The Very Soul of Mathematics — Isaac Barrow C17th — Part 2

0:00 It is an alternative presentation of the doctrine of ratio and proportion falls short of the mark set by you. So, well, it's late enough that we can listen to a conclusion. I began by noting the great paradox of Barrow's mathematics, that he venerates the methods and style of the ancients, while at the same time making substantive contributions to the development of the infinitesimal calculus. How, one might well ask, can someone who lavishly praised the rigor and methodological purity of classical geometry, along with its explicitly financietic standpoint, how can such a person go on to embrace infinitesimal methods? And the answer, I think, can be found in Barrow's commitment to geometrical foundationalism. For Barrow, the basic concepts of geometry include space, time, and motion, and in the geometrical lectures, which were published after the mathematical lectures some years, and were, these are where Barrow employs these concepts and in fact uses elements that would become elements of the infinitesimal calculus, and there he develops a kinematic concept of magnitudes that he sees as generated by continuous motions. And significantly, in the comparison of compound motions that Barrow undertakes here, he gets the construction of tangents and the determination of areas, in the same way that Newton, who, as it turns out, was Barrow's most successful student, that Newton would use a substitute later in his work on the calculus. So really, the doctrine of prime and ultimate ratios that you find in Newton, about 80% of it is found in Barrow, and it's founded on this notion of kinematic notion of magnitude being generated. Ratios are determined in a finite case, and then you just sort of take the limiting case and claim knowledge behind the limit. This program, I think, can be glimpsed in Barrow's treatment of ratios, and specifically in his defense of Euclid against Wallace on the one hand, and also Hobbes and Borelli on the other. By rejecting Wallace's contention that algebra is the proper foundation of all mathematics, Barrow defended the relational concept of ratios and made the case that the considerations of space, time, and motion are foundational in all of mathematics. His rejection of Hobbes' materialistic approach showed Barrow's commitment to the view that the abstract truths of geometry

2:30 are certain, independent of the structure of the material world, and grasped by the intellect through demonstrations that confer non-empirical universality. Finally, Barrow's reply to Borelli shows him to have accepted the principle that properly founded demonstrations permit the intellect to reason about infinite cases. A result established in accord with his methodology must, Barrow thought, hold in all cases, including limit instances where ratios of nascent or evanescent magnitudes determine an exact value for a tangent or quadrature construction. So in the end, Barrow's veneration for the ancients was not simply a matter of sheer antiquarianism, but it gave him the conceptual resources and determine ratios in the broader context that became the infinitesimal calculus. The details of that new employment of an old theory of ratios are an intriguing story that I'm going to leave for another day, probably for somebody else to tell. Thank you. Questions, comments, Andre? In Euclid, there is another, I don't remember, definition or not in the seventh book. In seventh, yes, where quotients rear their ugly head. Definition, but also there is, how say, the beginning of 10th book. Kind of spurious theorem which compares. Yeah, I mean, the whole business of compounding ratios, the stuff from six, right? I mean, that is a spurious definition there that makes reference to what appears to be magnitude to ratio. But we have good reason to think that's a post-Humanity and something slid out of the door. I mean, I don't mean to say that the numerical theory sort of was, if Walls was the first person to propose it, that'd be far from it. People have been working on this notion of denomination for ratio. I'm not sure. But the question is, because you mentioned that definition of his book in the discussion of 7th century. Was it really limited to that definition of this whole problem of 7th book? Well, in the case of the 7th book, I mean, Borelli, for example, says, gee, you know, Euclid must have been really confused because he goes back over the whole thing in 7, and the poor man, let's help him out. Let's just give him a new set of definitions. Clavius spent an enormous amount of time working through all of this.

5:00 And, I mean, I think he's the sort of definitive understanding from late 16th century understandings of the theory of ratios. Everyone seems to have signed on to the program that Euclid could not have been in error. He's not retracting something or changing his mind. And the usual presentation goes something like this, The theory of ratios is general enough that it can handle both commensurable and non-commensurable cases. Now, as it happens, there are two sciences. There's arithmetic and there's geometry. They have separate, distinct objects. There's no question of which is more fundamental. You don't need to reduce one to the other and the other to the one, so they'll develop autonomously. As it happens in the case of ratio, we can give definition of ratio and proportion for the arithmetical case that turns out to be a special instance of a broader definition that we could give to cover commensurable and incommensurable cases. We need that second one when doing geometry, because the objects of geometry are infinitely divisible, continuous magnitudes, and can't be thought of as multiple parts, etc. But also, there's a kind of purity that's needed, because the definition of ratio and the way that it's handled in 7 presupposes this idea of a unit, but there's no geometrical unit uh if you think i mean that the principium of the beginning of the rk of geometry is the point perhaps but the point isn't a unit and if you think it is the earth will open up and swallow you and all you know terrible things will happen uh but uh so that the thought is that well if you employ the unit concept then you're really doing arithmetic conceived of as in and that's a work seven project. But the theory of ratios can be seen as indifferent to whether it's applied to the continuous or the commensurable or the incommensurable case. Is that an answer? Yeah, but you don't find that, right? He does say that that definition is general and that particular case. I mean, it's a hard That's important. question that always crops up in doing this sort of business is that, well, what did the actual historical Euclid think? And then what did commentators from, you know, this or that other century take Euclid to have meant? And, you know, that's not obviously the same question, and there can be pretty radical differences. And then you have what are clear innovations,

7:30 and people who really want to get away from the Euclidean program, but with the exception of Hobbes, obligation to pay at least lip service to Euclid. Wallace is a fine example. He goes on about how Euclid's done everything wonderfully. We're going to give him a little hand here and there, but everything is really just fine. And then he walks away from the definitions. And when Hobbes, for example, says he violated a fundamental canon of Euclid. I have not, never, never considered Euclid. So it's all very entwined and entanglement. I think that's the general line about this. just a comment about Newton of course yeah say Wallace is a very beautiful Newton quite explicitly in his lectures arithmetic propounds he says they thought by a number I shall understand it's by a number I shall understand not a multiplicity of units, i.e. the Greek notion of arythmus, but the abstracted ratio of one quantity to another taken for a unit, which is the most transparent statement you could have, but I'm extremely interested in how far this tension between the geometric, the kinetic origin of this program, the geometrical And this point about the way that the notion of megathons in the point being, as you said, that the RK of geometry is precisely that it doesn't which doesn't rest on an arithmetical foundation it seems that what Newton's doing with that statement of his definition of number is circumventing that whole issue just going straight for we shall extend the field of our operations

10:00 that we can make sense of so we can set aside this problem of the the the commensability of the species of quantity right just he simply brackets that whole question yeah i'm just wondering whether he has a purely kind of methodological motive for doing that or whether there's something even deeper going on to do with the way that he thought about living well i mean newton seems to have uh his thoughts have undergone something of an evolution because there There are the later geometry treatises of the draft geometry, who never actually saw the light of day in his lifetime of day yet, where he says, ah, you know, this whole Cartesian thing, what a mistake. The ancients could do it all. The ancient theory of loci could be developed in a way that required merely the drawing of lines. and all of this algebra is really a distraction from the underlying geometric concept. He seems to come back to some idea of generation figure. But you'll notice that what, I mean, now maybe Newton is just really deeply confused, but what he offers is, he says, look, the foundation, I mean, Newton seems to have embraced a pretty strong version of geometric foundationalism, especially later on, because he says that, you know, I don't consider quantities as composed of into small parts, but as generated by continuous magnitude. Absolutely, yes, it's actually a generating figure. And that looks like it's something that he picked up from Barrow. Geometrical lectures. Yeah, very good reason to believe that. And he also, this suspicion that he has of algebraic methods as getting in the way, distracting us from the underlying geometric content. I mean, I think that he is very much on the side of geometric foundationalism. And in fact, the complaint that was raised against geometrical foundationalism was that it was mere mechanics, that by contaminating the first principles of mathematics with notions like space and time and motion and thinking of magnitudes as produced by some kind of process that, you know, some overly physical process, that's not what we mean by mathematics. Mathematics is this entirely pure subject that has no dependence upon the structure or contents of the material world. And algebra offers this

12:30 sort of super-duper science that can cover any possible case, and it's not restricted to continuous motion. Which has a kind of purely equational semantics to use. Exactly. But that goes for Newton. And in fact, if you look, I mean, there's a famous point where Newton says that, you know, it is thought that mechanics is this inexact science, but he says, well, no, God could describe any curve. And, you know, the constraint that we have in geometry is that the curve is described exactly. So the distinction between mere mechanics and genuine geometry is entirely spurious, and we ought to just say that, you know, there is this underlying science of motion and time and whatnot. Curiously, I mean, Hobbes embraced that sort of view, and the fact that Hobbes endorsed the mathematical view was taken as a decisive reputation in whatever view that happened to be. Yeah, but that's obviously all connected with Hobbes' atheism. Yeah, and it's, well, whether or not he's an atheist, but also, you know, 24 failed quadratures in the circle rushes in print, all inconsistent with one another, it tends to undermine one's street cred on a mathematical street. Paula? Yes, thank you very much. a distinction between the use of the term quantitas and the term multitudinal in these cultures because there is a huge problem concerning the interstation before and then it is somehow connected to the way of considering originally as a universality or universalism the short answer is no the longer more interesting answer is that in the case of Barrow most of the time He thinks that quantitas is this very general term, and then magnitudo and multitudo are the two versions of it. But then, as he works out his geometric foundationalism, he thinks that, well, really, multitudo is just magnitudo sliced into equal parts. so there is and of course he was also a professor of Greek so he gives you these long etymological accounts Wallace is completely so far as I can tell

15:00 he uses, depending on context he uses rather different terms I don't see in Matthäus' Universales it's like what the Germans call It isn't, things are kind of jumbled together, and, you know, he is, I mean, Hobbes actually goes after some of these etymological points as well. I mean, he thinks that a lot of the subject is poorly put together. In the case of Borelli, I don't think he is that interested in these sort of questions about what the ultimate foundational interests are. but depending on which author you take you will get I mean I think the standard model is sort of the barrel model there's quantitas which is then the two species and then you get various Greek ways of doing that Megathos Megathos Pelikon and Thorson Thorson is not in the U.S. No, no, it's... Marko, it's your turn. Yes, yes, I cannot resist before making my remarks to say one thing about Newton. I think that we should distinguish two things in the relation between the car and the Newton. The fact that the Newton in a certain moment of his life decided to object algebraic mechanics is a fact that it is certainly not true. But it's also a fact that this notion of general numbers is exactly based on Descartes definition multiplication. Because if you cannot say that a magnitude has a ratio with another magnitude, the result of the division is in the number of five divisions and the definition of division is Descartes 1 so the very essential foundation of Descartes' algebra is not rejected by you

17:30 starting from there, the idea that you can read on the geometrical slide line all the numbers is accepted because Descartes you can't define multiplication or magnitude and so we're doing that in numbers I think I agree almost completely with what you're saying I think it's important to remember that in the predesigned universal I mean what Newton is interested in is this idea of universality and what he points out a perfectly reasonable sort of problem that would arise from the tradition, namely that arithmetic, as traditionally understood, had no place for dividing one continuous magnitude by another. And his solution is to say, well, that goes back to a misunderstanding of what a universal arithmetic would be about, namely, you don't need this idea of collection of units. As he says right at the beginning, in the classical sense. What I mean is an abstract ratio of one quantity to another, with one quantity taken as unit. The objection that's raised is that kind of thing. There are two essential Descartes ideas there, that you cannot give division without both of them. The first one is that you have ratio, division, on magnitude, so you have to define And the second is that the choice of unity is arbitrary. Now, an objection that is raised by some to this is that if the choice of unit is arbitrary, then there's no fact of the matter before we choose the unit, and so there isn't any real, this is Barrow's kind of point, that there's no real arithmetical content to anything before we specify the unit. And so, I mean, his point, take it or leave it, is that that means that there's an arbitrariness in arithmetic that makes it insufficient to ground any genuine science. I complete a dream. My point is, if you look at Wallace's treatment, I mean, Wallace is going entirely the other direction. And Wallace has this, what I think is an almost insane notion, namely that if you frame the

20:00 geometric problem in the right sort of way, any specific geometric content just sort of melts away, and all you're left with are equations to be solved. And this gets back to a theme that we've heard time and time again, that there really does seem to be something fishy about that idea, that these spatial intuitions or concepts figure seem pretty darn important in the theory of geometry. And the idea that, well, once we've got the right sort of specification of the unit and laid down the appropriate equations, that all of that content is simply swept away. That seems insane. So my remark, now, you say, and I completely agree with you, because it's a matter of fact, that for Rawls, it's based on an algebra, and you say that in order to understand what algebra is for Rawls, or algebra that is 93, 93, no, 93 is the initial, 85 is the initial, so the point is that natural in algebra is a clear definition, algebra is an algebra, the algebra is a story about a lot of things, but you don't really find the key definition of what algebra is So the way you put algebraic, the word you say, has no clear. And historically, yes, algebraic all day. So, but, but I think that's enough. That's two remarks at least. So, Sebastian... And I love to begin my remarks. I love to begin my remarks. Okay, that was a long introduction then. Sebastian.

22:30 I think it's a very difficult issue to issue it in terms of such a way of our life. And, in a sense, I don't. To not agree with you on the fact that this is a little strange, and it is very specific to this paragraph. One can say that behind this kind of situation there will be several matters of works like those of Descartes in a sense. For instance, when nobody read the two first pages of natural matrix, because just after what they said was written in the Italian style, so this kind of demarcation between I would say considerations about the philosophy of mathematics, and specifically about the elements and the definitions, and on the other hand, the mathematical practice, they but in the works of different authors and should be seen as raising a problem or something like that. I would say that they are, it's a way they're thinking. And about the fifth, about the fifth definition, in order to end my remarks, I would say that when they're not able to do that, I mean, Barrow and Wallis are talking about that, is it a theory of pressure, or what do you think about pressure, because they can make mathematics, but when Barrow, in Wallis, is doing mathematical practice, and not utopia of mathematics, And here, in fact, I thought that it's hard and it's worth the fact that we have to differentiate these two things. There's no problem with that. There's mainly a problem for us, but I think it's not so heavy. But we have to be shocked by the fact that we have a different position in the mathematical practice as in, I would say, the mathematics. Well, look, there's certainly something in this. There's a theme that runs through a lot of early modern mathematics, which is this veneration for the ancients. And with the notable except in Hobbes, very few people are willing to propose sort of wholesale revisions of classical notions.

25:00 And even someone like Leibniz is saying, you know, forever and ever. They did really great stuff. and nothing that I do really differs from what Archimedes did. Be careful, because you are on the one hand about this, and you're on the other hand about all these and Archimedes. Do not speak in the same way of that when you speak about Archimedes. Well, I was thinking of the ancients as sort of, you know, of all one. But I do think that in the case of Barrow, the split, if you will, between the philosophical theory and the mathematical practice is probably more pathological, are than in many other cases. I mean, Barrow, the defense of Euclid on Raisha, I mean, point by point by point by point by point and insisting in the most lofty, flowery language that the great father and prince of geometry never said anything, that everything he did in this subject was as good as it could possibly be. That seems pretty out there. And the incredibly casual way in which Barrow introduces finitesimal considerations in the geometrical lectures. I mean, it's really pretty mind-boggling that he just, all of a sudden, eh, well, let's divide by zero when we need to and move along. I mean, that really, for someone who has insisted elsewhere that the only way forward in geometry is with rock-solid definitions and, you know, very, very careful logical derivation from that, I mean, the procedure in the geometrical lectures at several points is really quite surprising. And I don't know of anyone else in the period who has quite that extent of, you know, maybe, I don't know. Paul McKinney. So I just wanted to ask you a little bit more about the different theories of ratios in terms of different prospects. So you talked about the natural problem of universal complexity, for numbers that may be the system of what's going on, or even if they say that we have, that's certainly more of the issue of being.

27:30 Another thing that arises with people's definition is the line coming to what's going on, doesn't try it. So for many, for what these, we have to evolve and evolve. You need a big universe, right, to model it, or something like that. Well, I think that what you get is probably not surprising that the two theories are going to have complementary strengths and weaknesses, and how successful they are depends to a great degree on sort of what you think you're entitled to assume and what you regard as standing in need of proof. And people will not, generally speaking, have the same set of agreements about that going in. The relational theory, which is the Euclidean theory, has the great strength that it applies indifferently to commensurable and incommensurable magnitude, the same kind of argument for workers in all cases. It doesn't need, once you accept that kind of machinery, it gets all the work done. And it is true to this classical understanding of a distinction between arithmetic and geometry that it doesn't require you to compare, as it were, lines with circumstances. It doesn't, you know, you can have the same logos or old between two instances of one kind of magnitude and two of another, so you get the analogia, the repetition there, and that's a sort of very lofty and useful way of doing it, and you don't need to raise these difficult questions about, well, what is it to divide, say, the square root of seven by the square root of two, which has no solution in sort of the classical way of thinking about the quotient song. The strength of the numerical conception of ratios is that it gives us this very simple of ratio. One ratio is the same as the other when the quotients work out to be the same, having expanded our definition of quotient, end of story. And it also has this strength,

30:00 at least as Wallace and others saw it, that it doesn't require this infinite comparison of the pairs of ratios, arbitrary equimultiples and preservation of order relations. So on their view, then this epistemological point is much easier to find out whether one ratio is the same as the other. You just compare the quotients. You don't require all of this long, complicated, additional, annoying machinery. Now, whether one is ontologically simpler and committed to less than another is a question I don't really know how to answer since I've never been able to figure out what the ontological commitments in mathematics are in the first ways. And I'm not sure that authors in this period were too deeply concerned with questions about whether space is infinite or not, or whether there are an infinite number of things of a distinct type in the universe, because the thought is that, well, the space of mathematics is by definition infinite since we can, or at least indefinite to the Cartesian sense, there's as much of it as we need, so we're not requiring anything as more ontological than suspicious. Assuming that you can get past any worries you have about abstract objects and if you're worried about, you know, are there circles or are there triangles in a sort of platonic realistic sense, you can accept some version of it. Is that an answer? Yeah. You said at the start, you took this thing, you know, one of the things I was trying to do, and then that gets you brought to a report from some of the new considerations, and then you have complete things, you just have an idea of numbers of the graph. Yeah, I mean you could, that's a road you could take. something rather like that, where he says that there really is no abstract science of quantity, all there is material bodies, and geometry, and all of that. And how that all seems to work out is a very complicated story. Thank you. Yeah. I think what we're talking about is the intellectual quality, the extending of the arguments in these discussions and that of course is always very difficult because it is

32:30 merely the question of who is the right and that's a very that's a dangerous question because then and i raise this point because on one of the conclusions that come to a completely different My idea is that the position which Barrow outlines is actually the best possible at the time, and it is actually the basis of all mathematics in the 18th and in the 19th century to halfway the 19th century. And this thing, which was also mentioned, due to actually being also formulated, namely that the only way to understand the basic thing outside of geometry is the ratio of a magnitude to a unit magnitude. Then you can calculate this. This was the concept of what the basis of analysis was all through the 18th century. There was no problem in Euclidean. And if you then see that this is based on magnitude of one, it is the general concept, which is the concept of Euclidean geometry, which was a standard of rigor, it was about, and which was generally a book of pieces accepted. And if you then take that the antidenticals ratio, which however complicated the Euclidean zero ratio was, was a, understood by that, it was very difficult to understand, you know, it was very difficult to understand the difference in the data. And it was a bell-based theory, and it was also recognized as the kind of glitter which was the best that NASA man could offer. There was no better. Of course, there was other practice when people worked with concepts which were still human and could never reach any level of glitter comparable to that. Especially in the points you mentioned, the decimals have not reached and have not reached only 20 centuries. So it becomes, and that's the basis of why I come to a different conclusion.

35:00 because if you take a very broad view of the spreadings, then we have the Greek period where you have two kinds of things, they determine the language, each with a reference on a mission. Then the next accepted phase, after an intermediate period in the 17th century, which all these discussions were there, comes the continuum with the unit and the ratio as carrying the constant number. And when you see Newton doing this, that he gives up the aristatical number of constant was there, and chooses that. And then becomes, by David King, for example, that began the miracle-based continuum. I think he did quite well. I'm certainly not trying to say that Barrow was completely in error, and I think actually Barrow's approach is quite interesting, but what interests me is the notion of geometric foundationalism, which he and Newton seem to accept that I find a real going program, one that has a great deal of, I mean it has a serious amount to offer. I do find in Barrow, however, the rejection of algebra, the claim that there is nothing to be had in algebra other than this empty formalism from the examples, that seems to miss something quite fundamental. But then, when we talk about, say, the equality of argument, we can have to ask what was the And then people talk about how to commit, you see this as a target for us. But after some time, this concept, which we have, I just gave you a century, and that's quite nice. Yes, I can certainly grant you that. I think what's a little concerning for one who reads Barrow is the thought that, good all the books of algebra, right, and, you know, redoing all of the theory of equations essentially has this purely geometric theory, leaving no room for, I mean, if he took his

37:30 theory very seriously, I mean, as pointed out, however, the practice is rather different. I mean, if you look at Barrow's actual mathematical practice, he's quite happy to employ algebraic methods here and there, depending on the demands of the moment. Now the question that then arises is how seriously does Barrow take the philosophical claims, the sort of epistemological, methodological, metaphysical claims that he's making on behalf of geometry against arithmetic and algebra, how seriously can we take that in the face of his seeming willingness to use what we could, with some shorthand called Cartesian there's no I think there's no a priori answer to that but it is I mean the point you make is I think is valuable I wouldn't want to be thought to be saying that one or the other of these people has completely you know missed the boat and they're all working on you know interesting alternatives. I think Wallace, I mean, the quality of Wallace's argumentation is not always up to the standard of Barrow, but I mean, he makes a point, which is that, you know, we need to confine our reasoning to things that appeal merely to intuition, to sort of spatial intuition, that there's a lot to be said for a quite general way of treating ratios, for example, or treating geometric magnitudes algebraically. And the to have a guarantee that every algebraic manipulation that you employ in the course of reasoning can, in principle at least, be reconstructed or reconstrued as an admissible coherent geometric operation. I'm sorry to say this, guys, but we have to be out of the building in less than five minutes. That brings us to an abrupt end. Thank you very much, Doug. Oh, yes, the meetings will be here tomorrow. So don't go to the other place, please. We're going to have a guide tomorrow at 12 o'clock. Yeah, there will be. I'll meet you at 9 o'clock. And tonight, everybody is welcome to come to dinner.

40:00 Okay, those who are staying in a part city, please meet me in the lobby of the hotel at 7.30. And for those who are not going to go back to the hotel, the dinner is at the Excelsior, which is about a block from the train station. Just ask somebody.