What is a number?

Euler’s death in 1783 was followed by a period of stagnation in mathematics. He had indeed solved everything: an unsurpassed treatment of infinite and differential calculus (Euler 1748, 1755), solvable integrals solved, solvable differential equations solved (Euler 1768, 1769), the secrets of liquids (Euler 1755b), of mechanics (Euler 1736b, Lagrange 1788), of variational calculus (Euler 1744), of algebra (Euler 1770), unveiled. It seemed that no other task remained than to study about 30,000 pages of Euler’s work.
The “Theorie des fonctions analytiques” by Lagrange (1797), “freed from all considerations of infinitely small quantities, vanishing quantities, limits and fluxions”, the thesis of Gauss (1799) on the “Fundamental Theorem of Algebra” and the study of the convergence of the hypergeometric series (Gauss 1812) mark the beginning of a new era.
Bolzano points out that Gauss’s first proof is lacking in rigor; he then gives in 1817 a “purely analytic proof of the theorem, that between two values which produce opposite signs, there exists at least one root of the equation”. In 1821, Cauchy establishes new requirements of rigor in his famous “Cours d’Analyse”. The questions are the following:

– What is a derivative really? Answer: a limit.
– What is an integral really? Answer: a limit.
– What is an infinite series a1 + a2 + a3 + . . . really? Answer: a limit.

This leads to:

– What is a limit? Answer: a number.

And, finally, the last question:

                                        What is a number?

Weierstrass and his collaborators (Heine, Cantor), as well as Me'ray, answer that question around 1870–1872. They also fill many gaps in Cauchy’s proofs by clarifying the notions of uniform convergence, uniform continuity, the term by term integration of infinite series, and the term by term differentiation of infinite series.