Conceptualizing ratios

0:00 Transcription by CastingWords Sous cette circonstance, la proposition B3, le premier enfant de George Kissinger, a une part dans la fortune familiale, est analytique eut égale à la représentation du premier. in spite of the fact that this is something that is purely contingent, all these variables are true, and here I have the list of all these variables, all these other variables are not objecting, they are so rejected. All these variables are true. This is not the case for B8, And then the second child of George Bush, Pauline, was born in the family family. Why? Because Pauline was born, when she was born. And this would not have been the case if George Bush had been through her second child, Dorothy. or if George Bush had not had an infant, because in this last case, there would be no institution of the premier in B3 who would give propositions objectuelles, because he would not have an infant. What this shows, is that the balsanian analysis has to do with what is true, actuellement sous certaines institutions et pas avec ce qui peut arriver dans d'autres mondes possibles. L'analyticité bosonienne dépend des états de choses, des états de faits, pardon. Et ça, ça ne vaut pas seulement pour le concept large d'analyticité, mais aussi pour le concept étroit d'analyticité logique que j'ai évoqué. Applied Barcaic, c'est effectivement une notion qui est tout à fait fiable et non équivoque lorsqu'il s'agit de définir un critère qui identifie les inférences qui préservent la vérité.

2:30 But it also includes some inferences who are simply matériellement validates, which depend on the classification of the terms. And what is even worse is that it also includes some inferences who are empirically general, like, for example, the B9 to B8, Jeb, I don't know, 3 m, and applied bar to Jeb, the Jeb external Sunshine State, Florida. but it's true. I don't know what to say about it. I don't understand at all this story of the analytical theory, of the scientific theory, of the analytical theory, of the analytical theory. It seems to me that the concept that I have presented uses the notion of analytical and scientific theory in a bit of a metaphorical way. very strict, there I have the impression that it is a bit of a theory. If I understand well, a proposition for an analytical or synthetic should be in the form A index, but it should be in the form of an object. Otherwise, our definition does not apply. So if A plus B plus C equals A plus B plus C is an analytical, On doit la passer sur la forme, si je comprends bien. Le x qui est a plus b plus c est a plus b plus c. Oui, écoute, qu'est-ce que je peux dire? Non, mais laisse-moi finir. Alors, à ce moment-là, chaque variante sur le x que j'ai fait, ou bien x n'est pas un nombre, donc c'est une importante phrase, ou bien x est un nombre, alors c'est très bien, So it is analytic that is in the same sense that the man caduce is exactly in the same case. But when I take the definition of the somme,

5:00 how do I take the definition of the somme? Like a proposition of this formula where I can apply the notion of the analyticity, the only way I can see is that the x, which is a somme, is a collection, but then it is analytic. Mais... Ou égard à la variation sur les X. Dans quel sens elle est synthétique? Il faut que je te montre que la proposition est synthétique, n'est-ce pas? Voilà. Il faut comprendre dans quel sens la définition de la forme, quelle est encore plus? Dans quel sens une définition de la forme synthétique? OK. La connexion dans laquelle l'ordre des parties n'est pas pris en considération, and in which the parties and parties are considered as the parties included. D'accord. That's the concept of Somme. And the reason for which... Okay, I think that's the reality of the way I'm going to talk about. The reason for which it can't be analytically, because it's not even a proposition, it's a concept. Nous, on a défini la définition atomique synthétique pour les propositions, donc maintenant on ne peut pas s'en sortir en disons quelque chose qui n'est pas une proposition de patron analytique. Mais dans un système climatique, il y a des définitions, etc. Oui, bien sûr. Donc, ou bien j'interprète apostolant une définition comme une proposition, alors après je peux dire que c'est synthétique. Ou bien ce n'est pas des propositions. I don't know if it's not a scientific effect. You can't say that it's a scientific analytical. It's a scientific scientific method. It's something that's something that's something about it. So if I define a scientific scientific, like Paulson l'a fait, as he gave us this proposition, then, very good, to say that a scientific definition I can't do it as a proposition. My problem is that the proposition is equivalently to the definition of a son. That's what I understand. And then if I tell you, a sum is a collection of... I can't because there is no component objectual. I have to have a component objectual. But why are you saying that it's not a component objectual? What is it? What is it? The validity, the production is valid, all these components objectuals

7:30 A collection, okay. Une somme est une collection dans laquelle l'ordre des parties blablabla... Mais une somme, ce n'est pas une représentation objectuelle. La somme X, c'est une représentation objectuelle. Une somme... Mais non, mais il y a un problème ici avec... Franck, je ne sais pas pourquoi tu dis qu'une somme, ce n'est pas objectuel. De toute façon, une somme, c'est objectuel. Mais de toute façon, la question, ce n'est pas objectuel, pas objectuel. but it's in the phrase, the x, which is a sum, is the collection, which gives us the two conditions. That's a proposition. The x, which is a sum, is, etc. Why is it not analytic? Well, if you think about the sum, then you have the impression that it is analytic. But it's you who want to eliminate the quantificator. If you want to eliminate the quantificator, then you have to think about a general proposition, So you have to think about the x-t, but not in other ways, otherwise you have a quantificator. So with the quantificators, it's all our other thing, we need to give a definition which concerns the quantificators, but there, it's the individual which are definitively, which are specific to the propositions. The quantificators, we eliminate them. So we take the universe and existentity propositions, and we substitute them with the schemas. It's very good. But then, when you study your journey, you don't have a sum, you have a sum x, with x but you have a sum x, otherwise you don't have a sum x, otherwise you don't have a sum x. For each sum, it follows, but for each sum it follows, it follows. And you can't have a general definition of what is a sum in general? Well, yes, but it doesn't enter into the definition of what is a proposition that you can do in the theory. Yes, of course, you can have it. Okay. What do you mean, that the definition of a sum is neither analytic nor scientific? Yes, if the definition of a sum is, then it's not the same form. If the definition of a sum is, which is a sum, or the X, which is a form, is a collection telle qu'elle est analytique exactement comme l'homme failleux est mortel. C'est le même exemple que le célibataire, plutôt. C'est-à-dire qu'on ne passe pas par la définition... Il faut juste regarder Desanneaux, parce que ce que Desanneaux me dit, au paragraphe 305.6, c'est que la loi de l'associativité, qui est logiquement analytique,

10:00 est fondée par la définition de la somme, qui est synthétique à priori. OK, donc moi, il faut que je fasse sens de ça parce que j'essaie de laisser son corps. Sandra, moi, je comprends bien qu'il le dise et je pense bien qu'il a raison. Mon problème, ce n'est pas que quand on lui dit ça, il a tort. Moi, je pense qu'il a parfaitement raison. D'ailleurs, je pense que c'est un point de vue très conscient. Donc, il n'est pas très anti-conscient. Mais bon, je pense qu'il a raison. Mais tout simplement, mon point est que pour dire ça, il a besoin d'une autre définition de la mythique. C'est de la scientificité. Non, non, parce qu'il peut tout simplement dire ou ça, or, tu veux que ce soit une proposition, la propriété… D'être aussi une collectionnelle. D'être aussi une collectionnelle. Et puis là, ben, dans cette proposition-là, il n'y a pas… Alors que j'ai créé la composante objectuelle et je peux varier. Mais il n'y en a pas pour ça que c'est sympathique. C'est la loi de la sociabilité qui est analytique. Ce que vous avez dit, c'est que la somme, la définition de la somme est synthétique à priori. Elle est synthétique à priori parce qu'elle n'a pas de composantes objectuelles. Elle n'a pas de variantes… Elle n'a pas de composantes variables. But if we define celibataire as being a non-marié, the celibataire is a non-marié. So that's synthetically. Non, but it's not exactly the same thing. Exactly. It's exactly the same thing. Attends, attends. Si on dit un célibataire est un non-marié, ce qu'on a, c'est, ce serait à première vue une proposition qui est synthétique, à posteriori, mais ce que Boisano nous dit, c'est, ben, il n'y a pas le terme, le terme, je pense que les exemples qu'il donne, c'est des exemples, I said, everything is done in a cause. When he said, everything is done in a cause, it's not analytically. He said, in fact, it's analytically in a way caché. Why? Because we define it by what is in a cause. So, everything is done in a cause, it's all what is in a cause, in a cause. That's the sense of the question of Marcos. Why, in this case, it's not analytically

12:30 caché? Because there is nothing caché. and then there is no... I would say that what is caché is that the definition of the type 1x which is a sum is... No, but it's not possible Well, it's not possible No, no, but I don't know I don't know what the problem It means that if we introduce here a class of substitution which is a sum I suppose that sum is already defined Yes, that's the question Effectivement, on va avoir quelque chose qui est analytique. Et en fait, ce qui est absolument intéressant, c'est la chose suivante. Ce que vous me montrez, c'est que... Bolzano, on va dire, ce qui est important chez Bolzano, effectivement, ce qui est important avant l'analyté, c'est bien entendu la validité, donc la notion inférentielle. I'm trying to find something... because I haven't even introduced the notion in front of me. But what we're going to see is that, in Bolzano, the notions of the science deductive are a priori, in the sense that the propositions that we have to prove are effectively that they don't contain the constitution, blah, blah, blah. But what he tells us is that... And in fact, what he tries to show is that the scientific science is defined by a notion that is the fundamental objective. So, what we have in the scientific science, It's not an analytical order that would be defined by the institution and that would be defined by the notion of the plaque barcate, but it's a foundation that is defined on the basis of this notion of adfolge, which is a notion very aristotélican, and I'm going to talk about it.

15:00 Par contre, ce que Bolzano nous dit, c'est qu'à chaque fois qu'on a une relation de fondement à conséquence, on a une relation de la plaque barcaille. Comment, justement, en faisant ça? Oui, c'est vrai. Mais le contraire n'est pas le cas. Est-ce que tu me laisses un résumé pour qu'on se voit si j'ai compris ? Donc, si j'ai bien compris, ce qui s'applique à la somme associative s'applique aussi aux mariés et aux célibataires, donc c'est certainement la même chose. Tous les mariés, tous les célibataires, aucun célibataire n'est marié doit être traduit par X qui est célibataire n'est pas marié. So this is analytical with regard to the variation of X, but it is analytical with regard to the variation of X, only because we have decided that the line special in celibataire is married, which is fixed, independent of this variation. The variation leaves a variable, it's a special line. X, who is celibataire but not married, is analytical, X, who is Italian, is not married, is not married, because between Italian and married there is no special lien between celibate and non married. But no, this lien special can be established. This lien special can be established via another proposition alimentive, because otherwise we would have to do something to do with it, because it is always based on a special lien that we have to establish by another proposition, and this other, a special lien, is based on another proposition, a little bit of a definition, which can't be read as the x which is a somme and the x which is a celibate, because at this moment-là, we would still have to define the somme, so we would still have to define the somme a priori, as we say, the x. Another definition could be an other type, it's the definition. This x is the special line between the non-variable components and the non-variable components. And that's what we call the definition. Yes, that's what we call the definition. But if it's that... If it's that, it's an inclusion. Because in the case of celibataire and non-marié, there's no reason for that the definition is different from for SOM. But no! You should have decided

17:30 that celibataire and SOM are not the same type. No, no, no, why not? I understand that. That's exactly the idea of the answer that Karnap gives to her question. So, it depends on the significance that you have established. It depends on what you have chosen. It depends on what you have chosen. It depends on what you have chosen. I think it's important to see what Boise Amos said, because, in fact, the definitions, what we will see is that So if you have a theory of marriage and of not marriage, you have this postulat-là. Seulement in this case-là, not in the case of celibataire. There is absolutely a difference between definition nominale and definition real. It's a theory of celibate. It's a theory of celibate in which, in definition, the celibate is not married. And then it's a theory in which you are married. In fact, for the term... That would be scientific, a priori, if you are married and you are married. Yes, exactly. In fact, in the theory of celibate... No, it's not scientific, a priori, it's analytic, cachet. It's analytic. Yes, the celibate are not married, it's analytic, cachet. yes, we do everything, there are to do with the resources, technology… D'accord, d'accord. Sandra, I'm sorry. I have a fundamental fundamental to phone and I will absolutely do what to do, so you can continue, I will do it in 10 minutes. I will absolutely do it, because if I have an estudiant-marcant, I can't do that. So I will absolutely do it. I can do it. On can do a pause? Ah yes, I can do it. We can do it. Yes, I can do it. Yes, I can do it. On can do it again. On can do it again. His numbers, I quote, Barrow, not even in thought can be extracted from magnitude.

20:00 By this means that a short number, say the square root of 2, has no meaning to a number, since no number, integral fraction, is the root of the abstract number 2. that a mean proportion between the abstract numbers 1 and 2 does not exist for a number. Only by conceiving 1 and 2 quad magnitudes can we conceive the mean proportional between them, which we call the square root of two entities, according to the appropriate sign. But nothing is that... No, square root is not a number. It's not a number. It's not a number. It's a mean proportional between two magnitudes. So, a mean proportional between two magnitudes is a magnitudes. Notice that Barrow's geometrical understanding of magnitudes entails that Euclidean Arithmetical Numbers become part of the whole family of numbers, and in that sense, arithmetic becomes subordinated to geometry, upon which Barrow highlighted it explicitly. For this, Barrow used to be presented as a conservative mathematician, defending the philosophy of mathematics of the ancient, but I would like to stress that he is departing from the Euclidean understanding of numbers and magnitudes that the world kept strictly separating. So, now we can go over aggression. One of the first common, the Barrow has been learning a long philological discussion about logos and earth. I will not stop here. The first

22:30 thing that Barrow stresses, Barrow acknowledges that geometricians, including the most competent ones, cannot agree on contemporary geometricians, cannot agree on what is the nature of oppression. And he points to the abstract quality of that notion, and he points to the fact that a ratio is a relation of the index, and that makes it very difficult to define or to characterize. And the essential point here is, the essential point from Barrow's point of view, is that ratios signify, ratios designate something about the comparison of homogeneous quantum, So, the abracio, by nature, falls within the Aristotelian category of relationships. And that will be a crucial point in his subsequent discussion, as we will see. One of the things that he does first is to warn against the misuse or the too much use of numbers in investigating and dealing with ratios. To deal with the thinking of ratios, essentially, mostly in terms of ratios expressed by numbers, or compulsion of terms through numbers, this is a careless restriction of the theory of and in fact obscures and distorts that tradition. And in that account he criticizes, he takes up Wallis' heroes and he quotes Wallis Arithmetica, where Wallis says that geometrical ratios

25:00 are all homogeneous to one another because they all belong to the genus of numbers. And Beryl disappears from two counts. First, because numbers in his account, in his conceptualization, numbers are just names and symbols. They do not make up by themselves a genus of quantum. not be thought as proper mathematical objects. They are not quantifiable things, properly speaking. Secondly, ratios are not quantities, and that's an essential one that you discuss in a whole lecture, you put it completely to that. Secondly, because ratios are not of quantities, and merely relations between human beings and quantities. And that's what allowed this thinking of pressures in terms of cost, of relations, of comparison, that's allow Barrow to counter any suggestion that ratios adding themselves in some way or other adding themselves analogous to or can be taken to be one case. Barrow acknowledges that ratios to one another, that some ratios are equal, others are not. We think of ratios that are bigger than others or lesser than others. That's true. They can be compounded, they can be added, they can be subtracted, multiplied, divided. So, he rhetorically asks, are ratios a peculiar kind of quantity. And that's what he meant. Now, let me stop for a moment and

27:30 to stress that there are three issues that are often confused or mixed up when we talk about the status of Russia in the 70s. The first question is, a Russia's plan for some some sort of quantities. This is an annexed question. If they are quantities or some sort of quantities, then what kind of quantities, what sort of quantities. But this does not match, does not match. This is one question. And this is the question that Barrow is now is, where the ratios are the numbers that, or the numerical quantities, that denominate things. That is to say, the second question is, is the same thing as Albert Bay? And the The third question is whether the equality of ratios is legitimately defined by the cross-product of antecedents and consequences, instead of the Euclidean hecumultiple definition. So, the second question would be, can I define proportionality like, according to If you answer that A, B is the same thing as that, then for this fraction, then you have this.

30:00 In number 2, you have A, A over B. In number 2, you have A over B, which is not defined. It's not defined, so I don't know what it is. The ratio of A to B. I don't know that, but to the right, what's that? It's a fraction. Of what? A fraction. But that would mean A is a number. A and the term B. Yeah, but that would mean K is a number if that is a fraction. That's completely right. That's completely right. That's the problem. That's the problem. The problem is that the secondary literature looms all of these together. And they take that this question is the same as that question and is the same as that question. And that's wrong. And that's wrong for the reason that you pointed out. That, strictly speaking, mathematically speaking, generally speaking, that has no sense. I will come back to that. I just wanted to make clear that there are these things. Sorry, the quality in C is equivalent to A over B is equal to B, C over B. The quality in C is what? Is equivalent to A over B equal C over B? Can you write the same using the other notation? A, B? Yes. Well, let's let's let it be that way now. We'll come to the equivalents later on. But it is taken, it is understood, that if you can say so, then you can say that the ratios are the fractions, and that means that ratios are quantities. But those were different questions in the 17th century, of course, to anticipate. this was the main source of this agreement. Here, Barrow said emphatically, no, ratios

32:30 are not quantity, are not some sort of quantity. That question, is the ratio about to be the same thing as the fraction. This question was answered by most mathematicians even when that fraction existed, that question was answered by most mathematicians in the negative. ratios cannot be taken to be its denominator, or the quotient, the fraction of the numbers that the ratio compares. And this was accepted by the other body. When that was possible, when the operation was possible, when those magnitudes would be operated in that way, That was accepted by Barrow, that was accepted by, well, I'll go back to that. And with the three conditions also to... Ah, I will have a second. It is the difference between two and three. The difference between two and three. In Europe. Thank you very much. This... Come on. The answer was, most of the mathematicians was, that's not true. Most mathematicians who are almost every polychaic that I know of accepted that you could do that. Every time that you had for... It's a proportionate in one case of the rest of the other? You could do that, you could deduce that they were proportionate. In one case you conflate the ratio with quantity, and in the other case you give having the same ratio? I'm sorry? The proportion which I read to me is having the same ratio. The proportion is the time of the value to subtract, not the value to ratio.

35:00 How do you read the thing on the left? Before a proportion. Which can be said that we have the same ratio. On the right-hand side, it's not the ratio, it's the pressure. On two. No, I don't think it's two. Well, let me go ahead and we will discuss, I will return to that thing when I finish with Barrow's understanding of ratio. So Barrow poses himself the question, so are ratios a peculiar kind of quantity, are ratios quantities in some sense, some peculiar sort of quantities. And I can start by acknowledging that most, or many modern thinkers, modern mathematicians think that ratios are some sort of quantities, and he specifically mentions he mentions Gregoire de Saint-Densin and Taquet, he mentions Hobbes, he mentions Wallace, he mentions Borelli, Mersenne and Nipponch, Nipponch. But then comes Barrow's argumentation of why these people are wrong. The first thing he does is to acknowledge that, of course, there are ratios and quantities and some quantities related to the terms that the ratios compare, that it exists an important relation there. I mean, the ratios are not completely separated or have something to do with some quantities. But ratio, in Darrow's formulation, ratio only agrees with quantity

37:30 by metonyme, which for those of you that don't remember right now what metonyme means, is a kind of metaphor. So when we say that this lounge, that these are the lungs of the crown, We mean these are the lands of the king. The king has a crown, but it's not the crown, but then we use the term in that sense. So, quantities are to ratios, like the crown is to the king. Quantities are something that belongs to the ratio, but you cannot say that ratios are quantities. And there are several reasons, logical reasons, there are mathematical reasons that she discusses. Do you mean that ratios have quantities without being quantities? Ratios have quantities, but they are not quantities. That's more than saying that it's a metaphor. Perhaps he's talking a little bit, but he doesn't say more than that. He uses that technical word. Quantities belong to ratios by methanol. But if you then say that they are quantities, you... Quantities is one of the, say, one of the facets of ratios. It's one of the properties of ratios. Yeah, so I would be careful of saying they are... But they are not quantities. No, they are not. They have. Also, no, so, okay, but have is also quite a strong word. Well, have is a very strong word. You might say then, okay, where is the quantity? If you have it, you can give it to me. No, they have it. No, no, no, but what I repeat is not the same thing.

40:00 But if they have it, that doesn't mean that you can get it. Yes, but what does it mean that you have quantities? I'm just translating it to you. If you can say that they have quantities, but then what appears in Baro's subsequent discussion those quantities that we have are not uniquely determined. That is to say, well, I was going to talk about that later on, but in the same way that the relation between a number and a line is contingent, in the sense that you can say this line is one or this line is one half, But this is you that decide. There is something similar going on with ratios and quantities. You have ratios and you have quantities than many, but there is no intrinsic connection. Well, I would suggest then if you don't find examples of the sources which you have in this connection, that you wouldn't use it. I would suggest that if you don't find in the sources that the word have is used, then I think it might lead to misunderstanding with some people and so I would avoid it. Well, let me finish. I know there are slippery terms, and so I'm trying to exhaust all the things that Harold says about them, eventually we may have a more clear idea, perhaps not a perfectly clear idea. So, relations are pure, in his words, are pure, perfect relations between concrete things, and this is one of the important oppositions that Darrow brings in to denying that relations are one.

42:30 Ratios are a relation or comparison of quantum, and, says Barrow, a confusing world would arise if the objects compared and the comparison of them were the same kind of objects. So So this is one of the reasons why the ratios, which are comparison of quantities, cannot be quantified according to Barrow. Barrow puts that in the term of the opposition, concrete and abstract. He doesn't define what he takes for granted, what abstract and concrete is, that ratios are a comparison of concrete things, and ratios are abstract things. So you cannot have ratios in the same genus as the things that we have compared, because they are of different categories, one is abstract and the other is concrete. And the same happens with respect to absolute and respective things. Absolute things, things that stand in themselves, like magnitudes or quantities, and the pressures are not. The pressures depend on the absolute that they are relational. Logicians, says Barrow, using the word magicians, teach that relations depend upon absolute things and so you cannot have relations upon relations. Then there is a kind of very abstract argument saying that if ratios are, if ratios would be some kind of quantities different from the quantities that we know, the magnitudes that we know, but they were, for some reason

45:00 they were allowed to be, to make a new genus of quantities, let's call that R1, then the ratios between these magnitudes would be also quantities, and that would make another second or third layer of objects, that would be R2, and the relations of R2 would generate, and says Barrow, it is absurd to have, I quote, infinite kinds of quantity he felt to never dreamt of. Such a liberal and easy implication of beings is deserved to be rejected by philosophers. That's one more reason. And one more reason comes from, let's say, his empiricism. Barrow I add that no quantity of any ratio can immediately be estimated of itself. It occurs not to the sense. It shows not itself by its effects. It is not demonstrated by any certain argument. Therefore, it is supposed and affirmed gratis. Here comes the Barrow's argument that ratios can be compared, and are compared, and can be added, subtracted, etc., but only by way of the objects that they themselves compare. Pressures have no real meaning if they are severed from the things that they relate. Here the argument is similar to what she has said about numbers. Numbers are nothing in themselves. We need to know what is the unit to be able to limit the operations and comparison of numbers. a similar argument to point out, to stress that like number ratios, people themselves

47:30 cannot compare or compare. It is required to assume a permanent unit if we want to say that the ratio, the six dupla, is twice tripla. Otherwise, the unit is not saying that. So, the point is the comparison of ratios is not intrinsic, is not given universally. It depends on finding a common consequence. And that common consequence Barrow allows and discusses that common consequence introduces or allows one to think in terms of the denominator of of the ratio. The denominator is either, if the ratio is between numbers or magnitudes that can be operated numerically, the denominator is the fraction, or the number that comes when that fraction is resolved and you have a number and a unit. That's the denominator. And once this is done, the pressures can be assigned a quantity, and can be given a quantity. And this quantity is nothing else but the quantity of the denominators. And in dealing with, in that context, Barrow shows that if you have, for magnitudes that are proportional dense, And he makes a nice demonstration of that, assuming, obviously assuming, that these things exist, that these things can be, that we know what are those things. So Barrow proves that.

50:00 He makes a nice demonstration saying, well, if that's true, and let's assume that that's then it will be that A plus X will be that, and then he shows easily that you found an eckimultiple that does not fulfill the condition. Of course, assuming that you can write these things, so when you can write these things you can produce that result. And then, to conclude with his discussion, he turns to what he thinks are the most serious critics of Duclid's definition and the two mathematicians that propose more interesting approaches to ratio and proportionality in the 17th century, which were Takei and Borelli, according to Paro. And, well, of course, Paro points out that the problem here, the problem using that, the problem here is to know what does it mean, A divided by B. Because if A and B are inconmensurable, A over B, that only means the ratio of A to B. So we are not moving out of the problem, sorry, if he makes a demonstration, he has no answer to the question, what OVD is, but certainly an answer to the question, how can I operate with OVD, because otherwise he cannot make a demonstration. So what are the operational

52:30 is applied to other people. Fractions. Fractions. Fractions. The general rule of course. Of course, that every mathematician. Perhaps they are not fractions, but they are... And they are dealt with the properties that every mathematician applies to a number. But they are not normal. No, they are not normal. So, the thing is that Barrow then turns to Takei and to Borelli, Takei argues that Euclid's definition of the equality of ratios, the multiple definition, that Loupier's definition does not declare the nature of equal ratios. And Barrow answered that there is no such the nature of a ratio. That's interesting because both Takei and Borelli, they are saying that what they want to know, they want to take for granted that A over B, implicitly assuming that A and B are numerical things, A over B is the division of A over B and we know that is. Their criticism and their proposals to substitute the commutable definitions goes in that direction, although perhaps the words are not exactly that. Let's say what Takeh argues that Euclid's definition does not show the absolute or true equality of ratios. That is to say it is not obvious, according to Takeh, it is not obvious that

55:00 the equimutable definition is connected with a true equality. True equality is Takeh's through equality of reasons. That is to say the equality, the multiple definition is far from showing what are the quantities involved, what are the numbers involved, that the numbers are really the same. And so Takei suggests as definition, I quote, the antecedent of the one ratio contains or is contained in its consequence in the same manner as the antecedent of the other ratio does contain or is contained in its consequence. So the crucial thing is the same manner, and Barrow explicitly attacks or takes up this notion and to argue that there is no manner in which the antecedent contains the consequence. that the ratio cannot be conceived otherwise than as such or else as compared to other ratios through the rigorous definition of the multiples given by Euclid. When you say, do not show the numbers involved, the definition by Euclid multiples doesn't It doesn't show the number involved, which number? Take says it does not involve the nature of the numerical quantities involved. It does not declare the nature of equal ratio. But when we can say that two wretches are equal, what is the nature, according to the case, that the case is interested in, what is the nature of equal wretches? And Carol's answer is that's not a correct question. There is no such thing as the nature of equal wretches. And something similar goes on with Forelli. Forelli also questions that in Duclid's definition, I quote, does not declare the nature of incommensurable proportions.

57:30 that does not declare that thing. What does it mean? It does not declare, I take it to me that does not declare what's that one thing for. Of course, Artaro's answer is there is no such thing as the nature of proportionals. And, of course, he then criticizes that someone who has proposed that the definition of reasons are equal when the intestines divided by the make the questions, and Barrow comes out all these, all the inconveniences that this definition has. So, let us, I would like now to turn to how recently the government has dealt with that, with all of this, with Barrows' views and his opposition to Wallis' views. Perhaps the most recent extended treatment of that problem is found in Joseph's book on the quarrel between Wallis and Hobbes. And there is also a discussion in Paisio's book on the 17th century English Algebra, on English Algebra from the Renaissance to the 18th century. And I think both of them is very clear that these three questions are mixed up, are confused, are messed up. Joseph introduces a distinction that I think was important in the 70s and 80s to deal with medieval pressure.

1:00:00 It was the distinction between what was called the relational theory and the numerical theory of ratios. Yes. According to Joseph, and Sela and Sasaki, and there is quite an agreement, I couldn't say all, that all the mathematicians that have along with the problem say the same thing, but Jezef Silla-Paizio certainly do. According to them, there were in the 17th century a relational theory of ratios and a numerical theory of ratios, and the differences rest in whether ratios are or are not quantities, and by that it is assumed that, I quote, I would just say, every ratio has a size or a denomination, and two ratios are the same if and only if they have the same denomination. Now denomination, I don't know, denomination means that denomination in the 17th century. That's not true in the Middle Ages, the notion There was an interesting article in the 70s by Mike Mahoney suggesting that the medieval notion of denominations had been important in helping to introduce numerical understanding of questions in the 16th and 17th century. So it was claimed that there was some sort of continuity. I don't think that this can be sustained, at least my study of ratios in the 16th century showed that there is no such thing as a way. There is a clearly... In the 17th century there is a clear discontinuity. Barrow defines Barrow and Take and the people that were using denomination

1:02:30 was not making references to the evil theory of denominations. They were justified by denomination of a ratio diffraction, that you could divide, you could take for granted that the two terms compared were numerical and you could divide them. So, one of these, according to this bibliography, one of the advantages, one of the strengths of the numerical theory is to avoid the elaborate Euclidean definition. This is one of the main attractions. So, let me make a long quotation of that view, of that understanding of the problem of ratio in the 17th century. I quote, in the numerical theory, it is natural to assume that the ratio of A to D is equal to the ratio of C to D, just in case A times D equals B times C. Joseph adds that, I quote, such a criterion makes no sense under the relational theory, because A and D may be incapable of being compared directly. Thus, a fully developed numerical theory of ratios must require that all ratios be homogeneous and capable of mutual I think there are all mistakes here, and first, because ratios are necessarily homogeneous in numerical theory, they are numbers, they are all numerical quantities. And secondly, because what you need to take the cross-product role as the definition of equality is that the four terms involved belong to the semi-numerical field. You can operate

1:05:00 them, you can multiply them, and the multiplications can be super-convert together. So, this leaves The problem of distinguishing point 1 and point 2, and point 3, of course, as I said, even Farrow demonstrated that you can use that when that is possible, so there was no such At the third point, it is important to discriminate between a relational theory and a numerical theory of pressure. What's interesting is to look how lollicists' views about pressures are presented. And before that, of course, Joseph's book presents Tarot as a, I quote, a purist, a purist worried about the, I quote, corruption of the traditional relational theory, and to Jesus. It's a very rhetorical, very effective way of putting him as an old-fashioned guy. And then he adds, and I think that that's a distortion of Barrow's discussion, he adds that Barrow was convinced that his defense of the relational theory, of classical theory of racial proportionality, I quote Jess, it was the only philosophically defensible account of racial. And I don't think that that's true. I think Barrow was making a case against the numerical understanding of the four mathematical reasons, so, er, er, derimutors mathematical

1:07:30 Let me turn to the, the, the last point, the comparison between the point one and two. It's very interesting that Joseph quotes Wallis, saying that Wallis, always, I quote Wallis I got just a quote in Wallis. Observe a distinction, Wallis says, I always observe a distinction between ratios and quotients, the quotients, the denominator, the quotients of the terms. And I think Wallis' quotation is quite transparent. Wallis is complaining about how Hobbes has misquoted him, Wallis, and Wallis says, the ratio is to be estimated according to the quotient, or the quotient gives us the measure of the ratio. He says he agrees with that. The ratio, I repeat, the ratio is to be estimated according to the quotient, or the quotient gives us the measure of the ratio. And now Hobbes has distorted his views, Wallace's views, by saying, by turning that, the last two sentences, by turning proportion consists in the quotient. And that's not true. Wallace makes a strong case. He has never said, I quote Wallace, I have never said that ratio is a certain quotient, that it is a number, that it is an absolute quantity. And so the denial of them, there is another

1:10:00 another place where Wallis says, another quotation from Wallis, no, I do not make proportion for ratio, I do not make ratio a quotient or an absolute quantity. That was by Hobbes' inference. I say indeed that ratio depends upon the quotient. It is determined by the by the quotient, is estimated by the quotient, and denominated by the quotient, but not that it is the quotient. So, what is he saying is, I strongly like that. That's not true. Ratios are, depends on the quotient, is determined by the quotient, is estimated by the quotient, is denominated by the quotient, but the ratio is not the quotient. And so it's the, what is his view, and these are repeated over and over again, and so it's It is so clear that Joseph acknowledges that there is a problem here with his turning Wallis into a defender of all these things. And so I quote Joseph now. It may perhaps be true that, strictly speaking, Wallis' works do not contain the explicit formulation open quotations, the ratio of two magnitudes is the quotient arising from the division, end of the first quotations, just as still. But it is clear that Wallis does indeed take ratios to be quantity, da da da da. So Wallis does not take seriously Wallis denial and says this is a verbal doctrine, a verbal trick. But it's not true. And why this trick? Because Wallace, according to Joseph, Wallace, I quote, was not willing to appear to depart too radically

1:12:30 from the authority of beauty and classical geometry, knowing Wallis does make sense. Paisior does something similar, although his emphasis is put here that, according to her, this is the algebraic understanding of proportionality to that triumph in the 17th century, and that implies the other things, that means that the qualities and the non-traditional semi-traditions would have adopted that theory. So, to conclude, I think there is another interesting evidence that we must take into account. Up to the first decades of the 18th century, the notion of proportionality, the Euclidean notion of proportionality and the relation of theory of ratios was widely spread and widely use. To say something, take Bernoulli, the definition of the logarithmic spiral. This is defined just by proportionality, you know, old-fashioned proportionality. Actually, the graphical evidence, I think, is highly conclusive. Only through the 17th century and through the age of the 18th century, you found all mathematicians, including Hobbes, including Paltrey, including all the heroes of the numerical understanding of ratio and proportionality, all of them, they write things like that to compare, things like that to symbolize that

1:15:00 are proportional, and they use a different symbol from fractions and from ratios, and that's true after the first decades of the 18th century. And not only that symbol, but in fact there are many different symbols through the 17th century and the third decades of the 18th century of people using a specific symbolism to make reference to ratios and proportion. So I think that that's a strong indication that the classical understanding of ratios, that is to say, an understanding in which the ratio is not identified to the fraction, I think that was alive and kicking through the first decades of the 18th century. I don't think it is correct to say that secondary literature presents that the 17th century saw the trio of the numerical understanding of proportionality. That's not the case. We have declarations of Wallis and all these people, we have graphic evidence, and I think it must be recognized that there is something too subtle, too complex in the notion of rational proportionality to explain away in the way that it has been. Two comments, two very quick comments to finish. The first one is, if we want to play that, if we want to go to set was at the question, why those questions that to us are so close together, and why they

1:17:30 maintained for so long. Why was it so important, the understanding of ratios and proportionality that couldn't be easily conflated with fractions, while his notion of relation proportionality was so important in the mathematics of the 17th and the first decades of the 18th century, when we know that during the 18th century those notions disappeared, seems to be important in the works of creative mathematicians. I think one of the reasons is, an important reason is that proportionality with the classical understanding of Russia that brings, that has, that is incorporated into it. Proportionality was the only way in which the 17th century mathematics could explain function, what we now call functional relations, inter-functional relationships. You can only compare together things that vary according to the same law by putting it in past proportionalities. And so, I would say that it is when the notion of function emerged in the first decade of the 18th century that the notion of proportionality can become obsolete or can become a more, an instrument of more elementary mathematics. And the second comment is why we have these of seventeenth-century ratios from highly competent historians of mathematics that makes

1:20:00 these things up. So why we have this, why there is this interest in presenting the evolution of creation and proportionality as a consequence of arthematization or algeatization. I think that we are still, in some way, in some way or other, we are still dominated or possessed by that old idea that 17th century mathematics progressed or evolved or became modern thanks to the introduction of a numerical understanding of magnitude, the introduction of algebra. And, well, it's true that 17th century mathematics this new process of argumentization and argumentization. It is one thing to say that there is this process and there is another thing to understand argumentization and argumentization as the driving process of progress of modernization in 70's and 70's. I don't know. I may be wrong in many questions, as I said, but I would like to... Don't worry. No one can be wrong in anything you say. No, it's a paradox of a liar. He said, I am wrong in all the things I say. It's a paradox of a lie. Ok, so we open the discussion. Sabine. I can speak French because it's a bit late, so I'm going to translate it in English. I'm trapped by the similarities between the treatment of Wallis... Excuse me, I was in my English mode. You can't speak French. So I'm frappée by the similitude between the question of the question of knowing if the

1:22:30 rapport is a quantity and the treatment of the Faiblesse de Parme of the same question, so at the end of the XIV and the début of the XVII. It's exactly the same thing. First, I would like to say that there are two definitions of the denominator in the Middle Ages. The definition that we know is the denominator of a rational relation is put in the form N plus P sur Q, so an entier plus a fraction. So here, we specify the denomination with a very particular form, which allows us to give a number in relation with the nomenclature of O.S. And then there is a very general definition which is, for every rapport, whether rationnel or irrationnel, it is not specified, where the denomination is the result of the division of the precedence par le concept, without necessarily expressing that result. This is what makes us think about the fraction of Parme. So, Blaise de Parme, with the idea that in any relation, which is rational or irrational, to associate like that a denomination to a quantité, it's the question of knowing if we can read that the relationship is a quantité. The relationship is not. The relationship is a relationship, so it's the same way to Wallis. And he said the same thing that Wallis, a little bit of stress, it's not the ratio of the quantitudes, but it's not the quantitudes. Blaise de Parme dit que le rapport est comme une quantité grâce à cette domination, mais ce n'est pas la même chose. Je me demandais si, alors ce serait peu probable que Wallis ait connu, enfin Barreau pardon, je dis Wallis au début c'est Barreau, que Barreau ou Wallis ait connu le traité de Blaise de Parme, but did they have known Alvarice Thoma, the Detroit Pilotsman of Alvarice Thoma? Alvarice Thoma, who was known in Angleterre.

1:25:00 I don't know, I don't know what to do with the discussions about this one. I don't know, I don't know what to do with it, but I don't know what to do with it. But I don't know what to do with it. Yes, it was Portuguese. Yes, it was Portuguese. But his father was known in Angleterre, because he was cited by... Jim Bakker, a contemporary Galilee, who worked on the mechanically. Galilee, who worked on the mechanically, in English. He left only a few names. Harriet. Harriet is known. If I hear you, Blessed Pape would remember the idea of Barrow and Wallis. Barrow. It's just Barrow. It's just the same idea that Barrow. I said Wallis at the beginning, but it's Barrow. Ah, d'accord. The fact that we can express it as a quotient, it's Wallis. The fact that we can express it as a quotient, it's Wallis. Barrow doesn't agree with that. Barrow thinks that there are moments where it doesn't work. It's more the position of Wallis that you describe. Ratio has quantity, but they are not quantity for the Moroccans, the quotations. Ratio has quantity, but they are not quantity, but they are not quantity, and the ratios are linked to quantity, but they are not quantity, this is barren. But no, they are linked to quantities and they are not quantities. They are not the denomination, they are not the fraction, they are not the portion.

1:27:30 So, if the rate of Tharou doesn't say that ratios are quantity? No, no, no, no. I was very happy with all the things you told because it's a very important thing and it's It's one where one has to treat very carefully, and you get often in terrible mix-ups so that you can't understand what historians are writing, because this goes all through each other. So that's what I want to say first and I agree with most of what you say and the one thing I want to ask about I think where I don't quite agree may be also a case of misunderstanding or making things a bit more precise it is about how you use the cross and the bar meaning a multiplication and division. Now multiplication and actually the cost is called the times symbol and that means that multiplication one says 6 times 5 for instance. That 6 is a number. That 6 is a number. 6 times. You can say six times a piece of Leinen. You cannot say a piece of Leinen times six. So the only natural definition of multiplication can only be given for natural numbers or if you want rational numbers, but not for any others. The same applies with division, which is the turn. So if you take that seriously, then of course everybody knows that you see people do different. They say line, sequence, times, they do that. That means that in such cases, either there is a new

1:30:00 definition of multiplication and together with that a new definition of division, which is the case which Descartes did and others, or there is a not very well fixed transfer of the terms to non-numerical terms. The latter is quite acceptable in mathematics if it's done with the trick one has for that, namely that one says there is a set of axioms which fit here, and we assume for the rest that these axioms work. The other way to do it is to, when you, for instance, go to the continuum, as we know, in Dedekind, is to use all the tricks of the book from the arithmetrization to extend multiplication to a continuum. And that last distinction, I think, is what mathematicians have mostly in mind when they get into their most careful and rigorous thinking. And so they know the real meaning of multiplication can only be done with numbers as precisely as people in the 17th century also for exactly the same reason would know that for other magnitudes, multiplication can only be used if it has been somehow defined or if you assume that it can be done. I think that also gives the explanation of the question you said, why is there so much misunderstanding?