## Friday, August 14, 2009

### How much credit should the conjecturer get? Is Conjecturer a word?

Theorems are often named after who proved it. The ones who conjectured it are often forgotten.
1. Mordell's conjecture was solved by Falting. It is now called Faltings' Theorem.
2. Vazsonyi's conjecture was solved by Joseph Kruskal. It is now called The Kruskal Tree Theorem.
3. Baudet's conjecture was solved by van der Waerden. It is now called van der Waerden's theorem . Even though van der Waerden's original paper has as its title (roughly translated) On a conjecture of Baudet, Baudet is not well known.
4. Fermat's last theorem was solved by Wiles. If you type Wiles into Wikipedia you get as options Wiles Theorem which goes to a page whose web address is http://en.wikipedia.org/wiki/Wiles_theorem but whose title on the page is Fermat's Last Theorem. This one may still be in transition from being someones conjecture to someones theorem. It may be for a while. This one is so tied to Fermat that it might always have his name on it somehow.
If you know of other examples please comment. Is it unfair that the original conjecturers are forgotten? Alexander Soifer thinks so. In his book The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators (reviewed in my latest SIGACT NEWS Book Review Column) he suggests that we should name a theorem after both who conjectures it and who solves it. So what I call
Van der Waerden's Theorem
Soifer calls
The Baudet-Schur-Van der Waerden Theorem.
(Baudet is known to have conjectured it. Soifer argues convincingly that Schur also conjectured it.) Reading over van der Waerden's own account of how the theorem was discovered (included in Soifer's book) it seems to me that Artin contributed some to the solution of Baudet's conjecture. If standards for co-authorship were weaker then he may have been a co-author. In this alternative universe what I would call
The Artin-Van der Waerden Theorem
Soifer would call
The Artin-Baudet-Schur-Van der Waerden Theorem.
This is odd since you have prover-conjecturer-conjecturer-prover in the ordering. Perhaps another convention would arise. Perhaps it would be called the ABSV-theorem or ABSW-theorem. Perhaps we are better off, just for the sake of simplicity, using just the prover's name. There have been some fierce battles over who PROVED what. Do we really want to have fierce battles over who CONJECTURED what? I conjecture that we do not.

#### 8 comments:

1. Too late, there already have been fierce battles over who conjectured what when. For example how a Weil conjecture became the Taniyama-Weil conjecture and then either the Taniyama-Shimura or the Taniyama-Shimura-Weil conjecture depending on who is writing.

Serge Lang tells the story (from an anti-Weil position) here

2. Wow, Serge Lang comes off as a complete nut-job.

3. I wouldn't say that the conjectures were solved, I'd say they were proven.

4. It seems like naming the conjecturer is even worse, because it's quite likely that some obscure paper conjectures a given result. It seems the name should be awarded not only for stating the conjecture, but also realizing its importance (e.g. Riemann).

Also, in some situations, the conjecturer's name does stay attached to the theorem. Mordell's conjecture is still referred to by name, as are the Bieberbach conjecture and the Weil conjectures. Mertens conjecture is false and is still usually referred to by that name. Other things, like the Kepler and Poincare conjectures, may be too recent to have name changes yet.

I conjecture that being famous for other results, as well as having your conjecture open for a long time, will help your name stay. As such, I think the Riemann Hypothesis will keep its name.

5. The Poincare Conjecture might not be too recent to provide evidence. The Poincare Conjecture for higher dimensions was proven decades ago (dimension 4 in the 80's, 5 and higher in the 60's), and those results are still called the Poincare Conjecture.

For example, people just as often say "the Poincare Conjecture for dimensions 5 and higher," etc. rather than Smale's theorem, Freedman's theorem, etc.

On a related note, sometimes conjecturers get too much credit by getting conjectures named after them that they didn't even conjecture. These can be extensions or generalizations of the original. The Poincare conjecture in higher dimensions is one example, as is the smooth Poincare conjecture, etc.

6. Main Conjecture of Iwasawa Theory

7. Regarding Serge Lang coming off as a nutcase, the linked article is actually a mild example. He wrote extensively on his disbelief that HIV causes AIDS.

8. Here's a more slippery slope. What credit to give to a theorem's Announcer? What I mean is this: someone announces in public "X is true". He/she may even have given a talk outlining his/her proof. It may get referenced in a paper or two. But the proof never appears (e.g. the announcement was made 10 years ago, nothing happened since).

Now you find your own proof. It is a significant piece of work. Where does the credit go?