Jules Richard
Encyclopedia
Jules Richard was a French
mathematician
.
, 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 Méray
.
In a more philosophical treatise about the nature of axioms of geometry Richard discusses and rejects the following basic principles:
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.
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:
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.
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:
Let E be the set of real numbers that can be defined by a finite number of words. This set is denumerable. Let p be the nth decimal of the nth number of the set E; we form a number N having zero for the integral part and p + 1 for the nth decimal, if p is not equal either to 8 or 9, and unity in the contrary case. This number N does not belong to the set E because it differs from any number of this set, namely from the nth number by the nth digit. But N has been defined by a finite number of words. It should therefore belong to the set E. That is a contradiction.
Richard never presented his paradox in another form, but meanwhile there exist several different versions, some of which being only very loosely connected to the original. For the sake of completeness they may be stated here.
(B) Berry's Paradox, first mentioned in the Principia Mathematica as fifth of seven paradoxes, is credited to Mr. G. G. Berry of the Bodleian Library. It uses the least integer not nameable in fewer than nineteen syllables; in fact, in English it denotes 111,777. But "the least integer not nameable in fewer than nineteen syllables" is itself a name consisting of eighteen syllables; hence the least integer not nameable in fewer than nineteen syllables can be named in eighteen syllables, which is a contradiction
(C) Berry's Paradox with letters instead of syllables is often related to the set of all natural numbers which can be defined by less than 100 (or any other large number) letters. As the natural numbers are a well-ordered set there must be the least number which cannot be defined by less than 100 letters. But this number was just defined by 65 letters including spaces.
(D) König's Paradox was also published in 1905 by Julius König
. All real numbers which can be defined by a finite number of words form a subset of the real numbers. If the real numbers can be well-ordered, then there must be a first real number (according to this order) which cannot be defined by a finite number of words. But the first real number which cannot be defined by a finite number of words has just been defined by a finite number of words.
(E) The smallest natural number without interesting properties acquires an interesting property by this very lack of any interesting properties.
(F) A loan of the Paradox of Grelling and Nelson. The number of all finite definitions is countable. In lexical order we obtain a sequence of definitions D1, D2, D3, ... Now, it may happen that a definition defines its own number. This would be the case if D1 read "the smallest natural number". It may happen, that a definition does not describe its own number. This would be the case if D2 read "the smallest natural number". Also the sentence "this definition does not describe its number" is a finite definition. Let it be Dn. Is n described by Dn. If yes, then no, and if no, then yes. The dilemma is irresolvable. (This version is described in more detail in another article, Richard's paradox
.)
wrote in a letter to David Hilbert
:
Here Cantor is in error. Today we know that there are uncountably many real numbers without the possibility of a finite definition.
Ernst Zermelo
comments Richard's argument:
Zermelo points to the reason why Richard's paradox fails. His last statement, however, is impossible to satisfy. A real number with infinitely many digits, which are not determined by some "rule", has an infinitely large contents of information. Such a number could only be identified by a short name if there were only one or few of them existing. If there exist uncountably many, as is the case, an identification is impossible.
[This seems to be the only original source, used by all other biographers.]
Final remark: The mathematician Jules Richard is not identical with the publicist (* 1810, † 1868) and also not with the manufacturer of scientific instruments and founder of the lycée technique Jules Richard in Paris (* 1848, † 1930). In the big encyclopedias and diaries of scholars the name Jules Richard is missing - even in the French ones. Therefore his biographical data are rather scanty.
French people
The French are a nation that share a common French culture and speak the French language as a mother tongue. Historically, the French population are descended from peoples of Celtic, Latin and Germanic origin, and are today a mixture of several ethnic groups...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
.
Life and Works
Richard taught at the lycées of ToursTours
Tours is a city in central France, the capital of the Indre-et-Loire department.It is located on the lower reaches of the river Loire, between Orléans and the Atlantic coast. Touraine, the region around Tours, is known for its wines, the alleged perfection of its local spoken French, and for the...
, Dijon
Dijon
Dijon is a city in eastern France, the capital of the Côte-d'Or département and of the Burgundy region.Dijon is the historical capital of the region of Burgundy. Population : 151,576 within the city limits; 250,516 for the greater Dijon area....
and Châteauroux
Châteauroux
Châteauroux is the capital of the Indre department in central France and the second-largest town in the province of Berry, after Bourges. Its residents are called Castelroussines or Castelroussins....
. He obtained his doctorate, at age of 39, from the Faculté des Sciences in Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...
. 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
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
, von Staudt
Karl Georg Christian von Staudt
Karl Georg Christian von Staudt was a German mathematician born in the Free Imperial City of Rothenburg, which is now called Rothenburg ob der Tauber in Germany. From 1814 he studied in Gymnasium in Ausbach. He attended the University of Göttingen from 1818 to 1822 where he studied with Gauss who...
and Méray
Charles Méray
Hugues Charles Robert Méray was a French mathematician. He is noted as the first to publish an arithmetical theory of irrational numbers. His work did not have much of a role in the history of mathematics because France, at that time, was less interested in such matters than Germany.-References:...
.
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
Immanuel Kant
Immanuel Kant was a German philosopher from Königsberg , researching, lecturing and writing on philosophy and anthropology at the end of the 18th Century Enlightenment....
.
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
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
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
Giuseppe Peano
Giuseppe Peano was an Italian mathematician, whose work was of philosophical value. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in...
and Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
. 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 MathematicaPrincipia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913...
by Alfred North Whitehead
Alfred North Whitehead
Alfred North Whitehead, OM FRS was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education...
and Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
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
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
and Alan Turing
Alan Turing
Alan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst, and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a...
to their famous works. Kurt Gödel considered his incompleteness theorem as analogous to Richard's paradox which, in the original version runs as follows:
Let E be the set of real numbers that can be defined by a finite number of words. This set is denumerable. Let p be the nth decimal of the nth number of the set E; we form a number N having zero for the integral part and p + 1 for the nth decimal, if p is not equal either to 8 or 9, and unity in the contrary case. This number N does not belong to the set E because it differs from any number of this set, namely from the nth number by the nth digit. But N has been defined by a finite number of words. It should therefore belong to the set E. That is a contradiction.
Richard never presented his paradox in another form, but meanwhile there exist several different versions, some of which being only very loosely connected to the original. For the sake of completeness they may be stated here.
Other versions of Richard's Paradox
(A) The version given in Principia Mathematica by Whitehead and Russell is similar to Richard's original version, alas not quite as exact. Here only the digit 9 is replaced by the digit 0, such that identities like 1.000... = 0.999... can spoil the result.(B) Berry's Paradox, first mentioned in the Principia Mathematica as fifth of seven paradoxes, is credited to Mr. G. G. Berry of the Bodleian Library. It uses the least integer not nameable in fewer than nineteen syllables; in fact, in English it denotes 111,777. But "the least integer not nameable in fewer than nineteen syllables" is itself a name consisting of eighteen syllables; hence the least integer not nameable in fewer than nineteen syllables can be named in eighteen syllables, which is a contradiction
(C) Berry's Paradox with letters instead of syllables is often related to the set of all natural numbers which can be defined by less than 100 (or any other large number) letters. As the natural numbers are a well-ordered set there must be the least number which cannot be defined by less than 100 letters. But this number was just defined by 65 letters including spaces.
(D) König's Paradox was also published in 1905 by Julius König
Julius König
Gyula Kőnig was a Hungarian mathematician. He was born in Győr, Hungary and died in Budapest. His mathematical publications in foreign languages appeared under the name Julius König...
. All real numbers which can be defined by a finite number of words form a subset of the real numbers. If the real numbers can be well-ordered, then there must be a first real number (according to this order) which cannot be defined by a finite number of words. But the first real number which cannot be defined by a finite number of words has just been defined by a finite number of words.
(E) The smallest natural number without interesting properties acquires an interesting property by this very lack of any interesting properties.
(F) A loan of the Paradox of Grelling and Nelson. The number of all finite definitions is countable. In lexical order we obtain a sequence of definitions D1, D2, D3, ... Now, it may happen that a definition defines its own number. This would be the case if D1 read "the smallest natural number". It may happen, that a definition does not describe its own number. This would be the case if D2 read "the smallest natural number". Also the sentence "this definition does not describe its number" is a finite definition. Let it be Dn. Is n described by Dn. If yes, then no, and if no, then yes. The dilemma is irresolvable. (This version is described in more detail in another article, Richard's paradox
Richard's paradox
In logic, Richard's paradox is a semantical antinomy in set theory and natural language first described by the French mathematician Jules Richard in 1905. Today, the paradox is ordinarily used in order to motivate the importance of carefully distinguishing between mathematics and metamathematics...
.)
Reactions to Richard's Paradox
Georg CantorGeorg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...
wrote in a letter to David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
:
- "Infinite definitions" (i.e., definitions which cannot be done in finite time) are absurdities. If Königs statement was "correct", according to which all "finitely definable" real numbers form a collection of cardinal number
, this would imply the countability of the whole continuum; but this is obviously wrong. The question is now what error the alleged proof of his wrong theorem is based upon. The error (which also appears in the note of a Mr. Richard in the last issue of the Acta mathematica, which Mr. Poincaré emphasizes in the last issue of the Revue de Métaphysique et de Morale) is, in my opinion, the following: It is assumed that the system {B} of notions B, which have to be used for the definition of individual numbers, is at most countably infinite. This assumption "must be in error" because otherwise we would have the wrong theorem: "the continuum of numbers has cardinality 
".
Here Cantor is in error. Today we know that there are uncountably many real numbers without the possibility of a finite definition.
Ernst Zermelo
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem.-Life:He graduated...
comments Richard's argument:
- The notion "finitely definable" is not an absolute one but a relative one being always related to the "language" chosen. The conclusion according to which all finitely definable objects are countable is only valid in case that one and the same system of symbols is used; the question whether a single individual can be subject to a finite definition is void because to every thing an arbitrary name can be attached to.
Zermelo points to the reason why Richard's paradox fails. His last statement, however, is impossible to satisfy. A real number with infinitely many digits, which are not determined by some "rule", has an infinitely large contents of information. Such a number could only be identified by a short name if there were only one or few of them existing. If there exist uncountably many, as is the case, an identification is impossible.
Papers and Books by Jules Richard
- Thèses présentées à la Faculté des sciences de Paris par M. Jules Richard, 1re thèse: Sur la surface des ondes de Fresnel..., Chateauroux 1901 (126 pages).
- Sur la philosophie des mathématiques, Gauthier-Villars, Paris 1903 (248 pages).
- Sur une manière d'exposer la géométrie projective, L'Enseignement mathématique 7 (1905) 366-374.
- Les principes des mathématiques et le problème des ensembles, Revue générale des sciences pures et appliquées 16 (1905) 541-543.
- 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. Harvard Univ. Press, 1967, p. 142-144.
- Lettre à Monsieur le rédacteur de la Revue Générale des Sciences, Acta Math. 30 (1906) 295-296.
- Sur les principes de la mécanique, L'Enseignement mathématique 8 (1906) 137-143.
- Considérations sur l'astronomie, sa place insuffisante dans les divers degrés de l'enseignement, L'Enseignement mathématique 8 (1906) 208-216.
- Sur la logique et la notion de nombre entier, L'Enseignement mathématique 9 (1907 ) 39-44.
- Sur un paradoxe de la théorie des ensembles et sur l'axiome Zermelo, L'Enseignement mathématique 9 (1907) 94-98.
- Sur la nature des axiomes de la géométrie, L'Enseignement mathématique 10 (1908 ) 60-65.
- Sur les translations, L'Enseignement mathématique 11 (1909) 98-101.
- Contre la géométrie expérimentale Revue de l’Enseignement des Sciences (1910) 150.
Literature and Links for Biographical Data
- J. Itard: Richard, Jules Antoine, Dictionary of Scientific Biography, 11, Charles Scribner's Sons, New York (1980) 413-414.
[This seems to be the only original source, used by all other biographers.]
- S. Gottwald: Richard, Jules Antoine in: Lexikon bedeutender Mathematiker, Harri Deutsch, Thun und Frankfurt (M) 1990.
- J. J. O'Connor, E. F. Robertson: The MacTutor History of Mathematics archive http://www-history.mcs.st-andrews.ac.uk/Biographies/Richard_Jules.html
- :fr:Jules Richard
- http://www.answers.com/topic/jules-richard
- http://www.central-france.com/famous_people_jules_richard.htm
Literature and Links for the Paradox
- H. Meschkowski, W. Nilson: Georg Cantor - Briefe, Sphinhubyringer, Berlin 1991, p. 446.
- W. Mückenheim: Die Mathematik des Unendlichen, Shaker, Aachen 2006.
- A. N. Whitehead, B. Russell: Principia Mathematica I, Cambridge Univ. Press, Cambridge 1910, p. 64. http://www.hti.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full%20text;idno=AAT3201.0001.001;didno=AAT3201.0001.001;view=pdf;seq=00000086
- E. Zermelo: Neuer Beweis für die Möglichkeit einer Wohlordnung, Math. Ann. 65 (1908) p. 107-128. http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D38183&p=125
- Proof of impossibilityProof of impossibilityA proof of impossibility, sometimes called a negative proof or negative result, is a proof demonstrating that a particular problem cannot be solved, or cannot be solved in general. Often proofs of impossibility have put to rest decades or centuries of work attempting to find a solution...
- :fr:Paradoxe de Richard
- :fr:Paradoxe de Berry
- http://www.dpmms.cam.ac.uk/~wtg10/richardsparadox.html
- http://c10-ss-1-lb.cnet.com/reference/Richard%27s_paradox
- http://c10-ss-1-lb.cnet.com/reference/Berry_paradox
- http://orcmid.com/readings/logic.htm
Final remark: The mathematician Jules Richard is not identical with the publicist (* 1810, † 1868) and also not with the manufacturer of scientific instruments and founder of the lycée technique Jules Richard in Paris (* 1848, † 1930). In the big encyclopedias and diaries of scholars the name Jules Richard is missing - even in the French ones. Therefore his biographical data are rather scanty.