Is mathematics the Christmas present of the year?

by Anders Sandberg

Is mathematics the Christmas present of the year? TheoryMine is a company that uses automatic theorem discovery and proof to generate new theorems via computer, which customers can then buy the naming rights for (for a paper describing the method, see The Theory behind TheoryMine). Is this a scam? Or does it devalue pure mathematics? Or is this a great new way of acknowledging its beauty?

Buying theorem naming rights might sound similar to buying naming rights for stars. That *is* a con, since real astronomers explicitly do not use or acknowledge the registries of star-naming companies. However, there have been cases where discoverers of asteroids (who do have naming rights) have named them after benefactors or even held auctions for what name they would suggest to the IAU. Since minor planets may not be named after their discoverer (unlike comets), pairs of asteroid-hunters sometimes propose each other’s names quid pro quo. Similarly, species naming has also been auctioned off. The relevant astronomical and zoological societies have final say.

However, there is no formal system for naming theorems. It is considered bad form to name a theorem after yourself, but others commonly name theorems after the first person to prove them. This tends to get confusing when dealing with the truly great mathematicians. Many named theorems are not due to the person who gave their name to them – Pythagoras was not first with his theorem, Fermat’s Last Theorem is due to Andrew Wiles, and Napoleon Bonaparte probably did not discover his theorem. Pell got his equation named after him by mistake.

There have also been real cases of theorems being sold, including the well-known L’Hôpital’s rule, which was likely due to Johann Bernoulli. In a 1694 letter L’Hôpital offered an annual payment of 300 Francs if Bernoulli would inform him of his latest mathematical discoveries and withhold them from correspondence with others. No reply is known, but he seem to have achieved a deal: he wrote the first textbook on calculus in 1696 where he mentions the rule, forever linking his name with it.

Hence it is not that odd that one could buy the right to name a newly computer generated theorem. The real challenge is to get enough people to use it so that the name sticks. Given that the TheoryMine works in the domain of recursive theories generated to be deliberately obscure, this might prove to be hard. I suspect the recipients of a nice theorem ought to prove a few other theorems in the theory, and then write a paper about them. Sounds like a good activity for the after-Christmas holidays.

Is there anything bad with this trade? The aim of naming a theorem after somebody is usually recognition of who did the hard work. Buyers get their name on a product they have not worked on and are very unlikely to understand. But as we have seen, this may be true for many theorems. Most theorems are also utterly minor in importance. They are trivial, or applicable only within an exceedingly narrow domain. Their names (if they even have one) are unimportant. The theorems that matter are one’s that crop up everywhere: not only do they need names that can be recalled, but morally their discoverer’s should be honoured. But TheoryMine is unlikely to find anything of that magnitude (if it does, then mathematics is in for a much bigger revolution than any naming conventions can cover). The appearance of Aron’s theorem will hardly dilute the honour system of mathematics more than the appearance of the Futurama theorem (properly that theorem ought to be known as Keeler’s theorem, but in the TV show it was proved by “Sweet” Clyde Dixon, so maybe it should be called the Dixon-Keeler theorem). It is the important theorem’s that will be recalled, not because they are named, but because they matter.

If mathematics had a central body organising the ~250,000 theorems that are proved every year, things might be slightly different. Then it might decide on the appropriateness of suggested names like the IAU. But that wouldn’t change the possibility of giving a discoverer money in exchange for the discoverer suggesting a particular name. The only ethical issue would be for the discoverer (or in the case of a machine, its owner) not to bring people’s hopes up too much that their theorem is ever going to be useful or cited. (This is true for academic advisers to young academics too.)

No doubt many may think this is against the dignity of mathematics (not to be confused with the dignity of carrots or humans). But while much of mathematics has a solemn air, it has also many down-to-earth applications (Thomas Scoville: “Math may well be the Queen of Sciences, but she’s slumming in the Silicon Valley.”). There are silly theorem names like the Ham Sandwich Theorem (which in 2D becomes the Pancake Theorem, not to be confused with Pizza Theorem), the Hairy Ball Theorem, the ASS Theorem, not to mention the Frivolous Theorem of Arithmetic. It has many downright silly aspects, from recreational math to abstract nonsense. The strength of mathematics is that unlike art, religion or parts of the humanities, it *does not need* solemnity to work. You can laugh all you like at a proof, but it will still remain valid.

In fact, buying naming rights might be a way of sponsoring science. Rather than to pay a company that runs a computer to spit out random theorems (although this particular company seems to be run by the people who did the nontrivial and possibly useful work in constructing an automatic theorem generator, so they deserve some money too), why not hire a mathematician to name her next theorem after you? Or for mathematicians to auction off naming rights for their recent theorems on eBay? Or why not just go for the classic patronage model and give some money to researchers, in the hope that they will discover something great and maybe gratefully name it after you? If you think the possibility of Branson’s Theorem, a Lloyd Webber Theorem or an Adidas Theorem are too much, the best solution might be to work at getting more funding into mathematics.

Of course, for us with a more limited budget, there is another possibility: handmade theorems. Prove something nice for your loved one’s! A neat geometrical construction for the kids to ponder, some topology for your aunt, some category theory about symmetric relations for your partner…