Cavaillès as a Reader of Husserl: Two Versions of the History of Mathematics

Jean-Paul Cauvin

Emory University, Atlanta

Abstract: 
The paper addresses Jean Cavaillès’ (1901-1944) reading of Edmund Husserl’s (1859-1938) Formal and Transcendental Logic (1929) in the closing pages of Cavaillès’ posthumously published Logic and Theory of Science (1947.) The paper proposes a comparative reading of Cavaillès and Husserl on the epistemological significance of the history of mathematics. I argue that Cavaillès objects to Husserl’s transcendental phenomenology of logic from two inter-locking methodological imperatives. The first concerns Husserl’s inability to formulate an epistemology of mathematical experience which grants a proper intelligible content to mathematics in its own right. On the contrary, so Cavaillès claims, Husserl transforms mathematics into the organ of “formal ontology” – what Husserl calls, borrowing the technical vocabulary of Bernard Riemann ( 1826-1866), a theory of manifolds or multiplicities – a theory of the structural possibilities of objects in the abstract. The second concerns the epistemological implications of this reduction of mathematics to formal ontology for the historiography of mathematical concepts. According to Cavaillès Husserl’s phenomenology is incapable of perceiving the actual historical development of mathematical concepts because it refers the objectivity of mathematical objects to the acts of a transcendental consciousness. Cavaillès’ argument is that if logic is transcendental and if mathematics is formal ontology then the historical dimension of mathematical development disappears because everything that it is possible to know about the contents of mathematical science must have already been predetermined by the acts of some antecedent consciousness. The paper argues that Cavaillès’ overarching critique of Husserl is therefore two-fold: Husserl, like the logical positivists, denies that mathematics has an intelligible content of its own, consequently he dismisses the possibility that the history of mathematics is capable of disclosing anything epistemologically significant about the nature of mathematical intelligibility which could not also be discovered through a phenomenological bracketing of the acts of consciousness. The paper concludes by demonstrating that Cavaillès’ encounter with Husserl is also a more general repudiation of any epistemology of mathematics which refuses to grant conceptual autonomy to mathematics while at the same time being a spirited defense of the necessity of an authentically historical methodology in the epistemological analysis of mathematics.