Stud. Hist. Phil. Sci. 34 (2003) 29–43 www.elsevier.com/locate/shpsa

Transcendental philosophy and mathematical physics Michael Friedman Department of Philosophy, Stanford University, Stanford, CA 94305-2155, USA

Abstract This paper explores the relationship between Kant’s views on the metaphysical foundations of Newtonian mathematical physics and his more general transcendental philosophy articulated in the Critique of pure reason. I argue that the relationship between the two positions is very close indeed and, in particular, that taking this relationship seriously can shed new light on the structure of the transcendental deduction of the categories as expounded in the second edition of the Critique.  2003 Elsevier Science Ltd. All rights reserved. Keywords: Kant; Mathematical physics; Transcendental deduction

The first edition of the Critique of pure reason appeared in 1781. Soon thereafter, in January 1782, a highly critical review contributed by Christian Garve and revised by the editor J. G. Feder was published in the Go¨ttinger Anzeigen von gelehrten Sachen. This review, as is well known, maintained that what Kant had produced was simply a new version of an old doctrine—a version of psychological or subjective Berkeleyean idealism. Kant, not surprisingly, was displeased, and his next statement of the critical philosophy, the Prolegomena to any future metaphysics of 1783, was clearly intended, at least in part, to answer this charge of subjective idealism. Indeed, the Appendix to the Prolegomena, ‘On what can be done to make metaphysics as a science actual’, is almost exclusively devoted to an explicit reply to the Garve– Feder review. In particular, Kant attempts conclusively to differentiate his view from Berkeley’s by focussing on the critical doctrine of space (together with that of time):

E-mail address: [email protected] (M. Friedman). 0039-3681/03/$ - see front matter  2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0039-3681(02)00085-7

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I show, by contrast [with Berkeley], that, in the first place, space (and also time, which Berkeley did not consider) together with all of its determinations can be cognized by us a priori, because it, as well as time, inheres in us prior to all perception, or experience, as pure form of our sensibility, and makes possible all sensible intuitions and therefore all appearances. It follows [in the second place] that, since truth rests on universal and necessary laws, as its criterion, experience for Berkeley can have no criterion of truth—for the appearances (for him) had nothing a priori at their basis, from which it then followed that they are nothing but mere illusion [Schein]. By contrast, for us space and time (in combination with the pure concepts of the understanding) prescribe their law a priori to all possible experience, which, at the same time, yields the secure criterion of truth for distinguishing, within experience, truth from illusion. (4: 375)1 The second edition of the Critique appeared four years later, in 1787. Here Kant extensively revised some of the most central—and most difficult—chapters of the book: the paralogisms of pure reason, the transcendental deduction of the categories, and the system of the principles of pure understanding. The chapters dealing with the first two topics were completely rewritten. In the case of the principles chapter the revisions are not as extensive, but Kant did add two entirely new sections: the famous refutation of idealism, and a general remark to the system of principles, which, among other things, serves as a ‘confirmation’ of the refutation of idealism (B293). In addition, Kant also significantly revised the structure of the transcendental aesthetic by separating two distinct lines of argument with respect to both space and time—one he calls a metaphysical exposition, which articulates the synthetic a priori character of the representation in question (space or time) by elucidating ‘what belongs to it’ (B38), and the other he calls a transcendental exposition, which demonstrates the synthetic a priori character of the representation in question by showing that only on this assumption is a certain body of assumed synthetic a priori knowledge possible (geometry for example, in the case of space). Many of these changes introduced in the second edition, as one might expect, are intended further to delimit Kant’s view from subjective idealism; and they do this, in particular, by emphasizing the importance of the representation of space in Kant’s system and that ‘appearances’ include—indeed centrally include—material physical bodies located outside us in space. This is especially true of the refutation of idealism, of course, which argues that even my knowledge of my own mental states in inner sense is itself only possible on the basis of my perception (my immediate perception for Kant) of external material bodies located outside my mind in outer sense. It is also true of the general remark to the system of principles, which, after pointing to the need for given intuitions to instantiate the abstract pure concepts of the understanding and thus to provide them with objective reality, goes on to emphasize (in

1

All references to Kant’s writings, except for the Critique of pure reason, are given by volume and page number of Kant (1902–). The Critique of pure reason is cited by the standard A/B pagination of the first and second editions. All translations are my own.

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a manner clearly reminiscent of the refutation of idealism) the necessity for specifically outer, that is, spatial intuitions: It is even more remarkable, however, that, in order to understand the possibility of things in accordance with the categories, and thus to verify the objective reality of the latter, we require not merely intuitions, but always even outer intuitions. If, for example, we take the pure concepts of relation, we find, first, that in order to supply something permanent in intuition corresponding to the concept of substance (and thereby to verify the objective reality of this concept), we require an intuition in space (of matter), because space alone is determined as permanent, but time, and thus everything in inner sense, continually flows. (B291)

Many other passages in the revised second edition, although not quite as dramatic as these, also serve to underscore the fundamental importance of space and material physical bodies located in space. Between the publication of the Prolegomena and the second edition of the Critique, Kant published a less well-known work, the Metaphysical foundations of natural science, appearing in 1786. This work articulates what Kant calls the metaphysical doctrine of body, which examines the a priori principles governing the objects of specifically outer sense, namely, bodies or what Kant also calls matter(s). These a priori principles turn out centrally to include fundamental principles of Newtonian mechanics such as the law of inertia, the conservation of the quantity of matter, and the equality of action and reaction2. More generally, one of the main burdens of the Metaphysical foundations is to show how such synthetic a priori principles of ‘pure natural science’ play a necessary and indispensable role in the argument for universal gravitation with which Newton’s Principia culminates: the argument of Book III of the Principia which derives the law of universal gravitation from what Newton calls the ‘phenomena’ described by Kepler’s three laws of planetary motion and, at the same time, resolves the issue between the Ptolemaic and Copernican world-systems by establishing the center of mass of the solar system (which, as Newton demonstrates, is always very close to the center of the sun) as the privileged frame of reference for describing the true motions in the solar system—as, according to Newton himself, at rest in absolute space. I have considered the details of Kant’s interpretation of this Newtonian argument elsewhere3, and I will only sketch its essential features here. Kant, in the first place, rejects the Newtonian conception of absolute space as some kind of great empty ‘container’ existing prior to the bodies and matter that fill space4. Nevertheless, Kant 2 These same principles are prominently cited in the introduction to the second edition of the Critique at B17–20, as central examples of what Kant, in the Metaphysical foundations, the Prolegomena, and the second edition of the Critique, calls ‘pure natural science’. 3 See, in particular, Friedman (1992), Chs. 3, 4. 4 In his own words at A39/B56, Kant rejects Newtonian absolute space as an ‘infinite, self-subsistent non-entity’.

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also clearly recognizes that Newton has in fact determined the true motions in the solar system by showing us how actually to construct the privileged frame of reference in question. It is this construction, for Kant, that first defines the concept of ‘true motion’, and it does this, in Kant’s own terminology, by showing us how to transform ‘appearance [Erscheinung]’ into ‘experience [Erfahrung]’, so that the latter, in particular, is securely distinguished from ‘illusion [Schein]’ (4: 554–555). More specifically, we begin, as in Newton’s own argument, with the merely relative motions in the solar system: that is, the motions of the various satellites with respect to their primary bodies—the moon relative to the earth, the moons of Jupiter and Saturn relative to their respective planets, Venus and the other planets relative to the sun, and so on. These merely relative motions are, in Kant’s terminology, mere appearances, and we are not yet able to distinguish merely apparent motions from true motions (in particular, we do not yet decide, at this stage, whether the earth truly orbits the sun or vice versa). These motions, also in Kant’s terminology, are merely possible. In the second stage of the argument, we apply the law of inertia and Newton’s second law of motion to conclude that in each system of satellites together with a primary body there is an inverse-square force acting on the satellites directed towards the primary body in question. These same motions, now in the context of Newton’s second law, are, in Kant’s terminology, (provisionally) taken to be actual. In the final stage of the argument, we apply Newton’s third law of motion—the equality of action and reaction—to conclude, first, that the forces in question must act mutually (so that the satellites must attract their primary bodies in turn) and, second, that there is an inverse-square force, directly proportional to the product of the masses of the two bodies, between any two bodies (any two pieces of matter) in the universe. The motions described in this third stage—mutual, universal, inversely proportional to the square of the distance, and directly proportional to mass—are now declared, in Kant’s terminology, to be necessary. It is precisely these motions that yield both Newton’s formulation of the law of universal gravitation and the center of mass frame of the solar system—which, as Newton proves in accordance with the law of universal gravitation, is always located very near to the center of the sun. This Newtonian construction, for Kant, first defines what we mean by ‘absolute space’. Absolute space is not, as Newton thought, a mysterious metaphysical entity (or ‘non-entity’) somehow existing ‘behind’ the observable phenomena of motion; it is rather a privileged frame of reference serving to define the notion of ‘true motion’ for a given interacting system of interest (in this case, the solar system). Moreover, it is also this construction, for Kant, that makes the law of universal gravitation his model for a genuine empirical law of nature—a law which, despite its undoubted empirical status, also counts as universal and necessary. For, unlike Kepler’s laws of planetary motion—which, for Kant, are at this stage merely inductive regularities possessing what Kant calls mere comparative universality—the law of universal gravitation is determined from the initial empirical data (the Keplerian ‘phenomena’) by an a priori constructive procedure making essential use of synthetic a priori principles of pure natural science (the Newtonian laws of motion).

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It follows that the law of universal gravitation is necessary in precisely the sense of Kant’s official explanation of the category of necessity: 1. That which agrees with the formal conditions of experience (according to intuition and concepts), is possible. 2. That which connects with the material conditions of experience (sensation), is actual. 3. That whose connection with the actual is determined in accordance with universal conditions of experience, is (exists as) necessary. (A218– 219/B265–266) Since, as we have seen, this same progression from possibility, through actuality, to necessity is also instantiated, in the present case, by a progression from mere ‘appearance [Erscheinung]’ to ‘experience [Erfahrung]’, where the latter is securely distinguished from ‘illusion [Schein]’, we now see, at the same time, how the universal and necessary law of gravitation also serves, with respect to motion, as a ‘secure criterion of truth’. I now want to pursue is the question of how deeply Kant’s conception of the universal and necessary character of Newtonian mathematical physics is connected with, or embedded in, the transcendental philosophy articulated in the Prolegomena and the first Critique—especially in the second edition of the Critique, whose revisions, as we have seen, appear to be centrally motivated by Kant’s ambition to distinguish his views from subjective idealism. One indication that this connection may go very deep indeed is that, in the section of the Prolegomena corresponding to the transcendental deduction of the categories, Kant illustrates one of the central conclusions of the deduction—that, in the words of the Prolegomena, ‘[t]he understanding does not extract its laws (a priori) from, but prescribes them to nature’ (4: 320)—precisely by the law of universal gravitation. A second indication that the connection goes deep is that the passage we cited from the new general remark to the system of principles added to the Critique, where Kant emphasizes the need for specifically outer or spatial intuitions to instantiate the categories and secure their objective reality, is a very clear echo of a parallel passage in the Preface to the Metaphysical foundations: It is also indeed very remarkable . . . that general metaphysics [transcendental philosophy], in all instances where it requires examples (intuitions) in order to provide meaning for its pure concepts of the understanding, must always take them from the general doctrine of body, and thus from the form and principles of outer intuition; and, if these are not exhibited completely, it gropes uncertainly and unsteadily among mere meaningless concepts . . . [here] the understanding is taught only by examples from corporeal nature what the conditions are under which such concepts can alone have objective reality, that is, meaning and truth. (4: 478)

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The Prolegomena and the second edition of the Critique emphasize the fundamental importance of a priori principles governing specifically outer sense, and it appears that the a priori principles of outer sense in question are precisely those articulated in the Metaphysical foundations. The same point can be made in an even more specific fashion. For what Kant calls pure natural science or the metaphysical doctrine of body is, first and foremost, a doctrine of motion. Matter or body is defined, in the very first explication of the Metaphysical foundations, as ‘the movable in space’ (4: 480), and, as Kant explains in the Preface, ‘[t]he understanding traces back all other predicates of matter belonging to its nature to this one [motion], and so natural science is either a pure or applied doctrine of motion throughout’ (4: 476–477). An analogous emphasis on motion is also central to the revised passages on the necessity for spatial intuition added to the second edition of the Critique. Thus, the passage from the general remark to the system of principles continues as follows: Second, in order to exhibit alteration, as the intuition corresponding to the concept of causality, we must take motion, as alteration in space, for the example. Indeed, it is even the case that we can make alteration intuitive to ourselves solely in this way, as no pure understanding can conceive its possibility . . . How it may . . . be possible that an opposed state follows from a given state of the same thing is not only inconceivable to any reason without example, but is not even understandable without intuition—and this intuition is the motion of a point in space, whose existence in different places (as a sequence of opposed determinations) alone makes alteration intuitive to us in the first place. For, in order that we may afterwards make even inner alterations intuitive, we must make time, as the form of inner sense, intelligible figuratively as a line—and inner alteration by the drawing of this line (motion), and thus the successive existence of our self in different states by outer intuition. (B291–292)

It is even more striking, however, that this passage (from the general remark to the system of principles) is echoed in turn, in the second edition revisions of the transcendental deduction of the categories. In a crucial section on ‘the application of the categories to objects of the senses in general’ (Section 24), Kant explains how the understanding (and thus the pure concepts of the understanding or categories) obtains a necessary relation to sensibility (and thus to objects of our senses) by what he calls a figurative synthesis (synthesis speciosa) or transcendental synthesis of the imagination. Kant then goes on to illustrate this notion as follows: We always observe this [the transcendental synthesis of the imagination] in ourselves. We can think no line without drawing it in thought, no circle without describing it, . . . ; and we cannot represent time itself without attending, in the drawing of a straight line (which is to be the outer figurative representation of time), merely to the action of synthesis of the manifold, through which we successively determine inner sense, and thereby attend to the succession of this deter-

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mination in it. Motion, as action of the subject (not as determination of an object∗), and thus the synthesis of the manifold in space—when we abstract from the latter and attend merely to the action by which we determine inner sense in accordance with its form—[such motion] even first produces the concept of succession. (B154–155)

It would appear that a crucial step in the second edition transcendental deduction— the demonstration of how the understanding is applied to sensibility—proceeds precisely by the representation of motion. And it would therefore appear that Kant’s Metaphysical foundations of Newtonian mathematical physics, whose task, as we have seen, is the elaboration of what Kant calls the pure doctrine of motion, is centrally implicated in the argument of the transcendental deduction itself. But perhaps this passage and other related passages from the second edition are merely illustrative. The pure doctrine of motion, and thus, in the end, Newtonian mathematical physics as well are, to be sure, exemplary of the application of the categories. Here we have a body of necessary synthetic a priori knowledge, which, like all such knowledge, must eventually be grounded in the categories. Indeed, there is no doubt, for Kant, that Newtonian mathematical physics counts as a particularly central example of the application of the categories. But it does not follow that the main argument of the Critique, the transcendental deduction of the categories, is in any way dependent on this particular example. It would appear, in fact, that the argument of the first Critique must necessarily proceed at a much higher level of abstraction and generality, so that, in particular, the proof of the objective reality of the categories must be completed prior to and independently of all consideration of this specific example. I would now like to argue, however, that such a merely illustrative and exemplary reading is not sufficient. On the contrary, there is an important sense in which a consideration of this specific example is built into the very structure of Kant’s argument. Kant significantly altered the structure of the transcendental deduction in the second edition of the Critique. Whereas the first edition did not clearly separate the respective contributions of the faculties of understanding and sensibility, the structure of the second edition is deliberately constructed to emphasize this distinction. In the first half of the argument (Sections 15–20) Kant considers the role of the understanding alone, independently of sensibility. In the second half of the argument (Sections 22–27) Kant introduces the specific features of our human sensibility—that is, the spatiality and temporality of our forms of intuition—in order to show that there is a kind of necessary harmony between the understanding and our sensibility. It is this that shows that the categories—which have their origin in the understanding entirely independently of our sensibility—nevertheless necessarily apply to all objects of our senses and thus have objective reality. Between these two parts of the argument, Kant inserts a Remark (Section 21) explaining the distinction in question. Our understanding, like the understanding of any finite creature, must operate by combining and synthesizing sensory material given independently of the understanding itself: it cannot create objects of intuition but only think them. The first

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half of the argument, where we consider the understanding in abstraction from our forms of sensibility, does not abstract away this feature, but only the specifically spatio-temporal character of our particular forms of sensory receptivity. The first half, in Kant’s terminology, considers only a manifold of intuition in general (whether spatio-temporal or not), whereas the second half considers the relation of the understanding to our specific form of a manifold of intuition (that is, to a spatiotemporal form). In light of this, Kant explains the relationship between the two parts of the argument as follows: In the above proposition [Section 20] the beginnings of a deduction of the pure concepts of the understanding is made—in which, since the categories arise independent of sensibility merely in the understanding, I must still abstract from the manner in which the manifold for an empirical intuition is given, in order solely to discern the unity which is added to the intuition by means of the category through the understanding. In the following (Section 26) it will be demonstrated from the manner in which the empirical intuition is given in sensibility that its unity is no other than that which the category, according to the preceding Section 20, prescribes to the manifold of a given intuition in general. In this way, therefore, by explaining its a priori validity with respect to all objects of our senses, the aim of the deduction will first be fully achieved. (Section 21, B144–145)

It would appear, then, that we must invoke the specifically spatio-temporal character of our (human) sensibility in order fully to achieve the aim of the deduction. But there is a serious difficulty in understanding Kant’s actual execution of this strategy—that was first clearly articulated in a ground-breaking article by Dieter Henrich (1969). For the ‘proposition’ already demonstrated in Section 20 is that ‘[a]ll sensible intuitions stand under the categories, as conditions under which alone their manifold can come together in a consciousness’, or, as the conclusion of Section 20 puts it, ‘the manifold of a given intuition [in general] necessarily stands under the categories’ (B143). Kant has already shown, in other words, that all sensible intuitions in a manifold in general—whether spatio-temporal or not—must necessarily be subject to the categories. How, then, can Kant’s introduction of the specifically spatio-temporal character of our intuition in the second half of the argument generate anything more than a trivial instantiation of the more general claim already established in Section 20? All intuitions of a manifold in general stand under the categories; therefore all items in a specifically spatio-temporal manifold of intuition stand under the categories; QED. How, in particular, can the intricate considerations Kant actually presents in the second half of the argument—including centrally, the complex and rather murky considerations about the transcendental synthesis of the imagination Kant presents in Section 24—possibly add anything to this entirely trivial argument by universal instantiation? Henry Allison (1983) has made the helpful suggestion that there are two notions of object at work in the two parts of the argument. The first half (where we consider

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the contribution of the understanding alone) operates with a relatively thin, purely logical or intellectual notion of object—the notion of a mere object of thought, which, in particular, may or may not actually exist—and shows that the categories are necessary conditions for representing objects in this sense. The second half, in contrast, operates with a much more ‘weighty’ notion of ‘real’ or empirically existing object (in space and time) and attempts to show that the categories necessarily apply to such objects as well—in such a way that they thereby generate empirical knowledge of these more ‘weighty’ objects. Unfortunately, on Allison’s view, the second half of the deduction does not fully succeed in proving this last claim. As Allison puts it, ‘[e]ven if we accept all of Kant’s premises, including his doctrine of transcendental synthesis, his conclusion that the categories make experience [that is, empirical knowledge] possible and prescribe a priori laws to nature does not follow’ Allison (1983), p. 171. I agree with Allison that there are two different notions of object at work in the two parts of the deduction. I also agree that Kant’s own conclusion, in the second part, centrally involves the claim that the categories make experience possible by prescribing a priori laws to nature. Indeed, Kant himself states this conclusion (as that which is to be proved) at the very beginning of the crucial Section 26, entitled ‘transcendental deduction of the universally possible employment in experience of the pure concepts of the understanding’: In the metaphysical deduction the a priori origin of the categories in general was shown by their complete agreement with the universal logical functions of thought; in the transcendental [deduction], however, their possibility as a priori cognitions of objects of an intuition in general was presented (Sections 20, 21). What is now to be explained is the possibility of knowing a priori, by means of categories, whatever objects may present themselves to our senses—not, indeed, with respect to the form of their intuition, but with respect to the laws of their combination—and thus [how] they prescribe laws to nature, as it were, and even make nature possible. For without this it would not be clear how everything that may merely be presented to our senses must stand under laws that arise a priori from the understanding alone. (B159–160) I disagree with Allison, however, in so far as I believe that it is a condition of adequacy for a proper interpretation of the overall structure of the two parts of the argument—and, in particular, for a proper interpretation of the two notions of object at work there—that our interpretation show how Kant’s stated conclusion in the deduction actually follows in the deduction itself5. 5 Allison suggests that Kant does in fact succeed in his aims later in the Critique, in the principles chapter: ‘the problem [does] not lie so much in the argument of the Deduction itself as in Kant’s tendency to assign to it a task that should properly have been assigned to the Analytic as a whole’ (Allison, 1983, p. 172). Of course my condition of adequacy—that the conclusion of the deduction actually follows in the deduction—does not imply that the argument of the deduction is sound, nor that it is convincing or even plausible.

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Let us begin with the notion of object at work in the second part of the argument. From what Kant says in his statement of the conclusion in Section 26 it appears that objects in this sense are (mere) objects of sensibility: objects that may in any way be given or presented to our senses in virtue of affecting our receptivity so as to produce sensations in it. These objects, of course, are not things-in-themselves, but rather objects existing anywhere in space and time, which can affect our sensibility in virtue of our putting ourselves, as it were, into an appropriate spatio-temporal (perceptual) relation to them. Indeed, Kant’s argument (which immediately follows) hinges precisely on the fact that since space and time are our forms of intuition, all objects in the present sense are necessarily given or presented within space and time. I will return to this argument shortly. But I now want simply to observe that objects in the sense we are now considering—at work in the second part of the deduction— are merely specified as objects that may be presented to our senses and thus exist in space and time. This notion of objects, therefore, is a purely sensible notion of objects, which, in particular, has not yet been put into any kind of relation to the understanding—that is, any kind of relation to thought and knowledge. That objects in this sense are also (and necessarily) objects in relation to the understanding—that is, objects of thought and knowledge—is precisely what has now to be proved. If this is correct, it follows that the notion of object at work in the first part of the deduction (where the understanding is considered on its own, independently of our particular forms of sensibility) is defined solely in relation to the understanding. These objects are thus objects of thought and knowledge. But we have not yet shown, at this stage of the argument, that the only objects of knowledge available to us are spatio-temporal objects encounterable or presentable in our specific forms of receptivity. Again, that objects of knowledge are in fact restricted in this way is precisely what then has to be proved in the second half, specifically, in Section 22. So what are we able to say about objects of the understanding in general—objects, in Kant’s terminology, of an intuition in general (whether it be spatio-temporal or not)? Kant explains this notion in a well-known passage in Section 17: Understanding, speaking generally, is the faculty of cognitions. These consist in the determinate relation of given representations to an object. But [an] object is that in the concept of which the manifold of a given intuition is united. Now all unification of representations requires unity of consciousness in their synthesis. Therefore, the unity of consciousness is that on which the relation of representations to an object rests, and thus their objective validity and [the circumstance] that they become cognitions; consequently, it is [also] that on which rests the very possibility of the understanding. (B137)

Objects of the understanding as such—objects of thought and, as it were, of knowledge in general—are therefore defined (independently of space and time) in terms of that unity which results in a given manifold of intuition in general (spatio-temporal or not) by synthesizing or unifying that manifold under a concept. Objects in this purely intellectual sense, in other words, are whatever results from applying the

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discursive, conceptual, and judgemental capacities of the understanding to any given manifold of intuition (again, whether spatio-temporal or not). And it is precisely this most general and abstract form of purely intellectual synthesis or unity that Kant here refers to as the unity of apperception or the unity of consciousness. Given this interpretation of the two notions of object at work in the two parts of the argument of the deduction (purely intellectual in the first part, purely sensible in the second), our problem is now to understand why all purely sensible objects (objects presentable or encounterable in space and time) are also, and necessarily, intellectual objects as well (objects of thought and knowledge). And this problem is far from trivial. For, although it is certainly true that spatio-temporal manifolds of intuition, like all manifolds of intuition, are necessarily subject to the unifying and synthesizing functions of thought, space and time, as our specific forms of intuition, also have their own particular structure, which prima facie is entirely independent of the unifying and synthesizing functions of thought. Indeed, one of the main points of the transcendental aesthetic, in the metaphysical expositions of space and time, is that both space and time possess their own characteristic form of unity, and this form of unity is distinctively non-intellectual or non-conceptual6. Objects of our senses—objects existing in space and time—are thereby subject to the distinctive, non-conceptual unity belonging to space and time themselves. What still needs to be shown, therefore, is that they are also, and necessarily, subject to the purely conceptual unity arising from the understanding alone, entirely independently of space and time. What is it that ensures, in particular, that the synthesizing activities of the understanding penetrate all the way down, as it were, to whatever spatiotemporal objects may be presented to our senses—and do this, moreover, in such a way that the understanding thereby prescribes a priori laws of nature to these very same objects? It is now time to look at the argument of Section 26: We have a priori forms of outer and inner sensible intuition in the representations of space and time, and the synthesis of apprehension of the manifold of appearances [‘through which perception becomes possible’] must always be in accordance with them, for it can only take place via this form. But space and time are represented a priori, not merely as forms of sensible intuition, but as intuitions themselves (which contain a manifold) and thus [represented a priori] with the determination of the unity of this manifold (see the transcendental aesthetic∗). Therefore, unity of the synthesis of the manifold, outside us or in us, and thus a combination with which everything that is to be represented in space or time as determined must accord, is itself already given simultaneously, with (not in) these 6 The distinctively non-intellectual or non-conceptual unity in question consists in the circumstance that all spaces are necessarily given as parts of a single, unitary, all-encompassing space—and similarly for time. In the case of discursive or conceptual representation, by contrast, the parts (the constituent concepts or marks) must precede the whole rather than the other way around, and it may therefore be concluded, in the metaphysical expositions, that the representations of both space and time are sensible or intuitive rather than discursive or intellectual. See, in particular, A24–25/B39–40 and A31–32/B47–48.

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intuitions. But this synthetic unity can be no other than that of the combination of the manifold of a given intuition in general in an original consciousness, in accordance with the categories, only applied to our sensible intuition (B160–161)

The crucial point is that space and time are not merely forms of intuitions, in which all objects of our senses must be given, they are also intuitions themselves. And this means, in particular, that space and time are objects—which, as such, must themselves be represented a priori with the determination of the unity of the manifold7. Kant justifies this claim by referring us back to the transcendental aesthetic. In what sense of ‘object’ are space and time, represented a priori as ‘intuitions themselves’, objects? This cannot be the purely sensible notion of object at work in the second part of the deduction, for this notion applies only to empirical objects that can be given or presented to our senses in space and time considered as forms of intuition. (Space and time themselves do not affect our senses by producing sensations in our receptivity when we encounter them in space and time.) It follows, therefore, that it can only be the intellectual notion of object that is at work in the first part of the deduction. Space and time are (intuitive) objects in the sense that they themselves (as pure a priori formal intuitions) are objects of thought and knowledge. And it is precisely this we learn in the transcendental aesthetic—not in the metaphysical expositions of space and time, but rather in the transcendental expositions added in the second edition. In the transcendental exposition of space, for example, we learn that our synthetic a priori knowledge of the science of geometry can only be explained by the circumstance that our representation of space is a pure a priori intuition (that is, a pure formal intuition). Since space itself is thus an object of knowledge (and, indeed, an object of a priori knowledge), space, in its specifically spatial character (that is, its character as object of geometry), is thereby—by the first part of the deduction— subject a priori to the synthesizing and unifying functions of the understanding. Hence, all empirical objects encounterable in space are subject a priori to the synthesizing and unifying functions of the understanding as well—indeed, they are thereby subject to the understanding precisely in virtue of being subject to the laws of geometry8. 7 Thus, Kant’s footnote begins (B160n): ‘Space represented as object (as one actually requires in geometry) contains more than the mere form of intuition—namely, [it contains] the [act of] putting together [Zusammenfassung] the manifold, given in accordance with the form of sensibility, in an intuitive representation, so that the form of intuition yields merely a manifold, but the formal intuition yields unity of representation’. 8 This reading is confirmed by the footnote to Section 26, where Kant illustrates the notion of space as an object by the example of geometry (see note 7 above), as well as by the continuation of the above passage from Section 17, where the purely intellectual notion of object is also promptly illustrated by the example of space and geometry (B137–138): ‘in order to know anything in space, e.g., a line, I must draw it, and therefore synthetically bring about a determinate combination of the given manifold, in such a way that the unity of this action is, at the same time, the unity of consciousness (in the concept of a line), and only in this way is an object (a determinate space) known’.

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What corresponds to this in the case of the representation of time? What body of synthetic a priori knowledge is explained by the circumstance that time, too, is a pure a priori formal intuition? The main burden of the transcendental exposition of time, also added, of course, in the second edition, is as follows: Here I may add that the concept of alteration and, along with it, the concept of motion (as alteration of place) is possible only in and through the representation of time: so that, if this representation were not an a priori (inner) intuition, no concept, whatever it might be, could make an alteration—i.e., the combination of contradictorily opposed predicates (e.g., the being and not-being of one and the same thing at one and the same place)—conceivable. Only in time can two contradictorily opposed determinations in one thing be met with, namely, successively. Therefore, our concept of time explains as much synthetic a priori knowledge as is set forth in the general doctrine of motion, which is by no means unfruitful. (B48–49)

Time as pure formal intuition is thus the object of the synthetic a priori knowledge contained in what Kant here calls the ‘general doctrine of motion’—which is the very same science, I take it, as what the Metaphysical foundations calls the ‘pure doctrine of motion’ or ‘pure natural science’ and the Prolegomena calls ‘pure mechanics’9. And it now follows, just as in the case of space and spatial objects, that all spatio-temporal objects located in both space and time are necessarily subject a priori to the laws of the pure doctrine of motion or pure natural science as well; and it is in this precise sense, finally, that the understanding, as Kant says, does in fact succeed in prescribing laws to nature a priori. It is by no means surprising then, that Kant in the Prolegomena illustrates what he here calls the ‘seemingly bold proposition’, that ‘[t]he understanding does not extract its laws (a priori) from, but prescribes them to nature’, by Newton’s law of universal gravitation10. The law of universal gravitation thus becomes the central example of a genuinely well-grounded (and therefore strictly universal and necessary) empirical law of nature, which attains this well-grounded status precisely by being determined, from perception, by the understanding itself. And it is similarly no wonder, therefore, that Kant illustrates the crucial notion of objectively valid judgement in Section 19 of the second edition, transcendental deduction by the judgement ‘bodies are heavy [die Ko¨ rper sind schwer]’ (B142). If we take account of the fact that, in the Metaphysical foundations, Kant explicitly defines ‘heaviness’ or ‘weight [Schwere]’ as ‘the striving to move in the direction of greater gravitation’, 9 See Section 10 of the Prolegomena (4: 283): ‘pure mechanics can only produce its concept of motion by means of the representation of time’. 10 More precisely, as I argue in my (1992), Ch. 4, this proposition of the Prolegomena is instantiated by Newton’s derivation of the law of universal gravitation in Book III of the Principia, where the latter is derived from the Keplerian ‘phenomena’ by the application of both (Euclidean) geometry and the Newtonian laws of motion.

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where ‘gravitation’ is defined as ‘the action of the universal attraction that all matter immediate exerts on all [other matter] and at all distances’ (4: 518), it appears that ‘bodies are heavy’ is objectively valid precisely because this judgement is grounded in the law of universal gravitation—which is itself grounded, in turn, in what Kant calls pure mechanics or the general doctrine of motion. Since this latter science is then seen, in the second part of the deduction, as the indispensable means by which the categories ‘prescribe laws to nature, as it were, and even make nature possible’, it thus becomes clear how the judgement ‘bodies are heavy’ is grounded, and in this sense made possible, by the understanding itself11. If I am correct, therefore, Newtonian mathematical physics does not simply constitute one application of Kant’s transcendental philosophy among others. For, if my reading of the structure of the second edition transcendental deduction is accepted, we can only make sense of the application of the understanding to sensibility— whereby the categories acquire objective reality and thereby make experience possible—precisely by means of this particular application. It is not that the categories make Newtonian mathematical physics possible, and they also, by a different route, make experience in general possible. It is rather that the categories make experience in general possible only in virtue of their prior application in Newtonian mathematical physics. Kant, in Section 24 of the second edition deduction, maintains that the two previously independent faculties, pure understanding and pure sensibility, are now to be integrated or united with one another by a figurative synthesis or transcendental synthesis of the imagination, which is characterized, in turn, as ‘an action of the understanding on sensibility and its first application to objects of an intuition possible for us (and at the same time the ground of all other applications)’ (B150– 152). This pure synthesis is the bridge between the two parts of the deduction and thus between the two notions of object at work there. What I have attempted to show is that pure natural science and Newtonian mathematical physics are the concrete products of this synthesis. It is they, and they alone, that constitute what Kant here calls the understanding’s first application to objects of an intuition possible for us. It is they, and they alone, that first make clear, in the words of the Prolegomena, how ‘the understanding is the origin of the universal order of nature, in that it comprehends all appearances under its own laws and thereby first brings about experience (according to its form)’ (4: 322). References Allison, H. (1983). Kant’s transcendental idealism: An interpretation and defense. New Haven, CT: Yale University Press.

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It is at precisely this point that the differences between my approach and Gerd Buchdahl’s become most explicit and acute. For, in accordance with his ‘looseness of fit’ interpretation, Buchdahl views this crucial section of the second edition deduction as concerned with a purely contingent and commonsensical judgement having no essential relation to the specifics of Newtonian (or any other kind of) mathematical physics: see his (1969), pp. 628–641. Since, however, my own work on Kant’s philosophy of science was directly stimulated by reading Buchdahl (1969) in the early 1980s, it is a great pleasure indeed to have the opportunity to present these contrasting ideas here.

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Buchdahl, G. (1969). Metaphysics and the philosophy of science. Oxford, UK: Basil Blackwell. Friedman, M. (1992). Kant and the exact sciences. Cambridge, MA: Harvard University Press. Henrich, D. (1969). The proof-structure of Kant’s transcendental deduction. The Review of Metaphysics, 22, 640–659. Kant, I. (1902–). Kant’s gesammelte Schriften. Berlin: de Gruyter.