61. Was Sind Und Was Sollen Die Zahlen?: Dedekind By November of 1858 dedekind had resolved the issue by showing how to obtain the andarithmetical operations) from the rational numbers by means of cuts in the http://www.thoralf.uwaterloo.ca/htdocs/scav/dedek/dedek.html

Previous: Spectra Next: Cantor Up: Supplementary Text Topics What are numbers, and what is their meaning?: Dedekind Let us recall that by 1850 the subject of analysis had been given a solid footing in the real numbers infinitesimals had given way to small positive real numbers, the 's and In particular he was not satisfied with his geometrical explanation of why it was that a monotone increasing variable, which is bounded above, approaches a limit. By November of 1858 Dedekind had resolved the issue by showing how to obtain the real numbers (along with their ordering and arithmetical operations) from the rational numbers by means of cuts in the rationals for then he could prove the above mentioned least upper bound property from simple facts about the rational numbers. Furthermore, he proved that applying cuts to the reals gave no further extension. These results were first published in 1872, in Stetigkeit und irrationale Zahlen. In the introduction to this paper he points out that the real number system can be developed from the natural numbers: I see the whole of arithmetic as a necessary, or at least a natural, consequence of the simplest arithmetical act, of counting, and counting is nothing other that the successive creation of the infinite sequence of positive whole numbers in which each individual is defined in terms of the preceding one.

62. Dedekind Eta Modular Function: General Characteristics (subsection 04/05) Elliptic Functions DedekindEtaz General characteristics. Branch cuts(1 formula). EllipticFunctions/DedekindEta/04/05/0001/MainEq1.gif. http://functions.wolfram.com/EllipticFunctions/DedekindEta/04/05/

Elliptic Functions DedekindEta[ z General characteristics Branch cuts (1 formula)

63. Intuition And Rigor rigorous introduction into the theory of real numbers. The claimed theoremhas been indeed proven on the foundation of dedekind's cuts. http://www.cut-the-knot.com/fta/bolzano.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Intuition and Rigor One of the torch bearers of the formalization attempts in the 19th century was the Czech analyst Bernhard Bolzano (1781-1848.) In his critique of the attempts to prove the Fundamental Theorem of Algebra, he wrote The most common kind of proof depends on a truth borrowed from geometry, namely, that every continuous line of simple curvature of which the ordinates are first positive and then negative (or conversely) must necessarily intersect the x-axis somewhere at a point that lies in between those ordinates. There is certainly no question concerning the correctness, nor indeed the obviousness, of this geometrical proposition. But it is clear that it is an intolerable offense against correct method to derive truths of pure (or general) mathematics (i.e., arithmetic, algebra, analysis) from considerations which belong to a merely applied (or special) part, namely, geometry ... By which he of course meant that reliance on the geometrical intuition is an unacceptable tool in deriving analytic truths. He clearly accepts the statement as true but objects to the fact of its being used offhandedly, as a self-evident truth. In the article, Bolzano proceeds to justify the statement that is variably now known as Bolzano's or the Intermediate Value Theorem. His proof depends on the definition of continuity by Cauchy from which he derives the Sign Preserving Property of Continuous Functions . Assuming that at the left end of an interval the function is negative, he observes that it stays negative on a certain bounded set but not for the points near the second end of the interval. He continues

64. Cauchy Sequence -- From MathWorld converge in the reals. Real numbers can be defined using either dedekindcuts or Cauchy sequences. dedekind Cut. Author Eric W. Weisstein http://mathworld.wolfram.com/CauchySequence.html

Number Theory Sequences Math Contributors Lambrou Cauchy Sequence A sequence , ... such that the metric satisfies Cauchy sequences in the rationals do not necessarily converge , but they do converge in the reals Real numbers can be defined using either Dedekind cuts or Cauchy sequences. Dedekind Cut Author: Eric W. Weisstein Wolfram Research, Inc.

65. Maths@work - Famous Mathematicians Rigorously defined irrational numbers as classes of fractions using 'Dedekindcuts.'; Game a purely arithmetic definition of the essence of continuity. http://www.mathsatwork.com/famous_mathematicians/dedekind.html

Julius Dedekind 1831-1916 . Born in Brunswick, Germany on October 6th, son of a jurist, profession and corporation lawyer. . Studied at Gymnasium Martino - Catharineum in Brunswick. Main interests were physics and chemistry. . Studied at Collegium Carolinum. Study included analytic geometry, algebraic analysis, calculus and higher mechanics. . Private tuition in mathematics by Hans Zincke. . Became close friends with George Riemann Gauss . Became friends with Peter Dirichlet. . Became Director of Polytechnic. . Elected correspondent member of Berlin Academy. . Promoted to Professor Emeritus. . Elected correspondent member of Paris Academy. . Received many scientific honours on 50th anniversary of doctorate. . Elected foreign member of Paris Academy. Mathematics Provided one of the first precise definitions of the real number system by formulating Peano axioms. Rigorously defined irrational numbers as classes of fractions using 'Dedekind cuts.' Game a purely arithmetic definition of the essence of continuity.

66. Combinatorial Games (II) -Different Moves For Left And Right There are many approaches to this, including Richard dedekind's idea of dedekindCuts which, starting with the rational numbers, constructs the real numbers. http://www.ams.org/new-in-math/cover/partizan5.html

Combinatorial Games (Part II): Different Moves for Left and Right Feature Column Archive 5. Surreal numbers The surreal numbers were developed by John Conway, but the name universally used for these numbers, surreal , was coined by the Stanford computer scientist and mathematician Donald Knuth in his book Surreal Numbers (1974). In an oversimplified way one can view the real numbers as a way of filling in the "spaces" between the integers. There are many approaches to this, including Richard Dedekind's idea of Dedekind Cuts Introduction Hex Hackenbush Counting and sets ... References Comments: webmaster@ams.org Privacy Statement

67. The D List contributions to mathematics. His treatment of irrational numbers, Dedekindcuts, put analysis on a firm logical foundation. His work on http://www.siue.edu/~dcollin/Dlist.html

To view the photos of your favorite mathematician, just click on the photo. This will load up the image for you to view. To Save, right click on the photo, to return, you must click the "back" feature on your browser. Download all the photo's you wish, but please give proper credit to the publisher. Richard Dedekind was born on October 6, 1831, in Brunswick, Germany, the birthplace of Gauss. Dedekind spent the years 1858-1862 as a professor in Zurich. Then he accepted a position at an institute in Brunswick where he had once been a student. Although this school was less than university level, Dedekind remained there for the next 50 years. He died in Brunswick in 1916. During his career, Dedekind made numerous fundamental contributions to mathematics. His treatment of irrational numbers, "Dedekind cuts," put analysis on a firm logical foundation. His work on unique factorization led to the modern theory of algebraic numbers. He was a pioneer in the theory of rings and fields. The notion of ideal as well as the term itself are due to Dedekind. Mathematics historian Morris Kline has called him "the effective founder of abstract algebra".(1) Leonard Eugene Dickson was born in Independance, Iowa, on January 22, 1874. Dickson was the valedictorian of the 1893 class at the University of Texas. In 1894, he went to the University of Chicago and studied under E.H. Moore. Two years later he received a Ph.D, the first to be awarded in mathematics at Chicago. After spending a few years at the University of California and the University of Texas, he was appointed to the faculty at Chicago and remained there until his retirement in 1939.

68. Www.bath.ac.uk/~masgcs/book1/amplifications/ch8q8_2.txt {\it Well, almost. This is where we should start worrying about Dedekindcuts and Cauchy sequences. In fact we have disguised earlier http://www.bath.ac.uk/~masgcs/book1/amplifications/ch8q8_2.txt

69. Seminaire De Logique, Lambda-calcul Et Programmation Translate this page résumé. 18/12/98 Klaus Grue (Université de Copenhague) Dedekindcuts as a means for constructing kappa-Scott domains. abstract. http://www.logique.jussieu.fr/semlam/98_99/

98/99 (Responsables : V. Danos, C. Berline, J.L. Krivine, P. Rozière). 30/10/98: Jean-Louis Krivine (Paris 7) Lambda-calcul typé dans ZF. 6/11/98: Jean-Yves Girard (IML Marseille) Introduction à la ludique. 27/11/98: Thomas Ehrhard (IML Marseille) Sur les rapports entre définissabilité et prouvabilité dans MALL (travail en collaboration avec Antonio Bucciarelli). 4/12/98: double séance : 10h30: Tristan Crolard (Paris 7) Un lambda-calcul typé avec coroutines. 11h30: Juliusz Chroboczec (Université d'Édimbourg) Un lambda-calcul avec erreurs et sous-typage. résumé 18/12/98: Klaus Grue (Université de Copenhague) Dedekind cuts as a means for constructing kappa-Scott domains. abstract 15/01/99: Simone Martini (Université d'Udine) Local reductions in box-free proof-nets. 5/02/99: Paul Ruet (Université d'Edimbourg) Modeles de la logique lineaire non-commutative. 12/02/99: Jayanta Sen (University of Calcutta) Embedding Lukasiewicz aleph_0 logic in Linear Logic : an algebraic approach. 5/03/99: Andreja Prijatelj (Université de Ljubljana) From bounded structural rules towards linear modalities. 12/03/99: Serge Grigorieff (UFR d'informatique Paris 7 et Laboratoire de Logique, Algorithmique, Informatique de Clermont 1) Prédicats de vérité syntaxique pour l'arithmétique du second ordre (travail avec Loïc Colson).

70. The Arche Web Site Arché Research Project at the University of St Andrews. Description of the project, sponsors, researche Category Society Philosophy Philosophy of Mathematics...... The simplest proposalto which early attention will be given-is to mimic Dedekindcuts by developing an abstractionist theory of the rationals and defining http://www.st-andrews.ac.uk/academic/philosophy/arche/math.shtml

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