Arithmetization of analysis

History

Research program

• the various constructions of the real numbers by Dedekind and Cantor resulting in the modern axiomatic definition of the real number field;
• the epsilon-delta definition of limit; and
• the naïve set-theoretic definition of function.

  Legacy

• the widely held belief in the banishment of infinitesimals from mathematics until the creation of non-standard analysis by Abraham Robinson in the 1960s, whereas in reality the work on non-Archimedean systems continued unabated, as documented by P. Ehrlich;
• the shift of the emphasis from geometric to algebraic reasoning: this has had important consequences in the way mathematics is taught today;
• it made possible the development of modern measure theory by Lebesgue and the rudiments of functional analysis by Hilbert;
• it motivated the currently prevalent philosophical position that all of mathematics should be derivable from logic and set theory, ultimately leading to Hilbert's program, Gödel's theorems and non-standard analysis.

  Quotations

• "God created the natural numbers, all else is the work of man." — Kronecker