Jules Richard

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


Life and Works

Richard taught at the lycées of Tours, Dijon and Châteauroux. He obtained his doctorate, at age of 39, from the Faculté des Sciences in Paris. His thesis of 126 pages concerns Fresnel's wave-surface. Richard worked mainly on the foundations of mathematics and geometry, relating to works by Hilbert, von Staudt and Meray.

In a more philosophical treatise about the nature of axioms of geometry Richard discusses and rejects the following basic principles:

  • (1) Geometry is founded on arbitrarily chosen axioms - there are infinitely many equally true geometries.
  • (2) Experience provides the axioms of geometry, the basis is experimental, the development deductive.
  • (3) The axioms of geometry are definitions (in contrast to (1)).
  • (4) Axioms are neither experimental nor arbitrary, they force themselves on us since without them experience is not possible.

The latter approach was essentially that proposed by Kant. Richard arrived at the result that the notion of identity of two objects and the invariability of an object are too vague and need to be specified more precisely. This should be done by axioms.

  • Axioms are propositions, the task of which is to make precise the notion of identity of two objects pre-existing in our mind.

Further according to Richard, it is the aim of science to explain the material universe. And although non-Euclidean geometry had not found any applications (Albert Einstein finished his general theory of relativity only in 1915), Richard already stated clairvoyantly:

  • One sees that having admitted the notion of angle, one is free to choose the notion of straight line in such a way that one or another of the three geometries is true.

Richard corresponded with Giuseppe Peano and Henri Poincaré. He became known to more than a small group of specialists by formulating his paradox which was extensively use by Poincaré to attack set theory whereupon the advocates of set theory had to refute these attacks.

Richard's Paradox

The paradox was first stated in 1905 in a letter to Louis Olivier, director of the Revue générale des sciences pures et appliquées. It was published in 1905 in the article Les Principes des mathématiques et le problème des ensembles. The Principia Mathematica by Alfred North Whitehead and Bertrand Russell quote it together with six other paradoxes concerning the problem of self-reference. In one of the most important compendia of mathematical logic, compiled by Jean van Heijenoort, Richard's article is translated into English. The paradox can be interpreted as an application of Cantor's diagonal argument. It inspired Kurt Gödel and Alan Turing to their famous works. Kurt Gödel considered his incompleteness theorem as analogous to Richard's paradox which, in the original version runs as follows:

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