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

Jeremy Avigad in Computability and Incompleteness (2002)

... in a sense, computability is similar to the Supreme Court Justice Stewart's characterization of pornography, it may be hard to define precisely, but I know it when I see it."

Not quite from a 'paper' but floating around in the net:

"Who has not been amazed to learn that the function $y = e^x$, like a phoenix rising from its own ashes, is its own derivative?" -- Francois le Lionnais

Answer 2

Although the article itself is standard, I've always been fond of the title (and contents) of the Burstall & Hertrich-Jeromin paper Harmonic maps in unfashionable geometries (arXiv:math/0103162).

Answer 3

Two that I like can be found on p. 756 of Edgar R. Lorch's Amer. Math. Monthly paper "Continuity and Baire functions" (Volume 78, 1971, pp. 748-762):

[...] the reader is reminded of the fact that sets which are of type $F_{\sigma\delta\sigma}$ or $G_{\delta\sigma\delta}$ and not of lower type—with respect to any of the classic topologies—are very thinly scattered through the literature. In fact, looking for them is almost like hunting for unicorns.

IV. In order to penetrate further into this subject it is necessary to give an appropriate structure to T, the set of all coherent topologies. As mentioned earlier, this appropriate structure is itself a topology. This circumstance, that a collection of topologies is topologized, may seem a bit incestuous.

Answer 4

I just came across a paper of Waldhausen (On Irreducible 3-manifolds Which are Sufficiently Large) where he says "Frequently, a proof involves a sequence of constructions, each of which in turn involves alterations of some things. To avoid an orgy of notation in such cases, we often denote the altered things by the old symbols."

Answer 5

In the huge and austere book "Groupes algébriques" by M. Demazure and P. Gabriel we find in the last pages a "Dictionaire "Fonctoriel"", a dictionary of terms related to category theory where they have:

Satellite- Voir Cartan-Eilenberg et non Paris-Match.

Answer 6

"Now life is too short to work over the integers all of the time, ..."

J. Morava, On the complex cobordism ring as a Fock representation.

Answer 7

Chapter 2 ("Outline of the Upcoming Proof" by Arunima Ray) of The Disc Embedding Theorem book begins:

We present an outline of the forthcoming proof of the disc embedding theorem, to orient the reader before we begin. The nonorientable reader is requested to pass to their orientation double cover before continuing.

Answer 8

From the references of the Wikipedia page on large countable ordinals:

Wolfram Pohlers, Proof theory, ... (for Veblen hierarchy and some impredicative ordinals). This is probably the most readable book on large countable ordinals (which is not saying much).

Answer 9

In "Théorie algébrique des nombres" (in French and a great book about Dedekind rings and basic number field theory btw), Samuel frequently uses "Mézalor" as a phonetic replacemecont for "Mais alors". I guess you could translate it as "Butzen" instead of "But then". I think it was just a geeky "wink wink" at other mathematicians considering how much that locution was used in "dévissage" but I liked it anyway.

Answer 10

Pretentiousness is repulsive. (see page 9)

Answer 11

Sorry for blowing my own horn: if you read both French and English, you will probably appreciate the title of section 4 in http://archive.numdam.org/ARCHIVE/AIF/AIF_1997__47_4/AIF_1997__47_4_1195_0/AIF_1997__47_4_1195_0.pdf

Answer 12

There is the famous (and with contradictory interpretations) cry from Jean Dieudonné "à bas Euclide !", "Down with Euclide !". His books and prefaces are good sources for strong (and dated) opinions on what was "good" or "productive" mathematics and what was not.

Doron Zeilberger papers may contain also some colorful language.

Answer 13

No-one seems to have mentioned Joe Diestel (although "colorful" is maybe the wrong word-- perhaps because of my English interpretation of what this means-- but "lighthearted" is correct). For example, "Sequences and Series in Banach Spaces" we have the section on "Mathematical Sociology" when introducing Ramsey Theory (to talk about one set "accepting" or "rejecting" another). It's hard to pick out any particular quote, but the whole book is somehow far more lively and informal (without, somehow, even managing to be less than 100% accurate) than most maths books.

Answer 14

There is a hidden 4-letter obscenity on p. 95 of Set Theory and the Continuum Hypothesis by Paul J. Cohen. I wouldn't have noticed it even if I'd read the book, but it was pointed out by a dirty-minded reviewer. I like to think it was accidental, but who knows?

Answer 15

From the opening line of Fleissner and Kunen, "Barely Baire Spaces", Fundamenta Mathematicae Vol. 101, Issue 3, 1978:

If the reader will bear with us, we will bare the facts about barely Baire spaces.

Answer 16

Not really from a published paper but from A. Douady's state thesis. In the original:

Soit $X$ un espace analytique complexe. Le but de ce travail est de munir son auteur du grade de docteur-ès-sciences mathématiques et l'ensemble $H(X)$ des sous espaces analytiques compacts de $X$ d'une structure d'espace analytique.

Roughly translated to English:

Let $X$ be a complex analytic space. The goal of this work is to furnish the author with the degree of doctorate in mathematics, and $H(X)$, the set of compact analytic subspaces of $X$, with an analytic structure.

Answer 17

A new book on sieve methods is bizarrely called Opera de Cribro with chapter subtitles in an operatic theme.

Answer 18

Milne's web page contains a number of amusing anecdotes- https://www.jmilne.org/math/apocrypha.html

Answer 19

Growing Your Balls

A paper was presented at the FOCS '10 conference with title How to Grow Your Balls, see also the comments from the blog linked below; a tutorial was also given, with the more subtle title How to Grow Your Lower Bounds.

The whole story, and the reaction of the conference committee was priceless, here is just a taster quote from the blog of the first author, Mihai Pătraşcu (whose tragic fate is a story of its own):

If you read the paper, the algorithms repeatedly grow balls (aka shortest path trees) around vertices of the graph. After obsessing about growing these balls for more than a year, I found it natural to name the paper "How to Grow Your Balls". At least it allowed me to begin various talks by telling the audience that, "This is a topic of great economic importance; I receive email about it almost every day."

Answer 20

From Geoffrey Grimmett's monograph on Random Processes on graphs:

Within the menagerie of objects studied in contemporary probability theory, there are a number of related "animals" that have attracted great interest amongst probabilists and physicists in recent years.

Answer 21

How come no-one has mentioned Bloch's review of Milne's "Étale cohomology" yet?

Answer 22

After reading the bulk of "Smooth Manifolds and Observables" by Jet Nestruev I was very confused why I couldn't find any other work of Jet Nestruev. Then I eventually read the preface and this part put a huge smile on my face:

Unlike a well-known French general, Jet Nestruev is a civilian and his personality is not veiled in military secrecy. So it is no secret that this book was written by A. M. Astashov, A. B. Bocharov, S. V. Duzhin, A. B. Sossinsky, A. M. Vinogradov, and M. M. Vinogradov

Answer 23

The lecture notes "Introduction to Modular Representation Theory" by Zhiyuan Bai (https://zb260.user.srcf.net/notes/III/modrep.pdf) contain this gem:

$$\textbf{Lemma 5.1.} \text{Gal}(K/\mathbb{Q}) \cong (\mathbb{Z}/m\mathbb{Z})^\times. \\\text{Proof. Ask a toddler on the street.}$$

Answer 24

From Andretta's "Large cardinals and iteration trees of height $\omega$" (Annals of Pure and Applied Logic vol. 54, 1990):

We have tried to make this paper self-contained but we could not perform miracles.

Answer 25

I was reading Mac Lane (co-discoverer of category theory)'s paper today and it was very amusing how he lamented the irrelevance of modern set theory.

From To the Greater Health of Mathematics, The Mathematical Intelligencer volume 10 (1988) pages 17–20, doi:10.1007/BF03026636:

I doubt that set theory is the ultimate foundation of real mathematics. One friend puts it more pungently: A decision via large cardinals has the same ontological force as an explanation of excessive teen-age pregnancies by the axiom: "'Handsome Martian men in UFO's are frequent flyers in our friendly skies." Maybe to a never-never land?

The paper then listed 5 important questions that logicians and set theorists were neglecting.

Answer 26

On the preface of M. C. Irwin's Smooth Dynamical Systems (Academic Press, 1980) there's this little gem:

Similarly, there is not much emphasis on modelling applications of the theory, except in the introduction. I feel more guilty about ducking transversality theory, and this is, in part, due to a lack of steam. However, after a gestation period that would turn an Alpine black salamander green with envy, it must now be time to stand and deliver.

Answer 27

Given a homomorphism $f$, one must always salivate, like Pavlov's dog, by asking for its kernel and image; once these are known, there is a normal subgroup and $f$ can be converted into an isomorphism.

This colorful remark can be found in J. J. Rotman's An introduction to the theory of groups (last paragraph on page 35 of the 4th ed. of the book)... I recalled it recently as I was reading of the 172nd anniversary of Ivan Pavlov's birth.

Answer 28

I once reviewed a book titled "Applied Partial Differential Equations" by Ockendon et al. In the book, Christoffel's name appeared spelled as "Christawful." It may have been just a "Christawful" typo as I implied in the review. On the other hand, I always wondered if it was an intended joke which a junior author managed to sneak through.

Answer 29

Though not a paper, this line from page 51 of Robert Burckel's "An Introduction to Classical Complex Analysis: Volume 1" comes to mind often:

The theory to be erected here did not spring fully-armed from the head of Zeus, but condensed gradually out of the primordial vapors.

Answer 30

The writing in the book Hypergeometric Functions, My Love, by Masaaki Yoshida, has a lot of personality and is completely chock-full of colourful language. For example, the preface contains an extended metaphor where the author speaks about the modular interpretation of the configuration space $X(2,4)$ as his lover.

You might ask why this story attracts me so much. Before answering this, may I pose a question to you? Can you give a logical answer to the question of why your friend (wife, husband or some such person) attracts you so much? Your answer may be "I just like her/him." My answer is similar, but if you insist that I explain further, I (a man) would add "she has many nice friends, who make my life more enjoyable." I fell in love with the story of the modular interpretation of the configuration space $X(2,4)$. This story has many friends, i.e. it is related to various kinds of mathematics such as differential equations, differential geometry, configuration spaces, invariant theory, elliptic curves, K3 surfaces and their moduli, uniformization, geometry of bounded symmetric domains, arithmetic subgroups, modular forms, and combinatorics. This story was originated by Gauss and Jacobi. Other modular interpretations of $X(2, 4)$ were given by H.A. Schwarz. Terada and Deligne-Mostow later made several modular interpretations of the configuration spaces $X(2, n)$ $(5 \le n \le 8)$ of $n$ points on ${\bf P}^1$. These interpretations have been studied by a number of authors. I do not like too much (although I do not hate, and sometimes I enjoy) to share my girl friend with so many boys.