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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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closed as off topic by Loop Space, Felipe Voloch, Kevin Buzzard, Mark Sapir, quid Dec 25 '11 at 19:17

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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. – Will Jagy Apr 23 '10 at 5:09
Maybe I should expand the question to include colorful language cut from serious mathematics papers :) – John Stillwell Apr 23 '10 at 5:18
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." – John Stillwell Apr 23 '10 at 7:49
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. – Harald Hanche-Olsen Apr 24 '10 at 2:31
@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.) – Gerald Edgar Apr 24 '10 at 15:43

109 Answers 109

This quote is taken from the paper "How to write a proof" by Leslie Lamport. The paper is about a system to write mathematical proofs in a more formal way. (Of course I do not share the opinion expressed in this paragraphs.)

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In what way is this language colorful? It's a strongly expressed opinion, but that doesn't make it colorful. – Todd Trimble Dec 16 '12 at 15:18

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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Daniel Mathews, Chord diagrams, contact-topological quantum field theory and contact categories, Algebraic & Geometric Topology 10 (2010) 2091–2189. Section 2.2.2, Page 2122:

We give a baseball interpretation of the partial order $\preceq$. The $m$th symbol in a word $w$ is the $m$th inning. The sum of the first $m$ symbols is the score after $m$ innings. The relation ${w_1\preceq w_2}$ means precisely that after every inning, ${w_1}$ is not losing.
(Note that this is low-scoring baseball: every inning, each team scores $\pm1$ run. It is also fixed: the end result is tied. The lead changes precisely when words are not comparable; comparable words are uninteresting as spectator sport. Two words are comparable if and only if they describe a low-scoring, fixed, and uninteresting baseball game.)

Later in the paper, there is proof by skiing (with comparably colourful language) and various bypass shennanigans.

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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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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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> Doron Zeilberger papers may contain also some colorful language. Is this perhaps like saying that oceans are sometimes wet? – L Spice Apr 25 '10 at 4:38

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

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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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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? – Yemon Choi Mar 11 '11 at 1:10

In T.Y.Lams book "Lectures on modules and rings" there is a chapter on quotient rings. The three subsections of which are named "The Good", "The Bad" and - of course - "The Ugly". The three subsections are about existence and uniqueness of a "localization" with the universal property in the noncommutative case ("The Good" though nothing is good about this localization in general, everything nice is lost in the general case), Mal'cev's example of a domain that cannot be embedded into a division ring ("The Bad") and further theorems about which classes of rings can be embedded together with example that there need not to be a unique minimal such division ring ("The Ugly").

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There is an important theorem by Shelah in PCF theory which is known as "the trichotomy theory" in which three possible situations are described: The good, in which things act like we want them to; the bad, in which things behave the opposite of what we want them to; and the ugly, in which things are just messed up. – Asaf Karagila Jul 5 '11 at 16:19

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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Milne's web page contains a number of amusing anecdotes-

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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

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In French, Jolissaint is pronounced as "joli seins", which translates as "nice tits" in English. – ACL Aug 23 '11 at 6:44

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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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?

locked by François G. Dorais Dec 18 '11 at 20:37
The whole review is a must-read... – darij grinberg Dec 5 '11 at 5:03
I would like to upvote this for being outrageous, but that would be giving it praise it does not deserve. – Ryan Reich Dec 12 '11 at 22:41
Right...thanks, but I doubt I'd have any more fun reading the review than I did reading that quote. – Elizabeth S. Q. Goodman Dec 13 '11 at 4:47
I am a bit shocked that something like this was printed in BAMS as late as in the earlier 80s. – quid Dec 17 '11 at 13:09
It may well be colourful; it strikes me as crass. – Yemon Choi Dec 18 '11 at 3:22

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.

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I like "Let's take this guy" (in German: Bursche) when a Graph theorist picks a vertex. (it's not colourful at first sight, but think about it)

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My Intention was to say that (among others) mathematicians tend to anthropomorphize their subjects. – Hans Stricker Jan 20 '11 at 0:08
@Hans: Really? anthropomorphize? When I look at how many "monsters", "beast" etc. are out there, then I tend to think that at least the "official" termininology is more animalistic. – Johannes Hahn Jan 20 '11 at 10:56
I once followed a lecture of David Goss where he started calling his objects "guy", passed on to something like "unpleasant fellow" (when he was revealing some undesired properties of that object) and ended up calling it "sucker" - repeatedly and emotionally. – Peter Arndt Oct 22 '11 at 20:12

This reminds me of the little blue book by Swan... It must be "Theory of Sheaves", I don't have it on my shelf here. But I remember clever chapter titles. Maybe someone else here can tell us.

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From one of the papers on integrable systems

"The authors X.X and Y.Y took only a small peace of the integrability cake...."

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peace or piece? – David Roberts Apr 6 '11 at 6:01

protected by François G. Dorais Oct 15 '13 at 2:42

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