Jules Richard

Jules Antoine Richard ( born August 12, 1862 in Blet, Cher, † October 14, 1956 in Châteauroux ) was a French mathematician.

Life and work

Richard taught at high schools ( lycées ) of Tours, Dijon and Châteauroux. He earned his doctorate at the age of 39 years at the Faculté des Sciences in Paris with a theme to the surface of diffraction waves (of him Fresnel waves called ). It dealt mainly with the foundations of mathematics and geometry, he was referring to works by David Hilbert, Karl Georg Christian von Staudt and Charles Méray. In a philosophical treatises embossed on the nature of the axioms of geometry discussed, criticized, and he rejects the following guiding principles:

Richard came to the conclusion that the terms of the identity of two objects and the immutability of an object are too vague and require clarification. This should be done through axioms.

Although the non-Euclidean geometries were found at this time no application ( Albert Einstein 's general theory of relativity established in 1915 ), Richard already declared: " If the concept of the angle is set, you can choose the term of the straight line so that a or other of the three geometries is true. "

Become known over a narrower readership addition, however, only the Richardian paradox, mainly because Poincaré has extensively made ​​use of in order to disavow the set theory in vain, after which the advocates of the theory were forced to reject these attacks.

The paradox Richardian

The paradox was first developed in a letter from Richard Louis Olivier, director of the Revue générale des sciences et pure appliquées and published in the treatise " Les Principes Mathématiques et le problème of the ensemble " 1905. In the Principia Mathematica by Alfred North Whitehead and Bertrand Russell paradox with six other paradoxes is reproduced for self-reference. In one of the most important compendia of works on mathematical logic, collected by Jean van Heijenoort, Richards essay is also included. The Richardian paradox inspired Kurt Gödel and Alan Turing to their famous works. Kurt Godel undecidability looked at his set as an analogue to Richard 's paradox.

Richard made ​​use of for the construction of his paradox, a version of Cantor's diagonal method to construct a finite -defined number that is not included in the set of all numbers finally defined.

  • All finite definitions and thus all decimals finally defined form a countable set. These definitions can be lexically ordered and numbered the decimal defined and summarized in the form of a list. In this list, the n-th digit of the p n th decimal by the number p 1 is replaced, if p is equal to 8 or 9 is not; otherwise, p is replaced by the number 1. Behind the other written form the replaced digits with a decimal number.

This decimal number is not included in the original list, because it is different from every entry in the list to at least one point, namely from the n-th to the n-th decimal place. It has been defined by the previous paragraph but with a finite number of words, ie belongs to the set of all finite definable decimal.

Jules Richard did not publish another version of his paradox. However, it is often confused with the closely related Berry paradox, sometimes with the Grelling -Nelson's antinomy.

Writings

  • Thèses présentées à la Faculté des Sciences de Paris par M. Jules Richard, 1re thèse: Sur la surface des ondes de Fresnel ..., Chateauroux 1901.
  • Sur la philosophie des Mathématiques, Gauthier -Villars, Paris, 1903.
  • Sur une manière d' exposer la géométrie projective, in L' Enseignement mathématique 7, pp. 366-374. In 1905.
  • Les principes of Mathématiques et le problème of the ensembles, in Revue générale des sciences et pure appliquées 16, pp. 541-543. In 1905.
  • The principles of mathematics and the problem- of sets (1905 ), English translation in Jean van Heijenoort, From Frege to Gödel - A Source Book in Mathematical Logic, 1879-1931, pp. 142-144. Harvard Univ. Press, 1967.
  • Lettre à Monsieur le rédacteur de la Revue Générale des Sciences, in Acta Math 30, pp. 295-296. In 1906.
  • Sur les principes de la mécanique, L' Enseignement mathématique 8, pp. 137-143. In 1906.
  • Considérations sur l' astronomy, sa place dans les divers insuffisante degrés de l' enseignement, in L' Enseignement mathématique 8, pp. 208-216. In 1906.
  • Sur la logique et la notion de nombre entier, in L' Enseignement mathématique 9, pp. 39-44. In 1907.
  • Sur un paradoxical de la théorie et sur ​​l' ensemble of the axioms of Zermelo, in L' Enseignement mathématique 9, pp. 94-98. In 1907.
  • Sur la nature of the axiom de la géométrie, in L' Enseignement mathématique 10, pp. 60-65. , 1908.
  • Sur les translations, in L' Enseignement mathématique 11, pp. 98-101. , 1909.
  • Contre la géométrie expérimentale, in Revue de l' Enseignement des Sciences, pp. 150, 1910.
Recursive set Grelling–Nelson paradox Integrated Authority File
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