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

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?

• 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. Apr 23, 2010 at 5:09
• Maybe I should expand the question to include colorful language cut from serious mathematics papers :) Apr 23, 2010 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." Apr 23, 2010 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. Apr 24, 2010 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.) Apr 24, 2010 at 15:43

"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.

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.)

• In what way is this language colorful? It's a strongly expressed opinion, but that doesn't make it colorful. Dec 16, 2012 at 15:18
• 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. Dec 16, 2012 at 17:07

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.

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.

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).

• 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? Mar 11, 2011 at 1:10
• @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. :) Mar 11, 2011 at 2:11
• Fair point, David! Mar 11, 2011 at 5:42

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

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.

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."

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").

• 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. Jul 5, 2011 at 16:19

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

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

Pretentiousness is repulsive. (see page 9)

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.

• > Doron Zeilberger papers may contain also some colorful language. Is this perhaps like saying that oceans are sometimes wet? Apr 25, 2010 at 4:38

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.

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?

• He must have been a UNT alumnus. Sep 14, 2021 at 5:02

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.

• I tried putting something in a paper once about doing something Baire-handed, but they took it out. Sep 14, 2021 at 12:58

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.

• This was mentioned in this old thread, it's nice to have the full quotation here. Sep 14, 2021 at 11:25

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

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

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

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.

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

• @ZachTeitler, was that meant to be a link to the answer? Sep 14, 2021 at 2:56

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

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

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.

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.

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.

• 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. Sep 28, 2021 at 3:58

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.

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.

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."

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.

Here are several colourfully named concepts:

• perverse sheafs
• transgression
• schizophrenic objects

Errett Bishop wrote a paper on constructive mathematics which is titled Schizophrenic Mathematics and begins with a polemic against formalism in mathematics:

One could probably make a long list of schizophrenic attributes of contemporary mathematics, but I think the following short list covers most of the ground: rejection of common sense in davour of formalism; debasement of meaning by wilful refusal to accomodate certain aspects of reality; inappropriateness of means to ends; the esoteric quality of the communication; and fragmentation.

David Mumford wrote that algebraic geometry:

seems to have acquired the reputation of being esoteric, exclusive and very abstract with adherents secretly plotting to take over the rest of mathematics! In one respect, the last point is accurate ...

To which Vakil added in his book, The Foundations of Algebraic Geometry:

The revolution has fully come to pass ...

But warned:

Do not be seduced by the lotus-eaters into infatuation with untethered abstraction ...

And also quotes Atiyah:

"Should you just be an algebraist or a geometer?" is a bit like saying "should you rather be blind or deaf?"

Atiyah also calls spinors "the square root of geometry", which, though not colourful, is certainly esoteric.

From one of the papers on integrable systems

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

• peace or piece? Apr 6, 2011 at 6:01