The unification of mathematics. Geometric algebras, algebraic geometry and the philosophy of Langlands (Q2915747)

The purpose of the book under review is (reviewer's translation and paraphrase) ``to shed a philosophical and historical light on certain developments of mathematics'', in particular concerning the ``unification of mathematics'': how algebra became more geometric, and how the progress of analysis led to algebraic geometry. There are historical and philosophical discussions of several mathematical concepts, such as the complex numbers, Clifford algebras and spinors, algebraic geometry, reciprocity laws, the Langlands program and modularity theorems. Many specific mathematical notions (e.g. elliptic curves, modular forms,\dots) are explained in the book, but the reviewer cannot recommend reading these explanations as a first introduction to these notions. In the book's conclusion, the authors (reviewer's translation and paraphrase) ``hope to have put philosophy in touch with today's scientific realities''.