Or, how to hide the fact that you haven't understood a word of a mathematical seminar.

Recently, I attended a mathematical lecture given by a guest speaker where absolutely nobody, except possibly the speaker, had the remotest idea what was going on. Normally, one can absorb at least some of the preliminary definitions and follow, say, the first blackboard full of development of the theory, but on this occasion everyone was completely lost after the first definition. After the speaker had finished over an hour later to an enthusiastic round of applause, the chairman asked for questions, and, of course, there was a deathly and highly embarrassing silence. Then and there I resolved to put together a collection of universal questions for use in such situations. Such questions must sound sensible, but they are designed to cover up the total ignorance of the questioner rather than to elicit information from the speaker.

And what questions they are too! Here are just two examples:

- Can you produce a series of counterexamples to show that if any of the
conditions of the main theorem are dropped or weakened, then the theorem no
longer holds?
[

*The speaker can almost always do so - if not you may have presented him with a stronger theorem!*] - Isn't there a suggestion of Theorem 3 in an early paper of Gauss?
[

*The answer to this question is almost always YES!*]

I also like the following quotation at the end of the page.

Miscellaneous Methods [

for proving theorems] ... reduction ad nauseam, proof by handwaving, proof by intimidation, proof by referral to non-existent authorities, the method of least astonishment, the method of deferral until later in the course, proof by reduction to a sequence of unrelated lemmas (sometimes called the method of convergent irrelevancies) and finally, that old standby, proof by assignment.

- Rome Press 1979 Mathematical Calendar

The proof of their efficacy is left as a trivial exercise for the reader.

[Via a Geomblog commenter.]

But I think it might be easy to get busted with number 2 above if the speaker actually knows what he's talking about. If it IS in an early paper by Gauss, then he might reply "Which one did you have in mind?" and then the ball's back in your court. First one is pretty good though, and I've seen it used (although of course this is only a suspicion on my part).

Oh yeah other methods of proving theorems I've seen:

"Confuse the enemy" (i.e. your audience)in particular "Proof by Confusing Notation".

With easily intimidated undergrads there's also the Proof by Claim of Triviality. "Trivial! I'm not gonna bother explaining why."

Another frequent one amongs grad students, which does take a bit of talent to pull of right is "Proof By Assumption" - you sneak in an assumption in one of the steps which is equivalent to what you're trying to prove in the first place. Of course you want to make it look like it's an unrelated and innocous thing so you almost never say "assume that..." you say something like "since..." or "... implies that" and hope no one notices.

For the record - my knowledge of these devious methods comes from grading, observing and arguing with my fellow students. Not intentional personal practice (sometimes you do a Proof By Assumption even without meaning to).

(And I'm assuming the old standby "Proof by assignment" is the "It is left as an excercise for the reader...")

Posted by: radek | February 09, 2005 at 10:15 PM

Speaking of proofs by assignment, you might enjoy reading the following, particularly the sections on Bourbaki titled "Gaps everywhere" and "Even the theorems may be omitted":

http://www-didactique.imag.fr/preuve/Newsletter/000910Theme/000910ThemeUK.html

(Just triple click to select the whole line, even if you can't see all of it.)

Posted by: Abiola Lapite | February 09, 2005 at 10:23 PM

"Then a miracle occurs" is still my favorite fake mathematical proof: http://home.earthlink.net/~cdtavares/direcway/miracle.jpg

Posted by: Andrew | February 10, 2005 at 12:43 AM

How about:

"Could you just run that past us again?"

Posted by: Rafe | February 10, 2005 at 12:55 AM

That's a good article. Somewhat on topic I remember reading somewhere that Weierstrass always insisted on explicitly proving (almost) every step of a theorem, even if they really were trivial because he distrusted "intuition" so much. He called it rigor, others thought it was a waste of time. In the end the payoff was the discovery of everywhere-continuous but nowhere-differentiable functions, which others assumed didn't exist.

Posted by: radek | February 10, 2005 at 12:55 AM

"Could you just run that past us again?"

The rules of the game demand that one make no admission of fallibility; saying something like this would be giving everyone else an opportunity to snicker behind one's back: "Poor guy, it went completely over his head!" (Not that anyone else is necessarily any better ...)

Posted by: Abiola Lapite | February 10, 2005 at 12:59 AM

"In the end the payoff was the discovery of everywhere-continuous but nowhere-differentiable functions, which others assumed didn't exist."

This is one good reason to have all the "Counterexamples in XYZ" books on the bookshelf. I once had a friend who came to me because a professor'd set homework requiring that the students prove that a certain result was universally true; the thing was, after a while of thinking about the problem, I had the feeling that it wasn't so, and sure enough, "Counterexamples in Analysis" had a demonstration of this hunch of mine, completely confounding the expectations of the professor who set the problem. I chuckle to think of all the poor saps who managed to turn in "proofs" of the bogus result regardless.

Posted by: Abiola Lapite | February 10, 2005 at 01:02 AM

That "poor guy" though is a valuable public good. You want one in every classroom, seminar, etc. You just don't want it to be you. Clearly a market failure (insert toy model here).

Posted by: radek | February 10, 2005 at 01:22 AM

One of the plus sides of doing physics as a grad student is that all those tricks for not really proving things, first and foremost assuming your conclusion, fit so well. At least, the school I attend doesn't have physics professors up enough on their math to catch things like that. If they could, they'd assume it was the student's ability to see something without having to work it out.

Posted by: alek | February 10, 2005 at 07:13 AM