History
Kronecker originally introduced the term ''arithmetization of analysis'', by which he meant its constructivization in the context of the natural numbers (see quotation at bottom of page). The meaning of the term later shifted to signify the set-theoretic construction of the real line. Its main proponent was Karl Weierstrass, Weierstrass, who argued the geometric foundations of calculus were not solid enough for rigorous work.Research program
The highlights of this research program are: * the various (but equivalent) Construction of the real numbers, constructions of the real numbers by Richard Dedekind, Dedekind and Georg Cantor, Cantor resulting in the modern axiomatic definition of the real number field; * the epsilon-delta definition of limit (mathematics), limit; and * the naive set theory, naïve set-theoretic definition of function (mathematics), function.Legacy
An important spinoff of the arithmetization of analysis is set theory. Naive set theory was created by Georg Cantor, Cantor and others after arithmetization was completed as a way to study the singularities of functions appearing in calculus. The arithmetization of analysis had several important consequences: * 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 geometry, 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 Henri Lebesgue, Lebesgue and the rudiments of functional analysis by David Hilbert, 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, Kurt Gödel, Gödel's theorems and non-standard analysis.Quotations
* "God created the natural numbers, all else is the work of man." — Leopold Kronecker, KroneckerReferences
* Torina Dechaune Lewis (2006) ''The Arithmetization of Analysis: From Eudoxus to Dedekind'', Southern University. * Carl B. Boyer, Uta Merzbach, Uta C. Merzbach (2011) ''A History of Mathematics'' John Wiley & Sons.