Question

The popular MO question "Famous mathematical quotes" has turned up many examples of witty, insightful, and humorous writing by mathematicians. Yet, with a few exceptions such as Weyl's "angel of topology," the language used in these quotes gets the message across without fancy metaphors or what-have-you. That's probably the style of most mathematicians.

Occasionally, however, one is surprised by unexpectedly colorful language in a mathematics paper. If I remember correctly, a paper of Gerald Sacks once described a distinction as being

as sharp as the edge of a pastrami slicer in a New York delicatessen.

Another nice one, due to Wilfred Hodges, came up on MO here.

The reader may well feel he could have bought Corollary 10 cheaper in another bazaar.

What other examples of colorful language in mathematical papers have you enjoyed?

Answer 1

I don't even know if this is intentional or not. In his book Teichmuller theory, John Hubbard frequently references the category of Banach Analytic Manifolds. He adhers to the convention that a category be referenced by the concatenation of the first three letters of each constituent word, making the category in question BanAnaMan. This still cracks me up to this day.

Answer 2

Does merely transposing two words count? "It is also hard not to show that ..." [Arnold W. Miller, "Some Properties of measure and category," Trans. A.M.S. 266, 1981, p. 106]

Andreas Blass

Answer 3

From the ground-breaking paper: On the complexity of omega-automata by Muli Safra

Answer 4

The reader who makes it to the later chapters of M. N. Huxley's Area, Lattice Points and Exponential sums is rewarded with the following gem:

"If mathematics were an orchestra, the exponentials would be the violins. The $\rho(t)$ would be the flutes; they are introduced by the exponentials. The Poisson summation formula would be the tuba: powerful, but ridiculous when used too much"

Answer 5

Andre Weil (Oeuvres, vol. 2, page 558) purporting to be R.Lipschitz writing from Hades:

"Unfortunately, it appears that there is now in your world a race of vampires, called referees, who clamp down mercilessly upon mathematicians unless they know the right passwords. I shall do my best to modernize my language and notations, but I am well aware of my shortcomings in that respect ; I can assure you, at any rate, that my intentions are honourable and my results invariant, probably canonical, perhaps even functorial.But please allow me to assume that the characteristic is not 2"

Answer 6

Chang and Keisler's book on Model Theory is dedicated to all those model theorists who have never dedicated a book to themselves.

Answer 7

This is a little off the mark (from a textbook), but Exercise VIII.8.3 of Sarason's [Notes on] Complex Function Theory is:

Stand straight with feet about one meter apart, hands on hips. Bend at the waist, knees straight, and touch left foot with right hand. Straighten. Bend again and touch right foot with left hand. Straighten. Repeat 15 times.

Answer 8

Frank Adams was notorious for slipping little gems of humour into his paper and books. For instance, from his book, "Infinite Loop Spaces,"

(p. 128)

The reader may expect me to say something about "double coset fomulae." I shall indeed; I advise you to avoid them.

(p. 131)

Of course, this still leaves the question: what do you say to the algebraist who loves double cosets and insists that this is the same thing really? I suggest that you smile politely and say that you are maxmimizing your chance of finding a helpful and congenial interpretation of the double cosets. There is no need to say that the best interpretation is one which allows you to avoid mentioning the (expletive deleted) things at all.

For further entertainment, look at the entry [85] in the bibliograph, and look at "jokes" in the index.

Answer 9

At the risk of blowing my own horn, I will mention the line in the book, Category Theory for Computing Science" by Charles Wells and me. After mentioning the Russell paradox and how to avoid it, we say, "This prophylaxis guarantees safe sets." I caught at least one colleague rolling on the floor laughing, but only after reading it aloud.

Michael Barr

Answer 10

P. T. Johnstone's On a Topological Topos has some interesting choices of words. Sometimes the words are discussed in parenthetical notes.

([...] we are tempted also to introduce the term 'consequential space' for an arbitrary object of $\mathcal{E}$, apart from a slight reluctance to give the name 'space' to an object of a category whose underlying-set functor is not faithful—and, we must admit, the fear that somebody will at once invent a notion of 'inconsequential space'.)

Sometimes there is no more than a reference to existing literature.

The rest of the proof of Theorem 5.1 is a fairly straightforward woozle-hunt (Milne [27])

Reference [27] is, as you may have guessed, A. A. Milne's Winnie The Pooh.

Answer 11

The paper "Division by three" by Peter Doyle and John Conway has a wealth of colorful language including:

"If the arrows are good, straight, American arrows, it is very natural for each arrow to dream of marrying the arrow next door."

and

"Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel axioms for set theory are necessarily even consistent. Indeed, we’re somewhat doubtful whether large natural numbers (like $80^{5000}$ , or even $2^{200}$) exist in any very real sense, and we’re secretly hoping that Nelson will succeed in his program for proving that the usual axioms of arithmetic—and hence also of set theory—are inconsistent. (See Nelson [6].) All the more reason, then, for us to stick with methods which, because of their concrete, combinatorial nature, are likely to survive the possible collapse of set theory as we know it today."

Answer 12

A paper of David Bachman-Cooper-White describes a proof that a hyperbolic 3-manifold containing large embedded balls has large Heegaard genus. As they say at the end of the introduction,

a proper subset of the authors wish to subtitle this paper “Big balls imply big genus”

whch is indeed the best way to memorize the result.

Answer 13

From Ravi Vakil's notes "Foundations of algebraic geometry."

He says about spectral sequences:

"They have a reputation for being abstruse and difficult. It has been suggested that the name 'spectral' was given because, like spectres, spectral sequences are terrifying, evil, and dangerous. I have heard no one disagree with this interpretation, which is perhaps not surprising since I just made it up."

Answer 14

In the acknowledgment to Thomason and Trobaugh's paper on localization in algebraic K-theory, Thomason writes:

The first author must state that his coauthor and close friend, Tom Trobaugh, quite intelligent, singularly original, and inordinately generous, killed himself consequent to endogenous depression. Ninety-four days later, in my dream, Tom's simulacrum remarked, "The direct limit characterization of perfect complexes shows that they extend, just as one extends a coherent sheaf." Awakening with a start, I knew this idea has to be wrong, since some perfect complexes have a non-vanishing $K_0$ obstruction to extension. I had worked on the problem for 3 years, and saw this approach to be hopeless. But Tom's simulacrum had been so insistent, I knew he wouldn't let me sleep undisturbed until I had worked out the argument and could point to the gap. This work quickly led to the key results of this paper. To Tom, I could have explained why he must be listed as a coauthor.

Michael Harris has a rather interesting literary analysis of this quote on his webpage.

As far as I know, this was Trobaugh's only foray into mathematics.

Answer 15

According to http://en.wikipedia.org/wiki/Chandler_Davis, page 181 in Chandler Davis' "An extremum problem for plane convex curves" (in Victor L. Klee's "Convexity", Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 1963), one has

"Research supported in part by the Federal Prison System. Opinions expressed in this paper are the author's and are not necessarily those of the Bureau of Prisons."

The paper was written while its author was in prison for refusing to cooperate with the House Unamerican Activities Committee.

The quote can be seen in Google books.

Answer 16

Jon Barwise's Admissible Sets and Structures contains the following on page 69:

When used in a class or seminar, section 6 should be supplemented with coffee (not decaffeinated) and a light refreshment. We suggest Heatherton Rock 'Cakes. (Recipe: Combine 2 cups of self-rising flour with 1 t. allspice and a pinch of salt. Use a pastry blender or two cold knives to cut in 6 T butter. Add 1/3 cup each of sugar and raisins (or other urelements). Combine this with 1 egg and enough milk to make a stiff batter (3 or 4 T milk). Divide this into 12 heaps, sprinkle with sugar, and bake at 400 °F. for 10 — 15 minutes. They taste better than they sound.)

There is a response to this (with stronger ingredients) somewhere in Aki Kanamori's The Higher Infinite but I forgot exactly where. Later in that book, on page 289, Kanamori writes:

But first, a respite from the rigors: Instead of yet another recipe, we offer the following chess problem (M. Henneberger, first and second prize, "Revista de Sah" 1928):

White. King on b1, Rooks on b7 and c7, and Bishop on b5.

Black. King on a8, Rook on a3, and Pawn on f2.

White to play and win.

Send complete solutions to the author for a small prize.

Answer 17

From Strichartz's A Guide to Distribution Theory and Fourier Transforms:

(p.2) "You have almost seen the entire definition of generalized functions. All you are lacking is a description of what constitutes a test function and one technical hypothesis of continuity. Do not worry about continuity--it will always be satisfied by anything you can construct (wise-guys who like using the axiom of choice will have to worry about it, along with wolves under the bed, etc)."

Answer 18

I was always amazed that Clifford Truesdell could get away with a quote like this:

Nowadays, when the common student seeks a secure berth by grafting himself upon some modest little professor whom he regards as prone to foster painlessly his limaceous glide toward a dissertation not too strenuous or, even better, to draught it for him, tradition is moribund (...)

This is from his introduction to the selected papers of W. Noll. Admittedly, Truesdell was the chief editor himself, and could write therefore whatever he wanted, but it's still pretty strong. Felt too close to home when I first read it as a graduate student!

Answer 19

While this is not necessarily the meaning of "colorful" intended by the OP, there is probably no better way to find out what motivated the editors of the American Mathematical Monthly to reiterate a damnation by publishing the following erratum, than posting it here:

American Mathematical Monthly volume 77 (1970) p. 78

Answer 20

One of my favorites has always been Hermann Weyl's "... the gods have imposed upon my writing the yoke of a foreign language that was not sung at my cradle" (in the preface to his classic text `The Classical Groups: their Invariants and Representations') to excuse his supposedly poor English. This was a conceit of course---as the quote itself shows his command of English was impeccable.

Answer 21

From Tilman Bauer's "p-compact groups as framed manifolds:"

For our purposes, it is enough to work in the category of so-called naive G-spectra. I will drop the word “naive” since it will make this work appear so puny.

And in Tilman's paper with Natalia Castellana, "Adjoint spaces and flag varieties of p-compact groups:"

This comment is only meant to intimidate the reader and is insubstantial for what follows.

Answer 22

From Vector Calculus, Linear Algebra, And Differential Forms. A Unified Approach. by Hubbard:

When a matrix is described, height is given first, then width: an m x n matrix is m high and n wide. After struggling for years to remember which goes first, one of the authors hit on a mnemonic: first take the elevator, then walk down the hall.

Answer 23

You'll find a whole host of colourful language and allusions scattered throughout the works of Kato. To quote just one example from his Lecture on the approach to Iwasawa theory for Hasse Weil L-functions via $B_{dR}$:

Where is the homeland of zeta values to which the true reasons of celestial phenomena of zeta values are attributed ? How can we find a galaxy train to approach it, which runs through the galaxy of p-adic zeta elements and whose engine is the theory of p-adic periods ? I imagine that one coach of the train has the name 'explicit reciprocity law of p-adic Galois representations'.

Answer 24

A few days ago, some colorful quotes from Michael Spivak's A Comprehensive Introduction to Differential Geometry were posted here. Yesterday I noticed they were missing, which is a great loss, so I am attempting to restore them. The only one I remember immediately is

Bourbaki has apparently decided that the theory of manifolds has now entered that domain of "dead" mathematics to which he hopes to give definitive form. In this summary of results the corpse is laid out to public view; the complete autopsy is eagerly awaited.

(Volume 5, p.608, of the 2nd edition, 1975)

If anyone recalls some others, please add them.

Answer 25

In his article "Lectures on Mixed Motives" (Proceedings of Symposia in Pure Mathematics, Volume 62.1, 1997), Spencer Bloch writes:

"My experience with these lectures suggests that motives are like onions; they are complicated, multi-layered objects, and any attempt to cut too quickly to the heart of the matter can leave the audience in tears."

I've actually gotten some mileage out of this analogy in my teaching. When doing the first iterated chain-rule examples in calculus classes, for example, I advocate working "from the outside in" as opposed to the other way around, and employ a variant of Bloch's statement.

Answer 26

Masaki Kashiwara writes, in the introduction to his Systems of microdifferential equations:

Although this was a course at a French university, several examples of hyperfunctions are given just before Theorem 3.2.45.

and shortly after that:

The reader is adviced not to commit seppuku instantly if he feels he does not quite understand 2. of chapter 1.

Answer 27

From Donagi and Smith "The Structure of the Prym Map":

Wake an algebraic geometer in the dead of night, whispering: "27". Chances are, he will respond: "lines on a cubic surface".

Answer 28

I must post some more examples from Frank Adams. I recommend reading the last section of his paper "Finite H-Spaces and Lie Groups", which contains a letter to the reader written in the voice of the exceptional lie group E8. Two excerpts :

"This is as if one were to award a title for drinking beer, having first fixed the rules so as to exclude all citizens of Heidelberg, Munich, Burton-on-Trent, and any other place where they actually brew or drink much of the stuff."

"In the second place, to consider the question at all reveals a certain preoccupation with ordinary cohomology. Any impartial observer must marvel at your obsession with this obscure and unhelpful invariant."

Answer 29

The English translation by Kenji Iohara of Minoru Wakimoto's "Infinite dimensional Lie algebras" is as colourful as it gets, I think. For example on page 8

Namely, we can think of an element of U(A) as an element of A. But since U(A)and A are not isomorphic, this thinking is not an identification but a lonely unrequited love.

Or on page 26

An elegant shape of the left half of Mt. Fuji reflected in the surface of a lake, this is the proportion of the finite-dimensional representations of $\mathfrak{sl}(2,\mathbb{C})$.

Or on page 27

Since ancient times, it has been the charm of music that has soothed the fiercest warriors (or samurai). This law seems to be universal in the physical universe, and it is also true in the world of Lie algebras.

My personal favourite is on page 289

Moreover, the conformal superalgebra (CSA for short) has recently been discovered by Kac, and its definition is given in 2.7 of [K5]. This representation theory has been started in [CK], It is like a matsutake mushroom derived from a big tree called a vertex operator algebra, and it is a portable version of a super-conformal algebra and a vertex operator algebra. There is an experimental report saying that it is more delicious to munch a matsutake mushroom than its landlord- i.e. a Japanese red pine.
Let us munch it a bit.

Unfortunately perhaps, the language is not nearly as colourful in the original Japanese (it's just an outstandingly good book), and is an artifact of the translation. I've long had a dream of doing a more sober translation... but I suppose that Iohara's translation is not without its charm. Anyway, the colourful language is in my opinion is to be attributed to Iohara rather than to Wakimoto.

Answer 30

In this MO answer, I mentioned Arnold Miller's lecture notes, where he gives an entertaining account of the MM proof system (for Micky Mouse), having as axioms all validities and modus ponens as the only rule of inference. Although it is easy to prove the Completness theorem from Compactness in this system, it is nevertheless a kind of joke system, since the set of validities is not a decidable set, and so we would be fundamentally unable to recognize whether something is a proof or not in this system. Miller uses this example to illustrate the point as follows:

The poor MM system went to the Wizard of OZ and said, “I want to be more like all the other proof systems.” And the Wizard replied, “You’ve got just about everything any other proof system has and more. The completeness theorem is easy to prove in your system. You have very few logical rules and logical axioms. You lack only one thing. It is too hard for mere mortals to gaze at a proof in your system and tell whether it really is a proof. The difficulty comes from taking all logical validities as your logical axioms.” The Wizard went on to give MM a subset Val of logical validities that is recursive and has the property that every logical validity can be proved using only Modus Ponens from Val.

And he then goes on to describe how one might construct Val, and give what amounts to a traditional proof of Completeness.