{{Short description|Historical research project in mathematics}}
{{no footnotes|date=January 2015}}
The '''arithmetization of analysis''' was a research program in the [[foundations of mathematics]] carried out in the second half of the 19th century which aimed to abolish all geometric intuition from the proofs in [[Mathematical analysis|analysis]]. For the followers of this program, the fundamental concepts of calculus should also not make references to the ideas of motion and velocity. This ideal was pursued by [[Augustin-Louis Cauchy]], [[Bernard Bolzano]], [[Karl Weierstrass]], among others, who developed solid foundations for [[calculus]] and  analysis.

==History==
[[Leopold Kronecker|Kronecker]] originally introduced the term ''arithmetization of analysis'', by which he meant its constructivization in the context of the [[natural number]]s (see quotation at bottom of page).  The meaning of the term later shifted to signify the [[set theory|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 [[mathematical rigour|rigorous]] work. 

==Research program==
The highlights of this research program are:
* the various (but equivalent) [[constructions of the real numbers]] by [[Richard Dedekind|Dedekind]] and [[Georg Cantor|Cantor]] resulting in the modern [[axiom]]atic definition of the [[real number]] [[field (mathematics)|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 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 [[infinitesimal]]s 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 [[algebra]]ic 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]].

==Quotation==
* "God created the natural numbers, all else is the work of man." — [[Leopold Kronecker|Kronecker]]

==References==
* 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.
* [https://www.encyclopediaofmath.org/index.php/Arithmetization_of_analysis ''Arithmetization of analysis''] at [[Encyclopedia of Mathematics]].
* [[James Pierpont (mathematician)|James Pierpont]] (1899) "On the arithmetization of mathematics", ''Bull. Amer. Math. Soc.'' 5(8): 394–406.

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[[Category:History of mathematics]]
[[Category:Philosophy of mathematics]]
[[Category:Mathematical analysis]]