Geometrical Thinking — Minimal Surfaces — Part 1

0:00 Thank you. Thank you. So in the 20th century, to a much greater extent than you would like, I mean, basically the existence of Hilbert, Pash, created all this German school, and in the Italian case, because of Piana, and so, in the 20th century, he was also founded in France, because he couldn't build tradition in the 19th century, there was nothing in the 20th century. which I found quite interesting that there is such a clear national demarcation in the States. It got a bit of a bevel and so, and then it stopped just a short, maybe 20 years or so. But see, these guys weren't really axiomatic, they didn't, their stuff doesn't look like Pierre. No, okay. That's it, so, uh, but, uh... Yeah, but I was saying... What about, sir? You don't mind my saying, sir. I mean, what about Carthard? What about Carthard? This is not axiomatics and so on. Oh, I see, but you were talking about a... Strict axiomatics. Strict axiomatics, okay. Okay, so, by foundation, you... In France, in fact, two people who are on this list. One is Léon Lézier, and the other is Jean-Claude Petit. There was another paper by, but it was wrong.

2:30 Okay, okay. I wasn't understanding, as it were, the narrowness with which you were, the strictness with which you were drawing the criteria in the central general concept. Because obviously France contributed enormously in the 20th century to the conceptual organization of geometry, and I mean, especially algebraic geometry. That's true, but it's not the foundational enterprise, the axiomatic enterprise. It's not an axiomatic enterprise, no, not in the old experiment. I mean, some would argue it was nonetheless a foundational enterprise, but we weren't... It's clearly... And Klein, obviously, too. Klein obviously didn't court me to see the journal. Yes, but he wasn't again in the axiomatic research. So basically, lens puzzle is giving you the idea. Oh, interesting. Okay, interesting. At least number-wise, I'm not saying, you know, depth-wise or something, number-wise. This is what you're saying. This is fascinating. When is the book coming out? I'm still writing. First, I mean, it would be first is incidents, which I haven't written yet, and all the geometries I have written, and the third part is metric geometries, which I have written, so the incidents one is still to be done, but it was very interesting to me to not notice just how much, you know, the first people in one step. You have a tradition that you don't know, that you don't know. Okay, and we can say that that's what's happening. Definitely, yeah. But if it were only for Pashtun, nothing would have happened. If you didn't have the backing of him or the foundation, I would have taken on that. But I would say time for your class very long. Certainly, but you see, the existence of the Foundations of Geometry is only due to the fact that there was Gilbert and the students of Gilbert, several schools, were much broader than to do just how these were geometry,

5:00 they got a name for themselves and made themselves to support people until it completely here we are welcome to all of you to the Geometrical Engineering Workshop sorry we're late if I was able to do anything about it believe me I would have It's a pleasure to introduce Jeremy Gray, who's our first speaker, and his topic is Geometrical Thinking, the Case of Mental Services. Jeremy, take the mic. How's it going, Mick? Okay. Go ahead. Awesome. My first thing I want to say is thanks Mick for inviting us all here. I hope we'll be in a couple of days. I feel less eventful than this morning. Well, I'm going to be less eventful. I feel I'm the warm-up act. You go to concerts and stuff, music, or the crowd, you don't really want to hear. It's probably everybody else, so I'm very happy to be the person. I'm going to tell you what, I noticed as I came in, you're as bad as students and you're all professionals. You all sit at the back. Nobody, except the last people into the room. I don't know why this is, but I'm not a student. So, okay. Geo-rexable tensions, in case of minimal services. What I wanted to do was to start an idea about how we think, and take it in an area of researching And it's a hybrid area, it's a mixed area, in which we think in different ways, and I thought that would be helpful because it's clear, I think, that there are cases that have been cases in history, and there are cases, historically, even today, where people do geometry.

7:30 and it would be very hard for me to characterize exactly what that form of thinking is but in conversation, an example came up Victor and Michael can correct me if I get them up, I get slightly wrong but the situation in elementary projective geometry is that you have Pappas' theorem, and thus, if you're in favor of the coordinators of geometry, they have to tell you that the spew field you're working with is in line with commutative. Now, in the case of finite geometry, any finite spew field is commutative, that tells you correspondingly from the geometry will obey Paps's theorem, but there's no geometrical proof of that. There's an algebraic theorem. If I'm going to keep quiet and tell me I'm wrong later on, let's take this to the myth of some truth in what I'm saying, then it's probably kind of false, but I can see it's very polite. The point is, we lack a geometrical proof of a geometrical theorem. We have only an algebraic proof. This little story illustrates that mathematicians at least are capable of saying there are two kinds of thinking there's an algebraic way of thinking, a geometrical way of thinking, and a complex in which you would like, do not have a geometrical So that's kind of a clear case, as I thought of some months ago, to start this discussion with. I'm interested in the hybrid nature of mathematical arguments and how at all could we say that there were characteristically geometrical ways of thinking and that led me to this little story here because it's not enough to say there are certain We could list the concepts quite easily.

10:00 We can go to any geometrical book or any period we like, and we can pull out the key terms, and we can say, okay, these are the key geometrical concepts. I don't think that means that one is thinking geometrically. One might just be thinking mathematically, as it happens, about geometrical concepts. But I don't think it's being passed as clear, just so we could say people think when they do mathematics and if they think with words like periodistic they're thinking geometrically i'm sure you wouldn't have invited us here it'd be a really interesting thing to say i wouldn't expect it okay well good it seemed to me it was insufficient somehow after what's going on yeah so there are of course classics and this is just to show you that i too can download pictures from the web of examples of geometrical thinking right you could if we come away in two or three days time saying that there's no such thing as geometrical thinking then somehow what you do is doing has become even harder to understand um there's a sense until michael will tell us more about this in which Helbert demonstrates a certain way of thinking, right, which to just by the abstract is actually a little hybrid too. There are other ways of thinking. The one that we're going to be following is the differential geometric approach. And there is of course Klein's approach to emphasize the geometrical concepts are invariant under the behavior of certain groups that's what makes them geometrical so to think about groups and transformations is to think geometrically in a certain sense so could we argue that actually for example there is no geometrical thinking how am I point to say that we just don't do it now. We have something to get rid of at least. We've been invited to show to prove a negative. But, people might argue that we have used to be something called classical geometry, but it's dead. Really, nobody does classical geometry anymore, so it's a matter for some historians maybe. I don't like that point, I don't know if you have to say so in a minute, but there's another reputation of the idea that there's geometrical thinking, which is that, yeah, there are perfectly distinct ways of thinking mathematically, but geometrical isn't one.

12:30 Maybe it's not sufficiently foundational, topological is the way it means, or there's algebraic. But you might try arguing that modern mathematics is the study of concepts, a particular class of concepts of a mathematical kind and the conceptual thinking, but it doesn't work to break it down into any other kind of thinking. And you might even take the view of this logical, clear thinking. We just rethink. Some people are better at it than others, and if they're doing mathematics, that's mathematical. But really, it's just thinking. So you might just world, I don't want to go to neuroanatomy, but these days has fantasies of what people will tell you, and it might turn out that when you look at how mathematicians think, they're absolutely indistinguishable from any other kind of people doing thinking, they're legal, they're minds, or they're whatever. There's no bit of the brain which particularly operates in a different way when it's geometry from when it's social science. So, in all those different ways, it would turn out to be difficult or useless to say something called geometrical. Now, I'm going to not care about classical change. I don't actually care that there's a nine-point circle and some three points from these constructions lie on a line, and for me, I don't want to defend that. But I I do want to argue that you can think in different kinds of ways. Certainly, take the fourth point there. I don't think mathematical thinking is the same as just thinking logically. For somewhat the same reason I don't think mathematics is logic. But also because I think that it does involve a special class of concepts. and I want to argue that conceptual thinking is something that's happened, but it comes in different types. One of these types, I would argue, will be geometrical, but certainly in the modern international world, there are many occasions when this way of thinking is a hybrid way of thinking, and it might be more interesting to try and identify a geometrical way of thinking by comparison with something else than to take it on its simplest view.

15:00 So what I want to do is to move into an area that I've been looking at historically, which is the study of minimal surfaces and talking about the historical pale. You have not, in the other slides, the first point you have not presented it, but you want to say that it is whatever it is. I'm not bothered in this talk by the example of old-style classical geometry with drops and perpendiculars and these three lines meeting at the top. I'm definitely not defending classical geometry, that's for somebody else, but even if it was true that there was a specific way of thinking about those kind of theorems, because I don't wish to defend, I don't find out a relevant example for modern mathematics, I'm not going to defend it here. Thank you for the interruption. I love interruption. Please interrupt. Minimal services. I will interrupt in that case, Jeremy, just on a very quick point. On your point about neurocognition, as I understand from talking to cognitive neuroscientists, there's already a lot of negative evidence against your speculation. so they think there is very strong evidence that geometrical thinking in particular is geared to particular structures in the brain and they're quite distinct from those that are involved in when you do logic and algebra for instance I'm delighted to hear this obviously but all I wanted to say was we want a way of describing I wouldn't be satisfied with the answer that no we can't tell unless you put your head in a great machine It's exactly what they do. To me, the answer to somebody thinking geometrically is, well, bring him in the hospital, put him in front of a machine, and make him do it again. So we'll never know if Hilbert thought geometrically or not, because we can't put him in front of a machine like this.

17:30 We want an argument that says geometrical thinking goes like this and these people must have been doing it and find if it pans out that it's different from algebraic thinking that's very nice too but it wouldn't be for me that helpful as I'm working as a story No, I understand the point But finding minimal services and particularly the plateau problem curve in space and you wish to put a surface through it so the surface is spanned by the curve and of all the surfaces you can push through this curve you want the one of least airy this is the plateau problem differs from minimal surface problem you're given also a boundary curve so there are two ways of approaching this problem one is to make the curve out of a piece of wire and put it in soap solution and remove it and now have an answer it may not be unique but soap forms very clever the other answer is to go to mathematicians this takes forever okay it's a very very hard problem okay so eventually in the 1930s i still don't think well that's not quite true um it's not mathematical problem at all okay so for most of the theory that i'm talking about this daily part of the period there are very very few examples of solutions to the plateau problem solutions to a minimal surface passing to a given curve okay first person really to do anything with it is the branch who produced for the minimal surface problem a rather unpleasant partial differential equation so very much a piece of analysis you write down you wrote down Double-integral gives you the area of surface. You apply what we call now the method of Euler-LaGrange equation, and it chains for that. You've got a partial differential equation, solutions to which are minimal surfaces. The only one you can find is the plane. The plane is a massive surface. Thank you. But it shows you how hard the problem is, but that's the only example you can come up with. You can follow the next generation of biomarker which is called Meunier, died at some stage in the Florianic Wars, and I thought Revolutionary War, I can't remember. Meunier had what I regard as a geometrical approach to this. So here we imagine we have a surface that might need to be a minimal surface,

20:00 and we take plain sections of it through a normal. In this case, I think that the surface has been like a saddle, and at this point here, So we erect the normal through the surface, and we take plane sizes through the normal, and we get curves on the surface. And the picture shows here the two extreme cases of the most curved, steepy curve, the curve goes that way, and the other most steepy curve, one which is going this way. at right angles, these are curves, so they have a concept of curvature at that point and if you take the average of the curvatures you see here you get a mean curvature at that point and Mernier shows that minimal surface has zero mean curvature at each point, not just at this one we chose, but at each point, the two curves we get are going to be So the right angle is going to be equally curved, one curved upwards, one curved downward. So the average of their radii is zero. And these things are going to be minimal surfaces. So this seems to be a geometrical interpretation of what had been a partial differential equation before. You have an analytic object, you now have a geometrical object, and you have some between the two. Now this is what we have to hang on to. this is what a minimal surface will look like and if you skip back to that one you can see somehow or other that pretty much everywhere it does look like a saddle and he found two more minimal surfaces and he found the catanoid and the helicoid so there they are looking again everywhere saddle shaped and I love minimal surfaces so here's another that has absolutely nothing with the talk except I need the name of Enneva a bit later on but again you see how negatively curve they are they can do these weird bits of intersecting themselves and also you can see this is in some sense elementary minimal surface there is we really don't have a chance of saying if I come into the room the curve is this surface going to fit through that curve I mean it's not easy looking at these things to say what's the solution to the plateau problem these are minimal surfaces but what what is the boundary curve that it is the

22:30 solution to the plateau problem for and if you all draw curves in space for me I have no idea whether this or any other will pass you those curves Okay, we need to know a little bit more about how to construct minimal surfaces, and this, to my take, is a geometrical argument. It goes back to Gaussian. The surface over there, some kind of torus-shaped thing, at each point has a unit normal vector, a vector of perpendicular the surface and if you take the unit normal vectors and you translate them to a fixed point somewhere in space the end points of these give you a map from the surface to a sphere what example is minimized here if you haven't minimized anything yet listen this i want a gauss map i'm not minimizing anything yet we're going to make a connection in a minute this is how we get you is how you get minimal services easily. How do you find lots of minimal services? And in the last slide, when we didn't have this boundary condition, so... This is an example of a minimal surface. Okay? I'm going to show you how generally you would get minimal surfaces. In the story we have three. We have the plane, we have the We have the cathanoid, we have the helicoid. This is an example, and the surface is an example. For any boundary plane, it's for any boundary condition, right? No, it would be any plane curve. Yeah, for you. Which is not so interesting. Right, that's why Lagrange's partial differential equation is so hard. Anyway, you get solves a very elementary problem. So it's minimal with respect to any... Thank you for saying that. Any curve you draw on that, however weird, that would be a solution to plateau problems for. Of course that doesn't enable me to say that I've solved Scott's problem over there. His curve may not lie anywhere on that surface. That would be very easy to arrange. So how do we get min on the surfaces? The answer is via something called the Gauss map, so I want to explain what the Gauss map is. So it's a map that associates to each point on a surface, the end point of the unit normal

25:00 of that surface, both of course get all the base points of these unit normals over to a pixel orange. So what you can see is that the ellipsoid on the left, this is supposed to be a sphere, the ellipsoid on the left is pretty flat, the unit normals are not varying very much and the red region so the corresponding image of the sphere is quite small whereas the blue region the dark blue region here the unit normals are varying quite a lot so their image is quite big so this is the way the gauss map works And at some stage, when I skip a bit of history, Riemann decides that he will study the Gauss map of a surface. So he has a map from any surface to a sphere. And he notices that if the map, the two functions that make up the map, are separately harmonic, so the first coordinate and second coordinate are harmonic, and if the map as a whole is an anvil-preserving map from the domain surface to the sphere, then the domain surface is a minimal surface. So this is a striking result. You have, you could So if you imagine it's at work, any surface, put the surface on your desk, we construct the unit normals, we think of a big sphere, so I'm in this building that would be enough. Give a picture now of this surface out on this big sphere. And I'm gonna do the map from the surface there to the big sphere. If the map is harmonic, the purpose of the map of harmonic functions, if the map is homo-conformal, then that surface there is a minimal surface. Now, to Riemann, harmonic and conformal, says complex analytic. Once you have a conformal harmonic map from... So, conformal map with components of harmonic, you've really got a complex analytic map. And from this you get what are called the Weierstrass-Enerborough equations, for a curious historical reason, which is the Weierstrass-Enerborough and by the time this stuff was published,

27:30 it was handed over, and published awesomely, but it gives you a machine, and essentially says, okay, take the big sphere, and just run this map backwards, instead of going out along the norms, goes back down in a harmonic conformal way to a surface then it's a minimal surface so now every time you have a complex analytic map you have a formula which gives you a minimal surface so now suddenly you've got infinitely many minimal surfaces not necessarily solving the factor probably you want to solve you've got infinitely many pieces and now i leap forward in the modern period i want to tell you about uh and a little digression so the story just to sum at this stage is you have the capacity to construct infinitely many minimal surfaces practically no solutions to the plateau problem even because this requirement that they fit through a given curve is very tough you have two ways of thinking about the problem of minimal surfaces one is through the theory of partial differential equations. And the other is the theory of complex analytic maps, and especially that it's been harmonic and above all, conformal, which means angle-preserving, is obviously a geometrical tradition, and this is in the same family of ideas as the idea that mean, clear, and banish is everywhere. And we lean forward to this gentleman. This is Jesse Douglas. And he is, with Lars Alphurs, the recipient of the first fieldsman, 1936. And he gets the Fields Medal for his work on solving the plateau problem. And Mario and Tyler, my colleague at Warwick and I have for some time been trying to write up a story of Douglas and his work on the Normal Services because he's the only Fields Medalist you can find no information about. The only thing that's written about him really is a two-page entry in the ancient New South Africa. So there's some interest in China in this picture, which you won't have seen before, comes from the sun. It's a whole complicated historical story. So, first of all, Dante's looking for minimal surfaces and the definition of a minimal surface, the sensory value definition.

30:00 That is the image of a disk, or maybe some of it are made. You want your minimal surface to look like it's got a torus in it with a couple of holes that fit up with a couple of contours, Douglas will try and do that too later on. So, quite a generalization of a topological kind of the original problem. And all you want is a harmonic conformant map, so very much the real position. And this makes it the extremal of the area function. The usual thing for evaluating the area and trying surfaces of the nearby is the extremal, not necessarily the minimals. It's something Mario had to teach me 17 times before I got the point. These are extremals, not necessarily minimals. It shouldn't be called a minimal surface problem. what would Raj propose was the extremal service problem okay and not all extrema are minima okay so this is it turns out to be a particular issue that we will have to uh watch for but this hit upon a very interesting strategy for approaching this problem Those of you who get your stopwatches out will see something, and those of you who look away and turn back will see something else. Those of you who were stopwatches saw a parameterised curve. At each point, as T went from log to 1, my pen was somewhere. And those of you who looked away and turned back just saw a curve. So I wanted to imagine, rather implausibly, with this picture, this is somehow thinking of three dimensions. Well, actually, this would mean you're a rightful thing. So here we have a curve in three dimensions. All right. Now, it's a trivial problem in the 20th century to get a harmonic function which spans this curve. Okay, we can simply, by the plus on the integral, extend my continuous function on the boundary to a harmonic function on the interior, the famous Liebhsele problem. Okay, so we can do that much. And Douglas' idea was, okay, have I got a middle set? Of course not necessarily, but if I vary the parameterization of the curve, so maybe, you know, I speed up along this bit and then slowly go along here, and speed up along here, and slowly go along here, and so on, I'll get a different parameterization, and that will give me a different harmonic function for the end of the curve.

32:30 A solution to your state is normally unique. For each parameterization, with a bit of love, there is a unique solution. But it will vary with the parameterization. He goes looking for the parameterization that gives you a conformal harmonic map of the interior. That's what he claims he can find. And there's a whole interesting story about how he gets away from the boundary curve having to be smooth for example all these things which I didn't want to talk about today because they're more relevant to other aspects of the history story but he's varying the parameterization so that the harmonic function that's maybe analysis is conformed this is also an analytic thing and that's what's hard to find what he does is he works backwards he's very curious what he does decides to explain why he does it. When he starts with a integral equation which will tell him how to find the dramatization, he then realizes the integral equation is actually an Euler-Lagrange equation, or a functional. So that's the functional he works with, because with functionals you have the capacity to appeal to some anthropology and said oh this kind of interval equation must have solutions it must attain its lower bound you require technical stuff about the functional being lower semi-continuous and the space of all things on which is defined is is something like that so what he does actually he finds is functional now present this is quite mysterious functional I'm not saying it's the area But it does have, he checks, that's a wonderful problem. His functional, okay, is low semicontinuous. It's defined on the space of all parameterizations

35:00 of a given curve, and using Creche theory, you can compactify that space and everything works. So, on a priori topological analytical ground, his functional will attain its lower balance. not true of the dirigente functional and it's not true of the area functional but it's true of his functional unfortunately at this stage his functional has nothing whatever to do with minimal surfaces it's just a functional so then you prove so next slide okay he then proves that his functional takes its minimum when the area functional takes its minimum right so it's not a devious he has a strange functional he's coming up for a very natural reason varying parameterizations that seems very sensible to do what happens is that it leads into a functional which has on the face of it just put the functional up front which he does on his big paper nothing to do with area and then he shows that actually this functional can be understood, and it takes its minima when the area functional takes its minima. But this argument about the same minima is also this kind of topological, like in Bennett's process? No, at this point, the comparison between the two is, as it turns out, reasonably straight, it's a matter of manipulating inequalities. So it's not too bad as an argument, I don't want to do it, but it's not too bad as an argument, but as a functional on the face of it, it's completely mysterious, there's know that after the quran people say it's this mysterious functional they claim it's a cousin of the richly functional it isn't the only things in the literature about this functional are wrong okay because that was made but no attempt to explain it but he didn't want to abandon it he could have abandoned it okay but he doesn't want it that's not relevant to our theme here but it's an interesting fact about that but here at least there's a stated reason for liking this functional, which is this, you know, you have a theory of equations and functional equations, and this one is a very well-behaved one. Okay. Now, at this point, I just wanted to consult my notes for a second. It's this business of area. I mentioned before that the

37:30 The question when it's solving with the Sedley-Lagrange approach is getting an extremal for the area function. You're not getting necessarily a minimum and here are two things, one of which you know, one of which you may not. I want you to see two circles on the top, one at the bottom and in between them on the outside a rather fat capenoid and on the inside a very thin capenoid. These are the two the minimal surface problem in this context. You have a surface bounded by two curves. It's a catanoid. The one on the outside is the one you see. It's the stable one. If you make soap forms, don't do that. The one on the inside you will never see. It's another solution. Mathematically valid. It's an extremal for the area of function, but it's not a minimum. And it's unstable. If you vary these things ever so slightly, it will disappear. I can't resist this little digression. Why the Plateau problem, Joseph Plateau in the 1840s was working on this kind of problem, his eyesight had virtually degenerated to blindness by this time. He was a very distinguished developing scientist and he had foolishly decided to study sunspots by staring at the sun. And this, over the years after he done his work, essentially destroyed his life. he was such a distinguished scientist in the Belgian Academy of Sciences his idea of taking a secure position laboratory assistance and equipment and for some reason he starts to work on mineral sources soap films as he thinks they are and you have some assistance to describe what the soap films look like I can't imagine how the assistant would explain to Joseph that there is an unstable solution which we never see explaining this to someone who is seemingly partially sighted that there is also another solution which vanishes the moment you bring the social form out of the back of the bath. I mean, it's just extraordinarily difficult to believe this. This is exactly the problem we have with the area of extrema. We don't know whether we're getting a minimal surface or just an extremal surface, and mathematician is not the minimal one. Okay, so I mentioned that because it's going to come up a little bit later, did. You can construct curves of infinite length and you can construct curves which are spanned by surfaces, but every surface they span has infinite

40:00 area. Douglas decided that this was still kind of contour you could deal with, you could formulate a whole problem in this context. What's it mean? All the surfaces have infinite area. Well, what he says is, okay, let's imagine this is every surface spanning it has infinite area well still you can do this you can get a contour inside and for that one the area will be finite and the surface you get is the minimal surface is the solution to the plateau problem for every contour you can draw which doesn't go out to the blanket so you've got a very very crinkly curve here it's continuous probably nowhere differentiable but still you can formulate and solve a third problem for it, for Douglass, and you can specify the topological type of the surface in advance. So, for example, more than one boundary block goes back to the third problem for it, for Douglass. Non-orientable, you want to draw yourself a mervious band as a mineral surface, yes, you can do it. you want to have a torus in there by the end of the 30th Douglas is claiming that we could make a shopping list. You want a surface of genius 27 with 43 boundary curves, yeah, provided you don't think you're stupid with boundary curves like pulling them far apart, there'll be a surface that does that. It's an astonishing generalization of the original problem. At the same time as Douglas is doing his work But he turns out to have a rival, Thibaut Radeau. I like this picture because it kind of exemplifies the first thing everybody says about Thibaut Radeau. He is famous for being a mathematician with no sense of humor at all. Probably that's true if you look at this picture. So he's a rival. He's a Hungarian mathematician brought up very much in the German tradition of analysis and geometry, and he set himself a problem of working on minimal surfaces. He goes to Harvard in 1930, from time to do his work, before Douglas has published his brief paper, but to this stage only published lots of little announcements. And he gives a talk at Harvard

42:30 on how to solve the minimal surface problem and his idea is very nice. He says, okay I've got a curve, it's going to be spanned by a disk, we want the topography of a disk, we only want the surface of a minimal area of this kind. So his definition is area that's the usual double integral and he says here's how you solve it you take successive polygonal approximations to the surface you want you use a big theorem from convex analysis that these triangles you're filling the surface up with are mapped in conformally and then you have a limiting argument you get a limiting area and he gets a rough ride out of harbour and I would love to know if Douglas was there I don't think I think that was in Europe all the time. I think he gets a rough eye out of Berkhoff, but I really can't be sure. My argument is, by the way, that Berkhoff is a supporter of Drughness and what Drughness is doing. And what they ask right over is, well, why is this a minimal area? You might have an extreme one, but why have you got a minimal? And clearly, he hadn't thought about that. And I don't think anybody in this tradition thinks very hard about this distinction. It's somehow taken for granted. It hasn't been properly analyzed. criticism very seriously. And he decides that, actually, you don't need to work, you shouldn't work with the definition of area as a double integral. Area had been properly analyzed in a very general setting as part of the Bayes' theory. And in the Bayes' idea, you take polygonal approximations, because they're sort of the area wrong. You get all sorts of answers, depending on how you take the approximation. But you take the smallest one you ever get. That's the area. So, Rodeau's approximations by triangles was actually the right way to go. It was a mistake to pretend he was doing differential geometry. He, Rodeau, replies to a couple more papers, two more papers, that actually his approach to area is exactly Lebesgue's. So he will be getting a minimum of area that follows, I'm not giving the argument that follows from the way it's formulated, but its conformal comes from the use of the Cobas theorem to get the conformal maps of a little triangle screen, so in a minute you have a conformal area

45:00 preserving the map and there you are in the abdominal surface, but only for a disc, only for contours, single contours whose span is that topologically of a disc. So why does Douglas get the fields map? Because Douglas does and Rodeau does not. A little digression here, you will not find out by reading the citation. The citation by Karateodori is a complete mix of what Radon did and Douglas did. It's extraordinary. Douglas gets his understanding in Zipar's remarks by Douglas because of the novelty of the methods, the very much greater generality of the methods, the fact that you can do it for curves, all of them spanning surfaces of an infinite area, the fact that you can start to prescribe a topological type in advance, so the much greater generality of the methods is what? Guessing. So that's my historical story, and I now want to try and pull out from it some sense in which we can say that there was geometrical thinking going on, or perhaps there wasn't so much geometrical thinking going on. It's quite obvious that the very best way I'm going to be able to claim for this is that it's a mix of two things. there's plainly a lot of functional analysis there's this use of Kerber's theorem which says that any shape is a conformal can be a formal image of a disc or any of the triangles and on the other hand quite a lot of differential going on so I have a hybrid case to argue, and I want to argue that the way to think of these things is that mathematicians don't just have a bunch of concepts, but they have what I'm presently calling stories. So they have a theory that they're comfortable with, they may have criticisms of that theory, they may find some things that they don't like about it, and they have some problems And it's this dynamical connection between the list of concepts, as you can just pluck out and copy down, and you will find in the text of the books,

47:30 and the way they are formed into a theory and focused on particular problems that's going to matter when we want to say some things are analytical and some things are geometric. It wouldn't be necessarily the correct way to go to say, oh, look, 70% of this paper is analysis, so really it's an analysis paper. And the reason that would be, in my view, not always correct is that mathematicians, when they tell themselves these stories, regard certain steps in the story as more or less routine. And if it's more or less routine, even if it's bulk, if it's 20 pages to prove that some series of convergence is less than 1, that's 20 pages. But they say to themselves all along, now I have a convergence argument, do I need this number to be less than 1? Well, let's see, can I approximate it? Long, complicated. But it's what I've heard mathematicians describe as what a graduate student ought to be able to do. This is in some sense routine mathematics. So it may bulk barge in the paper, but it's not necessarily the most important or the most difficult thing in the paper. So if a large amount of analysis appears in a paper, it may or may not make that paper a paper own. of thinking analytically, it depends. Some of Douglass' work, I'm going to argue, is regarding routine, some of it as highly original. I'm not claiming Douglass' purely thinking geometrically. Another point to make is any study of area, any study of curved things in geometry is exceedingly likely to involve some piece of analysis somewhere, and we've learned that since often years. So unless you want to say thinking geometrically is just thinking about flat things, which is silly to me, then somehow thinking geometrically is going to involve thinking in other ways as well. So if we sort of compare to see what we get, then Douglas has a vision of mineral surfaces, which is quite some way away

50:00 from the partial differential equations where you're thinking. You think of those images of certain kind of math and that math is characterized by one analytic property that the components can be harmonic and one geometrical property that the whole thing can form. He knows that a minimal surface would look like and that's very much the mean curvature definition is a very straightforward mathematical transition points and he's really working with extremals of the area functional at this case this is his big problem this is what radeau challenges him on but the focus is finding extremals but not necessarily minimals radeau is much more when he after he recovers from being attacked from harvard he's much more of an analyst i think in many ways he's very comfortable using Kerber's theorem. He has Levesque's very, I must say, Geometrical Theory of Area. Levesque's Theory of Area is the solution to the following obvious problem. What do you think area is? Come on, let's cover it with things in their own area and add up. Okay, you can't cover it with things in their own area because it's curved and your triangles are flat. So we can take approximations. That's a very sensible way to go, as we've shown in the 19th century. It would be very dangerous because the way you make your choice of triangles can affect the arc that you get. And the Baye's solution is very simple. And he then proves that if you've got a certain amount of differentiability around the area you get is what the formula gave you. But he then needs that formula. and so in this way, Kerber gets genuine area minimisers and Douglas is forced to use Kerber's theorem he hates this, he's forced to use Kerber's theorem to show that his extrema are also minimal surfaces so I think don't argue that in a hybrid situation such as this one analysis so we should at least know we don't have to put a lot of weight but we can at least know because the problem is self-evidently a geometrical one it's about a curve in space and we want a

52:30 surface with a certain property and that property is in that area after all all of those words seem to belong to informal geometry that wouldn't tell us that working in this problem is thinking from this thinking geometrically at all because we might have to translate everything out of that language into some other completely different mathematical language in order to work with it and that's partly what I was alluding to as a reference to Tappas' theorem and Wedderburn's theorem earlier on, that there are occasions in mathematics where you just take the words and as Goethe said, make them mean something else um and you solve the problem over here and you translate back but there's no translation of the work in the middle I don't think we're in that situation I think what I've described is genuinely mixed or hybrid situation in which there is your virtual work and analytical work going on and I think that for instance insisting that these maps be conformal is part of it. So Douglass's way in is to vary the parameterization of the boundary and look for the harmonic infill. It is also conformal. So this is a geometrical condition which you handle of course by means of analysis. The conformality condition comes out as, you know, does some expression equals zero so yes you deal with this conformality question by analysis but i think that doesn't mean you're not thinking geometrically you're probably thinking both geometrically and analytically the question um is after all a geometrical one but above all the story is a geometrical one so you're the crucial thing to prove at this stage for these people is that this map is conformal and then you do some work and I think that's what I'm trying to call something that you then put in a box right section 3 of the paper this map is conformal section lasts as long as it has to last and then you've done it and you close up the box or you need it knows the map is conformal you do that somehow the story the big story says we need conformality and we have it. By the way, if you don't believe me, if you don't take the word of a gentleman anymore, then you'd better look at section three of the page. But really the analysis

55:00 is proceeding in a horrid geometrical spirit. And I want just to add a few more words than I have on the transparency there, that one of the seems to me might be a fruitful way of thinking about this is maybe there are several ways of thinking mathematically maybe I ruttled off a list of five algebraic, analytic, geometrical, topological on let's say theory there might be more I suppose so many if we want to say that people are thinking in one of those ways then maybe we all have an argument which is pretty much the same argument that we use for any of those ways it's the concept of different that's all so here's my I have sort of schema which would say so-and-so is thinking geometrically he's doing certain things which we need to elucidate and the concepts are the geometrical ones that you find in geometry books and then this person over here is thinking algebraically so the concepts are different but there might be some way in which the way they are thinking counts as mathematical this is algebraic thinking algebraically and it's very like thinking geometrically it's just that the concepts differ i would quite some for some reason i'm not sure about i would quite like it if we could say that these where there are all ways of thinking mathematically right they're like different cars you can buy okay they all do certain things that make some cars they've got different things under the bonnet that they can do slightly different ways but they're all cars so i mean i would be quite happy if we could say there was a geometrical way of thinking algebraic way of thinking, except that the concepts are different, but that this storytelling business, this focus on problems business, this organizational material might hold up from one domain of mathematics to another. But you wouldn't like it if that common way of thinking was just logical thinking.

57:30 You said you'd be happy, but you wouldn't be happy if that turned out to be just logical. No, but I don't know why. I mean, I think that I want to say there's some kind of specific, well, I think, I think my sense is that there's a difference between being able to say, A, to pass an exam where you're asked what is meant by the following terms. Define manifold, Riemannian manifold, tangent space, geodesic connection, you know, list 20 of them, right? And everybody in this room offered modest amounts of money, you would pass that test, right? It's not that difficult. We might forget the answer the next day, but hey, we passed the test, we got A's, okay? It doesn't make you a joker. Being able to organize those things and to say, now, you know, for this problem, I wonder if we can introduce a concept of length here, and then we get ourselves a metric space and we get a lot of theories and topologies. That's the thinking bit, and that would be thinking, I suppose, topologically, okay? Now, if someone says, you know, I wonder if we can define a concept of depth here and get a resolution of finite length and then say we've got, you know, the dimension of this ring. That's going way beyond being able to define the key terms in algebraic geometry. It's being able to say, I would like to have a certain structure here, to identify a certain structure which enables me to solve problems. I can see in some vague way how this happens if you're a structurally minded mathematician, which isn't all of mathematics, of course. And that's the difference I'm trying to get at. Now those two stories seem to me broadly similar, that in each case somebody said, How do I organize these concepts in such a way that my problem has, you know, some chance of being solved by a mixture of general theory, which I can avail myself of, and then some bare hands grappling with the specifics. And that's why I would like some kind of, I'd be happy if there was some kind of similarity. And yes, it's thinking logically, but all kinds of people can think logically and they can't deal with, you know,

1:00:00 they get an A on the first test and still will be able to tell you how do you think geometrically about minimal surfaces. I mean, all the stuff that I tried to demonstrate beyond being able to get an A on the initial test. And I think that's my answer to what's the difference between thinking but with some concepts and thinking geometrically, algebraically, topologically, whatever it might be. I also want to end on perhaps a slightly different maybe confusing one, I hope not too confusing it's very important I think in mathematics that there are these hybrid domains, two of the most important domains in mathematics have it in their name, algebraic topology, algebraic geometry admit to be hybrids straight away, that's their strength One of the things you sometimes see with these subjects is that the fact that you can go backwards and forwards from one way to another actually generates variously good ideas that in one domain haven't been thought of that way, but also problems and criticisms of that domain. And I don't think you can do that if you aren't, in some sense, thinking algebraically and then thinking geometrically, thinking analytically, thinking topologically, thinking analytically, thinking telepathy. There are these hybrids, it seems to me, in a funny kind of way, my final point, it's one of the pieces of evidence we have that in modern mathematics, there are these specific ways of thinking. the mathematics do carry uh questions from one way of thinking not just the concepts and way of thinking about them back to another one and find uh some advantage or just deeper problems than they otherwise would have encountered so that's it i rest my case Why don't we take a five-minute break, a strict five-minute break, and reconvene for a discussion today.

1:02:30 so now we'll have a discussion i should i should have said uh a little something in my way introducing one i hope it's the format we follow and that's and that is that it's hard to ask clarificatory questions during the talk but let's try to keep these to a minimum uh it can be distracting to a speaker to have to digress i'll let the speakers handle that any way they want to but my recommendation is try to minimize this but now it's time for discussion if anything is So, go ahead. Yeah, actually, two questions. One, really, more general to your last point, when you're talking about, like, least thinking, right? But I just want to say, if you're talking, could we come from this point in both states? When you are talking about mixing, somehow it seems that they exist in pure form, right? And actually your example, I think, that shows that it's not just mixing, because you're starting with what you say, by knowledge but then correctly there is again geometry which strikes back in functional analysis right and so it can be making point kind of i don't know can be which is just kind of two aspects It's not like you are going to kind of logical, theological thinking or something as a general framework where these things would be kind of, I would say, instances, right? we find kind of superstructures, it's not as good as that, but just you have kind of structure which has these two or more aspects, right, which you, and now we take it to be geometrical, but they just intertwine to begin with.

1:05:00 We can hear you from where you are, if you feel more comfortable, then. No, no, no, it might actually be easier, at least for a bit. At least to hear you from there. We could at least imagine starting the session with the speakers. What I couldn't do was characterize geometrical thinking a situation where it was clear and obvious that it was a geometrical thing. So I couldn't bring talk which said, here is a piece of geometry. It's geometry from the start to the finish, and every moment in it gets geometry. Nobody thinks it's anything else. So let's say what it is. Let's characterize that in some philosophical way. And I felt I couldn't do that, to be honest. I couldn't do much more than I've done. The concepts are all geometrical. They are organised in a way which suggests there is a story. And I felt that that takes 30 seconds to say, and Mick is never going to invite me anywhere ever again. That wasn't a good thing to do. But I could tell a story where the concepts are not so clear and closer to history, I guess, which is what I am, I thought. And I hope there's a relevance to the discussion, because sometimes it's easier to get what something is in a case where you have to make comparisons. I thought perhaps I could say something about, well, this is geometry, that is analysis. Maybe somewhere in this comparison maybe somebody else could say, okay, that helps us all to say what was geometrical about this, because

1:07:30 because you've got a comparison on it. One possibility, of course, is that, at least in some cases, well, one possibility is that this, again, avoids the question of what is it to think geometrically. It takes it for granted that we know what we see. that all those remarks about you know I don't know what mathematics is but I know it when I see it right aren't very helpful but they capture a certain feeling that people have so maybe I avoided the question your question suggests there's another possibility which is that these domains of mathematics are thoroughly hybridized, that there's, you know, one may come away thinking there isn't, never mind the neuroanatomists, anything other than a mathematical way of thinking. This bit happens to be rather geometrical, that bit happens to be rather analytic, but it's foolish to pretend that they're that separate somehow and that may also be true but there are some banalities that I don't want us to lose sight of when we do difficult questions in philosophy I guess there's a list of concepts we all feel are geometrical in a facetious example of an exam I define the following terms we can pull out a list of terms from a book and say okay this is the geometry test and we'd all agree these are geometrical words I have to wait a long time, unfortunately. Oh, I love you. It's teasing. Teasing. Um... Okay.

1:10:00 I'm nervous now. Um... It seems to me that one way forward is to say something about what the purpose or intention of the use of these concepts is, that somehow how these concepts are being used, independent of their kind of dictionary definitions. How these concepts are organized, how in particular, how they are thought about when the problem has not been solved, is some way of saying that this is thinking of a particular kind. So asking if these concepts, which every graduate student can give you a correct definition of immediately, how do these concepts enable me to go over here of this problem, that sense of how they argue or how perhaps they could be used is a way to think, is a way to analyzing how someone is thinking with those concepts and then we would want to make distinctions between thinking geometrically and thinking analytically because we might want to retreat to this is mathematical thinking. So that's why I run the hybrid as a clear case, a clear case of geometrical thinking. Well, maybe that's just mathematical thinking in geometry. There's no geometrical thinking going on, but mathematical thinking. So that was my attempt. You had to get into it, so Marco and Yvonne and Renau and Oliver. in that order, and no repeats, Marco. Sorry? Something bad against me? No, no, no, no, no. So I wanted to say something in the beginning of the discussion. I wanted to make a general point in the support of the discussion to be much more detailed about it.

1:12:30 So I think that there are three different, at least, at least I think there are more than that, but at least three different ways to make distinctions concerning the matter we are discussing. One is a disciplinary distinction, geometry, analysis, topology, and this intermediate in certain parts of your talk, there was this idea, the boxes, so you are working in a certain domain, and this domain has a theoretical but also sociological definitions, there is community working on that and another is a distinction concerning with the objects or the concepts you are working with that is not necessarily the same thing and i think that is there is a distinction on which you insist more your truth but it seems to me that there is also The third way to make the distinction is according to the way in which we make, I would not say inferences, because sometimes they are very informal, but connections. So you can make a connection using, for example, definitions, a review of logics, and then you collect. And then you can make connections because you see certain objects, and you see that certain objects have to be in a certain way, so the connection is of another source. So, it seems to me that to us, one interesting thing about the idea of geometrical thinking in the third one, not simply if you're working inside what is sociologically or academically or historical sense to be geometry, not simply to say you're working with certain concepts because, you know, in ancient geometry it's very quite easy to say that that are geometrical concepts and arithmetical concepts. In modern, geometry is not so easy. For example, the concept of continuity for functions to geometry, which is analytic, is very difficult to say. But I think that the more important difference is the way in which you connect. You connect using the view of information, or you connect using the projects or a sort of intuition of object in space,

1:15:00 So what do you want to do, mate? Should we take several? I'd be happy to take several comments rather than speak every alternate. You think I should reply? Okay. I think I agree with those two, obviously, but also with the connection. I didn't want to make the heuristic argument. I think your third point in some sense is the informal sense of intuition, and then you have a feeling about a problem you know i mean the way mathematicians make connections between things um i want to bundle that up into the story bit but it may be your comment made me think of something else that i think is a question here that other people may want to take out and it gets back to purity of mentality that do we want to say as mathematicians do, I remember being shocked the first time I heard it, but quite commonplace, this is the wrong proof. You go outside of the mathematical community and you say, well, it's an algebra, it's the wrong proof. And nobody knows what you mean. How can a proof be logically correct and wrong? And it's come from the wrong area of mathematics. It's making the wrong connection somehow or other, okay? the essence of so this mathematician say this and i suppose it's connected with whether the methods are in the narrowest sense connected to the problem the concepts of the formula the enabling formula problem are also the same concept and then you'll answer it or whether different concepts are being brought in if no new concepts are brought in the method is pure and if it's less and less pure the stranger the concepts are I suppose and that to my take is entangled with the question of whether the concepts you're working with are fundamental or whether you should write them out of the real explanation and replace them with something more fundamental and if you look for example at the three volume geometry by Novikov, Rubin and Frommin, which I think is probably as good a definition of

1:17:30 geometry in 900 pages as you'll ever see. Most of it's typology. Most of it's algebraic is typology. And that did make me wonder whether in some sense it's quite hard to say that people are thinking geometrically, because that isn't the fundamental of their subject you can find clear examples of the fundamentals of geometrical and actually just reminded me of for example so of course it's in some sense wrong but nonetheless it's a question for us i think can you adequately say that somebody's thinking in such and such a way if you also believe that way of thinking is not fundamental and rests on something else then you'd be saying your explanations were superficial that you haven't fully generalized your story I don't know so it's not a direct answer to your question because I agree I think with what you're saying but I just wanted to broaden it a bit if you start talking about the nature of the connections whether they are pure or impure then one complication we have is they might be pure but not fundamental and for that reason not properly analysed. So yes, it's thinking geometrically, but... What do you mean by fundamental? Well, there are many things you want to say that seems to me that are... Well, okay, to give an example, I might be able to construct an argument that was property of metric spaces of metric spaces and I might or might not be regarded as insufficiently fundamental because this is really a topological property right so there are many occasions when that conditioners say we don't have a full understanding of this it should be a topological theorem but we don't have those results yet that's what I mean by fundamental and then if you look in that three-volume work you will see that really the more geometry they do the more they talk about manifolds the more they talk about algebraic topology the more they want to locate this that or the other concept in some setting which

1:20:00 is actually really topological that's where I mean there are many many examples many of them if you know many times I'm sure so that's what I mean in some sense quite fundamental but all you're working at this level this is a special case of this should be here okay why don't we give you one a shot here and we'll proceed through the sequence Sorry, Ivan, it's hopeless, I'm afraid. A question about the notion of analytical thinking. Thanks, sorry. Not your fault, of course. Connected, of course, to this notion of keeping track to something of geometrical thinking. I wondered if, you dealt with this notion as if analytical thinking was one homogeneous kind of thing, but going back to the very rich and wonderful story you talk, I wonder if we could make differences between different kinds of analysis, different sorts of analysis, whose role is different. For example, you presented a definitely geometrical Meunier theory and the connection between minimal surfaces and . Well, when you look at the tools, when you make a list of the analytical tools, you have differences. You have some parts of an analytic, so to speak, overloaded with geometry, and some other parts which are not. So can this be connected to your notion of story? You presented things like that if I understood you well. If we try to put a hand on what is real geometrical thinking, we should not look at content or concepts or tools, but something more connected to the intent, the direction, the story.

1:22:30 Could we interpret this as a way to make a yelking between leading parts of analysis and some subordinate parts of analysis within one scheme? For example, when you look at all that you presented about Gauss now, it's obviously overloaded with geometry. The link with the curvature is like the nose in the middle of the face. But functional analysis, on the other hand, is subordinate, is answer, in a way, to the point. Could this be...this brings to the fore perhaps the notion of meaning. What is the meaning of the analytical tools we use? If it's the geometrical content or this notion of geometrically overloaded tools is a way perhaps to keep track from what you presented as obviously geometrical, and transitively I think it's a good idea I like the word overloaded a lot I'm not happy with hierarchy necessarily there might be three different things why it should be a hierarchy. But you presented this notion of detour. Yeah. Detouring is hierarchy. I'm happy to agree with myself. But I'm not so sure I want to imagine this would be a hierarchy of ways of thinking of humanity. That's why I like the word overloaded, genuinely a hybrid but at this moment the predominant way of thinking is

1:25:00 geometrical perhaps over here at this moment the predominant way of thinking is analytical but your comment makes me worry about something else one could make too many divisions I suppose it's interesting to imagine there's a analytical way of thinking and i might be willing we might all be willing to break the analytical way of thinking into two or three uh different methods i would worry a very great deal if it's hard to break into 30 or 40. that somehow now you're just everything's different and the way you've made these distinctions seems in a way to make the whole exercise of making these distinctions useless so that's another question for us I mean if we start characterizing a way of mathematical thinking is geometrical how many other ways are there for example I'm thinking oh what's it called the book by Lakoff which is a title like how mathematicians think or something I certainly gave up on it because there seems to be no shortage of different little mental mechanisms that mathematicians could use. And I felt in the end that this was not characterising anything. It just seemed to me there were lots of different things that mathematicians did. And I wasn't somehow informed about how this was mathematics. I like your word overloaded it. I also think you brought into the room a legitimate concern that if we make too many distinctions, we've wasted our time. there's a medium maybe we can make some useful distinctions but we discussed and we should certainly not want to make like it might be too many no um yes i have two related questions on the issue of extra balls versus let's just do one two short related questions yes one takes one One sentence and the other one is the real question. The difference between extremals and minimal surfaces, because as soon as you try to go for an analytic treatment in terms of

1:27:30 partial iterative equations of minimal surfaces, you start investigating extremals, and whether you are doing one or the other is a key part of your story. The short question of clarification, is there a clear there's a geometrical meaning to the x-moles that's not minimal surfaces and the big question is about what he said about RADO but maybe I didn't get the story right because he said RADO actually proved a theorem about minimal surfaces but with different tools and he hadn't thought about the difference between x-moles and minimal surfaces And so the way I made sense of that, perhaps, maybe I just misunderstood, is that you mentioned the idea of boxes and modularity, some extent, and the fact that you can have leading ideas and leading steps, which are geometric, and a huge load of analytical technicalities to fill them. The steps. Actually, in Rado's group, you mentioned three main steps, three main concepts. And some come from the analytic part, which are directed towards spectrums. And some come from the more dramatic part, which are directed towards minimal surfaces. And so the tools come with the story. And maybe you hadn't really understood what the whole story was about. well no you cannot see from an extremal surface if it's stable or not the helicoid i showed you is not stable it's only this one is stable if you start here and put a helicoid between these two lines and potentially looks like that that's stable once it wraps around normal once it's not stable because you wouldn't you couldn't tell by looking so there's that question is no The other point I want to make is this, he goes to Harvard with the belief that area is measured by the double integral, e.g. minus f squared, yes, that kind of differential geometrical thing, and confused, as almost everybody was at the time, between extremals and minimals, it somehow had gone into the collective way of thinking that this is finding minimals. they had stories and maybe not the right stories and maybe not quite the right story and and and

1:30:00 somebody i guess it's broken but i think we'll never know says extremely minimal which is it and that forces him to think about what area is he goes back to the big fundamental definition much more fundamental and different geometrical one and that way and i skipped the details says no mine are minimal services actually you know um and challenges douglas whose work is only now discovered or heard about to prove that those services and then all this they look extremely important too and that is a big problem for douglas now maybe this is by the way purely a historical point maybe it's a point of relevance to this meeting i'm not sure but that's that's But the fact that he had maybe not clearly understood what it had proved, or what his proof was about. Yeah, I think they just forget, and it's very strange, that this is a fairly elementary distinction, and the entire community of people working on this have just forgotten it. And there are probably good techniques, I mean, you do the second variation and all of the stuff that's been taught by Schwarz. In principle, maybe too hard to do a good case, but in principle there's a technique for sorting this out. But they still think the minimal surfaces is finding minimal surfaces. It should have been called extremal surface theory. They would never have made this misunderstanding. It's embarrassing actually. There's not a profound intellectual misunderstanding here. There's a foolish, collective, shallow misunderstanding. Thank you. It seems that you take for granted that the concept of thinking is essential in order to understand mathematics. And on the one hand, you're right, without thinking there wouldn't be any mathematics. So at least thinking is essential to the existence of mathematics. But on the other hand, the concept of thinking is so difficult, so obscure and perhaps even metaphysical. And there are alternatives to this talk of thinking. I would propose at first sight, mathematics consists in manipulating symbols according to certain rules. On the paper, on the blackboard, but not in the head.

1:32:30 And my question is, shouldn't we at least try to solve our philosophical and historical problems on this material level instead of using obscure concepts like thinking? Well, Jeremy didn't use that. I used that. I'm the one who picked the theme of that. Well, your thought. I'll give you the word thinking of my presentation there. As long as we have the same lawyer, one of us will get off. I did use the word thinking, of course. In a way, you're right. Absolutely right. I was invited to, well, I think you invited me to a talk on meaning. there's use I just talked about how these things are used and in a way I would be quite happy to say okay there are certain concepts and exactly what you're thinking there's a certain process who writes this up in a certain way, which ultimately is highly symbolical and the criteria for assessing these arguments are increasingly actually literally mechanical. Jeremy Avakad, for example, does exactly that. He spends his time talking to a machine until he agrees that he has approved the prime number theorem. The rest of us make do with friends. No, the machine is not better than our friends at this, that's a certain point of view. And there's no reason to believe that isn't because that machine thinks, but it manipulates the symbols forward very well. So, on the other hand, the distinction you made is that yes, that's not... there's no way in which we really believe that mathematicians work entirely exclusively at this symbolical level something else definitely takes place in all of us when we do mathematics and any mathematician that we've ever come across we cannot really imagine someone walks into a sits down at their desk and writes 20 pages of symbols and was doing nothing else but they haven't solved the problem at 9 o'clock in the morning and they finish by

1:35:00 No, that's not a good picture of this mysterious activity for thinking. So I could, I suppose, eliminate the word thinking. I could talk about the use of certain concepts in forming stories, forming questions, how these concepts appear to be manipulated in the course of moving and out towards the stage where it can be written down in simple ways. And that would be the question of what is it, of what is it thinking. and there might well be some genuine American doing that it's possible it's the other way around isn't it, though, that somehow thinking is very, very hard to understand, so let us take some clear case where thinking is taking place, like mathematics where the standards are very hard where we can make pretty sharp distinctions between various kinds of thinking. Oh, I'm having an informal feeling here, through to, I think if I go away and learn non-community wringing theory, I will, through to, oh, look, if you look at the ideal wall, then all these different kinds of things, maybe that's a better way to look at thinking, or at least a good way of thinking about thinking, than some others. So we shouldn't, you may you're right that we can eliminate the concept of thinking from the analysis of what I'm going to do it for a long period of time I just hold out prospects that possibly if you want to investigate what thinking is that thinking mathematics might be a very good way into thinking about thinking because it's a very easy but this is a different question sure I agree but if you use mathematics in order to understand what thinking is but here we want to of mathematicians thinking that's a good example of what thinking is but it may go the way meaning perfectly good collection of philosophical arguments i said that this meaning is a useless concept we shouldn't be talking about meanings we can do everything we want in philosophy without talking about meaning and maybe you're right we can do everything we want without using the word

1:37:30 thinking we have some sense explained in a way okay but for one of it i offer you the mirror image of what you said in the end okay uh we have two questions that's sebastian and paul my question will link it with the last question and we've got this question because i noticed in your abstract that you aim at distinguishing between the objects mathematical concepts and the way they are treated and so to take what Michael said between objects, concepts and from the other end. And I feel comfortable with that kind of distinction because if you make the distinction, an essential activity of mathematicians is to make inferences in order to give what i would call a mathematical analysis of objects and concepts for instance we can consider book three and book four of these elements as a mathematical analysis the concept of circle so if we just say that we make this distinction what is for us a mathematical concept i mean how do we grasp objects how do we compress these concepts and For me, it is not clear afterwards. And so it is also linked with discussions, mathematical descriptions, not about objects, but about practices, about inferences. For instance, if we take the Cartesian and the Pascalian solution to problems like the three tangent circles, the question is, do you provide us with a solution? The cart says, I give you an equation, it's a solution, and Pascal says not. So I feel uncomfortable with this tension and the fact that we are able to do this kind of system. I think you're right. I think it's, I mean in a way my little joke example of the exam shows exactly what's wrong in making that distinction. someone who knows how to write down the definition of the concepts but can't make inferences, certainly not elaborate inferences is not someone who really wants to call the definition so I think there are absolutely right I just wanted to raise and rule out of

1:40:00 an analysis of geometrical, it's simply thinking about certain concepts and all you do is you do for the key words and lots of geometrical terms. And to get some sense of what are the inferences and not just the line by line inferences but what for example would count as a good explanation, a right explanation with such examples, an equation, or a construction, or what is it that counts as an answer? And I think that's the right thing, it does give up, of course. Maybe we should be talking about the use of, maybe we can say there are geometrical concepts, because that's unproblematically acceptable, but then we talk about the way they are used. can we characterize the way these concepts are used, and now I don't know, I did suggest that I would be happier in a way if the way those concepts are used is very close to the way another list of concepts are used, or the concepts of an algebraic kind, or concepts of a topological kind, that these similarities between these ways of thinking, you know, you said they belong to a bigger family called mathematical, the use of mathematical concerns. And the only distinction is that this one has the words in it that this one has. But, I mean, when we say, for instance, that each theory is each way we can make inference to bring its own concept to an object, and at the end we discuss, I would say, the equation between this concept, for instance, algebraic, geometry, and political. originate objects or concepts like circles, which is a little bit surprising. Well, maybe it's not answered your question, but I mean, there is an object that consists of refining or generalizing or broadening those concepts. So we used to say a circle was this, but now we say it's that. probably don't change the mind about circles but we used to say a surface was this kind of thing and now we say it's that kind of thing um that might also in some way illuminate what we're doing

1:42:30 the uh the concept has changed the student has to give a different answer but it's the process of trying not to say thinking the process of using these geometrical terms has somehow resulted in us wanting to change what we take the concept so that would be again part of your process of inference but now it's a kind of reflective It doesn't take, it takes back to the starting point and changes the starting point and then starts again. Just wait if I can. Thanks. So, yeah, I think it's useful in this context to distinguish between thinking about concepts and thinking with those concepts. Basically different things. So, the naive view that geometrical thinking is thinking with the geometrical concepts isn't envisioned by the exam example. That's going to be thinking about the concepts rather than thinking with them. So, what do you think of that story? I tried those words out for myself. I didn't succeed as you just have in making such a clear distinction. I mean, there's some sense in which just knowing what the terms mean in some maybe naive sense of meaning and knowing how to use them and knowing how to change them, making allowable changes in what these concepts are. kinds of things and if it maybe that is what in some sense would be the would be a very good way of thinking of thinking about this um is to focus on those moments when uh the criteria proposed for saying what we didn't we used to mean this but now we mean this maybe there at that point somehow you could say there were different

1:45:00 characteristic ways of allowing concepts to change to some overarching view of why this adjective rather than that adjective is the appropriate branch of So, minimal surfaces is a good example of the story of between Laplacian and differential and it's a very good illustration of the intrinsic links between Laplacian and the matrix. But if you have a geometrical problem, you use the Laplacian like a tool to study your geometrical object. On the other hand, if you have an analytical problem, the geometry is not really a tool, but a kind of framework. And so my question, when you say that we have to make difference between thinking with geometrical concepts and thinking geometrically, thinking with geometrically geometrical concept it's to use geometrical concept as tool and the thinking geometrically is to adopt a geometrical framework work to analyze your problem. For example, if you take the statistician to do data analysis, I think it's a very, it's an example of thinking geometrically, because they traject all their problem in not, they don't really use geometry, they don't really, they, I don't, but all the problem is trajected in a geometrical way.

1:47:30 framework. And you can find, on the other hand, some statisticians that use very locally in the theory with geometrical tools. I agree. I think there are frameworks, I think there tools, it might be interesting to think of other ways in which we can recognize different aspects of a way of thinking and going on. I just offered two and you put them in very nice words. Yeah, I think that's right. I think there's probably more ways we can, there's probably more things we can do than state framework into that will nonetheless you know there was somehow geometrical or I'm not a big boat I might be tough for us Scott yes John I think you're the case that you've chosen here that the field of metal plateaus problems just absolutely lovely and when you mentioned that Cara Theodori wrote the report it made something because, of course, this is the Göttingen connection, and this whole idea of a boxing story. This is a Göttingen story. This is the way that Hilbert and Minkowski and other folks like those were actually thinking about physics. In 1905, Minkowski said, well, you know, physicists and astronomers can't solve their own problems. They need a mathematician, because the only creative element here is going to come from mathematics. mathematics as being composed of all these concepts and so on and then you've got this huge demeanor application that's just right for exploitation not by this as far as they've got the problems but it's the mathematicians that have the solutions in other words the idea is to apply these ready-made concepts that they've developed they just don't have any place to put them in right and so uh i'm i'm entirely comfortable with the idea this is a actually geometric thinking that's going on for uh that over for um i don't think anybody's going to be uh uh prompted to say

1:50:00 well these studies are just thinking physically they aren't why why won't we call these physical Well, it's because, of course, the way that they're solving the problem, the value added for Douglass or Rado, this is because they're making, they're putting concepts to use in a way they haven't been put to use before. Right? So this is, I just wanted to see if you, if really what, this goes back to the inference question, I think, is the concepts of theorem, we're actually rewarding these people, right, for putting concepts to use in a new way. This is going back to what you said about making the right theorem, right? This is the wrong theorem, getting the right theorem. Somehow they've actually made the right theorem, but how can we characterize then that they've actually done this in the right way, right? uh we have some way of of actually we have lots don't i mean one is efficacy and one another is purity um so in an application uh the right answer is liable to be something which can yield numbers if i'm going to work with a physicist i'm liable to be asked okay what is the energy on this thing what do i have to do to make it in the lab or whatever okay um and another answer is i mean all these different answers i guess to your question i mean whether i mean i suppose i might worry but it's not so much the right or the wrong film it's the right or the wrong proof that somehow the argument does or doesn't fit the statement of the problem. And actually there is an arena. I mean, any kind of transposition of physical problems or mathematical problems involves all kinds of approximations, naivete, whatever you want to say, mathematical problem. So these minimal searches aren't soap forms. I tell them some certain thickness. Soap forms will do things for you like have free boundaries which we don't have here.

1:52:30 On the other hand, soap forms will not do for you any of the unstable guys who brought them to the wrong. So we have to trick them with extra boundaries so yeah I mean there's another way in which we could look at the way the concepts are being used, which is like, into what story are we transcribing this sort of thing? Are we transcribing it to an analytical setting or transcribing it? So that would be another thing we could watch for. Yeah, yeah, of course. It goes into what we're wrong about that, right? The only creative element is not in mathematics, otherwise it's stuck, right? Mathematical theory doesn't flow, as you call it, yes. Victor, did you have a question? Yes, you mentioned in the beginning it wasn't that relevant to the rest of the talk, the problem of Benjamino Segre, whether there is a geometric proof of the fact that finite disarxian planes are popular. And, well, there is a proof by Heidegger Tecklenburg in the 90s. There was some question that it is purely geometric, but at any rate, it's not algebraic in the sense of, you know, a fight I still feel there's a commutative all good