61. Dedekind, Richard 12, 1916, was a German mathematician known for his study of CONTINUITY and definitionof the real numbers in terms of dedekind cuts ; his analysis of the http://euler.ciens.ucv.ve/English/mathematics/dedekind.html

Dedekind, Richard Richard Dedekind, b. Oct. 6, 1831, d. Feb. 12, 1916, was a German mathematician known for his study of CONTINUITY and definition of the real numbers in terms of Dedekind "cuts"; his analysis of the nature of number and mathematical induction, including the definition of finite and infinite sets; and his influential work in NUMBER THEORY, particularly in algebraic number fields. Among his most notable contributions to mathematics were his editions of the collected works of Peter DIRICHLET, Carl GAUSS, and Georg Riemann. Dedekind's study of Dirichlet's work led to his own study of algebraic number fields, as well as his introduction of ideals. He developed this concept into a theory of ideals that is of fundamental importance in modern algebra. Dedekind also introduced such fundamental concepts as RINGS. Author: J. W. Dauben Homepage e-mail: webmaster@euler.ciens.ucv.ve © 2000 Mathematics School Science Faculty, Central University of Venezuela

62. Dedekind's Real Numbers set of rational numbers ; Maddy1992, p. 81 ``by identifying real numbers withcertain sets (called `dedekindcuts ), dedekind misinterpretation. http://www.phil.cmu.edu/dschlimm/texts/reals.html

Dedekind's Real Numbers Dirk Schlimm Department of Philosophy Carnegie Mellon University Pittsburgh, PA 15213 email: dschlimm@andrew.cmu.edu February 22, 1999 Abstract Richard Dedekind's characterization of the real numbers as the system of cuts of rational numbers is by now the standard in almost every mathematical book on analysis or number theory. In the philosophy of mathematics Dedekind is given credit for this achievement, but his more general views are discussed very rarely and only superficially. For example, Leo Corry, who dedicates a whole chapter of his Modern Algebra and the Rise of Mathematical Structures (1996) writes: ``Dedekind defined the system of the real numbers as the collection of all cuts of rationals'' ([ ], p. 73). In this paper I will present Dedekind's own views of his ``definition'' and ``creation'' of the real numbers, and elucidate what he meant by saying that the real numbers ``correspond'' to the cuts. The upshot of this discussion will be that Corry's statement will be revealed as an obvious, but not uncommon (cf. [ ], p. 55: ``And every such cut, that corresponds to no rational number, defines an irrational number'' (my translation); [

63. 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.

64. Yet Another Outline/kludgefest (fwd) system 11.1.1 real number as cut -1, 1 containing infinite cuts I thinkthat what you are designating as a cut is _not_ a dedekind cut, but I ll http://www.asifproductions.com/aleph/Jul93/msg00034.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index yet another outline/kludgefest (fwd) To Aleph@pyramid.com (Aleph) Subject : yet another outline/kludgefest (fwd) From Mitchell.Porter@lambada.oit.unc.edu Date : Wed, 30 Jun 1993 23:07:06 -0700 Cc Aleph@pyramid.com Prev by Date: [no subject] Next by Date: angels (fwd) Previous by thread: [no subject] Next by thread: angels (fwd) Index(es): Date Thread

65. Lista Keywords Boundary value problem. PA KRUTITSKII The skew derivative problemfor the Helmholtz equation outside cuts in a plane. dedekind domain. http://www.dmmm.uniroma1.it/~rendiconti/rol/keywords.htm

Rendiconti di Matematica Parole Chiave (Keywords) (Parole chiave degli articoli pubblicati / Keywords in papers published) p -harmonic map MASASHI MISAWA Existence and partial regularity results for the gradient flow for a variational functional of degenerate type L. ESPOSITO, G. MINGIONE H. PESCE 3-Transposition groups M.M. VIROTTE DUCHARME F C -naturally reductive J.T. CHO, L. VANHECKE Hopf hypersurfaces of D'Atri- and C -type in a complex space form CC -solutions H. LESZCZYNSKI On CC -solutions to the initial-boundary-value problem for first-order partial differential-functional equations D -normal space F. CAMMAROTO, Lj. KOCINAC Developable spaces and cleavability D -paracompact space F. CAMMAROTO, Lj. KOCINAC Developable spaces and cleavability H -closed space F. CAMMAROTO, Lj. KOCINAC Developable spaces and cleavability H -type groups I. DOTTI, J. LAURET Curvature properties of solvable extensions of H -type groups L -minimal canal surfaces E. MUSSO, L. NICOLODI L-minimal canal surfaces L -operators A guide to L -operators M -ideal T.S.S.R.K. RAO

66. 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.org/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

67. Historia Matematica Mailing List Archive: Re: [HM] Good History--Correct Results admit to not seeing much of a distinction between determining an irrational bya dedekind cut (or pair of cuts), and then talking about cuts as irrationals http://sunsite.utk.edu/math_archives/.http/hypermail/historia/oct98/0030.html

Re: [HM] Good HistoryCorrect Results and Errors Roberto Baldino baldino@linkway.com.br Sun, 04 Oct 1998 22:51:09 -0300 Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Next message: Julio Gonzalez Cabillon: "Re: [HM] Rheticus" Previous message: Robert Tragesser: "[HM] Reference paper Leibniz/infinite" In reply to: Robert Tragesser: "[HM] Reference paper Leibniz/infinite" At 11:11 04/10/98, Gordon Fisher wrote: Dear Gordon I quote Dedekind ("Essays on the theory of numbers", Dover 1963, p. 15). "Whenever, then, we have to do with a cut (A1,A2) produced by no rational completely defined by this cut (A1,A2); we shall say that the number alpha corresponds to this cut, or that it produces this cut." Dedekind does not say that the cut IS the number. To say that the number is "completely defined" by the cut is not saying that the number is the cut. How do we proceed to "create" a new number? And how do we "regard it" as...? Moreover, this "number", created out of thin air, has the power of PRODUCING the cut, so that we can "say" that the creature corresponds to the creator.

68. Dedekind remarkable piece of work was his redefinition of irrational numbers in terms of Dedekindcuts which, as we mentioned above, first came to him as early as 1858. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Dedekind.html

Julius Wihelm Richard Dedekind Born: 6 Oct 1831 in Braunschweig, duchy of Braunschweig (now Germany) Died: 12 Feb 1916 in Braunschweig, duchy of Braunschweig (now Germany) Click the picture above to see five larger pictures Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Richard Dedekind 's father was a professor at the Collegium Carolinum in Brunswick. His mother was the daughter of a professor who also worked at the Collegium Carolinum. Richard was the youngest of four children and never married. He was to live with one of his sisters, who also remained unmarried, for most of his adult life. He attended school in Brunswick from the age of seven and at this stage mathematics was not his main interest. The school, Martino-Catharineum, was a good one and Dedekind studied science, in particular physics and chemistry. However, physics became less than satisfactory to Dedekind with what he considered an imprecise logical structure and his attention turned towards mathematics. Listing and Wilhelm Weber . The two departments combined to initiate a seminar which Dedekind joined from its beginning. There he learnt number theory which was the most advanced material he studied. His other courses covered material such as the differential and integral calculus, of which he already had a good understanding. The first course to really make Dedekind enthusiastic was, rather surprisingly, a course on experimental physics taught by

69. 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.

70. Rhonda Roland Shearer / Marcel Duchamp's Impossible Bed... Also note that the English translation dedekind cut (Poincaré) is incorrect.Duchamp s original notes in French read Poincaré cut. . http://www.marcelduchamp.org/ImpossibleBed/PartI/note19.html

A l'Infinitif A l'Infinitif

71. Documentclass{article} \usepackage{latexsym} \newtheorem {\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

72. MAT 300 Mathematical Structures. Matt Kawski. Spring 2004. Constructing Numbers To get to the reals is a major step. B.3.1 takes the classical route of Dedekindcuts. B.3.10 is the critical thm that distinguishes relas from rationals. http://math.la.asu.edu/~kawski/classes/mat300/construct.html

Constructing the numbers: I highly recommend to anyone who has some spare time (but first finish the regular homework!) to construct the number system all the way from the beginning. In some places, this process takes the place of our class MAT 300 here we chose to have a broader focus, but I still invite you to trace out some of the major steps. Our textbook provides a very nice outline of this process in the appendices. Below, I will give some pointers to some key-steps. If you would like to work this as an extra project, I will give you credit for this class, and will be ahppy to discuss progress and questions in office hours etc. Most may not want to do every step by hand so the usual rule is that you may use every definition and preceding theorem when proving the next step (but NOTHING else!). Defining the integers: A.2.2, A.2.3 and A.2.4. Then A.2.7 thru A.2.10. Defining addition: A.2.12 and A.2.22. A big deal is A.2.25 (using only the previous defns and thms!) Multiplication is a fun exercise: A.2.26 and A.2.27. On to substraction B.1.2. and the integers B.1.3. to B.1.5.

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