Abstract
Towards the end of the eighteenth century, at the height of the German Enlightenment, Immanuel Kant developed a revolutionary theory of space and geometry that aimed to explain the distinctive relation of the mathematical science of geometry to our experience of the world around us—both our ordinary perceptual experience of the world in space and the more refined empirical knowledge of this same world afforded by the new mathematical science of nature. From the perspective of our contemporary conception of space and geometry, as it was first developed in the late nineteenth century by such thinkers as Helmholtz , Mach , and Poincaré , Kant ’s earlier conception thereby involves a conflation of what we now distinguish as mathematical, perceptual, and physical space. According to this contemporary conception, mathematical space is the object of pure geometry, perceptual space is that within which empirical objects are first given to our senses, and physical space results from applying the propositions of pure geometry to the objects of the (empirical) science of physics—which, first and foremost, studies the motions of such objects in (physical) space. Yet it is essential to Kant ’s conception that the three types of space (mathematical, perceptual, and physical) among which we now sharply distinguish are necessarily identical with one another, for it is in precisely this way, for Kant , that a priori knowledge of the empirical world around us is possible.
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- 1.
Aside from the development of the new mathematical science of physical nature, the early modern conception of space as essentially geometrical has other important sources as well—notably, the development of linear perspective in the painting of the Italian Renaissance. For the latter see, for example, Edgerton (1991).
- 2.
I shall return in the final section of this essay to the sense in which Kant ’s conception represents the culmination these early modern attempts—and, at the same time, is also quite essential for understanding the later development of our contemporary (explicitly anti-Kantian) conception.
- 3.
I cite the Critique of Pure Reason by the standard A/B pagination of the first and second editions. All translations from Kant ’s German are my own. The passage just quoted reads in full (ibid.): “I call that in the appearance which corresponds to sensation its matter, but that which brings it about that the manifold of appearances can be ordered in certain relations I call the form of appearance. Since that within which sensations can alone be ordered and arranged in a certain form cannot itself be sensation in turn, the matter of all appearance, to be sure, is only given to us a posteriori, but its form must already lie ready for it in the mind a priori and can therefore be considered separately from all sensation.” (In B the words “can be” in the first sentence replace “are” in A).
- 4.
In the tree of Porphyry the highest genus Being is divided into the species Created and Uncreated, Created is divided into the lower species Material and Immaterial, Material into the lower species Animate and Inanimate, and Animate into the still lower species Rational and Irrational: Human Being is thus defined as Rational Animate Material Created Being. In the traditional Aristotelian classification of the logical forms of judgement the square of opposition depicts the logical relationships among judgements with respect to logical quantity (universal or particular) and quality (affirmative or negative), resulting in the four forms A, E, I, and O: A = Every S is P, E = No S is P, I = Some S is P, O = Some S is not P. Kant himself goes beyond the traditional Aristotelian classification, not only by adding the triads of categorical, hypothetical, and disjunctive judgements and problematic, assertoric, and apodictic judgements, but also by adding singular judgements to the two traditional forms of logical quantity (universal or particular) and “infinite” judgements to the two traditional forms of logical quality (affirmative or negative): see A70–76/B95–101. The resulting table of exactly twelve logical forms of judgement, together with the corresponding table of categories, has generated considerable scholarly controversy. I shall briefly touch on one such controversy below.
- 5.
See the discussion of substance and causality in the Schematism chapter (A144/B183): “The schema of substance is the permanence of the real in time, i.e., the representation of it as a substratum of empirical time determination in general—which therefore remains while everything else changes. The schema of cause and the causality of a thing in general is the real, upon which, whenever it is posited, something else always follows. It therefore consists of the succession of the manifold, in so far as it is subject to a rule”.
- 6.
These are the first two Analogies of Experience. I shall return to the Analogies, together with other principles of the understanding, below.
- 7.
It is puzzling, in particular, because Kant here appears to take back his insistence that sensibility and understanding are two quite different faculties with two quite different a priori formal structures. Indeed, an important line of thought in post-Kantian German philosophy, including both the post-Kantian German idealists and the Marburg neo-Kantians, explicitly appeals to what Kant says in § 26 to motivate an “intellectualist” reading according to which the forms of intuition become absorbed into the more fundamental unity of the understanding. And another important line of thought, culminating in the notorious “common root” interpretation of Martin Heidegger , insists on the radical independence of sensibility—leaving us, in the end, with no plausible reading of § 26. (For further discussion and references relevant to these two lines of thought see Friedman (2015)). The interpretation I am developing here aims fully to incorporate the ineliminable role of the understanding in the characteristic unity of space and time appealed to in § 26 while simultaneously preserving the independent contribution of sensibility.
- 8.
The paragraph reads in full (A24–25/B39): “Space is not a discursive, or, as one says, general concept of relations of things in general, but a pure intuition. For, first, one can only represent to oneself a single space, and if one speaks of many spaces, one understands by this only parts of one and the same unique space. These parts cannot precede the single all-encompassing space, as it were as its constituents (out of which a composition would be possible); rather, they can only be thought within it. It is essentially single; the manifold in it, and the general concept of spaces as such, rests solely on limitations. From this it follows that an a priori intuition (that is not empirical) underlies all concepts of space. Thus all geometrical principles, e.g., that in a triangle two sides together are greater than the third, are never derived from general concepts of line and triangle, but rather from intuition, and, in fact, with apodictic certainty.” The final sentence makes it clear that the science of geometry is implicated in the distinctive whole-part structure that Kant is attempting to delineate, a point to which I shall return below.
- 9.
This crucial difference in whole-part structure is emphasized especially clearly in the immediately following fourth paragraph of the Metaphysical Exposition of Space in the second edition (B39–40): “Space is represented as an infinite given magnitude. Now one must certainly think every concept as a representation that is contained in an infinite aggregate of different possible representations (as their common mark), and it therefore contains these under itself. But no concept, as such, can be so thought as if it were to contain an infinite aggregate of representations within itself. However space is thought in precisely this way (for all parts of space in infinitum exist simultaneously). Therefore, the original representation of space is an a priori intuition, and not a concept.”
- 10.
Compare note 7 above, together with the paragraph to which it is appended.
- 11.
I thereby attempted to build a bridge between the “logical” interpretation of Kant ’s theory of geometry developed in my earlier paper (an approach that was first articulated by Jaakko Hintikka ) and the “phenomenological” interpretation articulated and defended by Charles Parsons and Emily Carson . For further discussion of this issue see also Parsons (1992).
- 12.
Kästner ’s three essays on space and geometry were first published in J.A. Eberhard ’s Philosophisches Magazin in 1790. Eberhard ’s intention was to attack the Critique of Pure Reason on behalf of the Leibniz ean philosophy, and Kästner ’s essays were included as part of this attack. Kant ’s comments on Kästner , sent to J. Schultz on behalf of the latter’s defense of the Kantian philosophy in his reviews of Eberhard ’s Magazin, were first published by Wilhelm Dilthey in the Archiv für Geschichte der Philosophie in 1890. They are partially translated in Appendix B to Allison (1973), which also discusses the historical background in Chapter I of Part One. Kant ’s comments have played a not inconsiderable role in the subsequent discussion of space and geometry in § 26, and, after presenting my own interpretation, I shall touch on some of this discussion below.
- 13.
All references to Kant ’s works other than the first Critique are to volume and page numbers in Kant (1900-).
- 14.
This point becomes clearer in light of the final sentence of our passage from the Aesthetic—which brings Euclid’s geometry explicitly into the picture (see note 8 above: the example there is Proposition I.20 of the Elements).
- 15.
Immediately preceding this passage Kant illustrates the distinction by contrasting geometry with arithmetic (20, 419–420): “Now when the geometer says that a line, no matter how far it has been continually drawn, can always be extended still further, this does not signify what is said of number in arithmetic, that one can always increase it by addition of other units or numbers without end (for the added numbers and magnitudes, which are thereby expressed, are possible for themselves, without needing to belong with the preceding as parts to a [whole] magnitude). Rather [to say] that a line can be continually drawn to infinity is to say as much as that the space in which I describe the line is greater than any line that I may describe within it.” Thus, while the figures iteratively constructed in geometry are only potentially infinite, like the numbers, the former, but not the latter, presuppose a single “all-encompassing” magnitude within which all are contained as parts: i.e., the space “represented as an infinite given magnitude” of note 9 above (B39).
- 16.
This connection between Euclidean constructions and the operations in question is suggested by Kant himself (20, 410–411): “[I]t is very correctly said [by Kästner ] that ‘Euclid assumes the possibility of drawing a straight line and describing a circle without proving it’—which means without proving this possibility through inferences. For description, which takes place a priori through the imagination in accordance with a rule and is called construction, is itself the proof of the possibility of the object. However, that the possibility of a straight line and a circle can be proved, not mediately through inferences, but only immediately through the construction of these concepts (which is in no way empirical), is due to the circumstance that among all constructions (presentations determined in accordance with a rule in a priori intuition) some must still be the first—namely, the drawing or describing (in thought) of a straight line and the rotating of such a line around a fixed point—where the latter cannot be derived from the former, nor can it be derived from any other construction of the concept of a magnitude.”
- 17.
More fully (B132): “[A]ll the manifold of intuition has a necessary relation to the I think in the same subject in which this manifold is encountered. But this representation is an act of spontaneity, i.e., it cannot be viewed as belonging to sensibility. I call it pure apperception, in order to distinguish it from the empirical, or also original apperception, because it is that self-consciousness, which—in so far as it brings forth the representation I think that must be able to accompany all others, and in all consciousness is one and the same—can be accompanied in turn by no other.” Thus, the I think is the subject of which all other representations are predicated, whereas it can be predicated of no other representation in turn, and it is in precisely this sense that the I think cannot itself be a concept.
- 18.
The footnote reads in full (B155n): “*Motion of an object in space does not belong in a pure science and thus not in geometry. For, that something is movable cannot be cognized a priori but only through experience. But motion, as the describing of a space, is a pure act of successive synthesis of the manifold in outer intuition in general through the productive imagination, and it belongs not only to geometry, but even to transcendental philosophy.”
- 19.
The precise relationship between the representation of motion (“as action of the subject”) in § 24 and the representation of time as a formal intuition suggested in § 26 is a delicate and subtle matter into which I cannot delve more deeply here; I provide some further discussion in Friedman (2015).
- 20.
I observed that interpreters have appealed to Kant ’s comments on Kästner while discussing space and geometry in § 26 (see note 12 above): notably, Martin Heidegger , in his lecture course on Phenomenological Interpretation of Kant ’s Critique of Pure Reason in the winter semester of 1927–1928 (1977, § 9), and Michel Fichant (1997), published along with his French translation of Kant ’s comments. Both Heidegger and Fichant , however, interpret space as a “formal intuition” in the footnote to § 26 as geometrical space in the terminology of the comments on Kästner —so that, according to them, the formal intuition of space is derivative from the more original “form of intuition” within which geometrical construction takes place. But this reading is incompatible with Kant ’s claim in the footnote that space as a formal intuition is both unified and singular in the sense of the Aesthetic—and, most importantly, that it precedes and makes possible all concepts of space. Here I am in agreement with Béatrice Longuenesse : for her comments on Heidegger in this connection see Longuenesse (1998a, pp. 224–225); for her parallel comments on Fichant see Longuenesse (1998b/2005, pp. 67–69). I shall return to the relationship between my reading and Longuenesse ’s below. (I am indebted to Vincenzo De Risi for first calling the exchange between Fichant and Longuenesse to my attention and for prompting me to consider more carefully the relationship between my reading of § 26 and Longuenesse ’s).
- 21.
More fully (B202–203): “All appearances contain, in accordance with their form, an intuition in space and time, which lies at the basis of all of them a priori. They can therefore be apprehended in no other way—i.e., be taken up in empirical consciousness—except through the synthesis of the manifold whereby a determinate space or time is generated, i.e., through the composition [Zusammensetzung] of the homogeneous and the consciousness of the synthetic unity of this (homogeneous) manifold. But the consciousness of the homogeneous manifold in intuition in general, in so far as the representation of an object first becomes possible, is the concept of a magnitude (quanti).”
- 22.
- 23.
Kant here illustrates the generality of geometry by the Euclidean construction of a triangle in general (A164–165/B205): “If I say that through three lines, of which two taken together are greater than the third, a triangle can be drawn, I have here the mere function of the productive imagination, which can draw the lines greater or smaller, and thereby allow them to meet at any and all arbitrary angles.” (This is Proposition I.22 of the Elements; compare note 14 above).
- 24.
As we have seen, the pure intellectual concept of magnitudes in general abstracts from the structure of specifically spatial (geometrical) magnitudes and involves only “the synthetic unity of the homogenous in an intuition in general” (B162; compare note 21 above, together with the paragraph to which it is appended).
- 25.
I am here indebted to a very helpful conversation with Graciela De Pierris concerning the precise connection between the transcendental unity of apperception and geometry in my reading.
- 26.
Compare the paragraph to which note 22 above is appended.
- 27.
As I have said, I first arrived at my reading of the role of the transcendental unity of apperception in § 26 in the course of the ongoing development of my work on Kant ’s theory of geometry. In my critical study of Longuenesse (1998a), Friedman (2000b), I discussed her interpretation of the categories of quantity, but I did not consider her views on the “pre-conceptual” synthesis of space and time (which views were then opaque to me). Only after I arrived at my own interpretation of § 26, in Friedman (2012a), did I appreciate what I now see as her important insight. My differences with Longuenesse concerning the formulation and articulation of this insight will emerge in what follows.
- 28.
See Longuenesse (1998a, p. 224): “[I]f one reads ‘the understanding’ as das Vermögen zu urteilen, the capacity to judge, then one can understand, as I suggested earlier, that the capacity to form judgments, ‘affecting sensibility,’ generates the pure intuitions of space and time as the necessary intuitive counterpart to our discursive capacity to reflect universal concepts, concepts whose extension (the multiplicities of singular objects thought under them) is potentially unlimited. When this original intuition is produced, no concept is thereby yet generated. Everything, as it were, remains to be done. But part of the minimal equipment that a human being, capable of discursive thought, has at his disposal, is the capacity to generate the ‘pure’ intuitions of space and time as that in which empirical objects are instances of concepts (i.e., universal representations, representations whose logical extension is unlimited).”
- 29.
See note 4 above, together with the paragraph to which it is appended and the preceding paragraph.
- 30.
Longuenesse thus follows a well-known paper by Frede and Krüger (1970) in taking the correspondence between categories and forms of judgement to be reversed in Kant ’s published tables. This correspondence, according to Frede and Krüger , should align singular judgements with unity and universal judgements with totality rather than the other way around. (This is the controversy to which I allude at the end of note 4 above).
- 31.
See Longuenesse (1998a, p. 276): “For this to be fully clear, Kant should have said that the concept of number is not an ordinary concept, that is, not a ‘common concept’ that can be predicated, as mark or a combination of marks, of another concept. It is different from ‘common concepts,’ since it reflects as such (as multiplicities or, as Cantor will say, as sets having a determinate ‘power’) sets of objects defined by a concept.”
- 32.
See Longuenesse (1998a, p. 252; bold emphasis added): “I maintain that according to Kant , even the category of quantity is originally acquired insofar as the power of judgment, reflecting on the sensible given in order to subordinate representations to empirical concepts combined in judgments, generates the schema of quantity—that is, a successive synthesis of homogeneous elements (where ‘homogeneous’ means ‘reflected under the same concept’).”
- 33.
Kant discusses this situation in the Amphiboly in the course of criticizing Leibniz ’s doctrine of the identity of indiscernibles (A263–264/B319–320, A271–272/B327–328). Compare Longuenesse ’s discussion of the Amphiboly in her chapter on “Concepts of Comparison, Forms of Judgment, Concept Formation” (1998a, pp. 132–135).
- 34.
See note 22 above, together with the paragraph to which it is appended. For discussion of a variety of different notions of homogeneity—both logical and mathematical—see especially Sutherland (2004b).
- 35.
See Longuenesse (1998a, p. 265): “[T]he same capacity to judge that makes us capable of reflecting our intuitions according to the logical form of quantity also makes us capable of recognizing in the line a plurality of homogeneous segments, thought under the concept ‘equal to the segment s, the unit of measurement.’ To ‘subsume under the concept of quantity’ is to count these segments, that is, to reflect the unity of this plurality of homogeneous elements.” By contrast, the notion of homogeneity used in the traditional theory of proportion, as noted, has an essentially dimensional significance: lengths may be composed with lengths but not areas, areas with areas but not volumes.
- 36.
A specifically arithmetical development of the theory of proportion, devoted to commensurable magnitudes (numbers), follows in Book VII. See the “Introductory Note” to Book V in Heath ’s edition of the Elements (1926, vol. 3, pp. 112–113) for discussion of how the discovery of incommensurables is reflected in its structure.
- 37.
See Longuenesse (1998a, pp. 263–271 and 262–263, respectively).
- 38.
See note 21 above, together with paragraph to which it is appended. If, however, we take the traditional theory of proportion as Kant ’s model instead of arithmetic, we need to develop an alternative account to Longuenesse ’s of the correspondence between the categories of quantity and the (quantitative) logical forms of judgement (see note 30 above). Thompson (1989) has developed such an account, although I believe that it needs more work. Compare the discussion of Thompson in Longuenesse (2005, pp. 45–46)—which, in particular, suggests that she may be willing to revise her account of this correspondence accordingly.
- 39.
Longuenesse acknowledges—and even emphasizes—that the application of the categories of quantity to continuous magnitudes is most important to Kant . For example, in her section on continuous magnitudes (see note 37 above) she says that “the most important aspect of the category of quantity” is “the role it plays in the determination of a quantum” (1998a, p. 265), and she goes on to single out continuous quanta in particular (p. 266): “[T]he category of quantity (Quantität) finds its most fruitful use when it serves to determine the quantitas of a quantum, that is, the Größe, the magnitude of an objects itself given as a continuous magnitude, a quantum continuum in space and time.” Yet, because of her overriding emphasis on arithmetic and discrete magnitude in the original application of the categories of quantity, she is also led to the surprising claim that in the Axioms of Intuition “appearances are treated essentially as aggregates, namely discrete magnitudes” (2005, p. 50). One can see what she means by this from her earlier discussion of continuous magnitude. In particular, she there (1998a, p. 264) focusses on the notion of a “quantum ‘in itself’ continuum, but which I can represent as discretum by choosing a unit of measurement to determine the quantitas of this quantum—that is, quoties in eo unum sit positum, how many times a unit is posited in it.” However, by failing to emphasize here that this arithmetical “representation” of a continuous magnitude (as discrete) is limited, and is in fact impossible in the comparison of incommensurable magnitudes, her discussion may easily give the impression that continuous magnitude can be considered as a species of discrete magnitude.
- 40.
See again note 24 above, together with the paragraph to which it is appended and the preceding paragraph.
- 41.
As observed (note 27 above), I did not sufficiently appreciate the importance of Longuenesse ’s reading of § 26 of the Deduction in Friedman (2000b). In particular, I did not then sufficiently appreciate the way in which her account preserves the independent contribution of sensibility by viewing continuity as a de facto property of space and time as our two forms of pure intuition—and, as a result, I did not sufficiently appreciate the way in which Longuenesse can and does make a transition from considering what she takes to be the original application of the categories of quantity in the enumeration of discrete particulars to the measurement of continuous magnitudes in space and time (see notes 37 and 39 above, together with the paragraphs to which they are appended). So Longuenesse (2001/2005) is perfectly correct to rectify these oversights and to emphasize, accordingly, that a proper appreciation of her reading requires one “to pay attention to the distinct and complementary roles Kant assigns to the logical forms of judgement, on the one hand, and to the pure forms of intuition and synthesis of the imagination, on the other” (2005, p. 53). I now agree with Longuenesse in emphasizing the “distinct and complementary roles” of the understanding and sensibility, but, at the same time, I differ with her on two remaining issues: (i) the structure of the categories of quantity, (ii) the relation of the transcendental unity of apperception to space (and time) in the transcendental synthesis of imagination. With respect to the first issue, the main point is that I take the application of these categories, first and foremost, to be to continuous rather than discrete magnitudes, and so, on my reading, there is no transition from the discrete to the continuous case at all. For the second issue see note 44 below.
- 42.
Compare, once again, note 28 above, together with the paragraph to which it is appended.
- 43.
See note 17 above, together with the paragraph to which it is appended.
- 44.
I am not suggesting that Kant appeals to the understanding to explain the continuity of space and time. In particular, it makes perfect sense, from the point of view of contemporary mathematics, to assign the property of continuity to the original form of spatial intuition (as a manifold of perspectives) and to appeal to the possibility of continuous motions (translations and rotations) only to explain the resulting metrical structure. The point, on my reading, is rather, from a contemporary point of view, that Kant assumes a full Euclidean (metrical) structure for space from the beginning—in virtue of which all its parts (lengths, areas, and volumes) then counts as three-dimensional continuous magnitudes in the sense of the traditional theory of proportion. He appeals to the understanding, however, as that which is originally responsible for the status of space as a unified and unitary formal intuition (metaphysical space) within which all (Euclidean) geometrical constructions take place. The fundamental difference between myself and Longuenesse here is that it is precisely this pure geometrical structure, on my view, that most directly realizes the transcendental unity of apperception in sensibility. (I am indebted to Vincenzo De Risi for raising the particular question of continuity in this connection).
- 45.
See again the paragraph to which notes 25 and 26 above are appended. As I shall argue in the next section, it turns out that the mediation in question essentially involves the Newtonian mathematical theory of motion. In Longuenesse ’s reading, by contrast, there are two distinct aspects to Kant ’s grounding of experience—one involving the medium-sized sensible particulars of our ordinary perceptual experience, the other involving the objects of Newtonian mathematical natural science (2005, p. 54): “[I]f my reading is correct, Kant ’s argument is an attempt to account both for the pull of Aristotelianism in our ordinary perceptual world and for the truth of Newtonianism.” I, for one, find much less of a contrast in Kant between ordinary and scientific experience, and much less room, accordingly, for an independent grounding of the former. This does not mean, however, that the argument of the first Critique, on my view, simply collapses into that of the Metaphysical Foundations of Natural Science: see the Conclusion to Friedman (2013) for my account of the fundamentally different perspectives (on the same phenomenal world) represented in these two works.
- 46.
More fully (ibid.): “Time, as you correctly remark, has no influence on the properties of numbers (as pure determinations of magnitude), as [it does], e.g., on the properties of every alteration (as a quantum), which is itself only possible relative to a specific constitution of inner sense and its form (time), and the science of number, regardless of the succession that every construction of magnitude requires, is a pure intellectual synthesis, which we represent to ourselves in thought.”
- 47.
The crucial difference is that, whereas the science of number or arithmetic, for Kant , certainly presupposes the possibility of indefinite iteration (succession) in time, it does not yet constitute the determination of parts of time as mathematical magnitudes or quanta. As explained below, time only acquires what we would now call a metrical structure by means of precisely the mathematical theory of motion, whereby parts of time in particular can now be considered as quanta.
- 48.
- 49.
The quoted passage reads more fully (ibid.): “In phoronomy, since I am acquainted with matter through no other property but its movability, and thus consider it only as a point, motion can only be considered as the describing of a space—in such a way, however, that I attend not solely, as in geometry, to the space described, but also to the time in which, and thus to the velocity with which, a point describes a space. Phoronomy is thus just the pure doctrine of magnitude (Mathesis) of motion.” For further discussion of the transition from pure to empirical motion see again Friedman (2012b) and (more fully) Friedman (2013).
- 50.
Kant ’s three Laws of Mechanics are the conservation of the total quantity of matter, the law of inertia, and the equality of action and reaction; compare the discussion (and illustration) of the synthetic a priori propositions of pure natural science in the Introduction to the second edition of the Critique (B20–21). I (briefly) comment on the relationship between these laws and the Newtonian Laws of Motion in Friedman (2012b) and (more fully) in Friedman (2013).
- 51.
See, e.g., Friedman (2012c) and (more fully) Friedman (2013). The successive determination of true from merely apparent motions, for Kant , involves a nested sequence of ever more comprehensive rotating systems—as we proceed from our parochial perspective here on earth, to the more comprehensive perspective of the center of mass of the solar system, to the even more comprehensive perspective of the center of mass of the Milky Way galaxy, and so on ad infinitum. Kant thereby reinterprets Newtonian absolute space (and extends the Newtonian determination of true from merely apparent motions far beyond the solar system) as the regulative idea (which can never be actually attained) of the ideal limit of this procedure: the perspective (which can never be actually attained) of the center of gravity of all matter.
- 52.
Compare again note 47 above. In applying the Analogies of Experience to the mathematical science of motion, in particular, we determine the magnitudes of temporal intervals by reference to idealized perfectly uniform motions, which then set the standard for correcting the actually non-uniform motions found in nature. In Newton ’s famous remarks concerning “absolute, true, and mathematical time” in the Principia (1999, p. 408), for example, we thereby correct the common “sensible measures” of time such as “an hour, a day, a month, a year” (ibid.), and I argue in Friedman (2013) that Kant takes this procedure as his model for time determination in the above passage from the Analogies (A215/B262). I also argue that Kant has the same procedure in mind in his Second Remark to the Refutation of Idealism, according to which, for example, we “undertake [vornehmen]” such time determination from the observed “motion of the sun with respect to objects on the earth” (B277–278).
- 53.
Although the Analogies of Experience are thus not constitutive of appearances, they are (of course) constitutive of what Kant calls “experience.” Compare Kant ’s discussion of this distinction in the Appendix to the Transcendental Dialectic (A664/B692).
- 54.
See the paragraph to which note 51 above is appended.
- 55.
Friedman (2012c) is my most recent detailed discussion of this point.
- 56.
This is the famous response to Hume ’s “crux metaphysicorum” in the Prolegomena (§ 29; 4, 312). For a detailed discussion see again Friedman (2012c).
- 57.
The relevance of the law of universal gravitation in particular is suggested in § 19 of the Deduction, which develops an account of the “necessary unity” belonging to the representations combined in any judgement as such—“i.e., a relation that is objectively valid, and is sufficiently distinguished from the relation of precisely the same representations in which there would be only subjective validity, e.g., in accordance with laws of association” (B142). Kant then illustrates his point by the relation between subject and predicate in the judgement “Bodies are heavy” (ibid.). This discussion continues the discussion of “judgements of experience” developed in the Prolegomena, and the example Kant chooses invokes universal gravitation as discussed in both § 38 of the Prolegomena and the Metaphysical Foundations of Natural Science. See Friedman (2012c) and Friedman (2015) for further relevant details concerning the relationship between the second edition Deduction and the Prolegomena.
- 58.
That the space in question functions as a foundation for both the mathematical science of (Euclidean) geometry and the modern conception of universally valid mathematical laws of nature sheds light on the sense in which the original act of the understanding responsible for the necessary unity of this space is more general than the unifying activity expressed in any particular category. We are thereby not confined, for example, to either the categories of quantity (as realized in the science of geometry) or the categories of relation (as realized in the universal laws of nature); rather, the original act in question insures the application of all of the categories together.
- 59.
This conception is explicitly developed in Newton ’s unpublished De Gravitatione, but it also surfaces in some of his best-known published works, such as the General Scholium to the Principia and the Queries to the Opticks. See Stein (2002) for a detailed discussion, and compare Janiak (2008) for a somewhat different perspective.
- 60.
- 61.
- 62.
I discuss the relationship between Helmholtz ’s and Poincaré ’s conception of geometry as based on the principle of free mobility and Kant ’s “perspectival” conception of space as our pure form of outer sensible intuition in Friedman (2000a).
- 63.
For the relationship between Kant ’s reinterpretation of the Newtonian conception of absolute space (compare note 51 above) and the concept of what we now call an inertial frame of reference see Friedman (2013, pp. 503–509), where I also refer to DiSalle (2006) in the same connection. For Mach and the concept of inertial frame see DiSalle (2002).
- 64.
My most detailed discussion of the conceptual development from Kant through Helmholtz , Mach , and Poincaré to Einstein is Friedman (2010, pp. 621–664).
- 65.
- 66.
See Einstein (1921, pp. 3–4, 1923, pp. 28–29; my translation): “In so far as the propositions of mathematics refer to reality they are not certain; and in so far as they are certain they do not refer to reality. Full clarity about the situation appears to me to have been first obtained in general by that tendency in mathematics known under the name of ‘axiomatics’. The advance achieved by axiomatics consists in having cleanly separated the formal-logical element from the material or intuitive content. According to axiomatics only the formal-logical element constitutes the object of mathematics, but not the intuitive or other content connected with the formal-logical elements.” Although Einstein does not explicitly mention Kant here, these famous words were clearly intended and standardly taken as a rebuttal of the Kantian conception that mathematics (especially geometry) is paradigmatic of synthetic a priori knowledge. They were so standardly taken, in particular, by the logical empiricists beginning with Moritz Schlick —who is in turn favorably cited in precisely this connection by Einstein . For further discussion see again Friedman (2002).
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Friedman, M. (2015). Kant on Geometry and Experience. In: De Risi, V. (eds) Mathematizing Space. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-12102-4_12
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