Elementary Geometry, Groups & Fields — Discussion / Problem Solving in Geometry C17th

0:00 Then you compare them in such a way, that you compare the blue and the red in such a way, that if you join them by a segment, none of these segments intersect. The question is, what do you need for this? This was a Putnam competition problem, and the proofs presented there had to do with the sum of the length, then you get the minimal sum of length, and then you show that they cannot intersect in that case. But there was no length thing in this problem. As you've noticed, all it was was betweenness. So you can, I could show that you can get a proof by just using betweenness axis for this thing. Yet, there is another proof that uses metric stuff that is again incomparable to this. In other words, there are models of that that don't satisfy all these between-ness actions that you need for the between-ness proof. Which one is to be chosen in that sense? I would say, well, you either suspend judgment, or if you have a preference for the purity of method, then you would choose the one based on between-ness. so which one do you choose out of these is open because most of the time if they are compatible then you choose the weaker, it's clear but most of the time you would end up with incompatible other questions or comments thanks a lot Victor so we'll take a break now and reconvene at 4 o'clock please to our workshop Hank is now living and working what she told me the other night in Denmark after having retired in Holland, so I guess that's what a retirement in Holland means, is to go somewhere else to work. The topic of Hank's talk is the role of problem solving in early modern metamorphosis

2:30 of geometry. Thank you. Thank you very much. As I told him for this last session is to well, the main thing is to come back to these messages from the time that so much they were still proud of existence, alive, and kicking. It's also very much I think concerned with the themes of this meeting, geometrical thinking, but then the early modern version that I talked about. And there you found very much that the boundary lines of geometry with respect to other mathematical pursuits came under pressure somehow. And then there were discussions about whether the activities were geometrical or not, and what could be the good things and the bad things to combine with. The most essential discussions were, and it's not surprising, in connection with the merging of algebraic geometry, which in the period was going on, and which of course borders is analytic geometry in the works of Descartes and Mach, of course. And what I've been coming to appreciate in studying this, mostly in Descartes, in the work of Descartes, and now moving to what happened before him. What I kind of appreciate is that essential innovations that were around, especially then the one of the creation of the analytic geometry, related to problems and not to theorems. This is actually a rather general aspects of 17th century mathematics that there was a strong interest in methods, but then methods for solving problems. There was a strong interest in problems themselves.

5:00 All kinds of varieties of problems were thrown up and solved. Actually, very few theorems. There are very few named theorems in mathematics and in geometry that have the names of particular 17th century people who solved them, whereas there are many problems, especially when the algebra and the analytic methods held by algebra were so strong that one could go over to the calculus and to involve problems for mechanics also, there are many, many, many problems that could be solved. So what I noticed is that essential innovations in the mathematics of that period had to do with problems and not with theorems. That actually is one of the motivations of a research program, which I have reserved for coming years for myself and for any other person who wants to join, of course. And that is to look at the context where these problems were around and to see how this fact that they came from problems around the theorems, whether that can show more light to all the changes that came, the metamorphosis that came into geometry through all kinds of boundary crossings that happened. So I present the remarks now a research which I can't even call in progress because it has hardly started. And so there will be also quite some tentative remarks, and that's why I'm so very pleased with the format of this meeting as a workshop with a lot of possibility of questions and discussion. and so this is the occasion for me to thank the organizer for making this possible and I can report from the workshop until now that it is very very interesting for me at least and especially the

7:30 former was very helpful for me I've learned a lot I should also say to the previous speakers So this is the program. At the end, when I discuss the research program, there will be some remarks on the nature of geometrical problems and of analysis and the concept of given, which plays a strong role in analysis at that time. So here goes. As I noticed, if we want to understand what geometrical thinking or geometry was, then it is important to look at the boundaries and the places where there was discussion, whether it was geometry or not. And as it happened in the period which I'm discussing, and that's when the early modern period, And as to the developments that happened in mathematics, it is a period up to the work of Descartes. Then something very important happened, and as I'm going backward, I stopped there for the present education. So it's, let's say, the century before Descartes, which I'm discussing. and I'm discussing geometry there it will be good to mention now that this is pure geometry that will be clear also from the boundary lines I'm going to discuss and actually there were quite a lot of boundary lines there there was the one with arithmetic, there was the one with mixed geometry there was the one with algebra well. Then there was the one you might say is more seen as more inside geometry, that is if you start with, and what I'm telling you about is geometry is very much grouped around the elements of Euclid and everything which grew out of that.

10:00 Of course, as we know, the constructions there were rule and compass constructions, straight line and circle constructions, but there was already from classical times an interest in what would happen if you have a problem which you can't solve with the rule and compass, and where you are forced, unless you want to give up the whole enterprise, means of construction, and that changes a lot in the geometry, and of course that leads to questions about is it geometry or is it not. So that is a boundary line, and finally there is a boundary line between synthetic and so-called analytic methods and the difference between these. In all these cases we had discussions, different opinions, people who understood very clearly, and people who understood the question not so very clearly, misunderstandings and understandings normally. So what you would expect from any active scientific community confronted with great conceptual problems along these frontiers. So that is why I want to say a bit more about each of these frontiers, because in the rest of it, these things come all the time. I also say it because there are terms like arithmetic, geometry, analysis, and all these terms have undergone changes in the course of history which can make it very confusing if you use them without fixing them in time fixing their meaning. Analysis in particular we will see underwent very strong differences of meaning. It's actually there are a few words which have had so many meanings in the history of mathematics as the word analysis. So I'll do my best to make clear mostly what I say. And if that is not clear, then please ask me to clarify. Also, I want to take this occasion to say

12:30 that in this work, I found myself more and more often confronted with problems which are, as I experienced them, really problems in the philosophy of mathematics. But I have mostly concerned myself with mathematics, but mostly with history of mathematics and not so much with philosophy of mathematics. And so in many of the things I notice, I'm not a helpless about the philosophical aspects, which I see, but I see that they're very important. And so then I look at philosophers of mathematics as a client and see what clarification I can get. And also, and that's one of the reasons that there's some, well but some, that's a good word for that. That I am in this with the experts of the field of philosophy and mathematics here. And I'm sure I'll benefit from all the hints and points where the philosophy, where I obviously and missing things out of the philosophy of mathematics. So that's for the problem and plan. And so we begin with the boundaries I was talking about. So there's one boundary between geometry and arithmetic, both taken to be pure, I guess I'll say pure, geometry. And there, the essential difference in the 16th century was actually quite clear, because that was their quite common Aristotelian veritable idea that a science is determined by the subject matter. and the subject matter of geometry is geometrical magnitudes and the subject of arithmetic is number and they are both abstract it's the pure part and so that is the difference between the two

15:00 to clarify with more than geometrical magnitudes are line segments, area, solids, angles and with number at that time as we have heard more in the previous lectures had in mind the things we call positive O or rational numbers. These were seen as well based as well founded in the number concept as it was found in Euclid's arithmetical books And there was, in the time, tentative tie-out with irrational numbers, but it was also quite clear among, at least, the well-informed mathematicians that the status of irrational numbers was unclear. They were actually sometimes seen as miraculous, as understandable, as unspeakable, and some people liked this very much and others less. So they were, as to their status, as to their status of purity, they were, as to the stages of foundations, they were suspect. So that's how the state was in that sense. Then we have the relation to mixed geometry. Mixed geometry is a tendency to translate the concept, and maybe the nearest concept we have modern times is applied mathematics. but actually it's a much nicer term because there the essential difference as in the early modern times it was was that the quantities I should say quantities and magnitudes I use in the same meaning if I want to make a distinction I will say so because as we heard earlier the sources do more or less the same When I say numbers, however, I use numbers in the arithmetical sense and not in the sense of what we now call real numbers, but only in the arithmetically based numbers, in describing the early modern activities. So,

17:30 as also here is a distinction about the subject of the science, then in geometry you have quantities, but in geometry these quantities are abstract, whereas in mixed geometry, the activities which use mathematics, these quantities, you meet these, you find these quantities mixed hanging into something material. So if you have if you do surveying, then you find the quantity of length onto this ditch along your field, which you want to measure in order to find how much field you have, et cetera. And so mixed mathematics was a very large category, where this involves surveying, and bookkeeping, and navigation, and many of the arcs of war had quite a continent of mathematics in it. And in all these cases, you find the abstract mixed to the non-abstracted material things. The dividing line there is two, actually. First, then, the abstract versus the hanging on the concrete. and of course in these fields you would use numbers as a matter of course to deal with the problems you had because you would introduce measurement you would have unit measures and you would measure and you would calculate a lot with these things the most important mixed geometric field was astronomy There you see the mathematical quantities mixed with the astronomical meaning. And of course, this was a hotbed of calculation with numbers and where there was a lot of experience in hanging numbers to geometrical things. It's not that that's unknown. It was also quite clear that when you did that, strometry, necessarily

20:00 you were unprecise. You would either have an approximation because you couldn't calculate, for instance, a square root to the utmost precision, because it might be irrational. And anyway, when you are measuring with a measuring stem, there is no guarantee that it will ever be abstractly, precisely, exactly dismeasured. And this was no restriction of all these activities, also not for the value that people had to this. It was realized that there was a difference between the pure geometry and the mixed geometry. Then we come to the line which will be the most important one in this lecture, namely the borderline between pure geometry and algebra. Now, we have to make clear what algebra meant in that time, and it meant a theory in the sense of a collection of methods to deal with the solution of equations, which were written in all kinds of different ways, but which involved unknown numbers, And this was, of course, the new thing and the special thing and recognized as something quite miraculous that you can have unknown numbers and you can calculate with them. And then in that process, they turn from unknown to known in some way. Since there were also indeterminates, although only in the time we were discussing, these indeterminates were indicated with symbols of letters in the equation, as for instance, when you have an equation, the coefficients are usually not the ones you're interested in, which you consider as given, and these are then the indeterminates, and they were used. There were ways to give these. Also, before one had letter symbols for these, one could quite well discuss equations in general. Up to the fourth of the equations, there were algorithms to solve.

22:30 So the essential difference was geometrical magnitudes versus numbers, so the same difference between geometry and arithmetic, and algebra was indeed seen as part of arithmetic, and the operations of algebra, therefore, work the operations of arithmetic. That's what they work. And these operations are not geometrical. Again, the difficulty is always with the multiplication, and the multiplication in Hellspray and arithmetic has the definition which it should have the involved multiplication because you have the multitude, and so one of the factors you multiply indicates the number of times you have to operate with them. That you can only do with numbers. You cannot do that with a segment. You cannot say that a segment indicates how often you have to take something. So if you want to transfer the operation of multiplication and the related one of division, of course, to geometry, then you can't claim that you are using the same operation as you would use in a riskative. And that, of course, would be precisely the thing to do which all these divisions of the sciences had worked against, namely that you would take methods which were trusted and worked out in one field to apply in other fields. That's a classic Aristotelian dictum that you shouldn't do that. That's not how science worked. And so the idea of crossing the border or throwing away the water between geometry and algebra would mean that you were doing precisely that. We also know that ultimately it could be very effective and successful, so let us just quickly say what that evolved. And actually there were several ways to do this, but they come to two basic ones namely that either you redefine the operations independently from numbers so you redefine them as geometrical

25:00 but then you have to do a lot of work to show that it means something and that they still behave the same as the ones in arithmetic or if they don't behave the same then you have to be very, very careful which kind of argument from the one size you bring over to the other. Of, or, and if you redefine that independently of numbers, then maybe that could be done. Or, you have to introduce numbers, but then you are sure, certain, that you have to deal with irrational numbers, and then you have to base those somehow not on arithmetic, arithmetic, because arithmetically, they are suspect. So this is quite a conceptual problem. And for instance, what was mentioned several times, the idea of a Martesius Universalis, which was picked up from Faulkner's, and which planned a science of mathematics which would cover both geometry and arithmetic, it would be the universal mathematics, and which in Broclus is mostly seen as the science of ratios, because ratios in Euclid also are ratios both of numbers, of numbers or of magnitudes, and so that is indeed a part of mathematics which covers both, Mathesius Universalis, but then one had seen the possibilities of algebra, and there were attempts to work out this, under the title of Mathesius Universalis, by several people. Actually, there are a period I'm discussing, and that's the only thing I want to say about it, there are actually five different attempts to build up a matthesis universalis. One is by Viet, who introduced algebra. He took the first mentioned way of solving the problem, namely that you redefine the operations completely independently of numbers. He also defined multiplication as making rectangles, that meant that he introduced a dimension, see, that

27:30 meant that he had to deal with the argument of what happens when you have more than three factors. He gave an argument for that. He based this all on axioms and developed a marvelously well-thought system of algebra which could do what with those new, geometrically defined operations. There were, so that was in the 1690s that I started publishing about his project there. Then there was a gentleman called Van Roemen, who actually also wanted to combine that. He did it twice. The first time he published it, that was before 1600, and his approach was also to give an axiomatic approach, but then in his sense the axioms are just a long list of the operations you can do. and he based the question of irrational numbers on his teaching by Clavius and Clavius had, on the base of the denomination of ratios which we have also heard about he had an approach where you could very, very easily hide that there was a problem about the irrationals when you go to Clavius you see that he has an argument where he makes the jump is not acceptable, but he is very careful in dealing with the result. Perlomen, not so much, and so to the modern eye, the difficulty there is quite clear, but it was there. Actually, Perlomen gave another one after he had learned about what Fiete had done, where he then introduces algebra because of the letter algebra which he works out. So that's three. Then we have in the 1620s Descartes himself in the Regulé where he works out how the algebraic operations are philosophically considered and he actually proves the certain tri-stacles, temps-stacles

30:00 certainty of the algebraic operations philosophically on the basis of this criteria for certain philosophy. That project failed, it was published, and I mention now the ones where people really wanted to work it out as it should be. the fifth stand is what Descartes did in his geometry. So this was a, well, as foundational work concerned, this was a very difficult one, but people went in very different directions and were, in very different levels, successful. Oh, yeah. Well, that was what I was doing. And also, this was where the action was. This was where most of our questions of what happened to geometrical thinking in that time is about. then there was the rule and purpose in higher order construction there the essential tension is that when you do problem solving you have to know when a problem is solved because otherwise you can't do problem solving and actually geometry gave the example of a highly formalized way of proceeding. So you couldn't just leave that question untouched. Now, in Euclid and all these interpreters explain that the constructions are acceptable when you do it with ruler and compass and in Euclid you don't learn what you should do if you can't do that. So that's restricted but it's clear enough. On the other hand, the of problem solving, or the attempts at problem solving, had met several problems, the classical problems already in antiquity, which couldn't be solved by Ruhle and Kompass, and there were various means suggested to do that, either with a generalization of the machines, so Kompass, or a generalization

32:30 of the products of the machines, so rather than circles, you would work with connex sections. a business which was in a way different from problem solving because not only did it solve problems but it also set the rules for the problem solving. This was when I studied that in the case of the current head called the interpretation of exactness because there the mathematicians were confronted with a problem which was a decision problem so they had to find their arguments from outside mathematics, but they were forced to deal with it because you can't just sit down and say, sorry, we can't solve the problem of the three-section, or at least they couldn't, or they didn't want to. So they had to then formulate new rules. And this is everywhere in the problem-solving business. you see that all the time, because, of course, when you start giving new rules, then you also find new problems which cannot be solved by these rules, and then you have to go on again, and so around this problem-solving, you get a whole foundational activity of setting the rules in order to keep a structure within which it is meaningful to solve problems. So that was certainly also an important foundational activity at the time. I think, isn't it the same problem with the mix problem? Do you know when you found a problem? Less so because, yeah, thank you. Less so because this distinction between the constructions, because in the discussions here, the assumption was for both sides that you would remain in pure geometry. this is construction in pure geometry there were of course approximate constructions from surveying and so on it was quite clear

35:00 although although it found it difficult to formulate the reasons but it was quite clear that those methods were called mechanical often and whatever however handy they were they were not for geometrical problem solving in this sense. That's why I've also said this is problem solving in geometry because only then you get the structure in problem solving. And so in mixed mathematics you've got approximations and then you can numerically approximate and that's the whole other goal game. Thank you. That is a good addition. And then we have the last That is the difference between synthetic and analytic methods. And there the essential objection was, and you will see an example soon, the essential objection is that when you reason analytically, then you set up an analysis argument. And then, as a mathematician, what you do is, and this was, there were classical texts which made this quite clear, how you do an analysis. You start with an almost magic formula saying, assume the problem solved. And then you would make a figure, an image of how the situation would look if the problem is solved. And then on the base of that figure, you would explore the possibilities of your constructions, and if you were lucky, you would find the answer. Now, there is an essential objection to that, namely that it starts from an invalid assumption, namely that the problem is already solved. So, in dealing with that, you might have the idea that analytic reasoning or successful analytic reasoning would be enough, then you were wrong because you still would have to check synthetically whether it was okay.

37:30 That was actually generally accepted. If people used analytic methods, then they knew that a synthetic proof was necessary. But we also see in the same period that when there were analytic methods, especially when the algebraic analytic methods came in, that people would tend to leave the synthetic proof out because often that wasn't particularly difficult, nor very enlightening about problems of it. But generally acknowledged was that it belonged, both belonged to pure geometry, I mean, the fact that it's heuristic was not an objection to pure geometry, but it had no proof value, it had no English proof value, and so you had to add it. became clear, not only at this point, but also the means of construction, they became clear when proper's work became generally available in 1588. In the handout you see 1582, so you see this is worked out until the last moment, and I I hope that's correct, the Latin version of the book where people also learned about the classical ideas of analysis. So, that's about that. Then, as I said, this is more the background and to show you that there was a lot of things going on in the way of not how you do mathematics, but why you do it and if you're allowed to do it. The action domain and most important action, I think we can safely say, was the analytic geometry, which then later proved data calculus. So this was quite a clear border, possibly, going off. And so you had all the controversies. And the controversies were mostly about analysis and about problems. And so they were very much about geometrical thinking, style early modern. And so I feel confident that this can be presented here.

40:00 And so let us say a bit more of what this meant, this merging of algebra and geometry, and what the role of algebra was. Now the role of algebra in these activities where algebra was introduced into geometrical problem solving was heuristic. So it was to find the way to construct the object which the problem asked you to construct in geometry. And again, it was from Papus 1588. Oh, 1882 is also there. Yeah, that's how you see how these things happen. Papus wrote there about analysis and what it was. that was a very difficult text it was not always it was not free of contradictions and rather it really classical classical practice is a question to be debated but then there was something that was quite clear and there was also a recognition after that book that in this analysis, classical analysis of the freaks where Pappas wrote about some things were known but not very much I'm sorry, just, it's too different translation of the same text, or it's... No, no, no, no, it's one publication date, which is wrong, and the other is probably right. And Papus explained what analysis was, and what people picked up of that was that analysis was a technique where you started from things which were unknown. you started to work and think geometrically about unknown things and it was not difficult to see the relation to algebra because there you worked with unknown numbers and so the idea that the two could be combined seemed to be quite obvious. I think it would be good if I give an example so why how did the name come in, what Poppins reported, Poppins didn't report about algebra. Poppins reported about

42:30 what he called analysis, and this was later, the term was taken over by algebra, but that was confusing, so in the 17th century there was a good idea to deal with that, namely the one you would call the new analysis, that's the use of algebra, and the other would be An ancient analysis explained by powers. And the ancient analysis didn't use algebra. So the interesting thing of the whole business is that when algebra came into problem solving, it's not so that it was entirely a new thing. It had a sort of a rival, namely the classical way of analyzing problems. And I now want to proceed to give an example of such a vessel analysis. It's a very simple example, but it shows you what it meant, how non-algebraically you could deal with the unknown. And it is a so-called triangle problem. Triangle problems, there were a lot of problems around the triangle, problems are problems of the kind. We want to have a triangle, and we give you three elements. So we give it, for instance, three sides, and you can construct it very well. But, for instance, when you give it the height and some others, it becomes difficult, and you can imagine how many combinations of three elements you can make, and how many problems you can get, and all of these, you can try to construct geometrically, as well as you could, like in astronomy, you could find the algorithms to calculate these results. And both were seen as different and discussed. And here is such a problem, and so we have given, and we have the length of a line which is going to be the basis of the triangle, and that is given. we have the length of a line which will have a function

45:00 in the whole business and we have an angle comma which will be the top angle of the triangle. So here are the three elements and the problem is to do the triangle. And then the classical analysis is the analysis which is a heuristic device to find the construction, as later it will be that if you translate this into algebra, you get the equation, and that will be your heuristic device also to be translated back into a construction. much. And the analysis starts with the famous magic formula, factum yam sit. Let it have have been done. And so you can now make a figure which would be the figure of how this would look if it had worked. And so, and I have, of course. This is always, I am very sorry about it because there shouldn't be, but the one is not, which is, is given, is not the basis, which is because it's one of the sides. It seems to be the side. I'm very sorry about that. So this is a triangle, and the triangle will be a triangle, and we can, so this is the side which is given. The B we will see what it is. Oh, I should have explained that also. And this one here is the comma. And what is this D, which is also given, that is the difference between the C here and the B there. C minus B. So the list of rhythms is one of the sides and the difference between the other side now the basis and the other side and the act. So here we have, we certainly have to have this what do we do with the D? It should be C minus B

47:30 we have B here and C there and then there are actually not that many possibilities to bring it in a figure and the of this one is that you extend the C here to there, to give it over, and this would be the d. And then it seems also wise to extend this figure to here, so that you have triangles and you can also use your triangle. And here ends the first part of the analysis. Now I have done Dr. Young-Sitt, And now I can start doing the other part of the analytic reasoning. And the other part of the analytic reasoning is the reasoning with the givens. And it tells you that you look at what is given. Now, you see what is given is an A. and A is given by and that is, how do we know it's given by these two points and I'll tell about that now I'm going to say this means that but it means that the two points are given and that means that you also can draw this line so that is given and also So, this gamma here is given, and the d is given, but the d is only given here, it's given in magnitude, because the a was given, but I've started putting the a. and then I can, this is the first thing in the given things, I can just place somewhere and I can build the figure around it. The D is given in magnitude, so somewhere you have it and you may use it, but I couldn't place it here yet because I don't know that it's other edges precisely there. So here we have the three givens, and then we start saying what do we know more? And then you have, you know, a series of things. It says if that is given, then something else is given too.

50:00 If this line is given where it is in position and its angle is given, then this line here is given in position to hold. and there is a construction which makes you make a line to this point there. That solves our problem with the T because now I have a line on which it's given and I have the size of that line so that means the next step is that this one is given. Then we look and we see that now these two are given and that means that this line is also given that means in fact that this whole triangle which is not the triangle we wanted to have but it is a nice triangle is totally given that means that we know the angle here and here of course and then we notice that the even bigger triangle interesting one because it is address less because it has here angle C and then you know that this angle is also given and that means that I have two points and the angle then I can make this one and I can have this this point of intersection which means I have the three points That was analysis, an example of the classical analysis of which there were examples, and which Poppers had in mind with this message about analysis. So, what was that? It was using unknown and indeterminate magnitudes. He gave them once, and the unknown once. It is a way that givenness extends over the figure, that's actually what it is. Givenness extends over the figure and you start with things that are unknown because you have said part of your signature, you can't argue about it. And so the same for Algebrados can be done in geometry.

52:30 The key concept obviously is given and there was no doubt about the strong geometrical status of these arguments. And they could be much more complicated. And so that was, as it were, the geometrically acceptable rival of algebra. and then there is a phenomenon which I already mentioned and which I find important namely that both the old analysis in classical times and the new analysis in early modern times appear to have functioned near exclusively for finding solutions to geometrical problems and not in finding properties of objects or proof of series. And this is what interests me in this sense that guys have been choosing a subject for research, namely, sort of the theses and the question, here at IDP, in particular, analytic geography, what was theorem proving, the context was problem solving, and then you start thinking, well, there are different activities, which is problem solving is, you might say, task oriented, rather than truth oriented, compared to theorems, and so the first question would be, is that significant about what happened with the introduction of algebra? That it for problems. And this made me more and more interested in this activity of problem solving. And so here, let's briefly and of course it isn't quite clear that I should think about this unit of mathematical activity, namely a problem as it was in Euclid and as it was certainly still alive and being used in the time we're talking about. We have propositions in geometry, but they come in two kinds, namely serums and problems. And these are very different. In both cases, you have, again, the concept of given.

55:00 It's not very, well, we can see whether we can formalize that. But the serum goes, if such and such situation is given, then something else is given as valid, as true. Then something else is true, too. And then you have to provide a proof of that. And if you have done that, you said, what are the most wrongdoing? And in the common cases, it's a bit more complicated. You have givens. The givens can be objects and relations between them, also there. But then the question is not a prediction that what will be, but it is a task. Do something. And the answer should be a construction, of earlier going to be valid construction steps, and at the end you should also give proof, namely that what you have made satisfies the requirements which were set in the do class. So here we have a different thing, we have a task, and this is my first introduction of terms to sort of catch two activities. And so my question is, what has happened to this division, considering that algebra came precisely in the task business and not so much in the truth business? I should then say, and then I thought what I want to do is I want to study these problems and to see what I can learn about these problems in order to see if I can pick up something about, for instance, the user analysis and how that works on the later developments of mathematics. And I'm afraid that I won't go through these problems, but here you see mean proportionals and standard problems and the three sections and all these kinds of problems for reasons of time. I'm doing this a bit quick, yeah, and also there is an enormous collection of problems and an enormous collection of texts, and I like to call that the early modern tradition of geometrical problem solving, it is essential context, and here's

57:30 my research program, I want to study that, and I did so by trying to set up a database for these problems, and that is a very inviting thing to do, because as I said these things are very structured, and so everything structured calls for a database, and you have very different instances of it, so I into a kind of special question asking about this problem. And I found out, and this is the question I want to ask, which traces did this leave in the later mathematical analysis? Because it came from that area. So what I want to do at the end of my lecture is actually not more than making a couple of remarks of things which in making this approach to this tradition of early modern problem solving with in mind the question, what about the special nature of problems? So I will have three remarks. One about nature in general, one about analysis and especially this concept is given, and one which is already called Icons. So here comes the nature. So what they are are formalized, very formalized items. Oh yeah, and that applies to all three propositions here. The problems, what they are is atomic units, sort of the smallest unit of geometrical, mathematical thinking. And in that sense, that's an important kind of element in the structure of this activity, the tradition, and the material on which it works. In Frohlos, you had an extra element concerning geometrical rigor. Of course, this was in pure geometry, and so there was a high requirement of rigor of the logical samples of the arguments, both in the theorems and in the constructions. But the constructions and the problems

1:00:00 had a sort of extra element about rigor, which is not in the framework. Namely, this thing about which construction steps are allowed. That plays there. And if you take a different assumption there, then you change a part of the logic. It is as if you would change the logic for all the propositions. That's what you can't dimension so much for Euclidean geometry, but the step of allowing new constructions is of, you would say, similar or comparable import on the whole system, the logic of the whole system, as if you would change part of the logic proportionally. so I'm sure that it would be it would be fair to look at these things if you then see for instance in the elements or in any structured set of problems and at the beginning of that always were the elements as you know the propositions of the elements you can organize them in the tree system where there is dependence the one theorem to the other, and so you can see these beautiful structures which gives the structure of the whole element. Dependent streets. Then, let me see what I'm going to say. Right. Also, you see that you have a sort of association with the serums and the problems, namely with the serums. And this is the truth. This is the static thing. What happens there is about the certainty of the truth. But the truth itself doesn't change. There seems to not much action there, whereas problems, you have an association with movement, with development, with change, and those are first the signs where I see differences.

1:02:30 So here come the remarks. First of all, the dependence structure. Obviously you have of the problems, for instance, in dupin, you can also make a dependency. Now, if you have the total dependency, then the problems are also dependent on the propositions, because as I show, when you have a problem, you do some construction steps, or then you go back to earlier cool problems, but then for your proof that what you have done is correct, you go back to propositions. But as you only look at the steps, well, and that gets you to the question, is there in any sense, meaningful sense, a subtree of problems or alternative of theorems in which they are consistent? And obviously, if you take, for the problems, if you take the proofs as well, then you can't only look at the problems. But if you look at the construction steps, then it's quite clear that if you say the independence is dead, the tree consists only of problems, and it also comes somewhere. It has a meaning. So this tree ends in the construction of the plutonic solids, and it is much less clear in the tree of the propositions. Whether that is independent of the problems? Probably not. And whether it quite clearly gets to a clear result is also not significant. association with this opinion that the aim of geometry is problem solving and that the aim of the elements is to solve the problems to construct the protonic problems and actually the argument that geometry's aim is problem solving you see in many many cases there

1:05:00 And also when you read from classical sources that there also traces there that in pro-classical geometry, so we just said, okay, you do all kinds of things, and what the serums are for is for problem solving. And by proposition, you just mean as you use a general name for problem solving? Yes, yes. So the question is, is it like subtree? Yes, that would be a subtree. So the question is, I need subtree. But it's a question which I have and which I use to look with other eyes to these objects. Then we have the ancient analysis with the concept of given. And of that I want to say before. So given in theorems and problems, you can have given two kinds of things. you can have given geometrical objects and geometrical properties. And in analysis, you have given objects, and you are also, well, you have a given property that the lines should form a triangle, but the constructions deal and base themselves on the given objects, and so what does it mean when objects are given? And there actually is a classical answer because, as Bobbler's also made clear, the activity of analysis, the treasure of analysis, and we will list the books, and one of the books was the data of Euclid. And the data contains a lot of theorems, of opposition and propositions, of the time, of the time, if you have this and that, if given this object and that object, then another object is given two, and if you then analyze then it means that you have a construction to it. So, arithmetically, it would be, there is an algebraic, and here it is, there is a construction, a set of constructions. And given objects and given properties, the result is a new true property, and problems, there is a new given.

1:07:30 So the tree of theorems is sort of the spread of truth through the whole business, and the tree for the objects is the growth of an object, what it describes. So now, but if the theorems were to represent eternal things, if the theorems were to represent eternal things as you mentioned a moment ago how are we to understand this spread yeah well that's my problem so um so he gives a a formal definition of given but only for objects also for ratios but he gives that by means of objects the ratio is given when you have two object which exhibits that ratio, but not for relations, or not explicit for relations. So here, so what it means, you have been given an object which you may use in your construction. I am led to ask this term because in mathematics it's still very alive, given. When you hear lectures in mathematics, you say, given this, given that, and then we argue on. I don't know of a formalization of it. Actually, it seems to be a fossa of a concept which, in Greek times, had a very strict formalization, but not any longer. A fossil. Fossil. Fossilized term. I'm sorry, I understood the short one. If you just, if we'd say, we are given, say, two parallel lines or something, yet more specific, it would say we are given object with particular relations. And what do you mean by saying that this given not defined for properties Maybe I given objects with such, such specific properties. Yes, sure. Because the things that are given are the objects which carry the relation. But yeah, okay. Okay, but the, so when you go to the definitions of what given means, there you only find the definition of given for the object, because that is where the construction works.

1:10:00 So we have... So the idea is that you're not given that relation as an abstract thing? No, not when you look for a formal definition, which actually, that's the notable point, these givens, they formalize a structure for heuristic argument. And for the heuristic argument, it is first you say, not from the opposite, and then you go through these given and then given and then actually you find the synthetic when you follow the same path as the analysis through the figure. relates to a puzzle with Pappas because he talks about the direction of the analysis and the synthesis and says that the one is inverse to the other. He's also right because it is inverse in the sense that you start with the end. Then maybe you say, fuck me, I'll sit. Then you start with the end of the whole thing and then you start arguing with your givens. But apart from that, what you do with the givens is precisely what you're going to do in the construction. So the synthesis and the analysis have a more complicated interaction than is usually assumed the different direction of the synthesis. And to remind you, these are very preliminary remarks. It's also because I think it's so very interesting, but I have not many things to ask you, please. The last is the one with the icons. and this is then a question for how we are now looking at these things and the question of course occurs, what has happened to the problems when the Euclidean thing had these two or less symmetric

1:12:30 atoms of geometrical reasons reasoning, you can go very far with that symmetry and if we now look at, if we ask well what is the Atomic