Synthesis, Intuition and Mathematics

  • Gary Banham


It is possible now, on the basis of the treatment I have provided of the nature of transcendental synthesis as expounded in the only viable deduction argument and extended in the account of schematism, to return to the questions about the nature of intuition that were canvassed in Chapter 1. It will be recalled that there is a major disagreement in the current literature on Kantian intuitions concerning the priority of the two criteria that Kant offers for the notion of an “intuition” with some favouring the view that the primary criteria is that of immediacy, others that of singularity. Provisionally in Chapter 1 we leaned to the view that the singularity criteria may well be the primary one due to the paradox that Caygill pointed to around the notion of “immediacy”. A further rationale for favouring the criteria of singularity in the literature generally has been that it is often taken to be the case that it is this criteria that is most important in Kant’s treatment of mathematics as a body of synthetic a priori truths.1


Manifold Brittleness Assimilation Posit Defend 


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  1. 1.
    The connection between the singularity criteria for intuitions and the treatment of mathematics has been argued by Jaako Hintikka in particular, the classic treatment of which is provided in J. Hintikka (1967) “Kant on the Mathematical Method”, The Monist, Vol. 51, reprinted in Carl J. Posy (ed.) (1992) Kants Philosophy of Mathematics: Modern Essays, Dordrecht and London: Kluwer Academic Publishers. In this piece Hintikka connects his account of mathematical construction to a description of Euclid that draws on a description of the structure of Euclid’s method in some detail.Google Scholar
  2. 2.
    See the treatment for example of Thoughts on the True Estimation of Living Forces in Michael Friedman (1992) Kant and the Exact Sciences, Cambridge, Mass. and London: Harvard University Press, “Introduction”.Google Scholar
  3. 3.
    Explication of the conception of philosophy that the young Kant held has been given serious attention in Martin Schönfeld (2000) The Philosophy of the Young Kant: The Precritical Project, Oxford and New York: Oxford University Press. Since Kant’s works of the 1790s however seem to return explicitly to the considerations set out in these early works it would be work well worth while to treat the nature of Kant’s accounts of the material possibilities of metaphysics in connection with his understandings of science and nature across the Critical/Pre-Critical divide in order to open for renewed consideration the question of the “doctrine” of theoretical philosophy, a question buried in most contemporary treatments of Kant.Google Scholar
  4. 5.
    For a description and critical discussion of the “pluralizing parts principle”, see Verity Harte (2002) Plato on Parts and Wholes: The Metaphysics of Structure, Oxford and New York: Oxford University Press, pp. 53–63.Google Scholar
  5. 8.
    Kant’s argument here thus implicitly reprises the one Leibniz gives in his fifth letter to Clarke. See H. G. Alexander (ed.) (1956) The Leibniz-Clarke Correspondence, Manchester and New York: Manchester University Press, pp. 69–72. For reasons that will be rehearsed in Chapter 7 below this interpretation of the account of space in the Physical Monadology is controversial but since the reasons for treating it as such have more to do with the understanding of substance and force than with intuition and mathematics I have left defence of this interpretation to the following chapter.Google Scholar
  6. 12.
    See the “Introduction” to Walford and Meerbote (1992) Theoretical Philosophy 1755–1770, Cambridge and New York: Cambridge University Press, pp. lxi—lxii for reasons for thinking that the Prize Essay in fact pre-dates, at least in part, Negative Magnitudes. Google Scholar
  7. 19.
    The key argument to this effect is made by Lorne Falkenstein (1991) “Kant’s Account of Intuition”, Canadian Journal of Philosophy, Vol. 21, No. 2.Google Scholar
  8. 27.
    The crude view of Kant’s account has been rectified in recent years to the point where many contemporary philosophers are prepared to suggest that Kant’s philosophy of mathematics is practically equivalent to post-Fregean views. For the basic argument to this effect, see Gottfried Martin (1938) Arithmetic and Combinatorics: Kant and His Contemporaries, translated by J. Wubnig, Carbondale: Southern Illinois University Press. The acceptance of a view that, whilst less pronounced than this, goes nonetheless some ways towards it has been facilitated in recent years by the translation of Kant’s philosophy of mathematics into logical formulae by Jaako Hintikka. For an elaborate demonstration of this, see J. Hintikka (1973) Logic, Language-Games and Information: Kantian Themes in the Philosophy of Logic, Oxford: Clarendon Press.Google Scholar
  9. 32.
    This notion that what the Axioms supply is only a “metric” is the basis of the account given in Gordon G. Brittan, Jr (1978) Kants Theory of Science, Princeton: Princeton University Press, Chapter 4 and Brittan’s view is endorsed by Paul Guyer (1987), Chapter 7.Google Scholar
  10. 33.
    Daniel Sutherland (2005) “The Point of Kant’s Axioms of Intuition”, Pacific Philosophical Quarterly, Vol. 86, p. 152. See also here for a view of much the same character Longuenesse (1993, p. 274).Google Scholar
  11. 34.
    So Norman Kemp Smith for example describes this definition of extensive magnitudes as involving “a view of space and time directly opposed to that of the Aesthetic. Norman Kemp Smith (1918) A Commentary to Kants “Critique of Pure Reason”, London and New York: Macmillan Press, p. 347. Robert Paul Wolff (1963) concurs with this verdict in Kants Theory of Mental Activity: A Commentary on the Transcendental Analytic of theCritique of Pure Reason” Cambridge, Mass.: Harvard University Press, p. 228. Wolff also prof esses considerable puzzlement as to why Kant should wish to treat in the Analytic mathematical principles at all, apparently oblivious to the difference in level between the account provided in the Aesthetic and that given in the Analytic of Principles.Google Scholar
  12. 35.
    Martin (1938) famously argues on the basis of this exchange and the resulting change in Schultz’s manuscript which he studied that Kant’s conception of arithmetic is axiomatic, a view clearly at odds however with the statement concerning the lack of axioms in arithmetic in our passage. Charles Parsons (1969) “Kant’s Philosophy of Arithmetic” in Posy (ed.) (1992), pp. 53–4, is attentive to Kant’s denial that arithmetic has axioms and derives from it a problem concerning how Kant would explain the commutative and associative principles that have long been recognized as cardinal for it. Parsons also presses a problem about the types of “objects” arithmetic is dealing with, which leads him to fault the notion that arithmetic is connected to intuition at all.Google Scholar
  13. 39.
    The dependence of Kant’s description of geometry on Euclidean assumptions is clear in terms of his connection of geometrical demonstration to construction, a procedure that non-Euclidean geometries appear to render otiose. Hintikka’s description of Euclidean procedures is now classic but see for a detailed treatment of the background assumptions of Euclid’s system and its connections to the School Philosophy of Wolff Lisa A. Shabel (2003) Mathematics in Kants Critical Philosophy: Reflections on Mathematical Practice, New York and London: Routledge, Parts 1 and 2. Unfortunately the third part of this book that concentrates on Kant has, as the author confesses in her preface, some serious problems in terms of its treatment of the nature of intuition. Michael Friedman (1992, Chapter 1) describes the historical backdrop to Kant’s account of geometry in great detail although in so doing tends to a view of transcendental imagination that somewhat diminishes its transcendental character. Strawson (1966, Part 5) presents an attempt to rescue Kant’s discussion of geometry by freeing it from the physical interpretation that was so important in the initial turn towards discussion of geometry by Kant in the Physical Monadology. Longuenesse (1993) indicates a more charitable interpretation: “Kant overstepped what he had actually deduced (but not the method of the geometry he knew) when he thought he could assert that the form of pure intuition, of which he had provided a transcendental genesis, necessarily possessed the features associated with the space of Euclidean geometry” (p. 291).Google Scholar
  14. 40.
    The classic statement to this effect is provided by Bertrand Russell (1897) An Essay on the Foundations of Geometry, Cambridge University Press: Cambridge, p. 63.Google Scholar
  15. 44.
    A minimal defence of the claim that Euclidean geometry is the form of at least our visual space is provided by J. Hopkins (1973) “Visual Geometry”, Philosophical Review, Vol. LXXXII but decisive objections to this very limited defence of the Kantian claim are made by Gordon Nagel (1983) The Structure of Experience: Kants System of Principles, Chicago and London: The University of Chicago Press, pp. 33–9.Google Scholar
  16. 46.
    I owe this argument to Robert Hanna (2001) Kant and the Foundations of Analytic Philosophy, Oxford and New York: Oxford University Press, pp. 276–9 and see his account also of the problems with Helmholtz’s objections to the Kantian thesis.Google Scholar
  17. 47.
    For some treatments of Kant’s philosophy of arithmetic that relate it extensively to post-Fregean and post-Russellian notions of logic, see Wing-Chun Wong (2000) “On a Semantic Interpretation of Kant’s Concept of Number”, Synthese, Vol. 121, Robert Hanna (2002) “Mathematics for Humans: Kant’s Philosophy of Arithmetic Revisited”, European Journal of Philosophy, Vol. 10, No. 3, Sun-Joo Shin (1997) “Kant’s Syntheticity Revisited by Peirce”, Synthese, Vol. 113 and Hector-Neri Castaneda (1960) “‘7+ 5 = 12’ as a Synthetic Proposition”, Philosophy and Phenomenological Research, Vol. 21, No. 2.Google Scholar
  18. 48.
    Martin Heidegger (1935) What is a Thing?, translated by W. B. Barton and V. Deutsch, Chicago: Regnery, 1967, p. 219.Google Scholar
  19. 51.
    Kemp Smith (1918, p. 349) and H. J. Paton (1936) Kants Metaphysics of Experience: A Commentary on the First Half of theKritik der Reinen Vernunft”, London: George Allen & Unwin and New York: Humanities Press, Vol. 2, p. 135.Google Scholar

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  • Gary Banham

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