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

In a paper by Stark where he proves Gauss's conjecture that there are only nine imaginary quadratic fields where the integers form a UFD, he writes that Heegner used "classified theory".

I once met Stark and asked him if he did not correct the misprint on purpose, but he did not even remember it.

Answer 2

S. S. Abhyankar's book, "Algebraic Geometry for Scientists and Engineers" is actually more for mathematicians, and algebraic geometers in particular. It has the following quip(meant for Andre Weil who wanted to eliminate elimination theory):

Eliminate, eliminate, eliminate, Eliminate the eliminators of elimination theory.

The whole lengthy polemic can be read at this google books link.

Answer 3

Masaaki Yoshida's book "Hypergeometric Functions, My Love" is packed with many colorful passages. For example, opening at random I find:

"(Do you think I should write $R^{(A)}_b =P^{-1}R^{(H)}_a P$? The notation would smother you!)"

But I think my favorite is:

"I believe that developments of mathematics are made by generalizations followed by specializations. You should jump and fly like an eagle and then fly down toward a game. To establish a story of modular interpretation of $X(3,6)$ we must jump at least as a grasshopper."

Answer 4

I once had to make the point that the theory of spectra-with-group-action which I was using was much simpler, more naive, than the sort of beautiful and elaborate equivariant stable homotopy created by Peter May and his school. In the preprint I described the latter as the "Chicago, or deep-dish" theory. I took those words out of the final version, thinking of international readers who might not get the pizza reference. (I substituted some other humorously intended words which were a gentle dig at Peter.)

Answer 5

Number theorist Andrew Granville wrote a paper called "Prime number races" in which he studies the "race" between prime numbers $\equiv$ 1 (mod 4) and prime numbers $\equiv$ 3 (mod 4). The introduction is most certainly a colorful one:

There’s nothing quite like a day at the races...The quickening of the pulse as the starter’s pistol sounds, the thrill when your favorite contestant speeds out into the lead (or the distress if another contestant dashes out ahead of yours), and the accompanying fear (or hope) that the leader might change. And what if the race is a marathon? Maybe one of the contestants will be far stronger than the others, taking the lead and running at the head of the pack for the whole race. Or perhaps the race will be more dramatic, with the lead changing again and again for as long as one cares to watch. Our race involves the odd prime numbers, separated into two teams depending on the remainder when they are divided by 4:

Answer 6

I've always marveled that the abbreviated terminology for "thickenings of the corresponding special Lagrangian" on the bottom of page 26 of this paper of Richard Thomas made it into print:

https://arxiv.org/pdf/math/0104196v1.pdf

Answer 7

From Jim Stasheff's Homotopy Associativity of H-spaces I, the magisterial-sounding

To study spaces which admit $A_n$-structures, we can work directly with the maps…. In the case of a topological group, this amounts to working only with the classifying bundle and never mentioning group operations. This would be an exercise in rectitude of thought of which it would be pointless to countenance the austerity, for not only would it eliminate a useful perspective on the subject, but, by disguising its own main point, it would place the reader beneath a cloud of unknowing.

Note 1: this is partly a subtle dig at Claude Chevalley's Fundamental Concepts of Algebra, whose preface ends, "Secondly, that one of the important pedagogical problems which a teacher of beginners in mathematics has to solve is to impart to his students the technique of rigorous mathematical reasoning; this is an exercise in rectitude of thought, of which it would be futile to disguise the austerity."

Note 2: Stasheff is exhibiting his awareness of religious literature (The Cloud of Unknowing is a 14th century work of Christian mysticism, written in Middle English).

Answer 8

In a paper of F.A.Muller — Sets, Classes and Categories (DOI: 10.1093/bjps/52.3.539) — Solomon Feferman is cited:

I realise that workers in category-theory are so at home in their subject that they find it more natural to think in category-theoretic rather than set-theoretical terms, but I would liken this to not needing to hear once one has learned to compose music.

Colin McLarty in Learning from Questions on Categorical Foundations does mention this, too.

^{[Feferman 1977] S., 'Categorical Foundations and Foundations of Category Theory', in Logic, Foundations fo Mathematics and Computability Theory, R.E. Butts & J. Hintikka (eds.), Dordrecht: D. Reidel, 1977; pp.149-169}

Answer 9

From page 329 of Carothers' Real Analysis textbook, where uses Fatou's lemma to prove Lebesgue's dominated convergence theorem: "Now we unleash Fatou!"

Answer 10

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.

Answer 11

The following is taken from The paper "Rational points near curves and small nonzero $|x^3-y^2|$ via lattice" by Noam Elkies It was discussed in a previous MO question.

Citing the Simpsons is rather surprising and I wonder what is the story behind it.

Answer 12

Math Reviews used to be much more colorful. In the 1950s, Haefliger was working on groupoids, developing a lot of what is now fundamental in the theory of stacks. Palais reviewed a 1958 paper of Haefliger's, concluding with,

The first four chapters of the paper are concerned with an extreme, Bourbaki-like generalization of the notion of foliation. After some twenty-five pages and several hundred preliminary definitions, the reader finds that a foliation of $X$ is to be an element of the zeroth cohomology space of $X$ with coefficients in a certain sheaf of groupoids. Holonomy, the Reeb-Ehresmann stability theorems, etc., are then generalized to this setting. While such generalization has its place and may in fact prove useful in the future, it seems unfortunate to the reviewer that the author has so materially reduced the accessibility of the results, mentioned above, of Chapter V, by couching them in a ponderous formalism that will undoubtedly discourage many otherwise interested readers.

Answer 13

I came across this little gem when preparing for a talk on Kakeya sets and the ball multiplier problem, found on page 437 of E. Stein's Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals:

We will use this process to generate our monster, which will have a tiny heart and many arms.

Answer 14

I like the following from the Introduction of Iwaniec-Kowalski: Analytic number theory (AMS, 2004):

Poisson summation for number theory is what a car is for people in modern communities – it transports things to other places and it takes you back home when applied next time – one cannot live without it.

This is not the only good one in that introduction, I let you find the others!

Answer 15

In Jacquet and Langlands' "Automorphic forms on GL(2)", page 154, they discuss a construction which uses some choices of intermediate objects -- of course the question whether the final result depends on those choices comes up ; here is how they treat it :

We prefer to pretend that the difficulty does not exist. As a matter of fact for anyone lucky enough not to have been indoctrinated in the functorial point of view it doesn’t.

That made me chuckle.

Answer 16

According to the book "King of Infinite Space" Coxeter, "tickled his readers with unexpected turns of phrase such as":

... dividing the product of the first three expressions by the product of the last two, and indulging in a veritable orgy of cancellation, we obtain ...

Answer 17

There is the following apocryphal dedication of a doctoral thesis:

"I am deeply grateful to Professor X, whose wrong conjectures and fallacious proofs led me to the theorems he had overlooked."

In fact this is a description of excellent supervision, in giving confidence to a student!

Answer 18

"quantization commutes with seduction"

Was it a typo? Or was it intentional?

Answer 19

More Weyl, all Mancosu's translation, all in his fierce days advocating Brouwer's mathematics:

Weyl (1921) On the New Foundational Crisis of Mathematics,

It must have the effect of a deliverance from a nightmare for whoever has maintained any sense for intuitively given facts in the abstract formalism of mathematics.

Weyl (1925) The current epistemological situation in mathematics:

At set theory's outermost borders, blurred in fog, crevices (i.e., flagrant contradictions) soon appeared.

and ibid, of the intuitionistic conception of the continuum:

The ice cover was burst into floes, and now the element of flux was soon altogether master over the solid.

Though these were published in mathematical journals, they are maybe not what the question was after, since they are not part of normal mathematical exposition.

Answer 20

Fulton and Harris's "Representation Theory" has a few examples of colourful language. Two of my favorites:

In recent work their* Lie-theoretic origins have been exploited to produce their representations, but to tell their story would go far beyond the scope of these lecture(r)s.

*: The finite Chevalley groups.

Any mathematician, stranded on a desert island with only these ideas and the definition of a particular Lie algebra $\mathfrak{g}$ such as $\mathfrak{sl}_n \mathbb{C}$, $\mathfrak{so}_n \mathbb{C}$, or $\mathfrak{sp}_n \mathbb{C}$, would in short order have a complete description of all the objects defined above in the case of $\mathfrak{g}$. We should say as well, however, that at the conclusion of this procedure we are left without one vital piece of information about the representations of $\mathfrak{g}$ ... this is, of course, a description of the multiplicities of the basic representations $\Gamma_a$. As we said, we will, in fact, describe and prove such a formula (the Weyl character formula); but it is of a much les straightforward character (our hypothetical shipwrecked mathematician would have to have what could only be described as a pretty good day to come up the idea) and will be left until later.

Answer 21

Here is a colorful rejoinder by D. Zagier (in his reprinted article on the dilogarithm) to colorful language by Ph. Elbaz-Vincent and H. Gangl:

[Ph. Elbaz-Vincent and H. Gangl] called these functions "polyanalogs," an amalgam of the words "analogue," "polylog," and "pollyanna" (an American term suggesting exaggerated or unwarranted optimism). Presumably the correct term for the case $m=2$ would then be "dianalog," which has a pleasing British flavo(u)r.

Answer 22

I am rather fond of Sylvester's "Aspiring to these wide generalizations, the analysis of quadratic functions soars to a pitch from whence it may look proudly down on the feeble and vain attempts of geometry proper to rise to its level or to emulate it in its flights." (1850)

Answer 23

This is perhaps more of a silly play on words than colourful, but I still got a laugh out of it. One page 58 of Conway's 'The sensual (quadratic) form' while discussing Kneser's gluing method a sentence begins:

To further illuminate the utility of glue, ...

Answer 24

André Weil uses some very colourful language in the introduction of his 1946 book Foundations of Algebraic Geometry. I recommend any mathematician to read it. Here are some excerpts:

"As in other kinds of war, so in this bloodless battle with an ever retreating foe which it is our good luck to be waging, it is possible for the advancing army to outrun its services of supply and incur disaster unless it waits for the quartermaster to perform his inglorious but indispensable task."

"Of course every mathematician has a right to his own language---at the risk of not being understood; and the use sometimes made of this right by our contemporaries almost suggests that the same fate is being prepared for mathematics as once befell, at Babel, another of man's great achievements."

"... however grateful we algebraic geometers should be to the modern algebraic school for lending us temporary accommodation, makeshift constructions full of rings, ideals and valuations, in which some of us feel in constant danger of getting lost, our wish and aim must be to return at the earliest possible moment to the palaces which are ours by birthright, to consolidate shaky foundations, to provide roofs where they are missing, to finish, in harmony with the portions already existing, what has been left undone."

"...it is hoped that these may be helpful to the reader, to whom the author, having acted as his pilot until this point, heartily wishes Godspeed on his sailing away from the axiomatic shore, further and further into open sea."

Answer 25

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.)

Answer 26

Two from Casselman's "A companion to Macdonald's book on p-adic spherical functions":

The word ‘´epingler’ means ‘to pin’, and the image that comes to mind most appropriately is that of a mounted butterfly specimen. [Kottwitz:1984] uses ‘splitting’ for what most call ‘´epinglage’, but this is not compatible with the common use of ‘deploiement’, the usual French term for ‘splitting’.) Ian Macdonald, among others, has suggested that retaining the French word ´epinglage in these notes is a mistake, and that it should be replaced by the usual translation ‘pinning.’ This criticism is quite reasonable, but I rejected it as leading to noncolloquial English. The words ‘pinning’ as noun and ‘pinned’ as adjective are commonly used only to refer to an item of clothing worn by infants, and it just didn’t sound right.

and

These phenomena are part of what Langlands calls endoscopy, a word that might be roughly justified by saying that endoscopy is concerned with some fine aspects of the structure of harmonic analysis on a reductive p-adic group. Langlands attributes the term to Avner Ash, praising his classical knowledge, but I was pleased to find recently the following quotation that shows a more vulgar intrusion of endoscopy into the modern world:

Jeeves: “ . . . I had no need of the endoscope.”

Bertie: “The what?”

Jeeves: “Endoscope, sir. An instrument which enables one to peer into the . . . interior and discern the core.”

From Chapter 12 of Jeeves and the feudal spirit by P. G. Wodehouse.

This discussion is about distingishing fae jewlry from real. Since the endoscope also has medical uses, one could imagine an even more vulgar usage.

He has modified the notes several times so these might not be there anymore, but I have the older copies =)

Answer 27

Edward Nelson, Predicative Arithmetic, p. 50:

The intuition that the set of all subsets of a finite set is finite -- or more generally, that if $A$ and $B$ are finite sets, then so is the set $B^A$ of all functions from $A$ to $B$ -- is a questionable intuition. Let $A$ be the set of some $5000$ spaces for symbols on a blank sheet of typewriter paper, and let $B$ be the set of some $80$ symbols of a typewriter; then perhaps $B^A$ is infinite. Perhaps it is even incorrect to think of $B^A$ as being a set. To do so is to postulate an entity, the set of all possible typewritten pages, and then to ascribe some kind of reality to this entity -- for example, by asserting that one can in principle survey each possible typewritten page. But perhaps it simply is not so. Perhaps there is no such number as $80^{5000}$; perhaps it is always possible to write a new and different page. Many ordinary activities are built up in a similar way from a rather small set of symbols or actions. Perhaps infinity is not far off in space or time or thought; perhaps it is while engaged in an ordinary activity -- writing a page, getting a child ready for school, talking with someone, teaching a class, making love -- that we are immersed in infinity.

Answer 28

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.

Answer 29

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").

Answer 30

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.