Kant on Imagination & Geometrical Certainty

0:00 Thank you. In space, the concept and origins of space, in 68, he becomes a Newtonian and embraces, excuse me, an absolutist point of view, and he doesn't become Kantian until 1770, when he proposes a pure form of spatial intuition. So some of this I'll be mentioning in the paper. in the critical period two themes dominate Kant's account of geometry first, the certainty or necessity of geometrical knowledge and second, the applicability of geometrical claims about space to the empirical state of affairs in the first critique, as well as in the prolegomena Kant famously tries to account for both these aspects of geometrical knowledge by proposing a pure form of spatial intuition that is both the object of our geometrical investigations and constitutive of all outer experience. In brief, the a priori status of this form of intuition guards geometry against the comparative universality that is characteristic of empirical knowledge, and the proposal that this form of spatial intuition is the form of all outer experience, or what he calls the form of outer sense, secures the applicability of our geometrical claims to empirical space. Yet beyond proposing a peculiar object of geometrical investigation, Kant also proposes a peculiar method by means of which the geometer discovers the features of our spatial intuition. Namely, the geometer is assigned a method of construction. For instance, as detailed in the Doctrine of Method, and this is, you can refer to passage A on the handout, when a geometer attempts to determine how the sum of the angles of a triangle might be related to a right angle, he begins at once to construct a triangle establishes a proof that relies on the particular triangle produced in intuition. Whether this triangle is drawn on paper in an empirical intuition or constructed by the mere imagination and pure intuition, the geometer must, according to Kant, establish a chain of inferences that is always guided

2:30 by intuition in order to arrive at a fully illuminating and at the same time general solution of the question. This example nicely brings to light the role played by a form of spatial intuition in mathematical reasoning, insofar as the form of intuition underlies the very process of construction required to complete a geometrical proof. At the same time, the example also points to the importance of the imagination for Kant's critical account of geometry. For the generality Kant ascribes to solutions generated by the geometer rests on the rules of construction shared by particular geometrical objects, and these rules, so what he calls schemata, are expressions of the synthetic procedures of the productive imagination in pure intuition. As he puts it in the schematism chapter, the schema of a concept is a representation of a general procedure of the imagination for providing a concept with its image. And in regard to the example of a triangle, Kant claims, the schema of the triangle can never exist anywhere except in thought and signifies a rule of the synthesis of the imagination with regard to pure shapes in space. On this characterization, the a priori constructive activity of the imagination proceeds in accordance with general rules that ground the generality of geometrical proofs, and in this regard, the imagination grounds the universality of geometrical knowledge. And this has been highlighted by commentators like Young, Michael Young, and Friedman. While I don't deny the importance of the imagination for establishing the universality of geometrical knowledge, suggests that the peculiar interpretation of the imagination that Kant forwards in the critical period also contributes to, and indeed helps secure the certainty he attributes to geometrical knowledge. For not only does the imagination synthesize the spatial manifold by means of general rules of construction, on Kant's rendering, the imagination is also fitted to accurately discover the properties of our spatial intuition. That is, the imagination does not itself contribute to or limit what can be constructed in intuition. Only on such a construal can Kant assert that the property of space that we discover via the constructions of the imagination and which are expressed in our geometrical propositions are in fact true of our form of spatial intuition. The most explicit statement of this position comes in Kant's 1789 correspondence with Reinhold, where Kant claims that the power of the imagination to grasp the manifold

5:00 does not contribute to the possibility of representing the geometrical objects in sensibility. Specifically, and this is passage B, Kant claims that when we consider whether an object is sensible, the question is just whether there can be an exhibition of the idea in a possible intuition in accordance with our kind of sensibility. The degree, the power of the imagination to grasp the manifold may be as great or small as it will. Even if something were presented to us as a million-sided figure we were able to spot the lack of a single-sided first glance, this representation would still be a sensible one. And only the possibility of exhibiting the concept of a Schleagon in intuition can ground the possibility of this object itself in mathematics. For then the construction of the object can be completely prescribed without our worrying about the size of the measuring stick that would be required in order to make this figure, with all its parts, observable to the eye. In what follows, I want to help clarify reaches his critical position on the imagination, and in turn shed light on how the B edition imagination contributes to the certainty forwarded prior to 1787. One account is from Kant himself, as presented in the 1764 prize essay, and the other is from John Locke, as presented in his 1689 essay concerning human understanding. While neither Locke nor the pre-critical Kant appeals to the imagination per se, each does appeal to the power that the mind has to perceive the geometrical objects presented before it. And each, as we will see, proposes a peculiar form of maker's knowledge that is intended to secure our perfect insight into the mathematical objects we construct. Locke's account serves as an instructive starting point because on his score, the ideality of our geometrical objects, their status as voluntary collections of simple ideas, that the mind joins together as it pleases, grounds the certainty of geometrical knowledge. In the 1764 prize essay, Kant echoes many of Locke's claims, though unlike Kant, and this might be surprising, he, Kant, embraces an empirical form of space as the ground for geometrical space in order to secure the applicability of geometry to the empirical state of affairs. And now my goal in the paper, which is passage C on the handout. By attending to the problems that emerge from the methods of geometrical construction proposed in and by giving due attention to the progression in Kant's thinking about the connection between

7:30 space and geometrical certainty from the pre-critical to the critical period, we're in a better position to appreciate the role that the imagination plays in the B edition first critique. In particular, as I'll clarify below, these pre-critical treatments grant us important insight into why Kant in 1787 is intent on dissolving any connection between the imagination's the power of perception, what he calls our degree of sensibility, and the certainty of geometrical knowledge. So the next section, Locke on maker's knowledge and geometrical certainty in the essay. The notion of geometrical certainty that Locke forwards in the essay is grounded on his account of the origin of our geometrical ideas. As we shall see, it's because Locke characterizes our ideas of spatial figures as ideal, that is, as voluntary creations that are not modeled on things existing in nature, that the geometer is able to establish demonstratively certain propositions about these figures' properties. According to Locke's classification of ideas, and here you can refer to passage D on the handout, ideas of spatial figures, the objects of geometrical investigation, are simple modes of our idea of space. That is, they are, according to him, different modifications of the same idea of space, which mind is able to make within itself, without the help of any extrinsical object or any foreign suggestion. While our idea of figure is initially gotten from sensation, our mind can create more figures than learn from experience by varying the idea of space. Locke claims in particular, and this is passage E, besides the vast number of different figures that do really exist in the coherent masses of matter, the stock that the mind has in the idea of space and thereby making still new compositions by repeating its own ideas and joining them as it pleases is perfectly inexhaustible, and so it can multiply figures in infinitum. As I've claimed elsewhere in the paper from a couple years ago, the justification for Locke's claim that we can multiply figures in infinitum and without any foreign suggestion rests on his account of space as that which does not resist motion. In particular, we get our idea of space from experience by considering the area through which bodies can move unimpeded and without resistance. Where there is resistance to motion, we get our idea of solidity. It is because Locke understands space in this way

10:00 that he can maintain that we're able to construct an endless variety of spatial figures. From his account, when we create geometrical figures by combining lines and curves, we can create as we please because space itself offers no resistance elects to complete. The only constraint on our constructions is internal, in so far as we are constrained by our mind's power of construction, that is, by what our mind is or is not able to construct. Locke underscores his claim that our ideas of spatial figures are the product of the mind's free activity by asserting that these ideas are not modeled on things existing independently of the mind. As he puts it, the spatial figures of geometry are voluntary collections of simple ideas which the mind puts together, and when we generate these ideas we do so, quote, without reference to any real archetypes or standing patterns existing anywhere, quote. With this characterization of the ideality of our geometrical ideas, Locke emphasizes their essential difference from our ideas of substances. While our ideas of substances are intended to correspond to really existing bodies, the simple modes of geometry refer to nothing else but itself by any other original but the good liking and will of him that first made this combination. From the ideality that Locke ascribes toward geometrical objects, two things follow, which together I take to characterize his account of maker's knowledge in geometry, and which serve to underwrite the certainty of geometrical knowledge. First, given that there are no external constraints, whether spatial or otherwise, that restrict which figures the mind opts to create, we have complete and perfect knowledge can produce. For instance, when we create a triangle, the understanding simultaneously establishes what the term triangle stands for and what it means to be or exist as a triangle. In Locke's terms, we stipulate both the nominal and the real essence of the triangle. So referring to passage F here, a figure including a space between three lines is the real as well as the nominal essence of a triangle, it being not only the abstract idea to which the general but the very ascentia, or being, of the thing itself, that foundation from which all its properties flow, and to which they are all inseparably annexed. Then he states more strongly, and this is passage G, Thus by having the idea of a figure with three sides meeting at three angles, I have a complete idea, wherein I require nothing else to make it perfect.

12:30 That the mind is satisfied with the perfection of this, its idea, is plain, in that it does not conceive that any understanding have or can have a more complete or perfect idea of that thing that signifies by the word triangle. As suggested by Locke's remarks, the very reason that our ideas of spatial figures are complete and perfect is because we have created them. We have, by the power of our own understanding, combined and manipulated our ideas of lines and curves, and once we grant a name to any one of the spatial objects we generate, we can be sure we know the object completely because its creation has relied on nothing but the power and also, of course, the limits of our human understanding. Second, given the ideality of our geometrical objects and our complete and perfect awareness of what these objects essentially are, we are able, Locke claims, to establish demonstratively certain propositions about their non-essential properties. That is, we're able to demonstrate that a geometrical object has properties in its real essence. For instance, though we do not know immediately and intuitively that the angles of a triangle sum to two right angles, since this property is not included in its real essence, we can demonstrate this equality by finding some other angles to which the three angles of a triangle have an equality, and finding those equal to two right ones, we come to know their equality to two right ones. In other words, we must locate an intermediate idea, to use Locke's lingo that is equal to the angles of a triangle and also equal to two right angles. Though Locke is not explicit in this context, the intermediate idea that would allow us to complete the demonstration is the idea of angles that lie along a straight line. Two angles are both equivalent to angles of a triangle, which follows from the properties of parallel lines, and equivalent to two right angles, which themselves are supplementary. Given the transitivity of the equality relation, we can thus establish that the angles of a to two right angles. It's important to notice here that successfully establishing a demonstration demands more than simply knowing the complete and perfect real essence we have assigned to any particular geometrical object. If the task of the geometer were merely to enumerate the essential features of geometrical objects, those essential features we assign to the object, we'd be left with trifling analytic truth such as a triangle has three angles. However, since demonstrations aim to provide

15:00 instructive knowledge to establish the non-essential features of a geometrical object in this case, the geometer cannot rest content with the complete and perfect definitions signified by an object's real essence. And this is pointed out in what I include as passage H on the handout. She must, the geometer must in addition appeal to visible marks that capture the essential features of these objects. Drawing an analogy between our sense of sight and our mind's ability to perceive, the specific suggestion Locke makes in book four of the essay, and I'm summarizing what's in passage I, is that the clarity of our perception of the properties of those geometrical objects we have created hinges on examining visible instances of them, on examining, quote, visible and lasting marks wherein the ideas under consideration are perfectly determined. offer a nice and accurate distinction of the differences between our ideas of quantity. In this regard, the demonstrative certainty of our instructive geometrical propositions hinges on our perception of the agreement and disagreement between our ideas, which he indicates in passage H. Without this element of perception, there would be no way to go beyond the essential features that we have assigned to our geometrical concepts. In light of the role played by our perception of visible figures in geometrical reasoning, We find that for Locke, the powers and limits of the human understanding play a central role in his account of geometrical knowledge. And so referring to passage J there. On the one hand, the power of the understanding limits which spatial objects we can construct. And on the other hand, the same power sets limits on what we can perceive of the objects we have constructed. For instance, if we ask why a two-sided plane figure is not a legitimate geometrical figure, or why geometrical space is three-dimensional, Locke would, it seems, appeal to the limits of our mental perception and maintain, for instance, that space has precisely three dimensions because the human mind is not fitted to perceive dimensions of any greater number. Space has this property, in other words, because the limits of the understanding determine it to be the case. Nonetheless, Locke can still maintain the certainty of our geometrical propositions for regardless of the constraints of our human understanding geometry remains an a priori study of ideal objects. Objects that we have created and that have no alleged correspondence

17:30 to things in the external world. As we turn now to the pre-critical account of geometrical knowledge presented in the prize essay, we see Kant grappling with the very connection between mental perception and geometrical certainty that I take to be the hallmark of Locke's geometry. While Kant captures these features of Locke's account of maintenance knowledge, there is distinction between the objective and subject of certainty of geometry. Kant, unlike Locke, also attempts to secure the necessary applicability of geometry to the world of experience. In doing so, Kant, unlike Locke, leaves the certainty of geometry in jeopardy and faces a problem that will not be remedied until his introduction of the imagination in the critical philosophy. So now, turning to section in three, Kant's 1764 account of maker's knowledge. In the prize essay, Kant offers his response to the 1761 prize question posed by the Berlin Academy of Sciences, which invited authors to explore the nature and degree of certainty attributable to metaphysics, specifically to natural theology and morality, by comparing metaphysical certainty with the certainty characteristic of geometrical truths. the question, Kant distinguishes the method of mathematics from the method of philosophy and elaborates on the certainty afforded by each. As he presents it, the mathematician creates her objects of investigation by using what he terms a synthetic method of definition whereby the mathematician electively combines concepts. For instance, and this is Kant's example, the mathematician defines a trapezium by first electing to think of four straight lines bounding a plane's are not parallel to each other, and then choosing to call the figure a trapezium. In so defining the object, the mathematician also thereby brings the object into existence, for according to Kant, the concept which she defines is not given prior to the definition itself. In philosophy, on the other hand, definitions are arrived at by a process of analysis, and in this case, the definition does not produce the object defined. Rather, the definition sought is the definition of an already given concept. For instance, in their investigation of time, philosophers accept the concept of time as given and then attempt to pinpoint those characteristic marks that belong to its definition.

20:00 That is, they seek to isolate the marks, the mathmala, or properties that together adequately define the concept. So as mathematicians begin their investigations by electively defining their objects existence, philosophers begin their investigations with a given concept, and the definition of the concept is the goal rather than the starting point of their inquiry. In his later discussion, the synthetic method of definition characteristic of mathematics is connected to what Kant terms the objective certainty of our geometrical propositions. For having defined their objects of study, geometers have at their disposal completely adequate definitions that allow allow them to say with certainty that what they did not intend to represent in the object by means of the definition is not contained in the object. On this score, it seems that Kant is following in the footsteps of Locke who, as we saw just a moment ago, grounds his account of maker's knowledge on the complete and perfect knowledge we have of the real essences we have stipulated for our geometrical objects. However, things get more complicated for Kant, no surprise, because unlike Locke Kant indicates that our concept of geometrical space is learned from experience, and as such, it acts as a constraint on the objects the geometer elects to define. And this is something I'm claiming is not present in Locke's account. Admittedly, Kant does not explicitly state in 1764 that geometrical space derives from empirical space. What he actually says is that he doesn't want to trouble himself with questions about the essence of space. Be that as it may, he makes several remarks in the 1764 prize essay that mirror claims made in the 1756 monodology, a text in which he asserts that the form of geometrical space is derived from the form of physical space. Indications that this view is still accepted in 1764 emerge as Kant considers how to apply his proposed method of metaphysics to an investigation of the nature of bodies. In this discussion, he indicates in particular that space does not consist of simple parts, that is, it's infinitely divisible, that the simple parts of body do not occupy space, that is, they're not extended, and moreover, that the elements of every body fill their space by means of the force of impenetrability. This view, that bodies are situated in space in virtue of their external relations to other bodies, is forward in the earlier physical momentology.

22:30 And following the readings of Friedman and Leywein, Kant maintains in the 1756 work that there are simple monadic elements that are essentially non-spatial, and he explains their presence in space through their action on other bodies. Specifically, the simple parts of bodies have no spatial relations, that is, they're not extended in space, until they act on other bodies through the force of impenetrability. The forces between bodies thus somehow give rise to empirical space. Put differently, empirical space is a metaphysically real space that emerges from the dynamical relations among bodies. In this sense, empirical space is ontologically prior to geometrical space, for space, as studied by the geometer, is empirical space with the bodies removed. Kant's suggestion that our concept of space is learned from experience, geometrical space is the form of empirical space helps us make sense of why in the prize essay he counts the concept of space among the unanalyzable concepts of mathematics, concepts whose analysis and definition do not belong to the science of geometry at all. Now referring to passage M on the handout, Kant's claim here is that we do not determine the fundamental properties of space by electively combining concepts as we do with the trapezium and defining the concept of space. Rather, as he puts it, the geometer accepts the concept of space as given in accordance with her clear and ordinary representation. And without a definition of space at her disposal, the geometer must appeal to the clear and ordinary form of space as given in experience, and more precisely, to a sensible representation of space in concreto, to determine the properties of geometrical space. And so Kant remarks, and this is the long passage under M, I set about the task of philosophically defining what space is, I clearly see that since this concept is given to me, I must first of all, by analyzing it, seek out those characteristic marks which are initially and immediately thought in that concept. Adopting this approach, I notice that there is a manifold in space in which the parts are external to each other. I notice that this manifold is not constituted by substances, for the cognition I wish to acquire relates not to things in space, but to space itself. And I notice that space can only have three dimensions, etc. Propositions such as these can be well explained

25:00 if they are examined in concreto, so that they come to be cognized intuitively. But they can never be proved. From what basis could such a proof be constructed, granted that these propositions constitute the first and the simplest thoughts I can have of my object I should note here that in the prize essay, Kant attributes the intuitive cognition of space described above to what he terms the understanding. It is the understanding that he claims initially and immediately perceives characteristic marks in the object. Since at this stage of his career, Kant has not yet proposed the critical distinction between sensibility and understanding, and given the way he uses the term understanding in 1764, to our faculty for perceiving and reasoning, much as Locke uses the term understanding in the essay. So returning now to the passage I just quoted. Notice that the mathematical intuitive cognition of space Kant describes depends on representing space in concretum. On my reading, this requires that the geometer consider the clear and ordinary representation of space gotten from experience, and as such, this mathematical cognition is an intuitive cognition we discover the features of an already given form of space. On this point, Kant is taking an important step away from Locke's account of geometrical knowledge, which rests on a notion of an ideal geometrical space that we have created. And by taking this step, Kant bypasses the problem of applying our claims made about geometrical space to the empirical state of affairs. However, by relinquishing the ideality of geometrical space, Kant, unlike Locke, is committed to the position that there are external constraints on which spatial objects a geometer can electively define and thereby bring into existence. The geometer cannot, for instance, synthetically define a two-sided plane figure because the properties of space, rather than the limits of the human understanding, do not allow her to combine the concept two-sided with a plane figure. But notice as well that given Kahn's account of the unanalyzable concept of space, the objective certainty attributed to mathematics cannot be applied to our cognition of space. But recall that the objective certainty of mathematics is wedded to the adequacy of the definitions proposed. And in the case of the concept of space, there is no definition at hand.

27:30 To make sense, then, of the certainty attached to our geometrical claims about space, we need to turn to Kant's account of what he calls subjective certainty. And he states in particular, and this is passage N on the handout, that taken subjectively, the degree of certainty increases with the degree of intuition to be found in the cognition of this necessity. He points out that based on the standard, mathematical cognition has a greater degree of certainty than philosophical cognition because mathematicians reason about sensible signs in concreto, somehow make transparent the features of the object signified and allow the mathematician to make knowledge claims that have an associated degree of assurance characteristic of seeing something with one's own eyes. He claims further that our sensible mathematical signs provide a clearer impression than the abstract concepts of philosophy and this difference between mathematical and philosophical objects underscores the different degrees of certainty possible in each domain. And so, I quote now, this is the second passage under N, the grounds for supposing that one could not have erred in a philosophical cognition, which was certain, can never be as strong as those which present themselves in mathematics. But apart from this, the intuition involved in this cognition is, as far as its exactitude is concerned, greater in mathematics than in philosophy. And the reason for this is the fact that in mathematics, the object is considered sensible signs in concreto, whereas in philosophy, the object is only ever considered in universal abstracted concepts, and the clarity of the impression made by such abstracted concepts can never be as great as that made by signs which are sensible in character. As suggested by Kant's remarks, when we reason about space and want, for instance, to establish its infinite visibility, we appeal to a sensible sign. In particular, and this is his example, will take a straight line standing vertically between two parallel lines. From a point on one of these parallel lines, he will draw lines to intersect the other two lines. Kant remarks that by means of this symbol, the geometer recognizes with the greatest certainty that the division can be carried out ad infinitum. Notice that though we cannot be objectively certain that space is infinitely divisible, again, because there's no definition of space at our disposal, we can be

30:00 subjectively certain that space has this feature because there is a clear impression of its infinite divisibility that we gather from our sensible representation of lines in space. In regard to this standard of subjective certainty, it's clearly the case that Kant wants the objects of mathematics to do the heavy lifting here. The objects about which we reason make a clear impression and have a sensible character that heightens the certainty of our geometrical propositions. However, notice that the understanding still plays a role in determining the degree of intuition that Kant associates with the certainty of geometrical truth. For as indicated by his remarks on subjective certainty, and as explicitly laid out in his previous discussion of the intuitive cognition of space, the understanding is endowed a sort of perceptual capacity by which it grasps the very sensible signs in concreto presented to it. In the case of space, the understanding perceives its three-dimensionality, and in reference to a drawing in space, it also recognizes the infinite divisibility of space. Even in the case of a circle, where we have a definition at hand, we must, according to Kant's discussion, still represent a particular instance of an object in concreto and draw inferences that are dependent upon the power of the understanding to perceive the object. In this respect, with our degrees of assurance connected to our perception of sensible representations of mathematical objects, we again find a common thread between Locke and Kant. However, there remains a crucial difference that separates Locke from the pre-critical Kant. Locke grounds the certainty of geometrical knowledge on our allegedly perfect insight into the spatial objects we have freely and voluntarily created. But Kant does not have this standard of prize essay due to the foundational role he assigns to the concept of space. As we just saw, the form of space restricts the objects we can possibly define, and more importantly, when we reason about the properties of space itself, introduce, for instance, its infinite visibility and three-dimensionality, we are discovering features of a form of space that we gather from experience. As a result, Kant cannot invoke the same standard of certainty as Locke, for the objects of geometry are not ideal in the sense required to have a perfect maker's knowledge of geometrical space. And the problems I'm raising here are summarized in section O of the handout.

32:30 Moreover, the second problem, as brought to light by Friedman, the empirical grounding of geometrical space poses a serious challenge to the certainty of geometrical knowledge. For since it is the case in the physical monodology, and I contend also in the prize essay, the existence of a metaphysically real empirical space in terms of the fundamental forces acting between simple substances, our knowledge of the essential properties of space are entirely derivative from our empirical knowledge of the laws of dynamics. He's a quote from Friedman. Thus, the claims made by the geometer about the general form and properties of space, which are claims about empirical space emptied of bodies and forces, are ultimately contingent on the dynamical laws governing the empirical state of affairs. As a result, and as Friedman so nicely puts it, quote, it is by replacing Leibniz's conception of the ideality of space with his own conception of the fundamentally dynamical character of space that Kant himself has first exposed geometry to the threat of empirical disconfirmation, end quote. And now a further problem looms. Although Kant emphasizes that the certainty of our geometrical claims about space is wedded to the particular kinds of object about which it reasons, sensible, spatial signs, and concreto, notice that the understanding must also be fitted to accurately glean the properties of these sensible objects. To put the point differently, just as standard empiricist accounts of sense perception must secure the correspondence between our perceptions and the way objects are in themselves, you can think male perception here, Kant must secure the correspondence between what the understanding perceives and the actual properties of space. However, Kant offers no guarantee that the understanding is fitted to discover the properties of the objects it is presented. And without a guaranteed neatness of fit between what the understanding perceives and the actual properties of geometrical space, we're left with the possibility that so-called geometrical truths are relative to and contingent upon our human powers of cognition. It's possible, for instance, to claim considered three-dimensional because the understanding cannot perceive the fourth dimension, or that a two-sided plane figure is considered geometrically impossible because such a figure cannot be perceived by the understanding. What Kant wants to say, of course, is that space is actually

35:00 three-dimensional and that a two-sided plane figure is impossible given the actual properties of space, that we somehow discover that such a figure is impossible through reasoning about about the properties of space. However, given the account presented in the prize essay, he has not yet provided the grounds that would make such an explanation necessary. For that, we will have to wait for the first critique. So now, to the final section. Kant's critical account of maker's knowledge. As we turn from the prize essay to the critical period, Kant initiates some crucial changes to his account of geometry that allows him to address the problem of certainty that lingered in the prize essay Most notably, he transforms the object of geometrical investigation from a form of space given by experience to an a priori form of spatial intuition. This revision alone brings with it two salient consequences, which I list under P.O. handout. On the one hand, by claiming that this form of spatial intuition is constitutive of all outer experience, Kant can maintain the applicability of geometry to the empirical status of the values, which he established in the Prizes. On the other hand, Kant has rid geometry of the threat of empirical certainty, for now having an a priori object of investigation, geometry is guarded against the comparative certainty of empirical knowledge. While the certainty of geometry appears be on more stable ground, Kant's proposal of a pure form of spatial intuition does not alleviate the problems surrounding the role granted to perception in the prize essay. For though the object of geometrical investigation is now given from an a priori rather than an empirical source, we are still in the critical philosophy aiming to discover the features of a form of space that has not been created by us. As such, Kant must, it seems, offer some means to guarantee that our geometrical propositions do in fact capture the features of our spatial intuition. Otherwise, he'll be left with the same problem of perceptual contingency that he faced in 1764. It is in the context of Kant's critical model of how geometers reason that we find the guaranteed neatness of fit between our claims about space and the actual features of our spatial intuition. in particular is an account of geometrical reasoning that rests on the activity of construction,

37:30 an activity that allows the geometer to establish certain and necessary propositions about space and spatial figures without the assistance of perception. The turn away from the passive perceptual awareness of the prize essay is brought to light in 1787b preface where Kant claims that geometry reached the secure path of the science because a revolution took place in mathematical thinking. this is passage Q a new light broke upon the first person who demonstrated the isosceles triangle that is who demonstrated the base angles of the isosceles triangle whether he was called Thales or had some other name for he found that what he had to do was not to trace what he saw in this figure or even trace its mere concept and read off as it were from the properties of the figure but rather he had to produce the latter from what he himself thought into the object and presented it through construction according to a priori concepts, and that in order to know something securely a priori, he had to ascribe to the thing nothing except what followed necessarily from what he himself had put into it in accordance with its concept. As Kant puts it here, the revolution in mathematics is credited to the geometer who realized that geometrical thinking should not rely on tracing the image corresponding to the concept of a figure and reading off from the image the properties of the figure. In other words, the geometer realized that geometrical thinking must be divorced from perceptual modes of knowledge. And I read this passage in that light also because of some remarks made in the prolegomena that are similar. The method of construction that Kant forwards in the critical period can thus be seen as replacing the sort of method endorsed in the prize essay, one that we just saw, relied on our capacity to perceive the properties of space spatial objects. The replacement succeeds in part because the constructions that characterize geometry are linked with our judgments of spatial figures rather than our perceptions. In brief, and as at least suggested in this passage above, the geometer produces spatial figures in accordance with a priori concepts. I want to focus specifically on the pure concepts of the understanding, the role they play for Kant here. As such, the maker's knowledge of the first critique relies on intellectual modes of knowledge whereby we both produce and judge spatial figures according to those concepts that we thought into the object, that is according

40:00 to our operator categories. Certainly though, a maker's knowledge that relies on pure concepts alone will not suffice because this knowledge is intended to supply insight into the properties belonging to our pure form of spatial intuition. We must thus complete constructions that proceed but that also somehow reveal the features of space itself. It is in this regard, as Kant attempts to account for the relationship between understanding and sensibility, that the role of the imagination in geometry comes to the fore. In the first instance, the imagination has set the task of producing the very signs in concreto about which geometers reason in pure intuition. I mentioned this in the introduction earlier. But moreover, the imagination completes its constructive work by mediating between the pure categories and our pure form of intuition. As spelled out in the B edition, Transcendental Deduction, and you can refer here to R on the handout, the construction of spatial figures, that is, the imagination's figurative synthesis of the spatial manifold, both belongs to sensibility and proceeds in accordance with the pure concepts of the understanding. In other words, in the framework of the critical philosophy, the imagination is assigned a mediating role that grounds the understanding's ability to make judgments about the sensible objects that it has itself helped construct. Or to put the point more strongly, the understanding can make judgments about sensible objects precisely because the constructive procedures of the imagination proceed in accordance with the categories. For instance, when the productive imagination generates a line, segment, and space, it begins with a point and then generates the segment successively, going from part to part. As Kant clarifies in the Axioms of Intuition, this process is guided by the pure concept of magnitude and allows us to recognize the segment as an extensive magnitude, an object extended in space. Importantly, the activity in imagination also belongs to sensibility, insofar as the constructions of the imagination proceed in accordance with the subjective conditions of our spatial intuition. For instance, when Kant considers the impossibility of constructing a two-sided plane figure in space, he claims that the impossibility of the figure rests not on the concept in itself, but on its construction in space, that is, on the conditions of space and its determinations.

42:30 The suggestion here is that the imagination is fitted to reveal the features of space accurately and without distortion. It reveals in particular that the bi-angle is impossible because the form of space renders the object impossible to construct. Notice that we gain from Kant's account of construction a more nuanced portrait of the sensible representations that stand at the heart of geometrical reasoning. These sensible representations, our signs in concreto, are constructed by the imagination in accordance with our form of spatial intuition and the categories of the understanding. thus moving beyond the account of the prize essay, according to which the understanding draws inferences about geometrical space based on a sensible representation that somehow grants us insight into the properties of space, and more specifically, where the assurance of our geometrical claims was wedded to a subjective certainty akin to seeing with one's own eyes. That was the passage from 1764. As clarified in Kant's 1789 letter to Reinhold, which I included this passage be on the handout. The power of the imagination to grasp the manifold, its own power to proceed, does not determine or at all contribute to what is or is not sensible, that is, to what is or is not a possible object in mathematics. As Kant says to Reinhold, the question of constructing figures in space is not dependent on our degree of sensibility, that is, on what figures can be made observable to the eye. Rather, the constructability of geometrical figures hinges solely on the features of space. And so my concluding remarks, and these are included in passage S. By granting the imagination a central role in his critical account of geometry, Kant places the certainty of our maker's knowledge in geometry on less subjective ground. For with the imagination, as an alleged neutral intermediary and understanding, such that its construction procedures are both guided by intuition and directed by the categories, the certainty of our geometrical judgments is a certainty with purely a priori grounds. While the objects of geometry are not ideal in the Lockean sense, insofar as they are not the product of the mind's free activity, they are nonetheless a priori constructions that by means of the imagination enable us to discover the actual properties of our pure

45:00 form of spatial intuition. Count of the Imagination, grants Kant a way of dissolving the ties between perceptibility and geometrical certainty that he forwarded in the prize essay, and thus, a novel and important way of securing the certainty of geometrical knowledge. Thank you. Thanks. Thanks, Mary. Okay, questions? Doug? Make your knowledge to me, probably because I think it's pretty critical for Hobbes' attitude. I wonder if on your reading of Locke and or of Cronk, does it fall into the fact that we have maker's knowledge of geometrical objects, mathematical objects generally, that every property of those objects is in principle knowable and demonstrable? In principle, yes. I think in law, it is. In the earlier works, would I say that of the prize essay? I would, of the prize essay, definitely, because... So there couldn't be any that were unsolvable problems. In principle. But then the contingency is on the visible instances we have to rely on to actually complete the demonstrations. so we have to still get the method right so to speak for how we get those, to use Locke's terms how we get those properties to flow but in principle even in the case even in Kant's case it seems like you've been committed to the view that since you construct these objects in intuition there can't be so to speak any surprises in the critical period Um, surprises in what sense? Well, things that would follow as consequences that you would have been unable to, surprises in the sense that there might be things that you imagine you should be able to derive and you just can't, there are problems that you simply can't solve without these things. They're squaring the circle being an obvious case. think that well I know what a circle is there's no hidden facts about what it is a circle, I know what area is

47:30 in some sense my own invention or construction so the thought that there must be a way of determining by a simple construction what the area of the circle would have to construct the square area of the circle, it looks like it should be a solvable problem, a trisection of an angle can it be? But it seems that there's no, it doesn't look like, it's certainly in lock, if possibly not, there's not a lot of room for this idea that maker's knowledge, which is supposed to ground our demonstrative knowledge of all these mathematical objects or two methodologies, is somehow going to be insufficient. There will be these cases where we simply, we have all the concepts that are available, and the problems we can pose for ourselves if we can't acquire solutions. Right, the insufficiency would depend on method, the sorts of method we use to try to solve these problems. So if we can't, you're right, in principle they should be solvable. They are solvable. But whether we actually come to a solution depends on whether we are using the right method. And this is sort of indicated in that deep practice passage about the revolutions of place, because there was this great discovery, that they didn't realize. Thanks. Mary, going back to the pre-cratical philosophy, 1756, if I've understood the position he's setting there, in the physical monotology he's presenting a picture of metaphysically real space emerging from reciprocal relations between effectively dynamical, foci of dynamical action, unextended exactly but they seem to be quite straightforwardly like monads my question is this do we know if Kant read Boscovich and if so when it just has a very this seems to be essentially a kind of tensional theory of dynamics and it seems to carry a rather strongly Boscovician flavour what do you mean by we? I don't know Do we, well, does anybody, does anybody know? Do you know? Does anybody know? I would say yes. There's a bell that's ringing, but I'm not sure if it's because someone has brought this point out that there's a similarity or it's a direct influence of Kant having read Boscovich.

50:00 Certainly would be worth pursuing. Okay, thanks. No, thank you. Marco? Yes, I tried to understand because it's not a, it's not a, yeah. For me, for me it's not easy. So help me to understand that. So what you call construction in the critical period of construction according to a priori concepts, in fact, can be understood as the sort of rule of introduction that correspond to the absolute postulate of Euclidean geometry. What rule of construction are? Rule of construction are the rule of Euclidean geometry. For example, if you have two points, you have a start line. And it is what you call a rule of construction corresponding to the concept. If you have a start line, you can have it you can produce it if you have a segment you can have a circle right yes is that yes what what are the rules of construction are yeah so for Kant um okay this gets really abstract so the fact that you can follow let's try to yes well there are rules you use in practice when you You do geometry, textbook rules. You open it up and you have to follow the rules as a student. The rules he's talking about are rules at the transcendental level, right? Rules that our understanding and imagination need in order to even have a sensible object at all. So in order for me to sense anything in space, there are certain processes happening, constructions, of his examples um well i kind of mentioned it but uh in order to recognize this as an object i need to first recognize that it's in space so what happens is you trace so to speak you don't realize it's not psychological you're not aware of this but what has happened at the transcendental level is you've traced the object and then recognize it as spatial and then there are many

52:30 more kinds of constructions like this that let you recognize it as very particular as water bottle beyond simple object in space and time. Yes, but they are the object you have before. In the case of geometry you have not object before in constructed so you cannot apply the same proportion. No, he actually wants to say the same general rules actually apply. When I recognize this as spatial there are certain concepts magnitude that I have to apply to actually recognize this as in space, for him the same rule, general very general abstract rule has to apply when I construct a triangle because otherwise just an impure intuition as he would say so construct he would call it a pure intuition as opposed to drawing one he would call that an empirical intuition if I drew a triangle there it would be empirical think platonic somewhat, you know, in pure intuition if I construct a triangle, if I'm thinking about it, then I'm applying the same rules. And the reason he's so adamant about this is because he wants geometrical propositions to apply to water bottles and things in the world. So for him the same rules have to be at play when you're perceiving an object as when you're generating a triangle, a pure triangle in thought. So these are at work, purely and empirically. I said that in geometry we have not only three angles, we have clues about three angles. So, you know, the interesting point is to understand that our clues work in Euclidean geometry. So what you are saying is that the construction you're speaking of is absolutely not the construction in Euclidean geometry that we apply or to prove something in Euclidean geometry. The sort of construction, Euclidean construction, is already belonging to imagination, according to you. It's already something empirical, in some sense. In some sense, yes. In the Kantian states, the Euclidean-specific rules would be considered empirical practical rules. But what matters for him is that underlying any empirical construction I would do on the board are these pure constructions that follow very general rules that map onto our categories of the understanding.

55:00 Sorry. Okay, Gerhard. Kant is a critique with the remark that there are two abilities, sensibility and understanding. And then with B161, he says, This is only a biological introduction. In fact, we have to introduce a form of intuition. For this form of intuition, not a form of intuition. And now this form of intuition, I think it was you to break by imagination. But now, if this is right, there is a tension now with a wide round of three abilities. and the beauty. And I understand. Okay, good. So, yes, there is this distinction at B161, this infamously difficult footnote, where he distinguishes a form of intuition from a formal. So the form of intuition would be space, just the form of spatial intuition. And you're right, when I'm talking about a formal intuition, having... I just need to draw a triangle. That's all I'm going to do. So this is the formal intuition, right? It is a specific object, a particular. We are able to construct it in the form, in the form of spatial intuition. So when you said the formal intuition on my reading relies on the imagination, you're absolutely right. In order to construct that, we needed to have the imagination mediate between understanding and sensibility. But the reason I wouldn't say there's a third sort of capacity is because how I read this is the imagination is allowing the understanding to judge that object at all. So there wouldn't be a third. understanding needs the imagination to do the work at all. So to say triangle or judge that as anything. Yeah. Thanks. Sure, Jacques. Did you consider that the problem of the question of

57:30 schematism is not in your topic? I'm sorry, I didn't hear the word. The question now... The question of schematism, schematism, schematism, yes, is not in the topic. The schematism, so, yeah, I didn't... The question is why you don't speak about schematism. Okay, in some ways I did, but indirectly, insofar as... So as I mentioned in the beginning, the schematism, and sorry, maybe you weren't here for the introductory part, But in the schematism, he mentions that the schema are rules of the imagination for constructing objects, for giving a concept its image, for instance. So a triangle, to construct a triangle, you need the schema. And so even though I don't explicitly at the end come back to it, I think I do discuss it implicitly, and it maps on. I mean, is there a way in which you see that what I've said is not consistent with the schematism? Because you are at the metaphysical level and not at the transcendental level. In which, when I'm talking about geometrical practice? Right. So, I mean, I guess I would see it this way. The understanding has concepts. Sensibility has this form of intuition. And the imagination has these bridging rules, the schemata, right, that give the, as he puts it, the sensible account of the pure concepts. And then the a priori. Yes, and the schema are a priori. Right, so at the transcendental level, these are all a priori across the board. So, yeah, my focus was on the activity of that faculty, less on its object, so to speak. Right, so understanding has concepts, sensibility has form of intuition, and imagination has the schema. So, you're right, I didn't focus on it, but it is part of the more general future. Yeah? Okay. I have a question about this. I never know what to make of the contribution of imagination or understanding of any of this, because the way Kant describes it, he doesn't use those notions.

1:00:00 He says, we adopt as our new method of thought that we can know a priori of things, only what we ourselves put into them. And the German is, of the last phrase, So it looks to me like he's conceiving of this as some kind of voluntary thing. It's not stemming from capacities that are out of the range of our explicit knowledge. We lay things. We put things into objects. If you're going to bring understanding, imagination, any other sort of capacity into the picture here, you have to give some account of what it means to put properties into things, okay, and to do it yourself. I mean, what I'm really concerned about is that there's a question of agency or ownership that's being overlooked here, okay? And maybe that's what it all comes down to, conditions under which it's proper for me to consider something to be my creation. My creation must be self-esteem, okay? And if that's the case, then it doesn't seem to me that understanding or imagination plays a major role. Okay? It's not a question of these kind of cognitive capacities. It's a matter of legislative power or authority or something. I can do with my concept what I want. Okay. And now you're talking about the level of geometrical practice. Yeah. So I guess one first point is that I'm reading the we ourselves put into it differently, obviously. I mean, I'm using it as a springboard to talk about the a priori transcendental stuff. Right, I realize that. I just wanted you to say why. Well, maybe I should draw more, though. It's a nice drawing. Sorry? It's a nice drawing. I was going to erase this one. Nice. Let me give you one of Kant's examples from the doctrine of method. Of an actual construction.

1:02:30 And then I'll explain how I interpret what's going on. So the question is, how do you prove that the angles sum to two right angles? Right 180 degrees. and he says the first thing the geometer has to do is extend one of sorry, this is a little off but extend one of the sides then you construct a segment that is parallel to the other one and then through parallel lines you can kind of get these three are equal to those three I won't go through the whole thing so I think the point you're making is when I even construct a triangle and I say this is a triangle And then I've said a three-sided, three-angled plane figure is called a triangle. And that's what I've done. I've made the triangle as I've named it or defined it. But the point I'm trying to bring out here is that this for him is successful because underlying this at the transcendental level, not this level of empirical practice, is the form of spatial intuition and the same sort of constructive procedures that I did here are mimicked, so to speak, or this is a mimic or imitation, a cheap imitation, of the transcendental processes of construction by the imagination. And so the fact that I can make this judgment and do this successfully depends on the fact that at the a priori level, the imagination is doing something similar and generating things in space. But it doesn't seem to me, Mary, that it's right to describe myself as putting into a concept what my imagination does. I mean, my God, my imagination could do all kinds of things that I have no idea of. Oh, I'm not reading imagination. So I'm trying to, the imagination I'm talking about, I'm sorry, maybe I should have been obviously more clear about this, but it's the technical sort of faculty. I know that. the empirical imagination. So he has the productive imagination, which is what he says underwrites geometrical certainty. And then he has the reproductive imagination that allows us to reproduce figures empirically. So, I mean, when you say, you know, you can imagine sort of anything and come up with all sorts of... No, I mean it that way. I meant that I'm not familiar with all of the operations that

1:05:00 my imagination can perform. So my imagination might put all kinds of features into objects that I'm not aware of. I don't understand then, if that's the case, how it would be proper to describe that putting in of properties as myself putting them into the object. I had nothing to do with it. It's my imagination. Yeah, because for me, I'm not reading that as, when you say me, I would still say you are, though, it's reason. I mean, for Kant, this is sort of his way of suggesting, I mean, even in that passage about, that you just quoted about putting only what he himself puts in, I mean, the ownership is, yeah, the ownership is reason itself. The ownership doesn't require knowledge of ownership? I wonder about that because going back to Hobbes what his creativism was supposed to establish was this kind of transparency because what God created I don't have any special knowledge of. Question, who was it that created my understanding or my imagination? It wasn't me it was God so how how in God's name is this supposed to any special command over things. I don't know. God made my imagination. It may lay into things properties all the time, but I don't have any special knowledge of that. Right. If I did, I'd be God. He wants to... In another term, if I understand the world, it seems that the solution of certainty is simply to say that geometry is certain. is to say that the imagination is the solution of the problem of certainty. Imagination is what you are describing. So simply the solution is to say, okay. It's not a matter of discernment, it's a matter of legislation. It's not a postulation of certainty. Imagination is the name of the postulation of certainty. Maybe. And yes, that's what I thought I would say. Well, in my sense, it's a solution to the problem that Kant was facing with perception

1:07:30 because he kept, and he and Locke, you know, are worried about what the mind, the mind's eye, so to speak, and what we can perceive. And so in that sense, my emphasis was Kant recognized that was a problem and did exactly what you just said. use the imagination in 1781-1787 to solve that problem of perception, or to put it in your terms, the postulation of the imagination is the postulation of geometrical science. Yes, so yes, you do understand. Better than I do. Paula? Yes, I was very impressed by what you said about the impossibility of a figure. It's not in the concept, it's in the fact that you cannot realize it in space. And I was interested in the connection between intuition, imagination, and the realisability in space. Because there might be figures that you cannot construct in three dimensions, but you can construct in four or five. And so the idea is, what would Kant say about that? Is it that it's totally impossible because of how the imagination works? Or is just the intuition that is connected to the three-dimensional space and would imagination have the possibility of going somewhere else? Yeah, good. Yeah, my sense is that it's not, any restriction on four dimensions or beyond wouldn't be about the imagination per se, because we can't imagine beyond what even our three-dimensional space allows maybe, in some sense. But it would be intuition, a form of facial intuition that for him, you know, is form that underlies experience. So because he's so worried about the match between pure form of spatial intuition and empirical space, on my reading it would be spatial intuition itself, and not the imagination's restriction. I mean, there are some, and maybe I should note at this point, because it's somewhat connected, that I focus primarily on the second edition because the first edition in 1781, the imagination has a different spin. It seems to be a third faculty that stands on its own. I mean, he made some remarks that suggest this. And there, I think, he is trying to give it as much freedom, so to speak, as possible. But then he has problems trying to get things to work with understanding and sensibility.

1:10:00 So in the B edition, the imagination, as I'm emphasizing, is in the service, so to speak, of sensibility and the understanding. So, in the imagination, as I read it, if the imagination is doing its job right for experience, it's spatial intuition that's restricting it, not its own inherent limitations. So it could be applied to things other than what we have as transcendental faculties, is how I read it. Other questions or comments? Jeremy. Yeah, Jeremy. Well, part of this question, I'll pursue a little bit because I'm very confused about this. Supposing you talk Kant through some piece of four-dimensional geometry, the symmetries of a hypercute, let's say, what has he got knowledge of? Because it's not knowledge of space. Or is it? It's a knowledge of space, then, or hypothetical space. It seems to suggest there's only one space. And we somehow, by these transcendental algorithms, have knowledge of it. We can have the capacity for knowledge is rooted in some transcendental argument and gives us knowledge of the one real existing space. That's what the certainty of geometry is. It's true knowledge about real space, so far as we can ever have that kind of knowledge at all. So now you go back in your time machine and you instruct Kant in four-dimensional hypercubes, which I'm sure you can do. And what's he got knowledge of? What would he say to his not knowledge of space? Draw hypercubes. Yeah. Give me a few days. I'm on your computer. Speak a little bit. Well, I mean, it would be, the way I would read it is it's no different than saying, for instance, you can have knowledge of the rules of chess, right? putting in place certain restrictions. I'm going to start with four dimensions rather than three, and then you can go and examine, you know, what would follow from your starting assumptions. So it wouldn't be knowledge in his sort of strict sense, right, of the world or experience, you know, in his technical term, cognizing, you know, what is here in reality.

1:12:30 But you could still, I mean, you can learn this different, this alternative form of mathematics. You see, what always puzzles me, well, one of the many things he seems ever so certain about what space is, somehow or other. And this hypercube ought to scare him a little, it seems to me. It seems to me that it would make knowledge of geometry rather more hypothetical and rather more business of obeying certain rules. Oh, it just so happens in three dimensions it sort of clicks in with reality. but that's not the way it reads at all I always wind up thinking that it's confidence in what space is never mind the later history of geometry is actually naive it's a curious thing to say about Kant and suggest that I don't understand so the four dimensional case I don't think would count as geometry in the rich sense because there's only one for him right but I mean the way about this is, I mean, you call it naive, the view of space he has and is so confident about, but it's the view of space underwriting the Newtonian natural philosophy. So given that his project, more explicitly in the V edition, is to explain the certainty of these two things, these pillars of science, geometry and the Newtonian worldview, then it doesn't seem so outlandish, right? I mean, he's so certain because that's the geometry you need To have all the architectonic work for him. Okay, I don't want to hog the discussion, but if you can persuade Kant, there are other geometries, just say a high-dimensional one. In some sense, figures have been constructed in the imagination, or some way, if not in the imagination, then at least somehow you can draw a hypercube and do stuff. Then there are rules, we make instructions, we've got this symmetry group and all this stuff, right? You know, you can put that face here and that cube there and all this kind of stuff. It seems to me he also said our ability to do geometry is rather more elaborate and distinct as a way of acquiring something, even though I'm not going to call it knowledge, than he wants to say that it is. And your examples have always been straightforward generalizations of what textbook you did in geometry tells you to do.

1:15:00 and now we're saying know them that some capacity of the mind can do pretty reliable stuff there's way beyond that and I would have thought this sort of a cold geometry however can't conceives it a rather awkward relationship with knowledge and it doesn't seem awkward it seems to be explaining how natural it is and I want them to say something more about how there isn't a perfect overlap somehow not so many decades later this becomes finality I'm not saying anything surprising it just yeah but as related to the point earlier I hope I'm not talking past you or missing the point the imagination can do more it's not restricted to just this form of space per se, but as applied to experience, it is. So, you know, I mean, I'm gonna use a stupid example, but imagining unicorns or something would I think fall under the same category for him as the sort of thing you're describing. I mean, it's beyond what experience can offer, but it's not, and therefore it's not an object of knowledge in a technical sense. It's just a sort of mental play thing, you know, something that, you know, is a curious object, but not necessarily, you know, for him, a challenge to the boundaries that he's putting in place. That's my best case. Okay. Scott? Yes, my question concerns your passages M and N, and it's essentially a size issue, but I just have difficulty understanding this. When you write here about the concept and the relation to the sensible signs in . Right, and so there's a distinction being made here between mathematics and philosophy. So I'm thinking, why is it just philosophy? What's happening with philosophy? we had this idea that you presented that indeed philosophy has a concept, and then we're trying to identify properties that correspond to this pre-given concept of a thing, which is kind of like natural philosophy, if you think about it. We have a concept, we have a phenomenon, we're trying to understand them, we're trying to get our minds around this by defining various properties and states.

1:17:30 But of course, if we do that, in natural philosophy, we have sensible signs. I guess maybe my problem is, what is actually a sensible sign in concreto? Is it just the fact that we can write down symbols? In 1764, it's not as clear as it becomes. But in 1770, he's much more explicit that sensible is linked with spatial. So you can draw something in space. In the passages that this come from, the 1764 piece, more or less he's contrasting the sensible sign with the abstract sign, where the abstract sign is the word, as opposed to something, the words of philosophy, the concepts well the words actually he says explicitly it's the words that are the signs of philosophy okay, then most natural philosophy is mathematics most of the natural philosophy is mathematics in the sense of, yes, yeah, and I think you would agree with that, that it has the sensible spatial basis, and it's built, but there's, I mean, on Kant's critical reading, at least, the more mature understanding that Kant has a natural philosophy is, to space, you have to add mass or matter, you have to have something there, and then you can have natural philosophy. We can do this without mass at all, it hurts his mechanics, so mass now becomes, it's not even there. Right, but for the Newtonian view that he's trying to defend here, I mean, or, well, I'll just finish that thought if you want to challenge me on this, But for the Newtonian view that he's trying to underwrite, so to speak, the laws of motion, and bodies, et cetera, mass is somewhat fundamental. And it comes out most explicitly in the metaphysical foundations of natural science, which he writes in between the two editions of the first critique. Yeah, so what the example of Hertz gives us is that, and this is something that Michael Ray pointed out But then we have these different hypotheses that enter in, such as, in the case of Hertz, hidden masses. So these are just signs with no possible,

1:20:00 sensible component attached to them. Right, which actually for Kant would probably be akin to Leibnizian monadic natural philosophy, right? You begin with the non-spatial thing, which is what he comes to reject in the critical period he's a believer in the early years but then that sort of notion that there's massless sensible stuff in nature he rejects Michael yeah I want to go back it seems to me to I hope this is bit with what Jeremy said but I also want to go back to your reply to Doug about maker's knowledge it seemed to me you were saying that he he asked you this and his specific example was about the squaring of the circle here's the circle I have a perfectly good geometrical concept of that area so here's a question can I construct a square with exactly the same area as this circle has. So the general question is, are there questions that are going to come up if you really believe in this maker's knowledge idea? Are there questions that are going to come up that you just can't answer? And I think in the reply to that you said, well, well, no, but it will depend, the methods of solution will change, or something like that. I said in principle, yes. Oh, you said in principle, yes. But whether we actually do find the solution will depend on the methods. Right, well in these sorts of cases it turns out that you have to embed these problems in some sort of higher framework to get a solution out and it turns out in that case to be negative you can't do an elementary construction that will produce a square with the same area but you know that only because the problem gets embedded into some higher analytic framework where you can carry out the analysis. Now, to tie that up to Jeremy's question, I can't think of an example.

1:22:30 Jeremy probably can, but somehow to get, I mean, he was asking you about knowledge of this hyper-queue, you know, four-dimensional geometrical object. Now, I guess you can imagine Kant saying, well, of course there is a kind of formalistic, right but it's something like a formalism and it's not clear that it's it's not certainly not clear that it has a what the word would be a concrete meaning or whatever but you can produce formal results and it's not clear what those formal results are about but suppose you take a case you said I can't think of one but suppose you take a case where you can actually get a result ordinary geometrical things, but the only way you can get it is by invoking facts about, we call them facts, about things in hyperspace. Can you pick in a jar form? Not offhand, but there must be a bizarre type trick. I mean there must be a complicated three-dimensional configuration, which are nice. Okay, yes, there are certainly singular surfaces that are non-singular in spaces of high dimensions, but an AC trick. So you call them up for high dimensions, do them there and push them back down. So let's go with the idea that there must be one, even though I haven't thought one. What would Ken say about a case like that? So it would be impossible, then. I mean, the assumption is, the claim would be, you know, The method of solution would necessarily require recourse to a higher dimensional. I mean, you can think of examples. I mean, I can think of examples in the case of number theory. where, I mean, just think of an example like Fermat's Last Theorem. It's a perfectly concrete statement about ordinary natural numbers, right? But to get a proof of it, it turns out to be correct, but the proof of it involves... Telomology, incredible amount of abstract machinery from algebraic geometry. You know, what we can say about a case like that. Well, I guess there'd be the initial, I mean, my initial question would be something to the effect of do you know necessarily it's only solvable by appeal to those higher methods? Yeah, sure. Yeah, that's a natural question. You just didn't find the right solution.

1:25:00 But let's say that you do carry out some analysis that says that yes, it's necessary for you to use these higher dimensional, you know, higher mathematical routines. I mean, is this kind of thing a challenge to this whole picture, or is it... If it is a kind of solution, it is a challenge if the solution you're suggesting would be one where it didn't at all rely on construction procedures or the properties of space generally, then yes, I think it would. Because what he's saying is at this transcendental level and this crazy abstract stuff going on is the very ground for the empirical possibility of coming up with solutions to any geometrical problem. So if it turns out that the solutions actually these abstract, a priori, crazy, transcendental construction procedures, then there's a little bit of a problem for him. Because it's all supposed to fit together, right? What happens transcendentally is the grounds for him, for what happens in practice, if it's done right. I mean, this is why the B preface is so important, at least for my story, because there had to be certain ways of thinking put in place. And what he's really hinting at in the history of science, Celli and Bacon on my reading, is what they hit upon without knowing it was a method that mapped on in some nice way to what he is proposing happens at the a priori transcendental level. So if we now go askew from that and say, well, actually, they were wrong, that's the wrong science and the wrong methods, then for him, I think it would be a big challenge. Because either he'd be caught in a position where he'd say, well, this new science isn't you have to rework the whole system that you put together. I mean, would it be in the end that if you had held on to this Kantian view, you'd have to say that that's, we don't know that Fermat's last theorem is true. What we've shown is if blah, blah, blah, blah, blah, then Fermat's last theorem holds. But that's not in itself a demonstration of the truth of Fermat's last theorem. The demonstration of the truth of Fermat's last theorem would only come by some elementary number theoretic.

1:27:30 If you hold on to this, yes, project of the critique, then I would say yes. Oliver. I want to insist on a little detail of your talk. So, you said that insofar as the objects are in space, they are already conceptualized as extensive magnitude. But extensive magnitude is not a term used by Kant. Kant can't distinguish between quanta, quantum and quantitas. And Kant's pointed that insofar as things are considered in space, they are conceptualized as quanta, as measurable things, but not as quantitas. We conceptualize them as measurable, but we have no concrete idea of their actual size. Right, right. And my question is just a suggestion of a reasoning and I wanted to hear your opinion about this. If the proper object of geometry are the quanta and not the quantitas, then non-Euclidean geometry should be impossible because in non-joupledian geometry the properties of geometrical figures depends on their actual size. The angular sum of a triangle in a non-joupledian space depends on its actual size, but the actual size is not an object of geometry. It's quantitas, not quantum. It should be, but just to clarify, it should be impossible as a basis for the nature, not impossible as a proposed formalism or system, right? Yes. I agree. I mean, this historically is what happens, right? The first initial proposal of non-Euclidean geometries doesn't really bother Kantian so much, but it's when Einstein then uses it, you know, for the basis of understanding nature that it becomes a real flow because now you're incorporating it into our understanding of nature itself. But the impossibility question, I mean, it goes back to that Jeremy was talking about before.

1:30:00 I mean, it's still something you could think about, this system, right? I mean, but to know it in the Kantian sense or to use it for reality, for Kant, that's a no, that's a big no-no. Okay, yes, but one of them says, non-Euclidean geometry is not compatible with Kant because it is not intuitive. But I think this is nonsense. It is really ruled out by Kantian philosophy. As a basis for understanding nature, yes, not as a system, so yes, I agree. Can I say something? I think that you don't need to go so far to the distinction between one of those quantities in order to realize this conclusion. the simple fact that for Kant, the geometric theory where a triangle is a form of the integral object needs in order to be coerently defended the idea of similarities. So you need it. So it's much more specific on the bottom. You need it to have the idea that similarity is a linear geometry. So otherwise you cannot say that the triangle is a form of in the world and be with the naturalism of that. Same to me. The simple idea that Germany is Africa, that the quire or exclude the possibility of genealogy. That's that. Same to me. I'm perfect. You mean, thank you. Other questions or comments? Thanks, Mary. Thank you. All right. we have lunch at 12 o'clock the same place we had it yesterday so it's just a short walk but you've got plenty of time and back here at what time and back here at two o'clock