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Jordan Curve theorem - STUDIES IN LOGIC GRAMMAR AND...

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STUDIES IN LOGIC, GRAMMAR AND RHETORIC 10 (23) 2007 Jordan’s Proof of the Jordan Curve Theorem Thomas C. Hales University of Pittsburgh Abstract. This article defends Jordan’s original proof of the Jordan curve theorem. The celebrated theorem of Jordan states that every simple closed curve in the plane separates the complement into two connected nonempty sets: an interior region and an exterior. In 1905, O. Veblen declared that this theorem is “justly regarded as a most important step in the direction of a perfectly rigorous mathe- matics” [13]. I dedicate this article to A. Trybulec, for moving us much further “in the direction of a perfectly rigorous mathematics.” 1 Introduction Critics have been unsparing in their condemnation of Jordan’s original proof. Ac- cording to Courant and Robbins, “The proof given by Jordan was neither short nor simple, and the surprise was even greater when it turned out that Jordan’s proof was invalid and that considerable e ff ort was necessary to fill the gaps in his reasoning” [2]. A web page maintained by a topologist calls Jordan’s proof “com- pletely wrong.” Morris Kline writes that “Jordan himself and many distinguished mathematicians gave incorrect proofs of the theorem. The first rigorous proof is due to Veblen” [8]. A di ff erent Kline remarks that “Jordan’s argument did not su ffi ce even for the case of a polygon” [7]. Dissatisfaction with Jordan’s proof originated early. In 1905, Veblen complained that Jordan’s proof “is unsatisfactory to many mathematicians. It assumes the theorem without proof in the important special case of a simple polygon and of the argument from that point on, one must admit at least that all details are not given” [13]. Several years later, Osgood credits Jordan with the theorem only under the assumption of its correctness for polygons, and further warns that Jordan’s proof contains assumptions. 1 Nearly every modern citation that I have found agrees that the first correct proof is due to Veblen in 1905 [13]. See, for example, [9, p. 205]. This research has been supported by NSF grant 0503447. 1 “Es sei noch auf die Untersuchungen von C. Jordan verwiesen, wo der Satz, unter Annahme seiner Richtigkeit f¨ur Polygone, allgemein f¨ur Jordansche Kurven begr¨undet wird. Jordan beweist hiermit mehr als die Funktionentheorie gebraucht; dagegen macht er Voraussetzungen, welche diese Theorie streng begr¨undet wissen will” [11]. ISBN 978-83-7431-128-1 ISSN 0860–150X 45
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Th. C. Hales My initial purpose in reading Jordan was to locate the error. I had completed a formal proof of the Jordan curve theorem in January 2005 and wanted to mention Jordan’s error in the introduction to that paper [3]. In view of the heavy criticism of Jordan’s proof, I was surprised when I sat down to read his proof to find nothing objectionable about it. Since then, I have contacted a number of the authors who have criticized Jordan, and each case the author has admitted to having no direct knowledge of an error in Jordan’s proof. It seems that there is no one still alive
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