History of mathematics and profound mathematical results: the post-Cavaillès debate in French epistemology
Abstract
How can we define a profound mathematical result, and which epistemological criteria should we apply to evaluate the depth of mathematical analysis in the history of mathematics? This article provides a comparative analysis of the works of Gilles-Gaston Granger and Alain Michel on this subject. We argue that the differences in the authors’ perspectives highlight a more fundamental disagreement about how the history of mathematics should be understood. Like Cavaillès, Granger seeks to identify transcendental structures of a kind of creative necessity that are nevertheless incompatible with any form of predictability regarding the evolution of mathematics. In contrast, Michel, along with Brunschvicg and Canguilhem, views mathematical progress as the exploration of a broad range of possibilities. He believes that necessity is merely a retrospective illusion that pays no attention to the details of historical analysis.
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