Leibniz' Archimedean infinitesimals (last part) / discussion / closing ceremony / evening conversations

0:00 So, he lets dx be a fixed finite line segment. For all finite dx and dy, dy may now be defined by that proportion there. The fixed dy over fixed dx is equal to dy over dx, where dy and dx are variable points. Okay, so now what you do obviously is you let dx and dy get smaller and smaller. But the big question is what happens then at the limiting point? Well, that means that the same fixed dy can be given an interpretation of the limit when the variable dx is equal to zero, namely through this proportion that dy over dx is y equal to sigma, where sigma is the sub-tangent to the curve. So you can give an interpretation of it if you find this interpretation of the ratio in that limit. So what you have here, even though that's not exactly the same as all the other values of dy over dx, you can get to it. It's basically completely consonant with his law of continuity. Although different in species, you can get to it by getting arbitrarily close to it. The fixed dy over the fixed dx can be substituted for dy by dx in any formula. But since the resulting formula is still interpretable even in the case where dx equals zero, according to this idea about the sub-tangent, the law of continuity asserts that this limiting case may also be included in the general reasoning. dy by dx can be substituted for the fixed dy over fixed dx in the resulting formulas even for the case where dx equals zero, with dy and dx in this case interpreted as fictions. If a third variable, v, is involved, which varies, you can define fixed dv to fixed dx in an entirely analogous way. So to see how this works, let's just have a look at proof of this crucial form of the derivatives. We're looking at proof here of... The bottom of the a, d, y is equal to x, d, b, plus b, d, x, and a, y is equal to x, d, and we start out with b.

2:30 So if a, y is equal to x, b, you start out and you again put x instead of x, b plus b, b plus instead of b, and y plus b, y equals y. And you get that formula, subtracting from each side, it equals a, y and x, b. This is actually now a quote from Leibniz. This gives that formula, or that, and transposing the case as far as possible to lines that never vanish, this gives that. So in that one in the middle, A, fix dy over fix dx, equals x, fix dv over fix dx, plus v, plus dv, the v and dv is still variable, the other things are fixed, so that the only term remaining which can vanish is dv. In the case of vanishing differences, since dv equals zero, this gives that, as was asserted. Whereas also, because fixed dy over dx always equals dy by dx, one may assume this in the case of vanishing dy by dx and put a dy equals xdv plus 3dx. So, in similar instances, the use of the law of continuity is justifying exactly the same kind of procedure as we saw in the quadratures. This approach to securing the foundations of the calculus is clearly very similar to the fundamental solution for which Cauchy became famous, one of our accomplishments. As Watt argues, it leads very naturally to the concept of the function, and even to the introduction later in the history of calculus of the concept of the derivative. Indeed, Leibniz's definition Coupled with the stipulation in 2 that when dx equals 0, the c-counter becomes a tangent, implies that dy, the fixed dy, is identical to the differential of Cauchy, defined as the derivative of x times dx, but Cauchy defines f prime x as the limit of, well, I won't go through all that, anyway, so it leads very much to the... So even though Leibniz's difficulties and mistakes in the dispute about second-order differentials with Mumentheit stems from his confusion on this, as evidenced also in his own attempt to extend the above type of reasoning to justify second-order differentials,

5:00 Whilst we've shown that a correct justification for second-order differentials can be given along these lines, it's just that they will depend on a certain choice of the progression of the variables, i.e., on a choice of which differential is taken to be the constant, and that is what Leibniz didn't understand and got wrong. So, returning to the criticisms of Leibniz's and Newton's infinitesimals with which we began, it should not be clear that Ohm's allegation of a contradiction is not warranted. The above proof, for example, of the rule for differentiation of a product does not depend on an interpretation of dx or dv as actual infinitesimals, nor are they fixed finite terms. Leibniz's differential is finite, arbitrarily small, and variable, and given the Archimedean foundation of its methods, this is enough to ensure the rigor of proofs involving them, as well as to justify treating them as if they are infinitely small and as if an infinitive term can sum to a finite quantity. All right. In closing, I'd just like to make some suggestive remarks about the connection between Leibniz's use of the axiom of Archimedes in his mathematics and his more general philosophical principles. A couple of years or so after he wrote the Decoratura, the thing in 1676, he makes mention in an unpublished manuscript of what he calls his Hoculean argument. And here it is. But here there is a place for that Hokeyvian argument of mine that all those things which are such that it is impossible for anyone to perceive whether they exist or not are non-existent. In the context in which he introduces this, Leibniz writes, this puts an end to all inquiry about the infinitely small which cannot be perceived. What's interesting is that this points out the profound link between the Archimedean axiom and mathematics, and Leibniz's more famous principle of the identity of the discernibles. Since both of them may be derived from this Hoke-Illion argument, so I'll just give a sketch of that now, so I think a better name for the Hoke-Illion argument would be a principle of the non-existence of imperceptibles, so if you apply that now to differences, any difference between two things which is in principle imperceptible is non-existent, so that's just an application of the Hoke-Illion argument to differences.

7:30 Therefore, since indiscernible things are those whose difference is in principle imperceptible, and identical things are those whose difference is non-existent, if we apply that instead to mathematical quantities, we obtain the principle of undesignable inequality that both Leibniz and Newton incidentally used. Newton, you may want to ask me about afterwards, but the foundations that Newton uses for his calculus are, at this level, identical to Leibniz. Despite all the other differences on the surface and all the controversy about calculus and everything else, it's a wonderful historical irony. I've written on this and I'm actually writing a paper with Nicola Bucciardini, who's a noted Italian Newton scholar, and he agrees with me on this, so we're going to write a joint paper on it too. Any two quantities whose difference may be made smaller than any that can be assigned, and thus, in principle, are equal. And another way of saying that, of course, is We have imperceptible but still sizable enough time for one or two questions. Frank? I'm not a mathematics scholar, so this is just a request for information. But I'm already thinking that mathematics also works with apparently infinitely large numbers. Apparently infinitely large numbers. What numbers? Round numbers. Well, that was what I was referring to in the introductory bit about his actual influence. Oh, yeah, the same kind of grammatic actual influence. Oh, okay. And it is in the code. So it's again a fiction, right? So it's going to be, you can act as if there are infinitely large numbers, but only under the constraint that you're always going to be able to justify that. Okay. And is it the case that his actual practice in this regard... I had to leave various pieces out of this paper, it was going to be much too long. One of the things that Knobloch does in his analysis is to articulate 12 rules which he said pretty much constitute the infinite.

10:00 So there are various things, various rules such as infinite plus finite plus infinite, that's what he calls finite. And you can get the Mark-Median justification just like these ones of every single rule. I have a vague recollection that he talked about infinite expansions where there are actually terms that come after all the final terms. No, I don't think that recollection is true. One of the interesting things about some of these theories, again in the paper in 1976, He says, after going through some really nice philosophical reasoning, he says, well, I think what we have to say about an infinite sum is that given the rule of the series and any, I'm not going to say how he says it, but you can get arbitrarily close to the sum by taking a finite number of terms. So he basically gives the model definition quite nice and fine. I'm also thinking that I can reproduce it. The limit of quantum science is basically what it is. But again, that is consistent with the same type of language. Would it be interesting to get into this one, for example, you mentioned a book from the 19th century. Oh, right, yeah, the book's called The Labyrinth of Continuum by Leibniz. He didn't write it, I wrote it for him, but it's a copy of all the bits and pieces that he wrote about it. I didn't get questions if I want to respond to these two quotes. Yes, these ones, the Herculean Arguments are in that compilation, the one that the boss analyzed that I gave the quotation for is in Gerhardt. The language. Yes, so that's an old one.

12:30 What else? And the other one, the day quarter tour of it is in an edition, but not all of it. Do you have the world and the rest of the world? Yeah. Actually, if you look at my website, you might be able to download the paper. I haven't. Most of my papers are on there. Richard, can you just say something about how the Hercules argument, Which has a kind of oddly empiricist flavor to it, how, I mean, the whole page, I mean, it's sort of very ironic given Barclay's complaints about life, there's a lot of Barclay principles here, how these connect up with the principle of sufficient reason, or that's, they can't really be connected. I'd have to think about that. I wouldn't be surprised if there's a connection. I mean, I'm really playing with this stuff here, I'm just very fond of generalizing and generalizing, you know, so I'm taking, this just occurs in the, and he's talking about the reality of bodies, that's the, that's the context in which this, I mean, so I could, I could see given this sort of appearances, planets and lightness that end up, that Newton and lightness end up having. Similar foundations for the calculus. They have very similar foundations. That makes sense, but I'm just wondering to what degree this is really Leibnizian. Well, I have a very non-standard interpretation of Leibniz that is not the standard idealist one, although of course there is all of this stuff in Leibniz that, you know, to be realist is to just be in agreement with what can be sensed and so on and so on, so there is all that side to Leibniz. You know, one has to say, well, he has a kind of double aspect theory, you know, the phenomenon is both what you see from inside, but also has an objective existence outside, and that I just think is the key, it's in his theory of representation, but that gets us into a whole other thing. Just on geometry, doesn't it imply there are no models? Well, because no one sees models. Oh, I see, well, um... But monads perceive, I mean, it depends what you mean by perceive, but monads are able to perceive.

15:00 But certainly, well, you know, monads are not infinitely small things, that's for sure. That's not what they are, right? So they're not to be thought of as like actually infinitely small atoms. No, no, no. God is supposed to produce a prune like that. The engine of the composition is quite involved, and in short... This is related to all the issues you brought up. He takes a very hard line on this thing about the unity of the body and basically there aren't any gluons. Something that gives an identity through time. The gluon is a monad for life. So a monad is that which gives the unity. And the body, of course, is changing. It's a place. The body is different at every single instant. And so the body can't be self-identical. It's changing all the time. So why is it that we say it's the same body, that the body is the same thing? Well, then the teleology and so on comes in, the law of the series that gives you the monad. So the monad is the way, it's in the sense of presupposition that makes sense of a lot of empirical things. It's quite interesting as to what this is. The question here is just whether you can see the monad. No, you can't see monads. You can see the bodies that the monad is in. But then, by the way, the principal, they don't exist. Oh, yeah. I give an interpretation of Leibniz that maximizes the consistency of this position, but I'm not going to eradicate every single one of those. But he has got a theory of the consilience of perceptions, hasn't he, which brings in God as the supreme monad. I don't think there is contradiction systematically. Can I ask one quick question about the minimum, the principle of minimum... It's just a purely historical question. You're probably aware the first, not fully worked out, but the first sketch of a modern 20th century synthetic differential geometry was Schelmslev, the Danish geometrist. Well, Schelmslev had a geometry of imperceptibilia, which he worked out in the 1920s and 30s, which He was cited as a source for some of the intuitions which went into the later development of synthetic differential geometry, specifically cited in the first chapter of Cox's compendium on synthetic differential geometry.

17:30 I just wondered if there was any evidence that he was aware of this text of Leibniz. No, probably not. But he could have been aware of other texts where Leibniz said somewhat similar things, but not the sexual principle, I think. In its initial stages, combinatorial topology was conceived by its founders as anti-Cantorian. They didn't agree with the transfinite and they didn't agree with composing. Things out of point. So they saw in Leibniz, so Weiler was pretty smart, right? He was able to see that Leibniz had some structural continuity in terms of transition, continuous transition that he was going to. It was quite clear that combinatorial topology is, well, certainly in the tradition, as we say, is particularly important. It's conceived precisely as an anti-Cantorian project. They don't think general topology, general poinsettia topology ought to be the correct foundation of topology at all. There's an irony now that you won't find accounts of combinatorial topology in poinsettia topology textbooks. Pretty well, yes. There is an irony. And I don't know what led to the demise of it. I don't know enough about that period. We have time for one last question. The name that popped into my head when I saw that was Fermat. I mean, somebody says, you know, I have an argument, and he calls it a Herculean argument, and I wonder whether it goes somewhere else, that is, is it Herculean because it does Herculean work, or is it Herculean because he had to put Herculean effort into establishing it, and in that case, where's the argument that leads to, because it states a conclusion, in fact. Yeah, it does. I've not found it anywhere else. Maybe it's just because he thought he was cleaning out the Orgian stables. Maybe so, but I think what happened in terms of the development of Enlightenment is that how it comes out is as the law of continuity. I mean it develops into the law of continuity, so does all this Archimedean stuff. That's the point.

20:00 Thank you Richard. So, I'd like to say some closing words with respect to the conference, and I could run over the different talks and try to summarize them, which is what I... Okay, do you have any time? Yeah, I'm tired and you're probably too tired to listen. So what I will do is say something about what didn't happen, and then technically say something about what did happen but wasn't said. So that's two things. And then lastly, thirdly, give a kind of general characterization of the kind of project that is behind the conference. So let's start with what didn't happen. According to Hardin, we're always within a situation, and the situation arises because of uniformity and unifying. There's always a structure there, and the structure is necessarily classical, Boolean. But it's carried by, so I'll be going very slow here, I'm not very subtle, and I also haven't grasped entirely what you've positioned, but we'll get to it later, this situation is carried by reality in itself. So outside of the situation you have reality itself, and that consists of inconsistent structures. So the consistent structure of the situation in which you always already are, the reality of it is inconsistent structures. Okay, so this is kind of like, well, and this inconsistent structure, because you are always already within the situation which is consistent, the only way you are in contact with reality itself is in a way it has already escaped something like that. Okay, and then there's somebody else, there's somebody like Graham Preece. So again, I haven't entirely crossed, so I hope I'm referring to his youngerness of thought, and I have to go very quickly here, but so I think part of the way Graham Preece could agree, I think, with what you've told, he talks more about totality, but maybe he couldn't agree with that.

22:30 The terminology of Badiou, the due situation, it's all already consistent, but according to Badiou, there necessarily arises contradictions that we can't get rid of, and because of paraconsistency, this contradiction doesn't necessarily destroy the situation, destroy the consistency of the situation, no, it allows you to be both inside and outside of the situation. So, it allows you, now I'm interrupting a lot here from you, it allows you a certain contact with the reality in itself. Well, this is, so you can see that there's like similarities and differences and it would have been interesting to have, for example, a consultation between these two thinkers. There are also dichotomies and distinctions, and oppositions like analytical and continental, or consistent and inconsistent, or historical and philosophical, and philosophical and conceptual. Both are, I refer to them based on the beyond the limits of the ballpark, are both examples of where there is an entanglement. Both historical, philosophical, work, conceptual work, and the use of formal machinery from recent formal machinery, from logic and mathematics. So, this is, so absences can be as pertinent as presences, and so, to be on a more general level, we would like to have, for example, also transcendental idealist presence, like Michel de Beauvoir. We could get on there, we could end up, yeah, I don't know, for example, certain delusions, certain delusions in realists, like we have in England, and to enrich this discussion.

25:00 So, so that concludes my point one, the characterization of what we didn't have, which isn't as much part of the situation in which we are. What wasn't said, or what was only said, wasn't said here, if you like, it should be said here, so which was said on margins, or in the margins. So as one speaker said, it is important to sense for a few from where a certain, from where a speaker is speaking, especially if that speaker is... Most of the presentations remain within the boundaries of Protestantism. What you're allowed to say in a situation like this and well this is always a gain and a loss so the rigor of specialization is of course a necessary component of research and obviously any congress would be lost without it but in our present situation you know general situation philosophy where fragmentation and specialization have been pushed to the extreme you know a congress without a grand scheme This just is lost. But luckily, Congress does not just insist on what is being said here in this situation, there's also a lot that goes on after, you know, a few bottles of wine at the dinner table, after the dinner table, so that's why I mean with, you know, all stock margins. And to give two examples, so I just gave the example of, you know, I call it Protestantism, kind of... The installation that exists here and that already determines what you can say or cannot say, and which doesn't allow you to show from where you are speaking, what your stance is, as Bob Krausen would say.

27:30 And so, well, to give you two examples, yesterday somebody, after all these bottles of wine, started characterizing in a rather interesting manner. In a conceptually and historically interesting way, how this process came about and how this really constituted, so I thought, well, this is just as interesting as what you said is that here. And another example is somebody who gave a talk on logic and the philosophy of logic, very normal, very regular talk. You know, he was already drunk before dinner and got even more drunk after dinner and then when I was sitting at the table his eyes were falling closed and then somebody was saying something unfair about his favorite French thinker and, you know, suddenly he was awake again and was producing very interesting arguments, not necessarily the kind of arguments that sit within Protestantism, but just as interesting. So, what do you mean by Protestantism here, I'm not telling you. Then you can... Fill in the gap. Yeah, I can't say that here. So that would be like my... So you have the one half, which is what's being said here, then you have the other half. And what didn't happen, is also part of how you should assess or... You know, think about what happened, and I'm trying to make it part of you. So, you have what happened, what was very explicit in the presence of this situation, and then the things that didn't happen, and stuff that was set at the margins. Maybe put those people together, you know, I'm pretty upset with that. And then there's my third point. So I'm from Ghent, I'm going to see Alex from Ghent and this is coming from Brussels. So this is kind of like the Ghent-Brussels axis and well, it didn't fall out of the sky the other day, it has history and the history has a name... We studied in Brussels, but became a professor both in Brussels and again, and did a lot of work, but apparently did a lot of work in a lot of domains, but ended up coming out of the closet as the philosopher of nature and metaphysicist he always wanted to be, and really started up the Center of the Apostles in Brussels, a world-famous project.

30:00 You know, there's a lot of interesting stuff that gets derived on, very broadly speaking, ontology, quantum mechanics, and metaphysics. And, you know, I think that the thing we did here today, and we hope to continue our work, is in this mission. So, Apostle is certainly more, somebody you could, better, I mean, is somebody with a grand scheme, so in that sense he... He's more on the side of a Badiou than a specialist or a pragmatist, but he's certainly not, you know, we don't take him as a master thinker, but nevertheless, he's somebody that could be entirely representative. So that's like my third point, and that just leaves me with a word of thanks. So, a word of thanks to... The audience, which was more numerous than I had expected, to the speakers for having invited, for having accepted our invitation, to Karina Segers of the P.E.B. and Gideon Monarch, again to the speakers for having given some majestic support, and I want to thank also St. Louis Law School for covering the members of the group. Yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, And then I think Karim is going to say some more practical things, two more practical things. I hope you have won. Up to isomorphism, I completely subscribed the last time.

32:30 It is with respect to the proceedings that I want to say something. So there is actually two kinds of results that will come out of this conference. And maybe you can try to bring in your paper something of what cannot be said nevertheless. If you would feel any way of transgressing this boundary of this unspoken protestant policy. Prohibitions? Then go ahead. We will, what? Boundary conditions? Go ahead. We will certainly appreciate that. So there is two things. There are the proceedings which will be edited or published by the Royal Academy here. These are due in relatively short time. This is why we insisted that you prepare as much as possible, and I appreciate that many people did it, or having a draft. We would like to have the versions of the draft as quickly as possible. Why? It is not like a real refereeing system, I mean we have certain autonomy in deciding what we do here with this contact form for the proceedings, but there is of course some referee control, I mean in contact with the Royal Academy. And also, if you want to start this as quickly as possible, also part of our funding from the Royal Academy depends on the fact that there are more series, so I ask you please, after the conference, to finalize as quickly as possible the working version of your paper, send it to us, and you can still finalize it a little bit before the final publication. Secondly, there will be a book for 2009, more or less, where much more elaborated versions of what you did here, or even when you came to new ideas and you want to turn what you did here around in North Africa, as possible as well, there will be extra invited contributions, like for instance the Badiou contribution that Norman should have been presented here, will be in the book if everything goes well, and others. So they're much more in-depth, much more big, I would say, because here I would say 15 pages is a good, reasonable amount, but in the book you can write, if you want, 50 pages, if it makes sense at least.

35:00 So the book will be something like the... The long term or the middle long term continuation of the work that we hope nevertheless, even if I agree that this is not complete to succeed, but it was also very ambitious to think that it would succeed in this governance alone, but this gives a long time perspective and I hope that this also can bring some extra value to this idea of bringing together people from the different traditions and nevertheless provoke something. So the proceedings are short term, are just, let's say, academically acceptable and responsible versions of what we presented here. Fifteen pages in the more or less. The book You can take a lot of more freedom for the book, you can make a lot of more work, put a lot of more work in the paper, you can use as many spaces you want within reasonable limits, and you have a lot of more time. So keep this in mind, please. To finalize, I want to thank Will for being such a great companion and inspiration, we are actually, I think that the fact we are more or less succeeding in doing this, going over this boundary in our mutual discussions, because we are so very different in a certain respect, but we find each other always when it comes to the basics. And I want to thank all of you. You have been great participants, great audience, great speakers. I at least did not hear anything which was not at least very good. I have heard nothing which was mediocre and this is already a great realization. And voila, I hope to see you again. Thank you very much. It's my very bad writing. And that's double zero. And that's M-P-B.

37:30 And I declare I'm attending a contemplation involved in the connections of calculus. I'm very impressed, yeah. And of course what he means by structurism in that context is Dedekindian, because he means Dedekind, no such structurism. I don't know, in any way I was kind of racking that issue. But clearly what he means by that is Dedekindian. Well, no, it is what he means in the literature. Well, I mean, this is not a, you know, to express these things in poetic metaphors, I mean, so did Grothendieck, so does Auber. I mean, to me, those are the two greatest living minds in mathematics. So I don't take the fact that you might get somebody who says, oh, but don't you realize that geometric and kinetic intuition has been completely expelled from anything but a very kind of marginal heuristic role in the presentation of calculus. We'd understand that foundations are completely, you know, completely understood. We have the ultimate, you know, the ultimate ingredient to the definition of all mathematical concepts completely in focus ever since Dedekind and Cantor and, well, you know, this is crap. Well, I mean, I haven't read all of it. I haven't read the whole book, but I do remember having seen some stuff in there on the campuses. But there's a profound dynamic, and there's a profound dialectic clearly at work within mathematics itself in the last half century, which calls into question its entire, if you like, static set theorization of foundations. The only people who don't know are the philosophers of science. Well, actually, no. Let me be fair. There are some very good philosophers of physics who do know it. There are philosophers of mathematics who don't. The philosophy of mathematics, I'm sorry to say, because in the English-speaking world it's almost entirely uninformed, except for a handful of outstanding exceptions of whom you are clearly... Some of whom are standing in front of me now. Colin McClarty. No, no, but the whole point is that except where it's informed by the history of mathematics, which it is. But unfortunately, in only a pretty small handful of cases, I think it's just about the most moribund area of analytical philosophy. It has been for a long time. It's only where it has been thoroughly informed by material from the history of mathematics that it's coming alive again. Yeah, that's right. Generally, the history of all of these subjects is what brings them alive. Yeah, that's right.

40:00 Did you talk about the non-standard analysis? What do you think about it? No, no, I have a lot to say on all of this. The non-standard analysis is... The interpretation of infinitesimals is non-mathematical, and there's a lot to be said about this, because Russell managed to dupe everybody into thinking that infinitesimals had been completely banned from mathematics, and Robinson was quite happy to jump on the bat. ...and say, okay, now I'm rehabilitating them, and what I'm doing is rehabilitating Leibnizism. It's absolute rubbish, because what he was essentially doing was rolling out the compactness theorem, producing a hatbox of, well, pretty bluntly, logician's tricks, which were entirely parasitic on his standard foundation. I mean, not that it wasn't stupid, you know, you've got shorter proofs, I mean, I'm not saying that what he did was without value and importance, but the idea that what he did was... The significance is that it undermines Kantor's own argument against the use of Kantor's own. Oh, mathematics didn't disprove it, did it? Oh yes, it undermines that art. That's certainly not a concept. It's... Yes, oh yes. I taught calculus for a number of years, and I'm... It's pedagogically quite a useful device in teaching. But it is conceptually parasitic on the... Well, what I was going to say is in both logic and mathematics, a much better pedagogy than the one that everybody has gone for is to recapitulate a kind of slightly reconstructed history of the subject and to introduce at each stage what's not adequate about it to motivate the next thing. And that's far better for students to understand and to get a feel of the whole thing and to understand how the whole mathematical... but if instead... You take the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know, the, set the, you know,

42:30 I think I know where we can find statistics. Small walk, is it a problem? No, not a problem. I started with first degree in math at Cambridge, and then when I realized I certainly didn't have it in me to be a research mathematician, I did a retread and did a thesis in history and philosophy of math. I got very interested in category theory and topology theory, which has still remained my principal focus of interest. And the foundational ramifications thereof, or disputes as to whether there is any fundamental foundational significance in category theory, all the issues around that. That was my chief area of interest and has remained so. And then I kind of dropped out of academic life altogether in the 80s, because there were no jobs in England, ran a travel firm for 25 years, made a little bit of money, invested in some property, sold it five years ago, moved to France, bought a nice old townhouse in Boucher, and meantime, since right back since then, no, no, no, alas, alas, alas, I'm now a regular visitor, oh yeah, yeah. Yes, I'd love to. I'd love to. I'd love to. The ring is a regular visitor. We've got our fellowship. I have a room for you. The answer to this, of course, the answer to all these things is yes, I'd absolutely love to. And we'd be happy to do so tomorrow, except for the fact that I don't have the money to do it with. No, it's like I'd love to go and dine in the Ritz. No, well, I had just enough to buy myself this. And the point is that since 1973, I've been 35 years now, almost 35 years, I have been building up this archive of interviews, little mini-conf, although that came later. Which our conference will end up in. Oh yes, which your conference will end up in, as I said. And the one we just had in Boston with Bill O'Byrne-Cartier and Ted. But, you know, for 35 years now I've been recording stuff, symposia, colloquia, meetings, interviews, as I say. We now have 26,000, we now have 26,000 recordings in our archive. Are they searchable? Well, right now I am desperately trying to get together the funding to start to put the whole thing online and make it searchable. We have a catalogue, but...

45:00 And I hope in about a year's time you'll be able to download and listen to about three or four hundred of what I call the Crown Jewels, which is, in my own subjective estimate, the most important, the most interesting material that's been recorded in those thirty-five years. But that will still only be something like 3 or 4% of the total. And these are mostly practitioners? Oh, very much so, yes. Or people that are studying the practitioners? Both. I mean, with an archive of that size, you've got a pretty good selection of history and philosophy of science, but also, of course, mathematicians and physicists themselves. The idea, well, for instance, three times a year... This is actually something Tom Kuhn wanted to have done, but it really didn't happen. No. Well, three times a year we bring somebody who... we spent two years preparing for these interviews. We bring somebody who... I say our because, you know, I have like a panel of five or six people who I consult about with this. We bring together... Somebody whose work we think has been particularly important for the mathematical conceptualization over the last 30 or 40 years, some really key figure, together with an interview panel of five or six people. All of them also practitioners? Usually, but we, for instance, the first time we did this it was in 2004 with Lorvier and we had... Pierre Cartier, certainly a practitioner of appeals medalism, Angus MacIntyre, but we also had a couple of philosophers in mathematics. We had Colin McLarty, we had Leo Corrie from Jerusalem. And Jean-Pierre Marcus and a couple of other guys. And we spent three weeks doing this intensive debrief, which we planned for almost two years. Since then we've done it with six other people, and we're aiming to try and do it three times a year. Oh, in Fougères, which is where I'm based, which is a nice old medieval town in Brittany, about 25 kilometres from Rennes. Yeah that's what I decided I'd do with my life. I sold off and decided I'd invest every penny I had in doing this. It seemed like a real, well, fun thing to do and it's what I wanted to do with my life. Of course it left me totally skint from having to live on a shoestring and now I'm, well the problem is, I underestimated it, I funded it, I sold these properties and provided some endowment for the archive but of course we got through it in the space of five years.

47:30 Apart from the little bit of money I've got there. Did you get the right institutional backing? Well, that's what I'm trying to do now. That's not the kind of project that funding agencies would like, because you actually can show something for the money. Yes, you can put a huge archive online, which anybody can access, exactly. Well, I'm still out there plugging it at the moment. We do have a little bit coming in. There are two foundations, one in Sweden, one in Russia, that fund us to a small extent each month, just enough to keep our head above water. I do still have a small... the travel business still kicks over, so I'm able to provide a small income from that, but nothing like the scale I'd like to, and not unfortunately on the scale to fund the project of putting 26,000 recordings online. I mean, that would involve... I can show you the funding proposal if you like. A ballpark figure is probably... Over three years, it's probably about $125,000, $130,000. $125,000? $125,000, $130,000. Because we have to have... $80,000? No, that's dollars, that's dollars. So 80,000 euros? Yes, doable, doable. Doable for that because there's technology now called CEDA, which stands for Computer Enhanced Digital Audio Restoration, which will allow you to compress and digitize up to 32 tracks simultaneously in real time. So in theory, you could digitize the whole thing in only a little over a thousand hours of computer time, but it would actually take at least three or four times that because you're going to have to listen to a hell of a lot of the stuff in real time. You've got to have somebody writing the digital soundboards, which means that I'm going to need a full-time assistant for three years. I was here the other night. Okay, this is a... That of course was not an answer to the question. I do like Asian food, but I didn't think it was the greatest Indian restaurant. There is a very good Indian restaurant.

50:00 That is traditional, very good. There are a lot of Thai and this is not Thai, this is very good. Anything more bistro of Belgium is not going to work? Because I'm just thinking I didn't come from Brussels. No, I wasn't, this is fine. I must admit, I'd rather eat. The problem is, these places in general are smaller, except in the big thing where we were yesterday, but it would be stupid to go back to that. There were two, four or five. We're only five, aren't we? Yeah, but it's Saturday evening. Oh, how full are they? I don't mind, I'm easy, I'm easy. Well, if I really have to express a preference, I'd rather go for a bistro, for something more... Ah, OK, yeah, but... No, no, OK, but... I'm not that bothered, I'm not that bothered. The food is good, I'm sorry. We don't mind waiting 15, 20, 25 minutes. No, I'm not that worried. Well, as long as I don't end up eating in a grotty pizzeria. No, no, no, yeah, yeah, that's what I was going to bring you. Now that's wonderful, only Shamrock, a smart Indian restaurant. This one is really excellent. Irish Indian, what they have, you know. To answer your question, whatever Deeks can do will do better than I can. OK, but all the same, I'd like to stay in touch, obviously. The big problem I have at the moment is completing the catalogue because 26,000 items. You can imagine that's a year's worth just to catalogue it. I've managed to get it back to 1998, you know, it really needs to be taken all the way back to 1973. We started recording interviews with Penrose and Atiyah and these guys in 73. And also John Stachel, who I'm sure you know of, has given me a... He has a set of about 80 recordings which he made in the Chapel Hill conferences in the late 60s and early 70s, which include several hours of Dirac and Wigner. It's absolutely amazing stuff. I saw these last few years, there were a series of conferences in Rhode Island in the 50s for a lot of these characters. Yeah, I know about those, but we don't have anything earlier than 1968 in the archive. Apart from that, and everything we have before 73 was recorded by other people anyway.

52:30 In Bologna in 1998. Thank you very much indeed for giving that to me. Anyway, you must come down to Bouger sometime and just see the, you know, the physical, physically see the archive, and the next time we're having a meeting down there, we usually have three or four a year, oh yes, yes, we've got them on the web, we have a website, and we did one, for instance, two years ago on topological and geometrical concepts and foundations of physics, which I said I think was a very nice meeting, we did, we had this big session with Lorvier and Karkier on foundations, you know, category theory and set theory. And, you know, issues about the role of categories in the conceptual organisation of mathematics. That was a very, very interesting meeting. That's, of course, always been so much my own chief area of interest, that, obviously, the archive is slightly skewed towards foundations of math in that sense. But, you know, we've recorded a hell of a lot of stuff with mainstream mathematical physicists over the years, too. Chris Ice, you've got a lot of material with Chris and Penrose and people over the years. Not a bridge too far in principle, just that there are only 24 hours in the day. I've probably got three or four recordings of talks on mathematical economics. A very interesting woman, in fact, who, I can't think of her name now, Mihaela Ektini, she's Lebanese, did a PhD about the impact of stochastic analysis on financial economics. She gave a very nice talk in IHP, no, Lebanese. I think she finished. Oh, sorry. Has she finished? Sorry. Yes, yes, yes, yes, yes, and submit it. I know, but it happens. Actually something very similar. Very interesting. Well, what's the name of the Moroccan lady you know?

55:00 You know, I've actually met her. I've met her. I've met her. I know. It's very similar. But this girl was in... where was she? I can't think now. I think she was in Oxford. But... I can find her. Yeah, yeah, I can... No, Marco is an old friend. I mean, incidentally, I've recorded all the seminars and the recites for the last four years. Well, I say all, but probably not every single one, but I would think probably about 80% are in our archives, which is kind of beautiful. The most interesting history of philosophy of math seminar in Paris, of course, is the one which Cartier runs at the École Normale, which they record themselves in the service of diffusing on the savoir, so I don't have to worry about that one. But on the other hand, Cartier has come and talked several times in our archive, and in fact I just went with him ten days ago to this conference which we sponsored, which we spent two years putting together, on reference to, well... It's going to be published, in fact, by the University of Chicago in the series which Luke Kauffman edits, a book that's going to be called Representations of Space. And it was really to mark the 50th anniversary of Groenke's Tohoku paper. Originally it was going to be philosophy of geometry, particularly the possession of algebraic geometry within mathematics. Certainly, probably the area where the concepts have been created that have had the most far-reaching impact on other fields of mathematics, but of course it's spilled over into physics, so we had Luke Crane and John Glass and other people talking about the impact of broken geeks working in physics. I just got back from that, it was an extremely good meeting, really looking forward to editing the transcripts of that. I did actually, yes. I'll send, I'll send you a CD of the meeting. Yeah, this looks ok. Me not, but it looks ok to me. It's Italian? No, not really. No, is it? They look more sort of, you know, Franco-Belgian to me. I don't know if I've got a problem with Italian, but I can use Italian anywhere. Hello!

57:30 Anyway, we're most definitely staying in touch. Thanks for giving me that. I mean, I'm not, I should point out that I'm not a particularly good mathematician or mathematician. Nor am I, obviously. I'm very interested in the intersection of physics. I don't know if you can tell me. Do you know about this meeting in London in January in Imperial with Chris Itam and Bob Cooker and the Ukraine and various other people? I'm going to go and record that. That would be a very interesting session. You think we might have some more coming to join? That would be great. You think we might have some more coming to join? They don't know where we are though, do they? Oh, why would they do that? Thank you for your attention. That looks interesting. Yeah, I know, I know, I know, it's just teasing. Very wise. Do you want to pass them along, please? Thanks, we can have them. Thanks. This speech is linked to an article in the Journal of Humanities, an abstract number, so I can't remember the name of the article, but it's a good one, it's a huge article, I can't remember the name of the article, but it's a good one, I can't remember the name of the article, but it's a good one, I can't remember the name of the article, but it's a good one, I can't remember the name of the article, Just as a piece of advice, this is, yeah, it's not, of course, a high-end restaurant, but the food is very good, very traditional, not expensive, and it's really the Belgian stuff.

1:00:00 And what you can have here is, like, I think even moon, mussels, but you can also have, like, stew. Stump is like mashed potatoes with vegetables in there. And there are two kinds of meat. That's a very good one. You can also have rabbit with beer. That's also a very good one. Rabbit with beer. And the sorghum. This kind of stuff. Yeah, I'm sorry, there could be nothing for me. Probably not the ideal season, Bill. Ah. Salad. Salad. You do have to keep kosher. You have to keep kosher. Yeah, yeah, of course. Well, okay, I have to. I mean, have being in... Well, I thought you might be vegetarian as well. There could be an additional dimension as well. No, but I mean in practice. There's no way. Are you not... I'm sorry, I ask out of ignorance and a genuine desire to learn about the Jewish spread, but do you not... Are you not glad to eat fish? I'm allowed the attention, not here. It depends on the preparation. It has to, it's the preparation, okay, right, right, right, okay. It depends on the utensils. Oh, yeah, yeah, yeah. Yeah, yeah, of course, yes, that's always a problem. Yeah, yeah, of course, thank you. You wouldn't have had a problem if we had them meeting in Antwerp. You would have had much less of a problem. So in Brussels we could not go because Eric told me that we were trying to find some individuals. We found a place and it's open but they said they were completely worked up there. And where is that place? It's behind the... It's about a mile up the mountain. Do you remember the name of the school? Well, I think I'm going to go for the... I think I'm going to go for the Urfa La Ruz, or possibly the Filiano, and then the Russian, you know, the Urfa La Ruz, and then...

1:02:30 What was the kind of potatoes you mentioned earlier? You said they were very good with the, you know, with the meat and stuff mixed into them, in the Belgian... That's it, stop, stop. Which page is that? You know it's good? Yeah, we eat that. Octonis is chicot au gratin. Chicot is a kind of white vegetable, and it is filled with... Ham, ham. And then there is a cheese sauce over it. Thank you very much for your time, and I look forward to seeing you again next time. I'll go for the shtump with these, and maybe with a Russian egg before that. Well, I used to have a relationship with somebody who was, I was living with a boy who was a strict vegan, so I do sympathize with the, you know, the problems. Particularly tough whenever we meet, especially like Italy. Thank you for your attention. Thank you for watching.

1:05:00 Ontario certainly seems to be the place which has the biggest concentration of philosophy, especially philosophy of science, in English speaking Canada, by a long way. So I've been having a little bit of a concentration in English. Yeah, so drag me to the countdown. London is one of the most great places in Canada. This is not just a kind, it is the, in turn, this is Karen. It is the Yameda Lama. That's right. Correct. Given to me by Karen. This is her gift to me, which is why I'm wearing it. It's a present from Karen, which is why I'm wearing it. I don't think you are one of the only two or three people who ever got the t-shirt without participating in the seminar. Well, thank you very much. I do appreciate it. That's really a privilege. Special, I know. That is why I was wearing it, in your honor. And I wore it in Boston, too, for Lorvier and Cartier. Yes, yes, he liked it, too. He liked it very much, yes. Very much so. Very much? Very, very much so. What did he say? He just said, well, Lorvier is still active. You bet he's still active. I just spent ten days with him in Boston. I mean, my God. From dawn till dusk, active from about seven in the morning till nine. I mean, I was absolutely exhausted. The man just never stopped. I know, I was there as well. He was in Paris with, in fact, I picked him up from the airport, and it was during the strike. He was here, but the problem is, it was during the strike, so you would have had a problem getting there. But he came, I recorded everything, so I could send it to you. I know, it's not quite the same. He came for a meeting for Christian Moselle, you know, the French. I guess he was in the seminar which we always send out in the evening. Well, no, that was a separate thing which we organised for him. He'd actually come for a conference at the IHP in honour of Christian Husserl's 70th birthday. But then the previous day we arranged for him to give this seminar to Erich von der Leyen as well. But the meeting on Husserl was totally fascinating. That was one of the most interesting talks I've ever had in years. There are some other very good folks there as well. The guy who gave a very good talk in the DQ hours, himself, gave a very nice talk.

1:07:30 That was in the dispute on the chair. No, no, that was at IHC, after Giugnani and Parcherini. I know that a great friend of Villovir is Pertolongo. Oh, I know Pertolongo also, yes. I remember Pertolongo once... Pertolongo is a great guy. I remember the talk of Bill O'Gear about monoi satisfying the law, xy equals xyx. I remember, I was impressed by that. Must have been about 1990, I remember hearing him give that talk at the Comer meeting. This is his way of thinking of the natural numbers. What is this? This monoi construction, it's to do with the way you're supposed to think of the natural numbers in... And what they call, which is sometimes called objective arithmetic, which is this to do with the winding number. He did indeed. He did indeed. Which is what this is all about, yeah, yeah, yeah, yeah. Ha ha ha ha! What? You know, what could possibly be going on here? One second, yes. I'm going to go back to the past. Back to the past. And... Speck and Peinz. Speck and Wurz. Thank you for your attention.

1:10:00 Thank you for your attention. Is nobody else having a start? As if they aren't, then I won't have one either, oh he's not. What did I say? Did I change it? No, he went to ask... We were in Brussels, I mean he's a Frenchman in the city, I'm sorry. Fuck you, don't... He went asking whether you can ask what you want. Oh okay, and is there? Can you be so stupid? How can I be so stupid? Because I don't have superman hearing, I just wondered what was happening. Sorry, was that... Thank you. In that case, we'll have Vasili Aran. Vasili Aran? Vasili Aran? Okay, I'm sorry, I'm afraid I didn't speak Flemish. Can you tell me what is available on that list? Here's your menu, after all. Okay, I thought Quaternary Day might be a general thing. Okay. Well, I didn't mean to offend the man. He's being pretty obnoxious. But why are you so aggressive? I'm not being aggressive. He's the one who's being aggressive. Look, I'm sorry. I asked perfectly quitefully if he had an earful of roots. And he went immediately to the kitchen to ask whether it was true. Go off and leave now if you want me to. I'm getting really fed up with this, Karen. I have to be honest. I am getting fed up with this. Yes, and then virtually spat in my eye. The man is obnoxious, end of story. I will have these, sorry, after you finish. I'm sorry, but this is ridiculous. I'm not prepared to be preached to that way. I'm sorry, this is ridiculous. I'm simply trying to order. He said he doesn't speak French.

1:12:30 He speaks French more than Flemish. Then why is he... Oh, never mind. Because we start to think that, what's the point? It's not your business. Speak whatever you want. He went to the kitchen to see... Well, that's what I was doing. He went to the kitchen to see that what you were doing was still there. Okay, okay, okay. Could we just draw a line under it and move on? Thanks very much. It really would be a good idea, wouldn't it? Right, well, in that case, I'll just have the Stumpf. The sausage. I prefer to have the sausage if he's available. Yes, what a good idea. I'm sorry, I don't think, I really don't think I started that, whatever anybody thinks. Okay. Stumpf. Okay, well let's stumpf with sausage and pasta and wine. Thank you. Thank you very much. Thank you for your attention. What are you guys having? Oh, the left, okay, I'll join you. Yeah, yeah, I know, I know, I know. Sophia is okay. I'm going to... I'd rather have a Chimay, if that's what you're saying. Do you... Sorry. Never mind. Never mind. Well, yeah. The Chimay, I thought they might need it. I'll go for Chimay first, when I'm watching it. You can give us a chance to order, that's the whole point. I mean, you know, you just turn around and talk about it. I'm sure I can drink a glass, if I have a shimmy or not, no big deal. Do you have a question for you about Lohir? Sure! Is it true that for Lohir the non-standard reals are not filled, but they are on a linge and it's... They are precisely, well, he has a completely different way of constructing the non-standard reels from the Robinson construction, and what he is, he constructs them over the topos of smooth spaces, which has a ground ring which has nil-potent elements in it, because it doesn't obey classical logic.

1:15:00 So you can represent the infinitesimals as square zero infinitesimals, as completely geometrically, if you like. Think of this space, this smooth space, as not consisting of points, because if it's consisting of points, then it's not a point. Those would be the decidable of the topos, and you would have a condition which is technically known as the weakly decidable sub-object condition, which topologically means that all coverings localize and that the minimal covering of any space is separated, which means that you have, for any element in the ring, it's decidable whether it is equal to A or not equal to A, but of course with the... With a ring over a completely smooth topos, you don't have this condition. So you can make sense of the notion of... These are quantities, the square of which is zero, but which are themselves not equal to zero, which was the original kind of intuition, if you like, the kind of kinetic and geometric intuition of the infinitesimal, which Euler and indeed Fermat and other people had in the 17th century. But it's a completely different and much more geometrically motivated construction than the Robinson construction of the non-Archimedean field. Yes, of course it connects with all the work on point-free topologies and sheaths, because the way that Ian Grogensieck always thought of sets was as sheaths over a one-point space. You know, the notion of a sheath is more fundamental than that of a set, and you think of a set just as sheaths, which happen to have a base space which consists of one point. Therefore, of course, all the sections of the sheaths, you have global sections of the sheaths, you can go from local to global sections, and all the elements... All of the sections consist of decidable objects.

1:17:30 No, but the point is, I'm not, normally I can deal with other things, because it's a kind of thing that's going on in China. Well, I, I, we both are, so please forgive me, I didn't mean to bite the guy's head off. I don't. Let's, let's change the subject. No, but go on with your lecture. No, no, no, that's exactly right, let's, let's go back to this. No, no, the point was not because... It was actually he and Anders Koch who worked this out in the 1960s. But he gave the original lectures on the topos of smooth spaces, because he was very interested, in fact... Well, thank you, Raoul. Oh yeah, this is the strongest beer. This is the strongest kind of beer you can buy in Europe. And this is the strongest beer you can buy in Europe. I mean at the same time. Well, I didn't ask them at the same time, but I'm not going to try and drink them at the same time. This beer would actually be illegal. This beer cannot legally be sold in the United States. Oh, because it has more than 11% proof. It's far too strong. This is actually illegal in the United States. You'll never understand. Only nine, I'm sorry. Even so, I think it's still illegal. I think anything over six degrees is still illegal. But anyway, the motivation for this construction was that Lanvier was fascinated by the idea of Absolutely smooth parameterization of variation because his field before he went into mathematics was continuum mechanics. He studied continuum mechanics under Clifford Truesdell. I was very impressed in that whole program for the revival of 18th-century Eulerian-style continual mechanics and the idea that smooth variation was absolutely fundamental and one should see the topos of sets as really the purely static case, that one should think of the static case as modeled within the variable rather than the notion of variables. ...as things which had to take values at points of a domain which were somehow designating objects which were there, the same or different absolutely, to be the points of them, to be the values of the variable.

1:20:00 I mean, that's a conception, if you like, more or less a kind of fregean conception that he's always rejected. He has what, in his view, and I think he's right, is a much deeper way of thinking of the structure of domains of variation. In terms of lattice homomorphism, it's from parts of a domain of variation into parts... There are lots of what he calls either intensive or extensive quantities, which are functions, if you write maths in categories, which are varying either contra-variantly or co-variantly, depending on whether they're maths into or out of the domain, which is loosely the space over which they vary, and which...