Abstract
I argue that Kant's claim in the “Transcendental Aesthetic” of the Critique of Pure Reason that space and time are immediately given in intuition as infinite magnitudes is undercut by his general theory of mathematical knowledge. On this general theory, pure intuition does not give objects of any determinate magnitude at all, but only forms of possible objects. Specifically, what pure intuition itself yields is the recognition that any determinate space or time is part of a larger one, but it requires an inference of reason to go from that to the claim that space and time are infinite. I further argue that this result is consistent with Kant's claim in the second-edition “Transcendental Deduction” that the unity of space and time are the products of synthesis, but also means that the unity of space and time as objects cannot be used a premise in the Deduction but can only be regarded as a conclusion of the deduction and the following “System of Principles.”
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Notes
- 1.
All quotations from the Critique of Pure Reason are from Immanuel Kant, Critique of Pure Reason , edited and translated by Paul Guyer and Allen W. Wood (Cambridge: Cambridge University Press, 1998). Quotations from Kant’s inaugural dissertation, On the Forms and Principles of the Sensible and Intelligible Worlds (“ID”) are from Kant, Theoretical Philosophy 1755–1770, translated and edited by David Walford in collaboration with Ralf Meerbote (Cambridge: Cambridge University Press, 1992).
- 2.
I believe that the Kästner comments answer the objection made to me in comments on an earlier draft of this paper by both Rosalind Chaplin and Lisa Shabel , who each take the claim in (4) of the B-edition metaphysical exposition of space that space is an infinite given magnitude to be an independent and self-evident premise for the conclusion that the representation of space is an intuition rather than a concept; the Kästner comment shows that Kant derives this premise from the containment thesis of point (3) plus the assumption that the containment thesis applies to a space of any size. In support of this interpretation it might also be noted that in the inaugural dissertation, Kant’s first version of the Transcendental Aesthetic , he lumps the containment-thesis and the boundlessness-thesis together in a single argument (ID, §15.B, 2:402 in the case of space, §14.2, 2:399 in the case of time). I would like to thank both Shabel and Chaplin for their comments.
- 3.
See Michael Friedman , “Kant on Geometry and Spatial Intuition,” Synthese 186 (2012): 231–55, at pp. 240–1, and his quotation there from Kant’s response to Eberhard at Ak. 20: 420–1.
- 4.
Colin McLear , “Two Kinds of Unity in the Critique of Pure Reason ,” Journal of the History of Philosophy 53 (2015): 79–110, at p. 95.
- 5.
Friedman , “Kant on Geometry and Spatial Form,” pp. 237–8.
- 6.
Christian Onof and Dennis Schulting , “Space as Form of Intuition and as Formal Intuition: On the Note to B 160 in Kant’s Critique of Pure Reason ,” Philosophical Review 124 (2015): 1–58, at p. 15.
- 7.
Kant, Reflection 4673, 17:641; from Immanuel Kant, Notes and Fragments, edited by Paul Guyer, translated by Curtis Bowman, Paul Guyer, and Frederick Ruascher (Cambridge: Cambridge University Press, 2005), pp. 156–7.
- 8.
Onof and Schulting , “Space as Form of Intuition and as Formal Intuition,” p. 19. They make this claim in criticism of Béatrice Longuenesse.
- 9.
In this connection it could also be pointed out that in at least some regards what is represented in pure intuition does not have any determinate magnitude at all. We can see this by considering one of Kant’s typical examples of the representation of a geometrical figure in pure intuition , namely the construction of a triangle in pure intuition in order to prove that the sum of the interior angles of any triangle equals 180° (A 716/B 744). Kant asks us to imagine a triangle and to imagine extending one of its sides and then dividing the external angle thus created with another line parallel to the opposite side of the triangle; we will then see that the two angles thus created plus the adjoining internal angle together occupy one side of the straight line constructed by previously extending the side of the triangle, thus that they equal 180°; but we will also see that each of the two newly divided angles is equal in size to one of the other original interior angles, because the line dividing the two exterior angles is parallel to the opposite side of the original triangle, from which it follows that the original interior angles also add up to 180°. Now, one could argue that in this case too the result is not immediately given, but is rather inferred from or on the basis of what is intuited, but that is not the point I want to emphasize. Rather, the point is that while in this example one (and in fact only one) magnitude is determined, namely that the sum of the interior angles of a triangle equals 180° (although what the magnitude of each of the three individual interior angles might be is left undetermined), other magnitudes are left undetermined, in particular the length of the sides of the triangle and thus the size of the triangle as a whole is left undetermined – as indeed must be the case if the proof is to be valid for all triangles. And this suggests that pure intuition never gives objects of any determinate size, although it can present one figure as contained in another, thus the latter as larger than the former. This in turn suggests that the pure intuition of regions of space as contained in a larger space does not present the former as having any determinate size, thus neither does it present the latter as having any determinate size; it presents it merely as indeterminately larger than the former. This puts the burden of insight into the unlimitedness of space as a whole back onto the idea of iteration, that is the recognition that the intuition of a smaller space as contained within a larger one can be indefinitely reiterated, and thus that the larger space must be unlimited. But this just reinforces the objection that the infinite magnitude of space is inferred, not simply given in pure intuition .
- 10.
Onof and Schulting , “Space as Form of Intuition and Formal Intuition,” p. 15; McLear , “Two Kinds of Unity,” pp. 86–93; Friedman , “Kant on Geometry and Spatial Form,” pp. 240–1.
- 11.
Onof and Schulting , “Space as Form of Intuition and as Formal Intuition,” p. 21.
- 12.
Onof and Schulting observe that “there is no location specificity to constructions in space: when I construct a triangle, there is no reference to where it is constructed”; “Space as Form of Intuition and as Formal Intuition,” p. 40. That is correct, and it means that when I construct another triangle in pure intuition there is no determinate spatial relation between the two triangles; thus Onof and Schulting’s prior assertion that geometry requires a single (token of) space is false.
- 13.
See Allison , Kant’s Transcendental Deduction , pp. 46–7, and Kant’s Transcendental idealism: An Interpretation and Defense, revised edition (New Haven: Yale University Press, 2004), pp. 12–16, 27–8.
- 14.
This passage could seem to support the “diagrammatic” interpretation of Kant’s philosophy of geometry by Kenneth Manders that Friedman attacks in “Kant on Geometry and Spatial Form.” But I am not disputing Friedman’s point that the possibility of the abstraction that Kant describes in this passage presupposes the pure form of spatial intuition, only Friedman’s supposition that (repeated) construction of figures in pure intuition presents us with an actual singular space.
- 15.
See for example Reflection 6315, 18:618–21, at 18: 619; Notes and Fragments, pp. 361–3, at p. 361.
- 16.
Dieter Henrich , “The Proof-Structure of the Transcendental Deduction ,” Review of Metaphysics 22 (1969): 640–59.
- 17.
Henry E. Allison , Kant’s Transcendental Deduction : An Analytical-Historical Commentary (Oxford: Oxford University Press, 2015).
- 18.
Allison , Transcendental Deduction , p. 406.
- 19.
Onof and Schulting , “Space as Form of Intuition and as Formal Intuition,” especially pp. 12–16.
- 20.
Allison , Transcendental Deduction , p. 412.
- 21.
Allison, Transcendental Deduction , p. 415.
- 22.
It will be recalled that Onof and Schulting had included infinitude among the marks of unicity; they assumed that infinitude is given, while I argued that it can only ever be represented as the product of an incompletable synthesis , thus as an idea of reason. Allison does not employ the assumption of the infinitude of space or time; his argument is rather that an a priori synthesis of the unity of space and time precedes empirical synthesis of the world of objects in space and time, and that the necessity of the categories for the latter is guaranteed by their role in the former. Thus I might say that my real target here is my old enemy, the idea of an a priori synthesis that precedes empirical synthesis ; see my “Kant on Apperception and A Priori Synthesis,” American Philosophical Quarterly 17 (1980): 205–12.
- 23.
On the use of the categories for synthesis of the unity of space , see Onof and Schulting , pp. 42–4. The argument for the use of the full set of the categories depends on their role in the construction of geometrical figures, and not just on their use for the supposed synthesis of the unity of pure space itself.
- 24.
Onof and Schulting , “Space as Pure form and as formal Inution,” p. 21.
- 25.
Anthony Quinton , “Spaces and Times,” Philosophy 37 (1962): 130–47.
- 26.
It might be objected that Quinton’s thought-experiment depends upon the assumption that the subject must at least be able to represent himself as a single subject in both of his sequences of his experiences, thus at least remember having the night-time experiences while having the day-time experience, in order to even ask whether he is in fact the same subject through both series of experiences. But if this just takes the veracity of memory for granted, when in fact memory cannot function without a framework of objective knowledge to support it, all the better for my argument: the subject could not recognize himself as one without the representation of a single spatio-temporal world, but a single world of empirical objects in space and time, not some single pure intuition of space and time, which as I have been arguing is only a representation of a single form but not a single object. Thanks to Rosalind Chaplin for this objection.
- 27.
On this point, see also my paper “Space, Time, and the Categories: Kant’s Project in the Transcendental Deduction ,” in Ralph Schumacher and Oliver R. Scholz, eds., Idealismus als Theorie der Repräsentation? (Paderborn: Mentis Verlag, 2001), pp. 313–38.
References
Allison H. E. (2004). Kant’s transcendental idealism: An interpretation and defense (Revised ed.). New Haven: Yale University Press.
Allison, H. E. (2015). Kant’s transcendental deduction: An analytical-historical commentary. Oxford: Oxford University Press.
Dieter, H. (1969). The proof-structure of the transcendental deduction. Review of Metaphysics, 22, 640–659.
Friedman, M. (2012). Kant on geometry and spatial intuition. Synthese, 186, 231–255.
Guyer, P. (1980). Kant on apperception and a priori synthesis. American Philosophical Quarterly, 17, 205–212.
Guyer, P. (2001). Space, time, and the categories: Kant’s project in the transcendental deduction. In R. Schumacher & O. R. Scholz (Eds.), Idealismus als theorie der repräsentations? (pp. 313–338). Paderborn: mentis Verlag.
Kant, I. (2005). Notes and fragments, edited by Paul Guyer, trans. by Bowman C. Guyer P., & Ruascher F. Cambridge: Cambridge University Press.
McLear, C. (2015). Two kinds of unity in the critique of pure reason. Journal of the History of Philosophy, 53, 79–110.
Onof, C., & Schulting, D. (2015). Space as form of intuition and as formal intuition: On the note to B 160 in Kant’s critique of pure reason. Philosophical Review, 124, 1–58.
Quinton, A. (1962). Spaces and times. Philosophy, 37, 130–147.
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Guyer, P. (2018). The Infinite Given Magnitude and Other Myths About Space and Time. In: Nachtomy, O., Winegar, R. (eds) Infinity in Early Modern Philosophy. The New Synthese Historical Library, vol 76. Springer, Cham. https://doi.org/10.1007/978-3-319-94556-9_11
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