Kant on the Acquisition of Geometrical Concepts

Abstract

It is often maintained that one insight of Kant's Critical philosophy is its recognition of the need to distinguish accounts of knowledge acquisition from knowledge justification. In particular, it is claimed that Kant held that the detailing of a concept's acquisition conditions is insufficient to determine its legitimacy. I argue that this is not the case at least with regard to geometrical concepts. Considered in the light of his pre-Critical writings on the mathematical method, construction in the Critique can be seen to be a form of concept acquisition, one that is related to the modal phenomenology of geometrical judgement.

Keywords::

• Kant
• concepts
• geometry
• construction
• phenomenology

Notes

^1. E.g. A95, A129, B167, A310/B366, A 720/B748. Kant uses ‘a priori’ to modify a number of terms, such as ‘intuition’, ‘representation’, ‘principle’ ‘judgement’, ‘truth’, etc. I don't explore the relationship between this and other uses of the modifier here.

^2. I use the angled brackets and italics to indicate mention of the concept. I don't discuss here the concepts < Space> and < Time>, which also fall under the heading of pure sensible concepts, and which have their own acquisition procedure, as they relate to the pure intuitions of Space and Time.

^3. E.g., On a Discovery, 8:221. See also Inaugural Dissertation, §8, 2:395, §15, 2:406; Metaphysik Mrongovious, 29: 760–763.

^4. E.g., see the Blomberg Logic, §254; Jäsche Logic, §3.

^5. At A729/B757, having claimed that mathematics proceeds through the particular process of construction, Kant states that only mathematical concepts are apt for this process. As shall be discussed, only mathematics contains definitions, Kant thinks, because only mathematical concepts can be acquired through being defined in a construction procedure. Kant does speak of the acquisition procedure for categorial concepts as being of a specific kind, and he refers to this procedure as ‘original acquisition’ (On a Discovery, 8:221). That a priori concepts are acquired at all might seem surprising, since whatever else it connotes, ‘a priori’ surely connotes some sort of ‘independence from experience’. One can point out first of all that for Kant, a concept is a priori if and only if it issues in an a priorijudgement when deployed, where the latter is understood as one whose truth conditions are not provided by sensory experience (even if sensory experience nevertheless serves as a necessary enabling condition for the possibility of such judgements – B1-2). This condition does not by itself place any putative restrictions on a priori concepts’ possession conditions. Nevertheless, the very idea of categorial concepts as acquired might seem to be precluded by the fact that those concepts are for Kant's necessary conditions of the possibility of experience – for discussion see my (2011).

^6. I focus on the case of geometry in this paper. Although there are of course notable and important differences between Kant's handling of geometry and his handling of arithmetic and algebra, though there are grounds for thinking that geometrical knowledge was for Kant the paradigmatic case of mathematical knowledge (his mathematical examples are primarily Euclidean examples (e.g. Bxi–xii, A164/B205, A716-7/B745).

^7. There are well-known complications here with regard to interpreting the nature of the ‘synthetic’ and ‘analytic’ methods supposedly deployed in the Critique and the Prolegomena, respectively, whereby only the latter is supposed to be pursued upon the assumption of some well-grounded body of knowledge. However, I take it to be well established that Kant frequently argues from the basis of the assumption of some a priori knowledge in the form of mathematical knowledge (most notably in the argument for transcendental idealism in the Transcendental Aesthetic) – see Bx, A4/B8, B4, B20, A38-9/B55-6. For a recent discussion of the meaning of Kant's synthetic method, see Merritt (2006).

^8. That Kant is interested in securing epistemological (and not merely psychological) results does not entail that his inquiry cannot be construed as one into the literal origins or sources of our concepts, although it does require re-consideration of the type of epistemic normativity that might be at stake here, as I argue in my (2011). I take it that the account presented here is in some ways supportive of the picture presented in Longuenesse (1998).

^9. Henceforth, the ‘Inquiry’.

^10. Kant's insistence here on looking to actual practice in order to determine proper method is repeated in the Inaugural Dissertation, where he states that ‘in natural science and mathematics, use gives the method’ (§23, 2:410 – emphasis in original).

^11. The essential character of mathematical concepts as ‘elective’ is noted by Sutherland (2010).

^12. See Jäsche Logic, §§4–5.

^13. However, caution is required here because Kant is not explicit with regard to what in fact determines the parameters for givenness in this sense.

^14. As Kant puts it:

In mathematics I begin with the definition of my object, for example, of triangle, or a circle, or whatever. In metaphysics I never begin with a definition. Far from being the first thing I know about the object, it is nearly always the last thing I come to know. In mathematics, namely, I have no concept of my object at all until it is furnished by the definition. In metaphysics I have a concept which is already given to me, although it is a confused one. My task is to search for the distinct, complete and determinate concept. (Inquiry, 2:283)

This is a claim that Kant retains in the first Critique. There he holds that it is still the case that mathematics begins with definitions, although it is unclear as to whether or not he now holds the achievement of definition in metaphysics to be possible at all. (A729-32/B757-760).

^15. It is clear then that in 1763 Kant still viewed the proper method of metaphysics to be that of analysis, a view famously rejected in the Critique.

^16. This distinction corresponds to types of representational vehicle. Both types of representation can serve to express general content – e.g. < triangularity> can be expressed through a picture of a triangle or through the tokening of the word ‘triangle’ – the difference consists in the manner in which each type of representational vehicle expresses that same representational content. When considered with regard to our performance of mathematical operations, the distinction broadly corresponds to the contemporary one in developmental psychology and neuroscience between non-symbolic and symbolic numerical cognition, i.e. the respective products of our abilities to represent quantities through dots, strokes, etc. on the one hand and to symbolize those representations in terms of Arabic or Roman numerals or natural language on the other, e.g. see Ansarib, Chee, and Venkatramana (2005), Fias and Verguts (2004), Lipton and Spelke (2005), Spelke (2011). Parsons (1983) gives an illuminating discussion of the possible role of ‘concrete tokens’ deployed for the purpose of ‘verifying general propositions’ (136). Though my emphasis on concept acquisition of course differs significantly from the approach pursued there, my account of the epistemic role of signs in concreto (which is performed by intuitions in the Critique) is broadly in accord with Parsons’ account of intuition. For differing accounts, see Hintikka (1969), Howell (1973) and Thompson (1972).

^17. See Friedman (1985), Shabel (2010), Young (1982).

^18. I discuss this theme more in my ‘Kant on Signs in Concreto in Geometry’ (submitted for publication).

^19. The notion of intuition itself was made in the Inaugural Dissertation – the recognition of its necessary co-deployment with concepts for cognition, i.e. the Discursivity Thesis, was not made until the first Critique.

^20. This point is stressed in Warren (1998) and Waxman (2005).

^21. Either individually or conterminously, as when we can access spatial information through both touch and sight.

^22. Although Kant is not clear on this point, I see no reason to take him as claiming that our imaginational access to intuitional content does not involve sensory content, but rather that it involves an indirect reproduction of such content.

^23. When Kant is discussing the importance of the distinction of mathematics and philosophy, he notes that mathematicians have rarely philosophized regarding the nature of their own practice. He chastises them for neglecting that task, which he then characterizes as that of accounting ‘[f]rom whence the concepts of space and time with which they busy themselves … might have been derived’ (A725/B753).

^24. Typical statements of the dangers of invented concepts can be found at A222-B279.

^25. Cassirer (1981) claims that Kant read the Nouveaux Essais sometime between its publication in 1765 and the writing of the Inaugural Dissertation in 1770 (97–99), though he offers little justification for the claim. Tonelli (1974) adduces evidence for thinking that any familiarity Kant had with the work could not have occurred second-hand through its reception by his contemporaries. I think a case can be made for Kant's first-hand familiarity with the Nouveaux Essais (e.g. he refers to the ‘Essays of Locke and Leibniz’ (4: 257) in the Prolegomena 2 years after the publication of the first edition of the first Critique) though to do so would be beyond the scope of this paper. In what follows I present one example of the similar themes and modes of expression to be found in both the Nouveaux Essais and in Kant's Critical writings.

^26. See Leibniz (1996).

^27. Ibid.

^28. The speaker here is Philalethes, who is Locke's representative, though Leibniz does not have Theophilus – his representative – quarrel on these points. The conception of inference expressed by Locke would have been a common one within Cartesian and Port-Royal Logic. Hume too would have subscribed to it, challenging not the conception of inference at stake, but rather the scope of the knowledge that might be attained through it. For an excellent discussion of these topics, see Owen (1999).

^29. The passage is worth considering not least because it gives one likely contender for the source of Kant's focus upon the word ‘intuition’ (Anschauung, but which Kant also refers to with the Latin intuitus) that connects it with the epistemic sense of intuitive knowledge found in the rationalist tradition.

^30. Whereas Leibniz describes the demonstrative reasoning employed in proving Proposition I.32 as a ‘chain of items of intuitive knowledge [enchaînement des connaissances intuitives]’ Kant's reasoning with regard to the same proposition is held to proceed through a ‘chain of inferences [eine Kette von Schlüssen] that is always guided by intuition’.

^31. E.g. Friedman (1985), Shabel (2003), (2006).

^32. I have presented the critique here as if an empiricist account of geometrical concept acquisition is Kant's target. However, I argue in my (2014) that the primary target of the critique is in fact Mendelssohn's rationalist approach. Both positions are, I would claim, effectively criticized in the passage. Shabel (2004) argues that one of the targets here is one who employs empirical methods of proofs with regard to Proposition I.32 and that this target was in fact Wolff, who presented an account whereby the geometer proceeded with particular claims regarding the management of the compass, etc. (209–212). Perhaps Kant does have Wolff's proof in mind here, as it would give a plausible alternative of what it would be to ‘read off’ properties of a figure. Similarly, Dunlop's (2013) account of Wolff's theory of geometrical concept acquisition might suggest that he is the target here because that account seems to imply the adequacy of the acquisition of the concept < triangle> from occasions of perception of triangle instances (462).

^33. See my (2014) for discussion.

^34. ‘Über Kästner's Abhandlungen,’, (20: 411), translated by. D.R. Lachterman, quoted from Lachterman (1989, 53).

^35. For discussion of Kant's notion of analyticity and containment see e.g. Anderson (2004) and (2005), de Jong (1995) and Proops (2005).

^36. This passage is traditionally thought to be indicative of Kant's familiarity with Berkeley's criticism of Lockean abstract ideas in the Principles [e.g. (Guyer 1998, 165)].

^37. I have not attempted to give anything like a complete account of the relationship between geometrical concepts, geometrical schemata, and spatiotemporal intuition. Nor have I attempted to adjudicate here with regard to how this account might figure within recent debates in Kant's philosophy of mathematics. However, it is perhaps worth noting some potential relevance in regards to one such recent debates, that between Michael Friedman, and Lisa Shabel concerning the status of diagrammatic reasoning in Kant's philosophy of geometry. One of the points of concern is how generality might be expressed via a particular image contained in a diagram. Friedman's claim is that it is clear that for Kant the generality is contributed by virtue of the conceptual representations the subject possesses prior to the construction procedure involving particular diagrams:

In particular, whereas such diagrammatic accounts of the generality of geometrical propositions, as we have seen, begin with particular concrete diagrams and then endeavor to explain how we can abstract from their irrelevant particular features (specific lengths of sides and angles, say) by relying only on their co-exact features, Kant begins with general concepts as conceived within the Leibnizean (logical) tradition and then shows how to “schematize” them sensibly by means of an intellectual act or function of the pure productive imagination. (Friedman, 2012, 239)

On the interpretation suggested here, Kant's rejection of the Leibnizean logical tradition is more thoroughgoing than Friedman envisages. This is so, I claim, because for Kant the concept is not possessed prior to the schematization process. Rather we acquire the explicit discursive representation in the course of schematizing over intuition. This is the sense in which the concepts are ‘originally acquired’. My reading thus supports Manders's (2008) account of ‘conceptualization via the diagram construction conditions’ (74).

^38. For comments on earlier versions of this paper I am grateful to audiences at Humboldt University, University of Amsterdam, University of California at Berkeley and Clare College, Cambridge

Additional information

Notes on contributors

John J. Callanan

John Callanan is Lecturer in the Department of Philosophy at King's College London. His research is primarily on Kant's theoretical and practical philosophy.