Philosophy of mathematical invariants
Summary of the research project
The notion of invariant is the common thread of this project, both for its importance in mathematics and for the various conceptual possibilities, it offers from a philosophical point of view.
In a general and intuitive way, a mathematical invariant is an object property which is preserved even though certain transformations have been performed on the objects. Transformations, which are inseparable from invariance, are generally thought out at the same time as the properties that will be preserved, and this is one of the crucial points that this project wishes to shed light on.
It is therefore a question of working on the idea that the mathematical notions of transformation, the structures that support them and the theorems targeted, or more broadly the progression towards the general, are thought out and built together, in an entanglement where intuition and thinking strategy.
Go beyond the structuralist horizon. From a philosophical point of view, concentrating on the notion of invariant is also an attempt to go beyond a structuralist horizon which has flourished in philosophy as much as in mathematics, by grasping that the very notion of structure cannot on its own to account for important architectonic movements that accompany the progression of mathematical thought.
Seminar “Philosophy of mathematical invariants”. The main purpose of the monthly seminar is to explore the role that the notion of invariant can play in the progression of mathematics. Among other things, it will be a question of understanding how the idea of invariant makes it possible to say something about what remains invariant as well as about what acts, that is to say, in the broad sense, transformations.
It will also be a question of understanding what can be the relationship between notions of invariant used in various parts of mathematics that seem distant from each other, or their effectiveness in fields such as logic, physics, biology, but also literature, music or dance. This seminar will therefore be resolutely exploratory and will reserve a place for fields from various disciplines that we will try to create a dialogue around the notion of invariant (philosophy, mathematics, logic, physics, music, dance, among others).
Frédéric Jaëck is a professor of the philosophy of mathematics, and a researcher attached to the SPHère (CNRS-University of Paris) and ADEF (Aix Marseille University) laboratories.
He holds a doctorate in the history and philosophy of mathematics (Paris Diderot University) as well as a doctorate in fundamental mathematics (University of Bordeaux). His current research focuses on the conditions that make possible the progression of mathematics, and seeks to account for the nature of this progression. His approach is situated at the crossroads between a tradition of critical philosophy and a descriptive philosophy adapted to the mathematics of the 19-21st centuries.
Among other things, he directed the volume Lectures grothendieckiennes (Mathematical Society of France and Spartacus-IDH, 2021) fruit of the work carried out during a seminar he led at the École Normale Supérieure in Paris, edited the Gösta Mittag- Leffler and Vito Volterra. 40 years of correspondence with L. Mazliak, E. Sallent Del Colombo and R. Tazzioli (European mathematical society, 2019) and participated in the work carried out on the notion of generality in mathematics: The Oxford handbook of generality in mathematics and the sciences edited by K. Chemla, R. Chorlay and D. Rabouin (OUP, 2016).