## Sunday 3 September 2023

### Updating the History of the Pigeon Hole Theorem

The Pigeon Hole Principle......The basic idea behind this mathematical principle is what students would call common sense; if there are n objects to be placed in m receptacles (with m less than n), at least two of the items must go into the same container. While the idea is common sense, in the hands of a capable mathematician it can be made to do uncommon things. The late Alexander Bogomolny used the principle to argue that there must be at least two persons in New York City with the same number of hairs on their head. This "counting hairs" approach dates back to the earliest version of the principal I have ever seen.

The same axiom is often named in honor of Dirichlet who used it in solving Pell's equation. The pigeon seems to be a recent addition, as Jeff Miller's web site on the first use of some math words gives, "Pigeon-hole principle occurs in English in Paul Erdös and R. Rado, A partition calculus in set theory, Bull. Am. Math. Soc. 62 (Sept. 1956)" (although they credit Dedekind for the principle). In a recent discussion on a history group Julio Cabillon added that there are a variety of names in different countries for the idea. His list included "le principe des tiroirs de Dirichlet", French for the principle of the drawers of Dirichlet, and the Portugese "principio da casa dos pombos" for the house of pigeons principle and "das gavetas de Dirichlet" for the drawers of Dirichlet. It also is sometimes simply called Dirichlet's principle and most simply of all, the box principle. Jozef Przytycki wrote me to add, "In Polish
we use also:"the principle of the drawers of Dirichlet"
that is 'Zasada szufladkowa Dirichleta' ". I received a note that said, "Dirichlet first wrote about it in Recherches sur les formes quadratiques à coefficients et à indéterminées complexes (J. reine u. angew. Math. (24 (1842) 291 371) = Math. Werke, (1889 1897), which was reprinted by Chelsea, 1969, vol. I, pp. 533-618. On pp. 579-580, he uses the principle."

He doesn't give it a name. In later works he called it the "Schubfach Prinzip" [which I am told means "drawer principle" in German]

The idea has been around much longer than Dirichlet, however, as I found out in June of 2009 when Dave Renfro sent me word that the idea pops up in the unexpected (at least by me) work, "Portraits of the seventeenth century, historic and literary", by Charles Augustin Sainte-Beuve. During his description of Mme. de Longuevillle, who was Ann-Genevieve De Bourbon, and lived from 1619 to 1679 he tells the following story: