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]]</ref> | 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> | ||
| − | + | ||
| + | ==Summary== | ||
| + | |||
| + | The efforts that we today name "arithmetization of analysis" took place over a period of about 50 years, with these results: | ||
# the establishment of fundamental concepts related to limits | # the establishment of fundamental concepts related to limits | ||
# the derivation of the main theorems concerning those concepts | # the derivation of the main theorems concerning those concepts | ||
# the creation of the theory of real numbers. | # 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."<ref>Vojtěch | + | "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 et. al.</ref> |
| + | |||
| + | Prior to these efforts, analysis rested on two pilars: the discrete side on arithmetic, the continuous side on geometry. <ref>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>Hatcher</ref> | ||
| + | |||
| + | ==History== | ||
| − | + | The history of these efforts is complicated by delays in translation and publication of results. Some authors were very slow to publish. In fact, some important results were not published at all during their author's lifetime. As a consequence, some results were achieved multiple times, albeit in slightly different forms or using somewhat different methods, by different authors.The history of these efforts is complicated by delays in translation and publication of results. Some authors were very slow to publish. In fact, some important results were not published at all during their author's lifetime. As a consequence, some results were achieved multiple times, albeit in slightly different forms or using somewhat different methods, by different authors. | |
| − | + | . . . | |
==Notes== | ==Notes== | ||
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* [[Arithmetization]], Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Arithmetization&oldid=31486 | * [[Arithmetization]], Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Arithmetization&oldid=31486 | ||
| − | * 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 | + | * Hatcher, William S. (2000), ''Foundations of Mathematics: An Overview at the Close of the Second Millenium'' 3.2 Aritmetization of Analysis, Switzerland: Landegg Academy, URL: http://bahai-library.com/hatcher_foundations_mathematics |
| − | * Stillwell, John Colin (2013) " | + | * Jarník, Vojtěch; Novák, Josef; Folta, Jaroslav; Jarník, Jiří (1981). ''Bolzano and the Foundations of Mathematical Analysis''. (English). Praha: Society of Czechoslovak Mathematicians and Physicists, pp. 33--42, URL: http://dml.cz/dmlcz/400082. |
| + | * Stillwell, John Colin (2013). "Arithmetization of Analysis," ''Encyclopaedia Britannica'', URL: http://www.britannica.com/EBchecked/topic/22486/analysis/247690/Complex-exponentials. | ||
Revision as of 14:15, 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]
Summary
The efforts that we today name "arithmetization of analysis" took place over a period of about 50 years, with these results:
- the establishment of fundamental concepts related to limits
- the derivation of the main theorems concerning those concepts
- 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 pilars: 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]
History
The history of these efforts is complicated by delays in translation and publication of results. Some authors were very slow to publish. In fact, some important results were not published at all during their author's lifetime. As a consequence, some results were achieved multiple times, albeit in slightly different forms or using somewhat different methods, by different authors.The history of these efforts is complicated by delays in translation and publication of results. Some authors were very slow to publish. In fact, some important results were not published at all during their author's lifetime. As a consequence, some results were achieved multiple times, albeit in slightly different forms or using somewhat different methods, by different authors.
. . .
Notes
- ↑ Arithmetization
- ↑ Vojtěch et. al.
- ↑ Stillwell
- ↑ Hatcher
References
- Arithmetization, Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Arithmetization&oldid=31486
- Hatcher, William S. (2000), Foundations of Mathematics: An Overview at the Close of the Second Millenium 3.2 Aritmetization of Analysis, Switzerland: Landegg Academy, URL: http://bahai-library.com/hatcher_foundations_mathematics
- Jarník, Vojtěch; Novák, Josef; Folta, Jaroslav; Jarník, Jiří (1981). Bolzano and the Foundations of Mathematical Analysis. (English). Praha: Society of Czechoslovak Mathematicians and Physicists, pp. 33--42, URL: http://dml.cz/dmlcz/400082.
- Stillwell, John Colin (2013). "Arithmetization of Analysis," Encyclopaedia Britannica, URL: http://www.britannica.com/EBchecked/topic/22486/analysis/247690/Complex-exponentials.
Arithmetization of analysis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetization_of_analysis&oldid=31783