Editing Arithmetization of analysis

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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 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 considered that [[Isaac Newton]]'s calculus lacked rigor.

==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." &mdash; [[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 (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]]