Arithmetization of analysis

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."

Summary

The efforts that we today name "arithmetization of analysis" took place over a period of about 50 years, with these results:

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."

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

History

The history of these efforts is complicated by delays in publication of important results. Some authors were very slow to publish. In fact, some important results were not published at all during their authors' lifetimes. As a consequence, some results were achieved multiple times, albeit in slightly different forms or using somewhat different methods, by different authors. For example, only two papers of Bolzano dealing with the foundations of analysis were published during his lifetime. Both of these papers remained virtually unknown until after his death. A third work of his was finally published in 1930. It was based on a manuscript that dates from 1831-34, but that remained undiscovered until after WWI. This work contains some results fundamental to the foundations of analysis that were re-discovered in the 19th century by others decades after Bolzano completed his manuscript. "We may ask how much Bolzano's work could have changed the way analysis followed, had it been published at the time."

References

• Arithmetization, Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Arithmetization&oldid=31486
• Fundamental Theorem of Algebra, Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php/Algebra,_fundamental_theorem_of
• "Fundamental Theorem of Algebra," Wikipedia, URL: http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra
• 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.