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Enquête sur les modes d'existence des êtres mathématiques (version augmentée) [An inquiry into the modes of existence of mathematical beings (expanded version)]

This essay inquires how mathematical beings could be inserted into the architecture of modes of existence proposed by Bruno Latour in the framework of his pluralist and renewed ontology of the modern world. After a description of the problem, the work of Reviel Netz on the emergence of Greek mathematics, and of Charles Sanders Peirce on the diagrammatic dimension of mathematical practice are presented, as well as their impact on our essay. Its central part is the development of an empirical conception of mathematics that plays a central rôle in the sequel. Our analysis is based on the notion of experience according to William James; it is also inspired by certain aspects of Per Martin-Löf's philosophy. It provides a way of thinking the firm certainty with which proofs endow theorems, while invalidating the interpretation of this certainty as the mark of a direct access to an absolute and transcendental truth. The sequel of our essay builds on this analysis for defining a sort of quasi-mode of existence appropriate for mathematical beings that respects the principal features of modes of existence according to the latourian ontology. In the conclusion, the way this quasi-mode might be integrated into this ontology is discussed, in particular with respect to the mode of reference that prevails in many other sciences.