September 2005

Naming theorems

The July 19 edition of The New York Times carried an article profiling the actress Danica McKellar, the former child star who played the part of Winnie Cooper on the television series "The Wonder Years", and who is currently seen from time to time in the TV series "The West Wing", playing Elsie Snuffin, a White House speechwriter.

As an undergraduate at UCLA, Ms McKellar majored in mathematics, and in her senior year engaged in an undergraduate research project with fellow student Brandy Winn, supervised by Professor Lincoln Chayes, that resulted in the publication of a paper in a professional research journal. The main result in the paper is now known as the "Chayes-McKellar-Winn theorem".

Although she returned to acting after she completed her degree, Ms McKellar has maintained her involvement in mathematics through a math Q&A section on her website danicamckellar.com, in which she answers questions sent in by readers.

Reading the Times article made me reflect on how certain theorems acquire names. Since the majority of results are not given names, I suspect that Ms McKellar's theorem is so called in large part because of her television fame.

Although it is generally regarded as bad form for mathematicians to try to attach their own names to theorems they prove, occasionally a theorem of mathematics does become named after the person who first proved it, but there is no general rule as to how this happens. Most mathematicians prove many theorems in their lives, and the process whereby their name gets attached to one of them is very haphazard. For instance, Euler, Gauss, and Fermat each proved hundreds of theorems, many of them important ones, and yet their names are attached to just a few of them.

Sometimes theorems acquire names that are incorrect. Most famously, perhaps, Fermat almost certainly did not prove "Fermat's Last theorem"; rather that name was attached by someone else, after his death, to a conjecture the French mathematician had scribbled in the margin of a textbook. And Pythagoras's theorem was known long before Pythagoras came onto the scene.

Another example of a theorem ascribed to entirely the wrong person is "Wilson's theorem", that (p-1)! + 1 is a multiple of p for any prime number p. This result was not proved by Wilson. Wilson guessed it might be true, but a chap called Waring subsequently proved it. In fact, the result was known to Lagrange before Wilson or Waring got into the act.

Sometimes a theorem gets named not after its discoverer but for its content. The famous Four Color Theorem is an example. It says that any map can be colored with at most four colors.

Occasionally, the naming can be quite whimsical. There's the Ham Sandwich Theorem, which says that the volumes of any n n-dimensional solids can always be simultaneously bisected by a (n-1)-dimensional hyperplane. For n=2 it is known as the pancake theorem. It gets its name because if you take a ham sandwich you can make a single cut with a knife so that both slices of bread and the ham are cut into exactly equal halves. (This would be easy if the sandwich were made perfectly out of rectangular pieces, but most sandwiches are irregular.)

Or there's the Hairy Ball Theorem. This says that, given a ball with hairs all over it, it is impossible to comb the hairs continuously and have all the hairs lay flat; some hair must be sticking straight up! A more formal version says that any continuous tangent vector field on the sphere must have a point where the vector is zero.

Most theorems in mathematics are proved by one or at most two mathematicians, but occasionally a larger group is involved. The HOMFLY theorem gets its name from the initials of the six mathematicians who proved it: Hoste, Ocneanu, Millett, Freyd, Lickorish, and Yetter.

The Darling-Erdos theorem (about random variables) sounds delightfully intimate until you realize that it was actually proved by two mathematicians, one of whom was called Darling.

Besides theorems, names also get attached to axioms, lemmas, equations, algorithms, and methods. For instance, we have Playfair's Axiom, Zorn's lemma, Pell's equation, and so on.

Actually, Pell's equation was not due to Pell but to Fermat. Pell simply copied it down from one of Fermat's letters. Euler read it in Pell's writing and mistakenly credited it to Pell. (Pell's equation is actually not a single equation but a KIND of equation, namely a Diophantine equation of the kind y^2 - Ax^2 = 1. So it is misnamed in two ways.)

Though no one has as yet named a theorem after me, I believe I am the only mathematician whose name is attached to an angle. Though perhaps that's cheating, since it depends on a pun on the word "angle".

Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin (email: devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and The Math Guy on NPR's Weekend Edition. Devlin's newest book, THE MATH INSTINCT: Why You're a Mathematical Genius (along with Lobsters, Birds, Cats, and Dogs) was published recently by Thunder's Mouth Press.