E Artin's 'Über die Zerlegung definiter Funktionen in Quadrate' (part 2)

0:00 All right, well, I'm skipping ahead to where I left off. As you know, I had proved mostly that almost all of the proof, given almost all of the proof of Arden's theorem solving Hilbert's 17th problem, and he reduced it to this one big theorem, namely that if R is a given subfield of the real numbers and K is the partial functions in N variables, variables over R, and if you furnish some arbitrary fixed ordering on this field of rational functions k, then given any system of rational functions, phi 1 through phi m, you can find rational numbers a1 through an, which will specialize those given rational functions to have the same signs that they had in the given ordering. and so I just got through with the proof of that last time that was involving the specialization arguments specializable properties well scroll to where to use that theorem to prove the Gilbert 17 problem result that the part of the guy. Sorry, I guess maybe you would rather me scroll like this, less flashy. Okay, here it is now. Now we're going to prove theorem four, which is what he's famous for. Theorem four says, let R be a real number field, namely a subfield of the real is that can be ordered in only one way, such as, for example, the field of rational numbers or that of all real algebraic numbers or all real numbers. Then, every definite rational function of x1 through xn with coefficients in r is the sum of squares of rational functions of xi with coefficients from r.

2:30 And the proof of that is now reduced to just one paragraph, thanks to that big theorem 3, namely this. Let r be a real... Oh, yeah. field that can be ordered in only one way, then every ordering of k, this field of rational functions, is automatically an extension of the ordering of r. Furthermore, let capital F of xi be a definite rational function in k or with coefficients, well, sorry, in the field k, i.e., one such that for all rational number values ai at which f of xi is at all defined at the point xi equals a i f of xi possesses a non-negative function value f of a i greater than or equal zero by theorem three f xi cannot turn out negative in any ordering of k since otherwise it would be negative for suitable rational number values so you see that was theorem three which originally was stated for several rational functions but here it's actually applied only to one single rational function so if f i is positive in every possible ordering of k then it is by definition it is totally positive in k and then his theorem one in the first part of the paper shows that it must be a sum of squares so that's the conclusion of that theorem and then he gives a few variants or improvements of that result You said that it was proof for several functions, and it was proof for one. Right. There is a reason for that, which is the process of the induction. It started from one. That's right, I didn't say that, but it was necessary that he do the theorem 3 for multiple functions. Okay, excuse me, I didn't get your point. Yes, so this is the time 0.3 for one function. The process of the proof of 0.3 is by induction and the number of variables. And even if you start with one function, when you make this elimination one variable, you prove many.

5:00 Okay, so that was his main theorem right there. Now, you might remember from Hilbert's original presentation and his list of 23 problems, he said it would also be interesting to know if the rational functions can be chosen to have coefficients from the same field from which the given coefficients come, or the coefficients in the original polynomial. And that, by itself, is stated badly. Maybe I can give a simple example I just thought of. But basically, the answer to Hilbert's original 17th problem is no, it's impossible. You take R to be Q and join the square root of 2, and then take F to be, well, there's no variables, just the constant function, the square root of 2, which is positive semi-definite but not a sum of squares with coefficients in that field. So not a sum of squares like that. So I guess you could say, in that sense, Hilbert's 17th problem has a negative answer, but Arden put a few extra hypotheses here that make it true one of which is that the field must be uniquely orderable and the reason the square of two is not a sum of squares is that it's it can be negative it's not totally positive there's another ordering of that field so uh let's see art goes on with the theorem five it bars an algebraic number field then to every totally definite rational function with coefficients in r well every totally definite and a rational function with coefficients in R is the sum of squares of rational functions with coefficients in R. By totally definite, he means that, he defined it right before, let us call a rational function with coefficients from R totally definite if at rational points where it is defined, it takes only totally positive number values. So that's the kind of theorem. Of course, that function over there I gave you is not totally definite.

7:30 And there's another theorem he gives with weights. He's getting close to the ideal formulation here. If R is an arbitrary real number field and F is definite rational function with coefficients in R, then F can be represented like this as a weighted sum of squares of rational functions with non-negative weights in the field R and the coefficients of B in R. It's still not perfectly ideal. Alena wants to talk about it, but what was later found, I think, to be an ideal formulation of the Hilbert's 17th problem is not to, well, let me see, what is it, yes, here he's still using an unnecessary hypothesis that, excuse me, I don't know what that was, that the field must be a subfield of real numbers, and namely that it must be Archimedian. Now, that assumption is necessary in the way he's formulating it, because for him, the word definite, rational function, means it's non-negative at just those elements of the field itself. And it's nowadays, I think, more common to say that the word definite really means it's definite on the real closure of the ordered field. And he gave an example in his first paper with Schreier, I think it was mentioned maybe by Alain, of a function that is positive definite on some ordered field, but not positive semi-definite on the real closure. Because on the real closure, the function takes a small dip into the negative zone. So if you make that stronger hypothesis on the word definite, it, then you can drop the hypothesis on the number field being Archimene, but anyway, that maybe reflecting the emphasis in those days upon the rational numbers being this kind of the ultimate and absolute basis for all algebra, I guess, and not, I don't know, maybe that was just the way they looked at. Let me see, he goes on and makes some comments, One comment about the need or the lack of the need for the well-ordering theorem, namely, he says we have used well-ordering repeatedly, especially in the first paper with Schreier, because he always enumerates the elements of the field and builds up the real closures.

10:00 he says that if you well you can get away with just a countable subfield because all you need is the subfield containing all the coefficients of the given rational function there's only five and many so it's explicitly innumerable and so you don't need the full strength of the well ordering theorem if you care only about theorems 4, 5, and 6 and then still however he admits on the other hand our proof is no explicit method for the decomposition. However, one may well expect that the proof can be completed in this direction. So I have a question, because in the French translation it says we can nevertheless hope to complete the proof in the direction. No, this is man. This is impersonal. This is mandat. Aha. So it's a different translation? Yeah. Because it seems to indicate that acting himself is hoping to do it, I mean, in the French translation. Well, I guess I'm not responsible for it. No, no, no, I'm asking because I'm preparing the French in the French translation. Because I think it's an important point because it's a really difficult problem. But if he says he is expecting to do it himself, I think it's very difficult. In German text, it is Manta, so that's it. And as long as it's over, thank you. We can't wait. We can't wait. Yes, wait. Wait. Wait. We can't wait. We can't wait. How do you pronounce the, or translate it to Bohl?

12:30 Since we're digressing on this point, I bring up one of my comments on this constructivization. In 1955, Artin was at Princeton, I guess he was there for a long time, I don't know, But Kreisel was there for one or two years in 1955, and he, Arten asked Kreisel whether Arten's proof, Arten's own proof, could be made constructive. Kreisel did so within a month. A sketch of the method was published in 1960, and I wrote a review of that in 1996 in his Kreiseliana volume. Anyway, Kreisel reported that Arten was satisfied with such a sketch, or that sketch, and Arten grasped Kreisel's ideas quickly. So you said, I'm satisfied? Something like that. And that Arten had such a good command of algebra that he grasped those ideas quickly. And Kreisel also reports that Aachen was interested in whether his own original proof, and I guess not some new proof, could be made to construct it. And then Lombardy, one of our colleagues here, has constructivized the stronger form of the Aachen theorem, known as the real Stellens, et cetera. So that's a digression on the subject of constructivizations. And that leads to many other topics, you know, the great complexity of all of that, but anyway, maybe now I go back to the Arden paper, which is here. So that takes care of part one and part two of his paper. Now he has a part three in this paper called the Decomposition of Definite Polynomial Function, or Ganze Rationale. It just means polynomial, and what he is saying is, can you sharpen this result? If the original rational function is also a polynomial, then perhaps you can arrange for those squares also to be polynomials. And he says, however, that Hilbert showed that's not possible, and Landau showed that And this does hold for functions of one variable, even with rational coefficients. It's trivial for one variable over real coefficients, but with rational coefficients it is not trivial to show that you don't need denominators, I guess.

15:00 Mondau did that in 1903. And in the case of functions of two variables, which Gilbert had handled over the real numbers, the basis of the squares, that is, rational functions that are being squared, can be chosen to be polynomial in an arbitrarily given one of the two variables, if the field of all real numbers is laid down as coefficient field. And then Arten just extended Landau's method to more than two variables, 1903 there was something problem was known only for two variables and so there was no reason for landau to extend this to more variables with art and did and it was just an obvious extension there's nothing creative about it so here's what art showed let k be an arbitrary real field and I think it might be true for a non real field and and k of x the field rational functions in one variable of x with coefficients in k, then if the polynomial function f of x in the field kx, and he could have, I guess maybe in those days they didn't have the notation k bracket x. They just had to say k parenthesis x. So you say a polynomial function in the field of rational functions. If it can be written as a sum of squares, then there is also a decomposition of f of x in the sum of squares of polynomial functions. So, one variable over an arbitrary view of k, at least an arbitrary real view, that was his theorem 7, and I skipped the proof here. It's very straightforward and down to earth. It's just induction and division and equating terms and things like that. But I scroll down to one or two applications he gives of that theorem. Fairly long proof, I guess. Although, as usual, he says it's easy. And it is. But, okay, here. Theorem 8. Let R be a real subfield of K with a fixed ordering. if a polynomial function f of x in, sorry, that's maybe, ah, yeah, here, let's go to theorem nine. This is what I really want to do, theorem nine. The decomposition is claimed

17:30 in theorems four, five, and six, which I gave earlier today, can be arranged so that in the case of polynomial functions, the bases of the occurring squares are polynomial in and arbitrarily given one of the variables, but not in all of them, just in one at a time. You pick one and the others will have the nominating, so that's one theory. Proof. And then there's a final section in the paper called The Decomposition of Algebraic Functions. Anybody have a question? What is the word basis here? What is the basis? I believe the German word is basis, and I think he means that which is being squared, The exponential function has a to the b, so a is the base, b is the exponent. Nowadays we wouldn't say that, I think we would say just the square, so-called. The functions being squared are... And I had a little comment... Base, square. The base and the square. Oh, I see. I have a few comments on some of these things, and one of them was, let me see, oh no, down here, eliminating one variable from the denominators. In modern notation, you might express it like this. If you have a polynomial in n variables over, I guess, a real field k, and if it is sum of squares of rational functions then it is also a sum of squares polynomials in the last variable with coefficients that are rational in the first and minus wonder like that and then i reviewed the fact that mondown had already done it and since that time uh 1964 castles sharpened this result which you might call the landau argument result for any field k even one of characteristic two if f is in k bracket x1 if it is the sum of m squares in k parenthesis x1 then then f is the sum of exactly m squares or no more than m squares in k bracket x1 so the point of castles is there's no increase in the number of squares whereas uh in art and landau

20:00 there was potentially an increase in the number of squares and castles used this sharpening to time that this thing here, 1 plus x1 squared plus xn squared, is not the sum of n squared of rational function. And Cassel's, I forgot how this is done. It's an easy trick, I think, once you have Cassel's lemma. Maybe someone can remind me, and I don't have my books here. And Cassel's sharpening also helps establish lower bounds on the length of various sums of squares of other elements, for example, over the rational numbers, x square plus seven is not the sum of four squares of rational functions, thanks to Castle's lemma. And then Vister generalized Castle's lemma not just for pure sums of squares, but also weighted sums of squares, so an arbitrary quadratic form, a diagonal quadratic form. although Arten himself had also done this kind of arbitrary diagonal form back in his results. He just didn't control the number of squares. So now it's called the Castles-Fister theorem, but when I use it, if I don't need to know how many squares, I call it the Landau-Arten-Castles-Fister theorem because it really came from those people. Do you have a bound on the number of names you need? Well, I, you know, I guess every time you transform it, it potentially gets longer. So, okay, so that, those are some of my comments. I might have more comments later, but I pause and yield to my colleagues and the others. Anybody? Okay, so I'm not sure it's really useful because I think many of you really know this thing. But I have been asked to present Artem Tsuren with the hoop using Tarki. Just to see the difference, you switch from specialization to tasking's tasking principle. So it's very plain proof and tasking is very efficient, so it's almost nothing.

22:30 Mainly for people who have never seen that. So after that, I wrote it again. So you take R to be a field, a subfield of the true R with a unique order that's fundamental. And you take S to be a polynomial. You know it's sufficient to make a sub-square with a polynomial. And you have the hypothesis H, which says that the polynomial takes only positive value everywhere on Rn. And the conclusion of Artin was that it was a 7 square. That's a 7 chip just to prove using specialization and Artin's paper. So the alternative proof we use as that is of term principle instead of specialization. but it makes use also for the characterization of some of square as an artich fire and it will start exactly like artich fire. And the fact is that the proof like that is valid for any real closed field and also if you take any field with only one order but which is dense in its sphere enclosure, then the proofs work. So, now you take, the proof is like nothing, you take... Suppose that f is not a solve of squares in the field of rational functions. Then, if not totally positive, then there exists some order, where K is the field of rational function, making F strictly negative. And this order on the field of rational function, of course, is an extension of the order on the given field, L. Now you take P, the real closure of the rational function field, which has of course only one order, and F is also negative in P for this unique order of course. And now you turn to model theory, and you consider the formula written in the language for the remains and the parallelizing the small field F exists as X1, X2, and Xn, such

25:00 that F of X1 and so on, Xn, is strictly negative. So this formula holds in the field B because you just have to take for the value Xe the variable Xi itself. Just take the variable It's true in P. So you have a formula of star which is true in P. And then you turn to the real closure of the field L, the original field L, which has only one ordering, which extends the only ordering of L. But because you started with a subfield of air, you have that air is dense in airplane. And the hypothesis that it's positive everywhere goes to airplane too, it's all in airplane too, so it's positive everywhere in airplane. And now you have just to... I think it's a little bit of a sense. Now, using Tasker's principle from P to R prime, both are well-coached kids and all the real extension of S, we get that there is a formula star which holds in P and it holds in R prime. But it's in the contradiction with the hypothesis H, which falls also in A3. So, the stopping point was close, of course, so it's in some respects. So it's very short proof, and it's valid for any real closed field, and any field is in only one order, and then it's in its real closure. So the question of the density, right, maybe it's not common. The question of the density has been considered in many things. Because if you have a field with several orders, you can think of decomposing functions which take only totally positive value in sum of squares. And this has been considered by values process. And it involves questions about the density into the reflosure. But there are works from Trestell, McKenna, and also I've done some work.

27:30 So the density is somehow important. You really want some square. Otherwise you have to wait. Like a chip. Make sure if there is something you can just have a look and wait. Or to find definite over the real problem. Over the real problem. So, that's all I was asked to present when I was there. So, if the book is in Dasky, from Dasky's end of the... I think it's a mansion of Robin Sunwalk, I think. But, not very late after. You mentioned the last section, the one which was not yet translated. Sorry? You don't present at all the last section, the one which was not yet translated. It's been a long time since I've read that. Do you see Section 5 about the algebraic functions? So it's, yes, well, no, because it's interesting to see the learning of Section 4. Because the introduction has been numbered. So it's the case of positive functions. but on us now, we see our natural rights, we express ourselves, and of course, the wording of that is different, but, so, just to mention that he was already considering this function that were, that were is it not only about the but about the other varieties the English one the English one I'm still doing

30:00 and I think this algebraic result result in algebraic functions was like I said I don't remember what the results actually are there's 10 and 11 but this paper of 1934 was a similar subject he proved something about it I forgot, algebraic functions of one variable, wasn't it? It's basically a one variable theorem. Yes. So here theorem 1 is a totally positive element, so this field K of T is something of some function of the the reducible algebraic horizon, say 200,000. Which theorem? Negative 0 and 11. So it's a characterization of total positive elements, which is field K of the side, which is a function of field of the reducible algebraic side. I'm reading very quickly and then it says that it's a sum of square, if in every point AI it becomes positive for every branch, CI will go to the point. So it's not enough to be positive in every point, it can also be positive for every branch And some exceptions are authorized from variety to smaller dimensions. Smaller than? Yeah. So this idea that when you have something like a smaller dimension, you can have functions, you have some of the smaller dimensions, and they're going to be negative in the other. So, I think it's just interesting. I think also, your intent is very important. Of course, it's very important. Connects the existence of a regular, a real variety with a sort of function with having what to be real, or having what to be, and which necessarily what was real, has to be . So, this is an important point where orderings of non-arcomitian, non-arcomitian orderings are really needed.

32:30 And I think this is a sexually also argument of this. No more questions or remarks? Because we can go on this discussion. My points are general historiographical remarks, so if you want to add some remarks or comments on those two papers, you're free to do it now. For the weight of the version of the 17th program, only in our chances have remained, I could say, you developed a theory for re-extension of an outdoor field, There is a natural extension of the 17th program, although I've never found a way to some of those crafts. Yes, I think it's one of the remarkable points of this paper is the older paper we have, which seems so close to us. There is no problem to release that, quite remarkable. And it's not the same for all the papers at this time. Thank you.

35:00 Well, I don't think you need something as difficult as the Whitney Umbrella, just one point, one isolated point system. No more remarks on questions? Well, so I will try to make some historiographical remarks, some kind of naïve historiographical nonsense. I would have liked to elaborate them more, but each time I tried to work Fabrizio came to me to explain the carton paper so it was difficult for me to elaborate more so all the thought is to Fabrizio so it's perfect it's perfect that's perfect okay everything goes okay So, Charles starts with some allusion with the history of movie, with his screen going very fast. I would like to continue in this direction with a kind of allegory or parabola. It is, I guess, a well-known fact in the history of cinema that, in the beginning, when you want to, in a movie, to film a bank attack, you are first to have a sequence on the bad guy outside the bank. Then, a sequence on the bank. And then, a third sequence inside the bank. Okay? So the storyboard is something like that. First sequence, the bad guy with And third sequence, the bad guy in the bank.

37:30 Okay? And it appears that the same story won't be shown the same way today. you will have the bad guy outside, and directly the bad guy inside. And maybe more dead. But that's something else. That's not the point I'm concerned in. Okay? So, it seems to be an historical fact in the history of Moody. Okay? So, first point, that's a fact. Let's see some consequences. It seems that we have a kind of progress here. The film is shorter, Oh, today I say also because you have to make the continuation on the two sequences, and so you have an overlap of the sound with the... OK, there's something else. Well, you have a kind of progress, OK? Look at my notes, not to forget a point. So it seems that's an important point. Why it is like that? Because two would have been incomprehensible at the time. Okay? If the film would have been made this way, nobody would have understood the story. Okay? So, to un-understandable, un-understandable.

40:00 On the contrary, for us, when you look at the whole movie, it's very slow and much too explicit. OK? We don't have to be told all those things. We can guess that if the guy was there and then there, he'd go through the door. Okay? So, this sequence today appears to us much too longer and too explicit. We don't need those explanations. And it's a bit boring, as it appears that reading some of our text is sometimes a bit boring, because they are much too explicit. Well, now, this is the sequence which is removed by the progress. It is not this one, which is removed. Okay? So, the sequence which is removed is a transitional sequence which we make the transition between two situations. So, we have a kind of historical process, and the historical interest is there, in this remove the forgotten sequence. Okay? And which appears to be a traditional one. Not every kind of sequence is forgotten. And, in a sense, this is a source. Okay? Because when you are in the bank, here or there. You don't forget that you are in a bank. Even you can mention it. OK? But you don't have to read it. OK? So that's a kind of allegory of the process

42:30 and the kind of simplification we can have. Well, those who are asking questions. Well, that's my first parabola for history. Now, I would like to make a short and of a complete different kind of remark about art in paper. You can have various visions of the progress in mathematics. You can see mathematics, the dynamic of mathematics, in a problem-solving way. That means mathematics go on solving problems, and that's the main reason of their development. You can use Artin Schreyer paper for that, and you can find in the literature this kind of presentation that those papers were written for solving 17th Hilbert problem. It is true that Artin paper is solving 17th Hilbert problem, but it's not the only thing he's doing. doing. Another way to see the history of mathematics is to think that you have some program research or another way to say that some school, typically, Göttingen. In Göttingen, you are doing some kind of mathematics and you cannot do any kind of mathematics when you are at Göttingen. Okay, so you can see the paper of Artin and Schreier exactly as an example of this kind of history, as the one of Carton. Carton is a typically paper written at the Seminaire Carton, and you cannot imagine to be written elsewhere if you write this kind of paper at this time.

45:00 I know where you are. OK? So, you can see Artin Schreier and Artin Pfeffer as a complete illustration of the second view on history. You can also think history of mathematics as a progress in abstraction and see classical algebraic number theory and then modern algebra and then Schreier is a perfect illustration of this progress through abstraction. You have Hilbert Landau kind of text, and then real algebra as presented in Artin Schreier, and you have a kind of progress in abstraction. So you see, we can use the same paper to illustrate perfectly three kinds of history, three different kind of history and it fits perfectly okay so this paper can be used can be seen in various kind of history and it's okay for each one that's my second point And now I come to an even more historiographical nonsense. But maybe before you can have some remarks or reaction to this, to comment, Katrin Levy, as you are laughing. Well, I just wanted to say that this talker is a geolonger, and it's true, we're not talking so much. So, now... Are you saying that the history of the United States is not compatible? Just if you want to argue that you are in one because it fits, the argument is wrong. So, because it's a very common argument that you take one aspect, it fits, so that's the whole story. Okay, so here we have an example which shows you that at least those three, we can have many others, are satisfied with the same paper.

47:30 So, we have to be more careful in choosing the way we are going to make history. Because it's the argument to say, well, this paper shows how mathematics develops by solving problems. It's not fault, but it doesn't prove that it is the only dynamic. Okay? Okay. Well, now I would like to try to see more carefully the relation between modern algebra, logic, and mathematics at this time. I would like to try to to give some hints of the situation. So first let's start by one of the dynamic historical progress are present. You can see the history of mathematics. You have mathematics there, and particularly algebraic number theory, as done by Hilbert, Longo, and many others. People are dealing with things like Q, so polynomials. They are proving theorem, and they are giving them proofs and producing definitions. Here, let's imagine that mathematics at the end of the 19th and beginning of 20th century. We can see, we can have an idea of the progress of mathematics, and now we are around 1920,

50:00 20, 30. And we have modern algebra. And here we have a name like Steinitz. We can, that's very catchy. Steinitz, Neuter, and I think. And we can see the story of something like abstraction, going through more abstraction, formalization, something like that. OK, that's a common representation of the historian, in many respects, absolutely accurate. So here we have a kind of time direction. Now I would like to—and you can even be more precise from an epistemological point and more in the explicitation of the ideas. We are going nearer and nearer of the ideas contained there. They are expliciting the ideas which are there. Well, now I would like to use a kind of Wittgenstein duck rabbit trick. So, know the duck rabbit of Wittgenstein, you have something like that. And you can see both as a duck or a rabbit. So this is the duck version of the story. I would like to turn to the rabbit version of the story, which is Arsene's paper, Arsene Schreier's paper. But it's the same for both. Artin Schreier's paper is dealing... Artin Schreier's paper is referring to all those things

52:30 and is using all those things. So, it's not passed, it's there. OK? It's not forgotten. It's not elsewhere. It's in the paper. And he's developing, typically, modern algebra mathematics. He gives the...he's dealing with fields and those kinds of things. and isomorphism between doting is typically doing modern algebra. But more precisely, remember, the paper has three parts. We can put very precisely each part in the picture. The first part of the paper deals with Artin-Schreier. The first part of the Artin-Schreier's paper deals with to give a translation of those theorems into modern algebra. The introduction is the first part of the paper? The first part of the paper is the first part of the paper, as in the German paper. So, for example, here we have Sturm as a theorem, and the concern, and the explicit concern of Artin in the paper, mentioned in the introduction, is to build a correspondence of classical real algebra in the naive way, that corresponds between both theorems, to modern algebraic ones. Okay? So, that's precisely the work to build this correspondence, is the work of the first section. Okay? So, we can put it precisely in this picture. Okay? The second is entirely there. And a quick way to show it is that in the second part, section,

55:00 you don't have any mention of the rationale. OK? And you have theorem which are there. In this respect, it's a typical modern algebraic paper. So you have some new kind of theorem which are there, and which are dealing with extension proving that there are several or just one, and so on, and so on. Sorry? In the second part? Yes, because it's the one. Okay. So, typically, here, if you look at theorem 7 and so on, you have those typical statements. Now, you have a third part in this paper. The third part, so, we are agreed that here we have mathematics. That means we have statements, we have proof, we have definitions. Here, no doubt, it is also mathematics. We have statements of this kind, proof of this kind, So this is some mathematics, OK? But there is a third part, a very interesting third part, which deals with the relation between these notions and those there, OK? If you look, for example, to theorem, the theorem of the third part, theorem 8A and theorem 10 of the third part, they are satisfying the first condition mentioned in the introduction of the paper, it is that you have only one real, real algebraic field

57:30 there when you have absolute extension of Q, absolute extension corresponding, well, You have this statement, huh? Look at the two theorems. And those two theorems, you have, in this picture, to put there... To put there, they are giving relations between those notions and those ones. So we are, in a sense, in the middle, if we come back a moment to the dub version of the the picture, we are in the middle of nowhere. Elsewhere, it is a kind of meta, if we want to make a step in the direction of logic. It's a kind of meta-statement between modern algebra and, let's say, classical mathematics. So, in a sense, a kind of extension of the in the area of mathematics, because now we have definitions and statements with proof there. So I think it's one of the remarkable aspects of this paper. It is not only that it develops modern algebra, it's not only on, it's also over at the same time. OK? Because the third part is making a transition. And if you remember, in a sense, it's the bank. It has to do with the bank, the sequence of the middle picture removed. But more or Well, is this picture clear? Yes, more or less. Is this picture clear? What I mean by that is that in what sense it has to do with the bank affair is that Artin has something to do to articulate the sequence in the street and the sequence in the bank. He has some

1:00:00 work to do at this time to go from one aspect to the other. Of course, here it's even a technical article in the statements, because there are real and real, which are fractional. So he has his claim on the two words. Absolutely. Yeah. And typically, this distinction, you won't find it anymore. Okay? You will find it in Van der Varden with Formally. Okay? We are in this kind of vision of the stuff. Now, to make it not too long, if you see this picture, it's very close to the situation of the relation between logic and mathematics, reached by Tarski. We come to this picture in the history of logic and mathematics, but with Tarski, not before. That's the point I would like to make a bit clearer. OK? So, this is Tromsberg Principle there, OK? But this picture, this relation between logic and mathematics is not at all the situation at this time. The relation between logic and in mathematics, the picture is not this one. So I would like to try to give you very quickly some indication of that. Typically, if you look at Boole, Schroeder, Lovenheim

1:02:30 with many distinctions to be introduced, for sure. Their concern, if you have mathematics, is to develop a mathematical logic. So you are inside mathematics. And they are using mathematical expression. Boole, its expressions are of this kind. Okay? We are inside mathematics. That's the interest of the Boolean logic versus Aristotelian logic, but that's one of the aspects. It is mathematical. But to be mathematical, he chose to use mathematical, the same mathematical object. That's a simple way to be mathematical. It's to take mathematical expressions. Okay? But the story is a bit more complicated, because in the same way, one of the statements of Gould is to prove that we are not, in mathematics, only dealing with numbers. So, also, is an extension of mathematics, because it's not only number. His point is also to extend the not only on numbers. Well, so we are in mathematics. The point of logician as Fregue, for example, or Russell and Whitehead is to separate logic and mathematics. For Srege, it's very important to him to create a new, independent language of the language of mathematics. No doubt, his big list with stuff like that, it is not mathematical expression. And for him, it's very important that logic has to be independent of mathematics. So here, we have a picture. Mathematics is there, and logic is there. And, in very different ways, we have Fregeux, and we have, for example, Russell and Whitehead, or Whitehead. But of course, their picture was that mathematics would then be inside the logic box. So, you have this arrow. But this arrow is not a mathematical arrow. It's a kind of epistemological arrow. It's, we can rewrite mathematical there. Then, once it is done, you have no mathematics here.

1:05:00 mathematics are there. So you don't have, at the same time, mathematics and logic. If you want to see both at the same time, you have a kind of formalization arrow, but it's not, it doesn't represent a mathematical transformation. You don't have any statement of any kind of this arrow. You a Russell and Whitehead way to deal with logic, OK? So the work, and we could go on looking at Hilbert, but I don't want to be too long, which I think I already felt. But now the work of Tarski is precisely, and it is a work, it has to create many new concepts to do that. It's to make this arrow a mathematical arrow, and to make this one a mathematical arrow. And like that, you have mathematics here, and it is, I would say, the common situation today to do logic is a part of mathematics as algebra, and so on and so forth. And you can have relation between various parts of mathematics. logically, with arithmetic, you can, and so on and so forth, okay? So, just to give you an example of, in my view, there are two things, two things very important to realize, that But it's the idea of language. The idea of language, it's introduced by Karski, is one of the things. Because now, before, one theory, the only variable, if you want, were the axioms. OK? And after, it is axiom and language. That means you can variate the language. You can

1:07:30 play with the language as you are playing with extension of fields. As you know, we can enrich your language and have theorems about what happens when you have a richer language in the logic. For that, you have to recognize something which is language to be able to make it richer. So to put on the front explicitly the idea of language gives you the possibility to have some mathematical statement regarding it. And last point, if you look at the second important point, it's the notion of terms and atomic formula. We have seen in the paper of Artin that he is able to write those kind of things, pure polynome and pure polynome. That's amazing. This mixture of logic and mathematics, this mixture of logic and mathematics, it's completely from before, OK? Because you will never find this kind of expression. And to that, you have to build those correspondences, which is a major concern of Tarski in all its papers. And its concern is, again, something like the bank picture, the one we are no longer interested in, because we go through the situation, OK? Now we are in the bank. No problem. But if we look at Tarski, it is a fact, as it seems to be a fact, this story of removing the middle picture. It is a fact, if you look in Tarski paper, that is talking a lot and a lot about of those points. And when we are of the generation of the kind two of movie, it's verbose, it's and so on and so forth. It's too much explicit what, why he's talking so much of that? Because it is a bank attack. And we have to go to enter the bank.

1:10:00 Otherwise, people of the time won't understand nothing If I could say something, I think there are some key names that you're omitting and making it look as though Tarski appears out of nothing. You have to mention Post, you have to mention Hilbert. Sure. I mentioned that. And even just to look at Pressburger, although the Pressburger procedure is a triviality compared to Tarsky's, but conceptually it's exactly the same framework. And the concept of language is there in Pressburger just as much as in Tarsky's. I do agree that Tarski is, for me, a representative of more than Tarski. It was not my point. My point is that if you look at post-paper about completeness of proportional calculus, you will find those statements referring to Lewis, a part precisely which is not re-edited in the second edition of Lewis, which now we are going to proceed completely formally contrary to Whitehead and Russell. So you have those banks in the same way. So I absolutely agree that we have to go more into your... You'll even see it in C.I. Lewis's survey of symbolic logic. It's the idea of looking at syntax from the outside, which Frege certainly did not have and that Russell and Whitehead certainly did not have. They were a step backward from Frege. Yes, but more than 80% of these books, the presentation of Boone and Schroeder logic, don't forget. The book of Lewis survey is 80% devoted to Boone and Schroeder. So we don't have to remove that. I think you give a personal reconstruction of the history of man. Do you think that the true truth is given by Tarski with a kind of fusion between logic?

1:12:30 I've never you introduced twice a truth as square and didn't use this word once but it is your implicit no no my concern is that when I say that I give you some arguments there to explain in what sense my, the structure I give of art in paper in a sense, objective. It is personal in the sense, I decide in what I'm going to be interested in. Okay? But once you admit that it is what you are looking for, which you are not obliged to, okay, just a way to deal with, you will have this picture. Okay? But that's true. You can, And I start, remember, my presentation was not as naive as maybe it seems. Remember my point two, telling that you have three possible histories. I just give you a first one. But without stating that the fourth was the good one. It's just a fourth one. That's all. It's a fourth one, and I just give arguments as other way to deal with history, on and so forth, are able to do. But it's not because it is the one I present. It is, I think, the only one, the good one. A new interpretation of the . Sure. It's mine, and I'm supposed to do a . Right? So, for example, you see that Bobaki is because they have a kind of a posterior reconstruction of what happened and I think everybody who makes history makes a kind of reconstruction everybody is a constructive statement you are able to find a piano function to go from each one to the other one to state this kind of

1:15:00 quantification this kind of reconstruction of the story actually sorry your forced interpretation So from a constructive point of view, it should be possible to say, as Bobaki says that Cauchy's proof are false, it is possible to say that Martin's proof is false, but in fact what What is interesting is not ... What is interesting, coming back to a kind of absolute statement, what is in general interesting or interesting for you? No, I think that when we do a story of mathematics, we are trying to understand what is the evolution of the idea. And it is not... You say that it has nothing to do with that? What I... Yeah, yeah, yeah. So, I think the point of view of the guy who is doing this story is unheard of. Yeah. I just try to make it explicit. There are two things. It's the first point... Yes, but I try to be conscious of what is my point of view, and two, to give you elements to verify it. And that's important. I make reference to text. You can check. And when I'm making a calculus, I don't change notation. But this is a combination. Okay. Maybe. Maybe. You have to prove it. If you don't do the calculus in first, in the Hermit way, you cannot be sure that it's just a convenience. You have to do both, and then to... To place the name of the roots, giving an analysis... It seems a kind of picture of this way, of this time... This is a serious discussion.

1:17:30 It's a kind of necessity, it seems to be a kind of necessity, as in my history of movies. But how you can be sure that the ideas are the same if you didn't do both calculus, in the Hermit way and in your way? Maybe, but we have to do it to know it. Maybe you're right. I don't think that's wrong. I just say, to know it, you have to do it in both ways. And only after that, you'll be able to say, OK, I checked and it's the same. It's the same as in mathematics. OK? We have to make some verification. And we cannot state foreground without looking at that it's the same. Maybe it's the same, but we have to check. Yeah, we have to read two texts. We have to read two texts. Sorry? We have to read two versions. Yeah? The English version and the French version in order to compare. What? Every version without indexes and the Lombardi version with indexes. Yeah, but we have to do the same. Yes, but you, to make your statement, you have to read both texts and not only Lombardi's text. Yeah, yeah. I agree, I agree. First of all, we ask that, just to explain, it's a quite personal point, and then we ask that, you know, information and so on. So, here I am on this point. On the other point, I will not see on this point. One of the problems is what is a mathematical idea? And one of the problems is how to recognize what are all the mathematical working ideas in a mathematical project. And in some respects, I think, at the point of Alain, whatever he does, the popularization, the tricks he tried to use to speak to an audience, mainly a mathematical audience, I won't discuss the efficiency of that, but I think his point was really that there are some mathematical kind of ideas in this text which are normally not stated into account in the practical history, even in our various versions of this history, because they concentrated on a certain kind of mathematical ideas, for instance concepts, or theorem, or explicit tools, and there are other kinds of, one could say mathematical ideas,

1:20:00 which are connected with how the language operates in several texts, how we can construct the relationship between the certain types of mathematics and the northern languages, which are in fact very operational at some times because people are trying quite consciously to make this gap or construct such a connection. but we are not in a time where we are very aware of that sort of mathematical work, and I think, well, in part I'm wrong, but one of the, what Alan was trying to explain is how it's important in search text to understand also this type of mathematical ideas, which are not ideas in the, they are active, you said they are operational in this text. So I think it's what he was really trying to come there. So on that point, I would say that it's true for this text, I don't know, because you have to do serious historical work, and I trust the complete people who have done that, and you can't shake that like this. But certainly I don't think it's a forced reconstruction in the sense that we put sort of arbitrary ideas on what should be in the text to this text. I mean, the question is to make explicit certain things, exactly as you would make explicit certain concepts or certain tools or certain things in the text, which are not perhaps obvious at the first time. We have seen that in all the text we have looked at, especially what was obvious, but it was not. So, I think it's part of the story. I mean, I don't know if you agree with my description by now. Yeah, I just want to make a very mundane comment about the relationship between Arden-Schreier and Tarski's theorem. In the various applications of Tarski, one is making use of the fact that it's an axiom principle theory, and of course the axioms come right out of the Artin Schreier, come

1:22:30 right out of the Artin Schreier paper, as I'm sure will be realized, but from that point of view I find it curious that Artin's actual definition of a real closed field is something which is second order, namely that there is no extension which remains real close, which of course can't be stated in the first order, because that's all. The same kind of remark, just a point in continuation of that, I was struck by the fact that in the definition of Archimedean's field, He has no integer. His definition is, in Hilbert you have integer, in Trevin you have integer, but in his definition he has removed integer. And there is another point that you made, complementary to the fact that the fact that we have both class groups and the field, also the intermediate values, which appear is replaced by the action of the finable continuity, the finable self bound in the world as a super, as a list of about. And only much later it becomes an explicit task that we have. Yes, I wanted to, in the interaction of Martin Bernard, for example, in, so when we were writing in our book, in French, in the original edition, it was, it was looked by Jean Kerser, Jean Kerser And one of his remarks is, why do you call the core-tail-flow field, why it's well-known And this, so that they were sticking to this definition in terms of high-order. And that was exactly the answer that, if you want from the system,

1:25:00 And so it remained in French as . In German, and their definition is . Now, I want to ask you to . So you said you define very, say, in a few words, the three first approaches to the school or by progress and abstraction, and then you give... No, you were problem solving. Problem solving. Oh, sorry, sorry. And, yes. And, but you're... So you gave an example of what you... Can you describe it in a few words? like foreign-serving, or school, or other semiotical approach of history. Which means what? Which means that you are looking at how it is possible to write mathematics as they are. Now, mathematicians, maybe, I'm sure, have ideas. I haven't access to those ideas. If, in my agenda, I have to want to make some history in a quite rigorous way, I have to have an object which is more real than ideas. So the texts are real, and they are, to be more precisely defined, it is composed of expression, but that's to be think of a technical term. And those expressions, you can describe their relation. They are some kind of necessity to use some kind of expression in a certain way, and so on and so forth. Here is something which is really existing and really contrarian for the imagination. You can do anything with any kind of expression.

1:27:30 So if you can find the structure, this kind of structure in the text, it is because of that. So I don't know if I've been clear. those relations between algebra, logic, and so on and so forth implies things in the kind of expression you are using. Typically, this one in the blackboard is very interesting for me, and it's objective. You won't find this. You can check in the same way in other papers. Or, I'm right with Martin Davies, you will, but that's because it's the same story. My point is not that you can find, my point is not, it's a simplification. I do agree with you, it's a simplification, but the dynamic is the same. I stress, for simplification, Tarski, because we mentioned it and we have read one paper of Tarski, But, well, the distinction is real. Well, I don't know if I answered your question. But, well, it's as complicated as for Fabrizio to explain the reason to your shift, you see. And you have the bank, the middle picture. to remove this middle picture. So I need a lot of time to build this middle picture. May Matt make one other comment which is really peripheral to the interest of this group but maybe has a little perspective. There is an ongoing quarrel among logicians between two points of view. There is the one really which animated Frege, Russell Whitehead, and later Hilbert, the idea that mathematics was to have some kind of firm foundation and it's the business of the logicians to provide it. The other is the idea that logic and metamathematics provide one more tool that mathematicians

1:30:00 can use, as in model theory, and the whole business of foundations is a lot of old-fashioned nonsense and we should forget about it. And if you look on the archives of the FOM, Foundations of Mathematics, email list of about three years ago, you will see fights that are as intent as you can want between Harvey Friedman on the foundation side and people like Pillay on the other side. So I just mention this for whatever it's worth. So there is a metaphor, I guess, by Gerard, in one of the papers, who says, uh, those Christians are like the knights of the knights who were going to Jerusalem and stating that Jerusalem was decided to attack something else. And the Christians were walking towards foundation of mathematics and stating that they were to Carl and Enric Shehler decided to move to Iran. Another question is that they were looking for the Holy Grail. Another quote of Yannick Shehler meant to think that they were searching for the Holy Grail When I was a graduate student, some of the other students who wanted a piece, young logicians, would tell the following anecdote, that there was a magnificent castle on the banks of the which had grown over the centuries, and everyone admired it very much. In the basement of this same castle there was some spiders who over the years had built a great network of spider webs.

1:32:30 one day somebody opened all the doors and windows