# Philosophy of Mathematics

4 ECTS, semester 2, 12 weeks

Program requirements | CC+examen |

Teacher | Brice Halimi |

Weekly hours | 2 h CM |

Years | Master Logique Mathématique et Fondements de l'Informatique |

Metaphysics claims to be the theory of "all things in general", formal ontology claims to be the science of "something in general". Are the notions of "things in general" or "something in general" primitive and self-evident? This course would like to explore the contrary hypothesis, and in particular that philosophical generality is not separable from the forms given to it by mathematics.

The course will consist of three main parts. After distinguishing between the two dimensions of generality, i.e. integrality (the consideration of all things) and genericity (the consideration of any one thing), we shall begin by examining the first (the "absolute generality", i.e. the consideration of all things without exception), by showing that, as much as its rejection, it gives rise to paradoxes. We will then introduce the solidarity of the great registers of use of generality that are philosophy, logic and mathematics.

We will then focus on the notion of genericity, i.e. the notion of any object, and its formal counterpart, the notion of variable. Metaphysicians presuppose the possibility of referring to things in general, without realizing that the form of "something in general" that seems to deliver this possibility is an instrument borrowed from formal logic, and in fact elaborated by logic in connection with thematics. The second part of the course will be interested in the plural forms of the generic found in mathematics and in their link with the philosophical figures of the general. It will defend the idea that the former partly under-determine the latter, and will argue for the priority of genericity over completeness.

The third and last part of the course will deal with the notions of variable and variation. If they have been disjoined by modern logic in order to avoid any confusion of generality with a real process, more recent developments, reassociating logic and geometry, allow to join these two notions in a new way. We will give some illustrations, by describing the way generality can be thought of in terms of deformation, in modal logic and in logical semantics. This last theme is directly linked to a study day " Logic and space " organized in the SPHERE laboratory, on Monday January 10, 2022.

Evaluation methods : a final exam in the form of an informal essay.

-[1] K. Chemla, R. Chorlay & D. Rabouin (éds), The Oxford Handbook of generality in mathematics and in the sciences, Oxford University Press, 2016.

-[2] Leon Horsten, The Metaphysics and Mathematics of Arbitrary Ob- jects, Cambridge University Press, 2019

-[3] A. Rayo & G. Uzquiano (eds), Absolute Generality, Oxford University Press, 2007.

-[4] François Rivenc, L’Universalisme logique, Payot, 1993.