Arithmetization of analysis

The highlights of this research program are:

• the algebraic construction of the real numbers by Dedekind, resulting in the modern axiomatic definition of the real number field;
• the epsilon-delta definition of limit; and
• the naive set-theoretic definition of function.

An important spinoff of the arithmetization of analysis is set theory. Naive set theory was created by 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 banishment of infinitesimals from mathematics until the creation of non-standard analysis by Abraham Robinson in the 1960s;
• the shift of the emphasis from geometric to algebraic reasoning: this has had important consequences in the way mathematics is taught today;
• it made the development of modern measure theory by Lebesgue and the rudiments of functional analysis by Hilbert possible;
• it motivated the more extreme 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 integers, all else is the work of man." -- Kronecker

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