*There are no sects in geometry.*

*He who has heard the same thing told by 12,000 eye-witnesses*

has only 12,000 probabilities, which are equal to one strong probability,

which is far from certain.

In 1764 a collection of 73 articles on a variety of topics was published anonymously under the title *Dictionnaire philosophique portatif*. The writing was informal and often in the first person, and the work would be more accurately described as a collection of essays rather than as a dictionary. Despite being proscribed by the Catholic Church as well as burned in Paris and other cities, the work proved extremely popular: it was reprinted five times and expanded to include additional articles. The final version was published in 1769 under the title *Dictionnaire philosophique*. It was years later before it became widely known that the author of the work had been François-Marie d'Arouet, better known as Voltaire.

Lithograph of Voltaire. Public domain, Wellcome Collection.

It was in the opening lines of the article “Sect” that the first quotation from the beginning of this column appears:

Toute secte, en quelque genre que ce puisse être, est le ralliement du doute et de l'erreur. Scotistes, thomistes, réaux, nominaux, papistes, calvinistes, molinistes, jansenistes, ne sont que des noms de guerre. Il n'y a point de secte en géométrie; on ne dit point un euclidien, un archimédien. Quand la vérité est évidente, il est impossible qu'il s'élève des partis et des factions. Jamais on n'a disputé s'il fait jour à midi [Voltaire 1821, p. 129].

Every sect, in whatever sphere, is the rallying-point of doubt and error. Scotist, Thomist, Realist, Nominalist, Papist, Calvinist, Molinist, Jansenist, are only pseudonyms. There are no sects in geometry; one does not speak of a Euclidian [*sic*], an Archimedean. When the truth is evident, it is impossible for parties and factions to arise. Never has there been a dispute as to whether there is daylight at noon [Voltaire 1924, p. 267].

While it was true at the time it was written that “one does not speak of a Euclidean,” of course it is now commonplace to discriminate between Euclidean geometry and non-Euclidean geometries such as hyperbolic geometry. However, the actual meaning of Voltaire’s quotation still seems to hold true today: unlike religious sects, modern mathematicians do not argue that Euclidean or non-Euclidean geometry is the one “true” geometry, but only that each is equally consistent as a mathematical system.

Voltaire came back to this same point about sects in geometry at the end of the article “Tolerance,” generalizing to include other branches of mathematics:

C'est ainsi qu'on en usa long-temps dans une grande partie de la terre; mais aujourd'hui que tant de sectes se balancent par leur pouvoir, quel parti prendre avec elles? toute secte, comme on sait, est un titre d'erreur il n'y a point de secte de géomètres, d'algébristes, d'arithméticiens, parce que toutes les propositions de géométrie, d'algèbre, d'arithmétique, sont vraies. Dans toutes les autres sciences on peut se tromper [Voltaire 1821, p. 310].

It is thus that a great part of the world long was treated; but today when so many sects make a balance of power, what course to take with them? Every sect, as one knows, is a ground of error; there are no sects of geometers, algebraists, arithmeticians, because all the propositions of geometry, algebra and arithmetic are true. In every other science one may be deceived [Voltaire 1924, p. 304].

The second quotation at the beginning of this column came from the article “Truth,” which opened by relating the Biblical story of Pontius Pilate asking, “What is truth?” and then immediately leaving the room. Voltaire lamented Pilate’s lack of intellectual curiosity, and proceeded to imagine a lengthy training regimen for him: at least six months of logic, three or more years of geometry, a year of physics, as well as additional years of theology and metaphysics. Once this training was complete, Voltaire concluded:

I should then have said to Pilate:—Historical truths are merely probabilities. If you had fought at the battle of Philippi, that is for you a truth which you know by intuition, by perception. But for us who dwell near the Syrian desert, it is merely a very probable thing, which we know by hearsay. How much hearsay is necessary to form a conviction equal to that of a man who, having seen the thing, can flatter himself that he has a sort of certainty?

He who has heard the thing told by twelve thousand eye-witnesses, has only twelve thousand probabilities, equal to one strong probability, which is not equal to certainty.

If you have the thing from only one of these witnesses, you know nothing; you should be sceptical. If the witness is dead, you should be still more sceptical, for you cannot enlighten yourself. If from several witnesses who are dead, you are in the same plight. If from those to whom the witnesses have spoken, your scepticism should increase still more.

From generation to generation scepticism increases, and probability diminishes; and soon probability is reduced to zero [Voltaire 1924, pp. 306–307].

(*Note*: the original French for this longer selection can be found on pages 369–370 of [Voltaire 1821].)

The *Dictionnaire philosophique* contains many other mathematical references besides the two most common quotations that this column has traced. As an example, I’ll conclude with my personal favorite, which appears in the “Contradictions” article:

Je ne connais que deux sortes d'êtres immuables sur la terre, les géomètres & les animaux; ils sont conduits par deux règles invariables, la démonstration & l'instinct: & encore les géomètres ont-ils eu quelques disputes, mais les animaux n'ont jamais varié [Voltaire 1784, p. 124].

I know only two kinds of immutable beings on the earth, mathematicians and animals; they are led by two invariable rules, demonstration and instinct: and even the mathematicians have had some disputes, but the animals have never varied [Voltaire 1924, p. 83].

##### References

Voltaire. 1784. Oeuvres Completes de Voltaire. Vol 39. [Kehl, Germany]: Société littéraire-typographique.

Voltaire. 1821. Oeuvres de Voltaire. Vol. 8. Paris: Garnery.

Voltaire. 1924. Voltaire’s Philosophical Dictionary. Translated by H. I. Woolf. New York: A. A. Knopf.

“Quotations in Context” is a regular column written by Michael Molinsky that has appeared in the *CSHPM/SCHPM Bulletin* of the Canadian Society for History and Philosophy of Mathematics since 2006 (this installment was first published in May 2011). In the modern world, quotations by mathematicians or about mathematics frequently appear in works written for a general audience, but often these quotations are provided without listing a primary source or providing any information about the surrounding context in which the quotation appeared. These columns provide interesting information on selected statements related to mathematics, but more importantly, the columns highlight the fact that students today can do the same legwork, using online databases of original sources to track down and examine quotations in their original context.