Question: What's purple and commutes? <newline/>Answer: An abelian grape.

Get it? If you don't, relax: you are perfectly normal. Mathematicians, you see, are different... and their jokes prove it.

Question: What do you get if you cross an elephant and a banana? <newline/>Answer: |elephant| **|banana| **sin &thetas; <newline/>

This is, of course, precisely the kind of thing that makes people suspect that mathematicians inhabit another world: not necessarily a rich and ethereal world of platonic beauty, rather a world of nerdy, asocial introspection and bad haircuts. That is probably no different from suspecting that footballers inhabit a world of tacky, mock-Tudor mansions and ex-model wives, or that all engineers have train sets in their bedrooms. The stereotypes of a profession should never be confused with its individuals; but they do reveal something of its culture.

That's why the 'sampling of mathematical folk humour'^{1} collected by Paul Renteln, a physicist-turned-mathematician at California State University in San Bernardino, and folklorist Alan Dundes of the University of California, Berkeley, is fascinating. It is also a little alarming. To judge from this selection, mathematicians embrace and even revel in their own caricature.

The 'abelian grape' is simply the kind of word play that many jokes employ; but a mathematician could scarcely tell it without being conscious that only fellow mathematicians would understand. All the same, there is nothing particularly unusual about the playground exclusivity of the in-joke. But what about this one:

“None of the jokes are about the usual things that people find important.”

Paul Renteln

California State University, San Bernardino

Question: How many Bourbakists does it take to change a lightbulb? <newline/>Answer: Changing a lightbulb is a special case of a more general theorem concerning the maintenance and repair of an electrical system. To establish upper and lower bounds for the number of personnel required, we must determine whether the sufficient conditions of Lemma 2.1 (Availability of personnel) and those of Corollary 2.3.55 (Motivation of personnel) apply. If and only if these conditions are met, we derive the result by an application of the theorems in Section 3.1123. The resulting upper bound is, of course, a result in an abstract measure space, in the weak- *topology.

This isn't just a dig at the way mathematicians turn a real problem into something abstract and then express it in a formalized, pedantic and bloodless language. Like many other mathematical jokes, it expresses an apparent discomfort at the artificiality of the whole enterprise. The joke seems to suggest that mathematics remains precariously suspended by its own conventions, and that if you took them away there would be nothing left.

## Telling the proof

The central convention is surely the notion of proving theorems. So the basic anxiety of the mathematician is whether he or she can find a proof, and whether that proof will withstand peer scrutiny.

If, therefore, Freud was right to claim that jokes relieve anxiety by releasing suppressed feelings, mathematicians' jokes betray a deep-seated worry about whether their proofs are, so to speak, big enough. A humorous 'guide for lecturers' outlines the many different kinds of proof available:

Proof by omission: <newline/>"The reader may easily supply the details."

Proof by appeal to intuition: <newline/>Cloud-shaped drawings frequently help here. [A jibe at fractal geometry?]

Proof by eminent authority: <newline/>"I saw Karp in the elevator and he said it was probably NP-complete."

Proof by intimidation: <newline/>"This is trivially true."

And the ploy that worked for Fermat's last theorem and Kepler's conjecture: <newline/>Proof by exhaustion: <newline/>An issue or two of a journal devoted to your proof is useful.

As a window into the psyche of the mathematician, this is revealing stuff. But "it seems unlikely to appear in stand-up comedy routines anytime soon," Renteln admits. "None of the jokes are about the usual things that people find important, such as money, sex and power," he says. "Everyone laughs at the joke of the lawyer whose arm is ripped off by a speeding truck but who is only concerned about his Rolex, but no one cares about the mathematician who is worried about the validity of his proof."

All the same, Renteln feels that "Some of the jokes reveal the profound creativity of mathematicians, which runs counter to the common stereotype of mathematics as dull, routine drudgery."

## Self-referential

As well as proof anxiety, mathematical jokes reveal a kind of utility anxiety. Engineers believe equations approximate the real world. Physicists think that the real world approximates equations. Mathematicians are unable to make the connection.

And there is a host of jokes about how mathematicians consider their job done when they have reduced a problem to a previously solved one. A mathematician awoken from his hotel bed by a fire sees a fire hose in the hall, exclaims, "Ah, a solution exists!" and goes back to bed.

All of this suggests that even mathematicians are a little disturbed by what seems to be the sheer unnaturalness of their task. The pedantic dissection and debating of scripture by the medieval scholar-monk, in many ways analogous, was at least motivated by a belief that it brought humankind closer to God. In contrast, although some mathematicians work on problems with practical applications, the ideas that circulate in the upper tiers of the maths universe seem to refer to nothing but themselves.

Even scientists are sometimes forced to wonder if this isn't just some glorious 'glass bead game'. But perhaps mathematicians should take heart from the fact that one may find maths humour in surprising places.

The Simpsons, whose writers include several maths graduates from Harvard, pokes fun not at the geekiness of mathematicians but the ignorance of the rest of the world. Lisa is the sympathetic genius confronted by Homer's bottomless stupidity:

Lisa (showing a picture to her baby sister): Ooh, look Maggie! What is that? Dodecahedron! Dodecahedron! <newline/>Homer: Lisa, I don't know what you are doing, but it's very strange, and your father is trying to worry.

Or how about this:

Michael Jackson: Homer, this is Floyd. He's an idiot savant. Give him any two numbers and he can multiply them in his head, just like that. <newline/>Homer: OK, five times nine. <newline/>Floyd: 45. <newline/>Homer: Wow!

At what some might consider the other end of the scale, Thomas Pynchon's classic book Gravity's Rainbow contains not just equations but the only maths joke I know that is based on integration:

∫ 1/(cabin) d(cabin) = log cabin + c = houseboat

If that makes you snigger, perhaps you should be worried.

California State University, San Bernardino