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IN THIS ESSAY I WANT TO RECONSIDER a point of contact between Poincaré and Kant, that of mathematical intuition. In the Critique of Pure Reason, Kant holds that ordinary empirical cognition reflects two fundamentally different relationships between whole and part. To take one of Kant's examples, in seeing a house, I cognize its outline as one figure composed of a plurality of parts, just as I would cognize the same figure in pure geometry. Whether in a context of geometrical construction or of empirical cognition the figure is a totum, an object whose parts are "possible only in the whole" and whose apprehension occurs through what Kant terms mathematical synthesis. But, seeing a house involves more than seeing its outline. It also involves seeing a whole made possible by the sum of its parts - bricks and mortar dynamically interacting with one another and with the surrounding substances. Kant refers to such an object as a compositum; its cognition requires a dynamical synthesis.1 Besides applying to geometrical shapes, Kant holds that arithmetic, too, involves mathematical synthesis, because we cognize the number line as one limitless succession of homogeneous units.2 Finally, mathematical synthesis is also supposed to apply to space and time, which are given, again, as a single individuals composed of a plurality of parts.
Now it is well-known that, in Science and Hypothesis (1902),3 Poincaré takes a generally Kantian line in the case of arithmetic but not at all in his treatment of geometry and of space and time. In a series of important monographs and articles, Michael Friedman has helped us to appreciate this disjunction as philosophically deep and as historically momentous. Thus, in Dynamics of Reason (2001), Reconsidering Logical Positivism (1999), and elsewhere, Friedman argues that, as a matter of intellectual history, Poincaré's refusal to extend Kant's account of mathematical synthesis from arithmetic to geometry contributed to the collapse of Kant's distinction between forms of intuition and concepts of the understanding - a collapse crucial to the formation of both the continental and analytical traditions in philosophy and to the development of his own version of neoKantianism.4
While we can now better appreciate its historical importance, I do not think we have yet a clear picture of the philosophical issues at...