Poincaré, Kant, and the Scope of Mathematical Intuition

TLDR: In this paper, the authors consider a point of contact between Poincare and Kant, that of mathematical intuition, and argue that the latter's inability 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 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.

Abstract: IN THIS ESSAY I WANT TO RECONSIDER a point of contact between Poincare 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) Poincare 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, Poincare'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 stake in Poincare's "yes and no" stance toward Kant's picture of mathematical synthesis. My aim here is to effect some of that clarification. Partly at issue will be what has been a controversial topic in Kant studies, namely, the contribution of the so-called "argument from geometry" to Kant's treatment of space, and here I will be relying on a body of scholarship emerging over the last two decades that tends to downplay its significance in several respects. (5) Today it is no news to point out that Kant's doctrine of space as a form of intuition is motivated by epistemological considerations independent of his views about geometry. However, it is not widely appreciated that these same considerations survive the 19th century and reappear--almost certainly without Poincare's own recognition--in Science and Hypothesis, the very work that played so large a role in turning scientifically-minded philosophers away from the Critique. On my telling, neither Kant nor Poincare were fully in control of these epistemological considerations. Indeed, from a purely philosophical point of view, the attention to geometry served, I think, to distract them both from the deeper, prior epistemological issues which, nevertheless, are never far from the surface. I will not be writing as a scholar of Poincare's philosophy, nor will I be defending Kant's distinction between forms of intuition and pure concepts. I want to reexamine the issues underlying Poincare's treatment of spatial representation in Science and Hypothesis and to suggest--contrary to history, so to speak that they support rather than undermine Kant's distinction. …