Problem Solving, Metamorphoses, Geometry C17th (contd.)

0:00 I don't function like that any longer. That's not so surprising because it is often quite easy to translate a problem into a related theorem which you can look at as a theorem. But as a separate thing, they are no longer there. So the theorems are there. And even they became more or less iconic for mathematics. Mathematics without theorems, mathematics without proofs, mathematics without statements. You can't imagine what it is. And still we have seen in this that there are problems which have a separate identity, which even gives you to a structure of a tree which has more clearly, well, maybe not more clearly, but quite an independent entity of life. The theorems, they also carry the associations of the truth, not of the task. They carry the associations, but now cover all mathematics. Maybe mathematics is about truth and truth, not about problem solving. So, somehow, mathematical thinking lost the aspect of task or doing. Of course, it's still here, there still here. But in the apparatus of concepts with which we think about mathematics, It is not so clear. And also, it deals with this kind of heuristic undefined thing that is given. What we see, if you see this green spreading, that's the task being performed. And what you see is that this task being performed is that the objects themselves of geometry change in property. That's very strange. Here it is, here it is, this one, yeah this one was not given in the beginning and in the end it's given. So if we want to deal with there's reverse to find properties of points which change that relates to the

2:30 fact that data has been studied quite extensively by many people, not too many, much less than the elements and there's always when people report on that and And people have, particularly to mention the work of from Denmark, he has written a book about the data and what he has not seen about the data, that's not much. I mean, he has seen a lot and there's a lot in there. But there's still there the attempt to interpret this givenness in terms of how we interpret mathematics now, which would be such theory, properties which are fixed. And there's the acknowledgment that somehow you have the feeling that you miss the point when you do it. And so I think for analyzing these things, somehow, and here I turn to the philosophers of mathematics, you would need another logic than the usual logic, namely a logic which describes this movement of not truth as it is with the theorems, but of givenness, which which is more of an epistemological thing. It's the context that you have. You have it in your hand. So it seems that the whole business of formalizing, which we have had actually after the 17th century. The 17th century was a period where there was freedom of methods, you did the algebra, you left a bit of the precise logic, and later then the 19th century came and the formalization came through, whether built on numbers or otherwise, the axiomatization was taken over, and this thing of a heuristic activity taken so serious that you have a structure in it, a heuristic structure, maybe in sort of inverse logic, but like that, that seems to be not to have, let's say,

5:00 not have a fixed place in the image, the icon of mathematics. And then so might, So maybe we can gain to looking at that kind of .. And then I have my personal metaphor. So somehow, if we don't take the tasks in the semantic series, leave something out, which is comparable to me, you say, music is only the word of the composers. and we leave out the people who do it and the steps there and the actual performance. That is like every metaphor, it doesn't fit precisely, but I want to show that it's not, it is a point of mathematics which I experience as strange, but even more strange that I never noticed it before so strong. And so Okay, thank you. Questions? Okay, Andre? Thank you very much for this wonderful lecture. I very much agree, I have some remarks that probably I leave it out. I actually told you concerning meaning of theory. The principle point exactly that there is kind of aspect of the mathematics that deals with truth and other aspects, which you call task, I would say it's like action, but that's less important, and the two are really intertwined, so actually it does split into independent parts, like even if you have problems for one, action for the other, they intertwined in every, what rather misleadingly is called proposition, misleadingly because they not proposition and say logical sense right they don't always have the truth value but at the end you still okay we should think about different logic of this kind of thing and then I know how important for you in this context is the

7:30 word logic or just you use it but I would say why we should call it logic that if we okay I'm not going to define logic now but I think Frege remarked somewhere, that logic is a kind of science about truth, not to bring some truth, right, but truth in general. And if, which I think quite rightly, we say, okay, that's some aspect of mathematics, which is not, which is linked with truth, of course, but it's not by itself about truth. So I would say we would, wouldn't be logically, just would say logical analysis in a sense, leaves out something very essential in mathematics, not just informal stuff, but which is identified by problem so we would meet other probably count them logic yeah thank you and I fully agree I think this is precisely the point where I feel I'm not well prepared to do it because what I mean is logic is not also, I have, it is a formalized way which can work to deal with structured argument. But then I see the people who, in the studies on this kind of mathematics and on the data, for instance, that with, let's say, the logic which covers the proofs and the theorems in That doesn't seem to work that well. And then I have, because sometimes you hear about temporal logic, something like that, where things over time change. And then I think, oh, maybe for my database, I should learn about that kind of logic in order to see what's happening with the heuristic arguments or not. So it's really a question, is there a, because I could, if I now see logicists who want to study structured human arguments by formalizing them up to a sum of, then my question is, and if I pull that logic, is there a better word? That was exactly the arguments. What do you say by arguments? By arguments, we say to form some true belief, say, right? Something.

10:00 Ah, yeah, but the truth is, I agree with you, it's there. No, you can also have formalized arguments without talking about the truth. Or, at least, the people who did the downtime showed you, yeah. And, of course, that is probably an argument where sometimes you have to have luck. Synology may be also the case. And you shouldn't put that there because then you don't prefer it. But it's still a very structured argument, and there are people who make it their job I paid for that, to do precisely that kind of thing. So there might be some kind of tool for me to get that control. Because one of the things I'm interested in finding out is how much of this classical analysis was actually used in the 16th century, because that's not so clear. and I'm interested whether whether or not it was could I try to put words in your mouth I mean I actually thought what he was saying was more interesting than the way you understood it because he pointed out these givens are arranged in sequences okay and maybe the sequencing is driven by logic of truth maybe it's driven by logic of some other notion he called it the logic of givenness so and that does seem to me to be an open question need not be the logic of truth oh it must be quite related because so it is it is actually a simple argument of putting all these steps together in the well, actually in the correct way, because in this, in this, on the figure, you go from the, from the givens back, and that is also what you do, so the, I think the choice is either we don't care much what we call logic, or we study how it works, it may be perfect, or we are more serious about the term logic, is trying to delimit it,

12:30 It might make sense for logic here, not try to be too large. Marco? Yes, sir. I would like to make a deal with the president. I could have at least two remarks, but I will do only one. And I reserve it for the end of the discussion, if there will be time to meet the second. If there will be time. strictly connected with Andrei's talk. It seemed to me that, in fact, the question of the role of given and the fact that given is no more And the fact that early modern geometry that the geometrical theorem and the question of logic are very connected and it seems that there are at least two ways to connect them and one is to say that the essential problem of early classical geometry is the problem of expanding the euclidean geometry so in a sense extending the objects of Euclidean geometry, the rule and compass of construction, so the object that can be constructed using the rule and compass are clearly not enough in order to solve some difficulties. So I think that one of the points there is there is a problem concerned with the extension ontologies, for example, course, et cetera, et cetera. But the other way, in a sense, in a way to say the same thing, is that there's more than a logic there. I think that the problem is to understand that the structure of the ontology of the geometry is essentially different than the structure of geometry to which we are normally thinking. objects are not defined and given at the same time.

15:00 There is no an axiomatic definition that, in a sense, provides the object on which people work. There is no a sort of iterative activity that give one and then in the same way. The object on which we work are objects that are to be given after a definition that is, in a sense, independent of that. So there is nothing as the domain of geometry in which we can say there is an activity of producing the object on which we are. And it is a completely different way of conceiving the ontology with respect to modern mathematics. This change is largely due to Descartes, so it's not a case. is exactly there to understand that the thesontology is a structural and essentially different one. That's a power one. I think that that's a very, very important point. And I think there are two ways to look at it. In a way, there is a way to look at it as a very exact description of what the working mathematician does and the working mathematician doesn't look at all the objects but the working mathematician comes to his desk and finds a problem and there are objects and some of them are given and some of them are not that capable yeah what these objects yeah those and And somebody has given these terms, and you work with it. And if you want to know what given means, then they tell you what you can do with it. And that's exactly what Yuki tells you. Now, I think that is basically what it is. But I think many people react to that by saying, well, too trivial. That's too trivial. So it can't be that mathematicians don't think about who gives these things, how they come there, and something like all things. But I think he is, maybe naive

17:30 is actually the most valuable thing to look at this. And there's also one part which I would like to keep, if I would find a formalized way of doing this thing, namely that we only work with different things. And how they come to be given, we don't know. They work with them. In that case, I wouldn't use your term extension of geometry, because what happens when you do? You find a given thing, and in the end, it's still given. you can give it to somebody else and he can then meet and say oh it's on my table now I can go on not that today and so and we don't talk about the every meeting so the nature doesn't change because if you have something which is not yet given and then after that it's given but to see this activity as that the total of all the given things so no but when I think of extension of course I'm not referring to that because in this game you can give a segment and a circle and okay So when I speak of extension, the fact that you give new rules in order to be able to introduce in the game other sort of objects, it's not an extension of code, because... Yeah, well, I was thinking, like, field extensions or something. No, no, no, it's not that that I want to say code. Okay, yeah. Okay, Michael. Yeah, I had, well, first I want to ask an information question while it's still on my mind. Who is the author of the book about nuclear data? I missed the name. Christian Taisbach. Christian, T-A-I-S-B-A-K. T-A-I-S-B-A-K. Okay. Thank you. I have two questions. I don't know which one to ask.

20:00 Another deal. Another deal. Nothing else, since it's just come up, talking about given, and actually I have a couple of questions about given too. Well, one thing you just said puzzled me when you said I can take these givens and give them to somebody else. I mean, I'll give it to somebody else. I mean, that's not, that can't be true here because how could you give this deed to somebody else? Forget the other, how could you forget the other things and give deed to other people? Well, you couldn't. I mean, D is essentially bound up with the nexus of what's given here, right? No, the givenness of D is independent. It is. So it's just a magnitude. Yeah. It's a line segment. I mean, so the objects are idle points. But in what sense, then, are you given that? Okay, given a magnitude that's the difference between two sides of a triangle. I mean, these sides could be anything. You're going to get that. Given these two segments and an angle, and you're told that. Yeah, but that's what I was asking. He said, no, Matthew, you should take one element and give it to somebody else. Certainly, because the elements you're given, here is a segment, here's another segment and an angle. and you're told that the second segment is a difference of some unknown thing. That doesn't matter. Here is the segment. And this can be taken and given to someone. It is. But then you're not just given the segment. You're given, you're told here. You're given this segment and information. You're given some other. You're told it's the difference between two sides. Correct. Yes, yes. but so that could be those two sides could be anything but yeah but it turns out that because a is also given yeah but in this problem because that's that's because you have the nexus of games well yes and no the person will say the givenness as far as it is formalized only applies to the object not to the properties. There I haven't found a formalization of it, but maybe the logic there is no different

22:30 from the logic you also have in the zeros, where you also say you have a triangle and the difference is flat. So you get the impression that this constructional thing, no, you get the impression that what we are missing, that we have the logic which deals with the theorem and which deals with all the properties and and relations you can have. That goes very well in the series. But when you are constructing, you can also, because you ask, for instance, that the relation is OK, but apparently the only problem is how you deal with the different objects. For that, there is this whole business. Now, you might think, this is strange, is because why should one particular part of the geometrical enterprise not fit into the logic which, for the rest, serves very well in guiding you through the geometry? And I don't know. Because you might look at it as you And in there is a strange activity with construction which somehow falls outside the logic, outside the user logic. Or you say, apparently, we have a too small view of the real logic of maturity. And it should involve these things. But then it should also involve what happens when you allow more means of construction. So then suddenly there's not one Euclidean geometry, many more. And that's why I think there is a problem with the logic which is used and which is not used and the logic which is, let's say, used naturally and we are not worried. that I can, the other question you had, can I give a magnitude d? I think yes,

25:00 because I would again say that the only thing which you tell about this thing is that it is given. Now, I can tell somebody, hey, wouldn't you be able to find out a set of serums about what you can do when you have five magnitudes, three of which are given? I would say, okay, it goes squirting. I have given him these given things, because precisely what Euclid says, and I've precisely told what he can do with it. Namely, he can assume that he can start with that one in his constructions. Well, that sets up the scene very well. But that's okay, that I also found out with this database, but I have a problem. First I thought, okay, I list the things which are given, and the relations which are given, and the things which are required, and the relations which are required. It doesn't work. What you have to do is to set up what I call the scene, which is sort of the theater play, which at least have some elements of the end already, although of course you shouldn't tell it. So what you first say is all the magnitudes which are around first independently of whether they're given or not. And then I think I should do that because if I want to compare problems then I'll probably have to classify about how many points, how many lines of names there are, if this would work anyway. So you have a C and then in the problems you can then say what is given and those are always, until now, the objects. And the relations, they are, yeah, sorry, the

27:30 relations will be in the scene. Somehow I found out. And if you then want to you don't pin down the problem, then you say which of the objects are given, but not which of the relations are given, because if you would not give a relation, then you can't solve the problem, because there's so much of the things that they're focused. So there is a very strange difference there, and the strange thing is that I mean, I learned this at school, and then I found it not a problem, but it's a problem. You know, what's happening is that some kind of argument which is not strange, which is natural enough to you. And then I thought only philosophers did that. But now also in the story, and then you start asking questions, and suddenly it gets very, very strange. Hank, here, remarking towards the end of your talk, there's a question here about how we write history of mathematics, quite generally. And the largest gap we have now in history of mathematics is the history of differential equations. Ordinary differential equations, and especially part of the differential equations. And it's very, very like this. Absolutely. You look at Euler's, especially the integral calculus book. problems. You have a body of theory that you would extend it. That's a different thing. The character is a dynamic activity. But it's in the language of problems. Can we solve it this way? Can I reduce this question to that one? Must I resort to some ugly method like our series? And I think in a way the moral of what you're saying is that, more grandiose than you would want to say I might risk in saying that it's tempting to write history of mathematics in the way that reflects what we think mathematicians do and there's been a structural view of mathematics and I carry it around my head certainly but actually if you talk to the partial differential equations community so that's very different thing and then the way they are perceived that the way they

30:00 perceive their subject is really much more akin in a kind of way here to this They simply do want to solve problems. The theory they have lying around is huge. But it's a problem-focused kind of mathematics in a way that, I think, at least resembles this. And we should perhaps use the mathematics. Yes, but there is also that algebra has been very, very effective in dealing this, what Jolimona-Dolotus tells you very explicitly, if you want to use algebra, you translate your geometrical problem in equations. And then you do that by using the given relations. And then you use all kinds of transformation work. That is, you solve the equation. And then one should involve the echo of what Katruss had about, E6. He didn't like that. He said, of course you can translate problems in equations. But what is an equation? It's also a problem. And yeah, you can approximate the root, but that is not real knowledge. The point I want to make is that if you in this setting what you had to do with the analysis was that you had to translate the result back into constructions. One of the things you see is that people for that reason stated not solve the equation, not always, but they put it into standard forms which for which they had constructions and as you know when you do the actual solution of a quadratic or certain equation you get the square roots so etc they're very nasty they don't look very very nice for the equation solution you hardly ever see it with them so so they translate this equation into you would say an algorithm of constructions and if you have a good classification of the equations then you

32:30 recognize them quadratic equations is you have three different constructions for the quadratic equations and for that what you don't do is you write x is a plus of min square root of that because that doesn't reflect the construction actually. What reflects the construction is that your x times a minus x is b squared because you make a rectangle x times a, etc. So it gives you another view of what an equation is. Now what has happened with a lot of mathematics is that the point of how the problem gets to become the differential equation and how the different the solution or whatever you do with the differential equation becomes an answer has been put in the hands of others what the mathematicians do is they have all the differential equations in hand and they try to find what what the current And the current orthodoxy is what you should do with it. And this whole thing that, and that is because it was geometry, it had to come back. But what you see happening is the algebra gets a life of its own. And this life of its own gives another aspect to the equations. Actually, it's actually, why should be expressible as an equation in a geometrical sequence of one x, x squared, x third, that should be. Other kind of analytical things, it's actually quite a surprise that this works. And there are also problems where it's difficult. But then the algebra has its own charm and its own seductiveness, And of course, then, if it's precisely, then geometrical thinking can no longer be kept in its bounds, or it loses its identity, in a sense. And you can be happy with that or not, but what it has left in you is this idea, oh, that's geometrical, that's not geometrical, that's not, and that is no doubt true, it will change a lot. But it is very much, because we want to, because these differential equations, they are very

35:00 powerful means, and so does the mathematician do, he makes theories about these things as means, and so they become objects. And then you get a normal adult name, and then it is no longer geometry, because you're no longer interested in the geometrical question. You're interested in what is the standard dealing with analysts. I wouldn't argue the word except that in the domain of mathematics, especially as part of the differential equations, often it is the equations and the problems that are iconic. It seems to be the way in those domains. Well, there you are. It's a very small, slightly pepper-flavored footprint to your talk. It's nothing more than that. Well, yeah, so they have become objects, but those are not zeros. Very true, yeah. Sebastian? Thank you for your talk. We have two ways in order to address the question you raised. First is to, well, problems disappear. And the other, maybe more optimistic, is to wonder when neurons appear in that kind of geometry and why. And so I know that we find them in the, I was wondering, and I remember that we find them in the tract of Van Schuyden, the Cochinendis, and I said three or four or five examples of theorems in geometry in a very particular context, because in this tract, Van Schuyden tried to explain, gives, I would say, a dictionary with 30 or 40 problems, And each time he gives what he calls an algebraic solution and then the translation in the language of classical geometry. And as far as I know, it is one, it is the only example except another example you know of, another Dutch mathematician, Stempion, when he finds difference.

37:30 So my question would be, do you think it's merely contingent or does it signify something Is it linked with tradition, or it's in practical mathematics, or everything else? Do you have any insight on that? Yeah. And this relates to the remark that most of the algebraic analysis was in connection with problems. Yeah. And that is what also echoes the remark of many people who deal with Greek mathematics. analysis of this, we mostly have examples of analysis of problems and other theories. But against that is that Poppers talks actually about an analysis of problems. No, sorry, he says the problematic analysis of problems but also the theorematic analysis. As far as I understood Knorr, for instance, in his book on problems, mostly deals with that, and he comes by a convincing example of an analysis leading to zero. Who knows the norm? No, no. Yeah. And so, I go on that. On the other hand, I know about Van Schoten, who has a book, De Concinendi's Theorematies. Ex-calculate algebraic. Yeah, from the geometrical calculus. So, algebra to give us zero. And, of course, you can see a more orthogonal, A squared, B squared, C squared. It's letter algebra. There's not so much an unknown, but there are indeterminates. And indeterminates gives you a range, and then it's a theorem. So he did that, and I have the feeling that Tom Guillaume was also in this business. there I would think that the property mark might have an influence and also by that time a letter algebra had been developed so much and also when when

40:00 you are in the business of using this algebra to solve problems and then you have found your final equation and you want to translate these things with a construction, then you have to translate equations into geometrical situations. And then you see that if you have an expression, for instance, as the last coefficient of your equation, they have only indeterminates there, and you have to make that into an object. And for instance, if there it says square root of a square root b squared, then you know that if you want to have that thing you have to use Pythagoras. So it's also understandable from the practice which by that time had developed of retranslating your algebraic results into geometry that people could see that any equality equation with indeterminacy is in a way a theorem because it applies for all the values that the indeterminacy can have. By that time, it's a bit of a trivial remark at that time. So, what I'm saying, it doesn't seem to be an unexpected or surprising development that one would ask this question. But that's always a pity, because surprises are much better than not. So if you can make it a surprise, then you can do it. Well, sometimes it's connected to something, which is important to the fact that when Princess Elizabeth derives a theorem, and Descartes speaks about this theorem, it's linked with a possible practice of Stamptscher. Sure, sure, yeah, yeah, yeah, so, okay, so I also think it was practice where these things come up, yeah. Doug? First of all, thank you very much for an extremely interesting talk. I, in reflecting upon this notion of given and construction, it struck me that, and you didn't really attempt to go through too much of it,

42:30 the special curves such as the quadratrix could be taken as given and enabled solutions to all of these higher problems. And there are authors, Clavius comes to mind, who offer point-wise instructions of the quadratrix, for example, and seem not the least bit bothered by the fact that the point-wise construction doesn't yield everything. But it does, as Claudius says, at any rate, it's twice to solve a problem relative to a set of givens. So I guess the question, maybe perhaps for your database, is that how many times, how often do we find this resort to point-wise construction of otherwise unacceptable? Well, it is actually, for the case of Clavius, that's about the quadratrix. And he gave a beautiful point-wise construction. But he actually said, you just construct very many points, and then you connect them by a fluent line without wobbles. I don't know the legend, but it's beautiful. And then, of course, he also knows that this is a bit fishy. And then he says, and if that's not precise enough, then you can always do more problems. That's also true. And then, actually, when you accept that, then something very strange happens, because then a whole bunch of problems with your problem, B-section, B-section, B-section, B-section, B-section, is as easy as LBRs. There seems somehow to be something small. Something is wrong. It's spoiling to me. There was probably a discussion about it because there are two sessions. And in the first one where it comes out, that is in one of the editions of the Euclid elements. There he says, this construction is geometrics. And in the next one he says, is geometrics a quota mode? And this is one of my examples for the interface I'm expecting.

45:00 Here people are answering questions, which is a very good question, of which you can't give an answer. And so what you find is that they say things. And they have seen things, and sometimes they always say, sometimes not quite. And I say you can't give an answer because you can't give any answer. But you can't get from mathematics itself a ways to give the answer. because you have to decide in, it has to come from outside. And then, yeah, then you are in a whole other pain. So, and there are more, and the one who really struggled is that was Descartes. Because what he wanted to do in geometry, he would have liked to find for every equation of curve a means to trace that curve. And in some ways, you can do that. But, and he had the hope that he could somehow show that every equation in two nodes was a Pappers equation, and if he then, so solution of Pappers problem, and if he could then get the Pappers problems under control and find a tracing mechanism on the basis of of the position of given lines, then he would have solved his absolutely fundamental problem. Because then he would say, I can construct by the intersection of curves. These curves I can trace. Therefore, I really get the points of intersection. And so everything is . But he couldn't . But he goes in that argument, he goes to point-wise construction, and then he has to have an argument why that is still acceptable. Then he has also an argument why the Clavius one is not acceptable. Very interesting. The Clavius, he says, this is not acceptable because the points he gets are not essential for that. Because what he does is, he takes all three sections,

47:30 or three sections of angles, you get a lot of points, and then he resects again, get a lot of points. So you get only these things which you can get by continuous resection. And he wants to have all the points, also for the other sections, but So then you would have an over-countable input. That doesn't work. That was not his argument. So the guy says, that's not OK. But when you, for instance, construct a tonic section with a point-wise perception, which means you choose one ordinate, and you make the abscissa, and another one, and another one, and then you do point-wise, that is okay because the first choice is arbitrary. So somehow he then equates the arbitrariness of the first choice with the fact that it could come everywhere along the curve with or at least that's how far I can go with him and I can see the seductiveness of the argument. But it's easily killable. So this is one of the open questions which actually nobody cared much about because this point-wise construction remained until the 18th century, the official way of constructing a curve given by an equation. And there was no question of whether that was feasible or possible. But it's simply precisely in the business. Okay, we have time for two last questions, both of them seconds. Marco first and Michael last. So, in fact, a question about one point that has not been discussed yet. It's your slides about relation between algebra and geometry. I completely agree with what you say, insofar as we take the term algebra to refer to algebra before Viet and Descartes. And it seems to me that in fact something essential changes, and with Viet and overall Descartes, you...

50:00 What do I say? You say pure algebra against algebra, pure geometry against algebra, and in fact, your slide number six is concerned with the relation between, I want to say, geometry arithmetic more than geometry and algebra. the basic point is that algebra is a part of arithmetic and so the difference between algebra and geometry is in fact the difference between a theory of geometrical magnitude and a theory of arithmetical magnitude, not this before, but this one this one but already I think that in Arabic algebra there is not only an arithmetic part So let's set this apart. It's true that in the end of 17th century, algebra was essentially a part of algebra. But what happens that is quite relevant with the overall is there is no position between algebra and geometry. There's not the fact that algebra is applied to geometry. It's that algebra became a way to make geometry. it is a distinction is no more a good one because there is no discipline distinction between algebra and job there's no more there's no algebra and on the one hand and geometry on the other one algebra become a technique in order to make job yeah so in a way would you say is that after the attempt to make a Matthias Universalis either in his way or, in fact, in his way. My friend David Rabouin would not be happy to say that he's a Matthias Universalis. That is another thing. I'm so sorry. Then, let me say, then I agreed by then that what they did was that they took the operations

52:30 of algebra, which were, in Erasmus-Petik, they redefined it for geometry, and they still called it algebra, and, uh, but it's not another, it's not another theory, it's not another branch of mathematics. The algebra, once you have defined an operation of algebra in a geometry, algebra is part of geometry. Well, yeah, it's a question of what you call it. For instance, Viette made quite a distinct part. This is the analysis. And there, the algebra was, what do we call it, the middle of the analytic arcs. Before there was a translation into algebra and afterwards it was translation back to geometry or back to arithmetic in this case. So if you say algebra is geometry, that seems a bit quick because you take it as a recognizable part of arithmetic and then you would hope that it remained recognizable is in geometry it means that the term algebra changes with me you cannot use the term algebra to speak about the part of arithmetic before after Descartes and after that also what is another let's take after Descartes yeah yeah in as far as the fact you that in those two cases algebra was no longer based on the understanding of numbers and some basis was the continuum of geometries. So what you want to say is that the most interesting part of early modern mathematics is not based on the position between geometries and algebra. More important thing I want to say after the cup, it's not this opposition. I will say after the fact in the future. Yeah, just a quick question, do you, to what extent, I mean, it seems clear, modular what Jeremy said about there still being areas in mathematics in which this concern with

55:00 problems appears to be there. But there has been, to me, I take your point, there has been a general diminution, as it were, of concern with problems. To what extent has the disappearance of concern with problems connected with the concern with proofs and we because then in existence proofs you have a division between what we what logicians call pure existence proofs and constructed and constructive existence proofs looks somewhat like the solution to progress it's giving giving a recipe if you like for the production of some object which satisfies certain is different, why you would do it, because you want to prove existence for some reason. Yeah. The process is just the same. Yeah. Yeah. The question of whether the classical constructions were existence proof, that's long debated for a classical for the classical geometry, and also up to the 16th century, I don't think so. And I don't think the question come up, because as Marco said, existence was no problem, because you meet the things you deal. Right. And you make them. So what happens when you get existence proofs is that you want to have something that is that you have to, that you are in a situation that you have a property of that thing, a definition, an implicit definition. It is a point, and it has this and this property, and I don't know whether it exists. Those things came when there was a mass of analytical implicit definitions of things,

57:30 and you wanted to know whether they existed. And these were the equations. Because equations, like for instance, the polynomial equations, there you want to know whether it has roots. So you get the main theorem of algebra, it's precisely that. Yeah, and to do that, you have either a constructive proof. That is why it was so important to have constructive constructions, geometrical constructional roots of equation. But if you don't have that, then the only thing you can do is that you know a bit more about the continuum than what you usually have. Because what you then get is that you get an argument that the point is a limit of a sequence and then there comes an argument, oh, and that sequence is a sequence and it's convergent, so it has a limit and therefore that you can't do on the basis of a classical conception of the continuum because there's no nothing said about it. Sometimes people say and actually there is a kind of continuity when you get says if you have two lines you have its point of intersection, but that's in a way for constructional purposes you have still the same point because then you have to to find that line which does it. And that's 19th century, beginning 19th century. And you can also have very small fields in which all these points exist. It's still countable fields. Yeah, but that's even later. Yeah. Yeah. It was with equations and also with the existence proofs of solutions of differential equations. Then it became absolutely essential that you had your limit things under control and that you could make the step saying this, the properties which I'm looking an object for are precisely the properties of the limits of the sequence.

1:00:00 And now I have the property of the continuum, which I pick from the blue air or I take as an axiom or whatever, or you construct it like Data King, and then you know you have your existence pool. The other thing that occurred to me that changes what Jeremy said was the notion of what it counts to give an object. And give an object by some sort of infinite series was perfectly acceptable for somebody like Euler, but presumably would not have been acceptable back in this time. Well, it depends on whether they could have an exhaustion procedure so far in hand. Yeah. But that would be constructive. But in the quality of the parabola, there is a process of polygons, which come nearer to the parabola. And he shows that the areas which they have come nearer to a particular ratio, which is maybe two to three, compared with a triangle, and then he says that's it. So there were some ways of dealing with these processes, but then it was by means of the exhaustion method. And all the other methods were, you could see what happens and But that was known to be not really vigorous to work with. Okay, thank you very much. I have an announcement that I hate to make, but I need to make. So an accountant somewhere has determined that my chair cannot pay you for your Wi-Fi service at the hotel. If yours was as bad as mine, you probably don't feel like you ought to pay for it at all. But anyway, they're expecting it, and Fabian will take up a collection or tell you how to do this somehow.

1:02:30 I'll second announcement, dinner tonight at 8 o'clock at a restaurant named Cap Marine. You're all invited to come. Those who are staying in a part city, let's meet in the foyer at 7.40. Okay, and I'm still a guy. I'm hopefully better than I was this morning. I'm sure you'd be glad. Can I make, in which case, sorry. Can I just say one other thing before I leave? I think I'd like to say on behalf of the speakers that I think all six of the speakers believe that at least five good talks were given. I think probably everybody thinks that all the discussions were extremely well focused, directed and helpful. We would all like to thank you and Fabien for the wonderful hospitality that you can do this. and Philippe was the co-organizer thank you very much Mick Mick sorry can you just give me the address of the restaurant because I went