Appears in collection : History of mathematics, Philosophy of mathematics, and mathematics: which interactions? / Histoire des mathématiques, Philosophie des mathématiques, et mathématiques: quelles interactions?

Grothendieck talked about “algebraic geometry in its most fascinating aspect for me—the “arithmetic” aspect, apprehended by intuition, concepts, and techniques all of “geometric” nature.” (ReS, p. 372) This has become a very prominent development in recent mathematics, and the new scenario for the interplay of key ideas (and tensions) within pure math, like the relations geometry/number, continuous/discrete. It seems to us that our grasp of mathematical ideas (even basic ideas like number, space) can be transformed when a very innovative practice is established. Otherwise said, they can be transformed by immersion in, or interplay with, new structures. The traditional antithesis between discrete and continuous can be bridged, smoothed out, and integrated by means of rich hybrid structures such as schemes. But such developments raise basic questions, in particular – why is arithmetic geometry geometry? The evolution of ideas of space and geometry is an extremely complicated subject, covering as it does all of human history. In this talk, we will reflect on the meandering evolution of these ideas, which accelerated with 20th century mathematical abstractions, and attempt some answer to the key question.