On Physical Mathematics: an approach through Gilles Châtelet's philosophy

TL;DR

This work argues for a philosophically grounded understanding of Physical Mathematics by adopting Gilles Châtelet's physico-mathematics as a framework. It identifies three core gestures—inversion, differentiation via splitting and augmenting, and the emergence of virtualities—as the mechanisms by which mathematics, physics, and philosophy inform one another. The paper differentiates Physical Mathematics from traditional Mathematical Physics while stressing their deep connections, notably through gauge theories and the evolution of foundational concepts. It then uses mirror symmetry as a concrete exemplar to illustrate how physical intuition can drive new mathematics and how cross-disciplinary insights can align with Grothendieck's unifying program and potentially model-theoretic approaches. Overall, it advocates for a global, triadic epistemology to advance understanding of the universe’s structure through the interplay of mathematics, physics, and philosophy.

Abstract

Starting from Greg Moore's description about Physical Mathematics, a framework is proposed in order to understand it, based on Gilles Châtelet's philosophy. It will be argued that Châtelet's ideas of inverting, splitting, augmenting and virtuality are crucial in the discussion about the nature of Physical Mathematics. Along this line, it will be proposed that mirror symmetry is a natural study case to test Châtelet's ideas in this context. This should be considered as a first step in a long term project aiming to study the relations among mathematics, physics and philosophy in the construction of a global understanding of the structure of the universe, as it was envisioned by Grothendieck in the late 80's of the last century and it was started to be developed independently by Châtelet in the beginning of the 90's. The main suggestion of the essay is that it is in the relations between mathematics, physics and philosophy that new knowledge arises.