Kant on geometry and spatial intuition Article First Online: 07 March 2012 Received: 14 January 2010 Accepted: 27 July 2010 DOI :
10.1007/s11229-012-0066-2

Cite this article as: Friedman, M. Synthese (2012) 186: 231. doi:10.1007/s11229-012-0066-2
10
Citations
819
Downloads
Abstract I use recent work on Kant and diagrammatic reasoning to develop a reconsideration of central aspects of Kant’s philosophy of geometry and its relation to spatial intuition. In particular, I reconsider in this light the relations between geometrical concepts and their schemata, and the relationship between pure and empirical intuition. I argue that diagrammatic interpretations of Kant’s theory of geometrical intuition can, at best, capture only part of what Kant’s conception involves and that, for example, they cannot explain why Kant takes geometrical constructions in the style of Euclid to provide us with an a priori framework for physical space. I attempt, along the way, to shed new light on the relationship between Kant’s theory of space and the debate between Newton and Leibniz to which he was reacting, and also on the role of geometry and spatial intuition in the transcendental deduction of the categories.

Keywords Geometry Diagrammatic reasoning Space Intuition Schematism Transcendental deduction An earlier version of this paper was presented at the second meeting of the Stanford-Paris workshop on diagrams in mathematics in the Fall of 2008 from which the present special issue is drawn, and it was originally inspired by a paper presented by Marco Panza on diagrammatic reasoning in Euclid at the first meeting of the Stanford-Paris workshop in the Fall of 2007. Panza’s paper in the present issue is based, in turn, on his earlier presentation. Since Panza’s paper, as it now appears, has since been substantially revised, I have taken the opportunity substantially to revise my paper as well, and, in particular, I have chosen to take as my main target work of the Kant scholar Lisa Shabel that is very much in the spirit of Kenneth Manders’s original discussion of the Euclidean diagram (note 1 below). I am also indebted, in this connection, to comments on the earlier version of my paper from Jeremy Avigad. For helpful comments on the penultimate version of this paper I am further indebted to Daniel Sutherland and to an anonymous referee for Synthese .

References Allison H. E. (1973) The Kant-Eberhard controversy. John Hopkins University Press, Baltimore

Google Scholar Allison H. E. (1983) Kant’s transcendental idealism. Yale University Press, New Haven

Google Scholar Carson E. (1997) Kant on intuition in geometry. Canadian Journal of Philosophy 27: 489–512

Google Scholar De Pierris G. (2001) Geometry in the metaphysical exposition. In: Gerhardt V., Horstmann R.-P., Schumacher R. (eds) Kant und die Berliner Aufklärung, Band 2. de Gruyter, Berlin, pp 197–204

Google Scholar Friedman M. (1992) Kant and the exact sciences. Harvard University Press, Cambridge

Google Scholar Friedman M. (2000) Geometry, construction, and intuition in Kant and his successors. In: Sher G., Tieszen R. (eds) Between logic and intuition: Essays in honor of Charles Parsons. Cambridge University Press, Cambridge, pp 186–218

CrossRef Google Scholar Friedman M. (2003) Transcendental philosophy and mathematical physics. Studies in History and Philosophy of Science 34: 29–43

CrossRef Google Scholar Friedman M. (2005) Kant on science and experience. In: Mercer C., O’Neill E. (eds) Early modern philosophy: Mind, matter, and metaphysics. Oxford University Press, Oxford, pp 262–275

Google Scholar Friedman M. (2009) Newton and Kant on absolute apace: From theology to transcendental philosophy. In: Bitbol M., Kerszberg P., Petitot J. (eds) Constituting objectivity: Transcendental perspectives on modern physics. Springer, Berlin, pp 35–50

Google Scholar Kant, I. (1902-). Kant’s gesammelte Schriften . Berlin: de Gruyter.

Manders K. (2008a) Diagram-based geometrical practice. In: Mancosu P. (ed) The philosophy of mathematical practice. Oxford University Press, Oxford, pp 65–79

CrossRef Google Scholar Manders K. (2008b) The Euclidean diagram. In: Moncosu P. (ed) The philosophy of mathematical practice. Oxford University Press, Oxford, pp 80–133

CrossRef Google Scholar Newton I. (2004). Isaac Newton: Philosophical writings . In Janiak A. (Ed.). Cambridge: Cambridge University Press.

Parsons C. (1992) The transcendental aesthetic. In: Guyer P. (ed) The Cambridge companion to Kant. Cambridge University Press, Cambridge, pp 62–100

CrossRef Google Scholar Shabel L. (1998) Kant on the ‘symbolic construction’ of mathematical concepts. Studies in History and Philosophy of Science 29: 589–621

CrossRef Google Scholar Shabel L. (2003) Mathematics in Kant’s critical philosophy: Reflections on mathematical practice. New York and London, Routledge

Google Scholar Shabel L. (2006) Kant’s philosophy of mathematics. In: Guyer P. (ed) The Cambridge companion to Kant and modern philosophy. Cambridge University Press, Cambridge, pp 94–128

CrossRef Google Scholar Sutherland D. (2004) The role of magnitude in Kant’s critical philosophy. Canadian Journal of Philosophy 34: 411–442

Google Scholar Sutherland D. (2006) Kant on arithmetic, algebra, and the theory of proportion. Journal of the History of Philosophy 44: 33–558

CrossRef Google Scholar Tarski A. (1959) What is elementary geometry?. In: Henkin L., Suppes P., Tarski A. (eds) The Axiomatic method, with special reference to geometry and physics. North-Holland, Amsterdam, pp 16–29

Google Scholar © Springer Science+Business Media B.V. 2012

Authors and Affiliations 1. Department of Philosophy Stanford University Stanford USA