# 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.

** Places **

**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.

** Open Problems **

Where is the strangest place for you? Or the place you have heard about?

On the crapper. Seriously!

Very true.

i do my best thinking on the bus and the train. not very unusual, i suppose, since there isn’t a whole lot else to do other than people watch. oh, and lying on the floor in my office.

i proved a lemma on the plane, but later found out it was already known, so that’s not too exciting. a long plane ride simply *invites* working on theorems.

s.

I spend all of my idle time (meetings, waiting for the bus) thinking about a handful of problems for which I can hold the definitions of in my head, ones that are easy to visualize, like factoring and the lonely runner conjecture. Number theory problems are perfect for this.

I haven’t had any positive results yet, but have come up with a couple of interesting and possibly novel tacks.

The real benefit is that it’s been years since I’ve found myself feeling bored.

Not so strange, but I had a pretty nice NP-completeness proof during running recently.

For me personally: laying in a tiny enclosed space, in a capsule youth hostel (think capsule hotel, only smaller and cheaper) in Tokyo.

I find the bathroom to be productive almost regardless of activity (showering, brushing/flossing, etc…). I think the small space, lack of outside distraction/noise, and mundane tasks make for a good situation.

I remember Subhash Khot solving many problems while traveling standing in the packed Bombay local trains.

Pneumonia is no joke … fighting it takes most of a person’s energy … and so please stay in bed for as long as it takes to get completely well. And if it should happen, that during this quiet time, some theorems float into your mind … well … that would be good … and we will all look forward to hearing them!

Lying in complete darkness waiting for small children to go to sleep, so that I can get up and write stuff down.

God bless your recovering, Professor Lipton.

Sorry, but I wouldn’t say “inventing […] laws of optics, and the law of gravitation” but rather “discovering …”.

Thank you, and ditto! Nice point about “inventing” versus “discovering” which I didn’t catch—whether the former applies to [a particular style of doing] the calculus is the difficult question here, IMHO.

I agree about the calculus!

Both seem to be ‘ok’ for me.

As with many other people, a lot of ideas come while taking long bus rides. But I’ve also had some of my best ideas while doing the dishes🙂

Also there is that thing where I fall asleep thinking about a problem, while drifting into dreams I feel like I have a genius idea, then on the next day when I recover the idea it’s usually nonsense.

To the authors of the blog: why don’t you share the stranger places you’ve proved something?

Prof. Lipton, get better!

“Give us enough chalk. ”

Ooh, sore spot — as a matter of fact, the college where I teach took away our chalk at the start of this semester (and not the chalkboards, mind you). Previously janitors had chalk and would replace it or give you some through the day; as of January that was prohibited. In theory, we were supposed to ask the department daily for 1 or 2 pieces of chalk — even us night instructors, after the office was closed. I wondered if someone might get shanked like in prison, over chalk.

So perhaps there are things fancier than chalk, but even this base-level requirement (more generally, being able to write in a classroom) can indeed be in doubt in this day and age, granted priorities of administrators.

Come on, all of you – you should know much better.

In fact, the STRANGEST place where you can get new ideas is at your office desk, seated on a chair. Seriously. There is plenty of psychological evidence to support this. I am able to think a lot better when I am standing, or pacing up and down in my office. If you want me to compare working on the whiteboard (or blackboard) versus working on paper at a desk, definitely I prefer the whiteboard. To be more precise, the “big picture” ideas go on the whiteboard and the “low level” workout of the details are done at the desk.

NONE of the above comments (by you people) is surprising. In fact, I’d be VERY surprised if anyone gets a brilliant idea sitting at one’s office desk. As I said, there is a lot of evidence in psychology to support my statement.

I have heard that Richard Stanley also has some good results from lying in bed with back problems, but I can’t remember the details.

I’ve certainly done math while having sex (she was also an academic and insisted on mingling her work and play). Frankly, though, I didn’t agree that the two went well together.

I did also correctly solve (and frequently incorrectly) a difficult problem I’d been working at for weeks in a dream.

The office is the worst place (except when collaborating). Home is ok but walking somewhere is always the most productive. If only my dog wouldn’t pull every time she sees a goose, dog, squirrel or (especially) a duck. We might just have come back from the dog park but apparently seeing another dog requires going nuts and trying to meet it.

Bed is where I end up doing most of my work. What’s strange about that. You need to relax to think about math and lacking the money for a lazyboy type recliner the bed is the best place.

Once, for a Moore method style class, we were tasked with proving the Fundamental Theorem of Calculus. The proof of it came to me while I was in the shower.

i proved some nice results once on an airplane, stuck on the runway for an hour or more. went into “the zone” and sketched out some work on applications of euler calculus.

In the car, driving home after a 2-3 hour meeting with collaborators brainstorming a problem. Not really strange, just my pattern.

Turan solved his in a concentration camp!

I tend to get my best ideas while asleep. I’ll wake up in the middle of the night and have to write them down before I can go back to bed. Lately because of a busy schedule I’ve been doing a lot of thinking on a crowded subway. It helps me from getting annoyed by the pushing and shoving.

I hope you’re feeling better, Dick. Being sick is no fun, although a while ago I had some great, deep thoughts about P and NP while laying on the couch all day with a cold… It wasn’t quite right, but it was a big leap in understanding at least.

In shower, in the bathroom, while doing dishes, and immediately after waking up. Though none of the results have been even moderately big.

The roof of Franz Hall at UCLA. I wasn’t in psychology (the department there was in that building), but the rooftop there had a beautiful view of the Wilshire Corridor in LA, the UCLA campus, and so on. I wonder if they still leave it unlocked.

Warm wishes towards having a restful break and getting well.

I found this thread on mathoverflow quite insightful — though it is about communication rather than problem solving.

“In one extreme case, I ended up rolling around on the floor with my eyes closed in order to understand the effect of a gauge transformation that was based on this type of interaction between different frequencies. (Incidentally, that particular gauge transformation won me a Bocher prize, once I understood how it worked.) I guess this last example is one that I would have difficulty communicating to even my closest collaborators. ”

— Terry Tao

http://mathoverflow.net/questions/38639/thinking-and-explaining/38882#38882

One of my earliest math thoughts (I was 8) was sitting in the bathroom looking at the tiles and

how the 1×1 sq, the 2×2 sq, etc all were sums of the odds. What I was thinking, expressed in

terminology I did not have at the time, was

1+ 3 + 5 + … + (2n+1) = n^2.

It helped to have a visual aid.

The smiley above appeared because Bill Gasarch wrote “I was 8” inside parens, and 8 next to closing paren gives the sunglasses-smiley even though (IMHO) the posting script should recognize the parse.

Dick is feeling better today.

That’s very good to hear! 8) 8) 8)

>I was was sitting in the bathroom looking at the tiles…

The exact place where I (re)invented Cyclic Cellular Automaton, about five or ten years too late to claim any credit. It was nice to have a few days thinking I had found something though.

My insights usually occur lying in bed at night before going to sleep, though.

Maybe a myth, but I once heard that Cristian Calude from Auckland proved a theorem while on the operation table (waiting the anesthesia to make effect). The story says that he acknowledged the doctor in the resulting paper.

The daycare called to tell me I needed to provide more diapers for my son. On the way to the drugstore (15min walk each way), I realized the key idea that unlocked the problem I had been thinking about.

Professor Lipton, did you know that you can vaccinate against pneumonia? ( my doctor suggested I do because I have a lung condition which can prove very harmful if I were to catch pneumonia ).

Robert Thomason had a paper in the _Grothendieck Festschrift_, in which is the so-called Thomason-Trobaugh theorem. Is has been described thus

>On January 22 1988, he had a dream in which his recently deceased friend Thomas Trobaugh told him how to solve the final step…. Awaking with a start, he worked out the argument for the missing step. In gratitude, he listed his friend as a coauthor of the resulting paper.

This result ended up with an invitation to the 1990 ICM.

Get well soon, Prof. Lipton!

I’ve had the pneumonia vaccine, but my doctor said that you can still get it even if you are vaccinated.

Probably in sleep. There was few times when I spent hours trying to solve one math problem without success, then went to bed and solved the problem in my dream. I also like to think about problems while driving NYC Buffalo.

Probably the most popular place to prove theorems for mathematicians is at a cafe or coffee shop. Worked for Stan Ulam, Godel, and Ken Ribet, just to name a few.

Great post. Interesting read. Thanks, you always come up with these gems of posts.