Hume on Space & Geometry — Part 2 / Kant on Geometry & Spatial Intuition

0:00 ...plans can have a common segment, but if we consider these ideas, we shall find that they force is sensible, that is perceptible, inclination of the two lines, and that where the angle they form is extremely small, we have no standard of a right line so precise as to assure us of the truth of these propositions. End of quote. The case of the extremely small angle is analogous to the case of the ink spot in the following sense. When we distance ourselves successively from the ink spot, it eventually appears as a divisible minimum, and thus no longer as extended, at the threshold of perception immediately before it vanishes. Similarly, if we begin from a small sensible angle formed by two intersecting perceptible lines, and And we successively acquire visual images of the lines gradually approximating one another. Then we are eventually acquiring visual images where the lines appear to coincide with one another. We have now gone below the perceptible threshold of the original sensible angle, right? However, no minimum sensible angles continue to appear and then vanish in turn as the lines continue apparently to coincide with one another along every larger segment. For this, it would be nice to write PowerPoint. The perceptible lines are like a pair of scissors, right, gradually closing around the original point of intersection, so then the angles, what was before an angle disappeared, so at the threshold, before it disappeared, there is a minimum, right? So in the following paragraph, Hume contrasts the case of geometry with that of algebra and arithmetic. Again, let's read this in a different context, quotation 17, quote, there remains therefore

2:30 as the only sciences in which we can carry on a chain of reasoning to any degree of intricacy and yet preserve a perfect exactness and certainty. We are possessed of a precise standard by which we can judge of the quality and proportion of numbers, and according as they correspond or not to that standard, we determine the relations without any possibility of error. When two numbers are so combined that the one has always a unit answering to every unit we pronounce them equal. It is for want of such a standard of equality and extension that geometry can scarcely esteem a perfect and infallible science." End of quote. If we could actually determine the exact number of invisible sensible minima composing a given whole of extension, geometry would be just as exact as arithmetic, but unfortunately, we cannot. In the early modern mathematical tradition, as I have suggested, there are two notions of quantity, discrete and continuous, each with its own notion of size. A discrete magnitude is a collection or aggregate of discrete, indivisible units, and the size of such a magnitude is simply the whole number of units in the collection. A continuous magnitude is extended in space and or time, and its size or magnitude is determined in a column with a theory of ratios or proportion of root 5 of Euclid's elements. A central result of this theory is the existence of incommensurable magnitudes, such as the size and diagonal of a square, where there is no common unit by which both can be measured. The best we can do arithmetically, therefore, is to successfully approximate the magnitude of one in terms of the other. For Hume, by contrast, the only notion of exact size that makes phenomenological sense is that of discrete magnitude. We can phenomenologically determine that a sensible minimum is phenomenologically indelisible and therefore comes as discrete units in the traditional arithmetical sense. This priority Hume accords to arithmetic allows us to specify the sense in which all sensible minima have the same size as I pointed out before. They all have the same size precisely because they all count equally as units in the arithmetical sense. Thus, if an infinite

5:00 number of these units were to be added together, we would have an infinite sum of units, precisely as Hume's main argument against infinite divisibility, deriving from Zeno's metrical paradox of of extension contends. We were supposed to start with the final segment, right? We had this number of minima, which we call size, and that is an infinite extension. Nevertheless, no whole of extension compounded out of such minima can be precisely measured by determining an exact final number of reversible units. For Hume, the traditional conception of continuous quantity with its distinctive geometrical standard of exactitude is literally impossible, for it rests on the absurd supposition of infinite divisibility and also on the absurd supposition that we could have in that admiration for Hume constitutes this extension. Hume's geometry therefore contains really no mathematical blunder. On the contrary, Hume displays remarkable insight into the mathematical situation of his time by showing that a consistent version of empiricism and being strongly weathered to a phenomenological model of relying on images and diagrams, this consistent version of a principle of phenomenological reliance on given images implies that the traditional priority of geometry over arithmetic must be reversed. Certainty and precision in mathematics can only be achieved completely, if at all, in the science of arithmetic, in algebra. There is no exact standard of equality or congruence for continuous magnitude. According to Hume's Radically and Prince's model, phenomenologically presented continuous tension is compounded from discrete and divisible units whose sum, if we could attain it, but impossibly, would give its exact magnitude. Yet, because of the confounding of these units in any given phenomenological field, the sum

7:30 in question is not only unknown but completely indeterminate, only the size of discrete number can obtain the ideal of complete exactness, which geometry can only successfully approximate but never actually offset. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Welcome for questions. Well, this is very naive since I'm not a philosopher, but having heard your explanation of Coombe, I think he's just all wrong. He doesn't understand mathematics. Well, perhaps what he says may apply to the real world, that you can't measure a distance the notion of an abstract line, and in his statement about surfaces and lines being equal, he doesn't understand the equality of continuous quantity. Right, because, well, he is just that he's either in... If you are going to rely on the phenomenologically given image, and taking Euclid as having as a central part of the presentation, the diurnal, Then if you are consistent with what that phenomenology is given, then you cannot attain the idealizations or the exactitude of pure geometry. So I don't think he, he just said, we, there is no such thing as perfect certainty and exactness in pure Germany. And if we are consistent with a certain view, a certain piece of view of... Well, I guess that's his point of view, I simply disagree. Well, no, but what you may disagree is with the part that we have to be weathered to this, these phenomenological given images that's the part of the reception yeah but he said if we are whether if we are to be stuck with these phenomenological presentations with these phenomenological given images then it follows that we so much it cannot be precise but another point is i think it's a conditional he says that he says that the foundations that you put in geometry are based on perception and he says that's imprecise but i would say that the perception

10:00 gives rise to statements which we take as axioms which we then regard as precise and i'll notice we build the imagery right but he he he he's kind of refusing to take the step of idealizing of of abstracting from the appearances these abstract objects or these more precise And he says, well, we have all these definitions, we have these demonstrations, but if they don't really, there's a gap there, that's why he would say there is this platonic view, but he's skeptical about possibly obtaining, and so he doesn't call that mathematics, the is more precise than what the vulgar or the common sense, you know, but if he says, but I think the interesting point is, is that he says, but he rescues arithmetic and algebra, in other words, that if you want really precision, even consistent with my model, that is consistent with rhythm weather to the phenomenologically given, I can make sense by this standard, the so-called Hume's Principle, one-to-one, And it's some kind of finite. He doesn't have infinite... I don't think he has an infinite... That's a very curious reversal of the earlier historical point. Cardano, for example, he makes algebraic statements but he always has to prove them by geometry. He doesn't... He doesn't make a geometrical discussion to prove them. That's what he reverses, yeah. Very, very nice talk a very nice philosophical position. I did not know the point of view of Jung, but I would like to say three questions, but before one on Robin's observation, it seems to me that the things are not so simple, because I guess, no, I do not know Jung enough in order to claim that, The question is also concerned with the representative power of geometry. It seems to me that behind that should be something like the idea that the only way in order to say that the geometry as a representative power deals with quantity or it is, is avoid this sort of abstraction.

12:30 the idea that geometry is something that is simply an idealization. If it is idealization, it's a risk to lose the representative power. So if we want that geometry to speak about matters, we need to establish a link. and to establish this link is not so easy to say, okay, there is abstraction we need something more and so we need it I think that the thing and he's a skeptic on the possibility if there is exactness there is no representative if there is representative, there is not exactness it seems to me that I see I don't know, I'm not sure but I guess there is this problem here you are saying he's raising the right question the skepticism. Exactly. So, but my three points, three, no, three sonical questions. One is that, and I like very, very much this, I think that you have understood, this quotation six, the point that becomes indivisible. But, of course, how beautiful it is, it is circular, because it supports that time is indivisible. Because the moment in which, before it becomes indivisible, the moment before it vanished, the major impression was perfectly indivisible. But the moment is indivisible of time. It means that there is a moment in which... So, my question is, what about indivisibility of time? Well, okay. Well, he thinks that time is also composed of minimums. So at the end... Because the idea he has of time is that time depends on the perception of any perception of objects, images. So you only get the idea of time as parasitic on having perceived a sequence of changes. So, and the change is not, he thinks that, but this change appears as impressions and

15:00 ideas. And then this connects with his idea of the self. The self is a collection of ideas and impressions that are separate and distinct. So there is this jump from one idea to the next. Now there is an appearance of continuity of time, that there is no such thing. It's ultimately these inputs of ideas and impressions, which are separate and distinct, I think it causes, and they come and go, and, yeah. The other two questions I can give, I can put them before, because they're the ideas. So there are minimum time. Okay. The other two questions I've put in next. Before which we don't have any perception or sensation, right? I mean, below which. Thank you. There is a mathematician, a philosopher, a written philosopher and mathematician in the 17th century that had the same point of view, not really the same point of view, but the same conclusion about the purity of arithmetic on geometry, which for a reason, that is wallet. Is there some connection with that quotation, mention of a wallet and the other question connected to that? When you are speaking about arithmetic, a number, is it speaking of rational or real numbers? Is some sense in which it is speaking of real numbers and if yes, what real numbers are if they are not rooted in journalism for a number? No room for real numbers. No room for real numbers. No. And Wallis, yes. I think there is, in his library, recently was looking at what books were in the library of. Because the conclusion is exactly the same. The argument is absolutely not the same, but the conclusion is exactly the same. The argument is different. Right. The argument as well. Yeah. I think, yeah, I have to look into that. It's a very interesting question. Is Wallace skeptical of continuous quantity? Wallace says that I only say in order to explain quantity is to represent it in arithmetical terms. That there is a foundational purity of arithmetic under geometry. And also because of generality, et cetera, et cetera. So the argument is more a mathematical one. The conclusion is final decision. David is much more expert than your part, but it seems to me that you agree on that.

17:30 Another third question? I'd like to put you in the position of you. I'd like to make some kind of comments and ask you to say what you think Hume would have said in response. Uh-huh. Say it a little bit louder, yes. Okay, sorry, I think that Hume, and not just Hume, but also Barclay and Mell, the common empiricists in the state, taking the objects, the geometry, lines, points, circumference lines, I mean, this is very, very clear in Berkeley and even in the quotation that you have, you know, you talked about These are graphical ideas, but there's an alternative, which is that you can think of, and I think this does, I get this from reading the translations of the ancient Greek text. First of all, you have these solids, then you have surfaces of solids, and then you have the edges of the surfaces, and then you have the meeting places of the edges. And now if you think of these edges, if you think of the cube now, the edge is not aligned with thickness and a meeting point of two edges isn't a kind of little dot. So it seems as though, it seems to me as though what's happened here is that all these empiricists, starting with Barclay, they substitute these graphical things, a graphical dot, a graphical line, and so on, for the geometrical things, which are edges and meeting points of edges, meeting places of edges.

20:00 Okay. So this is one place where I think you usually go wrong. Let me see if I understand what the context will make it. For them, for the empiricists, each image is a fully determined particular. Fully determined. So in order to be fully determined, it is a token, right? So, it has to be this particular size, this particular length, this particular, as determined as visually, let's say, can be given only, right? So, so your contract is between that and some more, a type of, or identification? looking, we'll get to idealization in a second, but suppose you're looking now at a perfect looking cube, right? You can see by the difference in the shape of the slide, that there's an edge there. And if the cube is oriented in the right way, you can see three edges. And So the location, there's a location at which they meet. Now the location at which they meet doesn't need to have extension. The idea is that in order to be perceptible, there has to be some... Yeah, you've got these three... There has to be some thickness to be perceptible. Well, that's just a mistake. Because just imagine if you've got from a very touchy, which represents the faces, the three faces of a cube that you can see at the same time. By the way, one thing that Hume doesn't deny, I'm not sure about Barclay, Barclay will go with these two. They don't deny that when you see, when there are clear contrasts, so to speak, between this and that, this being bigger than that. There is no reason to doubt

22:30 that your judgment, you have an exact judgment insofar as you are making contrast with considerable differences. The problem is when you get to the points which are essential for mathematics where you require perfect exactness of the platonic senses you are talking about. So what you're talking about, an edge could be that they admit that, you know, of all appearances, it appears, it might appear. Yeah, but it doesn't mean that in relation to the background and so on, you have sufficient contrast to think that there is some, that you can talk about an edge there. But that doesn't mean, that doesn't secure the demonstration, that the demonstration based on the assumption that that particular image is exact, are going to be exact. It's just what you can tell as far as you can tell phenomenologically. Okay, well, okay, this brings me to the second thing. Okay. Which is really hard, back to Robin's point. which is there's just absolutely no reason why we should construe geometry to be about the phenomenology of experience. And I think it was a great advance when Kant said let's not think about the objects of geometry is being perceptible objects that focus our attention on space and regions of space. And so the real subject matter is that we're talking about these regions of space. So now you might be interested independently in the geometry of phenomenological experience, but that's not what including geometry is about. But the question is, how to make the bridge between phenomenologically given appearances

25:00 that are inexact and imperfect? And here is where humanity is skeptical about making the bridge between that and geometry as the science of geometry. How do you get the idealization, how do you get the generality, the sufficient generality with the only given image and so on. It's a great question. It's not a question, but it's about what the... It's kind of louder. It's not a question, it's about your objection to the idea that life can be a I put one face of a thing on one . I think the problem is that in yogic philosophy, any idea might have an empirical So, if you have the idea of the face of the cube, you must have incretions. So, if you have the idea of the edge of the face, there is an incretion at the original. And as the impressions are not exact, when you put one face on the other, you don't have an exact point to the floor, I mean, yeah. But I thought he thought that the impression was exact. I thought he thought that the impression of sensation was exact. It's not the problem that the line that you obtain is thick, but anyway, you don't have one line because of the edge of your... This is an empirical question, right? I mean, I don't accept your assumption. It's okay, that's energy, yeah, okay. It is in fact that my intelligence in that can't appear exactly.

27:30 But there's also a difference in appearing with that than being with it. Absolutely. Yeah. Isn't that enough to make the problem? Well, not for me, because we're talking about how you get the idea, and you can get the idea of experiencing that from its appearance. So you can say, well look, maybe I can consider an object which has the properties that this one appears to have. We're just thinking about how we get the idea now. So in certain that we accept the imperative to you about the origin of our view, which I don't accept the assumption that we couldn't get in a quite naturalistic way, Um, just to come to get, um, and, uh, um, Thank you. Thank you. little change in dinner plans because of the change in time of this session so it's at eight o'clock now instead of 7 30. and i will copy the final diagram

30:00 for all those who want to come to the restaurant and have a team What's the name and address? I'm sorry? No considerable errors there. Topologically correct. We shall assess the cognitive... What's the name and address? I will copy it properly. Okay, good. We will actually get information about where this is to be. So I'm very pleased to have our final speaker, Michael Friedman, talk about geometry and spatial intuition. Thank you very much, Saul. And let me again thank Marco for organizing this incredible conference and for everyone who said anything. It's a pleasure to be here in Paris for it. I just wish I could stay longer. Next time. So this, again, is a history of philosophy talk, so there are many, many quotations from Kant here. Does everyone have the handouts? So, in a way, this naturally follows the Hume thought, because Kant also gives an important role to spatial perception in geometry, and even more important consideration than Hume does of Euclid. is very important to Kant, although Kant interprets Euclid in his own way, and of course Kant, although he does give his spatial perception his role that he's not human's radical empiricism, with lots of room for idealization and rationalist elements. Kant's philosophy of geometry can only be properly understood against the background of two more general features of his philosophical position. His fundamental dichotomy between the two basic cognitive faculties of the mind, sensibility, or receptivity, he calls it, and understanding, which is an active faculty, so I'll indicate it that way, and also Kant's distinctive theory of space as what he calls the pure form of our outer sensible intuition. Kant's conception of space and time as our two pure forms of sensible intuition is central to his general philosophical position, of course, which he calls formal or transcendental idealism.

32:30 And, although a fundamental dichotomy between the two faculties of sense and intellect precedes Kant by many centuries and is characteristic of all forms of traditional rationalism, from Plato to Leibniz, Kant's particular version of the dichotomy is entirely distinctive of him. For, in sharp contrast to all forms of traditional rationalism, Kant locates the primary seat of our a priori mathematical knowledge in sensibility rather than the intellect. This is a very peculiar and dispensive view. In particular, our pure form of outer sensible intuition, space, is the primary ground for Kant of our pure geometrical knowledge, which he takes to be perfectly exact, perfectly certain, thing that rationalists like. Kant characterizes the distinctive role of our pure intuition of space in geometry in terms of what he calls construction in pure intuition. And he illustrates this role by examples, several examples of which we'll talk about, of geometrical construction from Euclid's elements. It is natural then, and some Kant scholars have explored this, to turn to recent work on diagrammatic reasoning in Euclid, originating in Kenny Manders, to elucidate Kant's conception. So on this view, when Kant says that spatial intuition plays a necessary role, we might take into mean that diagrammatic reasoning, in the sense that Manders and his followers, plays a necessary role. I shall argue that this kind of view of Euclidean geometry, as illuminating as it may be of an interpretation of the elements, it is very illuminating, is not adequate as an interpretation of Kant, and more generally that recent work on diagrammatic reasoning can at best capture only a part of what Kant's conception of geometry involves. Most importantly, it cannot explain why Kant took this conception crucially to involve a revolutionary new theory of space, the very three-dimensional space, it's not just paper or blackboard, in which we and all other physical objects live and move and have our view. We are in this space. The most fully developed diagrammatic interpretation of the elements with which I am acquainted is that of Marco Panza. And indeed, I first got the idea for this paper last year at our October recess meeting in Stanford

35:00 when Marco gave a version of his view of the elements, a diagrammatic view, and that he was kind enough to send me his long paper on that, which I recently read and studied. And although Marco is not primarily interested in the interpretation of Kantian philosophy, he's interested in the interpretation of Euclid, he refers to Kant and some of my own previous work on Kant, kindly, to motivate his interpretation of Euclid. Euclid's fundamental concern, however, is to argue for an Aristotelian rather than platonic interpretation of the element, according to which the objects of Euclid's geometry are not abstract objects in the platonic or modern sense of Platonism, entities apprehended by the intellect alone independently of sensual perception. The objects of Euclid's geometry, Marco argues, rather arise by a process of Aristotelian called it, from concrete, sensible particulars, namely from particular physical diagrams drawn on paper or a blackboard, of course, suitably interpreted and understood. This keeps going off. So here's just what Marco says at the very end of his paper, the twofold rule of diagrams in Euclid's plane geometry. The classical interpretations, which are platonic, can be contrasted and replaced by another interpretation, more Aristotelian, and then parenthesis, Kantian in spirit, according to which this geometry is or results from a codified practice essentially based on the production and inspection of physical objects like diaries. I shall argue that Kant's conception of the role of spatial intuition in geometry cannot a process of abstraction from concrete, sensible particulars in this sense, or really in any other sense that I can think of. Kant, as I said, diverges from traditional rationalism in locating the seat of pure geometry in sensibility rather than the intellectual understanding, and Kant thereby gives a central role in geometry to what he calls the pure, productive imagination. Perhaps the most important problem and facing interpretations of Kant's philosophy of geometry, then, is to explain how, for Kant,

37:30 sensibility and the imagination, faculties traditionally associated with the immediate apprehension of sensible particulars, how these faculties can possibly yield truly universal, necessary, and exact geometrical knowledge. For example, in a well-known passage from The Discipline of Pure Reason in its dogmatic employment in the Critique of Pure Reason, Kant contrasts philosophical cognition, which he calls rational cognition from concepts, with mathematical cognition as rational cognition from the construction of concepts. And Kant adds, famously, to construct a concept is to present the intuition corresponding to it a priori. And Kant concludes, after another page or so, philosophy can find itself merely to universal Mathematics can affect nothing by mere concepts, but hastens immediately to intuition, in which it considers the concept in concretos. Not, however, empirically, but merely in an intuition that it presents a priori, that is, which it has constructed, and in which that which follows from the universal conditions of construction must also hold universally of the object of the constructive concept. So that's perfectly clear, of course. It's typical in common. the joke. Exactly what, however, is the pure or non-empirical intuition corresponding to a general concept, a singular instance of this concept that is nonetheless presented purely a priori. Moreover, how can any singular instance of a general concept, no matter how it is supposed to be produced possibly be an additional source over and above purely conceptual representation of universal and necessary knowledge. It seems like universal necessity has to come from concepts. It has to be analytic. Immediately after the just-quoted sentence defining the construction of a concept of the a priori presentation of the corresponding and intuition. Kant adds, and I'm just continuing that, for the construction of a concept, we therefore require a non-empirical intuition, which consequently as intuition is a singular object, but which nonetheless, as the construction of a concept, a universal representation,

40:00 that is the concept, must express universal validity in the representation for all possible intuitions that belong under the concept, that is, since the instances of it. But how, once again, can an essentially singular representation, no matter how it is supposed to be produced, possibly express truly universal validity? Problems of precisely this kind underlie the contrary conviction, common to all traditional forms of rationalism, that mathematical knowledge must be conceptual or intellectual, as opposed The only place you can get the right kind of universality is from the intellectual realm. Kant illustrates his meaning in the continuation of our passage by an example of the Euclidean proof, Proposition 1.32 of the elements, which we've already seen several times before, where it is shown that the sum of the interior angles of the triangle is equal to the sum of two right angles. In quotation number three, that's the passage from Kant. You don't have to look at it. It's too long. but that's just to show that he's essentially summarizing Euclidean proof right after this passage. So now instead of doing that, I'm going to show you a picture of that, which is similar to pictures you've seen before. Given a triangle ABC... No dotted lines. No dotted lines. Some of the pictures you've seen before. Given a triangle ABC, one extends the side BC in a straight line to D, and then you draw the line CE parallel to AB. One then notes by proposition 1.29 that the alternate angles BAC and ACE are equal and also that the external angle ECD is equal to the internal and opposite angle ABC. But the remaining internal angle ACB added to ACE and BCD yields the sum of two red angles, And the two angles, A, C, E, and E, C, D, have just been shown to be equal to the first two internal angles, that is, A, B, C, and D, A, C, respectively. Therefore, the three internal angles together also equal the sum of two red angles. This construction and proof has universal validity for all triangles because the required inferences and auxiliary constructions, extending the line BC to D drawing a parallel CE can always be carried out within Euclidean geometry

42:30 no matter what triangle ABC we start with. In fact, that's part of this proof. It appears, in fact, that the proof procedure of Euclid's elements is paradigmatic of construction in pure intuition throughout Kant's discussion of geometry in the first critique, and mathematics more generally. I'm only really going to talk about geometry, which includes Kant's discussion, a fairly complete presentation of at least a good part of the elementary Euclidean geometry of the triangle. In the transmetal aesthetic, for example, Kant presents the corresponding side-sum property of triangles, that two sides taken together are always greater than the third, proposition 1.20, which we also have seen before. So here he's using that as an example, this time in the Transcendental Aesthetic, an example of an intuition from which you can derive a priori with apodictic certainty the relevant preposition. And he has in mind Euclid's proof in Prophecy of History 20 as an illustration of this apodictic certainty. The Euclidean proof of this proposition proceeds, just like Proposition 1.32, by auxiliary constructions and inferences starting from an arbitrary triangle ABC. So this time we extend the side BA to AB in a straight line, such that AB is equal to AC. We then draw CD, and note by proposition 1.5 on Isopsel's triangle, that the two angles ACD and ADC are equal, so that BCD is greater than BDC. Since by proposition 1.19, the greater angle is suspended by the greater side, it follows that BD is greater than BC, but BD is equal to the sum of BA and AD by construction, which is ADD equals AC. Moreover, Kant refers to the Euclidean proof of Proposition 1.5 itself, that the angles at the base of an isosceles triangle are equal, in a famous passage in the preface to the second edition of the first critique, praising the characteristic method of geometry. And here I just want to note, he attributes this to Thales, perhaps, and interestingly

45:00 he says here, it's not a matter of inspecting what you see in the figure or even in the concept. Rather, you have to bring forth, as he says, what the constructor has injected in thought and presented according to concept in order to know something a priori. He must attribute nothing to the thing except that which follows necessarily from what he himself as placing it by construction in accordance with this concept. The reference to 1.5 is made explicit in a correction later, so we don't have to worry about that. Okay. And you've also seen this, but I won't go through it dynamically. There's the construction of 1.5. Okay. Kant's reliance on Euclid is thus very clear. And once again, it is therefore natural to turn to recent work on the diagrammatic reasoning found in Euclid for elucidating Kant's view. In particular, we can appeal to what has been called the schematic character of Euclidean diagrams to explain how the appeal to sensible particulars can nonetheless result in necessary and universal and desired knowledge. We have observed, for example, that the specifically metrical properties of the triangle used in the proof of Proposition 1.32, the lengths of its size and the magnitude of its angles, don't play any role in the proof. It remains true for all continuous variations of these lengths and angles where you still have the triangle. That's Mander's reading of what schematic means here. Therefore, we have indeed proved a proposition valid for all particular triangles whatsoever. As we shall see, however, although the technical notion of what Kant calls a schema is indeed centrally important to his theory of geometrical reasoning, and moreover Kant models this notion of a schema, this technical notion for him, in important respects on central features of Euclid's proof procedure, Kant's notion of a schema or schematic representation is essentially different from the diagrammatic one in the sense of mandate. Kant's conception of spatial intuition, among other things, goes far beyond anything envisioned

47:30 in the diagrammatic approach, again, to embrace the idea that space, the very space in which we live in, has to be Euclidean, among other things. In the axioms of intuition, another part of the first critique, the principles corresponding to the categories of quantity, unity, plurality, and totality, Kant considers the Euclidean construction of a triangle in general from any three lines such that two taken together are greater than the third proposition 1.22 the restriction of course follows from 1.20 here is the relevant passage say by means of three lines such that two taken together are greater than the third a triangle can be drawn then I adhere the mere function of the productive imagination which can draw the lines greater or smaller and thereby allow them to meet at any and all arbitrary angles. So given that he puts the greater than the third condition in there, it seems pretty clear that he had 1.22 in mind, especially since we already saw him talking about 1.20 and other related propositions. This makes it clear, in my view anyway, that the construction in pure intuition of the concept of a triangle in general for Kant just... Oh, do I need... No, I didn't do it. Sorry. This makes it clear in my view that the construction in pure intuition of the triangle for Kant just is the Euclidean construction demonstrated in Proposition 1.22. Where in Kant's words here, I here have the mere function of the productive imagination which can draw the lines greater or smaller and thereby allow them to meet at any and all arbitrary angles. So here again, we've seen this too. Here's the construction. You put the three lines D, F, F, G, G, H all together on a line. you could do that by proposition, sorry, axiom 1.2, axiom 1.2, and then again you do this constructing of circles with the right radii, and there's the triangle F, K, G constructed from there. Moral, so that, I'm saying it's that construction that Kant means by the construction of pure intuition of the triangle. And eventually I won't say that's the schema of the triangle, but I'll do that in a second. In the chapter on what Kant calls the schematism of the pure concepts of the understanding, Kant carefully distinguishes what he calls the schema of a pure sensible concept, that is, a geometrical concept, from any particular image falling under the concept,

50:00 which may be produced by the general schema. He says, quote, I call the representation of a general procedure of the imagination for providing a concept with a image, that is, a particular instance, the schema. Kahn illustrates this idea once again with the example of the triangle. He loves the triangle. In fact, schemas are rather than images of objects that would lie at the basis of a pure sensible concept. No image at all would ever be adequate to the concept of a triangle in general, because it's a particular. It would never attain the universality of the concept, which makes it whole for all triangles, and so on. It would always be limited to some part. The schema of the triangle can never exist anywhere but in thought, and it signifies a rule of synthesis of the imagination with respect to pure figures in space. This rule of synthesis, therefore, appears to be nothing more nor less than Euclidean construction of an arbitrary triangle in the absence of intuition as a mere universal function of the productive imagination. Again, the lines can be made greater or smaller and allowed to meet an annual arbitrary angle, just like what he demands here. So from now on, I'm going to take that to be the schema of the triangle in cons-technical things. More generally, we can take the Euclidean constructions corresponding to the fundamental geometrical concepts, line, circle, triangle, and cell lines, means by the schemata of these concepts. We can understand the schema of the concept of triangle, as I've said, as a function or constructive operation, which takes three arbitrary lines, meaning the condition of adding together two greater than the third, takes that as input, and then as output in accordance with the 1.22 construction, yields the triangle that's constructed from there. We can understand the schema of the concept of a circle as a function, which takes an arbitrary point and line segment as input and yields the circle with a given point as center and a given line segment as radius as output in accordance with postulate 3. And here Kant talks about the construction of the circle in a very similar way. And interestingly, he says,

52:30 the construction of the circle, that is, sorry, postulate 3, cannot be proved, that's why it's a postulate, because the procedure it requires is precisely that by which we generate the concept of the figure. We give it an object and we generate the concept at the same time. Okay. Such constructive operations have all the generality or universality of the corresponding concept, circle or triangle. They yield, with appropriate input, lines, points, and so on, any and all instances of these concepts. Any circle can be constructed by appropriate point and radius, any triangle by appropriate initial lines by 1.22. So they're just as general as the concepts. Unlike general concepts, however, the output of a schema, these outputs are indeed singular or individual representations, particular instances, or what Kant calls images, which fall under the concepts in question. So the outputs of a schema are not conceptual or logical entities, like propositions or truth sounds, a la Russell or Frege say. The outputs are individual particular objects. nevertheless, they're just as general since they operate on any input and they give you all these instances. This last point is crucial for understanding why Kant takes pure mathematics essentially to involve non-discursive or non-conceptual cognitive resources, which nonetheless possess all the universality and necessity of purely conceptual thought. Characteristic of conceptual thinking for Kant is the logical procedure of subsumption whether of an individual under a general concept or of a less general concept species under a more general concept gene Characteristic of mathematical reasoning by contrast is the procedure of substitution by which, as we would now put it an object is inserted into the argument place of a function yielding another object the argument place of a function again, yielding another object, and so on. Reasoning by substitution, unlike reasoning by assumption, is therefore essentially iterative.

55:00 And it is precisely such iterative thinking for Kant that underlies both pure geometry in the guise of the platoon and proof, and the more general calculated manipulation of magnitudes in algebra and arithmetic. Kant's conception of the essentially non-conceptual character of geometrical reasoning is thus especially sensitive to the circumstance that, in Euclid's formulation of geometry, the iterative application of initial constructive operations represents the existential assumptions we would express by explicit quantification of statements in modern formulations following Hilbert. Thus, for example, whereas Hilbert represents the influence of visibility of a line by an explicit quantificational axiom, stating that between any two points there exists a third, Euclid represents the same idea in proposition 1.10 by showing how to construct a bisection function for any given line segment. And this is just like 1.1, really. Given a line segment, you have the two circles, and then you can bisect it by connecting the two points. And, of course, you can then iterate that on AC and then the new thing, AB, and so on. That's what infinitivitivity means in Euclid. More generally, Euclid constructs all the points in his plane by the iterative application of three initial constructive operations to any given pair of points. Connecting any two points by a straight line, plus to it one. extending any given line segment by another given line segment, positive 2, constructing a circle with any given point at center, and any given line segment as radius, positive 3. This constructive procedure yields all points constructible by straight edge and compass, which, of course, comprises only a small, denumerable subset of the full two-dimensional continuum whose existence is explicitly populated by Hilbert. In this sense, the existential assumptions needed for Euclid's particular proof procedure, the very assumptions needed to justify all these inner reconstructions needed along the way, are given by what we call Skollum functions, and you heard Saul talk about this, for the existential quantifiers we would use in formulating a Hilbert-style axiomization in modern quantification and logic. where in Euclid, all such Scollum functions,

57:30 they're not just written down as functions, but they're explicitly constructed by finite iterations of our three initial constructive operations laid down by the first three positives. So here you see that again. There's an iterative application of the Scollum function for bisection. Following Leibniz, Kant takes the discursive structure of the understanding or intellect to be delimited by the logical forms of traditional subject-predicate logic. In explicit opposition to Leibniz, however, Kant takes these logical forms to be strictly limited to essentially finitary representation. There are, for Kant, no Leibnizian complete concepts, as Leibniz puts it, which is like an infinite conjunction. Comprising within themselves, defining sets of characteristics, or partial concepts, an infinite manifold of further conceptual representation. Here's a passage from Kant in the second edition actually making this point and arguing on the basis of this point that space has to be an intuition, not a concept, because concepts can't be infinite in this way. But mathematical representations, such as the representation of space, way, that is, they do contain an infinite manifold of further mathematical representations within themselves, as in the representation of infinite divisibility, as part, as partial representations. So such representations, mathematical representations for Kant, are not and cannot be conceptual in this sense. Of course, we now have an entirely different conception of logic and of conceptual representation. Very different from Kant, one that is much more powerful than anything either he or even Leibniz ever envisioned. Nevertheless, we can still understand Kant's fundamental insight from our point of view if we observe that no infinite mathematical structure, such as either the space of Euclidean geometry or the number series, can possibly represent it within monadic quantificational logic as in only one place. Such infinite structures in modern logic have to be represented by the use of nested sequences of universal and existential quantifiers using polyadic logic,

1:00:00 not just relations with the universal existential theorem. These same representations that we think of as quantificational from Kant's point of view are instead made possible by the iterative application of constructive functions in the productive imagination, where, as we have seen, Scollum functions for the existential quantifiers we would use in such formula are rather explicitly constructive iterative things. Okay, so we now see, from Kant's point of view, why mathematical thinking essentially involves what he calls the pure productive imagination. and why, because those colon functions are functions of the pure productive imagination, why, accordingly, this type of thinking essentially exceeds the bounds of purely conceptual, purely intellectual thought in his Leibnizian sense, a kind of finitized Leibnizian sense. My first problem was using diagrammatic reasoning in the sense of Nadir's and his followers to interpret Kant's notion of construction is that it does not square with Kant's understanding of the relationship between conceptual thought and sensible intuition. It does not square, in particular, with Kant's developed view of the relationship between general geometrical concepts, their corresponding general schemata, and the particular sensible images, particular geometrical figures, which then result by applying these schemata. And then that's the same, this is the passage I have in the moment, mechanism chapter. Whereas diagrammatic accounts of the generality of geometrical propositions following members begin with particular concrete diagrams and then endeavor to explain how we can abstract from their relevant particular features, specific links to size and angles, for example, by an appropriate process of continuous variation or idealization or something, Kant rather begins with general concepts as conceived within the Leibnizian logical tradition. The concepts come first. They're given just by the definitions of circle, triangle, and so on. He begins with general concepts as conceived within the Leibnizian tradition and he then shows how to schematize these concepts sensibly

1:02:30 by means of an intellectual act or function of the pure productive imagination. there's that again there's the example of the triangle but the general concepts in question sorry, both the general concepts in question the geometrical concepts and their corresponding general simata are pure rather than empirical representations there's no perception of a sensible particular yet at all and a particular concrete figure a sensible particular occurs as it were for Kant, at the end of a process of intellectual determination in pure intuition. Nothing like Aristotelian abstraction from concrete, sensible particulars plays any role for Kant in this discussion of De Nata and Kant. My second problem, however, is even more serious than this and I think more interesting for it. For Kant's conception of geometrical reasoning is also essentially connected, as I've said, with our representations of space and time. Space and time, not with particular spatial figures drawn on paper or a blackboard, but space and time themselves, these things that we're in, and everything else is in, the whole physical world is in. And it is precisely here, as I have intimated, or hinted anyway, that Kant also engages Newton's conception of space and time as it figures in Newton's controversy with Leibniz, I'll articulate this a bit now. Space for Newton is a great ontological receptacle, as it were, for both all possible geometrical figures that we can draw and all possible material objects. And Kant's theory of space as a pure form of intuition is supposed to be an alternative, as we shall see, to precisely this Newtonian conception of the ontological containment. It is centrally important to Kant's philosophy of geometry that all possible objects of human sense perception, all objects of what Kant calls empirical intuition, must necessarily conform to the a priori principles of mathematics established in pure intuition. Where Kant says, for example, the synthesis of spaces and times as the essential form of all intuitions is that which at the same time makes possible the apprehension of appearances

1:05:00 and of every outer experience and therefore all cognition of the object of outer experience. And what mathematics in its pure employment demonstrates of the former, that is, pure intuition, necessarily holds also for the latter all the objects of pure intuition. And he has specifically in mind in this passage infinite divisibility. So, since we prove a priori that space is the opposite of the planet here, geometry proves a priori that space is instantly divisible, therefore physical space, with all of the things in it, planets and their motion, is instantly divisible and satisfies all the principles of Euclid and of Newton. And we know this a priori, so far. In order to appreciate the role pure geometry plays in our perception of empirical objects, Kant's view, we need explicitly to connect the functions of the pure productive imagination expressed in the construction of geometrical concepts, something like Euclidean construction, Kantian schematis, we have to connect that with the Kantian forms of pure intuition, space and time, as they are described in what Kant called the metaphysical exposition of space and time in the Transcendental Aesthetic. Kant in a later work his controversy with Eberhard in 1790 distinguishes between what he calls geometrical and metaphysical space geometrical space is the space of pure Euclidean geometry which for Kant is gradually or successively constructed by the two fundamental operations of drawing a straight line and constructing a circle in accordance with Paul Solitz 1 to 3 as a whole, it's always growing step-by-step in terms of finite configuration. Metaphysical space, by contrast, is the space of our pure outer sensible intuition, which for Kant is the pure form of all empirical intuition of any and all physical objects that may exist in this space. By the way, Kant thinks here that metaphysical space is actually infinite, and the reason of geometrical space that we can keep going with these constructions is because they take place in metaphysical space, whatever that means is there.

1:07:30 Okay. So to find out more about what he means by metaphysical space and its relation to geometrical space, I'm now going to turn to the first two arguments of the metaphysical exposition of space in the trans-moral aesthetic. That's what Kant means there by metaphysical space. So let's turn to these two arguments, which are intended to show, in particular, that space is a necessary a priori representation that precedes all empirical perception. That's the crucial thing. Yet all empirical perception is subject to it. Not a representation that can in any way be abstracted from our empirical perception of our spatial objects. The first argument attempts to show that space is an a priori rather than empirical representation by arguing that all perceptions of outer empirical objects in space presuppose the representation of space. And here I'm just going to point to a little part of it. Here's the crucial part of the argument. In order that certain sensations are related to something outside me, that is, that they're outer, outer. And then he says, that is, to something in another place in space than the one in which I find myself. That's what he means by outer. And similarly, blah, blah, blah, the representation of space may be presupposed. So I'm focusing on that thing in parentheses, actually. This argument emphasizes that space as the pure form of outer sensible intuition enables us to represent objects as outer precisely by representing them as spatially external to the perceiving subject, to me or to you. So that the space in question contains the point of view of the subject, the point of view from which the objects of outer sense are perceived and around which, as it were, the objects of outer sense are arranged. And that point of view is in the very same space. empirical spatial intuition or perception occurs when an object spatially external to the point of view of the subject affects this subject along a spatial line of sight, as it were, by, say, a light ray, so as to produce a corresponding sensation. That's an empirical intuition. And it is in this sense, therefore, that the pure form of spatial intuition expresses the manner in which we are affected by outer spatial objects,

1:10:00 So let us call this structure, I'm reading it in terms of perspectives, perspectival space. It has the structure of a point of view, objects around the point of view, and possible lines of sight between the object and the point of view. That's the first argument. The second argument goes on to claim that space is a necessary a priori representation which functions as a condition of the possibility of all other experience. Space is a necessary representation. One can never make a representation of the supposed fact that there is no space, although one can very well think that no objects are to be found in that space. The space must therefore be viewed as a condition of the possibility of appearances, depending on them, an aprile representation, and a stereological basis of outer appearances. The crux of this argument is that one cannot represent outer objects without space, whereas one can think of the very same space without the objects in it, and therefore as empty. What exactly does it mean, however, to represent space as empty of outer objects? And in what context do we succeed in doing this, having that representation? we do it. A very natural suggestion is that we think space as empty of outer empirical objects just when we are doing pure geometry, because we're not concerned with empirical objects, we're concerned with space. This would accord very well in particular with a concluding claim that space thereby functions as a necessary apiary condition of the possibility of outer appearances, for they would then necessarily be subject to the apiory necessary science of pure geometry, if that's what it means to think of the space independently and prior to the object. So something like Newton's Big Container, but not with the ontology of Newton. What is the precise relationship between the apiory structure attributed to space in the first argument, perspectival space with a point of view and a subject and all, and that attributed to space in the second, the structure of pure Euclidean geometry. It is natural in the first place to view perspectival space as itself in some sense a priori and

1:12:30 geometrical, since it does not depend at all on the particular empirical outer object actually perceived from any particular point of view. On the contrary, this perspectival structure is invariant under all changes in both the objects perceived and the point of view from which they are perceived. And in this sense, in Kant's language, it expresses the form rather than the matter or content of outer information. So it's a priori in that sense. And it's geometrical in the sense of the geometry of perspective. Moreover, and in the second place, changes in perspective themselves constitute what we now take on a modern understanding of geometry to be a geometrical object, namely a group of Euclidean motions or transformation, comprising all possible translations of our initial point of view from one point to another, and then all possible rotations of this point of view around some axis. In particular, any perceptible spatial object located anywhere in space can thereby be made accessible to this perceived subject, this kind of abstract subject, by an appropriate sequence of translations and rotations starting from any initial point of view and any initial associated perspective. But, here's the crucial point, there is a clear connection, I say, between this modern group theoretical geometrical structure and geometry in constant. For as Kant himself emphasizes, the two fundamental Euclidean constructions of drawing a line and constructing a circle are generated precisely by translations and rotations. As we generate a line... Oops, what happened? Generate a line from a point and then rotate that line around the point. I seem to have missed the quote, which was number, do I have it here, sir? Something got to the left here. Let me see if it goes. You were so worried with your motion that you forgot. Yeah, it got to. Well, there's that again. I'll show it to you yet another time, but no more than that. But, yeah, it's number 13 on the handout. This actually also comes from that correspondence related to Eberhard in 1790, but it's just

1:15:00 an interesting place where Kant explicitly says that the construction of a circle takes place by translation, generating the line from a point, and then rotating it around it. And that can be related to corresponding quotations in the first critique that, in In light of this, we can interpret that away. So in terms of group theory, here are the passages. They just got put in another place. These are the first constructions drawing and rotating, and here is that thing again about the posture of constructing the circle. If you read these two together and you think of what Kant means by describing, I think it's pretty clear he always means generating the line by a motion and then the circle by a rotation. Let me see if I have this again. I do. Okay, so if we think of it in terms of a group, then we think of the line and the circle arising as orbits of the group, right? You take some translation in the downward direction, you apply it to that initial point that I started from for a long certain time, and that you get an orbit. And then if you apply a rotation with the initial point as center to that end point, then you get the circle as a little bit of the group of rotation. On the present interpretation, therefore, it is precisely this relation between perspectival space, translations and rotations, and geometrical space, Euclidean constructions, constructions that links Kant's theory of space as a form of outer sensible intuition or perception to his conception of pure mathematical geometry in terms of the success of execution of equality constructions in the pure imagination. It is precisely in this way in particular that Kant is now in a position to claim that pure mathematical geometry is necessarily applicable to all possible objects of empirical perception, because all possible objects of empirical perception take place in perspectival space. That was the first argument of the metaphysical expedition, but perspectival space is, as it were, loosely isomorphic to the geometrical space, in some sense. So Kant can now claim the synthesis of space and time,

1:17:30 that is geometrical that is which operates on perception is the same thing that goes on in geometrical construction and if you put it together with something that comes right before it i can represent no line for myself no matter how small without drawing it in thought generating its points gradually from the point registering its intuition on this success of synthesis the generation of figures is based, the mathematics of extension, and so on. So here he puts together, I say, translations and rotations of this kind of generation and the science of geometry with the idea that everything that you can perceive empirically has to be subject to geometry. Now, the urgent need to establish this result, that geometry necessarily applies to physical space, as it were, to perceptual space, places Kant in a completely different intellectual world from that inhabited by Euclid, Plato, and Aristotle. For it is characteristic of the new view of mathematics arising in the 17th century that pure mathematical geometry is taken to be the foundation for all knowledge of physical reality. Pure mathematical geometry, beginning with Descartes, is taken to describe, and to describe exactly, the most fundamental features of matter itself. And in this sense, physical space and geometrical space, that is Euclidean space, are now taken to be identical. Kant's own understanding of this idea, as I have suggested, is framed by the controversy between Newton and Leibniz, where both took the geometrization of nature to be a now-established fact, not subject to controversy, but they reacted to this fact in radically different ways, Newton and Leibniz. Newton understood the situation quite literally. What he calls mathematical space and time, true or absolute space and time, constitutes the fundamental ontological framework of all reality. for Newton, is necessarily spatial and temporal, existing always and everywhere. And all physical or material objects are then created, and as Newton puts it, moved, even moved, within God's boundless and uniform sensoria. Here's a quotation from Prairie 31 for the Optics.

1:20:00 We have an all-powerful, ever-living agent who, being in all places, being omnipresent, is more able by his will to move the bodies within his boundless and uniform sensorium and thereby to form and reform the parts of the universe than we are by our will to move the parts of our own bodies. God is spatial. Everything is spatial. Space is the ontological framework for our reality. And space is exactly Euclidean geometrical space as well. For Leibniz, by contrast, world described by the new mathematical science, including the space in which bodies move, is a secondary appearance or phenomenon of an underlying metaphysical reality of simple substances or monads, substances which, at this level, are not spatial at all, but rather have only purely internal properties, like thought, and no external relations, including spatial. And this point in turn is closely connected with the fact that Leibniz self-consciously adheres to the idea that purely intellectual knowledge is essentially logical. For, although Leibniz appears to have envisioned some sort of extension of Aristothelian logic capable of somehow embracing the new algebraic methods of his calculus, there is no doubt that the traditional subject-predicate structure of this logic pervades his monadic metaphysics. Or in this sense, it's not a pun to talk about the relationship between monadic metaphysics and monadic logic, a point that was first emphasized early in the 20th century by Louis Kudera and Bertrand Ropper. So his metaphysics is deeply connected to this logic. and his logic, for Leibniz, is his model of what intellectual knowledge is, not just what the logic is. That's what the intellect does. It is precisely because ultimate metaphysical reality is essentially intellectual, in the logical sense, that's what the intellect, as opposed to the senses, grabs ultimate reality, that the entire sensible world, including space, is a secondary reality or phenomenon. Thus, although Leibniz, like everyone else in the period, holds that there are exact mathematical laws governing the sense of the material world, that is, the phenomenal world, he introduces a new kind of necessary gap, Leibniz, a new kind of platonic gap.

1:22:30 It's not a gap about exactness, it's a gap between the intellect and the appearances, between reality as known by the intellect, numeral reality, and the sense of the world. Kant's philosophy of transmissional idealism is also based on a fundamental dichotomy between reality as thought by the pure understanding alone and the phenomenal world and space and time given to our faculty. But Kant sharply differs from Leibniz in two crucial respects. First, mathematical knowledge for Kant is sensible rather than purely intellectual, as I said at the beginning. Indeed, mathematics is the very paradigm of rational and objective, sensible knowledge, resulting from the schematism of specifically mathematical concepts in our pure forms of sensible intuition. Second, and as a consequence, we can only have theoretical knowledge for Kant, any theoretical knowledge, not just mathematical, of precisely the sensible or phenomenal world. The noumenal reality thought by the pure understanding alone remains forever unknowable from a theoretical point of view. And we can only have purely practical knowledge, that is, moral knowledge, of the inhabitants of the intellectual world, namely God and the soul, by moral experience, not theoretically. Indeed, it is precisely the necessary limitation of all theoretical knowledge to the sensible or phenomenal world results from Kant's doctrine of the schematism, of the pure concepts of the understanding. Here I'll just give you a little bit of this. We can talk about it in the question period if anybody wants to. But the idea is the pure concepts of the understanding have, as it were, a kind of logical meaning independently of sensibilities and therefore independently of schematism, but they actually cannot relate to any object of knowledge through this purely logical meaning. get a logical meaning, you have to schematize the categories, whatever that means exactly, which makes the appearances subject to these universal rules of synthesis and suitable for experience. The categories without schematism represent only functions, logical functions of

1:25:00 the understanding for concepts, but do not represent any object. This meaning accrues to them from sensibility, which realizes the understanding in the very process of restricting understanding to sensible rather than purely intellectual knowledge. Okay, I have two more paragraphs on this one. Kant's philosophy of geometry, seen against the background of his more general transcendental idealism, combines central insights of both Ladnitz and Newton. For, in the first place, Kant's emphasis on the perceptual and intuitive aspects of geometry, and mathematics more generally, corresponds to Newton's approach. In Kant's for the logico-algebraic approach of Leibniz. And in the second place, Kant's sharp distinction between the faculties of intellect and sensibility, together with his parallel distinction between logical or discursive and mathematical or intuitive reasoning, arises against the background of the Leibnizian conception of the pure intellect. I said it was Leibnizian concepts he began with. And it is aimed more specifically at Leibniz's view that pure mathematics, including geometry, is, in Kant's sense, analytic, depending only on relations of conceptual containment within the traditional logic of concepts. Kant's point about pure mathematics against Latinx is simply that the pure intellect, characterized in this way, is not, after all, adequate for the text. Within such a predicate logic, you cannot really represent mathematics. It is for precisely this reason in Kant's view that the pure understanding must be applied to or schematized in terms of a second rational faculty modeled on Newtonian absolute space. No longer conceived, of course, along the lines of Newton's divine sensorium, that's too metaphysical complexion, but rather as a pure form of our human faculty of sensibility. Just a little bit about this, where Kant famously in the aesthetic contrasts the Newtonians and the Leibnizians. the Newtonians at least are able to ensure that all the appearances are subject to mathematical assertions namely geometry because we have as it were a pure space, a mathematical space on the other hand they get very confused when you try to understand beyond the field of the appearances

1:27:30 because then you get the idea that God is in space you can show with other texts that I think that's what Kant means here and of course that's theologically unacceptable Newton. The Leibnizians don't have that problem because they've separated the understanding and sensibility. They recognize correctly that representations of God and so on don't have anything to do with faith and are purely intellectual. However, they can neither give an account of the possibility of a prior mathematical intuition insofar as they lack a true and objectively valid a prior intuition nor can they bring empirical proposition physics is a necessary agreement with the mathematical proposition of geometry. Okay, we are now in a position, finally, to appreciate the deeper explanation for why no process of Aristotelian abstraction from concrete sensible particulars can play any role in our knowledge of equilibrium geometry or Kant. All perception of concrete sensible particulars for Kant, including particular diagrams drawn on paper or a blackboard or even shown with power plants, all of these presuppose that space as a pure form of outer sensible intuition is already in place. They're already in space. And in particular, they already presuppose for Kant that space already possesses the structure of Euclidean Geometry. Space acquires this structure for Kant precisely from the pure acts or operations in the generation of figures, the application of general geometrical construction, or Kantian schemata, resulting in what he calls the synthesis of spaces and times as the essential form of all intuition, including especially empirical intuition. All the constructive procedures of Euclidean geometry must therefore have already taken in place, as it were, in the space of pure intuition in order for any perception of concrete physical objects, like diagrams, the empirical intuition to be possible for Kant, including any perception of concrete physical diagrams. Thank you. Thank you. Thank you, Michael. Questions? Yes, it's a point that for clarification about the course of 11, because I never understood

1:30:00 this argument, so I take advantage of it. When he's talking about outer experience, and then he says that I need space to have because I have to distinguish between what is in another place and the place in which I find myself. What is the meaning of your I? Is it my body or is it my consciousness? Because if it's my body, of course it's important because an empirist can argue, and many empirists do argue today that I can have inner movement in my body of space without any supposition of an unpopular space of my body. If it's my consciousness, I don't understand what is the spatial relationship between consciousness and things. Okay. I was trying to express, as it were, my point of view. The point of view from which I perceive the things around me in the sense of perspectival geometry, if you like. So, you know, there's something like a point inside my head, you know, that has something to do with the binocular function of my two eyes, and where I am and where you are, and I'm perceiving you from a point of view, which is somewhere there. So it's not my body. Otherwise, it wouldn't be in pure intuition. But it's something associated with my body, namely a point of view. And there's also an orientation. There's not only a point of view, but there's a perspective and an orientation. Because it's not just a point, it's also the fact that I'm looking in this direction at you, not behind me. I understand, but you know that there are many research today that say that, for example, in my brain, I mean, not in this intentional relationship, but in my brain, many detectors of movement that can give me the sense of space outside of this intentional representation. See what I mean? So what would be the structure of deep, according to Kant? Well, Kant is not talking about the ground. Kant is just having a kind of a general, idealized, schematic discussion

1:32:30 of the relationship between space and perception of spatial objects in a very kind of idealized, quasi-geometrical way. So he's not asking about the sense organs, the brain, what are the biological and psychological mechanisms that allow me to have a sensation. He's not talking about that at all. What he's saying is that space, the space in which we perceive spatial objects, has certain abstract formal features that are common to all acts of empirical perception. So I may be perceiving you, I may be perceiving a bottle, both cases there's a point of view from which it's perceived and there's a point where it is and there's something like a line between those two things