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Arithmetization of analysis

From Encyclopedia of Mathematics
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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." [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 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 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]

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. Encyclopedia of Britannica, "Arithmetization of Analysis," John Colin Stillwell
  4. William S. Hatcher, Foundations of Mathematics: An Overview at the Close of the Second Millenium, 2000
How to Cite This Entry:
Arithmetization of analysis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetization_of_analysis&oldid=31747