Arithmetization of analysis
The arithmetization of analysis
was a research program in the foundations of mathematics
carried out in the second half of the 19th century. Its main proponent was Weierstrass
, who argued the geometric foundations of calculus
were not solid enough for rigorous work.
The highlights of this research program are:
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.
- "God created the integers, all else is the work of man." -- Kronecker