Difference between revisions of "Arithmetization of analysis"
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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> | ||
| − | Prior to these efforts, analysis rested on two | + | 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> |
"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> | "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> | ||
Revision as of 17:29, 15 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:
- the establishment of fundamental concepts related to limits
- the derivation of the main theorems concerning those concepts
- the definition 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
- ↑ Arithmetization
- ↑ 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.
- ↑ Encyclopedia of Britannica, "Arithmetization of Analysis," John Colin Stillwell
- ↑ William S. Hatcher, Foundations of Mathematics: An Overview at the Close of the Second Millenium, 2000
Arithmetization of analysis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetization_of_analysis&oldid=31761