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

I don't even know if this is intentional or not. In his book Teichmuller theory, John Hubbard frequently references the category of Banach Analytic Manifolds. He adheres to the convention that a category be referenced by the concatenation of the first three letters of each constituent word, making the category in question BanAnaMan. This still cracks me up to this day.

Answer 2

From the ground-breaking paper: On the complexity of omega-automata by Muli Safra (DOI: 10.1109/SFCS.1988.21948)

Acknowledgements

The author thanks his advisor, Amir Pnueli, for his encouragement and many fruitful discussions on this research.

Moshe Vardi initiated this research by a most illuminating mini-course on ω-automata he presented at the Weizmann Institute. He suggested the problems and helped in clarifying the solutions. Without him the work would not have started, progressed or ended.

Indispensable was the help of Rafi Heiman, whose signature at the bottom of a proof is more valuable than a Q.E.D.

Noam Nisan helped in the complexity evaluation of the determination construction.

Which leaves open the question of what is the author's contribution to the paper.

Answer 3

Does merely transposing two words count? "It is also hard not to show that ..." [Arnold W. Miller, "Some Properties of measure and category," Trans. A.M.S. 266, 1981, p. 106]

Answer 4

Chang and Keisler's book on Model Theory is dedicated to all those model theorists who have never dedicated a book to themselves.

Answer 5

The reader who makes it to the later chapters of M. N. Huxley's Area, Lattice Points and Exponential sums is rewarded with the following gem:

"If mathematics were an orchestra, the exponentials would be the violins. The $\rho(t)$ would be the flutes; they are introduced by the exponentials. The Poisson summation formula would be the tuba: powerful, but ridiculous when used too much"

Answer 6

Andre Weil (Oeuvres, vol. 2, page 558) purporting to be R.Lipschitz writing from Hades:

"Unfortunately, it appears that there is now in your world a race of vampires, called referees, who clamp down mercilessly upon mathematicians unless they know the right passwords. I shall do my best to modernize my language and notations, but I am well aware of my shortcomings in that respect ; I can assure you, at any rate, that my intentions are honourable and my results invariant, probably canonical, perhaps even functorial. But please allow me to assume that the characteristic is not 2"

Answer 7

This is a little off the mark (from a textbook), but Exercise VIII.8.3 of Sarason's [Notes on] Complex Function Theory is:

Stand straight with feet about one meter apart, hands on hips. Bend at the waist, knees straight, and touch left foot with right hand. Straighten. Bend again and touch right foot with left hand. Straighten. Repeat 15 times.

Answer 8

At the risk of blowing my own horn, I will mention the line in the book, Category Theory for Computing Science" by Charles Wells and me. After mentioning the Russell paradox and how to avoid it, we say, "This prophylaxis guarantees safe sets." I caught at least one colleague rolling on the floor laughing, but only after reading it aloud.

Answer 9

In the acknowledgment to Thomason and Trobaugh's paper on localization in algebraic K-theory, Thomason writes:

The first author must state that his coauthor and close friend, Tom Trobaugh, quite intelligent, singularly original, and inordinately generous, killed himself consequent to endogenous depression. Ninety-four days later, in my dream, Tom's simulacrum remarked, "The direct limit characterization of perfect complexes shows that they extend, just as one extends a coherent sheaf." Awakening with a start, I knew this idea has to be wrong, since some perfect complexes have a non-vanishing $K_0$ obstruction to extension. I had worked on the problem for 3 years, and saw this approach to be hopeless. But Tom's simulacrum had been so insistent, I knew he wouldn't let me sleep undisturbed until I had worked out the argument and could point to the gap. This work quickly led to the key results of this paper. To Tom, I could have explained why he must be listed as a coauthor.

Michael Harris has a rather interesting literary analysis of this quote on his webpage.

As far as I know, this was Trobaugh's only foray into mathematics.

Answer 10

Frank Adams was notorious for slipping little gems of humour into his paper and books. For instance, from his book, "Infinite Loop Spaces,"

(p. 128)

The reader may expect me to say something about "double coset formulae." I shall indeed; I advise you to avoid them.

(p. 131)

Of course, this still leaves the question: what do you say to the algebraist who loves double cosets and insists that this is the same thing really? I suggest that you smile politely and say that you are maximizing your chance of finding a helpful and congenial interpretation of the double cosets. There is no need to say that the best interpretation is one which allows you to avoid mentioning the (expletive deleted) things at all.

For further entertainment, look at the entry [85] in the bibliograph, and look at "jokes" in the index.

Answer 11

From Ravi Vakil's notes "Foundations of algebraic geometry."

He says about spectral sequences:

"They have a reputation for being abstruse and difficult. It has been suggested that the name 'spectral' was given because, like spectres, spectral sequences are terrifying, evil, and dangerous. I have heard no one disagree with this interpretation, which is perhaps not surprising since I just made it up."

Answer 12

According to https://en.wikipedia.org/wiki/Chandler_Davis, page 181 in Chandler Davis' "An extremum problem for plane convex curves" (in Victor L. Klee's "Convexity", Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 1963), one has

"Research supported in part by the Federal Prison System. Opinions expressed in this paper are the author's and are not necessarily those of the Bureau of Prisons."

The paper was written while its author was in prison for refusing to cooperate with the House Unamerican Activities Committee.

The quote can be seen in Google books.

Answer 13

P. T. Johnstone's On a Topological Topos has some interesting choices of words. Sometimes the words are discussed in parenthetical notes.

([...] we are tempted also to introduce the term 'consequential space' for an arbitrary object of $\mathcal{E}$, apart from a slight reluctance to give the name 'space' to an object of a category whose underlying-set functor is not faithful—and, we must admit, the fear that somebody will at once invent a notion of 'inconsequential space'.)

Sometimes there is no more than a reference to existing literature.

The rest of the proof of Theorem 5.1 is a fairly straightforward woozle-hunt (Milne [27])

Reference [27] is, as you may have guessed, A. A. Milne's Winnie The Pooh.

Answer 14

While this is not necessarily the meaning of "colorful" intended by the OP, there is probably no better way to find out what motivated the editors of the American Mathematical Monthly to reiterate a damnation by publishing the following erratum, than posting it here:

Erratum: In the article, "On the Ph.D. in Mathematics," by I. N. Herstein, on page 821, line 26, of the August-September 1969 issue of the Monthly, please read "damn" instead of "darn."

American Mathematical Monthly volume 77 (1970) p. 78

Answer 15

From Vector Calculus, Linear Algebra, And Differential Forms. A Unified Approach. by Hubbard:

When a matrix is described, height is given first, then width: an m x n matrix is m high and n wide. After struggling for years to remember which goes first, one of the authors hit on a mnemonic: first take the elevator, then walk down the hall.

Answer 16

The paper "Division by three" by Peter Doyle and John Conway has a wealth of colorful language including:

"If the arrows are good, straight, American arrows, it is very natural for each arrow to dream of marrying the arrow next door."

and

"Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel axioms for set theory are necessarily even consistent. Indeed, we’re somewhat doubtful whether large natural numbers (like $80^{5000}$ , or even $2^{200}$) exist in any very real sense, and we’re secretly hoping that Nelson will succeed in his program for proving that the usual axioms of arithmetic—and hence also of set theory—are inconsistent. (See Nelson [6].) All the more reason, then, for us to stick with methods which, because of their concrete, combinatorial nature, are likely to survive the possible collapse of set theory as we know it today."

Answer 17

What about Johnstone, in his introduction to Topos Theory (1977):

Finally, I have to state my position on the most controversial question in the whole of topos theory: how to spell the plural of a topos. The reader will already have observed that I use the English plural; I do so because [...] the word topos is not a direct derivative of its Greek root, but a back-formation from topology. I have nothing further to say on the matter, except to ask those toposophers who persist in talking about topoi whether, when they go out for a ramble on a cold day, they carry supplies of hot tea with them in thermoi.

That cracked me up. And for many years it was as far as I got into the book.

Answer 18

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.

Answer 19

Masaki Kashiwara writes, in the introduction to his Systems of microdifferential equations:

Although this was a course at a French university, several examples of hyperfunctions are given just before Theorem 3.2.45.

and shortly after that:

The reader is kindly advised not to commit seppuku instantly if he feels he does not quite understand 2. of chapter 1.

Answer 20

From Strichartz's A Guide to Distribution Theory and Fourier Transforms:

(p.2) "You have almost seen the entire definition of generalized functions. All you are lacking is a description of what constitutes a test function and one technical hypothesis of continuity. Do not worry about continuity--it will always be satisfied by anything you can construct (wise-guys who like using the axiom of choice will have to worry about it, along with wolves under the bed, etc)."

Answer 21

Jon Barwise's Admissible Sets and Structures contains the following on page 69:

When used in a class or seminar, section 6 should be supplemented with coffee (not decaffeinated) and a light refreshment. We suggest Heatherton Rock 'Cakes. (Recipe: Combine 2 cups of self-rising flour with 1 t. allspice and a pinch of salt. Use a pastry blender or two cold knives to cut in 6 T butter. Add 1/3 cup each of sugar and raisins (or other urelements). Combine this with 1 egg and enough milk to make a stiff batter (3 or 4 T milk). Divide this into 12 heaps, sprinkle with sugar, and bake at 400 °F. for 10 — 15 minutes. They taste better than they sound.)

There is a response to this (with stronger ingredients) somewhere in Aki Kanamori's The Higher Infinite but I forgot exactly where. Later in that book, on page 289, Kanamori writes:

But first, a respite from the rigors: Instead of yet another recipe, we offer the following chess problem (M. Henneberger, first and second prize, "Revista de Sah" 1928):

White. King on b1, Rooks on b7 and c7, and Bishop on b5.

Black. King on a8, Rook on a3, and Pawn on f2.

White to play and win.

Send complete solutions to the author for a small prize.

Answer 22

From the introduction of Model Theory by Wilfrid Hodges:

"Finally a dedication. If this book is a success, I dedicate it to my students and colleagues, past and present, in the field of logic. Many of them appear in the pages which follow; but of those who don't, let me mention here two thoughtful and generous souls, Geoffrey Kneebone and Chris Fernau, both now retired, who ran the logic group of London University at Bedford College when I first came to London. If the book is not a success, I dedicate it to the burglars in Boulder, Colorado, who broke into our house and stole a television, two typewriters, my wife Helen's engagement ring and several pieces of cheese, somewhere about a third of the way through Chapter 8."

Answer 23

From Tilman Bauer's "p-compact groups as framed manifolds:"

For our purposes, it is enough to work in the category of so-called naive G-spectra. I will drop the word “naive” since it will make this work appear so puny.

And in Tilman's paper with Natalia Castellana, "Adjoint spaces and flag varieties of p-compact groups:"

This comment is only meant to intimidate the reader and is insubstantial for what follows.

Answer 24

A few days ago, some colorful quotes from Michael Spivak's A Comprehensive Introduction to Differential Geometry were posted here. Yesterday I noticed they were missing, which is a great loss, so I am attempting to restore them. The only one I remember immediately is

Bourbaki has apparently decided that the theory of manifolds has now entered that domain of "dead" mathematics to which he hopes to give definitive form. In this summary of results the corpse is laid out to public view; the complete autopsy is eagerly awaited.

(Volume 5, p.608, of the 2nd edition, 1975)

If anyone recalls some others, please add them.

Answer 25

I like the following footnote that appears in a paper by G. Baumslag:

"I thank Graham Higman for allowing the dust of Oxford to rest on my unopened manuscript for thirty months."

Answer 26

In his article "Lectures on Mixed Motives" (Proceedings of Symposia in Pure Mathematics, Volume 62.1, 1997), Spencer Bloch writes:

"My experience with these lectures suggests that motives are like onions; they are complicated, multi-layered objects, and any attempt to cut too quickly to the heart of the matter can leave the audience in tears."

I've actually gotten some mileage out of this analogy in my teaching. When doing the first iterated chain-rule examples in calculus classes, for example, I advocate working "from the outside in" as opposed to the other way around, and employ a variant of Bloch's statement.

Answer 27

I was always amazed that Clifford Truesdell could get away with a quote like this:

Nowadays, when the common student seeks a secure berth by grafting himself upon some modest little professor whom he regards as prone to foster painlessly his limaceous glide toward a dissertation not too strenuous or, even better, to draught it for him, tradition is moribund (...)

This is from his introduction to the selected papers of W. Noll. Admittedly, Truesdell was the chief editor himself, and could write therefore whatever he wanted, but it's still pretty strong. Felt too close to home when I first read it as a graduate student!

Answer 28

This isn't so much a serious mathematical paper, but Miles Reid - Undergraduate Algebraic Geometry is full of bizarre sentences:

If $I(X)$ is defined as the set of functions vanishing at all points of $X$, then for any point of $X$, all functions of $I(X)$ vanish at it. And indeed conversely, if not more so, just as I was about to say myself, Piglet.

or,

The name of the theorem (Nullstelle = zero of a polynomial + Satz = theorem) should help to remind you of the content (but stick to the German if you don't want to be considered an ignorant peasant).

Answer 29

Yiannis Moschovakis, Notes on Set Theory (1994), p. 81:

6.26 About topology. General (pointset) topology is to set theory like parsley to Greek food: some of it gets in almost every dish, but there are no great "parsley recipes" that the Greek cook needs to know.

Answer 30

A gem of R.H. Bing:

Dimension 4 is the most difficult dimension. It is too old to spank, the way we might deal with the little dimensions 1, 2, and 3; but it is also too young to reason with, the way we deal with the grown-up dimensions 5 and higher.

Source here: https://www.ams.org/journals/bull/2011-48-03/S0273-0979-2011-01320-9/S0273-0979-2011-01320-9.pdf

Answer 31

You'll find a whole host of colourful language and allusions scattered throughout the works of Kato. To quote just one example from his Lecture on the approach to Iwasawa theory for Hasse Weil L-functions via $B_{dR}$:

Where is the homeland of zeta values to which the true reasons of celestial phenomena of zeta values are attributed ? How can we find a galaxy train to approach it, which runs through the galaxy of p-adic zeta elements and whose engine is the theory of p-adic periods ? I imagine that one coach of the train has the name 'explicit reciprocity law of p-adic Galois representations'.

Answer 32

The English translation by Kenji Iohara of Minoru Wakimoto's "Infinite dimensional Lie algebras" is as colourful as it gets, I think. For example on page 8

Namely, we can think of an element of U(A) as an element of A. But since U(A)and A are not isomorphic, this thinking is not an identification but a lonely unrequited love.

Or on page 26

An elegant shape of the left half of Mt. Fuji reflected in the surface of a lake, this is the proportion of the finite-dimensional representations of $\mathfrak{sl}(2,\mathbb{C})$.

Or on page 27

Since ancient times, it has been the charm of music that has soothed the fiercest warriors (or samurai). This law seems to be universal in the physical universe, and it is also true in the world of Lie algebras.

My personal favourite is on page 289

Moreover, the conformal superalgebra (CSA for short) has recently been discovered by Kac, and its definition is given in 2.7 of [K5]. This representation theory has been started in [CK], It is like a matsutake mushroom derived from a big tree called a vertex operator algebra, and it is a portable version of a super-conformal algebra and a vertex operator algebra. There is an experimental report saying that it is more delicious to munch a matsutake mushroom than its landlord- i.e. a Japanese red pine.
Let us munch it a bit.

Unfortunately perhaps, the language is not nearly as colourful in the original Japanese (it's just an outstandingly good book), and is an artifact of the translation. I've long had a dream of doing a more sober translation... but I suppose that Iohara's translation is not without its charm. Anyway, the colourful language is in my opinion is to be attributed to Iohara rather than to Wakimoto.

Answer 33

In this MO answer, I mentioned Arnold Miller's lecture notes, where he gives an entertaining account of the MM proof system (for Micky Mouse), having as axioms all validities and modus ponens as the only rule of inference. Although it is easy to prove the Completeness theorem from Compactness in this system, it is nevertheless a kind of joke system, since the set of validities is not a decidable set, and so we would be fundamentally unable to recognize whether something is a proof or not in this system. Miller uses this example to illustrate the point as follows:

The poor MM system went to the Wizard of OZ and said, “I want to be more like all the other proof systems.” And the Wizard replied, “You’ve got just about everything any other proof system has and more. The completeness theorem is easy to prove in your system. You have very few logical rules and logical axioms. You lack only one thing. It is too hard for mere mortals to gaze at a proof in your system and tell whether it really is a proof. The difficulty comes from taking all logical validities as your logical axioms.” The Wizard went on to give MM a subset Val of logical validities that is recursive and has the property that every logical validity can be proved using only Modus Ponens from Val.

And he then goes on to describe how one might construct Val, and give what amounts to a traditional proof of Completeness.

Answer 34

This is taken from the seminal paper "Quantum error correction via codes over GF(4)" by A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane.

Answer 35

I must post some more examples from Frank Adams. I recommend reading the last section of his paper "Finite H-Spaces and Lie Groups", which contains a letter to the reader written in the voice of the exceptional lie group E8. Two excerpts :

"This is as if one were to award a title for drinking beer, having first fixed the rules so as to exclude all citizens of Heidelberg, Munich, Burton-on-Trent, and any other place where they actually brew or drink much of the stuff."

"In the second place, to consider the question at all reveals a certain preoccupation with ordinary cohomology. Any impartial observer must marvel at your obsession with this obscure and unhelpful invariant."

Answer 36

From William Thurston's "Hyperbolic Structures on 3-Manifolds I: Deformation of Acylindrical Manifolds":

Let us stick to the case that $M$ is a compact, acylindrical manifold. Then $H(M)$ is a hard-boiled egg. The egg complete with shell is $AH(M)$; it appears to be homeomorphic to a closed unit ball. $GH(M)$ is obtained by thoroughly cracking the egg shell on a convenient hard surface. Apparently no material is physically separated from the egg, but many cracks are developed -- cracks are dense in the boundary -- and at the same time, the material of the egg just inside the shell is weakened, so that neighborhood systems of points on the boundary become thinner. Finally, $QH(M)$ has uncountably many components, which are obtained by peeling off the shell and scattering the pieces all over.

Answer 37

From S. Skewes's "On the difference $\pi(x)-\mathrm{Li}(x)$", Proc. LMS 5, 1955:

"I wish in conclusion to express my humble thanks to Professor Littlewood, but for whose patient profanity this paper could never have become fit for publication."

Answer 38

In the Book "Introduction to lattices and order", the authors (B. A. Davey and Hilary A. Priestley) talk about ordered sets with a bottom/top. In this context, they say the following:

Computer scientists commonly choose models which have bottoms, but prefer them topless.

Answer 39

John (Horton) Conway unrelentingly gets away with colorful, even whimsical language in definitions, in explanations, in paper titles, even in some book titles (The Sensual (Quadratic) Form.) Even in SPLAG, there is the following:

"...we earnestly recommend that you use

The Best Method: guess the correct answer, and then justify it." SPLAG, p. 302

On Numbers and Games is just rife with colorful stuff. (I'm surprised no one has pointed out this elephant in the room yet.) The next to last theorem of the book is

THEOREM 99: Any short all-small game G which has atomic weight zero is infinitesimal with respect to (double-up) and dominated by some superstar.

And the last words of the book are famously

"...a certain feeling of incompleteness prompts us to add a final theorem.

THEOREM 100. This is the last theorem in this book.

(The proof is obvious.)" ONAG p. 224

Answer 40

Not from a paper but rather from a book, the first page of the introduction to G. R. Kempf's Algebraic Varieties reads:

"Algebraic geometry is a mixture of the ideas of two Mediterranean cultures. It is the superposition of the Arab science of the lightning calculation of the solutions of equations over the Greek art of position and shape. This tapestry was originally woven on European soil and is still being refined under the influence of international fashion. Algebraic geometry studies the delicate balance between the geometrically
plausible and the algebraically possible. Whenever one side of this mathematical teeter-totter outweighs the other, one immediately loses interest and runs off in search of a more exciting amusement."

Answer 41

"Indeed, I wrote the outline of this book while wandering across India, so that, in my mind, Henkin's method is inexorably linked to the droves of wild elephants I met while crawling among the swamp plants of the preserves of Kerala; the elimination of imaginaries, to the gliding vultures above the high Himalayan peaks; and the theorem of the bound, to the naked bodies of the Mauryan women that the traveler saw on the bends of a jungle trail, before they had time to cover themselves. I dare hope only that this book will evoke similarly pleasant images in my reader; I wish it will be as pleasant a companion for you as it was for me."

From Bruno Poizat's "Model Theory". He also constantly belittles the readers of the English edition of the book. Highly recommended!

Answer 42

From Donagi and Smith "The Structure of the Prym Map":

Wake an algebraic geometer in the dead of night, whispering: "27". Chances are, he will respond: "lines on a cubic surface".

Answer 43

Diaconis and Efron wrote a paper "Testing for Independence in a Two-Way Table: New Interpretations of the Chi-Square Statistic" that was followed by 10 papers discussing their suggestion. The following is from Diaconis and Efron's rejoiner:

The critical paper that they refer to starts with a splendid colorful language:

Update: This is an additional answer too good to be missed.

Answer 44

Waldhausen once inserted a bit of music -- written out as notes on a staff -- in the final draft of one of his papers, explaining "This replaces an unnecessary axiom." The melody, by Grieg, was called "Fool's Morning Song".

Answer 45

The last paragraph of E. Artin's "Theory of Braids":

Although it has been proved that every braid can be deformed into a similar normal form the writer is convinced that any attempt to carry this out on a living person would only lead to violent protests and discrimination against mathematics. He would therefore discourage such an experiment.

Answer 46

There is a paper entitled Zaphod Beeblebrox's Brain and the Fifty-ninth Row of Pascal's Triangle.

Freudenthal made this quote about terminology:

A more imaginative nomenclature than one relying on overburdened terms such as "fundamental," "principal," "regular," "normal," "characteristic," "elementary," and so on is desirable. Inventors of important mathematical notions should give their inventions suggestive names. The disadvantage that good names might prevent the inventor's name from being immortalized as an adjective would be more than compensated by the advantage that this honor could not possibly be bestowed on noninventors.

(from twf:178)

Answer 47

Spivak, A Comprehensive Introduction to Differential Geometry, Volume 1, p.94,

Now that we have a well-defined bundle map $TM \to T\;'M$ (the union of all $\beta_x^{-1} \circ \alpha_x$), it is clearly an equivalence $e_M$. The proof that $e_N \circ f_* = f \circ e_M$ is left as a masochistic exercise for the reader.

Volume 3, p. 103, indexed under "Idiot, any,"

These normalizations are usually carried out with hardly a word of motivation, as if they are so natural that any idiot would immediately think of doing them—in reality, of course, the authors already knew what results they wanted, since they were simply reformulating a classical theory.

From Volume 5, p.59,

We are going to begin by deriving certain classical PDE's which describe important (somewhat idealized) physical situations. The word "derive" had better be taken with a hefty grain of salt, however. What I have really tried to do is give plausible reasons why the physical situations should be governed by those PDE's which the physicists have agreed upon. I've never really been able to understand which parts of the standard derivations are supposed to be obvious, which are mathematically simplifying assumptions, which steps are supposed to correspond to empirically discovered physical laws, or even what all the words are supposed to mean.

Incidentally, Spivak gave an entertaining series of lectures on the subject of classical mechanics, whence

I haven't the slightest idea what any of this means! But I'm almost certain that it amounts to the similarity argument we have given. Aren't you glad that you aren't a mathematician of the 17th century!?

Answer 48

A famous Sherlock Holmes meta-mystery is the identity of the giant rat of Sumatra. In The Adventure of the Sussex Vampire, Sherlock Holmes declares to Dr. Watson:

Matilda Briggs was not the name of a young woman, Watson, . . . It was a ship which is associated with the giant rat of Sumatra, a story for which the world is not yet prepared.

Sherlock Holmes fans have tried to figure out what the Giant Rat of Sumatra is, and how it might be related to a ship.
The solution to this greatest of all Sherlock Holmes mysteries is to be found in a mathematics book- one whose topic is Catastrophe Theory. On Page 196 of Curves and Singularities by J.W. Bruce and P.J. Giblin, we learn that the giant rat of Sumatra is in fact the family of functions $f_a(t_1,t_2)=t_1t_2(t_1-t_2)(t_1-at_2)$. Section 11.2 (Pages 196-200) of the book explains how we have established that this is indeed the giant rat of Sumatra, and elucidates why indeed the world is not yet prepared for its story. The relationship between the giant rat and the Matilda Briggs is not discussed, although we are led to suspect the worst, given that Catastrophe Theory is the book's theme.

Answer 49

I like George Kempf's succinct description in that same textbook, of the splitting of vector bundles on P^1 as a theorem "last proved by Grothendieck".

I never forgot Dieudonne's opinion in Foundations of modern analysis chapter 8, that defining a derivative as a number instead of a linear form, is "slavish subservience to the shibboleth of numerical interpretation at any cost."

In the section entitled "Woffle" of Miles Reid's Undergraduate algebraic geometry, he states delightfully that in the prerequisite algebraic chapter II, "the student who is prone to headaches could perhaps take some of the proofs for granted here, since the material is standard, and the author is a professional algebraic geometer of the highest moral fibre."

Answer 50

The AMS Memoirs 947 "Rock Blocks" by Will Turner is full of colorful lanuage. For example in the introduction one finds out that:

"Hannah Turner supported me financially (partly), and libidinously (entirely)."

Or:

"We choose not to spend time chomping on this old pie, since we have become aware of dishes with a more exotic, and alluring aroma."

and so on....

Answer 51

R.Coleman writing about the Dwork Principle in Section III of "Dilogarithms, Regulators and $p$-adic $L$-functions":

"Rigid analysis was created to provide some coherence in an otherwise totally disconnected $p$-adic realm. Still, it is often left to Frobenius to quell the rebellious outer provinces".

Answer 52

At the end of the introduction to Spin Glasses: a challenge for mathematicians, Michel Talagrand writes:

It is customary for authors, at the end of an introduction, to warmly thank their spouse for having granted them the peaceful time needed to complete their work. I find that these thanks are far too universal and overly enthusiastic to be believable. Yet, I must say that in the present case even what would sound for the reader as exaggerated thanks would not truly reflect the extraordinary privileges I have enjoyed. Be jealous, reader, for I yet have to hear the words I dread the most: "Now is not the time to work".

Answer 53

I always liked Edward Burger's A Tail of Two Palindromes. It begins as follows:

Upon a preliminary perusal, this parable may appear to be about pairs of palindromes, periods, and pitiful alliteration. In actuality, however, it is the story of a real quadratic irrational number $\alpha$ and its long-lost younger sibling, its algebraic conjugate $\tilde{\alpha}$ ($\alpha > \tilde{\alpha}$). How in the dickens are all these notions connected? We begin at the beginning...

Although the conjugates $\alpha$ and $\tilde{\alpha}$ are not identical twins, unlike the two zeros of $(x - 3)^2$, they do share a common family history: they each were born of the same irreducible parent polynomial having rational coefficients,

$$P_{\alpha}(x) = P_{\tilde{\alpha}}(x) = (x - \alpha)(x - \tilde{\alpha}) = x^2 - \text{Trace}(\alpha)x + \text{Norm}(\alpha),$$

where $\text{Trace}(\alpha) = \alpha + \tilde{\alpha}$ and $\text{Norm}(\alpha) = \alpha \tilde{\alpha}$. Perhaps not surprisingly, some conjugate pairs exhibit similar personalities. But how similar can they be? And how can we detect those similarities simply by looking at $\alpha$? As we will discover as our tale unfolds, the answer - foreshadowed in the title - is encoded in what can be described as the number theoretic analogue of the DNA-sequence for $\alpha$. However, before delving into $\alpha$'s genes, we first motivate our results by weaving a lattice of algebra.

Answer 54

I don't agree with this quote by Errett Bishop (a constructivist who developed real analysis along constructive lines), but I admire its brio:

Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself.

It's an odd spin on that famous Kronecker quote about the integers and God.

Answer 55

You might want to read http://www.ucs.louisiana.edu/~avm1260/lenstra.html for hilarious language during lecturing.

Answer 56

I always liked

$L$ takes on the character of a very thin inner model indeed, bare ruined choirs appended to the slender life-giving spine which is the class of ordinals.

from Kanamori and Magidor `The evolution of large cardinal axioms in set theory' (1978).

Answer 57

Does Serre's naming of the Pin group count as "colorful language"?

Answer 58

the book "Combinatorial optimization: algorithms and complexity" by Christos H. Papadimitriou, Kenneth Steiglitz contains the following exercise (19, pg 380):

The following is from the New York Times of November 27, 1979. Determine, when possible, whether each statement is (a) true, (b) false, (c) misleading, (d) equivalent to a well-known conjecture, the solution of which was probably not known to Mr. Browne.

Answer 59

In section 3 of

• J. Frank Adams -- Stable homotopy theory (3rd ed., LNM 3, 1969)

the author discusses two different attitudes towards what the "proper" definition of the stable homotopy category should be, which he personifies by the tortoise and the hare:

The hare is an idealist: his preferred position is one of elegant and all embracing generality. He wants to build a new heaven and a new earth and no half-measures. ... The tortoise, on the other hand, takes a much more restrictive view. He says that his modest aim is to make a cleaner statement of known theorems, and he'd like to put a lot of restrictions on his stable objects so as to be sure that his category has all the good properties he may need. Of course, the tortoise tends to put on more restrictions than are necessary, but the truth is that the restrictions give him confidence.

You can decide which side you're on by contemplating the Spanier-Whitehead dual of an Eilenberg-MacLane object. This is a "complex" with "cells" in all stable dimensions from $-\infty$ to $-n$. According to the hare, Eilenberg-MacLane objects are good, Spanier-Whitehead duality is good, therefore this is a good object: And if the negative dimensions worry you, he leaves you to decide whether you are a tortoise or a chicken. According to the tortoise, on the other hand, the first theorem in stable homotopy theory is the Hurewicz Isomorphism Theorem, and this object has no dimension at all where that theorem is applicable, and he doesn't mind the hare introducing this object as long as he is allowed to exclude it. Take the nasty thing away!

Answer 60

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 61

In Berger's "A panoramic view of Riemannian Geometry" :

"The Cayley projective plane $\mathbb{CaP}^2$ is beautiful. In the Riemannian zoo we like to call it the panda."

Answer 62

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 63

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 64

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 65

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 66

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 67

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 68

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 69

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 70

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 71

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 72

"quantization commutes with seduction"

Was it a typo? Or was it intentional?

Answer 73

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 74

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 75

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 76

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 77

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 78

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 79

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 80

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 81

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 82

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 83

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 84

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 85

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 86

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).

Answer 87

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.

Answer 88

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 89

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 90

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 91

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 92

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

Answer 93

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."

Answer 94

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.

Answer 95

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.

Answer 96

"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.

Answer 97

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.

Answer 98

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).

Answer 99

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.

Answer 100

Pretentiousness is repulsive. (see page 9)

Answer 101

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

Answer 102

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.

Answer 103

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.

Answer 104

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?

Answer 105

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.

Answer 106

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.

Answer 107

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

Answer 108

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

Answer 109

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

Answer 110

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."

Answer 111

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

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

Answer 112

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.

Answer 113

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

Answer 114

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.

Answer 115

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.

Answer 116

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.

Answer 117

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.

Answer 118

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.

Answer 119

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.

Answer 120

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.

Answer 121

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.

Answer 122

An excerpt from E. C. Zeeman. Seminar on combinatorial topology:

... choose a spine in the interior; expand each edge like a banana and collapse from one side; then expand each vertex like a pineapple and collapse from one face.

I find this a very good specimen of colourful language: the colourfulness is used to convey an idea in an elegant yet precise way. Halmos wrote `Clarity is what’s wanted, not pedantry; understanding, not fuss'. Zeeman's excerpt is a good example to go with Halmos' advise.

Answer 123

From one of the papers on integrable systems

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

Answer 124

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)

Answer 125

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.