Jules Richard (mathematician)

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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 the high schools (Lycées) of Tours , Dijon and Châteauroux . It was only at the age of 39 that he did his doctorate at the Faculté des Sciences in Paris with a topic on the surface of diffraction waves (which he called Fresnel waves). He mainly dealt with the basics of mathematics and geometry, referring to the work of David Hilbert , Karl Georg Christian von Staudt and Charles Méray . In a philosophical treatise on the nature of the axioms of geometry, he discusses, criticizes and rejects the following guiding principles:

  1. The geometry is based on arbitrarily chosen axioms - there are an infinite number of equally true geometries.
  2. The axioms of geometry are given by experience. A deductive development takes place on an experimental basis.
  3. The axioms of geometry are definitions (in contrast to (1)).
  4. Axioms are neither experimentally enforced nor chosen arbitrarily. They are a prerequisite a priori, because only through them is experience possible at all (a view also represented by Immanuel Kant ).

Richard came to the conclusion that the concepts 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 had not yet found application at this time ( Albert Einstein did not set up his general theory of relativity until 1915), Richard already explains: “If the concept of the angle is determined, one can choose the concept of the straight line so that the one or other of the three geometries is true. "

However, only Richard's paradox has become known beyond a narrow readership, primarily because Poincaré made extensive use of it to disavow set theory in vain, whereupon the proponents of set theory felt compelled to reject these attacks.

Richard's paradox

The paradox was first developed in a letter from Richard to Louis Olivier, the director of the magazine Revue générale des sciences pures et appliquées and published in 1905 in the treatise "Les Principes des mathématiques et leproblemème des ensembles". Bertrand Russell took it up in 1908 in his list of mathematical paradoxes, which he later adopted in the influential Principia Mathematica . Richard's paradox inspired Kurt Gödel and Alan Turing to write their famous works. Kurt Gödel regarded his undecidability theorem as an analogue to Richard's paradox.

Richard used a version of Cantor's diagonal method to construct his paradox in order to construct a finitely defined number that is not contained in the set of all finitely defined numbers.

  • All finite definitions and thus all finitely defined decimal numbers form a countable set. These definitions can be lexically ordered and the defined decimal numbers numbered and summarized in the form of a list. In this list, the nth digit p of the nth decimal number is replaced by the digit p + 1 if p is not equal to 8 or 9; otherwise, p is replaced by the number 1. Written one after the other, the replaced digits form a decimal number.

This decimal number is not included in the original list because it differs from every list entry in at least one position, namely from the nth decimal number in the nth position. But it was defined by the previous paragraph with a finite number of words, so it belongs to the set of all finitely definable decimal numbers.

Jules Richard did not publish any other version of his paradox. But it is often confused with the closely related Berry paradox , sometimes also with the Grelling-Nelson antinomy .

Fonts

  • Thèses présentées à la Faculté des sciences de Paris par M. Jules Richard, 1st 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. 1905.
  • Les principes des mathématiques et leproblemème des ensembles , in Revue générale des sciences pures et appliquées 16 , pp. 541–543. 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. 1906.
  • Sur les principes de la mécanique , in L'Enseignement mathématique 8 , pp. 137-143. 1906.
  • Considérations sur l'astronomie, sa place insuffisante dans les divers degrés de l'enseignement , in L'Enseignement mathématique 8 , pp. 208-216. 1906.
  • Sur la logique et la notion de nombre entier , in L'Enseignement mathématique 9 , pp. 39-44. 1907.
  • Sur un paradoxe de la théorie des ensembles et sur l'axiome Zermelo , in L'Enseignement mathématique 9 , pp. 94–98. 1907.
  • Sur la nature des axiomes 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 , p. 150. 1910.

literature

  • J. Itard: Richard, Jules Antoine , Dictionary of Scientific Biography, Volume 11. Charles Scribner's Sons, New York (1980) 413-414.
  • S. Gottwald: Richard, Jules Antoine in the Lexicon of Major Mathematicians, Harri Deutsch, Thun and Frankfurt (M) 1990.

Web links

Individual evidence

  1. ↑ Russell: Mathematical logic as based on the theory of types (PDF; 1.9 MB), in: American Journal of Mathematics 30 (1908), page 223 (6)
  2. ↑ Russell / Whitehead: Principia Mathematica I. 64 (7)
personal data
SURNAME Richard, Jules
ALTERNATIVE NAMES Richard, Jules Antoine (full name)
BRIEF DESCRIPTION French mathematician
DATE OF BIRTH August 12, 1862
PLACE OF BIRTH Blet , Cher
DATE OF DEATH October 14, 1956
Place of death Chateauroux