Discovery & Proof, Hilbert to Plucker / discussion

0:00 The restaurant is called Le Hoche, did you hear about the famous general? Yeah. So after this information, we announce the talk of Jeremy Gray, who is sitting near me, and who is going to... who teaches at Open University, and who is going to speak about discovering in mathematics from Finberge to Kutka. Thank you for the nice introduction. I have a number of thanks to offer to the organizers, of course, It's been a very smoothly organized conference, take great care of us, dinners and the lunches. We really appreciate that. And in particular, I'd like to thank you for the rich intellectual diet that you've had in the course of the last few days. In the next few days, it's a very, very congenial conference, to speak personally, for the number of different interests that are fitting together.

2:30 I find that very attractive. I'm particularly grateful to my namesake Jeremy for talking about algorithms this morning and to Paolo for talking about real algebraic geometry which I shall allude to ancient past and to Philippe for setting up the topic of projective geometry which is of course going to be where Flucke comes in but I'm particularly grateful to Jeremy I can say this because he actually outlined my talk you may remember that he said you usually get a talk which has some vague generalities ...followed by a case history, followed by a somewhat specious, well, he didn't say specious, but followed by a claim that the case history supports the vague generalities that went before. So all I can do is apologize that I go two-thirds of the way, right? But rather than making a vague, specious claim that my case history supports the vague generalities of Trump, what I claim is that the talk allows me to make some more vague generalities. So that's going to be the crucial difference of two sets of vague generalities, that being all that a poor historian can do by way of philosophy. So I'll do my best to reach the general audience that has gathered here and talk about a particular topic which comes in fact with four parts. There will be an introductory philosophical part, short, concentrating on Gödel, some I'll then be a repetition, in fact, of some things that were said yesterday by Hilbert's proof theory, then the case study, which will illuminate, I hope, some of the ways in which proof and discovery interact, and then I'll make an analogy, and it's going to be an analogy with some of the things that happen contemporaneously with us now in the history of science. In case I want to go in that direction, it's a route that some of us have been down, harder to be down, for other reasons before. So the thing that set me off of all of this in my reading was an article by Richard Cheatsen entitled Gödel and Quine on Meaning and Mathematics, and it came out in 2000, published by Cambridge University Press, a book called Between Logic and Intuition. And the paper, indeed, discusses Gödel and Quine, conveys what I think is the usual criticism

5:00 of Quine's views of mathematics, that somehow Quine failed to appreciate the way in which pure mathematics works. It adds some very nice comments, some due to parsons, on the way the fundamental hypotheses in mathematics do not resemble those on the theoretical sciences. Again, it's a rather nice little comparison. The fundamental hypotheses of mathematics are frequently taken to be elementary and easy to understand, but the fundamental hypotheses of science are quite the reverse. So it doesn't seem to be very easy to mesh, for example, history of science with history of mathematics if those are the different sort of starting points for the subjects of mathematics and sciences, or physics anyway. But what really caught my attention in this paper were the remarks by Gödel, and they come from the third volume of the collected works, and I think probably they were the ones I'm quoting, which quoted by Keatsen in the paper, but I found them rather eye-catching. In these unpublished essays that Gödel was working on, and they're part of the unfinished He objected to Hilbert's binary philosophy of mathematics, and he said, not on the grounds that I, Gödel, have shown it to be wrong, but rather in the hope that the certainty of mathematics could be secured by cultivating or deepening our knowledge of the abstract concepts themselves, and seeking to gain insights into the solvability and the actual methods for the solution of all meaningful mathematical problems. And he gave the usual argument for why clarification of meaning couldn't consist simply in giving definitions, and then said that he rather felt sympathetic to the phenomenology of Husserl. There's a long quote I'll try and read slowly. Now, in fact, this is good, there exists today the beginning of the science, The beginning of a science which claims to possess a systematic method was such a clarification of meaning, and that's the phenomenology founded by Husserl. Here, clarification of meaning consists in focusing more sharply on the concepts concerned by directing our attention in a certain way, namely, onto our own acts in the use of these

7:30 concepts, onto our powers in parrying out of our atoms, and so forth. But one must keep clearly in mind, he went on, that this phenomenology is not a science in the same sense as the other sciences. Rather it is, or perhaps in any case should be, a procedure or technique that should produce in us a new state of consciousness, which we describe in detail the basic concepts or grasp other basic concepts, hitherto unlandless. I believe there is no reason at all, Sir Gödel, to reject such a pre-procedure at the outset as hopeless. And I wanted to focus on the emphasis he places on our own acts in the use of these concepts and our powers in carrying out these acts. Now, I guess it's true that Gödel is probably the only out-Platonist to be widely cited in the 20th century, right? Martholidians are thought to be Platonists when nobody's looking, or not on Sundays, or only on Sundays. You have a choice, but Martholidians are supposed to be secretive about their Platonism, with the exception of Gödel. And I suspect that this accusation that he's an out-Platonist is made to marginalise them So I don't know whether there's been a flourishing tradition of investigating these views of his or not, but I somehow doubt it. But I thought in the spirit perhaps of the peacock from yesterday, I don't know. I would bring along a humble analogy, but I don't have the Danish for piano. I want you to imagine that you bought a piano. And what you want to do is to get the piano into your living room. So you do a certain amount of preliminary measurements, and you satisfy yourself that it's possible, otherwise I suppose you wouldn't report it, and you get it delivered to the doorstep. Alas, you have a problem. There is an awkward bend in the hallway, and you can't get it in, just that little gap between where the stairs go up and the door to the living room is, is somehow too narrow. Then it turns out, as somebody points out, you could take the front window out, bring

10:00 the piano in through the front window, and then put the window out, okay? And you now know, with genuine knowledge, that you can put a piano in your living room. And the proof of this was the doing of it, okay? An exercise you wouldn't have done for much smaller objects or for much larger ones by measuring with a supply. and my analogy my first analogy is this mathematical knowledge is often is of a piano moving time that you know as a mathematician or you would say of a mathematician he has learnt something about the object he's talking about because he can carry out a certain form of activity if you say to him how do you know that all finite abelian groups do this he outlines a proof which you hope will hold up against the or allow rules of deduction and the axioms of group theory. But he moves the mental object around. He decomposes it according to certain rules omitted by certain theories. He performs a series of acts on this mental object, and they are expressed, of course, in the form of proof. And I claim this is perfectly reasonably some sort of knowledge. Knowledge of mathematical objects can be understood as a way of talking about mathematical objects that makes them epistemologically accessible. You're getting genuine knowledge here, and this is kind of tongue-in-cheek, but I'm not sure what it is to say something about a mathematical object other than it's manipulable according to the rules. You can have your psychology as much as you'd like, it if someone claims to deliver a novel theorem, in group theory they claim that they can follow the rules just the same way that the furniture man claims he can get the piano into your living room. That's why you hire him to deliver the piano and it's why you have a group theorist telling you theorems. In group theory they can move the mental object around in this fashion. So I move now to Hilbert. This is going to be the kind of end summary of what In 1925, as I'm sure you all know, he gave expression to a philosophy of mathematics that had preoccupied him for some time. It doesn't completely survive the work of Gertner, of course, but his finitary approach to mathematics, I think, is one of the founding statements of what I call the syntactic view of mathematics.

12:30 He did this at the height of his confrontation with Brouwer over intuitionism, and it comes very close to saying that in many of the most advanced areas of mathematics, mathematicians follow the rules of whatever axiomatized system they are working with, and if they do this correctly, then the conclusions reached after any finite number of steps are proved, and that's all you can ask for. So this is a presentation of mathematics that seems to be contrived and partial. It's a very narrow view of mathematics and this is often discussed when controversy between Hilbert and Brouwer is discussed. And I don't want to go in the direction of or in what particular ways it would have been particularly offensive to Brouwer, but they're there. It's partial because we actually imagine mathematicians do lots of things. They don't just sit around and write up their proofs. It somehow seems, the way Hilbert describes things, that mathematicians do lots and lots of things, I suppose, which somehow aren't mathematics they read other people's papers they talk with each other whatever it might be and then finally they sit down and they write a rigorous argument and at that point they are doing mathematics that's what mathematics is is writing down of lines of a proof that ends up with the theorem you want it's not the only view Hilbert had elsewhere. This is somehow the stripped-down essence of mathematics. And I think it panders to a view which has persisted to this day, that there are two aspects to mathematics. One is this mysterious, quite possibly psychological, could even be irrational, but in any case in the last analysis, irrelevant activity of discovery. Okay, mathematicians do whatever it is they do to find these treasured theorems, okay, and we're invited not to worry too much about that. What the mathematician is doing with his hat securely on is a cold, ideally mechanically checkable, activity of proof.

15:00 And sometimes one sort of feels in a little bit of a corner that the rigorous writing of proofs is mathematics, or at least it's the only it, it could legitimately claim the attention of a philosopher. So Hilbert focuses on the written activity of reading. It isolates, his philosophy isolates a feature of mathematical work that is certainly essential, the establishing claims, and then it says that's all there is. And it plainly isn't. Hermann Weil, for instance, has very interesting views about about how doing what Hilbert does is not an empty game, because there's a complicated relationship between what Hilbert is describing and the workings of contemporary science. And that's what, for Herman Weill, legitimizes the activity, including, in Weill's case, the switch back from Brauerian intuition to a rather more classical position on mathematics. And I think Weill's position is different from Quine's, for example, on the relationship in mathematics and science, but I'm not expert enough to go in that direction. Now, what I want to take from Hilbert in my journey back to Pluka is this idea of mathematics as a rule-government activity. In the Hilbert formulation, of course, it's axioms, permissible rules of reduction, and something that's almost mechanically checkable, and these days, more and more, is mechanically checkable. Okay? I think this is Hilbert's kind of outgrowth of a perfectly standard view of mathematics, which is mathematicians should be writing down good arguments, good proofs. And as so often when Hilbert writes something, what he does is he goes back and raids the past and finds some little bit of the past which absolutely supports him, or it does by the time he's presented it. He's not a historian. He has no appetite for history. He's not particularly concerned to make his historical remarks any more accurate than they need to be. Okay? And what they need to be is a mathematical or philosophical insight into the nature of mathematical activity, which can conveniently be hung on the peg of, and in this case, it turns out to be, Weierstrass.

17:30 so Hilbert in his paper on the infinite 1925 begins with craze for Weierstrass Weierstrass offered, says Hilbert a firm foundation for mathematical analysis lacking a concept of the infinite which he, Hilbert, is going to supply how fortunate, that's one reason for choosing Weierstrass but the foundation of analysis, says Hilbert on this occasion was built by providing rules which clarify the notions of minimum, function, derivative, so that the most diverse combinations of superposition and juxtaposition and nesting of limits always yield unanimously acceptable results. Hilbert's making a twofold point here. He's setting up a story about the infinite, convenient for his own paper. is also suggesting that rule-governed activity put by him into an axiomatic setting is in any case a classical way of doing mathematics. And Hilbert is, of course, in his dispute with Frau concerned to preserve as much classical mathematics as he can. So I think the implication is that Weierstrass worked in an area where there were already some good ideas and some not so good ideas. And he sets to, and he refines these things, and he makes them truly precise, truly mathematical. He produces some absolutely standard definitions, solid techniques that make sense of the fundamental aspects of analysis. And in this way, Biostrass is not transforming analysis. You can think of it much more as bringing it into focus. People had quite good ideas, not quite good enough ideas here. Maybe they were wrong there. Maybe they're right here. the creation from nowhere of Herr Weierstrass, it's that he goes in and really tidies it up. You can work from this forever afterwards, and you're okay. So this is a kind of view of what he's doing, and I move now back to Plucke, and the theme here is the study of singular points of algebraic curves. So an algebraic curve is given you in the plane, plane by a polynomial equation in two variables. The context here, first of all, one should say, is that not much in the 1830s, we're in the 1830s, not much was known about cubic

20:00 and quartic curves, and very, very little about more complicated curves. A mathematician could perhaps have been embarrassed by how little was known about algebraic curves. And the presenting problem is that the equations just get very long. How are you going to deal with complicated curves? The first insight into how to proceed is the one that Philippe talked about this morning and it grows out of numerous pieces of work, but we could take Poncelet as a good name. The point where the two red lines cross, I will call a pole, and the blue line is its corresponding polar. So to a point in the plane outside the ellipse, I can associate the blue line, and the construction says you draw the two tangents to the ellipse from the point in question, and where they cross ellipse gives you two points, those two points define the line, that line is the polar of the original point, and then vice versa. Okay, so you can start with the blue line and recapture the point, which is its pole, and you can do this, Consulé has an odd scary argument about the principle of continuity here, but you can do this if your point is inside the ellipse, or even if it's on the ellipse, if it's inside the ellipse, the blue line is The outside is the point, on your lips the blue line is the tangent, okay? So you get a correspondence between points and lines, okay? And it's a very nice correspondence, I'm sure most of you know this, maybe all of you know this, but let me just remind you in that, okay? The virtue of this correspondence is that if you have three points on a line, you will always get three lines to a point. And if you have three lines to a point, you'll always get three points on a line. So you have a comparison between points and lines which preserves, in some sense, the linear structure of each type of space. They, in French context, called this method of reciprocal polars. We tend to call it duality, and there was a very modern expression of the fundamental nature of duality presented to us.

22:30 It very, very quickly became a fundamental idea of projective geometry. And the trick is this, a very nice trick, you have a curve. At each point on the curve, you have a tantrum. So what you've got is a whole bunch of lines. very well. Replace each one of those lines by its pulp. You now get a whole bunch of points. And that is called the dual curve of the one you began with. So you can take a curve and you can find that you have a dual curve. And notice immediately the way duality works, if you do it again, you get back the curve you began with. You don't go wandering around the space of all curves. You take a curve, think of it, all of its dualize each tangent by replacing it with its corresponding pole. The curve you've just filled up is the dual, and you flip backwards and forwards from one to the other. It's a very pretty thing. And if you do it to a conic section, you get a conic section. Now, to do it to a cubic, a curve of degree 3, it starts to get interesting. The reason is that from an external point, down at the bottom of the picture, you can see timing and diamond, is first of all a cubic curve. The cubic curve is shown in blue, purple, whatever colour Mr Gates has allowed me, and there are a number of points from the point at the bottom to the curve. I haven't drawn them, but you can see one is going more or less north-west, one pretty much north, one slightly north-east, and the fourth one, actually, which goes well up there, tangent to the curve, some 45 feet above our heads, okay? There are four real tangents for this thing. There are two imaginary tangents, this is a hint of what's coming. All of these things lie, the points of tangency of the four tangents, I'm talking about 3 and C, not the other one, that are the tangents out of the point down there at the bottom. Those four points of tangency lie on where the blue curve meets this red thing. Previously, you had a picture of a point, an ellipse, and a straight line. Now you've got a point, a curve of degree three.

25:00 Instead of a straight line, a curve of degree one, you have a curve of degree two. But it plays the same role. The red curve of degree two, instead of a hyperbola, meets the cubic curve in the point where the tangents from the point you're interested in are actually tangents. So the red curve is determined by blue one and your choice of point. I move that point around the red curve but it picks out the point of tangents drawing the diamond point there to any point which is both blue and red, you would be drawing the tangent to the blue curve. So here you can see four of the tangents, and we want to explain why we get n, n minus 1 in general. So I want you to imagine that the blue curve is a degree n, okay? We can continue to think 3. I'm claiming that in this case there, the blue curve meets the red one in the points of tangency, the red one is a curve between n minus 1, to two, and we get six points of intersection, three real and on the screen, one more real that, alas, fortified to eat up, and two complex, no escaping, you won't get all sticks, okay? The blue curve meets the red curve in the right number of points. This is to invoke another theorem from polynomial algebra, the curve of degree k and the curve of degree meeting k times n points, provided you count carefully, provided you allow points of infinity, provided you allow tangency to count more than once, and provided you count complexes. Now, if you start with a qubit of degree 3, and you think, well what does it mean through that point, through that point there are apparently six lines which are tangents to the blue curve very well, let's do a line what are those six lines becoming? six points what were those six lines? they were six collinear sorry, concurrent lines so the points are six concurrent points so the dual curve is cut by a line in six points corresponding to the sixth pandem. So what we're claiming here is that there are generally n minus 1 pandems

27:30 to a curve to a curve with n, that's the Zeus theorem, and that was saying duality allows you to interpret that. It says that the dual curve has degree n, n minus 1. Or, if you want n minus 3, the dual curve has degree 6. So you do it again. Dual curve, dual curve with a degree 6, has degree which is 30, and is not 3. This is a problem, okay? This is the duality paradox. Pansolet had ideas about how to solve it. It sits there as a chunk. Duality is this tremendously tempting idea, but as soon as you start applying it outside of the area or conic section, you have a problem. It, on the face of it, gives you a crazy answer. So, this is the claim, the first polar is the red thing, they meet in N, N minus 1 points. Now, how do we get round this? How do we solve this duality paradox? This is Plucke's argument in essence. I've changed the blue curve of it, I've moved it away from the middle of the screen and taking it towards the infinite of the branch, which is on your right. They're now quite close together. Look at what happened. The red curve, which is the first polar, which is picking out the points of tangency, is, as you see, one of the tangents is still safely off, pretty much northwest, somewhere like that, but there are two points of tangency which are coming close together. And if you go all the way and run into a curve with a singular point, those things will coincide. Now the line from the diamond, the point we're interested in, through the singular point of the cubic curve does not deserve to be called a tantrum. It simply crosses the curve, each branch gets crossed, nice and simple. Okay? What Plucco says is you have lost two tangents. They're not tangents. Okay? If you'd like a double point, if you'd wanted to say there were still six tangents, you'd have to say that two of the tangents were spurious. It's not tangency in the sense that for Paolo's elementary example of y equals polynomial and x, to draw a line through a double point and say,

30:00 Well, it's a double point, it's touching, meeting the curve twice, what's the tangent to do if it isn't that? That's kind of, there's one way out. Foucault's way out is to say, look, from that diamond point to the singular point, you can certainly draw a line, it doesn't deserve to be called a tangent. It's the fate of two tangents. There's clearly going to be two tangents from the diamond to that bit where the blue and the red curve are very close together, blue curve, very close together. Okay, but that's not a tangent. Two tangents have disappeared. A double point, says Kruger, drops the degree of the dual curve by two. Big sigh of relief. If you take a curve with a cusp, this is y squared equals x cubed, okay, and you ask about tangents, well, I don't want to do the maths here, but you can see actually the red post-polar is actually somehow tangent at the cusp, right? This suggests that something better will happen, and it does a cusp that drops to the degree of the dual by three. And now you're ready to do battle with the duality parabola, okay? If a curve has a double point, its dual has what's called a bi-tangent, a line that is tangent to it in two separate fact. Your curve starts with a cusp, its dual has an inflection point. Conversely, if you are giving what it is, your curve starts with a bitangent, the dual has a double point. If it starts with an inflection point, the dual has a cusp. Okay, so you think of double points and bitangent, cusps and inflection points. So what might happen to a cubic? Let's take a non-singular cube, okay, and dualize it twice. Okay, if it has a degree, and that degree is three, its dual has degree six, there's nothing getting in the way, there's no singular points to really do that. It doesn't have any bitangents, because a bitangent touches the curve here and here, that's four points of intersection already, if you count tangents at lofty speed, and you can't do that to occur with three in a straight line, so there are no bi-tangents. You're left hoping that the dual has a number of cusps.

32:30 I know you would have come from inflection points, non-singular inflection points. Well, how many c cusps c for the unknown number of cusps? Each cusp brings the degree of the dual down by three. So, if you want the degree of the dual of the dual to be three and not thirty, you better have a number of cusps such that three times the number of cusps brings down the dual from thirty to three. This gives you that c is equal to nine. Apparently, the dual curve has nine cusps, your original curve had nine inflection points. and you build on this, more problems. Kruger points this out. You can't have nine real inflection points on a cubic curve. You can have at most three. The others are complex. So this is another source of worry. You can then do a very similar argument. If you enumerate the number of double points, and you go through exactly the same argument before, you'll get the first of those formulas. I did that for a case of a cubic, a curve of degree n, it works out like this. There's another formula that you need. I'm not going to explain it, it's similar in type to the original one, and that gives you another formula, and from this you can, it takes a group or a page to write them down, do pretty much everything that happens if you start off with a curve of degree anything up to 10 and you only care if its dual is no more than 10. Now some of these curves have low degree up to high degree dual, don't bother with them. You do pretty much a whole page worth of different places. For example the non-singular quartic curve a curve of degree draw, known to have 24 inflection There's two things you can do at this point, if you're a mathematician of the 18th century. You can either say you believe the Pruca formulae. Nothing else is going on. They've been derived all right. There are no weird singularities. You haven't missed any other aspect of the curve. In which case, the argument I gave you for inflection times on a cubic will give you by-handed points on a quartic, and it will tell you there are 28.

35:00 Or you can say, if you're Jacobi, well, I don't know. human enumeration of the bi-tangents to a curve of degree n, because that will enable me to believe that Stuka has not missed anything, and the duality argument is okay, and that's what Jacobi is able to bash out, and very, very hard it was for him too. It was hard for him, it was hard for Eamon, I guess. Some criticisms of this before the most substantial I hope you can see that as what I intended it to be but maybe more than that this is just supposed to be two ellipses crossing one another no singular point, well, four no gaps, nothing one ellipse on top of another this is not for Flucke a quartic curve even though he uses it to construct a quartic curve having 28 bitangents this is excluded from his list Right? For the reason that it's not morally a quartic. Morally, it's two quadracents, right? They've just got together and pretended to be a quartic or something. Okay? It's a reducible curve. He does not, for some strange reason, want to allow reducible curves. He's only interested in the irreducible one, and that emerges from the way he's proving things, but he never actually says that. Okay? So that's missing. also the singularities I directed your attention to were the double points on the cusp there's no reason to believe that nature is exhausted or even mathematical nature is exhausted by that and here's just a zoom it's a triple point what I hope you can see there is a quadruple point if you go in a bit closer to that I think I've got one more yes this is a quintic curve of some kind you can get all kinds of strange looking singularities that are not discussed by Cooke. So you've got a whole story now, I don't want to go into, of can we make analytic sense of this? Can we make, which is different, geometric sense of this? Can we say that somehow these complicated singularities are the result of a coming together of ordinary things? It seems reasonably clear, I could pull this thing into sort of a trefoil knot shape and we should just see double points. So that's the kind of question that they got into.

37:30 But the real thing that strikes me is not the things that, as it were, Plucke are only gestured towards, more complicated. I try to draw your attention to the fact that very often these singular points and their dual are imaginary. We could actually write square root to minus one. There's a whole literature about what imaginary is in this period that believes people will feel differently. It simply does mean what we mean by x plus i y. These complex points are essential. The number of them is essential. You cannot rescue the duality paradox without them. You cannot dualize and dualize again and get back to where you started from unless these points are there. But they're entirely complex. For example, on the cubic, six or eight of the inflection points are complex. This struck Kukka rather forcibly, and he made the following remark about the mountain to see where I put my English translation, but I'm sure now if you needed immediate intuition he says, must at least take a new and more audacious flight than before in order to grasp what is, in every case, and remains imaginable. You're never going to be able to make all of those inflection points real. They are complex. Klein, who's a student, as you know, comments on this in the invictrum, that this intuition needs to be trained, and he then illustrates this point by finding immediately an error that Plutcher had made himself in the study of the Bicampion. So, this is the point I wish to dwell on. I don't want to go into two particular areas which occur out of this. Plutcher's theory is not projected. Each of the big books, the algebraic geometry books, opens with about half a book on what happens well away from the origin. He never says, oh, we can take any line in the plane to be the line at infinity, and therefore... So they're not truly projective books. Much more projective thinking comes in with Tesla. And they're not truly complex. That comes in with Riemann, and then Clebsch.

40:00 What you have in Pluka is real curves defined by polynomial equations with real coefficients. the points set to understand that I are in the real plane accept rescue some complex points turn up just at the last minute a rhetorical structure you would not accept in a detective story you really wouldn't allow help to appear in the last without ever having been mentioned before what are the complex points into this mess? How did Kluke get into this mess? Okay, and now we have a sort of story which I hope resembles some of the things you heard this morning. The curve is defined by a polynomial equation. Kluke's big idea is that you can go in near the singular point and try and take Taylor series expansions of what's going on. You don't need to do anything except partial differentiation. It's very elementary in a certain sense. If you want to understand the passage from a curve to its dual, you have to invent certain other curves on the way which are associated with it by the same sort of process. You're endlessly differentiating everything in sight. The key theorem is Bazoo's theorem. But out you come with a resolution of the duality paradox. And, I would add, an investigation of novel properties of cubic curves, quartic curves, curves of degree 5 and 6. You've opened up, if you could, a new domain in mathematics that hitherto people knew was there, but they had very little to say. Suddenly, there is a lot you can say about plain algebraic curves of higher degrees. But in particular, you have a resolution of duality. But at a very high price, it seems to me. And that's what I can't actually formulate exactly, and I want to direct your attention to. I imagine Plucco going through standard algebraic things of the kind that Weierstrass could have given Tixters, Hilbert could have given Tixters, power series expansions, all of this kind of stuff. It's entirely fine, okay? But look what happened. The problem was real, plain algebraic flows. No questions. Nobody ever said before Flucke, well, you know, if you're talking about polynomials in geometry,

42:30 you've got to mean complex curves. Nobody says that. When Flucke does this work, nobody says, ah, Julius, you've opened up the subject of complex curves, hitherto blocked to us. It used to be real geometry. It can't be any longer. have become complex. So my paradoxical formulation at the bottom is there. The means available to tackle the problem have changed the problem itself. This is not what I imagined when I read Hilbert describing what a proof is or describing what biostrust is. You knew absolutely what the problem was modulo some lack of clarity which you get to tidy up. The proof somehow expresses is exactly what, in some mysterious way, you already know. What I imagine is happening here is you go to the problem, which is about real geometry, and you come away with a different problem. It has become complex. Somehow the means you're using to study the problem have, and I don't know what to say. I don't want to say whether it's recast it, it's reformulated it, have we rewritten it. I don't want I don't want to say the curves really always were complex, it's just Blücher was the first person to face up to this, that seems wrong. I don't want to do the kind of idealist number of saying the curves have become complex, they weren't before, okay? And there's still too much of a Platonist in me to be able to say that the curves change because our attitude to them changes. I mean, I can't lie to the way I give this to this. But I don't think that discussions of proof express, and I would like them to see, the way that doing a proof, constructing a proof, right, putting the apparatus together, enough arguments together to say, now I'm going to write it down properly, amount to what the proof is. So I'm now two-thirds of the way through. I've given you, you can tell that was the case history, You had the vague generalizations before, and now comes the vague analogy. We're genuinely interested, I hope, in what it is sometimes. I don't think this is the only case in mathematics. Sometimes we find that the problem we're investigating is transformed

45:00 by the only means we have of epistemological access to the object. The acts we can do, the things we can do with them, force us to recognize that the objects aren't what we thought they were, which is different from saying we always knew they had this dimension to them, but we were never able to study them. The analogy I want to invoke is with experiments in science, and in particular with the history of experimentation, which has been a major theme in the history of physics in the last ten years or so, so Gallison's book on how experiments end, or the volume edited by Book Hall on Scientific Practice, which is where Gallison's trading zones first appear. it seems to me that there has been a very valuable shift in the history of physics onto the questions of experiments, and half of it the book Scientific Practice is focused on the word in fact on the word Hetz Hetz fails to discover that cathode rays are electrons I like cathode rays because Pruker is the person who first discovered them, okay, he got so driven out of geometry by Steiner that he went and did hard OK? Hertz fails to discover that they are electrons, and ten years later, J.J. Thompson does exactly that. And one of the themes of this book, Scientific Practice, is how could that happen? And how do you describe it? What, as a historian, do you do when you're looking at this kind of thing? And I want to suggest that an experiment is sometimes a useful analogy a proof. Doing an experiment is like producing a proof. That's the analogy I wish to propose. Okay? So notice an experiment isn't just messing around in the lab. I mean, they do that. Okay? All those photographs of, ooh, x-rays, isn't that spooky? Right? Us scientists messing around in the lab. But that's not what an experiment does. Okay? An experiment has a protocol. It says we're doing these things for these reasons. If we get this outcome, we'll be pleased, we think we're confirming some belief we've had. If we fail to get it, well, maybe we're wrong, maybe we need a better experiment. But it's not just messing around.

47:30 It's a deliberate attempt to go after something and to try and access it, to try and make nature do something which you predict. The literature is not necessarily our friend here. Guess which is more common. successful experiments in the literature or accounts of unsuccessful experiments, as they could show, right? Mostly, all you can find is accounts of somebody explaining why somebody else's experiment was unsuccessful, okay? And just failure, sorry guys, we couldn't find an experiment to clarify this one. It's probably only existed in people's diaries at all, okay? But then that's like proofs, right? Nobody goes into print on why they have or how well, they have totally failed to solve the Riemann hypothesis. Well, you might just think that's quite a good thing, but you wouldn't go with a mundane or ordinary thing. We're not going to find accounts of proofs which don't work, but we do find accounts of somebody saying that somebody else's proof doesn't work. We do find in the physics literature, in fact, this is the case in Point, different experiments to the same end very different what J.J. Thompson does not do is repeat Peirce's experiments, the same equipment the same experimental setup, but a better vacuum, he has a better vacuum and that's crucial, but he does actually a different experiment, he has a different idea of what you would have to do to show that electrons and cathode rays are the same kind of thing and when Jeremy was talking earlier today about the interesting fact that we do have multiple groups, I want to suggest that at least the history of science literature has accounts of why you would find multiple experiments at the same end, okay, and that that is perhaps one of the ways in which we can explore the ways in which we have multiple groups. Another point to make, when Paolo was describing Kitcher's he presented it correctly of course as an account of a scientific theory. Is this theory better than that one? So you have a whole ensemble of the thing and he described how you might try and evaluate it as better or worse and then gave reasons for suggesting that this approach is likely to be unsuccessful unless it's heroically modified. That's a very standard position from history of

50:00 it. It was about theory. The intervention precisely is to say, let's go in close and examine the role of experiments. And I want to suggest again that if you go in close and examine the role of experiments in science, you could go in close in mathematics and examine what it is to construct a proof. So I'm moving out of my concluding remarks. I sometimes at a very great distance, the siren call of the strong program, okay, and one of the strong programs is that we treat success and failure equally, okay, and I certainly don't want to go that far, okay, but I think that there's a sense in which what Kluger did, right, was and certainly a great discovery, is not the kind of success that one imagines is implicitly conveyed as the message by the kind of proofs that Hilbert would want you to produce. There's some difference, it seems to me, in kind between what is discovered, this element of transformation in Pluker's work that is not there in the standard account of a Pluker, okay? In scientific practice, the book edited by Pluker, there's some discussions of how to write this up. He imagines Petz's work as the 1890s. After all, what would somebody say from the 1830s? Oh, marvelous. A whole laboratory devoted to science. So, somebody from the 1990s who would be appalled by the smell, they would think you get that from the way the electricity sector produced. You think the whole laboratory setup is dismal, and you're amazed they can get any results at all. And the lesson he wants a historian to draw, and I'm a historian after all, is that there are going to be ways in which you can profitably account for what a mathematician does by being quite close to the period, okay? So instead of saying, well, you know, we have an example today, and it's entirely... well, you know, they lacked complex function theory. That's quite true, and without complex function theory, they

52:30 can't do certain things. It's not that that's the wrong account, you maybe just want other accounts which will explain why somebody does whatever it is they do, given the means available to them, okay? I also think that this analogy, when it works, and I think it'll work quite often, between an experiment in... I also think that that analogy is somehow closer to what's going on than trying to compare mathematical theory with physical theory. I have no problem with people trying to do that, okay? reasons why that's going to be, well, it's not going to be an exact comparison, okay, they're going to be overlaps, and they're going to be things that are distinct about science. I think it might be more proficious to inject into accounts of what a mathematical theory is, an explicit attention to how proofs are constructed and carried out, right? I mean, business where you say, I've done enough examples now, I think I've got something like a theorem. Now I think I'm going to conclude it. You shift your attention from calculating lots of examples and always getting the answer zero to saying, ah, okay there's a pattern here, I'm going to now do something different. That's the point I'm going to say, I think, you know, I understand these cathode rays enough to determine that they are or aren't carrying the electric current. I do the following things to eliminate the electric current, or focus it, or whatever. So I think it will be a useful analogy, at least for those of us who are historians of mathematics and working in a way which we hope will be proficious for philosophers of mathematics. So I'm optimistic that I think it will be. Thank you.

55:00 Well, it is a bit of a character here, but it's hardly his fault. In On the Infinite, he really is boxing himself in to a view of the essence of what mathematics is. He wants to say that in some sense classical mathematics is finite. So he concentrates on just one aspect of doing mathematics, but he declares that to be the real thing. Certainly in necessary conditions, you can't actually write down these kind of pretty good proofs. You're unlikely to qualify for that condition in any way of choice. But he does seem to me in that paper to want to say that that's what mathematics is, these written proofs and that conditions of the guy who write these things down. So take the second point of yours first. No, he doesn't say it has to be axiomatic. The whole point, I think, is referring to Weierstrass. He's doing what Weierstrass did. Biosos wasn't, of course, an activist sort of person, but you now should be eating, or Hilbert, but Hilbert's in a tradition of careful and accurate reasoning, thinking that he's like a written character. What I think happens, because Hilbert ignores all of, at that point, all of the soft stuff, in order not to give ammunition to Brown, he just is going to focus on the written stuff, aspect, right? I rather think of it as a bacterium and it excretes some things on a cell wall or something. And Hilbert is saying, that's the mathematician, not the whole cell, but this little bit. It's a very, very narrow view. The consequences, nonetheless, the idea of Hilbert saying this, of lots of people, is that I think we have come to the stage now in the last few years where we are beginning to rebel against the view that everything other than producing proofs is of no interest to the philosopher. It's psychology, maybe it's math day, you know, we really would love our students to be more original, but, you

57:30 know, it's, well, the ontology and epistemology aspect of mathematics is crucial, it's this formal aspect of it. I think, and I can say, I hope that he'll be quite creative. He panders to the need. I think there's mathematics, there's a, mathematical can be of a particular kind of philosophy, legitimately be instituted, and for the rest, well, you know, let's have a thousand flowers. If you happen to do mathematics in a darkened room, no one's going to stop you. I'm delighted you're grinding out a theorem. Because it's just a fact about you, not a fact about So that was the, um, Hilbert somehow contributes to an unduly narrow view of mathematics. Everything that is left out of that is somehow then dismissed as sociology, psychology, whatever. And I think that's a wide enough spread for you not to be entirely correct. You mentioned the analogy between experiment on the one hand and the construction of a mathematical proof on the other hand. I've always thought that the correct analogy is between experiment on the one hand and the calculation or computation on the other hand. The whole purpose of mathematical physics is how you, instead of performing an experiment and getting the experimental result, you model the empirical situation mathematically and perform a computation in your mathematical model to predict that result without having to support the physical experiment. But I think I would say that that process of modeling, which may involve new mathematics or some very clever mathematics, is certainly part of it. And so, I mean, I don't dispute what you've said, but I think that in getting to the stage you now have, something that numerically simulates nature and can test your theory, you have probably done quite some sort of conceptual work. And I see in the construction of experiments

1:00:00 a similar kind of thing. We're going to have to block out the Earth's magnetic field before we can measure this. And, you know, various quite conceptual arguments have to go on before or you can actually do an experiment with it, okay? And that's one reason why I think it's a sort of analogy. So when Jeremy was talking about algorithms, I thought, I don't have any quarrel with that, and I don't have any facility with, you know, hard mathematical logic, and he knows what an algorithm is, I don't. But there's a sense in which I feel that the, just as the understanding of nature is deepened, expressed by the scientists in the design of an experiment. I also think that the understanding of a mathematical complex of ideas is partly expressed through that bit where they clear the decks and say, okay, I now think I know what the theorem is. But I only want an analogy, but I do think it's there. If you tell the rest of the room, I know from your face I have a good piece. But it turns out that there are many remarks in Wittgenstein's writing, and many of his three words. Calculation and... The discussion is going to continue with Wittgenstein. Oh, yes. This is... There's so much to talk about with the philosophical remarks, too, but I wanted to just get clear on the Pluker story this is really just for clarification I'm puzzled as to how if the story is that Pluker started out with real curves that then sort of ended up finding himself with complex curves because he was forced to by the problems he was addressing how could he have started out with Bazoo's theorem which he took to be one of his basic methods, because that's going to require both complex numbers and points to infinity, right? So that was already built into the framework he was beginning with, right? So now I'm not sure. So what exactly is Plucer discovering in the...

1:02:30 I mean, what is Plucer discovering about the setting? You could argue as follows. No, I don't want to do that. No, no, no. You can do the following argument. Preliminal equated to zero can well have complex roots. No problem. Okay? That's the fundamental theorem of algebra. So, if you're messing around with curves, you're going to have to live with things that are going to get complex. Right? I still think that that change is forced on you by the package. Once Pluka says, look, I'm going to go for this talk about first polos, and I'm going to use Basu's theorem, he may have said to himself, at that moment, I'm bound to get stuff that's complex. He may have been crossing his fingers and hoping that he could separately prove that there weren't too many of these useful points and they all turned out to be real after all. Oh, well, I mean, look, if you consider two concentric circles, right, in the plane... Yeah, they need four points, two which are complex. Yeah. But... And I don't know, you know, Pluker is second only to Riemann in my estimation. I bet he saw that. But the fact is that nobody before him was saying to study algebraic curves, algebraically, you're going to wind up going complex. And it's... It takes a generation, it takes them to Riemann, really, to say, OK, that's what they are. Cayley, 1879, says, I had thought it was obvious and written down somewhere that a polynomial fxy equals 4 is to be understood with x and y convex, but I cannot find anywhere where it was written down. They have to be dragged into writing, I think, this. I mean, it may be that the difference here is between a historian He's quite happy for people to sit around for 20 years. And the problem, the way discussions of ideal theories carried out by ideal minds proceed, where, you know, this delay is intolerable. Isn't it obvious that this goes with that? Well, but I'm not sort of assuming that as a general principle, only in the specific case. I don't see how you can possibly think Pazou's theorem has a hope of being true.

1:05:00 Unless it goes complex. I know of no account which really grapples with the fact that you have a problem here in real curves and you wind up talking about complex points and complex coordinates before future. I don't find reasons in Euler who uses this a bit. and Cramer is another it's a kind of it happens off stage some of these numbers are complex and nobody cares about that nobody cares about the two points you can't see in that setting whereas you have to care that seems to be what's crucial here that's what forces the change if Plucke had come up with a theory which said, oh then there are some funny complex points of no interest at all I don't think it's because these things are what, say, duality, that people have to say, okay, how can you possibly have a real curve, made up of real points, plus, what is it, 2 plus 3i, comma, 4 minus 7. Well, you see, that suggests that they're making... In gentle disagreement with the government, I agree with your analogy. Actually, I wrote an introduction to paper, we joined paper with the physicist, which then spells it out, roughly in your terms. going a little more deeply in the analogy, namely, the myth, the ongoing myth for a while that the proof could be entirely formalized, this means that it could be taken out of the knowing subject and could have the objectivity of a potentially mechanizable proof outside the human construction of knowledge. This myth is a kind of a continuation, and a lot of that came out in history, to the of the absolute facts in experiments. So the proof for a physicist, of course, is the experiment as long as there was a belief that those facts were outside the knowing subject, this was really a belief in this externality of nature and knowledge from the knowing subject. And this was carried on with the myth of a potentially mechanizable proof where certainty was given and by being outside the human construction of knowledge.

1:07:30 So this absoluteness, which has the power, unfortunately, we went out of that in physics, materialism, and mathematics, but in physics it is no doubt, mostly because of quantum mechanics. And the part that may be going on with the notion of construction principles that I mentioned, which is the theoretizing mathematics where you propose conceptual structures and spaces and shapes or whatever, which is the analog in physics, which, again, theoretizing where you propose trajectory first and then you keep going, you propose electrons and elementary particles, which are conceptual constructions as well, which interact with experiments in the same way where they interact with proofs in that matter. I have to say that the way you set up the juxtaposition between the murky, mysterious enterprise of discovery versus the whole mechanical enterprise the group I think that had nothing to do with the way he works on things I think he had a classical view the classical view was to distinguish methods methods of investigation or invention or discovery all three terms and to distinguish those from methods of demonstration and the distinction was based on a legal concept, the old Latin concept of investigatio, which was to track down the criminal, but you track down the criminal, you apprehend the criminal, but that's not the end of the case. You've just now got your suspect. You, of course, had to know who your suspect was in some sense in order to get the suspect, but the trial, the whole enterprise complete till there's a demonstration of guilt or innocence. So that was the idea, quickly, that was classical between methods of investigation and methods of demonstration. I believe this was, of course, something that was emphasized heavily by the algebraicists from Vietnam.

1:10:00 And what he wanted to do, his novel idea was this. If I can have a mathematically precise way of describing the methods of investigation and the methods of demonstration, then maybe what I can do is to show you that every time a method of investigation, and complex analysis and stuff. Every time a method of investigation establishes or apprehends a real conclusion, you'll be able to get the same thing by demonstration. That was his idea. He wanted to do it on such a grand scale of generality. Of course he had to have a standard way of representing methods of investigation and methods of demonstration and that was worth formalization and at no other stage I think I should put my hand up and plead guilty to a minor offense, can I plea bargain? Or you can move my grasp No, I think I would say many things about Heldt that would contradict what I did say because I think he's a in some ways a contradictory person rhetorician from time to time. So he's a man who solves problems and he spends acres and acres of time bashing through difficulties that amaze other people, finding connections that people haven't suspected. This is what he really remembered for by mathematicians much more than the proof theory, particularly. So, I entirely accept that. I might have chosen to give a somewhat similar talk, which would have said just that methods of discovery are frequently disparaged as being just beyond who knows how these mathematicians do it. Yeah, and I perhaps shouldn't have particularly tried to pin it on Hilbert, though I was very struck that he singled out from, I think he sees himself as a kindred spirit of Biostrasse, just updated. And that spirit consists of making precise, writing things down exactly, and the emphasis is on writing. which in Brouwer, of course, would have found truly bizarre. The emphasis is on what? It's on writing.

1:12:30 On the infinite, there's a heavy emphasis on, you know, line by line deduction down to the end. I don't agree. Okay, I won't take any audience's time now with any more of my mistakes, but that's what I remember. And that's a particular kind of Hilbert, and I think it does, it's not hoaxly unfair to Hilbert, and he does pander to this idea that all kinds of funny things happened before, entrails and pigeons are coming before, I know, as long as it works. And then you do something different. You write it up properly. And even you in your argument said every time you have a method of discovery, you can find a proper proof if the discovery method is working. He thought that was what his monetary conservativeness proof is supposed to do. It's supposed to general who are mechanical you know transforming a method of investigation or a real result into a real proof of a real result and so he was vindicating methods of investigation by saying okay so my crime is to have missed transformation and I will I will I will I've pretty bothered with you in a I don't remember it in that paper, it's okay. Um, okay. Any sort of scary? Each object curve, not because of the complex points or the complex points, but because you understand that the singular points are essential. They are not simply particular things that can be left on the margin, but they are essential to the object. Yes, but that's not the only thing that happened. Yes, but since you stressed very much one aspect, it seems to me that this other aspect is also quite a shift in what you described. Yes, I agree. Nice exchange. I'm trying to show you that. Yeah, I want you to... This seems to me And what is the correct direction of my series, which applies to the series which I am building up.

1:15:00 I should say that it's eight seconds to put a contribution to that to see what other questions is going to be the bottom of the board. So I could allow you to say that this syntactical discussion and now it's filled by a reasonable thread. Can it not be discussed with the syntactical means? No, I didn't intend to say that the syntactical analysis can only do these things, but not those things. So, what I found, what I thought I noticed in the way proofs get written is somehow, they confirm what has already been suspected. And that's one of the things that Biosas is doing in this paper. You already have some suspicions, pretty shrewd suspicions, about what's going on. And you refine them. Sometimes, you have a set of things you can do to get access to the chemical object. And these activities actually, however you describe it, I'm aware this is clumsy. It seems to change the object. They don't allow you access to something you always knew was there but you're blocked. And that's all I want to say. I just wanted to say that, well, put it this way. In the slightly pejorative sense of the word mechanical, a mechanically checkable proof, a proof that you do mechanically, like following the rules, is surely unlikely to me to a genuine novelty. I'm trying to explore, if I want to bring meaning to the perfectly rational activity, exploring in the way that one constructs an experiment, the fact that the mathematical object may turn out to be other than you think.

1:17:30 I want to take that out of the soft, squidgy, psychological bit, put it into the philosophically accessible bit, by saying, think like experiments, that's all. Yeah, well, and you see, you say you can say something like that, but that means that you have restricted the discussion of groups, which is a very small part, and you can have a discussion about it, and, yeah, so we can get to allow the sort of people to do this part, I think. No, no, I'll take any of it. No, I'll take any of it. Go on. And what I just wanted to talk very about, what I really enjoyed about this project, in particular, and instead what we have sold our attention to, and why we have right at it, is that the activity of constructing a proof as creative processes, however mysterious they may remain, however dissatisfied you may be with the two options that I think is either you just make this project up and really have come along, or they were there all along. And so what I would like to hear from you is not necessarily now, perhaps in some future is more consideration as to what kind of third alternative or what kind of way you can bridge the gap between the human force choice between lightness and idealism and respect to the creative power of the act of truth. Because proofs clearly give us reasons to really think there's something to do, or, more than if you preach to our understanding, just need to have some serious, but fully understood, archaeological consequences. Well, thank you, it gives you some idea of what I want me to do next, which I'm speaking against.

1:20:00 I will have a pause. We are supposed to have double volume, but I do quarter past four, maybe 20 past four. Yeah. There's a giant one here. Thank you very much. Oh, right. Oh, this is for this stuff. Yeah, but it's got some stuff on a picture person. Okay. I saw some of his work on that. Oh, this is brand new. I know, but he has published all the other things. It's a value hardly accessible. Is it that...? He's published in the Danish network. Oh, right. Yeah, it's got some stuff on the diagrammatic resources. On the Euclidean stuff. I have, he's my good friend, so I have this entire library of Joniana. Anyway, but this is a new thing on inference packages. Okay. But, anyway. No, please. Oh, okay. I don't want to thank you for helping to sort of change things around. change things around, but I'm very grateful and I'm looking forward to hear it, so, but I'm going to leave it now. I have to talk to you one minute. Oh, sure. It's the same talk. Yeah, and I'm so sorry we got a little mixed up in. No, no, no, no, it's no problem. No, in fact, I wanted to tell you something that, in fact, that I told you earlier, that tomorrow morning, I have to make an injection to myself, And this will be the first time I do it on my own, so I suspect that I will be nervous during the night, so I might not come tomorrow morning. But you can also knock on our door tonight, if you need psychological support. No, no, that's very nice. But I took sleeping pills, so I should be able to overcome that, but you know, this is something a little bit, this is the first time I do it. So I may not come tomorrow morning. So it's during the Friday. Yeah, exactly. I completely forgot that, because you know for me tomorrow is something quite abstract, and then I realized that it was tomorrow, and you know I'm a human being, especially when I have to do that.

1:22:30 They are self-inflicted. Exactly, and this is the first one. Exactly. Well, you know, bon courage. But, you know, I want to stay easy, and I want to not put any pressure on the restaurant. If I manage... I'll go back. Yeah? Okay. So, I'm here tomorrow and Friday. I'm starting my own battle in Paris. And, actually, I'm going to be in Paris since that time. You have lots of ideas. Well, one thing I'm going to try to do this afternoon is get a chip for my GSM cell phone so that I will have a functional... Catherine without a telephone, it's like a fish without a tin. I'm very unhappy without a cell phone. Is there a chance we see each other today? Friday. And the rest. Yes, exactly. We shall see. Yes, and maybe you're around. Maybe you're busy. And Paris. That's probably because I'm thinking of for England. and I have to go for a defense. But thankfully not yours, right? Thankfully not yours. No, for once. Oh, you're an external examiner. So, I mean, I'm going to have a couple of days, but I'm going to be easy to stay. I see. Let's try to catch up here. Catch up while we're here. And then Thursday. I don't know what we're doing. bike. But I haven't given that up. I am going to rent a bicycle when I get to Paris. I've seen some of the places. I've studied the map. Paris, a vélo. No, no, no. In Lille, I've seen the Lille map. There are icons in the shape of bicycles. Yeah, but I think that means a vélo, but it may just mean passage. I don't think it's a vélo. But I thought I saw something that was a vélo. You sound right. I'm better. Well, it's moved down, So I'm better. It's not up here, it's down here.

1:25:00 But I actually can't stay for the afternoon session because I have to get back to Centreville before le pharmacie est fermée. Go do that. And I'm just asking you if you're going to join me. Otherwise other people can't use them when things go wrong. You can say that's me, but I'm actually quite really looking for Italy. I might be able to join you for dinner. Oh, did that also come about? Yes, it did. Not only the lunch, but the dinner tour. Yes, I think so. I think so. I wanted to see ways. Even doing that allows us, even instructing proofs, allows us. Okay. Are you going tonight to dinner? It's still on the side of that account. It's like yours functions. Yes, I'll have to make do with it. very philosophers. I'm sure you're going to have enough to be aware of it. But, certainly I'll be around tomorrow. How do you find out about, I don't know, the maximum focus about the world in age terrible things? Just talking about that. Do you have a copy of that paper? No, that's right. I don't have it here. I give you this on my webpage. As I wrote five papers, I don't remember which paper we had, the long introduction, just making up. Okay. Okay, brilliant. I shall write the long. If you write to me, I email you and say please send me the introduction. Which one of the five people, because I don't remember which one. It's very helpful. Thank you. Can we talk on? Okay. I want to go there. Because somehow I think maybe I'm talking to you. No, I think I'm mentally exhausted. I have a... I have a... We can't speak French. We can't speak French. It's to say, the passage of the polar when the point is at the interior, it's not the proximity of the polarity. It's the theory of the polarity. You show it like that, you trace the diameter conjugated to the polarity, and you show that A, B, the intersection, C, the intersection of the two conjugates A and C harmoniques.

1:27:30 And then, once you have the point inside, you trace the diameter, or you join the center of the tonic with the point. You have A, you have B, and you have C. It's not necessary to do the center. It's not necessary. We can do that. No, but it needs to be a diameter. No, no, but it's not necessary. No, but it's not the... It's not the... It's the diameter. After you do the diameter conjugated, and the polarity, it's the one. Yes. D'accord. And there is, in this case, an hyperbolic conjugated. Yes, so it's another way to generalize it. There is another very interesting thing here, Mr. Ponsolet, as you said. Chez Ponsolet, the center of the crony and the line of the crony. It's the soul card. Yes, yes, yes, yes. And all the other points, all the other points. Yes, yes. And then... That's a good one. For me. And what I want to say is that it seems that in Von Stott there is a part of the duality of Kuker which is demonstrated purely synthetically. Yes, that's it. It's formidable. I didn't understand it. I'm not sure that it's just. What's up everybody? Thank you.

1:30:00 Thank you. Thank you. Okay. Thank you. Thank you.

1:32:30 Thank you. Thank you. Thank you.