Enqu\^ete sur les modes d'existence des \^etres math\'ematiques (version augment\'ee) [An inquiry into the modes of existence of mathematical beings (expanded version)]

TL;DR

This paper explores how mathematical entities can be integrated into Bruno Latour's ontology of modes of existence, proposing an empirical, experience-based conception of mathematics that challenges traditional notions of mathematical certainty.

Contribution

It develops a novel empirical conception of mathematical beings, grounded in experience and inspired by William James and Per Martin-Löf, to fit within Latour's ontology.

Findings

Mathematical proofs provide firm certainty without implying access to absolute truth.

A quasi-mode of existence for mathematical beings is proposed, aligning with Latour's ontology.

The paper discusses integrating this mode into broader scientific reference frameworks.

Abstract

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 r\^ole 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-L\"of'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.