Diagrams for algorithms

0:00 Go on. It doesn't matter. You turn up. I'm very happy to introduce Karine Kemmler, who's going to speak to us about diagrams for algorithms. Thank you so much, Marcus. A pleasure to be with you as a chairperson. I am going to spend one precious minute thanking Marco for all the energy put in this workshop together. I am really very grateful to him. And I also want to express my gratitude to Saul and Michael, without whom we all know that this wouldn't have happened. It's a really wonderful experience to have this opportunity to convene and talk. and that was the minute for me to say. I've worked for years on the diagrams I've published and today I'm going to present something on a new bunch of material. I'm afraid that the like to say in the next hour is somehow the content of the book to be, but I'll do my best to give you at least a hint of what is going to be in this book. And I would like to put this talk under the auspices of a statement and assertion by Helbert, which you have here.

2:30 here. I put the German, and I also gave you the translation made into English by, that there may be one new thing, even though I suggest to change two words in that translation for reasons that may look obvious for those of you who read the journal. So that's the 1900 talk about mathematical problems, and Hilbert claims the arithmetical symbols, I first read the translation, are written diagrams, and the geometrical figures are graphic formulas, and no mathematician could spare these graphic formulas any more than in calculation, the insertion and removal of parentheses, or the use of other analytical signs. I suggest to modify slightly the translation and speak of arithmetical graphene as written diagrams and geometrical diagrams as graphic formulas to render better the play with words that Hilbert was making here. The reason why I'm interested in this statement is that in it, Hilbert emphasizes, despite what the practice of such people as Lagrange, Bourbaki, and so on, may lead us to believe, the indispensability of several kinds of inscriptions for mathematical work. Moreover, Hilbert recalls the mixed nature of any kind of inscription, that is, that

5:00 there is a diagrammatic nature in any arithmetical graphematic system, in the same way as there is something formulaic in any mathematical diagram. To the historian, in my view, this statement suggests several tasks. First of all, the task of analyzing the various regimes of inscription shaped by collectives of mathematicians in this slide, namely analyze the diagrammatic features of ways of writing, analyze the formulaic dimensions of diagrams. Indeed there existed, several such regimes of systems of inscriptions that were shaped for mathematical work, and we are interested in capturing them as such. If we are, which is the task I set to myself, to inquire into how mathematical activity has been carried out. And it seems to me that this is a key task if we are to establish bridges between history of science and the study of cognition. But to the historian, Hilbert's statement indicates a second task. one of interpretation. Indeed, many historians in the past have mainly focused on texts as if, and what I mean by text is the discursive part of the text, as if interpretation could

7:30 dispense with other scribal elements involved in mathematical activity and inserted in mathematical writings. A clear sign of that fact is the poor critical edition of diagrams and other non-discursive elements. However, and in that he exposes the limits of this way of reading, Hilbert's assertion reminds us of the fact that diagrams are not only ancillary to text, to the discursive parts of the text, but they can also be inseparable from discourse and convey meaning. So, in my view, and this is how I read it, Hilbert's assertion even opens up the possibility that a whole dimension in a text could unfold within diagrams. So the task one, which is identifying various regimes of mathematical inscription, is essential for the second task which is interpreting texts as horrors. I would like today to provide new type of evidence supporting Hilbert's claim. And I would like with the concepts he introduced, namely that of written diagrams and graphic formulas, I would like to approach a shift in the practice of algebra in China between the 11th century and, say, the 14th century. So, in an introduction, I will give you evidence that there was a shift. In the first part, I will focus on the later practice, stressing the diagrammatic features of the

10:00 algebraic graphies with which algebra was carried out in this later phase, and in the second part I will move backwards to the earlier practice in which I will argue the formulaic I think dimension of diagrams was essential and may have been missed by the previous historians. So I think that by now I have convinced you that this is a topic for a book. So, here is the introduction, in which I'm going to first introduce you to a kind of algebraic practice that took shape somewhere between, let us say, the 11th century and the 13th century. This is a text from the end of the 13th century, an introductory text by Jou Chédier. And I chose a very simple problem to introduce you to this kind of algebra, a very simple which is very important because Drew Chedier will have a comment on this problem that explains the shift in his view in the practice of algebra. So suppose, says Drew Chedier, that we have a rectangular field or figure, you note that field means figure as Agathe mentioned it in Sanskrit. We can come back to that in the discussion. And I want to stress the fact that here we have a very typical kind of problem. We have a rectangle, the array is given, and then it's followed by one only says, and there

12:30 is a second condition. Here we have the sum of the length and the breadth of the rectangle and the question is to find out the dimensions. So the new technique amounts to taking one One sample of what I translate as the celestial origin, the celestial origin in other contexts is the celestial unknown, the celestial element. I interpret this as origin, it's not an important point here. It is the unknown, and it's going to be used in a polynomial computation aiming at finding out an equation whose root is one of the unknown sought for. So, one takes this unknown as, in this case, the solution suggests, the breadth. And, in addition, Jouffredier gives column of number 1, 0, which is a place value notation for the polynomial that is going to represent the breadth. progress in this notation. Place value notation is a new thing to represent polynomials at that time. It's not completely new for equations. I will say a little bit more on that topic slightly later. Here you see that the polynomial is simply represented by a sequence of two 0,1 to represent the polynomial which we wrote here, 0 plus x. Other authors will add a sign close to the constant term of the polynomial or close to the x in order to allow having the unknown with a negative power, but let us keep such

15:00 So, the solution of the problem goes as follows. We have a polynomial representation of the unknown we chose, the breadth, subtracting it from the quantity set, you remember the quantity set is the sum of the breadth and the width. The remainder makes interpretation the length. Multiplying with the breadth, so multiplying the length and the breadth, generates the number product, the area, which is the polynomial you have 92x minus x squared. I translate the place value notation. You have one polynomial representation for the area and then you eliminate with the area that you have and you get the equation which is here. And in the lower part of the transparency, I gave a kind of modern explanation of what happened. So you see that, first of all, you see that this equation is interesting. it has a negative term for the power of the unknown of the higher degree as well as for the constant term. And you see that at the end of this reasoning the equation is obtained as a whole. I will come back to that. It's obtained as a formula, as a whole, as a list of coefficients. And from now on, the solution of the problem is to take this as an operation, as you would take a division, for instance, dividend and divisor, and to look for the roots with an algorithm that operates on this list of numbers and yields a root. This kind of technology to find out roots of the equation has a history, a long history in China.

17:30 We'll go back to that in a moment. So this way of yielding the equation as a formula, as a whole, at the end of a polynomial computation is new. Before that, we had problems solved by equations that were given in the following way. Each coefficient of the equation was yielded at the end of a sequence of operations. So you would describe how to find the constant term, how to find the coefficient for x, how X square and then you would prescribe finding out the root. And in correlation with the fact that the equation is obtained as a formula, you have a way of obtaining the equation by means of a polynomial computation. So, in a commentary on this problem, Drew Chesier is going to oppose two ways of establishing the equation. One, he will call the old way, and a second one he will call the new way. So, in the old way, you have a diagram. I inserted the diagram in that text. It's not a diagram you can find in the text. It's a very well-known diagram, and you can find it in contemporary text in China. But I inserted it for your understanding. We'll find, we'll see other texts with diagrams, but this one is just for you to understand the text. So if we elucidate this problem with the old method, then first follows an analysis of the diagram I inserted for you, multiplying the sum by itself yields four pieces like the area of the rectangle, you see the four pieces put around the triangle, and one piece like the square of the difference.

20:00 So first you have a diagram with its analysis, and then a reasoning, one places, placing means on the surface on which computations are carried out, the number product, the area, one multiplies it by four, which yields a given number, and also a given interpretation of that number. subtracting it from the magnitude above that remains the square in the middle you have for it a value and an interpretation and then you insert it, you interpret it as a dividend which is the name of the constant term of the equation and you take x square 1 with a coefficient 1 then you solve the equation. So in this way of establishing the equation, you rely on the diagram and you shape the coefficients of the equation by reference to this diagram. And then you complete the solution of the problem. And in contrast to this, there is is the kind of reasoning we saw at the beginning, which was the reasoning with polynomials. So I would like to first look at the algebraic practice that unfolded on the basis of representing polynomials as columns of coefficients and what was done later on. And I would like to look at it, this will be my first part, from the point of view of equations and polynomials as written diagrams." So, Jouchesier is someone who wrote a second book in addition to this

22:30 elementary book in which we found the kind of solution I just discussed. He wrote a less elementary book in which, in order to establish equations to solve problems, he used not only polynomials or equations with one unknown, but he used polynomials and equations up to four unknowns written on the computing surface, on the surface with which computations were carried out with rods. For the sake of simplicity, I replaced all the digits by Arabic numbers, imagine that they are written in a different way. So, we have a problem, I did not even mention the problem, and for this problem, we have established the two following equations. So I wrote on the left-hand side the way of writing that Jouchi-Jae uses, and on the right-hand side the modern notation. Yes. So, you see that vertically, when you go step by step to downwards, you increase the power of X, and when you move leftwards, you increase the power of Y. This is why, for instance, in the column which is on the right, under the Chinese character that indicates the constant term, you have 1 in the third position, which is x to the power 3, okay, and completely on the left hand side minus 2 is minus 2y squared. and then you use kind of Cartesian coordinates to identify to which kind of term the coefficient belongs.

25:00 So we have two equations, and now Jufvici is going to work on these two equations. So, the reason why I'm claiming that we have here written diagrams is that Jouches-Yé is going to structure this notation with geometrical elements. First of all, the table is going to be read in columns that correspond to inserting parentheses and expressing everything in function of the powers of y. So you have the insertion of parentheses that corresponds to structuring the notation in columns. And you will carry out the elimination between the two equations by a graphical work on these two tables. Normally, we would multiply the upper table by 2 minus x, which means first multiply by 2, then taking the original table, multiply it by minus 1, and shift everything downwards, so you see the graphical work on the inscription, and then adding the two tables together. So you have a graphical work on the notation in order to carry out delimination. In this case one can carry out delimination much more easily by subtracting the two tables, then you delete, you eliminate the only term in the right column and you simplify and you get what Juchy calls the right term. expression. Then you do exactly the same and you get the left expression. And again I put

27:30 modern notations and what they amount to in terms of a graphical work with the expression. So, you have left and right equation, and what amounts to the elimination of Y will be multiplying the inner columns, multiplying the outer columns, subtracting. So, you see how the tables are decomposed and used with geometrical elements, and the computation is done with these elements. And in the end, you have two polynomials. I put in grey what belongs to the inner part of the table, in white what is the outer side. And in the end, you subtract, and you get the equation as a formula. This is the first sense in which I will say that we have diagrams for algorithms. We have diagrammatic notation on which the computation bears using the geometrical elements to structure the notation and using it to carry out the computation. First meaning for my title. This leads me to a difficult problem. I showed you something, but somehow I cheated a little bit. Indeed, some of these notations I showed you, some written diagrams are in the book, and they are on paper. Most of these notations and operations are carried out on the surface on which computations were done. And this

30:00 This reminds us of the fact that, not only in China but everywhere, the diagrammatic objects that are used in mathematical practice are not only on paper or, say, on any kind of medium that would be kept, they are not only in documents that are intended to become public, but they are also, and more importantly, on ephemera, on erasable media. You can think of dust, wax, sand, rods, all kinds of media. And my claim is that it's impossible for the historian to deplore the loss and concentrate on what remains, because regarding diagrammatic objects, the writings and the ephemera are linked to each other, they are intertwined, They have diagrammatic objects in common, they are connected, and for instance if I focus on the case of 13th to 14th century China, even though from a material point of view they are on different media, the counting surface and the paper for the writing, these These notations have lots in common, even though on the one hand some can be changed, erased, moved, and so on, and on the other hand not, they have lots of structural properties in common. This is a comment on the synchronic situation. Now, another reason why the historian cannot just say, oh, it's too bad, we don't have sources, let us focus on what we still have, is a diachronic problem. them. This connection between the ephemera and the writings we still have, this connection

32:30 changed. And in the case of the Chinese writing, if I look at documents before, say, the 10th century, there is nothing in the writings but discourse referring to other supports, other media on which there are diagrammatic activities and drawings. Whereas if I look at writings after, say, the 10th or the 11th century, I start having parts of the configurations computing within the book. And the same holds true for geometrical diagrams. Before the 10th century, nothing is inserted, is illustrated in the writing. And all of a sudden, there is a shift in the composition of the text after. And we have some illustrations. And practices that we can reconstruct on, from the point of view of how the earlier texts refer to the computing surface or the diagrams, we see a continuity and there is a history unfolding. So we cannot just sit and say, let us concentrate on what remains. So I would I would say that in terms... Is the material surface in question the ephemera? Yeah. The dust, sand, rod? No, for us it's counting rods on the table or on the floor for us. But in other contexts you had, for instance, in the Arabic world you had dust, for instance perhaps in Greece you had sand, and so on. And I want to say that even though I'm speaking of China, I think this is something general. So, we have, as historians, a philological problem, which is, you know, it's not because we have diagrams in text that we know what the original diagrams are,

35:00 and we have a lot of work to do in order to restore something that looks like something less deceiving than what we have in the nose in our documents. But also if we are to develop an approach on the long durée and try to see conceptual evolution beyond the fact that our documents do not show us diagrams and notations and so on in the same way, We have a problem of restoring the diagrammatic objects for which we have, in the earlier period, only reference in the writing or evidence of a later type. And for me, this problem is absolutely essential because the diagrams on which I'm going to focus, whether they are on the surface for computing or geometrical diagrams, they migrated from being material objects somewhere, which we did not keep, to being illustrations in the writings. And in this migration, they retain something throughout the entire period. I'm not going to develop the arguments at the basis of how I restore and I edit the documents and so on, because I only have one hour. But do believe that anything I'm going to show you, behind it there is a reasoning based on documentary evidence to show you something as I decided to restore it. And what I would like to describe now is the migration from ephemera to the pages of writings in the case of the quadratic equation.

37:30 So, to do this, very unfortunately, I cannot dispense with going back to something on which I already spoke many times, but it is, unfortunately, the indispensable basis for what I want to show. And this is, as briefly as I can, the algorithm to extract square root that we find in the canonical work, which is the nine chapters compiled in the first century. And I will have to go back to the commentary on this algorithm to extract square root, the commentary by Liu Hui in the 3rd century. So I am going to go so quickly that you may not even see me, but I need this for the following part of my talk. So you have in the book a text of an algorithm which is completely discursive, nothing but words, nothing but words that tell you what to do, the interpretation is another problem. And this is how you can reconstruct what happened in the first century on the surface on which about. So the number, the number, the square root of which you are extracting is called dividend. The whole square root extraction is going to be put within the framework of division. And I'm going to draw something that I'm going to show you that belongs to your quiz commentary that explains why this algorithm works. So this is the square, the area of which is A. Let us put A in the dividend place, which is in the middle. The first digit, I'm going to forget about anything connected with orders of magnitudes. I delete orders of magnitudes.

40:00 You want to delete A square from A, once you have found the first digit, 2, you put 2 in the quotient first, at a given place, we are not interested in orders of magnitude. you put 2 again below in the divisor row, and you're going to set, to present, deleting A squared minus, from A, by saying that you multiply the quotient by the divisor, and you subtract it from the derivative. Now, in order to go on with the computation, you need to prepare yourself to multiply the next digit, let us call the next digit B. You need to prepare yourself to find out about the next digit. To do this you are going to prepare twice A that you will multiply by the next digit once A, the second time B, and you will delete B squared. And you do this by first multiplying the A you have here, and next, when you will have the value of the second digit, you will add it here, add it at the quotient, and multiply what is here by B to delete it from here. So, this is all that I want to say, except that this diagram is referred to by the commentator in the 3rd century with colors, so the A square is described as being yellow, this is described as being yellow, all squares are in yellow. These two are in blue-green, and then in vanillium, and then again in blue-green, and so on.

42:30 So you have a reference to the figure with colors, probably also grid, but I skip this dimension. Now, what is very important for me is that if we only look at what remains on the surface when we have deleted, when we have deleted the first digit, we have a figure which is or you can read it as a rectangle, which is the figure of the quadratic equation. So this is the first extraction. You extract from this figure a sub-figure that is going to be the figure of the quadratic equation because this figure writes when you look for suppose you forget that you had the first digit a if you look at the operation starting from there you are solving this quadratic equation so the gnomon or the rectangle is what I will call the graphic formula of the equation using Hilbert's concepts for that first extraction, and what will happen is that there will be on the counting surface the second extraction, which is the extraction of the two positions if we delete the digit we obtain for the quotient. For the quotient, we have here the constant term for the equation. We have here the term in x. And this is going to be the algebraic representation of the equation. And And when you go on with the algorithm of square root extraction on this table, what you do is solve this equation.

45:00 So my thesis in what time remains is that when an equation is going to be used to solve a problem, this equation is going to be established or written down as a gnomon or as a rectangle with a figure similar to what I showed here. This will be the graphic formula for my equation and it will be solved by a graphical process on the surface for computation which will bear on these two terms and use a certain numerical computation developing from this. I'm just giving you an idea. So I claim that that the object equation emerges from the proof that the algorithm for computing square root is correct, it emerges from the diagrammatic activity with respect to this algorithm, it forms an autonomous object to which will be attached an algorithm to solve the equation that is extracted from the algorithm to solve square root extraction. So we have here a framework to deal with equations and the theory of equation will develop within within this framework until the moment when we shift to the second practice I described in the first part of my talk. And this is the first type of algebra Drew Chedier was referring to in his comment in

47:30 in which he was describing the older approach to algebra. So I'm going to try to observe how this setting, how this organization around equations was transferred onto paper within writings. I'm going to examine how equations developed, how the knowledge on equations developed, with equations taken both as a graphic formula in this geometrical form and a written diagram in this form on the computing surface. First of all, to explain to you better what I mean by establishing the equation graphically and then solving it numerically, I would like I want to mention very briefly the first problem that is solved with the quadratic equation in the 9 chapters. My diagram, you remember that there is no diagram in any text before the 10th century, so it's my diagram because we are in the 1st century. So we have a square town, the side of which is a gnome. Someone walks at a distance southwards. The orientation is not like Agathe. We have north downwards, south upwards. We have an orientation on the diagram. So someone works southwards and then turns to the west. So we have S and W. There is a tree northwards to a certain distance and from the northern gate of the city. And the question is, what is the value of the side of the city? I'll just show you how the problem is solved.

50:00 Please don't read because it's not at all important. You see that we have a list of operations computing what is called the dividend, the constant term. Then you have another list of operations that is given to compute what is called the joint divisor, which is the coefficient for x. By the way, at that time, there is no identification of the higher coefficient, the one of x2. The term in x2 has not yet been identified. So, the problem is solved by an equation because the coefficients of this equation are computed in a given way, and this is how Liu Hui suggests that the equation was established. He suggests, I'm not going to tell it in the change terms, that this triangle is similar to the large triangle, and if this is x over 2, this is n plus x plus s, this is w, And if you write this rule of three, you obtain this. This is how I explain it and how I wrote it. What Yukoi says is that on the basis of this rule of three, I just indicated this area, The area of this rectangle is equal to the area of that rectangle. Then you multiply the area by 2. And what you get is a rectangle in which you have x squared plus s plus nx. So you obtain the equation graphically, you recognize the gnomon you had in the extraction of square root, and then you apply the extracted part of the algorithm for square root. So you obtain, you establish the equation as a rectangle which has a given structure,

52:30 and from this you have the data that you put on your equation as a graphic formula and you go on. Thank you. So now I'm going to see what happens on the equation with this organization of equation when the whole thing is transferred onto paper. And for this I am going to mainly observe a text by Liu Yi that dates from the end of 10th century or the beginning of the 11th century, a text by Liu Yi which is a key text in this story. It's a key text because we have key terms with respect to equations that are evidence in this text. First of all, the equation as a written diagram is going to change, the term in x squared is going to be identified and it will have all kinds of and not only one. Second, in this text we have evidence that the equation widened, widened in the sense that it was possible to have negative coefficients in the equation. In In fact, the scheme of the equation will remain A equal Bx plus Cx2, but in this scheme, either B or C will possibly be negative, and will have three kinds of equations.

55:00 And the third key term in this text is that we have the development within the text of the so-called Ruffini-Horner algorithm. So, in relation to the fact that there is a place value notation that emerges for equations, we have an algorithm that will treat all the coefficients of the equation in a uniform I don't want to enter into these details. So let us observe the equation as a graphic formula in Lui's text. So it's a text, I'm sorry, I didn't read it. It's a text in the preface of which we have that we have all kinds of problems, we can enter into them from all kinds of ways. out of them with the rectangular field. The rectangular field being the shape of the equation to which all the problems are going to be reduced. I go very quickly so I do want to remember. So, all the problems are going to be shaped on exactly the same format as what we saw with Druchedier, an area for a rectangular field always the same, 864 poux, a condition when only says that and then the linear condition on the breadth and the width, and then some unknown is asked for. And then, systematically, after the statements of the problem and the answer being given, you will have a diagram giving the equation as a graphic formula. I show you one here that is representing what we would write as E square plus 12x equals 868. So you see the square plus the rectangle, that is 12x, equals an area that is given.

57:30 It's very interesting that there is, there are problems given by analogy, and the same problem is going to be set in different worlds dealing with silver and people and so on, And the same diagram will be given to represent the equation which is the same, even though the nature of the quantities is completely different. This indicates a shift. In my view, these kinds of diagrams were only used within geometry up to a point. And this is indication of the fact that there is a shift. And geometrical diagrams can be used to interpret geometrically problems that are not connected with geometry. This is an important shift, but I cannot dwell on it. this is the geometric graphic formula for the equation x square which is the larger triangle minus 12x equals a given area so you have another diagram writing this equation with the captions I have no time to describe the caption, but you see that we are still within the standard form of equation. This is the next problem where the same kind of diagram writes the equation 60x, which is the larger diagram, equals the upper rectangle, which is 864, the constant term, minus x squared. square, and you see that in this caption on this square, which is not a square in my drawing, you have the area of the square with which one increases the area to obtain 60x.

1:00:00 And at the same time as this diagram is introduced, there is a new section that appears in the text, which is the section recording the geometrical reasoning that transforms the problem into to the canonical diagram that will represent the equation as a graphic formula. There is another problem which is interesting of the same kind, interesting in the fact that the difference is that instead of having that the square increases, you have the term negative within the square to mark the fact that you have a negative area that is added to 864 to make the rectangle 60 last equation that I want to show that is obtained to solve a problem. This equation that writes the equation that is below it, you see eight small squares in black. Coloring is an ancient technique used for drawings, yet here it is used to represent something different it will be used to represent negative areas and in the remaining of the text we will have this color used to be put on surfaces that are to be counted negatively so the first thing I wanted to show about this is the graphical representation of the equation, the graphical formula writing the equation. Very briefly, I would like to say that if we do not read these rectangles as expressing the equation, there is a whole

1:02:30 part of the meaning of the text that we miss. But I would like to show that this is not only the representation of the equation, this is also something with which operations are carried out geometrically. You're now eating into your question time. I will just use five minutes of my question time. Thank you so much. So the first thing I want to show is that you will have in the text, you will have in the text a representation of the computation on the counting board. Now they have appeared in the text. So I just want to point out the fact that in parallel to the upper thing is x squared, the lower thing is a rectangle, and the whole thing is the area, you have the area exactly in the same way as I wrote, a blank line in which things like here will be added, and the term in x. And the same diagram is going to be used to account for why the computations that are carried out to compute the root of the equation are correct. And you see that the graphic formula of the equation is the basis on which you analyze the parts and account for why the computations are correct. I'm not explaining to you that I'm going fast, but this is the case. The only thing I would like to indicate, even though this is going to be so brief that it's going to be difficult to follow is the kind of diagram that is inserted in order to account for the

1:05:00 algorithm that is going to be the Ruffini-Owner algorithm. I don't want to enter into the discussion of this diagram this is the diagram i drew to explain and this is the diagram we have in the text as i redrew it and translated it so you see that we have one black rectangle and in fact there were two surfaces that were exchanged and on the base of the diagram what explains the meaning of two subtractions that are carried out to compute the new constant term and a final subtraction that is carried out to prepare for computing the next digit. So there is a very subtle way of analyzing the surface of the equation, cutting it into pieces and rearranging the pieces that is involved with accounting for the numerical algorithm. I don't want to enter into the detail, but what I want to give as a hint is the idea that this rectangle is the basis of operation as such so we do not have operations on the equation as formula but we have operation on the equation as rectangle so we have a whole algebraic activity that develops on the basis of equation taken as, represented as, graphic formulas in the first part of my period, and there's going to be a shift towards a purely algebraic, only with written diagram practice of algebra. So there is a shift in the kind of diagrams with which one deals with equation. This is the end. Thank you very much.

1:07:30 Well, okay, I'm going to step in here before people have had time to put their hands up. with just a couple of questions of detail okay yes I noticed that in the In later diagrams, Bu was used for both area and line length, but in the very first thing that you showed us, you had Bu for area and Mu Fenley or something. with an eye yes so in fact you can ok you can take the pool as the unit that is going to be the basis for expressing lines surfaces as long as you take one unit as a determined thing and then either you use a given scale to express decimal expansion or you use units that are specific to surfaces in order to express a surface so you have two ways of three ways even of expressing the value of a surface yes that's right now there are I don't translate the I don't reasons that are not interesting but you're right that there is a connection between the units used for length and the units used for surfaces but it would be taken for granted that people know

1:10:00 the relation between who and yes yes yes yes it's it's in fact it's given by rule and the rule changes over time okay so I have abused my position Jean-Jacques so this text thank you for the question because that allowed me to see a little bit more Okay, positive and negative numbers were introduced in a very specific context in the first century, which is in the context of solving linear equations, it was attached to this context, And these were algorithmic marks to explain how to extend an algorithm for cases in which, in order to reach a solution that is a number full stop, you have to mark an opposition between numbers. So, positive and negative are marks that allow you to give one algorithm, and only one algorithm, despite the fact that you might go through all kinds of computations. But in the first century, and in this text, the result is neither positive nor negative, it is a number. I mean, it's important because you do not say that the result is positive, you say that the result is a number. There is a word to say that this number is positive, and this is very important to understand that these are really marks are algorithmic marks. So, in the text I commented upon, there is something very strange, which will not survive, and which is that only the term negative is used. Later, in texts dealing

1:12:30 with equations, you will have positive terms and negative terms, and the algorithm, the same algorithm used on any equation to find a root, but in this text you only have the term negative that is used, and it is used in some ways of describing the numerical algorithm. Instead of subtracting something, you increase it, so you have a shift, an operation that is transformed into the opposite operation because of something. And in other ways of describing the same algorithm, the term for which you invoke the operation as negative. So, it's still an algorithmic mark, but you have colors that are used to represent the area as negative on the figure. And that, I cannot tell you how to interpret because it's not within an algorithm, but it's a way of representing in the same shape an equation that can have a term that is inverted. So for the moment, I can simply put this like this, and I cannot go further in the analysis. Perhaps when I go deeper in the analysis of this text, I will be able to give a better answer. But the puzzling feature is that negative is used alone, and not in a pair with positive. All right, Agathe? I noticed that in the diagraph that there are some Chinese texts within the diagraph.

1:15:00 And there is a sort of connection in the construction of the diagram with the movement of an algorithm that can be considered alphabetically. And I was wondering what is the relation with the Chinese and how much, So, in fact, on the equation, you have two main families of operations. One is establishing the equation and the other one is solving the equation. When you establish the equation, it means establishing the rectangle, the canonical rectangle, that expresses the equation, and you establish it by a kind of geometrical work. work, then when you have the equation you shift to the numerical parts of the representation to work out the root. You have what is called detailed procedure to explain all the movements on the table on which the computations are carried out. But it's only for the first problem in which you set up the scheme that is going to be modified with negative numbers. It's only in the first problem that you have a diagram for the computing surface.

1:17:30 You won't have any longer diagrams for the computing surface after the first problem. What you have is a text describing how you handle this basic scheme to transform it when you have other kinds of coefficients. And what you have is the equation that is the rectangle that can be dissected with colors to account for the change that is described discursively about what happens on the surface for computing. The diagram that is established as equation will be the basis to show why you can transform the numerical algorithm, and the numerical algorithm is only described discursively, the transformation, except in the first problem when you have the basic structure of the equation. computation that is showed, as I showed you when I showed you the illustration, this one. So it's only in the first problem that you have this kind of scheme, and then a text, which is called the detail of the procedure in each problem, will describe the transformation. I put, I did not even find time to say that, I put in bold what was negative in the formulation of the text, commonality is written here. So this is described in the text except in in the first problem where it's described in detail, and the rectangle for the equation will be shown, decomposed in a certain way, accounting for the transformation of the basic computation. So it's a text that writes with all these ingredients. This is what is to be described. I mean, I don't want to say that there is the text and illustrations. It's a text that is writing with all these ingredients. That's my point.

1:20:00 There is also a sort of graphite algebra with rectangles for quadratic problems in Babylonian and Greek mathematics. What are the similarities and the differences with the Chinese point of view? Okay, the first thing is that, as far as I know, the quadratic equation dealt with rectangles is restored, and we have very little evidence about how it was carried out. but let us take Heurop's reconstruction as something for which we have real evidence. The algorithms are completely different, and I think that the function of the drawing of the diagram, if it existed is completely different from what I have showed. Namely, what he reconstructed was the fact that the diagram was used as a support to interpret the operations of the algorithm solving the equation and what I showed shows a practice in which the diagram is that with which you are going to establish your equation before you solve it. Do you see what I mean now? So I think that the equation as rectangle for the quadratic equation and for what we can evidence until 11th century China, the equation

1:22:30 as rectangle being something with which you establish the equation, is not something that I saw in what I hope reconstructed. But it's absolutely true that, and I'm completely I am fully aware of this echo of rectangles underlying the procedures solving quadratic equations except that they are used in accounting for the correctness of procedures that are are completely different from the procedures that are underlying this text I'm showing. To give just one example to make this point, all these things rely on the place value system for which the place value, I mean, the algorithm does not use the value of the basis of the place value system. You have a solution by radicals that are the approach explored in the text that I hope is dealing with. So these are different families of numerical approaches to equations. Marco. Yes, it seems to me, I'm not sure, but I guess that your first quotation from Hilbert was thrown from a context in which Hilbert was emphasizing, sorry, because the effect of my age, no problem, Hilbert is emphasizing the fact that there's a way to make mathematics that is like a sort of play on symbols. which is a contextual arithmetic, okay?

1:25:00 So, and it seems to me that his analogy that he makes between formulas and the diagrams is a way to infest of this possibility of pure formal transformation of many finitary mathematics. But I'm not sure that it is a really deep appreciation about the role of the respective function of formulas and diagrams. So my question is, what is the difference? Because in your thought, it seems to me that you are suggesting that, in fact, diagrams and the formula, at least in a certain sense of formulas, as something as similar as that you can in certain sense speak of formula as diagrams so my point is my question is what difference there is if any for you between a symbolic notation that admits manipulation according to rules of substitution or loop of transformation are in formal calculus and a diagram. I would like to clarify one point. I did not want to say that formulas were like diagrams because that would not account for the shift. And I wanted to stress that was a shift. What I wanted to stress, and this is why I made these two parts, is that, and it seemed to me that this was one point made by Hilbert, I did not quote what was before but that somehow summarized what came before, it seemed to me that he stressed the fact that you have a diagrammatic dimension in any symbolism in the same way as you have a written dimension in any diagrammatic device.

1:27:30 And that seemed to me, this morning, someone, I forgot who spoke of, perhaps it was on logic, can I call this a diagram or not, and so on. So it seems to me that you can speak of the diagrammatic features of some notations, and they are important to account for the fact that with different notations, you do not compute in the same way. With different mutations, you rely on diagrammatic features of the mutation, more or less, when you are working with the mutations. So I wanted to stress that we could analyze and distinguish various diagrammatic features of systems of notation. And on the other side, I wanted to insist on the fact that diagrams can also have, and I think that this is clearly what Tilbert stresses, and I find this opposition quite beautiful, that diagrams can be something on which you can have operations. So it's not only something that you illustrate and, you know, you move a little bit. You can have a rule system of operations, and in that sense, it is formulaic, it's a formulaic feature. And it seems to me that this kind of algebra gives an example of a formulaic feature of a system of diagrams. Because it's not one diagram to illustrate one thing, it's a systematic representation of a mathematical object with a system of variations and with some, again, with a system

1:30:00 of transformations and operations and functions. I understand this, but still, in the habitual interpretation of, at least, the formula is syntactic and semantic as a diagram of numbers so this difference is simply important when you there is no dancing as a syntactic and semantic distinction between when you write a free angle or a circle when you write a form I guess you are then you're substitution and then you are interpretation okay I don't know I I think that when you have rectangles that are structured systematically in a given way with colors and signs, this is the beginning of the diagrammatic syntax. I wanted you to say that. This is the last question solved. So I thought the quote from Milder was very striking, but in isolation, we don't know what the context is except that it comes from the list of mathematical problems, but where and how? It's page 2, 3, or 4 of the, he's still speaking about mathematical activity in general, and so on, and it comes after, I'm sorry, I did not read it yesterday, but it comes after a paragraph, it's the conclusion of the paragraph where he's discussing different kinds of activities on symbols and figures. I can give you, I have it in my computer at home, I can say to you, but I was trying to understand that they take it at face value, and what I say had nothing to do with your comparison with Chinese mathematics. But it seems to me that you can only say that the arithmetical symbols are written diagrams if they're interpreted symbols. That the letters in them have already

1:32:30 been given meaning in some way or other. They may have been given geometrical meaning, or they may be involved in, say, you take a differential equation, you have some interpretation in terms of functions and how they may solve this equation by satisfying various constraints. And on the other hand Bain talks about calculations with these things, so it's not just an isolated formula, but changes that you're making. On the other hand, the geometrical figures are graphic formulas, I interpret this meaning that in principle they're dispensable as means of reasoning, but in fact, so it comes back to this issue, whether diagrams are essential or all mathematics is simply something that can be formalized, and I think what he's saying here is in principle it can't be. Would it be fair to describe it as the logician's reading of this? One little question is, I mean, more specific, I mean, does the arithmetical sign for Huber here include variables and algebraic? That he does not... Or are we talking about notation for numerals, like with strokes? I think that when he says, you see, it was, there is also a French translation where it was translated, I think, signs and symbols in French translation. And I think, in my view, it's rather really meant, in my view, to refer to expressions with a given structure that gives them some special features on which you are going to rely when you operate. It could be the stroke notation for you. Absolutely.