Curves in Proofs / Hume on Space & Geometry — Part 1

0:00 Euclid elements, not in the neurons. not that he denies that there are neurons but after all he's trying to say there's some a priori basis for empirical perception so he naturally looks to a priori formal geometrical structures what the philosopher is trying to do is give a description of the a priori formal precondition looking at the brain you can tell anything about a priori formal No, I'm not going to develop too long, but the reason to look at the brain or not is the type of experiment. You have one type of experiment which is called trite perception, and another type of experiment which is by the word sensory impression, that you can detect results of sight and dimensional... Well, that's not space in Huster. I mean, by space, again, he kind of means physical space. He means physical space. He doesn't just mean any kind of spatial representation or spatial sense. He means perception of physical objects and physical statements. Well, I don't think it relates to what is written as not in content, but I think it probably relates to the kind of state or angle or perspective in which we see the perception. You said at the beginning that you somehow wanted to react to minders and fathers, followers, and what's going on in the museum is the form of interpreting Kant. Right, right. That I wouldn't want to interpret Kant. In particular, I follow Marco Ponson's very rich interpretation of Euclid, which I find very convincing as an interpretation of Euclid in the Greek mathematical context. What I'm saying is that if you don't think that Euclid can use that as an interpretation of Kant, because he's in a different context, a different intellectual context, with other

2:30 concerns and other agendas. So, in particular, Kant is not going to want to say that we start with concrete, sensible particulars that may have stress from it. I do understand that, Mark, and I particularly agree that there is a risk in saying that by doing certain things, one of these trying to interpret or give an interpretation of Kant, precisely I think it's a very difficult idea for us to speak on to all these transcendental stuff that you have and all these very abstract ways of looking at things. It's difficult? Well, you'd have to practice. I find it quite a thing that I've been doing this for 35 years. I mean, I think that the certain climate must be such that many people wouldn't find it confusing, either because they would go, especially because they would go straight away empirically. But you don't want to do that, you told them. And I was very happy to hear that. I think that that is just this idea of... So, one thing I want to say in order to somehow let... I mean, to make the conflict between what you're saying, Mandy, let's start, is that Russ is not offering any interpretation of Kant. Criticizing either Manders or Markov for giving an interpretation of Kant. That's not my point. I just wanted to contrast what he says with what they say in order to, I think, understand what Kant is saying better. I think it's a contrast that's useful for understanding what Kant is doing. I don't mean to criticize them for what they're doing, which is interpreting Euclid. Or we can threaten Euclid. Not at all. I find them very rich and interesting, especially Markov. I want to get him on my side before the question. I know this isn't going to work, but I'm trying to. Another thing which he may sometimes one might find difficult to understand is really what this rich notion to make people come downstairs, really, like the pure productive imagination, it's tough

5:00 to take. So since I've been practicing this conceptual scheme or framework for 30 years, can you say that's new something? Yeah, here it is. Here it is. Wait a second. It makes it possible for me not to collapse the pure productive imagination into this visual thinking. Because if we do that, then I think that all the actuality that we want to do that goes along. Okay, good. Excellent. So, two moves I mentioned in that respect. So, what I guess isn't adequate, because this could all just be visual thinking, you know, in anybody's sense, right? So, that's not enough. That's the pure productive imagination. So, my first move is to say, think of these constructions as like what we call Scholem functions. That is, you start with this general mathematical concept, say circle, or in this case, black sector. And then, what Kant wants to do is connect the general concept, a priori with intuition, in such a way that, and of course, it's in intuition that all the particular instances of the concept occur, particular figures. But there should be some kind of a priori connection between those two in order, for example, that geometrical reasoning with what looks like particular figures can be as general as pi, right? And also, sorry, as general as you please. And also not to reduce, because as you were suggesting, if it really is universal and necessary, then the modern view wants to say, oh, well, then it's just purely conceptual and analytic. So my first move is to say, no, no, we can actually find something for Kant here, right? First of all, it's not purely conceptual are not sufficient. Because the logic of these concepts is too weak. So, and what you need then is something like, you know, what we take to be universal existential quantification. Well, you don't have that, but you then have something, you have something like a Scholar function for those things, right? So, the function of the Scholar function is as it were purely logical from our point of view, right?

7:30 is not logical from Kant's point of view simply because what he means by logic is something much weaker than what we have that has no quantifier dependence and no polyadic logic in it. So, so far nothing about vision, actually. Nothing about spatial perception. But at least you understand from that point of view, I hope, why it's pure and a priori, but why it's still not analytic given what he means by analytic. It's not concept containment in the sense of subject bracket logic. It's logic, but non a priori. It's your point. It's logic in your conception of logic, because your conception of logic... No, no. It's logic in our conception of logic, but from the concept point of view, it's synthetic. But from your point of view, it's synthetic or aprior? Well, from my point of view, we first want to understand... So, is that okay? So that's my first move. But that move, which is kind of the way I started working in Kant, is not satisfactory because it actually has nothing to do with perception. Because you can think of these Skolen functions as just algebraic symbols, and you can represent this whole thing, as I think some of Hindicic followers have done. You just have some functions that you write down, function signs, and then you iterate them, and you substitute, and you actually don't need to draw anything anywhere. but then I say but Kant wants to this is the second part of it the second point for Kant it's crucial because of this Newtonian background and this modern science background to have it be the case that geometry is the basis for all of our cognition of sensible physical reality which of course starts with our perception of objects in space then applies mathematics to them geometry to them builds models, and so on. So Kant wants to establish an a priori link between spatial perception and the application of geometry to anything you perceive in space. Well, to do that, and I tried to say he does that, by connecting these Scolan functions, now thought as real Euclidean things, orbits of rotations and translations, not just functions on it. They really are these group theoretical things.

10:00 Something like that. And furthermore, those things are connected to the Euclidean construction. And so, therefore we can understand how Kant is able to say whatever pure geometry says, in pure intuition, necessarily holds for everything outside of us. Newton wanted, but without this idea of God's consortium, which is no good theologically. So, he has Leibniz and Newton as his background, Leibniz, you know, several times. First, Leibniz's conception of logic, Leibniz's conception of the intellect, then the debate between Newton and Leibniz about space. And he's trying to find a way, given that whole intellectual context, to understand geometry as a sensible activity, meaning it takes place in intuition rather than in the understanding, and something that makes perception possible rather than something that's abstracted from perception. We can't forget Martha. Thank you very, very much. You did definitely convince me that the diagrammatic account of Ken and me and others of Euclidean geometry is not Kantian, so I will hidden the parentheses in the quotation you made. You only have to change the paper again. No, I don't think so. I think I simply have to hit the . It's not very much. But moreover, you convinced me that because of that Kant is wrong. And even you convinced me why it is wrong. I want to say in his account Euclidean junk. You made a very good... So I would like to make two points. One concerning Kant, one concerning you. Kant first. You made a very good summary of your point now in Answers to Analyza. First, Scholem function, and second, space. Scholem function, I agree completely, we can reduce Scholem function to purely algebraic tools. The point is that if Scholem function

12:30 are reduced to poor algebraic points, some inferential necessities in the argument of Euclid are not disanswered, so we they are not enough. We can do that, but they are not enough to support the inferences, so we would need scalar function plus axioms, and we have no axioms, so we have diagrams in place. So this is the first Concerning the first part of your paper, the point where I think that Kant's account is wrong. Second, more important, I think that in order to account for what you say in the second part of the paper, in fact, there is something more deep here. because, in fact, Kant's explanation of universality is something connected with necessity. Something is universal because it is necessary. And this explanation of necessity is intrinsically connected with your exigency of accounting for the necessity of the legitimacy of Newtonian science, with something completely extrinsic to Euclidean geometry. So it seems to me, you can have Euclidean geometry without Newtonian science, perhaps not the vice-versa, not the vice-versa, certainly not, but certainly you can have Euclidean geometry without Newtonian science, for a long time, in fact. So, it seems to me that it's exactly this link between the account of universality and the account of necessity of Euclidean geometry that is somewhere wrong. We can account from universality of Euclidean geometry without accounting from its necessity. And I'm not sure if I'm able to do, but I think that we could, we should be able to do that. So this is the point concerning Kant. Now, the point concerning you is much shorter. Concerning me. Concerning you in your reading of Kant. You say something that for a lot of time had completely convinced me,

15:00 and was already the very starting point of my reflection at this point. Kant's logic is too poor and in fact it's only monadic logic and non-polyadic logic we have discussed a lot of that so I was convinced for a long time of that now I have a little doubt about that so I would like to put a question in this way I'm not sure that it's the criticism it's simply something that I see now that I did not see before Kant's logic Aristotelian logic is certainly not polyadic. But in Greek science, so, there is place for polyadic statements. For example, we take the theorem of infinity of prime numbers in Euclid. It's clear before any prime number exists something. So, in some sense, even if there is no polyadic logic, there is the idea that we can use and we can appeal to connected, quantificated statements. So, why not this possibility is, according to you, used in order to account for continuity, for example or for the feature geometrical objects that are essential in order to make in my point of view the real lack here is more than of the elements on which the quantificator applies the points of the element in a set because there is no idea that continuity is a property of something as a set or a collection of one So more than the quantification, seem to me now, but perhaps it's something very new in my mind, so perhaps I don't see it very well. It seems to me that more than the quantification is the element of which the quantification applies that are lacking. Okay. So with regard to the first point, were you thinking about this business about order, and that here, for example, to see that DCB is greater than ACB, you have to just first perceive that this angle is a part of this figure. Yeah. Okay. And so the Skollum functions by themselves, you need the common notions, and furthermore, you need the ability to see that one angle is a part of another one figure.

17:30 More generally, one figure after you see is a part of another figure. For example, yes. For example. Yes, that's true, and that's important. Kant, to my knowledge, doesn't ever talk or doesn't say anything very much about it. He focuses on a distance, I think. I don't think that's incompatible with having this role, too. He doesn't say much about it, and I confess I haven't thought much about what he does say or should say or could say about it. So that's just something that I haven't really thought enough about, of how that would fit into a constant view. So that's a point that's well taken. There's something essential about, and here it looks like some kind of inspection of a figure. I'm not sure, though, because you might argue that, well, look, it just follows by construction that this thing is bigger than that, because this is an extension, you know, by construction. B is a construction of B, A, and so the figure B, D, C contains the figure A, B, C. I mean, it doesn't seem to me, because you actually see, not only in this figure, but you see as it were, or you know as it were, because this is an extension, that in any such case, this figure contains that figure. But I don't know, I haven't thought about this, probably you've thought about it more. I don't think that it's more. I don't know. So that's an important point. Again, I think Kant, but the skull and function business about existence is not sufficient to take count of all the perceptual, perception-mediated inferences that you are interested in and you put in geometry. And that's an important point, of course. And historically, this is important because this is where the Hilbert and Posh said theory of order, we have to acclimatize it instead. Okay. So, second point. That has to do with the necessity of Euclidean geometry, and in particular, that Newtonian space is necessarily Euclidean, and that's not, as it were, that's not part of the universality and necessity of Euclidean geometry when it was first, you know, developed. Well, nevertheless, I mean, Euclidean geometry was, I think, I mean, correct me if I'm wrong, and traditionally thought to be, not just, you know, a discipline that is generated by

20:00 paper, it was thought to be a correct description of space, no? That space was the object of this theory. Or space, I'm not sure, or spatial objects, certainly. Okay, spatial objects, then, three-dimensional spatial objects. And it was thought that in describing spatial objects, at least in many cases, we can subject them to mathematics by subjecting them to Euclidean geometry. So there's at least still the idea of a link between spatial objects, objects perceived in space, and Euclidean geometry. It's not just that there's, as it were, a well-regulated practice of Blackboard discussion. It was also thought that that well-regulated practice has something to do with space. Okay. Well, but, as I said, in the 17th century, you know, that view was inflated to say, look, geometry, Euclidean geometry, is the basis for our knowledge of the physical world. In 17th century, certainly, yes. So everyone went crazy about that, right? But that's important, and Kant is one of those people, right? And Newton was one of those people. And Leibniz and Descartes and virtually everybody thought that. So it is an important and powerful view. That's the view that Kant... So it's very natural for Kant to link the necessity of Euclidean geometry on these two grounds to this new idea that, you know, we can come to physics and to empirical science presupposing geometry and go from there. It's natural, but drunk. Natural, but drunk. Well, we now know that. It seems to me that, I mean, I'm a little uncomfortable. After all, I mean, that's an happenings. No, it's both a joke. I mean, it took a long time to know why that's wrong, and if we had to... Ah, yeah, no, it's both a joke. it's not just some trivial matter that this is wrong okay so um there was a polyadic logic oh yeah that's a very interesting point um certainly you're right of course there is uses of polyadic quantification in mathematics right of course before polyadic quantification theory is invented

22:30 by Frege and Schroeder and so on. Let's say Frege, right? That's, you know, we're now in the late 19th century. You know, you've got all this reasoning with limits and so on, and you have Cauchy distinguishing uniform and point-wise continuity, which is a complicated polyadic formula, and reasoning, you know, pretty successfully with it, although there was a mistake there. So, that is obviously and certainly true, but Kant's point is about explicit articulations of what logic is, and what logical form is, and what that has to do with characterizing the intellect as opposed to the senses. There he's operating against this Leibnizian background, and he's just pointing out that of course he himself has every event has a cause he uses polyetic logic of course but the point is that what we officially and explicitly mean by logic has no clear place for it of course we reason polyetically but from Kant's point of view that shows that logic as explicitly and traditionally understood weak to be the engine of that reasoning. So among other things, we need intuition. I know there are more questions, but it's been a long day. And of course, Marcel has some. Robin has a question. Why don't we just quickly have the three main questions but I'll try to give you a quick answer. And then we'll have more champagne. We have champagne, and we'll have it the sooner you think it's a quick question and quick answer. So my question has to do with the existence of geometrical objects. And if I understood your talk correctly, Kant's point of view was that geometrical objects are continually constructed by means of the Euclidean constructions . And I would think that that was also Euclid's point of view because, for example, before he, it's 147 of the evaluated proposition, he has to show how to construct the squid. So my question is, take something like a regular heptagon, which cannot be constructed with the ruler compass. Would Kant believe it exists or not?

25:00 Good question. That's a very good question. I don't know if Kant's talking about the regular heptagon, but he does talk about conics. He, just following tradition, distinguishes, what is the distinguishing, I mean, construction from... Well, comics can be constructed as many points as... Point-wise. Point-wise as you can't draw the whole area. But he, of course, just following tradition, he takes them to be constructed solitude from the comics. And so Kant seems to embrace that traditional distinction. So, from Kant, I think Kant would agree that Kant... There's some reason to think that Kant would agree that conics are essentially different from lines and circles in terms of constructability. So Kant does not narrowly confine the existence of methodical objects to strange and practical structures. He doesn't have a systematic account, not surprisingly, I don't think he follows Descartes and just has a kind of algebraic generalization, because he also has the idea that curves can be generated by motion and he has Newtonian ideas and he can embrace transcendental quote unquote construction. He hardly says anything much about those, though. But I think a charitable view would be he would kind of embrace anything that Newton would call a constructive curve. He would take that to a good. But one footnote to all of this is doesn't think any geometrical objects exist because the only objects that exist for him are actually physical objects. And these, he says, this is just the mere form of an object. So when I construct a triangle in Kant's technical sense of object and existence, for Kant actually there are no mathematical objects. There are only physical objects. So what we construct are what he called the form of appearances, that is the forms of what the construction guarantees somehow is that we can subject all of the physical objects to what mathematics says about this quote-unquote object it is about the same project about the the poverty of the human humanity and

27:30 To our view of this, you use quotation line and you stress in the infinite columns, comparing I would agree with Marco, I think, and there is another aspect of this text is the difference between two, I would say, logical forms, the fact that representation, well, that the concept contains representation under itself and about space, that the state contains its parts within itself. And if we take this, I think that if we take, I don't know, predicate logic, and Coreatic predicate logic. The logical form is not a concept, it is a kind of generalization of the certain term. So, okay, the point is can you delve and predicate logic, give account of the relation between the parts of space and space. I would say no. And so I think that what Kant says here can, is also true if you say, if we don't say only And in a way, I think that you say that in your lecture when you say that if we use current or function and so on, it is not enough to get the link between geometry and

30:00 sensible representation so that it means that we like we need something my ambition no it's very good I think I just did for just a tiny quote right I mean I think these notions of and you can see in Kant's logic within itself and under itself actually have a clear meaning just in this traditional, intentional logic, right, I mean, human contains within itself the concept rational and animal as part of its intention. They are parts of this concept, they are partial concepts, okay? No, I think, you know, we said that to be short, so I think we should try to keep it short because otherwise we are going to blow up the problem. Okay, maybe we'll just talk over Shepin. I'm sorry. There will be nothing to be sorry to for me. So, thanks again for this splendid talk. I go for the champagne. Well, as I said before, I think we should thank all those, thanks to whom that was possible. Marco first. Thank you so much for having made it possible. What about you? And thank Kareem for her wonderful work. Thank you. So, if you have a glass, perhaps you kill it, and otherwise we are going to throw everything away which is... I'm there, I'm not thinking about it. No, no. Under this, the relation of rational to human as a member of the potential, it's a part of the human being. But human also falls under the relationship of rational. Human falls under rational. Rational is a part of human being.

32:30 It's a partial concept, not the whole population. This whole population applies to concepts. This is the difference of how the difference... ...forces an extra diagram property which represents another situation feature. Okay, so this is the situation, and what we really want to know in general, I mean, what would make inferring that the situation with these features has that feature is simply that the combination of these features entails Galatman. In other words, you couldn't have all 1 to k without k plus 1 as well. Okay, so now that's what we've got to look at here. So our situation features are these. And we want to know whether when you've got these two situation features, you've got to have this one as well. Excuse me, what do you say are the diagram properties in those cases? The diagram properties, in this case, it will be the third one that your graphical circle, your perceptual circle, appear to intersect. Okay. In this case, it will just be that your figure appears to be a circle with center A and radius AB. Okay. So nothing very complicated. So perceptual statements in all cases. Absolutely, absolutely. The diagram properties is at the perceptual level, the perceptual-graphical level, okay? Okay, now here is my argument. Okay, now I'm going to, what I'm going to do is I'm going to say in this particular case, yes, There is this entailment. In other words, it's not the case that circles would share a radius compared to intersect.

35:00 Okay, here I've got a little argument for it. Each circle passes through the other center. These are the circles that we're talking about now. The circles are instances of the geometrical concept of circle mentioned earlier. This is very important here. And what this means, if you think back to that concept, is that the circles have the geometrical properties that perfect looking perceptual circles appear to have another premise here different arguments to perfect looking perceptual circles each passing through the other center appear to intersect I can actually argue for this but I'm not because unless you press me therefore the questions I've got an argument now kind of justified the influence No, the conclusion is about the geometrical one, yeah, yeah, good, good question, yeah. So yeah, so I'm going to kind of leave it there, I mean, yeah, I'm going to leave it there for the, I just want to speak very quickly about the intermediate value theorem. Okay, now look, I don't claim that that case is conclusive, but I think there's a strong There's a strong case there for the claim that the inference from the diagram is valid. But now, let's turn to the intermediate value theorem. Okay, this is just reminding you what it is. Well, actually, this is the intermediate zero theorem. I've shifted because Jim, James Robert Brown, shifts to this when he starts talking about

37:30 the intermediate zero theorem and then he says, well, it's the same for the intermediate value theorem. And of course, you can. You know, once you've got this, it's trivial to get the intermediate value theorem out of it. So, so I'm just going to flip back and forth between intermediate zero and intermediate value theorem. Just to show you that I'm not getting Jim wrong, he says in his book, pictures can prove theorems. The intermediate zero theorem diagram is his first example, and this is what he says. He's quoting word for word here. We have a continuous line running from below to above the x-axis. Clearly, it must cross that axis in doing so. Using the picture alone, we can be certain of this result, if we can be certain of anything. Of course we cannot be certain. Right. Okay. Now, okay. But look, you know, I overheard a conversation between Robin and Peter Neumann earlier in the day, which kind of tended in this direction, let's see, it was kind of obvious. But, so, of course, nobody talked about certainty, but people do say it's obvious. Now, what's going on here when you think you can get it out of the diagram? I think what's going on, um, is that, um, one kind of, um, runs analytic continuity, epsilon-delta continuity, together with graphical continuity, what, which, uh, Jim calls, um, pencil continuity. Okay? Continuité du crayon, du crayon, yeah. So, because pencil, that word, is shorter than graphical and I wanted to get it on the screen, I'm going to follow his usage. Now this is how I think that he's going to do his reasoning. And actually, I put something in print to this effect, and he doesn't do that.

40:00 So, any function F, that's epsilon-delta continuous on closed interval AB, with F of A less than zero less of F of B, has a pencil-continuous curve from below the x-axis to above. And this is what we get from the diagram. any pencil continuous curve from the x-axis to, from below the x-axis to above crosses the x-axis. And then three, any function whose curve crosses the x-axis has a zero value. So, valid argument, any x1-delta continuous function, F goes goes into AB, the half of A less than zero, less than B, as a zero value. Okay, this is how I think you agree now. I'm not going to, I'm just going to grant two, okay? Although you might think, how can we get this generalization after a single diagram? Interesting question. I'm going to talk about this one, okay? But I'm going to talk about one and three, all right? Now one, premise one, I'm going to factor it into two parts. I'm going to read it as any function f epsilon delta continuous on a closed integral from, with f of a less than zero, less than f of b, has a tensile continuous curve. And, yeah, okay, this is, I'm going to, I'm going to read it, need to, any function And if f of a is less than zero, less than f of b, that curve runs from below the x. That's a good book. And I want you to look at one here, Roman 1 here, this part of it, where you're connecting, You're linking an analytic condition with a graphical or perceptual condition. And this is what I want to concentrate on. I'm going to dispute it. In fact, I'm going to say that there are counter examples to it.

42:30 Here is epsilon-delta continuity without pencil continuity. Now, the domain of this function is the set of rationals in the closed interval 0, 3. Right now, we're in rational space again. Epsilon delta, G is an epsilon delta continuous function here. It's this function that I've defined here, right? I mean, epsilon-delta continuity doesn't, isn't restricted to real spaces, okay? It's an analytic notion. It has more general applications, okay? And we can apply it here. And I like this example because it shows quite dramatically that epsilon-delta continuity doesn't mean the same thing as delta continuity. So this is my first map, you could say, well, look, this is all very well, but we're interested in functions on the real, okay? So what about counter examples with functions on the reals? Well, in my view, any function defined on the closed interval that is epsilon-delta continuous but nowhere differentiable does not have a visualizable curve. I know some people disagree with me about this. About what, sorry? If you've got a continuous but nowhere differentiable function, I claim that this does not have a visualizable curve. It doesn't really have a curve. Okay. Okay. I mean, the approximations to it have a curve. I mean, if you think of the functions that you get, that you iterate in the process,

45:00 the limit is the continuous no way different function, you can get a function that has a code. So it's going to have an intent for continuous code, do you think that has a visual output code? Right, right, exactly, exactly. So, I mean, that's another, yeah. I mean, but I would make the strongest claim. Okay, no, no, I don't want to discuss that. Okay. So, wait a minute. I'm missing something. Yeah. So that's my reason for, or those are my reasons for not accepting the premise one. Now I want to go on to premise three. Okay? Any function whose curve crosses the x-axis has a zero value, right? Now, crosses the x-axis, by that I mean, you know, I really am, can I say x-axis? I mean the line representing the x-axis. I'm sorry, the graphical line representing the x-axis. And the curve here is a visual thing, okay, a perceptible curve we're talking about, okay. And crossing, too, is a perceptible thing, okay. We can see whether it crosses it. I mean, here, that's the relevant notion. Okay, so, again, we're linking, we're associating an analytic condition, function having zero value with graph conditions, curve crossing the x axis. Okay, now, now first off, all I'm asking you to do is to look at the diagonal in close interval from minus one to one, where you take the point at zero, and you, and you define it so that it's moved up, but by an invisibly small distance. So, if your unit, if the distance representing your unit here is one centimeter, take epsilon to be 10 to the minus 100 million.

47:30 I mean, something that gets your distance well left from the flight length. Okay, it is not going to be able to see any difference in the curves representing them. Okay, this is continuous, but nonetheless it shows you that three is wrong. Okay, so you can't just equate these two. But now, I think, I'd like you to consider this situation. I suppose he replaces the argument with the premise three with the three star here. Any epsilon delta continuous function cross of the x-axis at the zero value. Even this, I think, has counter-examples. And so, what I'm going to do is again exploit the rational space here. So, I'm defining h on the rationals in the close interval 0, 2. And this is it. Okay, h of x just is x squared minus 2. Okay. This function is epsilon-delta continuous on the rationals, but it has no zero value. I claim curve crosses the axis. Now, crossing here is a visual thing, okay? It crosses . Now, why say that this is the curve? I don't say, look, I'm cheating here. I'm using the curve that I would draw if it was x minus x squared in the reals. Well, I am using it, it is the very same curve, right? Curve here is a graphical thing. And suppose you start with the function in the reals. What is the curve going to look like if you subtract all the irrational points? It's going to look the same because, I mean, it's going to look, you know, if you, gaps will not appear. Why not?

50:00 Because the rational, as you know, are dense in the real, in the sense that every real is a limit point of rational. So here, I think, is a dramatic case where you've got a curve crossing the x-axis, the graphical conditions, but it has no zero value. Okay. And this shows that you can't go from the graphical to the analytic in the way that you want. Also, I think it shows, really, that we're relying essentially on the difference between the reals and the rationalists. And what is that difference? The difference is that the reals satisfy dedicating completeness. So it seems to me that Balzheimer is right. You've got to get this result from dedicating completeness. You can't bypass it. So here's another, what am I doing with this? Right, this, this is again the function we've sorted before. Let's see, what am I doing with this, this stage? Let's go back to this one. Yeah, yeah, now what Jim says about dedicating completeness. Got five minutes? Okay. Okay. Now, what Jim says is, look, Dedekind completeness is less evident than the intermediate value theorem. The intermediate value theorem is obvious, Dedekind completeness is not obvious. Well, for the reasons that I've given, I don't accept his view that the intermediate value theorem is obvious in the way he wants. Okay, his view is that because you can derive the intermediate value theorem

52:30 from Dedekind completeness, Dedekind completeness gets evidential support from the intermediate value theorem, which is kind of empirical induction. And he cites Gödel and Russell, both of whom suggest at certain points in their writing that even when an axiom lacks intrinsic credibility, it can get support from more intuitive things that are derivable from it, but I think this is completely wrong. It doesn't work. First of all, in order to work, you've got to know that the things you derive from it are true independently, all right, and I argue that Jim is wrong to think that we know the intermediate value of zero and true independency, authentic incompleteness. And also, it does rely on this very contentious methodology in mathematics that Russell and Gerdel at one point in Devoisian gave expression to. Perhaps, you know, I would concede this much to Jim. You might say, look, suppose you put yourself back in the time of Dedekind, Bayer, Strauss, Cantor, and all those people who are formulating accounts of the real numbers. Then you might have felt, well, look, I want my account to come out so that the intermediate value theorem is true. Okay, so desideratam, you can say it's something that we would like to be true in our account of the real. So it is a desideratam. But that's a completely different thing from saying that it's something we know to be true independently. It's something that you have to get out of your real, and you're going to do it as we So let me just, okay, let me just draw these remarks to close. Now what I'm saying in the analysis case, these analytic concepts, I'm not saying there's

55:00 no relation at all. I mean Jim at one point, look, it's absurd to think there's no relation at all between conceptual concepts, the graphical concepts here, and the analytic concepts. I'm not saying there's no interesting important relation, but I am saying that the relation is not of a kind which enables you to justify reading things, reading intersection points off the diagram. That's what I'm saying. Whereas, if we go back to the geometrical concepts, the geometrical concepts that we were operating with are intimately linked with corresponding perceptual concepts, and the relation between them is such, when you look at it closely, to justify reading off the intersection point from a diagram. There are other considerations here too, but the take-home message is the difference in the kind of concepts that we get to analysis as compared to the kind of concepts that we're operating in pre-analytic geometry is what counts and crucial. It makes the difference. I think I've spoken enough. Thank you. I'm sorry, I forgot about this. Three, four, five. I think I'm going to take, abuse my role as chairman and start by asking a question. And that is that you're attempting to convince us whether whether or not it's justifiable to draw a conclusion from the diagram. And behind that, I sense the expectation that there's only one geometry and there should be an answer. But my point is that the answer depends on your hypothesis. And you've cleverly hidden your hypotheses under perceptual and human notions which are imprecise and therefore I think the question can't be answered.

57:30 It's true that perceptual concepts are vague, and that's the imprecision that you're referring to. I mean, that may be true, I mean, maybe I could be more explicit about my hypotheses. But I thought you had a stronger point, which is that they couldn't be mainstream. Is that right? I don't know if you could. Maybe I could, okay. But the answer can go either way, depending on which your hypotheses are. And as you pointed out in both cases, if you assume the rational numbers in geometry there's no intersection. And on the other hand, if you assume real numbers with that small and delta continuity, then the value is given here. Yeah, I mean, the way I look at it is we've got to feel our way to what is the appropriate space for Euclidean geometry. We try and look at their practice and I think, I mean, I kind of gestured towards this when I was discussing whether we should take seriously the possibility that space has Russian points as a base for Euclidean geometry. But what does that have to do with perceptual concepts? You can't perceptually distinguish a circle and irrational. No, it doesn't. It doesn't. But I mean, these are two different considerations that Robin was throwing up for me. One is the imprecision of the graphical concepts, the perceptual concepts, or the vagueness of them. Okay, and the other is that you could be assuming a rational space or you could be assuming a real space. I mean, I don't really see what the, how to put it together quite. Okay, we'll pursue that later, but I should also call everybody else that wants to talk.

1:00:00 Okay, yes. So the difference between the two cases in the linear values theorem and the Ukrainian diagram, the Ukrainian theorem, is basically based on what you call diagram discipline. In the first case, in the case of geometry, you were talking about at some point you said we don't start from definitions, we are perceptually, from perceptually given figures, and then how, and I guess that relates to your, what do you call that, discipline, that you can perfect your perceptual figures, or the drawing of your perceptual figures. Now, what is that perspective there? In relation to what? If you want to start from definition, what does it mean that perceptually you can try the circle more and more perfect. Perfect in relation to what? What is the standard? The standard here, I mean what we're talking about? I say that the geomotical field holds properties in relation to that. A perfect looking circle appears to have. Yeah, okay, this is a question. What is a perfect looking circle? It's a circle such that any actual improvement on it falls below the powers of visual acuity to the text. So, when we get to that stage where any improvements in the circle are indistinguishable to us perceptually, we've got a perfect look at the circle. Okay? But you are building in the national perceptual, I guess we go back to this, some exactness that cannot be achieved in an explanation for what? Improvement in what respect? In what respect?

1:02:30 I mean, you are not going to somehow be in accordance to those properties to find. You don't mean being more and more such that every kind of detradition. Right, so what question? No, no. Being more such that what? I think that we start off with a perceptual comparison as well, a relation, a perceptual relation. So not only can we apply our perceptual concept of a circle to the room of a cup, but we can also make judgments that one circle is a better circle than another. This is a perceptual thing, it's not defined, I mean, it's something that... Can you have a steady distance where, you know, perfect or not? But that's absolutely true, you can... Are they to say more perfect in this thing? Yeah, I mean, you can certainly have perceptual circles, one of which, both of which have unequal radii, on one where the inequalities are more gross than another. And that may cause you to make this judgement that the one circle is better than the other. This is a perceptual judgement thing. Um, I'll try to read the federal of, uh, Okay. But if there's something that's so far, it's really so far from the time that comes in the book, what it seems to me is that you're putting aside a strong class, because what he's saying correctly is that there's a way of being a realist about mathematics that knows realist about the future. And what he says in his way is that pictures are either the windows that will happen or also there are kind of stuff that A to B, I made it, how are you?

1:05:00 So in the end there are like tools. So what does he need to give a consideration of these tools? So, what I mean is that I don't think that this is leading up so strongly for people in this country. And so, I don't know if you have a point. Well, yeah, of course, countries aren't for a tool, you know, that is a bit hard. But you see, I think what's going on here is that he's making certain assumptions about the permissibility of moving from a graphical property to an analytic property and the other way around, from an analytic property to a graphical property. So my counter-examples were to these assumptions. So it's really the link between the graphical and the analytic that I'm calling into question. It therefore makes sense to ask whether the extension they occupy, a geometrical sphere of a given finite diameter, is very invisible. Quotation 7, quote, the system of physical points, which is another medium, is too absurd to need a reputation. A real extension, such as the physical point is supposed to be, can never exist without parts, different from each other. And whatever objects are different, they are distinguishable and separable by the imagination. This is also a semi-ological process of distinguishing in the separation of shapes. I introduce this because some people have interpreted all what he's saying as simply applied geometry, and say, well, okay, he's talking about applied geometry, he's really thinking about physical atoms, and so that makes sense of the text, but thinking of it that way, but he has said explicitly that that's not what he's really talking about, pure geometry. this. Sensible minima, however, are literally

1:07:30 un-extended for Hume, and they therefore occupy no geometrical space at all. Yet minima cannot be conceived on the model of geometrical points in the mathematician's sense either. And here is where he, you know, his sense of what geometries do is different from what he takes to be pure For this, the mathematician's sense of points, they are entirely imperceptible, are non-entities, as he puts it, incapable of adding up to any real perceptible extension. So these are ideal entities for him that wouldn't be ever presented as geometrical figures or diagrams. Quotation 8. The system of mathematical points is absurd, and that system is absurd because, by the way, absurd for him means that you can, if you continue, if you extract the consequences, you might reach a contradiction. Sorry? It's a technical sense. It's a technical sense, right. Some students say, oh, absurd, you know, it's not absurd, it sounds good. Okay, the system of mathematical points is absurd, and that system is absurd because a mathematical point is a non-entity, non-entity meaning, meaning it's not perceivable, it's not ever given as an immature, and consequently can never by its conjunction with others form a real extension, a real again, perceptible. We can clarify the sense in which all sense of minima then have the same size, fixed size for Hume by considering his conception, again something else, but his conception of the different degrees of certainty and exactitude in the science of geometry and arithmetic. As I suggested at the beginning, the traditional view in early modern mathematics took geometry, of continuous quantity to have its own standard of congruence or equality that is both independent from and more exact than any corresponding standard supplied by algebra in arithmetic, the sciences of discrete quantity.

1:10:00 Exact results about continuous magnitudes are established by geometrical demonstrations of equalities of proportion. In the case of incommensual geometrical magnitude, such as the size in the diagonal of the square, which cannot be compared by adding a final number of equal component magnitudes representing units, to obtain both original magnitudes, we can only approximate the ratio of their magnitudes arithmetically by a never-ending sequence of numbers, as in the decimal expansion of the curve 2. So, yet, Hume turns this conception completely on its head. All quantity for Hume is ultimately discrete, resulting from the addition of some number of discrete indivisible units. The object of the science of geometry in space phenomenologically appears continuous as a result of the confounding of the indivisible sensible minima, out of which geometrical extension is ultimately composed. The un-extended simple minima composing extension, just like arithmetical units, are indivisible and consist of no smaller magnitude, but unlike ideal mathematical points they are perceptible whereas geometrical units traditionally conceived such as a foot or a meter are arbitrary and can always be divided into further smaller geometrical units Hume's sensible minima are ultimate units divisible in principle in no smaller parts whatsoever Hume is thus working with arithmetical units, with ones, which always add up to finite whole numbers. That's the model he does. Ones that always add up to finite whole numbers. But that's at the basis of the continuum, and what we perceive as continuum confounded, that there is this finite number of units. In the entire appearance, however, they are phenomenologically presented as confused or confounded with one another, and we can attain an image of any single minimum only by phenomenologically isolating it from the original homogeneous extended whole.

1:12:30 Because we cannot simultaneously arrive at all of the ultimate minima composing our experience of extension, or even of two adjacent minima, determining the total finite number of minima comprising a given call of extension is impossible. First, you know, he seems to suggest that it's unknowable, but I think in the end, he makes a stronger claim that it's even indeterminate. So it's impossible in the sense that it's indeterminate. Hume devotes a large portion of section 4 of part 2 of the treatise to answering the objection that demonstrations in geometry prove the infinite divisibility of space. The outcome of this discussion is that geometry, unlike arithmetic, is not a perfectly exact science, because demonstrations in geometry are not perfectly exact. Quotation 9. Quote, none of these demonstrations can have sufficient weight to establish such a principle as this of infinite divisibility, and that because with regard to such minute objects, they are not properly demonstrations, being built on ideas which are not exact and maxims which are not When geometry decides anything concerning the proportions of quantity, we ought not to look for the utmost precision and exactness. None of its proof extends so far. It takes the dimensions and proportions of figures justly but roughly, and with some liberty. Its errors are never considerable, nor would it err at all, did it not aspire to such an absolute perfection. It's a very platonic position. In a way, paradoxically. I mean, it's just that he says, don't try to attend. Exactly. And of course. This is because geometers are skeptical. Yes, yes, yes. Fully skeptical about that. This is because geometers could only attain such an absolute perfection, an ideal exactness concerning proofs of dimensions and proportions of figures,

1:15:00 if the exact number of minima in each figure could be known, but as we have just seen, this is impossible. Proof of equality based on congruence failed for the same reason. When placing one figure upon the other, the supposition that we can determine whether all their parts correspond to and touch one another is fictitious, for we do not have a distinct notion of all the minute parts of the figures compared. We could only know that congruence exactly obtains if we knew the exact number of minima in each of the objects compared, where we cannot have a separate and distinct awareness of each of the minima composed in extension. Indeed, the exact final number of minima in a given whole of extension is completely indeterminate, not simply unknown. For the notion of a minimum is defined by a phenomenological process of successive diminution or division, which, as I pointed out above, only arrives at one single minima at a time. By the way, this process of successive division or division cannot be extended potentially. It doesn't have the potential infinity either. Phenomenologically, whatever you say potential you will arrive at is not given, hasn't been given, and you have to reach the final point at which you are, and you can go on, but that's another final point, and so on. Similarly, there is no precise boundary between curves and straight lines. Quotation 10, nothing is more apparent to the senses than the distinction between a curve and a right line. But however easily we may form this idea, it is impossible to produce any definition of them which will fix the precise boundary between them. When we draw lines upon paper or any continuous surface, there is a certain order by which the lines run along from one point to another, that they may produce the entire impression of a curve or right line. But this order is perfectly unknown, and nothing is observed but the united appearance." End of quote. As a result, we have no perfectly exact notion of a state line, quotation 11, the original

1:17:30 standard of a right line is in reality nothing but a certain general appearance, and it is evident right lines may be made to concur with each other and yet correspond to this the standard, though corrected by all the means, either practicable or imaginable." End of quote. Euclid gives an idealized definition of a state or right line in terms of the order of its points. Marcus was questioning whether that was used, but I can't hear Hume is using that definition of the order of the points, right? According to definition 4 of book 1 of the elements, a straight line is a line which lies evenly with the points on its south. For Hume, however, this standard, whatever other difficulties it may have, is simply not applicable. The order of the points is completely unknown because, as we have seen, we cannot reach all the minima but simultaneously we cannot reach the point, what we would call the point. We are left therefore with the in principle vague and exact general appearance of a straight line as an image of the senses or imagination, where Hume points out in quotation 11, which there is no reason at all that two different straight lines in this sense may not coincide in a common segment. Just as in diagrammatic interpretations of Euclid elements, Hume thinks that we must begin from the sensory appearance of particular lines and curves drawn upon paper or on a continued surface, as Hume puts it in quotation 10. drawn up on paper or any continuous surface. For Hume, however, there is no room for any intellectual process of idealizing these appearances or abstracting from them, right? So as to arrive at the supposed perfectly exact object of geometry, there is no path to there. We are stuck with the chronological problem. One might therefore suppose that Hume has nothing interesting to say

1:20:00 about Euclid's elements at all, as a theoretical system at the beginning of Book 1, Part 3, Hume summarizes the results of his previous discussion of geometry of Part 2 of the treatise by remarking, quotation 12, I have always already observed that geometry, or the art by which we fix the proportions of figures, though it much excels both in universality and exactness, the loose judgments of the senses and imagination, yet never obtains a perfect position and exactness. But how, for Hume can the science of geometry be much more universal and exact than the loose judgments of the vulgar, as he calls it, or common sense. They can achieve relative generality and exactness, but not the perfect precision and exactness for which is supposed to be held in her esteem. Hume addresses this question in the second following paragraph immediately after the introduction of his standard of perfect or exact equality, that is, of the one-to-one correspondence, and the resulting contrast with algebra and arithmetic. And this is in quotation 13 that will talk about this important a point of how geometry could be more universal and exact than the least judgment of the value of common sense. I will not read the quotation 13, but I will summarize. Geometry, as a demonstrative science, starts from a small number of original and fundamental principles, actions, from which it is then possible to demonstrate all the other propositions, the theorems, the fundamental principles, quote, depend on the simplest and least deceitful appearances. And of course, they have relative simplicity. It's likely that he was not thinking about the fifth postulate. He was not thinking about the parallel postulate, but the others.

1:22:30 He's thinking about the simplest and least deceitful appearances. degree of intuitive certainty of which geometry is capable, which, however, as I pointed out before, always falls short of perfect exactitude. Demonstrations consist in a number of intuitive steps such that the intuitive certainty of the premises is transferred to their consequences. In the case of geometry, in particular, the intuitive certainty of the simplest and least is therefore transferred to much more complicated geometrical proposition, such as Hume's example concerning the sum of the angles in the Chilean one. These more complex propositions are themselves incapable of intuitive phenomenological presentation. Hume's example is a special case of a well-known I don't know, as you know, colorary of proposition 32 of one of these elements, the sum of the angles is the angles equal to the angles, due to proper. The result in general is a much greater exactness in the comparison of objects or ideas than what our eye or imagination alone is able to attain. So we use these devices that appear to be very simple and intuitive and are given, phenomenologically given, to let us get results that are much more complex and not easily imagined or not given. But, and here there is a gap, right, uncertainty there, created by that. The function of the demonstrative science of geometry, as opposed to our cruder estimations and vision in common life, is, quote, to run us up to such appearances as, by reason of their simplicity, cannot lead us into any considerable error. That is the simplicity of the axiom. Four paragraphs earlier, Hume considers the sense in which we can have intuitive or demonstrative

1:25:00 knowledge of equality in quantity or number. In the case of geometry, he points out that we must confine ourselves to very limited portions of extension, which are comprehended in an instance that you can take in one simple apprehension. As to equality or any exact proportion, we can only guess at it from a single consideration except in very short numbers or very limited portions of the extension which are comprehended in an instant. And we perceive an impossibility of falling into any considerable error. In all other cases, we must settle the proportions with some liberty of perceiving more artificial manner. So you have to introduce some, stipulate, the use of names, here's some kind of nominalistic view too, the use of names can help you along, to help, which is very compatible with the and there is this view, the nominalist view, so I'll use the name, or stipulate certain conventions to help you along to continue with this artificial. Hume has here not yet distinguished between arithmetic and geometry, and he proceeds to do so in the following three-paragraph, where he reminds us that geometry never attains a perfect precision in acceptance, that algebra and arithmetic are the only sciences in which we can carry on a chain of reasoning of any degree of intricacy and yet preserve a perfect exactness and certainty. And finally that it is the nature and use of scientific geometry to run up to such appearances as by vision of their simplicity cannot lead us into any considerable error. As we have seen Hume begins this discussion earlier by remarking that quote art by which we fix the proportion of figures much excels both in the universality and exactness of the census. It is clear, therefore, that the more artificial manner for settling proportion Hume refers to in the immediately preceding sentence in quotation 14 is no other than the demonstrative

1:27:30 We are now in a position to appreciate the sense in which the demonstrative science of geometry, despite its unavoidably lesser degree of exactness and certainty in comparison with algebra and arithmetic, is nevertheless much more exact and certain than the loser judgment We begin with relatively limited regions of extension, neither too large nor too small, in which the immediate appearances in an instance or at one view present us with the simplest and least deceitful possible intuitive apprehensions of geometrical figures. This includes such cases as the proposition that we cannot draw more than one right line between two given points, where although errors are certainly still possible because of the necessary and exactitude of all geometrical appearances, our mistake can never be of any consequence. What we perceive with full intuitive certainty, therefore, is only the impossibility of falling into any considerable error. And the function of we have seen is to transfer the relatively high degree of certainty of the fundamental principles to all the other abstruse consequences. So, but in what sense is the proposition that we cannot draw more than one right line between two given points still susceptible to very small or inconsiderable errors, and how do such very small errors depend on the limited regions of extension that we can apprehend at one view. Earlier, in Part II, Section IV of the Treatise, Hume challenges the supposed geometrical standard of perfect exactness in connection with this very example. Quotation 15, I would fail to ask any mathematician what infallible assurance he has, not only

1:30:00 of the more intricate and obscure propositions of his science, but of the most vulgar and obvious principles. How can he prove to me, for instance, that two right lines cannot have one common segment? or that it's impossible to draw more than one right line between any two points should he tell me that these opinions are obviously absurd and repugnant to our clear ideas I would answer that I do not deny we are two right lines inclined upon each other with a sensible angle sensible means perceptible angle But it is absurd to imagine them to have a common segment. But supposing these two lines to approach at the rate of an inch in 20 leaves, I perceive no absurdity in ascertain that upon their contact they become one. That's sort of far, far far. And of course, Hume's point is that two straight lines that approach one another very slowly, over a very considerable distance, make an angle with one another where they intersect that is extremely small. When the angle in question is so small as to be insensible, that is, imperceptible, the appearance of the lines in a region very close to their intersection is phenomenologically indistinguishable, that is, we cannot perceive it as distinguish or distinct from that of a single line. In this region, the two lines have a common segment in the intuitive appearance, contrary to postulate one of Euclid, right, in that area. So, this example also shows in what sense such errors can never be considerable, so long as the region in question is limited. For so long as we are considering lines that are distinct in the appearance within a relatively small region capable of being apprehended on one view, then no sensible violation, there appearance in such regions which are comprehended in an instance with thereby perceiving an

1:32:30 impossibility of falling into any considerable error. Recall that I just quoted before that Hume remarks that the science of geometry much excels again both in universality and exactness They lose judgment of the sense of the imagination, yet never attains a perfect precision and exactness. And now he continues, and this is in quotation 16, I think I have it in there. Yeah, I have read the first part in another quotation. Now I continue, this is in the continuation, also in quotation 16.