A Theory Is Not Complete Until You Can Explain It To the Person in the Street

David Hilbert? Mary Winston Newson? Adolphe Quetelet? Joseph Diez Gergonne? Michel Chasles? William Spottiswoode? Henry John Stephen Smith? G. Carey Foster? Alphonse Rebière? Anonymous?

Dear Quote Investigator: In 1900 the influential German mathematician David Hilbert addressed the “International Congress of Mathematicians” held in Paris. He presented a collection of unsolved problems which has had an enormous influence in guiding research during the following century and beyond.

Some of the solutions discovered for these problems have been quite complex and intellectually demanding. Nevertheless, Hilbert indicated in his speech that he placed great value on simplicity and intelligibility. He credited an unnamed old French mathematician with the following remark:

A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.

Would you please explore the provenance of this saying?

Quote Investigator: The earliest match known to QI appeared in a letter dated February 25, 1825 from French mathematician Joseph Diez Gergonne to Belgian astronomer and statistician Adolphe Quetelet. Below is an image showing the part of the letter containing the quotation. The spelling in the letter differed a bit from current French spelling: “longtemps” was “longtems” and “raconter” was “racconter”. After the image and French text an English translation is shown. Boldface added to excerpts by QI: 1

Excerpt of Gergonne letter

Il y a longtems que je répète à mes élèves qu’on n’a pas encore le dernier mot de la science sur une théorie, tout aussi longtems qu’on ne l’a pas amenée au point de la racconter à un passant, dans la rue.

For many years I repeated to my students that one does not yet have the last word of science on a theory for as long as one has not brought it to the point of being able to recount it to a passerby on the street.

Below are additional selected citations in chronological order.

The crucial citations in this article were identified by June Barrow-Green and Reinhard Siegmund-Schultze in their wonderful 2016 piece in the journal “Historia Mathematica” titled “‘The first man on the street’ — tracing a famous Hilbert quote (1900) back to Gergonne (1825)”. 2

In 1827 Quetelet published an article in “Nouveaux Mémoires de L’Académie Royale des Sciences et Belles-Lettres de Bruxelles” (“New Memoirs of the Royal Academy of Sciences and Belles-Lettres of Brussels”), and he reprinted the statement from Gergonne: 3

Qu’il me soit permis de rapporter l’opinion de cet habile géomètre; elle a été pour moi une flatteuse récompense de mes recherches. « II y a long-temps que je répète à mes élèves, m’écrivait-il, qu’on n’a pas encore le dernier mot de la science sur une théorie, tout aussi long-temps qu’on ne l’a pas amenée au point de la raconter à un passant, dans la rue . . . »

Allow me to relate the opinion of this skillful geometer; it has been a flattering reward for my research. “For many years I repeated to my students that one does not yet have the last word of science on a theory for as long as one has not brought it to the point of being able to recount it to a passerby on the street.”

In 1837 French mathematician Michel Chasles published “Aperçu historique sur l’origine et le développement des méthodes en géométrie” (“Historical overview on the origin and development of methods in geometry”), and he presented a modified version of the saying that was probably derived from the remark of Gergonne: 4

Nous ajouterons, avec un des géomètres modernes qui ont le plus médité sur la philosophie des mathématiques, « qu’on ne peut se flatter d’avoir le dernier mot d’une théorie, tant qu’on ne peut pas l’expliquer en peu de paroles à un passant dans la rue. »

Et en effet, les vérités grandes et primitives, dont toutes les autres dérivent, et qui sont les vraies bases de la science, ont toujours pour attribut caractéristique la simplicité et l’intuition.

We will add, with one of the modern geometers who have meditated the most on the philosophy of mathematics, “One cannot flatter oneself that one has the last word of a theory until one can explain it in a few words to a passerby on the street.”

And, in effect, great and original truths, from which all others derive, and which are the true foundations of science, always have the characteristic attributes of simplicity and intuition.

In 1873 Irish mathematician and astronomer Henry J. S. Smith was President of the Mathematical and Physical Section of the British Association for the Advancement of Science. He employed a version of the saying during an address he delivered at the forty-third meeting of the group held in Bradford, England: 5

An old French geometer used to say that a mathematical theory was never to be considered complete till you had made it so clear that you could explain it to the first man you met in the street.

This is of course a brilliant exaggeration; but it is no exaggeration to say that the eminent writers to whom I have referred have given something of this clearness and completeness to such abstract mathematical theories as those of the electrical potential, the action of capillary forces, and the definition of absolute temperature.

In 1877 English chemist and physicist G. Carey Foster was President of the Mathematical and Physical Section of the British Association for the Advancement of Science. He referred to the saying during an address delivered at the forty-seventh annual meeting: 6

In an address, delivered from the Chair of this Section four years ago, Prof. Henry Smith quotes the saying of an old French geometer, “that a mathematical theory was never to be considered complete till you had made it so clear that you could explain it to the first man you met in the street.”

Very likely Prof. Smith was right to call this “a brilliant exaggeration;” at any rate I know of no reason for disputing his opinion; but I believe the exaggeration would really be very small if the dictum were applied to the theories of physics instead of to those of pure mathematics.

In 1878 mathematician and physicist William Spottiswoode was President of the British Association for the Advancement of Science. He employed another version of the saying during an address at the forty-eighth annual meeting: 7

Considering, however, our limitations of time, and the varied nature of our audience, it would seem not inappropriate to suspend, mentally if not materially, over the doors of our section rooms, the Frenchman’s dictum, that no scientific theory ‘can be considered complete until it is so clear that it can be explained to the first man you meet in the street.’

In 1889 French mathematician Alphonse Rebière published “Mathématiques et mathématiciens: pensées et curiosités” (“Mathematics and mathematicians: thoughts and curiosities”). He referred to the saying while pointing to Michel Chasles: 8

Chasles exagère un peu, lorsqu’il affirme, dans son Aperçu historique qu’on ne peut se flatter d’avoir éclairé et réduit convenablement une théorie, tant qu’on ne peut pas l’expliquer en peu de mots à un passant dans la rue.

Chasles exaggerates a little when he affirms in his Aperçu historique that one cannot flatter oneself that one has clarified and satisfactorily reduced a theory when one cannot explain it in a few words to a passerby in the street.

David Hilbert’s speech at the 1900 meeting of the International Congress of Mathematicians was translated into English by Mary Winston Newson and published in the “Bulletin of the American Mathematical Society”. Hilbert referred to the saying: 9

An old French mathematician said: “A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.” This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us.

In conclusion, this saying can be traced back to Joseph Diez Gergonne who penned it in a note he sent to Adolphe Quetelet in 1825. There are many versions of this remark because it has evolved over time.

There also exist other thematically related sayings such as the following:

An alleged scientific discovery has no merit unless it can be explained to a barmaid.

There is a separate Quote Investigator article about the above saying with citations beginning in 1955 which is available here.

Image Notes: Public domain illustration of a collection of overlapping digits from TheDigitalArtist at Pixabay. The image has been cropped and resized.

(Great thanks to James A. Landau whose message led QI to formulate this question and perform this exploration. Landau told QI about the statement from William Spottiswoode, and he presented an 1878 citation.)

Notes:

  1. 2016 November, Historia Mathematica, Volume 43, Issue 4, “The first man on the street” — tracing a famous Hilbert quote (1900) back to Gergonne (1825) by June Barrow-Green and Reinhard Siegmund-Schultze, (Letter dated February 25, 1825 from Joseph-Diez Gergonne to Adolphe Quetelet; authors of this article obtained a scan of the letter from archivist Olivier Damme), Quote Page 421, Elsevier Inc. (Accessed via sciencedirect.com on September 30, 2021) link
  2. 2016 November, Historia Mathematica, Volume 43, Issue 4, “The first man on the street” — tracing a famous Hilbert quote (1900) back to Gergonne (1825) by June Barrow-Green and Reinhard Siegmund-Schultze, Start Page 415, Last Page 426, Elsevier Inc. (Accessed via sciencedirect.com on September 30, 2021) link
  3. 1827, Nouveaux Mémoires de L’Académie Royale des Sciences et Belles-Lettres de Bruxelles (New Memoirs of the Royal Academy of Sciences and Belles-Lettres of Brussels), Tome IV (Volume 4), Résumé d’une nouvelle théorie des caustiques, suivi de différentes applications à la théorie des projections stéréographiques par Mr. A. Quetelet, Start Page 79, Quote Page 88, M. Hayez, Imprimeur de L’Académie Royale, Bruxelles, Belgique. (EuDML: The European Digital Mathematics Library) link
  4. 1837, Aperçu historique sur l’origine et le développement des méthodes en géométrie (Historical overview on the origin and development of methods in geometry) par M. Chasles (Michel Chasles), Quote Page 115, M. Hayez, Imprimeur de L’Académie Royale, Bruxelles. (Google Books Full View) link
  5. 1874, Report of the Forty-Third Meeting of The British Association for the Advancement of Science Held at Bradford in September 1873, Notices and Abstracts of Miscellaneous Communications To the Sections, Mathematics and Physics, Address by Professor H. J. S. Smith, M.A., F.R.S., President of the Section, Start Page 1, Quote Page 7 and 8, Published by John Murray, London. (Google Books Full View) link
  6. 1878, Report of the Forty-Seventh Meeting of The British Association for the Advancement of Science Held at Plymouth in August 1877, Notices and Abstracts of Miscellaneous Communications To the Sections, Mathematics and Physics, Address by Professor G. Carey Foster, F.R.S., President of the Section, Start Page 1, Quote Page 5, Published by John Murray, London. (Google Books Full View) link
  7. 1879, Report of the Forty-Eighth Meeting of The British Association for the Advancement of Science Held at Dublin in August 1878, Address of William Spottiswoode, President, Start Page 1, Quote Page 3, Published by John Murray, London. (Google Books Full View) link
  8. 1889, Mathématiques et mathématiciens: pensées et curiosités (Mathematics and mathematicians: thoughts and curiosities) by Alphonse Rebière, Quote Page 181, Librairie Nony, Paris, France. (Google Books Full View) link
  9. 1902 July, Bulletin of the American Mathematical Society, Volume 8, Article: Mathematical Problems: Lecture Delivered Before the International Congress of Mathematicians at Paris 1900 by Professor David Hilbert, (Note: Translated for the Bulletin with the author’s permission, by Dr. Mary Winston Newson. The original appeared in the Göttinger Nachrichten, 1900, pp. 253-297, and in the Archiv der Mathematik und Physik, 3d ser., vol. 1 (1901), pp. 44-63 and 213-237), Start Page 437, Quote Page 438, The Macmillan Company, New York. (Google Books Full View) link