CAVAILLÈS, MATHEMATICAL PROBLEMS AND QUESTIONS

Abstract

This paper concerns the role of mathematical problems in the epistemology of Jean Cavaillès. Most occurrences of the term “problem” in his texts refer to mathematical problems, in the sense in which mathematicians themselves use the term: for an unsolved question which they hope to solve. Mathematical problems appear as breaking points in the succession of mathematical theories, both giving a continuity to the history of mathematics and illuminating the way in which the history of mathematics breaks up into successive theories with different kinds of operations and, in a sense, different kinds of a prioris. We briefly compare mathematical becoming to the succession of episteme in Foucault’s Les Mots et les choses. We then come back to the necessity that Cavaillès attributes to mathematical becoming, and which the position of mathematical problems illustrates, in order to discuss its various consequences in Cavaillès’ later works but also in Canguilhem’s discussion of Cavaillès’ role in the Resistance. Finally, we study other types of problems, in Cavaillès’ writings: philosophical problems and what we will call “questions” rather than “problems,” and which contrary to mathematical problems, as Cavaillès uses the term, cannot be solved but pervade the whole of the history of mathematics. We will put these “questions” in relation to Lautman’s Ideas.

Keywords:

• Cavaillès
• Foucault
• mathematics
• problems
• a priori
• history
• becoming

disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 All translations of Cavaillès’ and Canguilhem’s work are the author’s.

2 In his remarkable analysis of Cavaillès’ Spinozism, Knox Peden reads in Canguilhem’s account of Cavaillès’ Resistance “the force of reason against history” (in particular Peden 22). It may express Canguilhem’s views, and Badiou’s, but there is still an irony in the fact that Cavaillès does not write as a philosopher about his action, which then de facto falls outside the domain of his philosophy. It is as if, contrary to the rationalist aim, philosophy discovered its limit: something which cannot be written about, something which then cannot be thought.