Ptolemaic Revolutions: Mathematical Objectivity in Jean Cavaillès and Gilles-Gaston Granger

I argue that Gilles-Gaston Granger (1920–2016) broadly incorporates the central affirmations of Jean Cavaillès’s (1903–44) philosophy of the concept into his own epistemological program. Cavaillès and Granger share three interrelated epistemological commitments: they claim (1) that mathematics has its own content and is therefore autonomous from and irreducible to logic, (2) that conceptual transformations in the history of mathematics can only be explained by an internal dialectic of concepts, and (3) that the objectivity of mathematics is an effect of the productive form of its rationality—that is, they claim that mathematical objectivity is the result of a generative process internal to mathematics. Mathematical objectivity, as I demonstrate, requires Cavaillès and Granger to modify the limits of epistemological critique as envisioned by Kant. For Cavaillès, this modification requires the elimination of the transcendental in favor of a dialectic of concepts. For Granger, the philosophy of the concept remains a genre of transcendental critique, but in a radically revised “Ptolemaic” style. I initially demonstrate how Granger extracts each of these claims from Cavaillès. I then follow Granger as he develops these claims in the service of a novel description of the content and structure of mathematical objectivity.