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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?

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    $\begingroup$ Latest paper, my co-author put in "but we will choose a more painful way, because there is nothing like pain for feeling alive" but the referee jumped on it. $\endgroup$
    – Will Jagy
    Commented Apr 23, 2010 at 5:09
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    $\begingroup$ Maybe I should expand the question to include colorful language cut from serious mathematics papers :) $\endgroup$
    – John Stillwell
    Commented Apr 23, 2010 at 5:18
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    $\begingroup$ By the way, your remark reminds me of another in a similar spirit that made it into the Princeton Companion. In his article on algebraic geometry, János Kollár says of stacks: "Their study is strongly recommended to people who would have been flagellants in earlier times." $\endgroup$
    – John Stillwell
    Commented Apr 23, 2010 at 7:49
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    $\begingroup$ I was actually rather surprised recently by a referee who did not know the phrase “red herring”, and had to look it up. He insisted that we change it to something more understandable. It makes me wonder how much “colourful” language is weeded out by referees, and whether the mathematical literature is poorer for it. $\endgroup$
    – Harald Hanche-Olsen
    Commented Apr 24, 2010 at 2:31
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    $\begingroup$ @Harald: If you intend your mathematical papers to be read by a wide range of readers, then write them in simple language, suitable for those who are relative beginners in English. I remember reading long ago some metaphoric phrase in a mathematics research paper, then imagining students all over the world getting out their English dictionaries, looking it up, and still not understanding what it meant. (I no longer remember what the phrase was, just this reaction to it.) $\endgroup$
    – Gerald Edgar
    Commented Apr 24, 2010 at 15:43

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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

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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).

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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."

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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.

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"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.

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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.

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.

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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.

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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).

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    $\begingroup$ Entertaining (and I'm sure we all know books like that in our respective fields)... but aren't we looking for instances of such language in serious math(s) papers, the point being to find levity defying gravity? $\endgroup$
    – Yemon Choi
    Commented Mar 11, 2011 at 1:10
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    $\begingroup$ @Yemon - you're right, of course, but the usually stuffy wikipedia (obligatory xkcd comic should be immediately obvious to the reader) doesn't have the freedom that an author has. The author is only constrained by personal adherence to social norms in writing, whereas wikipedia is Ahem controlled Ahem constantly edited towards improvement and encyclopedic style. :) $\endgroup$
    – David Roberts ♦
    Commented Mar 11, 2011 at 2:11
  • $\begingroup$ Fair point, David! $\endgroup$
    – Yemon Choi
    Commented Mar 11, 2011 at 5:42
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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.

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Pretentiousness is repulsive. (see page 9)

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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

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  • $\begingroup$ Veuillez expliquer le blague? $\endgroup$
    – Yemon Choi
    Commented Aug 23, 2011 at 0:51
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    $\begingroup$ In French, Jolissaint is pronounced as "joli seins", which translates as "nice tits" in English. $\endgroup$
    – ACL
    Commented Aug 23, 2011 at 6:44
  • $\begingroup$ Oh, for some reason I had "seins" and "reins" mixed up in my head earlier... $\endgroup$
    – Yemon Choi
    Commented Aug 23, 2011 at 9:47
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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.

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    $\begingroup$ > Doron Zeilberger papers may contain also some colorful language. Is this perhaps like saying that oceans are sometimes wet? $\endgroup$
    – LSpice
    Commented Apr 25, 2010 at 4:38
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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.

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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?

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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.

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    $\begingroup$ I tried putting something in a paper once about doing something Baire-handed, but they took it out. $\endgroup$
    – Gerry Myerson
    Commented Sep 14, 2021 at 12:58
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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.

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    $\begingroup$ This was mentioned in this old thread, it's nice to have the full quotation here. $\endgroup$
    – Robert Furber
    Commented Sep 14, 2021 at 11:25
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A new book on sieve methods is bizarrely called Opera de Cribro with chapter subtitles in an operatic theme.

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

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  • $\begingroup$ Several books of anecdotes and apocrypha also exist, with the imaginative titles 'Mathematical Apocrypha' and (if I recall correctly) 'More Mathematical Apocrypha'. $\endgroup$
    – Ketil Tveiten
    Commented Jan 12, 2011 at 9:13
  • $\begingroup$ The second one is called Mathematical Apocrypha Redux. $\endgroup$
    – Pandora
    Commented Jan 19, 2011 at 17:28
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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.

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

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    $\begingroup$ The whole review is a must-read... $\endgroup$
    – darij grinberg
    Commented Dec 5, 2011 at 5:03
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    $\begingroup$ I would like to upvote this for being outrageous, but that would be giving it praise it does not deserve. $\endgroup$
    – Ryan Reich
    Commented Dec 12, 2011 at 22:41
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    $\begingroup$ Right...thanks, but I doubt I'd have any more fun reading the review than I did reading that quote. $\endgroup$
    – Elizabeth S. Q. Goodman
    Commented Dec 13, 2011 at 4:47
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    $\begingroup$ I am a bit shocked that something like this was printed in BAMS as late as in the earlier 80s. $\endgroup$
    – user9072
    Commented Dec 17, 2011 at 13:09
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    $\begingroup$ It may well be colourful; it strikes me as crass. $\endgroup$
    – Yemon Choi
    Commented Dec 18, 2011 at 3:22
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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

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    $\begingroup$ @ZachTeitler, was that meant to be a link to the answer? $\endgroup$
    – LSpice
    Commented Sep 14, 2021 at 2:56
  • $\begingroup$ See also mathoverflow.net/a/62966 (an answer to a question about pseudonyms of famous mathematicians). $\endgroup$
    – Zach Teitler
    Commented Sep 16, 2021 at 14:46
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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."

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

enter image description here

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

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    $\begingroup$ The notes are typed by ZYB but I suspect this line comes from the original lecturer... $\endgroup$
    – Yemon Choi
    Commented Aug 12, 2023 at 22:55
  • $\begingroup$ @YemonChoi Ah, thanks for pointing this out. I just saw the footnote in the notes now: "∗Based on the lectures under the same name taught by Prof. S. Martin in Michaelmas" $\endgroup$
    – Loreno Heer
    Commented Aug 13, 2023 at 11:31
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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.

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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.

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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.

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    $\begingroup$ Rotman's group theory book's index has the following cycle of references: "Navel, Morris (see Pippik, Moishe)"; "Pippik, Moishe (see Nombril, Maurice)"; "Nombril, Maurice (see Ombellico, Mario)"; "Ombellico, Mario (see [a name in written in Cyrillic])"; and that last name refers back to Navel, Morris. My friend and I emailed Rotman to ask him about this and he said it came from a joke among his childhood friends. $\endgroup$
    – Zach Teitler
    Commented Sep 28, 2021 at 3:58
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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.

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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.

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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.

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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.

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