Generality in Leibniz

0:00 It is really important that you first put a piece of mathematics at the beginning. But I think that you have to be a mathematician at the beginning and not at the end. Ah but no, because if you put the analysis of a text at the beginning, it then allows you to send back what you develop. I think it would be better to put it at the beginning to enlighten a certain number of developments that you are doing. But it is also a matter of process. As far as I'm concerned, the approaches that I often follow are, I read the text, I start from the text, and then from the problems of the text, I will look for other authors who can help me with the analysis. Because here we have the texts of the authors before and the examples after, so we wonder if it is the NITS that serves you. I'll start with a few examples. First of all, there is the demonstration of the positive motion of the parabola. The parabola. Okay. That's right. First of all, and then in the second chapter. After the first chapter, we'll see the process. It's longer than the graph of the hand. Because this is a book and this is an article, so I asked myself the question also to know if I said it comes from the format. It's possible. It's possible to do it like that.

2:30 I think the introduction gives a false idea of what we are going to find, in fact, we have the impression that I had said that the plans are the same as those of Nancy Parchette and Berger. In the middle of the book, there is a chapter on mathematical physics as an example. I find that very interesting, but this structure that I have just said is therefore ... Apparently, I read it well, but it's not the one that was announced in the introduction. We have the impression in the introduction that we are going to read Caius Palli on Leibniz, and the general ideas, the link between these general ideas and Leibniz, that we just said, is not at its start, and we understand it only at the end. No, it's really at the beginning, and then after the introduction, it would be better to say that later.

5:00 I think what you are doing is more interesting than what you are announcing. It's better in this class than in the other. By the way, in my opinion, under the notion of object and object, there is a second term, condition. And that, I can put something under it. Yes, absolutely. Do you have another question? I wanted to ask you, physically, about the idea of an analyst. The general question is what is the condition of the subject in this case, according to the integral framework, the conditions of the subject are more general, more abstract than the subject, but in several situations, we find that What I wanted to say is, first of all, I wanted to say that I had a sense, a sense of what I understood, and there is already now a little confusion between these two phenomena.

7:30 There is a tension there that I have not yet predicted. So when she says that afterwards, she analyzes this. What kind of literature do you have? Which authors did you have in the 17th century, at Aristotle's? I ask this question because in Blanchet, Robert Blanchet, who wrote The History of Logics, he said that at Aristotle's there are different ways to understand this. You can understand it with a more general P and an S, but you can also understand it in a different way. The name of the net is called, right? Both of these concepts are related to mathematics and physics, and they say that when we have a abstract concept, as I said in the essay, we also need to have a more concrete concept, and at the same time, we need to have something that is clear. And I said, when I looked at that... It may also include a few things, and then the other idea. That's it. In the most sub-language, terms are used together to take it into account.

10:00 In relation to what you said, I would like to have an illustration in relation to the magnetic situation. And so that's where I think that if you had put an example that you develop and then you develop an analysis, it would be much easier for me to understand how it relates to the subject. Yes, but the problem is that I have an example, but it's a long chapter. No, but I thought it was an example at the end. This is a new chapter. No, but here you have an example. But here you have an example. Here you have an example. So, can you take... With this example, I tried to understand what to do with Kahn-Weil in this context. An example first, and then a contraction, and then the isolation of the concepts of genome. It would have been better to do it on someone else than Leibniz. It would have been a pretext. No, because he is at the same time a mathematician and a mathematician himself. Do you think... I think it's possible to change the order of 2 and 3, I don't think we can do it like 2 and 3.

12:30 I think it's possible. I think it's possible. I think it's possible. I think it's possible. I think it's possible. We can't put the chain in the first place because we need instruments that are already in the first chapter of the analysis. All of this is presupposed for the description of the chain, of the hybrid objects, etc. So we have to keep the first chapter in the analysis and say that we present it as an instrument to understand. In any case, the chain could easily illustrate the concept of Leibniz, without any problems, and end on a more methodological discussion. Because here, it's true that it completely overshadows the case of Leibniz. Me too, it caught me when I read it. As it overshadows the case of Leibniz, we start talking about banquets, freshness, we are a little lost. That is to say, as it was announced, it was something about Leibniz, so we say to ourselves... In fact, we use mathematics as a tool to think about mathematics. But as in Renaud's work on chemistry, we realize that mathematics is also an object of mathematics. So I think it's better to end with a methodological discussion, where we can generalize once we have shown that mathematics works as it says. We have to start with the idea of a mathematical philosopher, showing that his philosophy corresponds to his practice, and that what we find in both, in this echo, we find in the most general relations, we find at the same time in contemporary philosophical and mathematical philosophies, but that they themselves are the most contemporary philosophies, in the chain, in the university, etc. I think it's going to be more difficult. At this point, it will appear at the end. I wanted to say one thing about canonical objects, because I think that I'm using canonical objects in one sense. One sense, however, is a relatively concrete sense, you know, in analytic geometry, you can have a polynomial equation, and depending on where you put the axis, it will be more or less messy.

15:00 So you try to set it up so you put the attributes in the right place so that you will have the simplest form, and you call that the canonical form of the equation. There's also, when you use representation theory, and one of my examples is for a couple of scientific chapters, you're working with matrices and there's kind of a diagonalization. It's a canonical form for this big horrible matrix that you get. And when you put it in this canonical form, it exhibits along the diagonal the most important elements of information. So it's mathematically extremely useful to put it in this canonical form. The kind of thing I had in mind when I say, in mathematical practice, canonical forms emerge, in which, when you look at it from a geometrical practice, it involves triangles.

17:30 Well, I just think that's what used to be. I mean, there is a sense of this triangle. It's more than that of any other polygon. You know, the reason why I'm stressing this is that, in my view, this is what we notice in the ninth century, that the triangle is the most important. So, this is important to remember. Precisely, it's really not to be as clear before. It's just an idea that each process recognizes objects canonical and objects can be intrinsically canonical or canonical with respect to one understanding of mathematics. And that's somehow for one mathematical practice, let's say. Yeah, and maybe that's what I'm trying to say is that I think I'm using in both senses. And I'm also not sure what it means to say that an object is... Except that if there's any technically canonical object, I think it would be a circle. But probably the notion of canonical objects depends on how many canonical objects there are. So even though the circle may be an example of who you are, what that statement means may vary from the other.

20:00 And it seems to me, in which facts would you say that it's the same thing, an example that is used when you speak of economical form for an equation and economical function? So, I think it's also, for example, a polynomial and a polynomial equation, like a well-formed formula, become mathematical objects. You could say that it is also a canonical object. Well, in this, I'm sorry, I'm just challenging you to... No, that's great. I mean, as I said, you know, I'm like, that's fine. There are many different types of mathematical objects, because one is simply an object, because we do it. In other words, it seems to me that the fact that you have a canonical equation is suddenly kind of equation among equations, whereas when you speak of a circle as a canonical object, it's not the same... Okay, so let me go back to that then. A canonical equation exhibits certain information about when things occur in an especially simple way. It's epistemologically efficient. Now, circles, the circle usually plays the role of this, usually plays this role. And then we come up with, in every era, we come up with different... We use full ways of representing circles and inserting them into representations. My favorite example is the way the circle becomes the place where we generate sines and cosines and trigonometric functions and things that are longer and exponential now.

22:30 So it's like we keep finding new conditions of intelligibility. Of this canonical object, every new, really important condition of intelligibility that we find, it reinforces this canonical yet. I'm not convinced that this is in science. Do you think that we agree on the fact that we have two subjects? Do you agree that there are options? Well, that's kind of what I started out saying, in some ways, that I thought that I would be using the subject of canonical options. But in the text, I know you use canonical options, but in what context? In what? Does this ambiguity go back to, for example, the use of the name of Schmitt? So the idea of the canonical object is that you have something which will crystallize different points of view, and it resists, so it becomes canonical because you want to understand it, and you can't work with it. We just vary the point of view, and the example of the shape is great, because we have the physical point of view, we have the differential point of view, we can use proportion, we can use geometry, and so forth, but we don't, one thing is the discussion about what is the canonical object in this book, and another thing in this article, because in this article, if we put chapter three before chapter four,

25:00 I think we understand very clearly. Only one meaning of mathematical object. But only meaning? It is only one meaning, which is the... In fact, this is the... when you define a canonical object in the beginning... No, I don't know if this is useful. Page 4, when you look at the definition which is given of a canonical... so the general context is... Mathematical analysis lends itself to generalization in mathematics, and so the idea is that we have problematic items, mathematical things, so we want to understand them, and so we confront that, that some objects, we claim that some objects are exactly like this, they remain stable, and we try to understand it more and more. This is the only definition. That's enough. And if you have a problem, for example, with the right triangle, you can just get rid of the parenthesis because you don't need it. You don't even have to say, like, the right angle or the circle. You just claim that there are canonical objects. And then you give as an example and you say it's not to be constituted in any way. Some canonical objects appear at some times for different reasons, and the example that you see this time is the example that you treat in the chapter, and the chapter is clearly how Leibniz comes back and back on this example of Le Chesnet, and varying the different point of view to understand the different conditions of intelligibility and how it serves a certain idea of what is in general.

27:30 This is a very specific point of view on this object, which is not one point of view, but the fact that you have four of them, four of these components. So it's a very specific idea of what analyzing is. I found that quite clear. Because if we read the methodology, we say, okay, it's a very... General point of view on philosophy of mathematics. We have canonical objects. You know what? There's also quite a bit of discussion in my book about canonical forms. Canonical forms of this. Canonical forms which are chosen because of the way in which they... Here's another kind of example. Go on with this sentence because you just said something which seems very important to me. Let me explain to you why this is a hard proposition. I'm going to talk by caveats now. We only have access to mathematical objects via our representations. There is no best representation. I think even the term representation doesn't quite capture your point fully here. It is sometimes indeed pretty straightforwardly at the level of representations, but it's also, it seems to me, at the level of background heuristic perspectives, which are also kind of cross-fertilizing. There's a further level.

30:00 Which the very notion of representation is itself. Yes, yes, that's precisely my point there. Functioning in at least those two. So one does sometimes have to look at the background, the way that that's cross-fertilizing to determine when, you know, you're going to say that perhaps when you, I think this possibly connects with the question that Karine was asking about, the distinction between canonical forms and canonical objects. On the background, on the side of the project, whereas what you call a mathematical form is rather on the side of the project. Yeah, and I'm just saying that I don't think you can put them apart. No, you can't put them apart because there is always a multiplication, but that doesn't mean that they are the same. That's also what you said. It's not that you can take them apart, but you cannot take them one at a time. So it seems to me that you have a proportion between both sides of things that you introduced as impossible to distinguish.

32:30 But I would also say that there's something peculiar about mathematics in that it's representing all things. And when they do that, for example, some philosophers make the mistake of thinking that because it's precipitated by a character that it's transparent to you. And then it remains to be seen whether the object is going to become canonical. Is it an interesting object or is it a trivial object? A lot of times this, of course, generates trivial objects and everybody forgets about them. Yeah, which is exactly what I had in mind by drawing attention to the distinction between kind of heuristic perspective operating in the background, you know, concretized representations, that I think that's precisely where that distinction does become useful because it's salient to the question whether it is, does turn out to be interesting in the long run. It may well take quite a long-run research program before it's finally decided that this particular object is canonical. Exactly. That's a very good example of what I had in mind. ...partly internal to specific research programs in set theory, to the interests of a set theorist who works on large kernels,

35:00 but against a broader one when one's looking at the cross-fertilization of heuristic perspectives on the... It may be that it will be seen if there's a shift in the which brings about a kind of reintegration of mathematics in a way that makes the large cardinals program seem like a cul-de-sac which is precisely I think one of the possible consequences of what's going on at the moment in homological algebra won't go down that alley but that could be the kind of development which would certainly affect the status whether these things turned out to be canonical forms. There's another example, I think that one really has to do a lot of detailed case studies, that's another fascinating example because Hamilton thought he directly had, which was a strongly canonical form when he got the quaternions, but then given the way that late 19th century algebra developed, it seemed to be something very idiosyncratic and rather even perhaps slightly marginalised until Clifford came back in. And that's something, again, the status, the quaternions are a very interesting case because it seems to me that they're something which would have appeared to Hamilton to be a good candidate for a canonical object, but which later came to be seen as being much more a candidate for a canonical form. Well, I'm trying to understand exactly what it is that you... That's why, as I say, I think an awful lot of detailed case studies are going to have to go into articulating this distinction. I'm not saying it's not a very useful distinction. Another one, another very interesting case study, I think, would be the so-called objective number theory program, which brings back the treatment of arithmetic in synthetic differential geometry, which again brings back in the circle. Absolutely, it's a canonical object because the integers are reconceived as essentially winding numbers. Once again, the circle comes right back in.

37:30 I said that generality is in the variation at the topology. This is what is claimed in page 4, that is to say that dimension analysis differs from cartesian analysis precisely on that point of view that simple is not the fact that I reduce something to simple notion, but the fact that I try to find fundamental conviction of intelligibility of an object. If this object is the object of several points of view, I need to have all of these points of view. There is more about generalization than about generality, but it is particularly difficult for me that the two words, generalization and abstraction, are included from the beginning of the text, but it is really an explanation about the distinction between the two. Later in the text, in the second part, I think I'm pretty close to a count of what I understand for a count is in the Chinese text where you begin with a problem

40:00 A relatively restrained object and a relatively restrained algorithm. And then you see these kind of standards. And what I'm trying to say there is, in some textbooks, that is the way the presentation, the presentation of some of those textbooks proceeds like the literature. But in the other text, it says, here are the axioms. Now let's look at a recent data problem. You can find both textbooks, if I can. Both textbooks are really important and influential. And they seem to be important and influential in tandem, together. We found two ways to do it. In the works of the same form, for example, the foundations of differential geometry, and that's because one, you get a paper already on the abstract of the book, which appears next year, in the book, the abstracts, come up the bottom pages, examples, and slowly doing up the concepts. You know, it's probably in that book, but I don't have my name in that book. But I agree with you. Do you know what? Just a minute, sorry. Math book? Do you know? Actually, don't forget that that's very interesting. Oh, mathematics, it's an archive. Yeah. Okay. We'll check this. I agree with you. It is often two different moments of the... This is an exhibition of the theory, because what this thing calls part is giving us the elaboration of the generalization, and afterwards there is a mathematics collection group and all this theory.

42:30 You know, they're not completely, it's just, it is a way station. These are summarized in a very nicely organized assembly for that point in time, but we don't really know. So I don't think I've still answered your question about how canonical logic is related to generalization. If you think of the role in which, now I'm going to say something and it might sound trivial. To me, it's not trivial because it's related to the point I made earlier, which is when you prove the Pythagorean theorem, you have to prove it about that triangle. You can't prove it, you can't denote that triangle and prove it about that triangle if it doesn't have any form. At the same time, that triangle represents all other... Johnny West represents a specific triangle, iconical, and it represents an infinite number of other triangles that resemble it all in certain ways, symbolically, and has to do both at once in order for it to prove to work, which means that it's playing an essential role in the process against the...

45:00 If physics is what it is, then it's a natural moment for the state. Yeah, but here, in France, what you call the time for a different music? On which island is the representation of a time? Well, it's that representation taken iconically. Yeah, standing for all kinds of times, all kinds of different times. Yeah, I, I, yeah, you know, I mean, the problem is, I remember once having a conversation with Louis Riemann. I said, there aren't any individuals to come to this class. And he said, of course there are individuals in mathematics. They're just different from individuals in other fields. For example, you might say they don't really obey the principle of the identity of the soul. But you know what? Individuals in physics are different behaviors than individuals in biology. It works for the geometric example but what would you say about foundations of analysis and what would you say if somebody who wants to teach calculus to people from Spivak's text you start with the axioms of a complete audit field and you work better your individuals there I think there's a good answer to it by the way but I'd like to know what yours is the natural numbers Hmm, I've never understood what the definite article is doing there. I mean, isn't that just an isomorphism class of, you know, whatever satisfies the axioms for a structural, structural axioms for a progression? Well, see what I think about that.

47:30 Not at all, no. Sure, sure, sure. In fairness, my point did relate to the issue of generality because it's precisely the difference between the way that the object's terminology relates to generality in the geometric example. And the way that as far as I can see it, it is much more difficult to say how it relates to the notion of generality involved in the calculus example. So I didn't intend to take the discussion away from generality, but let's stick with generality. Generalities and supporting texts, or do only some of them involve generalities and do only some of them involve generalities or process the generalization to segregate them is something interesting. And I guess you could also say, are there cases where general, process the generalization represents a degeneration? I haven't thought about this.

50:00 I think that David was trying to say something. I think he said that even if the question is not very good for discussing geography, because I think that the version of generality that you are talking about is not exactly true. This is your general theory about how generality occurs in mathematics. What you need is a kind of hybrid approach. In this application, for example, this would work from the camp without any problem. And we wrote about that too. So, this is not specific to language. What you show in the article of the ExoPolitics about impressionism is that language conception of generality takes into account the degree of the city, and this I don't know, of the object. It will play on this many point of view of the object. And so, and you see this as a way to challenge Cartesian version of generality with a big reduction to the simple and so on. So this example is very good in a way because it works for Leibniz and Descartes at the same time. And the example of Lachenet is better because we have an object which is resisting and so we can see the hybridicity of the object and this would be a problem for... Because the way iconic and symbolic doesn't fit exactly together in this case of Le Chemin. When we can't stabilize it, we need different symbolic approach to the same object. Both in physical relationship, ratio and proportion, algebra, we use all of these differential equations. And so in this case, this is a very specific idea of generality. In the discussion, we should focus on this one, and not on your general answer, even if it's very interesting, but it's another problem to know if we accept to generalize this, generalize this to all objects. When you say that, it looks like it. And the fact that you have another point in the article which is more local, in a certain way, there is...

52:30 I would find it extremely important, because I found this very important in the first part of the paper, this point was to determine, and I find it very important to stress that, because there we are in a very interesting Contrast in terms of purposes and in terms of interpretation. So it seems to me that there could be one main line that you pursue via the exchange of parts, to switch between parts two and three. That could help understand much better what you want to say. And it causes different notions that we just discussed in the Nobler article, but that you don't develop, and it would be nice to have this kind of hierarchy. For example, you mentioned the Tentament Anagogicum, but you don't talk much about Harmony and Theological and so on. And it would be nice to have, to see all of these elements. What I read first of your article and then yours, as I said to Tain earlier, I had the impression to find all the justifications that we need to understand all of these things because we need to understand how it will crystallize in something, harmony, beauty, and so on. And the notion of can-be-need-for-a-play is nice for that because that allows the fact that all of these points of view, we can divide them and we can express generally.

55:00 And you just mentioned that like this because you know all of this, but I think for the reader it is better. You know what your remarks make me realize is that you could say that Descartes' conception of methods by reading the complex things in a highly deductive way is simple. So can I ask, because Corrine chided me rightly for not paying careful attention to your text, what you've just said in characterizing Descartes' view of generality. It is to be contrasted with Leibniz in as much as Leibniz is starting out essentially from a more piecemeal methodological approach. Generalization starts from a set of, so you say specifically this is generalization as exhibited in Leibnizian analysis. Generalization is on page three, no, it's on the third page after the beginning of the section on Leibniz, on analysis. I'm sorry, I haven't counted the pages in my text. It's page three after the section begins, or Leibniz analysis. It's the passage which begins, generalization starts from a set of solved problems and asks how successful procedures may be extended to new problems and how the success of the procedures may be explained. It's that paragraph that, just after footnote four. Well, you've just anticipated my question, sorry, which was going to be, isn't that just as true of Descartes as of Leibniz? Okay, yes, and the distinction is broadly that Descartes is pursuing a generalized search for the conditions of intelligibility of canonical items, in some sense ab initio, I don't want to say vacuo, but without...

57:30 There's not the same degree of attention to the conditions of solubility of the specific problems in which those items are involved, which is the stress that Leibniz has. It's the Leibnizian stress on the specific problem situations in which those items are involved in generating the canonical objects or forms. Well, as far as I remember, it's going to be a big reconnection. How does special informals begin to perform on models? Have I got it? No. Okay. What's the term she used? Not just models, but a number of neurological machines. By which she means not just a conceptual model, but she's talking about more something that's physically realized in laboratory, you know, like a highly constrained environment where you get to be able to behave in a certain way and other things that you can try to purify and kind of make nature do something physically.

1:00:00 I'm not sure, first, how you fit your use of complex within this context and I would be interested in your examples because I would then ask the question how That we may have greater generative than what you are talking about and that we need other uses of practices of other types of generative which mean practices of the use of mathematics only. So I'm not sure I understand how, with what we expect and what you expect, how we connect with what you are using and what you expect. If you can try to perhaps illustrate it for me, because it seemed to me that you were saying, that what she was saying seemed to me to be usable also in mathematics, and I understood that the idea of swapping part two and part three worked, made part three easier to understand, so I would like to... Can I put it very, very quick?

1:02:30 Very quick suggestion, is that okay? Isn't the connection that strongly between what Cartwright is saying and the specific model of generality that you find in Leibniz precisely the strongly adaptive component, the strongly adaptive component that they both share and which is not present, at least it's not present in Descartes. It's not present in his presentation of his method. It is arguably present in his practice, and couldn't be otherwise if he was actually going to prove things. The point that you're making, for instance, when you draw attention to Leibniz's criticism of Descartes in order to maintain the universality and sufficiency of his method. This is Leibniz in the De La Jeunette speaking of Descartes. Sorry, the universality and sufficiency of his method. He contrived for that purpose to exclude from geometry all the problems and the curves that one could not treat by that method under the pretext that they were only mechanics. This kind of exclusion, now this is you, this is no longer a quotation from Leibniz, this kind of exclusion however cuts off the process of generalization artificially and that is why Leibniz criticizes it. It seems to me that this is another point. Do you? I think it is the same point because it's precisely the adaptive point that Cartwright is, I think that's precisely what Leibniz is saying here, that it's precisely in concrete problem situations that you come to understand, you know, the way that, you know, the important. Distinctions between the different kind of geometrical and mechanical objects and whether they are candidates for being canonical objects or forms. I thought that was... I mean, I would like just to... Okay. Sorry, this may be a complete red herring, in which case I apologize. The truth is that when you have something in general, you're not... Something very relative that, if you think of N-couples of real numbers,

1:05:00 Comparative real numbers as representation of planes, and coupled real numbers as representation of entities. There are a number of important reasons why this representation doesn't actually represent these living planes. Let's just say that these living planes. But when I show at the beginning of a topology textbook, when the person is discussing, when someone's program is discussing what they have to discuss, they invoke essentially this set, it's just a set of ordered pairs, but to explain what they really need, they draw a picture of... And the whole chapter includes plenty of why they had to draw the picture. This representation and this representation, I identify this representation goes to Escher for Cartwright. This is the concrete representation, the more iconic representation. And this is the more abstract representation, but for them... What they have to do is fit out the presentation with the more concrete representation, which means that not only do you look for the conditions of intelligibility, you have to be able to know what you're talking about.

1:07:30 I have a question, but perhaps you can try to meet the rational bridge to Michael's question, and then I can go back. Sure, but just that it seems to me that the rational bridge, okay, well, just that I thought that the connection between Cartwright's remarks about models and Leibniz's remarks about generality was precisely that Leibniz was a bit inspired by the cogency, what you term in your text, the cogency and urgency of the specific examples, the problems that Descartes had excluded by fiat. As belonging to mechanics rather than geometry in order to maintain the purity and universality of his method. Whereas Descartes, sorry, I beg your pardon, Leibniz, precisely wanted to find in the conditions of solubility of those problems the right route to the understanding of the general conditions of intelligibility and the identification of canonical objects and forms, that it would go via those problem-solving strategies. That's what I saw as the bridge, very roughly. Back to the point about Leibniz having, as it were, a more piecemeal, bottom-up approach to the characterization of generality.

1:10:00 There are a lot of different types of problems that we have to deal with in order to solve the problems that we have to deal with in order to solve the problems that we have to deal with in order to solve the problems that we have to solve. He is something for which he is going to use concrete drawings and so on to work with. So this is how I would say that there are two meanings of concrete that are involved, it seems to me. But that's how I understand it. But I agree with you that the passage you read... And I quite accept that there's more than one sense of concrete going on here. I think concrete, as indeed Emily herself has drawn attention to it, and Nancy Cartwright as well, precisely the axis, abstract concrete, is one which allows many different dimensions. It's not just a simple spectrum. It's a highly polarized spectrum. I certainly make a claim that's what's happened with set theory.

1:12:30 The concept of generalization, as Descartes' concept of generalization, is both, and I suggest that perhaps you might be wrong, so evaluating seems to me that you seem to say that you have to be strict to those objects from which the lessons need to work, right? That's the concept you're putting in, and I have often, the question that arose was, you know, are there Are there processes of generalization, you know, that represent distortions or generalizations? Well, it seems to me that she'll change the result. Do you know who the person who should be here right now is going to be? Because she's really good about reminding work in general.

1:15:00 You know, in certain situations where you decide... To restrict yourself to certain representations, and you're very strict about it, but that can also be a proof. That can be proof from mathematics, too. But there's, you know, and that's to defend Jacobi. Let's see what we can do with these means. Now, of course, he does think about the answer. He forgets he said that. Just say, let's see what we can do with these representative means. With regard to the approaching of uniformly of these set of objects, they must be receptively contrived so as to fit with the method, whereas Leibniz tries to sum in general any particular object that is not any particular problem. I mean what of the more context-specific kinds of generality that arise in these restricted situations is exportable usefully to give more general methodological guidance and what is not? And that is a question which would... It doesn't come naturally as the background in Leibniz, but it wouldn't come naturally to Descartes, because Descartes is really assuming, because he has an overall reductionist program, that the method has to be universal. Yeah, sure, sure, absolutely. But he makes use of it precisely, as you say, for internalists.

1:17:30 But talk of the talk of being true to the nature of the object seems already to disclose a very strong ontological agenda behind your position, and that's precisely what you want to... As long as you don't say there's... Well, it's a nation of God's awareness for Leibniz and Descartes, not of ours. Well, this is much too far away from the text, but I'll get into a discussion with you about that later. I would dispute that. Two and three. One is the fact that he's looking for harmony in order to express generality, and I don't like it very much. And the other one, I think, what is interesting, and then there's the one in connection with the discussion of the topology, but what is interesting in the first part is the fact that we are using strategy to serve and to express generality. And this is the classic. I mean, it's not in a way. If you want to do it in an aristocratic way, you have to do it this way. You have to delimitate the domain of all gates, and the universal dream would be the fact that what you say is about all of them. So that's the first strategy.

1:20:00 Then the question is, is this strategy enough? And what is interesting is that because language is very sensitive to the fact that you are always different from your representation, you can't do this. Because as soon as you delimitate, there is another representation of the same object, which is, so then the question is, how can I express generality? And what Emily proposed is not to go to the harmonies, but to the objects. That is to say that if I find an object, and this is very like Nietzschean, if I find something which is stable to different representations of the kinds of objects... The general point of view will be the crystallization of all these points of view to the objects. And so I think that this is very clear. So the strategy of generality and harmony will be, it's the only name that we have to say different heterogeneous representations fit together. How can you say it the other way? It's harmonious. And then it's another discussion about CAP. It's the question, it's another, it's what can we do even in the CAP. And so this is... It's very well, I think you meant... You didn't say that because, I mean, it fits very well, but it fits very well for the CAP too and... No, that's exactly what George Dickens said. That's a problem.

1:22:30 No, I think it's... And this obviously brings with it a hierarchy of knowledge, a hierarchy of the sciences, and a hierarchy of being, which with all those... But what is interesting is that both Can I just say one thing very quickly? There's a supposition in the background here, which is perhaps too obvious for you to even call attention to it, but it's the supposition that there is just one way that generality has to be, and that certainly operates in the background. No, no, the supposition that there is one may be identified, that the way in question may be identified quite differently as between Aristotle and Descartes, it clearly is, and Leibniz, but there is still this supposition that there is one... There is no correct way for generality to be that is to be sought for. That's what we're searching for the characterization of. There are two strategies. I'm sorry, I haven't conveyed my point clearly. The point is there are diverse strategies but there is one You know, there is one target in view, which is the characterization of the way that generality has to be in order to be generality, and that seems to me to already build in a very strong ontological presupposition, which is problematic. Well, supposing one only ever arrives at partial unification. Well, that's my point. Sorry, I apologize.

1:25:00 We have a connection between the two kinds, right here, right here. That's what you mean, because what is it, it's very specific, the case of Dachshundi, is you write several articles and you vary the point of view. It's not the only fact that you have a specific point, because, for example, in Descartes' Ommetté, you have a specific problem, which happens. You have your method, all the specific, and in Descartes you write nothing. You give the solution, that's it. You give the complete solution. Maybe we could say that one strategy method sets it, and it entails a clover of the objects, and one strategy is object-sensitive, and the work of both strategies, that we worked with, describes the paths of mathematical geometry and the same strategies. My question was about examples and exemplifying and exemplifying in the book that we discuss in the conference, examples, and what you write as you speak.

1:27:30 You can also read this as an example, and that is not what you are saying, and you need to say it. So, first of all, that is the difference between picking out the meaning and what the examples are. Just small questions to... More questions? Because we have to sit down. And we need the excellent questions. The other thing that I find really important is that he assumes that the things we know are unified very precisely themselves, and so he takes us back over time and says, in college we're going to be required to divide them. They present themselves as unifying. And to be intelligent is to exist. So, when you say exemplary, you're going to have to prove yourself in common. The first reaction is, you know, you treat concepts intentionally. And I think if that's true...

1:30:00 I think that's really not going to mesh philosophically with what I'm trying to do because I have to take a couple of steps back because I think that's a lot of things to explain, but I wouldn't want to try to actually carve that out because I think that's too much of a scope because that's what I'm trying to do. There are other ways to approach example making. For example, if we work on an example group, and in fact that being an example about something is having a property and denoting the property. This is an approach that has to be made from three concepts. In the book, many other texts and articles, as you say, would be dealing with the question of example making and computation and the question of irrelevance, for example. So maybe, since you're doing something very different, it would be nice to maybe explain or point it out clearly, why this is interesting. The time has come to send emails rather than leave comments.

1:32:30 Thank you to Emily, who arrived directly from Australia. Thank you very much. Really, really a good discussion. Thank you for all your contributions.