239
$\begingroup$

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?

$\endgroup$
  • 88
    $\begingroup$ 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 Apr 23 '10 at 5:09
  • 17
    $\begingroup$ Maybe I should expand the question to include colorful language cut from serious mathematics papers :) $\endgroup$ – John Stillwell Apr 23 '10 at 5:18
  • 40
    $\begingroup$ 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 Apr 23 '10 at 7:49
  • 37
    $\begingroup$ 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 Apr 24 '10 at 2:31
  • 31
    $\begingroup$ @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 Apr 24 '10 at 15:43

110 Answers 110

7
$\begingroup$

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.

$\endgroup$
7
$\begingroup$

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.)

alt text

$\endgroup$
  • 5
    $\begingroup$ In what way is this language colorful? It's a strongly expressed opinion, but that doesn't make it colorful. $\endgroup$ – Todd Trimble Dec 16 '12 at 15:18
  • $\begingroup$ Hi Todd, my new constribution was this mathoverflow.net/questions/22299/… as for this on, it looked good when I posted it. One great colorful language I just learned from Barry Simon was that in Kelly's first edition of general topology he used "ways" instrad of "nets". His main motivation was to talk about "subways" rather than "subnets." However, Steenrod talked him out of this term. $\endgroup$ – Gil Kalai Dec 16 '12 at 17:07
7
$\begingroup$

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.

$\endgroup$
7
$\begingroup$

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

$\endgroup$
6
$\begingroup$

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."

$\endgroup$
6
$\begingroup$

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).

$\endgroup$
  • 4
    $\begingroup$ 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 Mar 11 '11 at 1:10
  • 3
    $\begingroup$ @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 Mar 11 '11 at 2:11
  • $\begingroup$ Fair point, David! $\endgroup$ – Yemon Choi Mar 11 '11 at 5:42
6
$\begingroup$

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").

$\endgroup$
  • 4
    $\begingroup$ 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. $\endgroup$ – Asaf Karagila Jul 5 '11 at 16:19
5
$\begingroup$

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.

$\endgroup$
  • 41
    $\begingroup$ > Doron Zeilberger papers may contain also some colorful language. Is this perhaps like saying that oceans are sometimes wet? $\endgroup$ – LSpice Apr 25 '10 at 4:38
5
$\begingroup$

Pretentiousness is repulsive. (see page 9)

$\endgroup$
5
$\begingroup$

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

$\endgroup$
  • $\begingroup$ Veuillez expliquer le blague? $\endgroup$ – Yemon Choi Aug 23 '11 at 0:51
  • 3
    $\begingroup$ In French, Jolissaint is pronounced as "joli seins", which translates as "nice tits" in English. $\endgroup$ – ACL Aug 23 '11 at 6:44
  • $\begingroup$ Oh, for some reason I had "seins" and "reins" mixed up in my head earlier... $\endgroup$ – Yemon Choi Aug 23 '11 at 9:47
5
$\begingroup$

Kleinfeld's paper On a short proof of my doctoral dissertation “On simple alternative rings without nilpotent elements” (J. Algebra, 2013) is a gem. After giving some background information, and a fourteen(!) line proof of the theorem which once upon a time earned him his doctoral degree, he writes:

Now what do we draw from this proof? First of all, I feel that Bruck was wrong to deny me access to our joint paper in claiming a dissertation. But OK, it didn’t harm me, so I can’t sue, but allowing a result which is so undeserving of a PhD dissertation puts shame on him and shame on me for not seeing how easy it is to prove this result after the Bruck/Kleinfeld result. More people to add on this list are the people at the University of Chicago, namely Kaplansky, Albert, and MacLane. Kaplansky and Albert, who had already published papers on alternative rings, had they seen such a proof, or imagined such a proof, wouldn’t have given me a post doctoral fellowship in 1951. MacLane didn’t think too much of my result because it was negative. It ruled out all examples except the octonians, and if he’d found something wrong with my thesis, he would have told me, too. Add to this list Herstein, who became a close friend. He was at the Cowles Commision at the time, but came over at any free moment to listen to lectures and talk to me at the University of Chicago. He, too, must never have seen how simple a proof there was. [...] To that list, add the editor of the Proceedings, because I published a lengthy paper consisting of my dissertation in the Proceedings in 1952[2]. Also add to the list several other algebraists who were going to put their students on writing a master’s thesis reproving my doctoral dissertation. I told them it was too easy. So shame on all of them. But no harm is done because those people I mention are not here any more.

$\endgroup$
4
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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 Jan 12 '11 at 9:13
  • $\begingroup$ The second one is called Mathematical Apocrypha Redux. $\endgroup$ – Pandora Jan 19 '11 at 17:28
4
$\begingroup$

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

$\endgroup$
4
$\begingroup$

The last paragraph of Chapter 7 of Amnon Neeman's Algebraic and Analytic Geometry book reads:

Note also that, even if the reader thinks coherent sheaves are for the birds and only vector bundles are natural objects worth studying, the proof forces one to consider coherent sheaves. The exact sequences we form in the proof inevitably will take honest, God-fearing vector bundles and make out of them Godless coherent sheaves.

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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

$\endgroup$
  • 1
    $\begingroup$ The whole review is a must-read... $\endgroup$ – darij grinberg Dec 5 '11 at 5:03
  • 3
    $\begingroup$ I would like to upvote this for being outrageous, but that would be giving it praise it does not deserve. $\endgroup$ – Ryan Reich Dec 12 '11 at 22:41
  • 6
    $\begingroup$ 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 Dec 13 '11 at 4:47
  • 4
    $\begingroup$ I am a bit shocked that something like this was printed in BAMS as late as in the earlier 80s. $\endgroup$ – user9072 Dec 17 '11 at 13:09
  • 1
    $\begingroup$ It may well be colourful; it strikes me as crass. $\endgroup$ – Yemon Choi Dec 18 '11 at 3:22
-2
$\begingroup$

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)

$\endgroup$
  • $\begingroup$ Brazilians say "a gente pega um cara" (more or less literally: let us take a guy) $\endgroup$ – Mariano Suárez-Álvarez Jan 19 '11 at 23:43
  • 1
    $\begingroup$ My Intention was to say that (among others) mathematicians tend to anthropomorphize their subjects. $\endgroup$ – Hans-Peter Stricker Jan 20 '11 at 0:08
  • 3
    $\begingroup$ @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. $\endgroup$ – Johannes Hahn Jan 20 '11 at 10:56
  • 6
    $\begingroup$ 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. $\endgroup$ – Peter Arndt Oct 22 '11 at 20:12
  • 3
    $\begingroup$ "Sucker" reminds me inevitably of Chuck Weibel, who used to say this all the time. $\endgroup$ – Todd Trimble Dec 13 '11 at 12:29
-2
$\begingroup$

From one of the papers on integrable systems

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

$\endgroup$
  • 4
    $\begingroup$ peace or piece? $\endgroup$ – David Roberts Apr 6 '11 at 6:01
-3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ I've just thumbed through The Theory of Sheaves, and I saw nothing in the titles (or in anything else there) that applies. The cover is blue all right, but I imagine you're thinking of something else. $\endgroup$ – Todd Trimble Mar 2 '11 at 2:15

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

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).

Would you like to answer one of these unanswered questions instead?

Not the answer you're looking for? Browse other questions tagged or ask your own question.