In the bizarrely-named “research excellence framework” bearing down on UK academics, some proportion of the credit will be given for the impact of the research. “Impact” is narrowly defined as economic and social impact; your department must have commercialised your discovery in such a way that either money changed hands or some social good was done.

Mathematicians have gloomily said that this is a game in which the rules are against us; the timescale given for the impact is far too short for the important and long-term impact that mathematics demonstrably has.

But have we been outsmarted by the computer scientists?

Jacob Aron reports in *New Scientist* on a company set up by computer scientists in Edinburgh. They use theorem-proving software to prove new “theorems” (I put this word in quotes since it may be unclear whether their product really qualifies as theorems), and they will sell you a theorem for fifteen quid. Once it is yours, you can name it after yourself, a friend, or a loved one.

The company name is the punning “TheoryMine”.

It seems to me that this provides both economic and social impact: economic obviously (it brings in money to the company, and to the country if they sell theorems abroad), and social because there are probably people who will be happier and more fulfilled to have a theorem named after them.

I know this because I had the chance of making similar impact once and turned it down. I was emailed by someone who was desperate to obtain Erdős number 2. He begged me to put his name on the next paper I wrote, no matter what it was about. Ah well, I didn’t reply, so there goes my chance of social impact. (He didn’t offer to pay, but I suppose I could have set a price.)

Allowing people to name a theorem after someone who had no hand in the proof has a venerable tradition in mathematics. It is said (though the story may be apocryphal) that Euler gave Pell’s equation its name because he knew that some English mathematician had worked on it but couldn’t remember which one. The equation should probably be named after Brahmagupta, who considered it a thousand years earlier; but even he was not the first.

When I was a student, the theorem asserting that, when a finite group acts on a finite set, the number of orbits is equal to the average number of fixed points of the group elements, was known as Burnside’s Lemma. The name arose because Burnside gave it without attribution in the second edition of his group theory book. (I believe that it was attributed in the first edition, which I have not seen.) When the theorem was taken up by combinatorial enumerators, they not unreasonably assumed that it was Burnside’s own. Peter Neumann pointed out that it was actually proved by Frobenius, while an important special case was due to Cauchy much earlier; he proposed calling it the Cauchy–Frobenius Lemma. (The title of his paper, “A lemma that is not Burnside’s”, led to many people calling it “not-Burnside’s Lemma”.) His solution was not universally adopted, and many people still use the old name. A recipe for confusion! My own view is clear. The lemma enables us to count orbits; call it the Orbit-counting Lemma.

If I recall correctly a story I heard in a lecture on the history of Lie algebras, the Cartan matrix and Cartan subalgebra were invented by Killing, and the Killing form by Cartan.

A different phenomenon is shown by the fact that we still talk about Fermat’s Last Theorem, even though it was proved by Andrew Wiles. This is perhaps more defensible; as Paul Erdős said, the purpose of life is to prove and to conjecture (and he put equal value on the two activities).

Apart from the very common problem of misattribution, there is another problem. It bothers me when people refer to Cameron’s Theorem; have I really only proved one theorem worthy of note? (Not quite true, actually. One of my proudest moments was when my name was adjectivised by Graham Higman, who subtitled a course of lectures he gave “A Cameronian Commentary”.)

Clearly there are complex psychological and sociological forces at work in the naming of theorems. (PhD topic, anyone interested?) But most likely TheoryMine have found a niche market and deserve the credit they will undoubtedly get in the REF.

But I don’t think I will be buying a theorem from them anytime soon.

Peter, let me bring a bit of social science to your rescue. As a mathematician, you obtain utility from having more impact than other mathematicians, not from having lots of impact per se. So, as long as you lot nearly all have little impact, you are fine. As a computer scientist, on the other hand, I now have to worry about keeping up with the TheoryMine Joneses.

IMHO the TheoryMine should actually pay you to get one of theirs theorem and publish it, with you and their prover as co-authors (surely you can add your stuff…)

This will put them in business of providing Erdős number 3 for cash…