What are some examples of colorful language in serious mathematics papers?

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

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 2

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 3

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 Completeness 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.

Answer 4

From S. Skewes's "On the difference $\pi(x)-\mathrm{Li}(x)$", Proc. LMS 5, 1955:

"I wish in conclusion to express my humble thanks to Professor Littlewood, but for whose patient profanity this paper could never have become fit for publication."

Answer 5

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 6

From William Thurston's "Hyperbolic Structures on 3-Manifolds I: Deformation of Acylindrical Manifolds":

Let us stick to the case that $M$ is a compact, acylindrical manifold. Then $H(M)$ is a hard-boiled egg. The egg complete with shell is $AH(M)$; it appears to be homeomorphic to a closed unit ball. $GH(M)$ is obtained by thoroughly cracking the egg shell on a convenient hard surface. Apparently no material is physically separated from the egg, but many cracks are developed -- cracks are dense in the boundary -- and at the same time, the material of the egg just inside the shell is weakened, so that neighborhood systems of points on the boundary become thinner. Finally, $QH(M)$ has uncountably many components, which are obtained by peeling off the shell and scattering the pieces all over.

Answer 7

In the Book "Introduction to lattices and order", the authors (B. A. Davey and Hilary A. Priestley) talk about ordered sets with a bottom/top. In this context, they say the following:

Computer scientists commonly choose models which have bottoms, but prefer them topless.

Answer 8

John (Horton) Conway unrelentingly gets away with colorful, even whimsical language in definitions, in explanations, in paper titles, even in some book titles (The Sensual (Quadratic) Form.) Even in SPLAG, there is the following:

"...we earnestly recommend that you use

The Best Method: guess the correct answer, and then justify it." SPLAG, p. 302

On Numbers and Games is just rife with colorful stuff. (I'm surprised no one has pointed out this elephant in the room yet.) The next to last theorem of the book is

THEOREM 99: Any short all-small game G which has atomic weight zero is infinitesimal with respect to (double-up) and dominated by some superstar.

And the last words of the book are famously

"...a certain feeling of incompleteness prompts us to add a final theorem.

THEOREM 100. This is the last theorem in this book.

(The proof is obvious.)" ONAG p. 224

Answer 9

Not from a paper but rather from a book, the first page of the introduction to G. R. Kempf's Algebraic Varieties reads:

"Algebraic geometry is a mixture of the ideas of two Mediterranean cultures. It is the superposition of the Arab science of the lightning calculation of the solutions of equations over the Greek art of position and shape. This tapestry was originally woven on European soil and is still being refined under the influence of international fashion. Algebraic geometry studies the delicate balance between the geometrically
plausible and the algebraically possible. Whenever one side of this mathematical teeter-totter outweighs the other, one immediately loses interest and runs off in search of a more exciting amusement."

Answer 10

This is taken from the seminal paper "Quantum error correction via codes over GF(4)" by A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane.

Answer 11

"Indeed, I wrote the outline of this book while wandering across India, so that, in my mind, Henkin's method is inexorably linked to the droves of wild elephants I met while crawling among the swamp plants of the preserves of Kerala; the elimination of imaginaries, to the gliding vultures above the high Himalayan peaks; and the theorem of the bound, to the naked bodies of the Mauryan women that the traveler saw on the bends of a jungle trail, before they had time to cover themselves. I dare hope only that this book will evoke similarly pleasant images in my reader; I wish it will be as pleasant a companion for you as it was for me."

From Bruno Poizat's "Model Theory". He also constantly belittles the readers of the English edition of the book. Highly recommended!

Answer 12

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 13

Diaconis and Efron wrote a paper "Testing for Independence in a Two-Way Table: New Interpretations of the Chi-Square Statistic" that was followed by 10 papers discussing their suggestion. The following is from Diaconis and Efron's rejoiner:

The critical paper that they refer to starts with a splendid colorful language:

Update: This is an additional answer too good to be missed.

Answer 14

Waldhausen once inserted a bit of music -- written out as notes on a staff -- in the final draft of one of his papers, explaining "This replaces an unnecessary axiom." The melody, by Grieg, was called "Fool's Morning Song".

Answer 15

The last paragraph of E. Artin's "Theory of Braids":

Although it has been proved that every braid can be deformed into a similar normal form the writer is convinced that any attempt to carry this out on a living person would only lead to violent protests and discrimination against mathematics. He would therefore discourage such an experiment.

Answer 16

A famous Sherlock Holmes meta-mystery is the identity of the giant rat of Sumatra. In The Adventure of the Sussex Vampire, Sherlock Holmes declares to Dr. Watson:

Matilda Briggs was not the name of a young woman, Watson, . . . It was a ship which is associated with the giant rat of Sumatra, a story for which the world is not yet prepared.

Sherlock Holmes fans have tried to figure out what the Giant Rat of Sumatra is, and how it might be related to a ship.
The solution to this greatest of all Sherlock Holmes mysteries is to be found in a mathematics book- one whose topic is Catastrophe Theory. On Page 196 of Curves and Singularities by J.W. Bruce and P.J. Giblin, we learn that the giant rat of Sumatra is in fact the family of functions $f_a(t_1,t_2)=t_1t_2(t_1-t_2)(t_1-at_2)$. Section 11.2 (Pages 196-200) of the book explains how we have established that this is indeed the giant rat of Sumatra, and elucidates why indeed the world is not yet prepared for its story. The relationship between the giant rat and the Matilda Briggs is not discussed, although we are led to suspect the worst, given that Catastrophe Theory is the book's theme.

Answer 17

There is a paper entitled Zaphod Beeblebrox's Brain and the Fifty-ninth Row of Pascal's Triangle.

Freudenthal made this quote about terminology:

A more imaginative nomenclature than one relying on overburdened terms such as "fundamental," "principal," "regular," "normal," "characteristic," "elementary," and so on is desirable. Inventors of important mathematical notions should give their inventions suggestive names. The disadvantage that good names might prevent the inventor's name from being immortalized as an adjective would be more than compensated by the advantage that this honor could not possibly be bestowed on noninventors.

(from twf:178)

Answer 18

I like George Kempf's succinct description in that same textbook, of the splitting of vector bundles on P^1 as a theorem "last proved by Grothendieck".

I never forgot Dieudonne's opinion in Foundations of modern analysis chapter 8, that defining a derivative as a number instead of a linear form, is "slavish subservience to the shibboleth of numerical interpretation at any cost."

In the section entitled "Woffle" of Miles Reid's Undergraduate algebraic geometry, he states delightfully that in the prerequisite algebraic chapter II, "the student who is prone to headaches could perhaps take some of the proofs for granted here, since the material is standard, and the author is a professional algebraic geometer of the highest moral fibre."

Answer 19

The AMS Memoirs 947 "Rock Blocks" by Will Turner is full of colorful lanuage. For example in the introduction one finds out that:

"Hannah Turner supported me financially (partly), and libidinously (entirely)."

Or:

"We choose not to spend time chomping on this old pie, since we have become aware of dishes with a more exotic, and alluring aroma."

and so on....

Answer 20

Spivak, A Comprehensive Introduction to Differential Geometry, Volume 1, p.94,

Now that we have a well-defined bundle map $TM \to T\;'M$ (the union of all $\beta_x^{-1} \circ \alpha_x$), it is clearly an equivalence $e_M$. The proof that $e_N \circ f_* = f \circ e_M$ is left as a masochistic exercise for the reader.

Volume 3, p. 103, indexed under "Idiot, any,"

These normalizations are usually carried out with hardly a word of motivation, as if they are so natural that any idiot would immediately think of doing them—in reality, of course, the authors already knew what results they wanted, since they were simply reformulating a classical theory.

From Volume 5, p.59,

We are going to begin by deriving certain classical PDE's which describe important (somewhat idealized) physical situations. The word "derive" had better be taken with a hefty grain of salt, however. What I have really tried to do is give plausible reasons why the physical situations should be governed by those PDE's which the physicists have agreed upon. I've never really been able to understand which parts of the standard derivations are supposed to be obvious, which are mathematically simplifying assumptions, which steps are supposed to correspond to empirically discovered physical laws, or even what all the words are supposed to mean.

Incidentally, Spivak gave an entertaining series of lectures on the subject of classical mechanics, whence

I haven't the slightest idea what any of this means! But I'm almost certain that it amounts to the similarity argument we have given. Aren't you glad that you aren't a mathematician of the 17th century!?

Answer 21

At the end of the introduction to Spin Glasses: a challenge for mathematicians, Michel Talagrand writes:

It is customary for authors, at the end of an introduction, to warmly thank their spouse for having granted them the peaceful time needed to complete their work. I find that these thanks are far too universal and overly enthusiastic to be believable. Yet, I must say that in the present case even what would sound for the reader as exaggerated thanks would not truly reflect the extraordinary privileges I have enjoyed. Be jealous, reader, for I yet have to hear the words I dread the most: "Now is not the time to work".

Answer 22

I don't agree with this quote by Errett Bishop (a constructivist who developed real analysis along constructive lines), but I admire its brio:

Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself.

It's an odd spin on that famous Kronecker quote about the integers and God.

Answer 23

R.Coleman writing about the Dwork Principle in Section III of "Dilogarithms, Regulators and $p$-adic $L$-functions":

"Rigid analysis was created to provide some coherence in an otherwise totally disconnected $p$-adic realm. Still, it is often left to Frobenius to quell the rebellious outer provinces".

Answer 24

I always liked Edward Burger's A Tail of Two Palindromes. It begins as follows:

Upon a preliminary perusal, this parable may appear to be about pairs of palindromes, periods, and pitiful alliteration. In actuality, however, it is the story of a real quadratic irrational number $\alpha$ and its long-lost younger sibling, its algebraic conjugate $\tilde{\alpha}$ ($\alpha > \tilde{\alpha}$). How in the dickens are all these notions connected? We begin at the beginning...

Although the conjugates $\alpha$ and $\tilde{\alpha}$ are not identical twins, unlike the two zeros of $(x - 3)^2$, they do share a common family history: they each were born of the same irreducible parent polynomial having rational coefficients,

$$P_{\alpha}(x) = P_{\tilde{\alpha}}(x) = (x - \alpha)(x - \tilde{\alpha}) = x^2 - \text{Trace}(\alpha)x + \text{Norm}(\alpha),$$

where $\text{Trace}(\alpha) = \alpha + \tilde{\alpha}$ and $\text{Norm}(\alpha) = \alpha \tilde{\alpha}$. Perhaps not surprisingly, some conjugate pairs exhibit similar personalities. But how similar can they be? And how can we detect those similarities simply by looking at $\alpha$? As we will discover as our tale unfolds, the answer - foreshadowed in the title - is encoded in what can be described as the number theoretic analogue of the DNA-sequence for $\alpha$. However, before delving into $\alpha$'s genes, we first motivate our results by weaving a lattice of algebra.

Answer 25

I always liked

$L$ takes on the character of a very thin inner model indeed, bare ruined choirs appended to the slender life-giving spine which is the class of ordinals.

from Kanamori and Magidor `The evolution of large cardinal axioms in set theory' (1978).

Answer 26

the book "Combinatorial optimization: algorithms and complexity" by Christos H. Papadimitriou, Kenneth Steiglitz contains the following exercise (19, pg 380):

The following is from the New York Times of November 27, 1979. Determine, when possible, whether each statement is (a) true, (b) false, (c) misleading, (d) equivalent to a well-known conjecture, the solution of which was probably not known to Mr. Browne.

Answer 27

In Berger's "A panoramic view of Riemannian Geometry" :

"The Cayley projective plane $\mathbb{CaP}^2$ is beautiful. In the Riemannian zoo we like to call it the panda."

Answer 28

In section 3 of

• J. Frank Adams -- Stable homotopy theory (3rd ed., LNM 3, 1969)

the author discusses two different attitudes towards what the "proper" definition of the stable homotopy category should be, which he personifies by the tortoise and the hare:

The hare is an idealist: his preferred position is one of elegant and all embracing generality. He wants to build a new heaven and a new earth and no half-measures. ... The tortoise, on the other hand, takes a much more restrictive view. He says that his modest aim is to make a cleaner statement of known theorems, and he'd like to put a lot of restrictions on his stable objects so as to be sure that his category has all the good properties he may need. Of course, the tortoise tends to put on more restrictions than are necessary, but the truth is that the restrictions give him confidence.

You can decide which side you're on by contemplating the Spanier-Whitehead dual of an Eilenberg-MacLane object. This is a "complex" with "cells" in all stable dimensions from $-\infty$ to $-n$. According to the hare, Eilenberg-MacLane objects are good, Spanier-Whitehead duality is good, therefore this is a good object: And if the negative dimensions worry you, he leaves you to decide whether you are a tortoise or a chicken. According to the tortoise, on the other hand, the first theorem in stable homotopy theory is the Hurewicz Isomorphism Theorem, and this object has no dimension at all where that theorem is applicable, and he doesn't mind the hare introducing this object as long as he is allowed to exclude it. Take the nasty thing away!

Answer 29

You might want to read http://www.ucs.louisiana.edu/~avm1260/lenstra.html for hilarious language during lecturing.

Answer 30

In a paper by Stark where he proves Gauss's conjecture that there are only nine imaginary quadratic fields where the integers form a UFD, he writes that Heegner used "classified theory".

I once met Stark and asked him if he did not correct the misprint on purpose, but he did not even remember it.