Mathematical knowledge

0:00 Thank you. Okay, so as you know, I suppose the title is Mathematical Knowledge and the Interplay of Practices, which is the main theme, or at least the way I am describing currently, the main theme of a book that I am writing. I've been writing a story for some time and I guess it's going to take a lot of time more because I'm beginning again actually, from scratch this summer. It's an objective point of view on your talk. Yeah, I like that. I like to cover some surveying, you know, control system up there. Anyway, it's interesting, actually, that there are many points of contact with what David was telling us about in terms of the understanding of inquiry, in terms of how I position myself also regarding this problem of, well, what I call before absolutism versus relativism. and in terms of, well, many different things. Also, the question, well, in an old-fashioned discussion or way of labeling, what David was telling us about is this idea of the introduction of new concepts in mathematics. David King was talking about that 150 years ago. People are talking about it today. And that's actually a central theme for me, too. Of course, there are also big differences in terms of the kind of, well, the state of the kind of mathematics I am discussing, and also because there is not this connection in my case with current developments, everything will sound to you very elementary.

2:30 Actually, in the examples I will be giving today, my perspective is that one can apply of situations, I think it is flexible enough, the kind of schematic way of thinking that I will present, so that in particular, of course, it can be applied to developments of the last 50 years in mathematics, and probably, why not, exactly the same way to today's developments. At the same time that I'm not making this attempt to make any contact with current developments, I am making attempts to make contacts with other researchers. I'm also very interested in the connection between mathematics and physics, that's one thing that I always like to keep in mind. But I'm also interested in, well, one very easy way of describing the kind of approach I'm trying to develop is to call it a naturalistic point of view on mathematical knowledge. Of course, I don't like the label, because the label is today totally polysemic, it doesn't mean anything naturalism, because it has meant too many things to too many people already. So, to describe it a bit more quickly, two elements that I find important for me is, first, to understand mathematical knowledge as knowledge produced by human beings, so living beings, like us. the connection to what is usually called cognitive science and biology, and stuff like that. Those are connections that I am interested in. And also, something that is not always the case in people who are interested in those connections, for me, I'm quite far from the absolutistic perspective that David showed us, because actually I argue that historicity is at the core of the development of mathematical knowledge. So, both of these elements, the cognitive and the historical. So it's one way of making a bit more precise the kind of naturalism that I'm trying to develop, is to say that it is some kind of cognitive historical perspective on mathematical knowledge. now I will of course that's a very as I say it's a relatively flexible

5:00 kind of framework it is more than anything else a program for doing all kinds of different lines of research there are all kinds of possible ways of trying to elaborate it and substantiate it and I of course it's impossible to even start a deep discussion of that in one hour but I will try to have some of some of the things I find more interesting. So, to begin with, I will propose a certain way of thinking about what I'm currently calling mathematical practice, but by the summer I may be using a different label, because I'm becoming dissatisfied with calling this mathematical practice. There are some historical equations for calling this mathematical practice, but they may be wrong original. So, anyway. Anyway, it's very clear, everybody knows, in philosophy of science today, the idea of scientific practice is prominent, and in philosophy of mathematics today, the idea of mathematical practice is prominent. But again, also, in connection with that, there is a diversity of viewpoints. And actually, this links with a talk I gave here a few months ago at the beginning of the academic year where we discussed the views of some philosophers on this topic of mathematical practice i didn't discuss my own views i had already thought of this talk of today as a follow-up of that one so in that one i was discussing other people's viewpoints i discussed something in lacatosh and picture in maddie also in a book that is coming out this summer, Mancosu. For instance, in the case of the Mancosu, Pablo Mancosu's edited volume, which is a collective piece of work, I would describe that as more partial studies of different aspects of mathematical practice. While some other people have proposed more global kind of perspectives on mathematical practice, in particular, that is the case with Kitschurch. To be honest, it is Kitscher that I find my own approach more linked, or easily linkable. So, maybe the best idea is to start by bringing to your mind again some of the Kitscher's ways of thinking about mathematical practice,

7:30 and then presenting my own perspective as an alternative, a critical response to that. And I will simply remind you that there is, of course, the main idea for him is to understand the development, the growth of mathematical knowledge in terms of a series of practices. And he has an explicit model of practice, of what a practice is, which is we have a, Actually, it's all very linguistic, which is something I'm not happy with, but anyway, language, in some, for instance, we can think about, I don't know, mathematics in the early 19th century, the language of quantities, the science of quantities, right? We are in the middle of a theory of quantities. It's a matter of mathematics. It's a matter of mathematics. It's the same. The language of mathematics. The language of mathematics. I'm talking about quantities. And you... Yeah, of course. There are statements that are accepted by the people who are doing mathematics that way. There are, let me see, to put it in the same... Yeah. Accepted forms of reasoning is for types of... reasoning procedures that are accepted. This is all trying to describe what the community is accepting. Typically, to be a member of the community, you are accepting and using. There is a set of questions, and there is something that seems to go to a different level, meta-mathematical perspectives. Not, of course, in the sense of Hubert, but in the broad sense of meta. blah, blah, blah about mathematics. Views on mathematics. They! Hilbert is preaching all the time to other mathematicians. Like historians do, he is also disciplining the others. So, set theory is the future. We have to pay attention to axiomatics. Problems are the source of everything. So, this kind of thing is metamathematical views. So, that was more or less teachers idea of all the practices. For a teacher, a mathematical practice is this

10:00 kind of thing. A very disembodied set of five different types of things, all of them linguistic in the end. So you have a language, you have the statements, the reasonings, the questions, and the metamathematical views, and that's it. Well, and then he also Thinking about mathematics as produced by human beings, he has stories to tell about that, and to tell about the cognitive roots of mathematics. I actually think he had better ideas himself than the ones he presented in his book, that he didn't dare to present his own, totally, his own views. And I think he blew it, and by doing that, the book would have been much better if he had actually told you what he was really believing in. Because the kind of mix-up, the kind of compromise argument for which he went is actually very unsatisfactory. Especially if he has some ideas about the ontology of mathematics, which are very bad, actually, and stuff like that. But there is this connection with cognitive, or potential cognition, cognitive studies of reasoning and knowledge development. And there is also, well, also there is a lot of influence of the 70s, philosophy of science, in the future. Because he has this picture of the development where we have a practice reigning, and then a new practice reigning, and then a new practice reigning. of Kuhn and revolutions, in a way he has that kind of picture, which I find also very mistaking to be used with mathematics. And I will not talk about the other side of things the more, but I will talk about this especially because actually one of the key in my title you read the interplay of practices. And that's at the core of what I'm trying to do. So that's a totally misguided kind of picture from my point of view. And there is no normal science in that sense. There is a plurality of practices at every historical period and even more than one. But before going to that, now, my own way of... I will modify this picture trying to avoid very strong modifications, because if you try to be very subtle, then you make everything very complicated, and it may not be the good idea for one hour discussion.

12:30 So, I propose the following. We have the first thesis against this kind of disembodied picture. In order to have practice, you need practitioners. So, that cannot be a practice. If the practitioner is nowhere, then we are not really talking about... We are talking about something that is involved in the practice, but we are not really describing the practice. to start with a couple where you have a framework which may well be very similar to that. The first element in the couple may already be a four-to-four action. And then the agent. And this is a practice. This is what makes it possible for you to start analyzing the practice. Let's just start with the agent and the framework. In my case, I would rather leave the M of Kitscher as something that goes into the agent, actually. M would be part of that. And the framework, I've written here, we have a language, we have symbolic means, we have problems, and we have a corpus of mathematics. Anyway, it could well be these four things. It could well be these four things. In the framework? Yeah. In the framework, of course, yeah. The reasoning is not something of adjunct? No, no, the accepted reasonings are typically what we call the methods. So, and of course, whenever you prove a theorem, you are offering a method that may well be applied somewhere else. As you know, many times it is more important the kind of method being used in the proof rather than the final statement itself. So this is the kind of thing, I think, at least the way I understand it, this is the kind of thing that we call reasoning, and then it is part of the framework. Of course, it has to be used, internalized by the agent, and so on.

15:00 Now, so let's hope that more or less it is clear what we are talking about when we talk about the framework. To use very traditional ways of talking, you have clearly a language and symbolic means. The symbolic means do not have to be the ones of logical theories today. They can be diagrams, for instance. Those are symbolic means. If you analyze the practice, the framework of Euclidean geometry, the diagram is at the core. It's really at the core. But anyway, you have a language and symbolic means, you have a theory, and also problems. so it's not merely established theory there's also some more things going on and then at the level of the agent I think this is interesting I will try to show when I, I mean, first I'm trying to describe the general kind of picture and then I will try to flesh it out with some examples and some proposals it's interesting that the agent you can, that's a very imprecise kind of idea, but we are talking about human agents, right? So, I'm not very interested right now in trying to apply this to martial mathematics. You could, but the agent is a human agent. And one of the possibilities is to think about normal agents. So, a normal human being in the sense of the humanities, or no, sorry, the social sciences, normal human being, being typically endowed with the typical cluster of cognitive abilities. So that's one possibility to make it, to substantiate this idea of the agent with the normal agent. Another possibility, which is of course of more interest to the historian, would be, less interest perhaps to many philosophers, would be you can always take actual historical agents, and then you can introduce a lot more information into what is going on. I will show you both possibilities later. And also, in between those two

17:30 possibilities, in between the normal agent and the actual, the real historical agent, you can also have something like a typical member of a community. So try to, which is actually that's something that historians are interested in. And, you know, if you're interested in early 20th century mathematics of some kind, you are typically, a historian will tell you, well, don't always pay attention to the, you know, big names. Try to think what it is to be a more normal member of the community. So this is the typical member of the community. So I propose those three different ways of working with the agent. The normal agent, just think to anybody with normal cognitive abilities, right? Of course, if they have some cognitive problems, there are typical cognitive problems that affect the learning of mathematics. We are not interested in that. That's very interesting, of course, for cognitive science. It is very interesting that we are here thinking about the more usual situation. The typical member of the community and their actual historical actors. Okay, so now, well, this is actually, I take it, of course, it's an absolutely general kind of framework, and still, it already suggests some possibilities, and can be applied, actually, to developments in mathematics. I can easily think of applying this kind of framework in the developments of mathematics from 10,000 years ago to 2,000 years ago to 5 years ago. So, apply it to whatever you want. Actually, also, if you think about, for instance, you know, work on the role of diagrams in mathematical knowledge, or understanding and explanation in mathematical knowledge, or stuff like that, what is that doing? normally it is narrower perspectives on some facets of what is going on in this couple of the agent and the frameworks that she is using so so it's I can think easily fit those different types of studies as facets as concentrating

20:00 and it's very important to do that of course because simply by reflecting on you know, you won't go very deep. So it's very important to do these more concrete and narrower partial studies, but I can think of them as fitting into the picture in many ways. Okay, now, so that's what I'm calling a mathematical practice, probably. What is being done by the agent with the framework, And, of course, sometimes agents are simply learning to do things with the framework, or sometimes they are trying to develop new frameworks, which is more interesting for Kitscher and for me too, but I'm interested in both, actually. Now, the other very important idea is that, in my opinion, we have to analyze things not in terms of one mathematical practice, as if there was just one big thing going on, but in terms of many mathematical practices that are interconnected. So a key thesis for me is that there are several different levels of knowledge and practice that are coexistent. and that they are systematically intervened. So the links are not arbitrary, there are some systematic links. And actually those links, in my opinion, are crucial to an understanding of mathematical knowledge, and especially to the development of new practices, new frameworks, etc. What do I mean by co-existing? And I mean, for instance, that historically, typically, in every historical period, you will have several different practices. It's never just one... Parallel? Parallel? Parallel, or whatever, alternative, of course, to... Yeah? Yeah. This, in many ways, it's already entertaining. Okay, now I need to use a diagram, I think you are better than using diagrams than you think about it. And you probably need, you know, bidimensional is going to be very bad, so think about it endimensionally at least. Now, what I wanted to say is that different practices coexist historically, but they also coexist within a single agent.

22:30 I think we can easily come to a mind. I am a very confused kind of mind. But I hope, I can easily convince you that everybody of us in the process of learning mathematics has learned to work with several different kinds of frameworks and has learned several different practices and has learned systematic links. Let me give you some examples. A very easy example. Actually, I should have prepared something for high-tech stuff, but I'm becoming more and more old-fashioned, so I decided not to do this. Anyway, I can still write. Counting. So, typically we learn that before going to school, actually. We learn that already at home. So, by the age of five, six, people know how to do counting. And I don't call that a mathematical practice, by the way. In counting, it's not really a mathematical practice. say that this is a technical practice. It is so important. Anyway, we can have arithmetic. Counting involves, the way I'm thinking about it involves, well either, you know, what the shepherd did, or one, two, three, four, five, how many of us saw oral language, but not written symbols. So this is, maybe one can even use the central role of written symbols as one of the key characteristics of mathematical practices. And also that allows me to say that this is not really a mathematical practice. Anyway, between counting and the part where you can have things like, I don't know, whatever you want.

25:00 15 plus 51 equals whatever. And all of these practices we learn about how to sum, multiply, and do stuff like that with This is a symbolic practice, right? Normally, the normal agent learns at school, right? So in primary school, people spend a lot of time doing this. And then the normal agent that goes into university mathematics will start learning, for instance, how to think about numbers in terms of structures, right? a set in with some successor mapping properties, the axioms describing what is going on. This is a very different practice. Both of these are followed by everybody and are completely mathematical practices. And of course we know very well how one is linked to the other. So, one has knowledge of two different frameworks, the same agent has knowledge of two different frameworks, and knows how to elaborate two different practices, but they are systematically interconnected, and actually are, well, one of the ways of thinking about this is that this new conceptual, we are thinking about numbers, shows light on what is going on when you were doing the other practice. Anyway, that's an example of how different practices coexist. Also, another example, if you wish, people learn first how to measure. Again, nothing to do with mathematics, really. Measuring practices are technical practices, so we learn how to do things with ruler and so on. So, next step, we may start to learn things about fractions, and you want to think about this historically, for example, theories of proportion, right, at the point when you discover that this is getting very involved, so you're eudoxus, and later on, maybe some theory of magnitudes, if I'm more elaborate, so we have measuring practices, well, fractions, proportions,

27:30 magnitude theory, are different practices that are interconnected. Sometimes, of course, we have a displacement of some practices by others, but typically it's the case with all of us that we know about the measuring practices, and we also know about practices with fractions, real number stuff. So there is again there a plurality of practices that are systematically interconnected. Or to give another example and to emphasize once again that this is making connections And then we start on with what is actually what I may call, take a survey, kind of label technical practices. First, practical geometry, how to do things, you know, how to draw, how to use drawings for building or doing whatever you need with them. Euclidean geometry, Cartesian geometry, for instance. That's three different practices, which actually, it used to be the case all the way up to the early 20th century, that learning mathematics involves the learning of the three of them. I mean, from, of course, from the point in which Cartesian geometry becomes part of the usual training. Okay, so this is the kind of picture that I'm thinking The different practices coexist not only within a given historical period, but already inside the very same agent, the mathematicians. Now, so, what I think is that if we think this way about mathematical knowledge, there's a lot, actually, that you can say, even if you disregard all of the subleties. So, even in this regard, all kinds of subtle ideas about, you know, yes, of course, I mean, when we are thinking about mathematical practices that they are historically given to us, the members of a community do share metamathematical ideas. If you are, I mean, too, probably many of you were here on Monday and Tuesday in the discussions we had, so to give examples from there, if you are a member of the Berlin School of Firestruck, you have learned that one should do things in some ways, or should be alithmetical,

30:00 and should prefer certain ways of presenting functions instead of doing things some other way. If you have been exposed to Riemannian ideas, which Hilbert said, puts you in a higher class at the time, then you have learned to think about things in different ways. So you have all kinds of subtleties here, like different images of mathematics, different criteria for a good solution to a problem, or different values, maybe. All of these things, let us forget. Of course, that belongs to the general picture, but I wouldn't forget all of that, because all of that is very interesting and makes it very complicated. And what I want to emphasize is that even by reducing a lot of complexity, we can still say interesting things. So let's forget all of that. And my claim is that with this kind of picture of a multi-layered situation in mathematical knowledge, and this kind of idea of practice, by taking into account that we have working knowledge of several different practices, and of their systematic interconnections, that there is interesting things to be said. What kind of interesting things? I will give you the idea I'll come to the second part of my talk, and I will elaborate some very simple, and I hope you will excuse me for talking about very simple examples of mathematics, but I think they are interesting enough for presenting the ideas. What kind of things? For instance, the links between practices, I believe, restrict what is admissible as further developments. They guide the formation of new concepts. Actually, one thing I'm very interested in is elaborating a theory of mathematical concepts as part of this kind of approach to mathematical knowledge. I really believe I'm definitely one of those who are much more thinking about mathematics in terms of concepts rather than objects. So I think this kind of approach will allow me to at least propose ideas that Marco will be able to criticise and show me that we need objects.

32:30 So they guide the formation of new concepts, and they actually lead the links between practices, lead to objective results. So let me go to the second part, and let me tell you my thoughts, maybe one of the more interesting things. instead of talking about how to elaborate these ideas in the context of elementary mathematics, this is more interesting for most of us, I hope, to talk about advanced mathematics, because there is where we find particularly difficult problems and puzzles for any philosopher trying to do philosophy of mathematics. And actually, in connection with that, I forgot to say I may be wrong in this, but anyway, this is the kind of perspective I am elaborating. In the case of elementary mathematics, there are what I call strong cognitive roots of the material and the frameworks that are being used, strong cognitive roots in the normal agent and her everyday practices, like the technical practices of counting or measuring and so. So that's a very different situation from advanced mathematics, in which I believe that there are hypothetical elements coming into the picture. This is not something I would like to discuss for a very long time, except if you tell me that I really should. Because I have written several times about this and maybe talked several times in conferences, I'm getting bored of this stuff. But anyway, the idea is that one of the key distinguishing features of advanced mathematics, and I'm tempted to say that Euclidean geometry is already advanced mathematics in this sense. One of the key distinguishing features from elementary mathematics is that there are hypothetical elements involved in the elaboration of the in the framework and in the assumptions behind the framework. So things are not rooted in the cognitive setup of the normal agent and her everyday practices, but we are

35:00 rather getting already into a sophisticated mathematical world, where, for instance, we maybe, what kind of hypothesis some typical examples of hypothesis, I always tend to give the same examples, but infinity, I will discuss this. There are That's a hypothetical idea. You may accept it, you may not. But what happens if you are there? Then you are starting to go some kind of analysis, some kind of advanced analysis. Or, of course, the action of choice in the same kind of field. But to give a totally different example, continuity. Continuous analysis. I may be totally wrong with it, but I am in good company. so together with Riemann and Deleckin and Cantor and Hilder and maybe a few others but not, there's also a very large group of people who think differently yeah, together with degenerate mathematicians I believe that the idea of continuous domains is hypothetical we are not even, contrary to Francaret, like to say continuity is not simply given to us from the world but rather superimposed by us trying to think about it. So I think of it as part of the hypothetical. And this is one of the reasons for saying that the gradient geometry is already... Yeah, that's hard. Hardness is given. But not continuity. Right? I'd like to do... Anyway... For the old... What Hilbert was calling ideas... Sorry? Yeah? There are only national numbers. Sorry? Yeah, there are only national numbers. Of course. I mean, the empirical results behind quantum physics tell you it's all discrete, right? Those days? And many theoretical physicists have tried during the 20th century to elaborate frameworks physics in which continuity is not basic. And yet, of course, there is a strong tension inside physics, because the basic framework of quantum physics contradicts the empirical side of quantum physics, because in the framework you are talking all the time about continuity, continuous domains, and all kinds of continuous processes, and pitting that with the other in a very funny way that actually people, when they tell you, when you open the black

37:30 of this. It happens that they are not very happy with it. Normally not said in popularization books about quantum physics, but you find it in the correspondence of physicists in their papers and their attempts to develop different things. Penrose, for instance, is one who is leaving people who have been trying for many years to learn some non-continuous way of thinking. So I think that this is not so strange. Hilbert was saying this in his paper That's one of the typical references for this kind of way of thinking, and he was talking about ideal elements in mathematics. I am using the phrase hypothesis, hypothetical elements, but basically to refer to that kind of thing. So this is a very important piece of the background, the general background. So when we are having this kind of way of analyzing mathematical practices, So, when we get to advanced mathematics, there is typically hypothetical assumptions behind the framework. So, I take this to be a very important result of actually all the foundational discussions. And actually, another important reference for this, Lakatos, the Renaissance of Empiresism in the philosophy of mathematics. So actually, a large part of what he was describing there was quasi-empiricism, and this was, I prefer to talk about the presence of hypothetical elements in a mathematical model. He was calling attention to the fact that you can find people in all of the different foundational positions, very relevant people, so from Vila to Brussels, all of them are talking about this, so this seems to be an interesting, common point. Anyway, questions maybe about this? Some clarifications about this kind of... Should I insist more on the idea of hypothetical conception of advanced mathematics? No? Okay, great. So then I can go to the second part of what I wanted to tell you about today, which is to flesh out this kind of viewpoint

40:00 concrete stuff. I believe that the most important objection to the naturalistic and practice-oriented approach to mathematics is precisely linked to the perceived objectivity of mathematics and mathematical results. Let me, for instance, see the quote that actually which is true from Motion Mathematics, a well-known set theorist, in a book of 1980, he says, the main point in favour of the realistic approach to mathematics is the instinctive certainty of almost everybody who has ever tried to solve a problem. He is thinking about real objects, whether they are sets, numbers, functions, or whatever. and that these objects have intrinsic properties above and beyond the specific actions about them on which he is taking his thinking for the moment. So this is in one of his books that is well known from 1980. Of course, so you have this perceived objectivity and this perceived existence of intrinsic properties, what the mathematicians are studying, and of course the simplest way of reacting to this is to explain it by adopting realism about mathematical objects. So, this is, in my view, this is to extend to mathematics the naive semantics of a large part of natural language. Maybe I have not taught natural language, but to extend in a naive way the naive semantics of natural language through the case of mathematics and then you become a realist. But this is too simple. And so when I contrast this idea of Moshevakis with what Wittgenstein wrote, he already was thinking about people thinking that way, traveling for a long time. And he says, what a mathematician, this is Wittgenstein, what a mathematician is inclined to say about the objectivity and reality of mathematical facts is not the philosophy of academics, but some people of people agree with me. And, of course, I am not a big and stunning, I am not, my point is not to cure mathematicians from this, so I'm not going to do that.

42:30 But, I, I, so I will understand treatment in a different way from videos. Treatment in the sense of, well, let's try to be more serious and try to elaborate a slightly more sophisticated understanding of what is going on. And my point is not going to be at all to deny the objectivity of mathematical results, but to suggest that we can understand it in a different way. And here I give I think, sorry about this. Of course, I happen to know something about this and actually when you are interested in in advanced mathematics. It is the case, I believe, that what is related with the foundational discussions and the foundational thinking of the late 19th, early 20th century was particularly relevant. Because there, this kind of question was at the center. Unlike the case, I believe, with categorically, it is about something different. Not about making explicit and trying to fix somehow the hypothetical assumptions classical advanced mathematics start trying to elaborate something different, new conceptual perspectives, unifying perspectives, and all kinds of... But it's about something slightly different. So I will give a couple of examples from the most basic things about sets and infinity. So let me distinguish two situations when we think about the production of hypothetical assumptions about infinity in late 19th century mathematics. One has to distinguish, here I'm talking very much, not as a historian, but this is going to be a philosophical kind of, and I will be talking about the normal agent, so I'm thinking about what happens when somebody learns normal agent. It's very important to distinguish this two of introduction of the infinites. One thing is to introduce a simple assumption of the existence of infinite sets, like this set of natural numbers. And this other thing is already going far beyond that. So I will distinguish these two levels, and I will try to say a few things about each one of them. And how the interconnections between practices

45:00 that, actually, mathematicians discover stuff about what is going on when we do this, right? The idea is simply, of course, this is to accept a hypothesis, right? Also, for instance, Penelope Madi agrees with me that this is hypothetical. But the point is that what you can say after introducing a hypothesis is not arbitrary. There are things that you are forced to say. So you discover stuff about stuff you are inventing. The dichotomy invention discovery is too poor. It doesn't help at all. We would need at least a tripotomy, so pure invention, pure discovery. And the typical thing that we do, we do mathematics, which is not... Neither one of those, right? And there is nothing mysterious about this. I believe by this kind of description I'm making it very clear and very non-mysterious. When you accept the hypothesis that there is a set of natural numbers, At the background you have a lot of knowledge about what is going on with the natural numbers. You have all the previous knowledge and practices about natural numbers. So, for instance, you may say, well, you know, we have the natural numbers, blah blah blah, dots dot dot, and we have a quotation. And we have the set of network numbers. And of course, this, I mean, people like Frege and Dedekin were very unhappy because they were logicians, they were very unhappy with the trick of the dots. But from this perspective it's not so bad as a trick. I mean, it simply refers, makes the point that we are introducing And it is linked with some practices that we already know how to go about producing numbers, and so on and so forth. The assumption we are going to introduce is that there is a set of all natural numbers,

47:30 so there is a completion of the process. Okay, so we introduce the idea of n. As soon as you introduce the idea of n, at the base you are forced to accept also definable sets of numbers, definable sets of natural numbers, like for instance the set of prime numbers. So if you are going to accept a set of all natural numbers, there is also a set of all prime numbers. There is the set of all numbers, even numbers, multiples of whatever you like. So 10 to the 10 to the 10, the multiples of 10 to the 10 to the 10 form an infinite set. And of course you are immediately forced to, like Galileo was, to start thinking, what do you find out here? we have a one-to-one correspondence between the set of all-natural numbers and subsets and so this is not something you are this is something that I believe is being forced by the systematic connections between the new kind of practice that is starting to be developed and the previous ones so as I say, the argument is when you introduce the set of all-natural numbers you are forced to accept Well, the sets of definable subsets of natural numbers, that's a very small one, definable subsets of the set of natural numbers, this is a lawful kind of infinity, or you can call it a tamed kind of infinity, which is very far from what makes set theory interesting. This is not very interesting. But you discover that there is a one-to-one correspondence between the set and its subset. Actually, maybe somebody would make a question about Galileo and I would be happy to give my opinion on that, but I will skip it right now. As you may well know, actually Derekin decided to turn this thing, which at the beginning seemed to be paradoxical, into the very definition of infinity. For a set to be infinite, there have to be 1 to 1 correspondences with its subsets, with proper subsets. So this is actually part of the definition of infinity. A second example of how the introduction of a hypothesis by connections with previous practices leads to objective results.

50:00 When we introduce, this is a totally different kind of hypothesis. This is the idea of, we have the domain of natural numbers, and now we are hypothesizing that there is also the domain of all possible subsets of the natural numbers, definable or not, right? I mean, if you know about the foundational discussions, this was at the core of the problem. If you read the famous letters between Béa, Bébé, Hadamard, this is what they are discussing about. You cannot do that. Anyway. But people did it, of course. People introduced this hypothesis. And so the idea is simply to remain in the easy example of the natural numbers, you have not only the definable subsets, the definable sets of natural numbers, like the set of primes, but you also have all possible subsets, arbitrary, so-called arbitrary subsets of the natural numbers. And you have the set of all such, right? Which is our set of n. Now, interestingly, once again, when you accept this hypothesis, it is not arbitrary but necessary to come to the conclusion that this is larger than this. So Cantor's first result in set theory that, of course, we can flesh it out in a different way if we go to the actual historical situation in which Cantor discovered this. But anyway, this is the first discovery of Cantor about set theory that, basically, this is essentially the same as the set of real numbers, and that that set is bigger, a larger infinity than the set of natural numbers. There is nothing arbitrary about this. It was forced by two things, of course. You can always deny the premise. No, no, I don't accept it. Then you are not forced to do anything. If you don't accept this, like for instance Morel or other people, you are not forced to accept that there are different cardinalities in the infinite. But if you accept the premise, then you are forced by connections to previous practices, to accept the conclusion as well.

52:30 Maybe now, to give another example. And actually to show you, and this is interesting because now I can make connections with the topic of the previous two days. Actually, I think one of the interesting things about discussing set theory, many people, or several people, are very uncomfortable with philosophers of mathematics discussing again and again set theory. I think it is normally because there is some kind of stereotype picture and again some kind of disembodied picture of what set theory is and what it is doing as a systematic foundation for mathematics. One of the ways of counteracting this is not simply to avoid completely to talk about set theory, but to actually contextualize it, to make it embedded in a richer history and to show, for instance, the impact of advanced classical mathematics also, right? People were not just introducing anything they were thinking about. There were actually big reasons, and some of them are not articulated by the practitioners, but as a philosopher I am tempted to say that this is still going on in history, right? So, let me give a third example, trying to show why this was actually almost inevitable to accept that at the time. Yeah, I would say inevitable, which is a strong given, of course, what people were interested in doing and preserving the mathematical knowledge. There's always the possibility that they have gone all constructivists. And everybody today would be constructivist mathematicians except for a small group of people who are crazy and accept power sets and stuff like that. Today is the opposite. To discuss why it is that way is a very interesting discussion, again involving mathematical and non purely mathematical practices, but that's a very long story. To keep things under control, let me introduce a typical community agent right now. Instead of a normal agent, we are going for the typical community agent in the second half of the 19th century, who accepts the real numbers and analysis.

55:00 It is a very central practice, or actually a set of practices, but at the core of late 19th century mathematics. One of the things that we were discussing the other day was how people were using at the time, by the late 19th century, the Dirichlet conception of functions, what it was called the Dirichlet idea of arbitrary functions. So this is part of the story, actually one can even argue that it was a natural kind of idea, given some background, and I'll give immediately the reason why I think so. The reason is in the connection with the real members, because in fact if you about what is going on with the real numbers when you are using the very old, by this time, representation of the real numbers by decimals, what you have is the real numbers in a unit interval are exactly the arbitrary functions the natural numbers ciphers, right? So all possible ciphers correspond to real numbers. This is the idea. So the mystery of the continuum, this was still, of course, has always been a very fascinating idea, but that was there. In a unit interval you have all possible decimal expansions. So all possible assignments of ciphers to the first, second, third, fourth place. So this is just a function from n, an arbitrary function, from n to the second function. So this was actually a good reason to accept arbitrary functions. Nobody argues this way. I would love to find a mathematician at the time arguing this. I have found no one, and still I think it's in the back of, it must be in the back of their minds. Now being very ahistorical.

57:30 The situation is exactly the same as here. Instead of decimal, you can go for just zero from once. and this is actually what Cantor uses in actually the one thing that I didn't mention on Monday when I mentioned Cantor but of course it is absolutely relevant for the discussion of Cantor and arbitrary functions, when he proves the Cantor theorem, he does it this way we take arbitrary functions from the set of natural numbers to 0, 1 and we can prove that the set of all these functions is actually larger than the set of natural numbers. So here he's doing this to use this set and show that the cardinality of this set is bigger than the cardinality of this set. And actually Cantor is aware, although he doesn't mention this in publications, but he is aware that actually this is the same as that. The power set of n correlates very simply with the set of all functions from n to 0, 1, because you can take whenever it tells you that a natural number is associated with 0, that means it is not invisible. When it tells you that a natural number is associated with 1, it means it is invisible. And then they are the characteristic functions of all possible subsets of n. So actually, he was aware of this. And Russell was the first to make it explicit in publication, but actually So for this kind of reasons, now I'm talking about a typical member of the community of mathematicians at the time who accepts at the very least the real numbers, so also bio-expression have to think this way somehow or another. They have good reason to accept the hypothesis of our domains simply out of what they know about the real numbers, and they can good reason to accept the idea of arbitrary function of the region. I have thought about, so the proof of this is the famous, well actually the famous Cantor theorem is simply, it's so very clearly also that you can apply this same result,

1:00:00 of the same kind of reasoning method, reasoning, when you have an arbitrary set, and you take the functions from s to 0, 1, and then you can prove that that set of functions is actually bigger than the third case. So, this is Kantor's theory. The proof is very simple, but it's diagonalization. So, the famous introduction of the diagonalization method. I have thought about giving you another example from Kantor. And now coming to the real historical agent. But I don't know if you may be tired already, you may want to start a discussion on what I have already proposed, or do I go on? I'm happy to do whatever you like. How long is it? Sorry? How long is it? It could be 5 minutes, 5 to 10 minutes. 10 to 20 minutes? I'm very much, I don't know why, but I have an internal clock. I normally know what I'm doing with time. Okay, good. So, fourth example. Now, up to now I have been talking about Cantor as a member, typical member of the community. Now let's go further into his metamathematical views. Let's flesh out more his ideas about how to do things, and his values, and his images of mathematics. actual historical agent here doing stuff with previous frameworks and new frameworks that he's developing in this case. And actually, once again, I link with things that I proposed to you on Monday because I mentioned in the handout I gave, there was a Thank you.