Spatial representation, magnitude and the two stems of cognition

Abstract

The aim of this paper is to show that attention to Kant's philosophy of mathematics sheds light on the doctrine that there are two stems of the cognitive capacity, which are distinct, but equally necessary for cognition. Specifically, I argue for the following four claims: (i) The distinctive structure of outer sensible intuitions must be understood in terms of the concept of magnitude. (ii) The act of sensibly representing a magnitude involves a special act of spontaneity Kant ascribes to a capacity he calls the productive imagination. (iii) Contrary to what is assumed by many commentators, it is not the case that the Two Stems Doctrine implies that a representation is either sensible or spontaneity-dependent, but not both. (iv) Outer sensible intuitions are both sensible and spontaneity-dependent – they are sensible because they exhibit the kind of structure Kant takes to be distinctive of outer sensible intuitions, and they depend on spontaneity because they are cases of sensibly representing a magnitude.

Keywords::

• intuition
• sensibility
• concept
• spontaneity
• magnitude
• productive imagination
• space

Notes

^1. ‘[…] there are two stems of human cognition which may perhaps arise from a common but to us unknown root, namely sensibility and understanding, through the first of which objects are given to us, but through the second of which they are thought’ (A15/B29). ‘Intuition and concepts therefore constitute the elements of all our cognition so that neither concepts without intuition corresponding to them in some way nor intuition without concepts can constitute a cognition.’ (A50/B74) – References to the Critique of Pure Reason use the A- and B-edition pagination; translations are from Kant (1998), tacitly modified where appropriate. References to other works of Kant's are by volume- and page-number of the Academy Edition (Ak. = Kant 1902ff) using the following abbreviations: Logik = Logik, ed. Benjamin Jaesche. Prolegomena = Prolegomena zu einer jeden künftigen Metaphysik die als Wissenschaft wird auftreten künnen [Prolegomena to any Future Metaphysics that will be able to come forward as Science].

^2. Conceptualist readings of Kant are advocated by, among others, Allison (2004), Ginsborg (2006, 2008), Griffith (2012), McDowell (1998, 2009a, 2009b), Pippin (1982), Sellars (1978, 1992) and Strawson (1966). Nonconceptualist readings have been proposed by Allais (2009), Hanna (2005, 2011), McLear (forthcoming-b), Rohs (2001) and Tolley (2013). See McLear (forthcoming-a) for an overview of the debate.

^3. See A68f/B93f, A320/B376, Logik §1 (Ak. IX: 91).

^4. Throughout this paper, I will be concerned with sensible, as opposed to intellectual, intuition and indeed with sensible intuitions of outer, as opposed to inner, sense, but I will not always make this explicit. When I speak of intuition simpliciter, this should be taken as shorthand for ‘outer sensible intuition,’ except where indicated. The same goes for ‘understanding’ and ‘discursive understanding’.

^5. See A68f/B93f, A320/B376, Logik §1 (Ak. IX: 91).

^6. See, for instance, the well-known dispute between Hintikka (1969a, 1969b) and Parsons (1982) on whether the singularity of intuition entails its immediacy. Furthermore, Wolff (1995) makes the case that not only intuitions but concepts, too, can relate to objects immediately, so that we need to distinguish between conceptual and intuitive immediacy.

^7. Perhaps the most prominent advocate of this position is Wilfrid Sellars, who argues that although having an intuition is distinct from making a judgement, an intuition must be understood on the model of a complex demonstrative referring expression (a ‘this-such’) comprising a demonstrative and a sortal term. See Sellars (1967, 1978, 1992).

^8. ‘The unity of aesthetic representation – characterized by the forms of space and time – has a structure in which the representational parts depend on the whole. The unity of discursive representation – representation where the activity of the understanding is involved – has a structure in which the representational whole depends on its parts’ (McLear forthcoming-b, 19f.).

^9. ‘Thus, if a representation has a structure in which the parts depend on the whole rather than a structure in which the whole is dependent on its parts, that representation cannot be a product of intellectual activity, but must rather be given in sensibility independently of any such activity’ (McLear forthcoming-b, 18f.).

^10. ‘Synthetic unity of the manifold of intuitions, as a priori given, is therefore the ground of the identity of apperception itself, which precedes a priori all my determinate thinking’ (B134).

^11. Kant's conception of analytic judgements highlights another respect in which discursive activity can be said to move from whole to part rather than part to whole. In Kant's view, analytic judgements ‘dissect [the subject concept of the judgment] into its component concepts, which were already thought in it (albeit obscurely)’ (A7/B11). Notice also that, according to the account offered in §77 of the Critique of Judgment (Ak. V: 407), the intuitive intellect proceeds from an intuition of the whole to cognition of its parts. This might be thought to support the contention that part-on-whole dependence is characteristic of intuition in general, whether finite and sensible or infinite and intellectual. By parity of reasoning, however, it would also show that whole-on-part dependence cannot be a characteristic of intellectual activity in general. This at least suggests (though it does not establish) that such dependence does not constitute the distinguishing characteristic of finite intellectual activity, either. Thanks to Johannes Haag for drawing my attention to this passage.

^12. That relation of representations which constitutes a judgement is ‘sufficiently distinguished’ from a merely associative unity by the fact that the former, unlike the latter, depends on ‘principles of the objective determination of all representations, insofar as cognition can arise from them, which principles are all derived from the principle of the transcendental unity of apperception’ (B142). Again, we can note the intimate connection between judgement and the doctrine of apperception, while leaving open the precise meaning of the passage. See B141 for the claim that associative unity is due to the reproductive imagination and B151f for the claim that this capacity is sensible rather than intellectual.

^13. See also B153, where Kant says that the transcendental synthesis of imagination is an act of the understanding, and B162, where he says that understanding and productive imagination are ‘one and the same spontaneity’ considered under different aspects.

^14. See e.g., A714f/B742f. Note that for Kant mathematical cognition is not defined as being cognition of magnitudes. It is defined, rather, in terms of its method, as rational cognition based on the construction of concepts in pure intuition. It is a consequence of this definition that mathematics cognizes magnitudes. For discussion see Sutherland (2004b).

^15. Compare the following passage from the Axioms of Intuition: ‘Empirical intuition is possible only through pure intuition (of space and time); what geometry says about the latter is therefore undeniably valid of the former […]’ (A165/B206). Consider also Kant's claim in the Aesthetic, that the representation of space ‘is the ground of all outer intuitions’ (A24/B38).

^16. See, in particular, Sutherland (2004a, 2004b), to which the discussion that follows is indebted.

^17. Cf. the following explanation from the Lectures on Metaphysics: ‘That determination of a thing through which one cognizes something as a quantum is quantity or magnitude’ (Metaphysik K3, Ak.XXIX: 991). As Shabel (2005) puts the point: For Kant, as for the moderns generally, magnitudes have magnitude.

^18. Compare the fourth and fifth of Euclid's Common Notions.

^19. Kant appears to hold that the holistic nature of outer intuitions is a consequence of their character as magnitudes and my point here is that the converse implication does not hold (see Sutherland [2004a] for discussion). To see why Kant might hold this, consider that if space exhibits strict logical homogeneity, then the identity conditions of any part of space can be given only in terms of its relation to other parts of space. There is nothing intrinsic about a part that could differentiate it from other parts. So necessarily any talk of a part of space presupposes reference to other parts of space, indeed to the whole of space. Kant seems to say as much at Prolegomena, §13 (Ak. IV:286): ‘[…] the inner determination of any space is possible only through the determination of the outer relation to the whole of space […], i.e. the part is possible only through the whole’. This holistic character of space should then also characterize the representation of space. Thanks to Daniel Smyth for bringing this passage to my attention.

^20. This is not to deny that objects which are considered as magnitudes can differ from one another qualitatively. The point is that insofar as their character as magnitudes is concerned, they are regarded as strictly logically homogeneous; see Sutherland (2004a, 199f.). I say more about this at the end of this sub-section and in the following sub-section.

^21. See A272/B328. See also A725/B753, where Kant speaks of space and time as the ‘only original quanta’.

^22. I am grateful to Tobias Rosefeldt for helping me clarify this point.

^23. The notion of a mark is discussed in Logik, Introduction, §VIII (Ak. IX, 58ff).

^24. For discussion see Anderson (2005) and Schulthess (1981).

^25. These roles are, of course, relative. If A is contained under B, and B under C, then B is a species relative to C, which is its genus. But B is a genus relative to A, which is a species of it.

^26. The division need not be dichotomous, but can comprise more members, as long as there is no overlap between them and they jointly exhaust the extension of the divided concept. The dichotomous case is simply the easiest for illustration purposes. For discussion see Anderson (2005) and Wolff (1995, 160–170).

^27. Consider an example: I begin with the concept 〈scholar〉, and the task is to represent a plurality of scholars. Because the only means at my disposal is the introduction of conceptual marks, I can represent, say, two scholars by introducing a mark that sorts scholars into short and tall. But this means that I have succeeded in representing numerical difference only at the cost of introducing further qualitative differences. Clearly, this would be the case no matter how many additional marks I introduce.

^28. This is not to deny that there are concepts of magnitudes. Mathematical concepts are of this kind. But they are not purely conceptual in the sense that their content depends essentially on the possibility of being constructed in intuition and so depends on intuition. Kant characterizes such concepts as ‘containing’ pure intuitions (A719/B747). This distinguishes such concepts from what I am calling purely conceptual representations.

^29. Thanks to Stefanie Grüne and Marcus Willaschek for pointing out the need for clarification here.

^30. Although this is a separate point, we should also note that Kant holds that sensations themselves, considered in abstraction from their spatial or temporal extent, are magnitudes; see A166/B207–A176/B218. Although sensations are intensive rather than (like space and time) extensive magnitudes, if the account of the general concept of magnitude presupposed here is correct, intensive magnitudes, too, are conceived as manifolds of strictly logically homogeneous parts. For discussion see Sutherland (2004a, 196f.).

^31. Because Kant recognizes spatial concepts (see Note 28 above), we should distinguish between a direct and an indirect dependence of spatial representations on acts of the productive imagination. Sensible spatial representations (i.e. outer intuitions) depend directly on exercises of the productive imagination. Conceptual spatial representations depend indirectly on exercises of the productive imagination of spatial magnitudes inasmuch as they depend for their content on being constructible in intuition and thus on sensible spatial representations. I say more about construction and the productive imagination in Section 6.

^32. See Land (2006) for discussion.

^33. Note that if I am right and spatial representation for Kant partly depends on acts of spontaneity, yet is still in a robust sense sensible rather than discursive, then there is a danger of using the term ‘sensibility’ ambiguously. For we might mean by this either a capacity that is merely receptive and so on my reading of Kant would not be a capacity for spatial representation of the kind under discussion in the Critique. Or we might mean by it a capacity for the latter kind of representation, hence, a capacity for representations that exhibit the characteristic structure of outer intuitions, yet depend also on spontaneity. So we should distinguish between a merely receptive kind of sensibility and a kind of sensibility that is not merely receptive but, as it were, conditioned by spontaneity. Part of the point of the doctrine of the productive imagination is, I think, to show that the kind of sensibility Kant is concerned with in the Critique is the second kind. Perhaps no-one has made the case for this more eloquently than Sellars (see e.g., Sellars 1967, 1978, 1992; see also the lectures published as Sellars 2002). For a defence of the idea see also McDowell (2009a, 2009b). We should also note that according to Sellars (1992) Kant was himself not entirely clear about the commitments entailed by his theory of sensibility and, in a sense, fell victim to the ambiguity just outlined. A related ambiguity is noted by Beck (2002) and Allison (2004).

^34. For discussion see, e.g., Evans (1985) and Rödl (2012); see also Sellars (1967).

^35. See Evans (1985).

^36. The passage is sometimes read differently. Thus, Tolley (2013) reads the claim that a representation contained in a single moment can only have absolute unity as marking one side of a distinction between a manifold's having a unity and a manifold's being represented as having a unity. However, because a moment is a temporal boundary in the same way that a line or a point is a spatial boundary (see A169/B211) and because, as is explicitly pointed out by Kant in the paragraph preceding the passage, the kind of manifoldness being considered is temporal manifoldness, an absolute unity cannot be the unity of a manifold, whether represented as such or not. For helpful discussion of this point see Henrich (1953).

^37. The qualifier is intended to allow for the possibility of kinds of spatial representation that do not depend on spontaneity. The kind of spatial representation enjoyed by non-rational animals is presumably of this kind, for Kant. In the Critique, Kant is concerned with finite rational knowers. What he says about spatial representation applies, in the first instance, to those finite rational knowers whose forms of sensibility include space. Whether, and to what extent, it also applies to non-rational creatures is not obvious.

^38. For further support of this point, consider that it is a premise of the First Space Argument that in outer experience I represent things as being in different locations in space from my own: ‘For in order for certain sensations to be related to something outside me (i.e. to something in another place in space from that in which I find myself) […], the representation of space must already be their ground’ (A23/B38). For discussion see Warren (1998).

^39. I say more about it in Land (2006, forthcoming).

^40. See A712/B740ff.

^41. Although this passage focuses on geometry, Kant holds that all of mathematics depends on construction. For discussion see Shabel (1998).

^42. For the recent debate see Carson (1997), Friedman (1992, 2000, 2012), and Shabel (1998, 2003).

^43. Kant's most extensive (though still rather abbreviated) discussion of mathematical construction is found at A713f/B741f.

^44. This idea of ‘reading off’ properties of the figure is most naturally understood in terms of the Euclidean diagram. The diagram allows the geometer literally to see that certain spatial relations obtain between different parts of the constructed figure. For discussion see Shabel (2003).

^45. See Shabel (2003).

^46. It might be thought that according to Friedman's (2012) account of construction, this is not the case. For Friedman rejects the view, propounded e.g., by Shabel (2003), that the Euclidean diagram plays an essential role in Kant's theory of construction. However, despite his reservations concerning diagrams, Friedman is still committed to the idea that construction involves the generation of determinate spatial representations. On his account, Kant's notion of construction is understood in terms of the notion of a function. For example the construction of the concept ‘triangle’ must be understood, according to Friedman, in terms of a ‘function or constructive operation which takes three arbitrary lines (such that two together are greater than the third) as input and yields the triangle constructed out of these three lines as output’ (2012, 237). Crucially, the outputs of such functions ‘are indeed singular or individual representations’ (2012), that is, what I am calling determinate spatial representations.

^47. As Kant puts it, the geometer proceeds by ‘determining [her] object in accordance with the conditions of […] pure intuition’ (A718/B746).

^48. This suggests that there is a sense in which the concept at issue is a practical concept; that is, a concept of a way of acting. Note that Kant characterizes a mathematical postulate as ‘the practical proposition that contains nothing except the synthesis through which we first give ourselves an object and generate its concept’ and gives one of Euclid's postulates as an example (A234/B287, my emphasis).

^49. See the discussion above, in Section 5.

^50. For example, B154, A163/B204, A164f/B205.

^51. This is not to say, of course, that these are the only salient doctrines. A full account would have to give due consideration to, for instance, the arguments put forth in the Transcendental Deduction of the Categories.

^52. For comments and suggestions I am grateful to audiences in Chicago, Frankfurt, Oxford and Potsdam, and in particular to James Conant, Stefanie Grüne, Johannes Haag, Till Hoeppner, Colin McLear, Tobias Rosefeldt, Daniel Smyth, Andrew Stephenson, Daniel Sutherland and Marcus Willaschek. Work on the paper was supported by a research fellowship at Corpus Christi College, Cambridge.