuous. And so on and so forth. This is my principle. No, no, no, no. It doesn't work. So, in a sense, he's, even more strongly than I was presenting it,

15:00 I have to confess that I do not understand. Is that a good sign? Possibly a very good sign. Let's begin with your definition of practice. A practice is, you say, a paean. Actually, yep. By the French word, and my first problem is there. One agent, two agents, a lot of agents. One, two, three, four. How many? Yeah, as I said, I am not very happy with the word practice. Actually, I started calling it a practice. Mostly because I was coming from Hitcher, going with the word, it may be much better to call it some other word, but if you have good proposals, I will be very happy to... Actually, the practice is what the agent does with the framework, what the agent does with the framework, what the agent does when elaborating new frameworks, and what different agents do with the framework, of course. So, I mean, those are the practices of mathematics that we are interested in. Part of the cognitive elements that you need to put into place in order to understand how the community of mathematicians or particular mathematicians go about their practices. In your pair, the agent is one. Yeah, in my pair the agent is just one. You can make it an idealized agent, then it is normal. as a sort of type of action of an agent with a framework. That's it. This is what you want to say. More or less, yes. With several frameworks, often. No, one practice is a framework. Then you say there are different practices at the same time, so one agent can interact in different ways with different frameworks. But one practice is a kind of action. No. I mean, what people call, what we normally call mathematical practice often involves different frameworks. So, still crucial for understanding what is going on is...

17:30 So why there are different practices at this moment? Yeah, because I was calling this a practice. Why different practices? Let's call this, let's call this, what? An FA. We need FA's. We need FA's in order to understand mathematical knowledge and mathematical practice. Now I can talk in a singular. Mathematical practice. It's important, in order to understand mathematical knowledge and mathematical practice, to realize that there is a plurality of FAs. And that the FAs are systematically interconnected. And that in mathematical practice, it is often the case that the links between different FAs are crucial to what is going on. The objectivity of the results, the restrictions of the neuro-relevant... Let's try to be legal. Well, you say that the probability of F A. I don't accept that. F A is a pair composed by an agent. An agent, you say, let's take, let's avoid the difficulty of the fact that there are concrete items. Otherwise, I can't pair with all A little terms. So it becomes a metric. No, that's very interesting. So, I'm taking an abstract argument. So, this is on the white way. Can I make a remark? Parathetical remark on that. That's actually very interesting for understanding different conceptual understandings in different mathematicians. So, communication between mathematicians. Ok. Let's take an abstract agent. Ok? So, we have an agent and a framework. Now, you can have different frameworks. So the difference of SAs is given by different frameworks. But you can also have different interaction between an agent and the framework. The agent cannot interact in the same way in the framework. So you can speak of priority in two senses. The sense of priority of framework, the sense of priority of action of the agent on the framework, and even the sense of priority of agent. Let's keep this on the side. are you speaking about? The fact that you have different frameworks, or the fact that you can act in different ways with the same framework?

20:00 Actually, I see that flexibility as a virtue of the approach. Depending on the question you may be interested in about mathematics and mathematical practice, you can work with it in different ways. For some kind of questions, like for instance in my very last example, and two frameworks at least two maybe more because there are actually more than two frameworks that Cantor is using and deploying while he's trying to elaborate a new framework actually what we will come to call a new frame but so there it's the variety of frameworks and what the agent is doing with them that counts. For other purposes, in order to recognize an agent as a well-established member of a community, actually the agents cannot do whatever they like with a framework. They have to learn some ways of treating the framework. Otherwise, people will say, well, you know, this guy talks about functions. He doesn't really know what he's talking about. Cantor about client functions. Right? Action. I can give you the quotation. He's talking about functions. He's not trained properly, and he doesn't really know how to do things. So, there are parameters like that. It's not that they can do whatever they like. Part of the picture is that they have to learn the trade. And then another part of the picture is, once they know the trade more or less, which is, of course, this is a fascinating situation, that once they know the trade more or less they can go in different ways to it. So here the plurality of agents is very important. You can use this fixing more or less a framework you may take Reynolds' work we are fixing a framework more or less. Let's talk about people who do Bayer-Strauss style mathematics or Riemann style mathematics and let's see how different agents are understanding what it is about and try to elaborate harder the framework. I don't see this as a problem at all. I mean, in knowledge, not only in mathematical knowledge, in knowledge, absolutely, plurality, the fact that there are no normal agents, there are

22:30 no normal agents. The normal agent is an idealization. None of the agents in the real agents is normal. We are all different. Differential psychology is at the core of an understanding of knowledge, right? I think. We should try to understand what we want to understand. If the point is that we have the framework and that there are different persons that can interact in certain ways, and you describe the different concrete persons that interact in different ways through the framework, that simply means making history. What difference is making history? You say, as Bajestras makes Bajestras mathematics, as the other mathematician, So what are the philosophical problems? If the problem is that if you want to categorize something, you want to say, ok, I need some categories that are general categories that allow me to understand something. So I don't see the necessity of the agent there, because we have a framework, a literate modality working with the framework, then the fact that there are not real people that are working with whoever is not, is a historical contingency that in this point of view is not really relevant. So the agent is not part of the pair, the framework is, and the way of working with people. I mean, frameworks do not live on their own. So what I don't understand is what do you do with this pair? If we use this pair, you use this pair in order to make history. You make this pair in order to understand the names of mathematics. You use this pair in order to describe the way which mathematical knowledge works. What's the aim? No, no, no. Absolutely not. My aim is to understand what mathematical knowledge is. What's mathematical knowledge? My aim is philosophy and mathematics. So why not, then I think, why not to rephrase that? But, at the same time that my aim, main aim as a philosopher of mathematics, is to understand what mathematical knowledge is, there is no problem for me that I can use the same framework in understanding history. Why should that be a problem? Is it bad that there is some contact between philosophy, mathematics, and history of mathematics? Is it really so bad? No, no, but there is no position to use that. The point is that I want you to understand what you want to make with a certain tool, because we have to construct a tool in order to aim it.

25:00 Well, I gave you an example. So, should you, could you, no, would you, sorry, should you agree with this way of describing your point, you have, you called it Frank, I would like to change the world, the world, and speak, and speak, that's okay, that's okay, I would be okay with that. My problem was not to change it from point of security, because it's the same difficulty there. Let's call it a domain of possibilities. Something that you can do in a certain way, with a certain tool. That's too vague. I want to have something slightly more concrete. It's not because you passed from the word domain of possibilities. in the word framework theory, you made this thing more clear. So, instead the domain of possibility is too large. Okay, so we should change the world. But we cannot only change the world, we have to change the concept. So the very big problem that we have, if you explain me what the problem is, if you don't explain me what the framework is, your discourse is... there is nothing to that. Because, exactly because of that, exactly because I agree with you that the domain of possibility is there. But it's not only because you change the domain of possibility, it's a framework. I understand, I understand. Very good question, in modern axiomatic spirit. I like it. Let me write it down, actually. So, now I'm British. No, of course, I hope you will accept that it's not very easy to explain everything. Absolutely not, absolutely not. It's also a problem, you remember the contract. It's actually the problem I have, so I think this is a good problem. I think this is a very difficult problem, but the thing is, this is the problem, not another one. No, I don't agree with that. I think the problem... Frameworks absolutely do not live on their own. They don't live on their own. They are dead. frameworks or theories or domains of possibilities are absolutely dead unless there is an agent. Learning them, using them, solving problems with them, trying to elaborate new methods for solving problems, or whatever.

27:30 So I mean, I am very much a naturalist. Okay. Let's suppose that we have to understand what the framework is or the domain of possibilities, the theory is... Then we have to understand the way in which we have to connect agents. It's not only sufficient to take an agent to connect. We need to explain what's the sort of relation that we have with frameworks and agents. So the other problem is what is the relation between the two parts of the real pair. So, you should explain what the first power is, what the framework is, and what the sort of relation framework is an agent. Then, the agent is simple, because you can say common agent, or in real historical terms. I think that is complicated also, but let's take it aside. At least these two things you have to do. To explain me what the framework is, and to explain what sort of relation a framework has this agent. Of course. But I'm not afraid. Sorry? I am absolutely not afraid. So, I am not Frege, yeah? Now in Germany it's better. So, you are not Frege. I am not Frege. I know that you are not Frege. No, absolutely, yes. I am not... Why is it relevant? This is very relevant. Because I have absolutely no problem with the fact that what I have here is maybe, if you are very nice to me, you might describe this as a promising way of thinking about mathematical knowledge that has many problems. I am absolutely happy with that. And I will publish a book about that. Because I don't believe that I need to have a perfect analytical description of what a framework is. And a perfect analytical description of what an agent is. And a perfect analytical description of the relations between one and the other to start doing work. And to start, no, no, I think you are going to a little bit and I am emphasizing the point. No, I am not interested in systematic philosophy in the old style. I don't think I am doing that. I am trying to show that with this kind of framework we can eliminate some problems and I hope that other people will become interested in it. And even further, one of the problems in the relation between the, and I definitely also, I think, my priority is not to go first for the framework and then to understand what

30:00 the agent, what the relation between framework and agent is, because actually I believe many of the deeper problems are just the other way around. One of the problems here is that we have counting with oral language normally, and mathematics with symbolic notation. What is the role of oral language in human knowledge? What is the role, the difference between oral language and symbolic notation and their cognitive impact. You want me to give an answer to that? No, I can't. And I'm happy with that. And actually, I think maybe people in psychology and cognitive science, after they become interested in analyzing mathematics in a certain way, maybe they can illuminate further what their role of symbolic notation is in human knowledge. So I don't have to wait for the cognitive scientists to tell me what I have to do here. I may be doing the opposite. I may be telling them, you know, you have to do something to study. And maybe you will solve your problems if you study this. So, I don't have a very naive kind of way of thinking about the relationship between cognitive science and naturalistic philosophy of mathematics or science. Can I ask a question on related question? What is the distinction between Lester Mathematics and Goss? It's very... I am always against the icon, so I should be against this one too. But it is useful, at least, for a fair separation. At the same time, you said that the quantum logist is not a mathematical practice which was a technique. I call it technique. In this case, what is the mathematical mathematics? You said that the quantum mathematics is characterized by hypothesizing about sort of not very national concepts like continuity and unity? Maybe fractions as an example of an elementary element. Fractions. So my first question is, what is an elementary element? And the second is, why do you think that concepts like continuity, infinity and fractions are not only the root piece in our world? Fractions, I think, that's elementary enough to be strong in the movement.

32:30 Actually, there is also a very interesting historical work that people in cognitive studies of this should pay attention to that. There is a very nice book that actually was edited by Chinese and other people, one is probably fractional, and fractional is lost, and you have defined fractions in different guises and ways in many different countries, civilisation and cultures, and so there's something, I, today I was using this contract between military and France in a very simplistic way, and I think, I believe I'm still pointing towards something that may be interesting, which is, Especially, the difference between the kind of knowledge that we may have in cases like the national numbers and fractions, which I tend to think is elementary, and the kind of knowledge we have to go far, far to the other side, with real analysis or receptivity or algebraic geometry. So I think that there is a big difference. I have actually written about this a couple of times already. But I suppose it's all, again, sketchy. I think there is uncertainty to use this very dangerous work in elementary mathematics. that you don't find in advanced mathematics because it's dependent on the hypothesis. So today, for me, the important thing was to simply point out to the fact that I believe in very many cases we are building on hypothetical grounds, but there are some cases in which we are not. Typical example, natural numbers, and basic knowledge about elementary properties of the natural numbers, The existence of prime numbers, of infinitely many prime numbers, if you use infinite in a non-sexual theoretical way, and so on, of course. I don't know if that's the idea. And what's called the forms that you continue to use? Those are, I believe, like Brehmann, that they are philosophical.

35:00 And I believe the whole foundational discussion gives you tons of arguments for arguing in that way, not only the whole foundational discussion, but all the results of physics in the 20th century. And actually, Hilbert argued this way about it. His paper on the infinite decided to start raising the point that there are problems with that both from inside mathematics and also if you pay attention to physics of his study, there is no reason to think that infinity is out there or continuity is out here, but it's in here. Like in our mines, whatever that is. Mines are typically agents with framework also. Okay, I'm supposed to do that. I don't like her wishes. Thank you. Thank you. I should set this up. Is it okay? Yes, of course. I wanted to propose. I'll bring another bottle. Oh, thanks. I appreciate it. Thank you so much. I have a proposal for a new definition, energetic, energetic questioning is a marco-prosance. No, no, no, no. I mean it in a good natured way.

37:30 We are the point, so we can ask him, whatever. No, it's not, okay, it's simple, of course. I mean, it's also entertaining because there's a problem that I hear. No, no, but you cannot avoid the problem. No, I'm not trying to avoid the problem. But I'm approaching it in a very difficult way. Thank you.