Strange Places To Prove Theorems
People can do research in a variety of unusual places
André Weil was one of the most talented mathematicians of the 20th century. He helped found the Bourbaki group, was a great expositor of mathematics, and was a great creator of problems, besides solving many hard ones himself.
Today I want to do three things: point out the nice post that Ken Regan did by himself the other day, second announce he will be doing several more since I am sick with pneumonia, and last raise a new topic: where is the most unusual place you or someone “proved something?”
Weil was once asked to prepare a budget for his math department, and simply requested:
Give us enough chalk.
I am not sure that is the right answer today, but it makes for a great quote. He had similarly unusual insights of where mathematicians could work on their problems—we will see that he made great progress in one of the strangest places of all.
I thought about writing all this precisely because I am home sick, unable to leave the house or even the bed much. Perhaps the limited environment will help me make some new contribution to theory, or at least write a fun post. We will see.
Prison: Andre Weil was in a military prison in Bonne-Nouvelle, a district of Rouen, during several months of 1940. He was charged with failure to report for duty. While in prison he worked on the Weil Conjectures, which later made him famous. At least one colleague remarked later,
Perhaps I should stay in prison a few months
Bed, awake: René Descartes discovered, or is that invented, the notion of coordinate systems.
Some mathematics historians claim it may be that Descartes’ inspiration for the coordinate system was due to his lifelong habit of staying late in bed. According to some accounts, one morning Descartes noticed a fly walking across the ceiling of his bedroom. As he watched the fly, Descartes began to think of how the fly’s path could be described without actually tracing its path. His further reflections about describing a path by means of mathematics led to La Géometrie and Descartes’ invention of coordinate geometry.
The quotation is from this article.
Bed, not awake: Atri Rudra is sure once he proved a lemma in his sleep, because he went to bed thinking about the problem, woke up just to write down something, and promptly went back to sleep. The something was indeed a proof.
Bed, recovering: Alan Selman proved a difficult theorem in the 1970s on polynomial-time enumeration reductions and diagonalization while flat on his back for several straight days at home recovering after a myelogram. (We might separately be interested in whether anyone has been creative while listening to “Spinal Tap”—or maybe not.)
Attic: Andrew Wiles spent years working on finding the proof on Fermat’s Last Theorem. During this time he spent much of it working in his attic.
Basement: Probably true of many people, but Ken obtained the self-lowness theorem in this paper while ensconced underground as his 8- and 5-year-old children ruled above. He has proved results during faculty meetings and while listening to various sports including 8-hour 5-day cricket matches, but none of that is unusual.
Beaches: I have already mentioned that Steven Smale famously said that he did some of his best work on “the beaches of Rio.”
Country Home: Issac Newton spent the years right after getting his degree in 1665 at his home in Woolsthorpe, largely to escape a plague outbreak in Cambridge. There he changed the world inventing: the calculus, laws of optics, and the law of gravitation.
Bathtub: Archimedes, Archimedes of Syracuse, discovered while taking a bath how to tell if a crown was all gold. Legend has it that he took to the street, without clothes, crying:
[“Eureka!”—“I have found it!”]
He would not have said this accented on the second syllable as “you-reek-a!” but rather on the first syllable with an `h’, like “heu-rika”—and this is related to heuristics for algorithms.
Queues: Mike Paterson is claimed, by Albert Meyer, to be able to think hard about problems while just standing in a long queue. Albert always told me that this really impressed him greatly.
On a Date: The man shall remain anonymous; the lady did not become his wife\dots
We will stop here, but you may carry on.
Where is the strangest place for you? Or the place you have heard about?