Infinitesimals and infinites in the history of mathematics: A brief survey

摘要

We will review the main episodes of the history of the infinite in mathematics, stressing the asymmetry between infinitesimals and infinites. While the actual infinite, after being avoided for so long, when it became a mathematical object it entered mathematics to stay, infinitesimals have had a troubled history of condemnations and resurrections. This has probably to do with the ambiguous basic act of mathematization as revealed by the passage from the Egyptians’ ropes to the length without largeness of Euclid’s definition of the line. The main characters of the narration are Aristotle, Archimedes, Cavalieri, Euler, Cantor. In the end we will give some information on the last re-emergence of infinitesimals, in nonstandard analysis, and on some quite recent and intriguing new ideas.