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Difference between revisions of "Arithmetization of analysis"

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The phrase "arithmetization of analysis" refers to 19th century efforts to create a "theory of real numbers ... using set-theoretic constructions, starting from the natural numbers." <ref>[[Arithmetization]], Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Arithmetization&oldid=31486</ref>
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The phrase "arithmetization of analysis" refers to 19th century efforts to create a "theory of real numbers ... using set-theoretic constructions, starting from the natural numbers." <ref>[[Arithmetization]]</ref>
 
These efforts took place over a period of about 50 years, which saw the following:
 
These efforts took place over a period of about 50 years, which saw the following:
 
# the establishment of fundamental concepts related to limits
 
# the establishment of fundamental concepts related to limits
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"The theory of real numbers is logically the starting point of analysis in the real domain; historically its creation marks the end of this period."<ref>Vojtěch Jarník (author); Josef Novák (other); Jaroslav Folta (other); Jiří Jarník (other): Bolzano and the Foundations of Mathematical Analysis. (English). Praha: Society of Czechoslovak Mathematicians and Physicists, 1981. pp. 33--42, http://dml.cz/dmlcz/400082.</ref>
 
"The theory of real numbers is logically the starting point of analysis in the real domain; historically its creation marks the end of this period."<ref>Vojtěch Jarník (author); Josef Novák (other); Jaroslav Folta (other); Jiří Jarník (other): Bolzano and the Foundations of Mathematical Analysis. (English). Praha: Society of Czechoslovak Mathematicians and Physicists, 1981. pp. 33--42, http://dml.cz/dmlcz/400082.</ref>
  
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Prior to these efforts, analysis rested on two pillars: the discrete side on arithmetic, the continuous side on geometry. <ref>Encyclopedia of Britannica, "Arithmetization of Analysis," John Colin Stillwell</ref>
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Prior to these efforts, analysis rested on two pillars: the discrete side on arithmetic, the continuous side on geometry. <ref>Stillwell</ref>
  
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"The analytic work of L. Euler, K. Gauss, A. Cauchy, B. Riemann, and others led to a shift towards the predominance of algebraic and arithmetic ideas.... In the late nineteenth century, this tendency culminated in the so-called arithmetization of analysis, due principally to K. Weierstrass, G. Cantor, and R. Dedekind."<ref>William S. Hatcher, Foundations of Mathematics: An Overview at the Close of the Second Millenium, 2000</ref>
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"The analytic work of L. Euler, K. Gauss, A. Cauchy, B. Riemann, and others led to a shift towards the predominance of algebraic and arithmetic ideas. In the late nineteenth century, this tendency culminated in the so-called arithmetization of analysis, due principally to K. Weierstrass, G. Cantor, and R. Dedekind."<ref>Hatcher 2000, 3.2 The Arithmetization of Analysis</ref>
  
 
==Notes==
 
==Notes==
  
 
<references />
 
<references />
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==References==
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* [[Arithmetization]], Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Arithmetization&oldid=31486
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* Hatcher, William S. (2000), ''Foundations of Mathematics: An Overview at the Close of the Second Millenium'', Switzerland: Landegg Academy, URL: http://bahai-library.com/hatcher_foundations_mathematics
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* Stillwell, John Colin (2013) "The Arithmetizaton of Analysis". ''Encyclopedia Britannica''

Revision as of 03:24, 16 April 2014

The phrase "arithmetization of analysis" refers to 19th century efforts to create a "theory of real numbers ... using set-theoretic constructions, starting from the natural numbers." [1] These efforts took place over a period of about 50 years, which saw the following:

  1. the establishment of fundamental concepts related to limits
  2. the derivation of the main theorems concerning those concepts
  3. the creation of the theory of real numbers.

"The theory of real numbers is logically the starting point of analysis in the real domain; historically its creation marks the end of this period."[2]

Prior to these efforts, analysis rested on two pillars: the discrete side on arithmetic, the continuous side on geometry. [3]

"The analytic work of L. Euler, K. Gauss, A. Cauchy, B. Riemann, and others led to a shift towards the predominance of algebraic and arithmetic ideas. In the late nineteenth century, this tendency culminated in the so-called arithmetization of analysis, due principally to K. Weierstrass, G. Cantor, and R. Dedekind."[4]

Notes

  1. ↑ Arithmetization
  2. ↑ Vojtěch Jarník (author); Josef Novák (other); Jaroslav Folta (other); Jiří Jarník (other): Bolzano and the Foundations of Mathematical Analysis. (English). Praha: Society of Czechoslovak Mathematicians and Physicists, 1981. pp. 33--42, http://dml.cz/dmlcz/400082.
  3. ↑ Stillwell
  4. ↑ Hatcher 2000, 3.2 The Arithmetization of Analysis

References

How to Cite This Entry:
Arithmetization of analysis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetization_of_analysis&oldid=31767
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