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The popular MO question "Famous mathematical quotes" has turned up many examples of witty, insightful, and humorous writing by mathematicians. Yet, with a few exceptions such as Weyl's "angel of topology," the language used in these quotes gets the message across without fancy metaphors or what-have-you. That's probably the style of most mathematicians.

Occasionally, however, one is surprised by unexpectedly colorful language in a mathematics paper. If I remember correctly, a paper of Gerald Sacks once described a distinction as being

as sharp as the edge of a pastrami slicer in a New York delicatessen.

Another nice one, due to Wilfred Hodges, came up on MO here.

The reader may well feel he could have bought Corollary 10 cheaper in another bazaar.

What other examples of colorful language in mathematical papers have you enjoyed?

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Latest paper, my co-author put in "but we will choose a more painful way, because there is nothing like pain for feeling alive" but the referee jumped on it. – Will Jagy Apr 23 '10 at 5:09
Maybe I should expand the question to include colorful language cut from serious mathematics papers :) – John Stillwell Apr 23 '10 at 5:18
By the way, your remark reminds me of another in a similar spirit that made it into the Princeton Companion. In his article on algebraic geometry, János Kollár says of stacks: "Their study is strongly recommended to people who would have been flagellants in earlier times." – John Stillwell Apr 23 '10 at 7:49
I was actually rather surprised recently by a referee who did not know the phrase “red herring”, and had to look it up. He insisted that we change it to something more understandable. It makes me wonder how much “colourful” language is weeded out by referees, and whether the mathematical literature is poorer for it. – Harald Hanche-Olsen Apr 24 '10 at 2:31
@Harald: If you intend your mathematical papers to be read by a wide range of readers, then write them in simple language, suitable for those who are relative beginners in English. I remember reading long ago some metaphoric phrase in a mathematics research paper, then imagining students all over the world getting out their English dictionaries, looking it up, and still not understanding what it meant. (I no longer remember what the phrase was, just this reaction to it.) – Gerald Edgar Apr 24 '10 at 15:43

109 Answers 109

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."

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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.

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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".

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I hate to comment like this, but for community wiki, I feel less bad, and it won't affect the ranking at this moment...could someone vote this one up? I'd rather like it to have 27 votes, but no more. It feels fitting. – Charles Siegel Jun 2 '10 at 13:07
Now we just have to get your comment to 27 upvotes as well. – Steven Gubkin Jun 8 '10 at 13:05
+1 but not voted up! – Abhishek Parab Jun 8 '10 at 13:27
28 votes are not a problem: just replace "lines on a cubic surface" by "bitangents to a plane quartic". I'm sure it works. – Laurent Moret-Bailly Oct 10 '10 at 9:54
Voted back down to 27, although it feels strange to downvote this. – Todd Trimble Dec 16 '12 at 13:32

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.

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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 Completness 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.

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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."

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"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!

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I think we could fill this entire question with Bruno Poizat (aka Johnny B. Goode) quotes. I highly recommend browsing the titles in his bibliography -… – François G. Dorais Apr 23 '10 at 17:11
«Quelques modestes remarques à propos d'une conséquence inattendue d'un résultat surprenant de Monsieur Frank Olaf Wagner» is amazing. – Mariano Suárez-Alvarez Apr 23 '10 at 22:21
«Deux ou trois choses que je sais de $L_{n}$» surely beats that one, though! – Mariano Suárez-Alvarez Apr 26 '10 at 4:47
Poizat coined the terms belle paire and the dope. (The first is often translated beautiful pair, but big rack would be a more accurate.) – François G. Dorais Apr 27 '10 at 2:52

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.


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.

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Miles Reid is an extraordinarily entertaining speaker. I heard him once call a theorem "Ice-cream of Tuesday" in a talk. More amazing still was that this was indeed a good, descriptive name, for the theorem! – Daniel Moskovich Feb 3 '11 at 22:47

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)

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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."

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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".

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The piece (Alborada del Gracioso) is actually by Ravel. It is now an obstruction to republishing the paper in question, since the line of music was put in to avoid a retyping of the following pages, which would be hard to justify if the paper were redone in TeX. – John Rognes Oct 3 '10 at 13:19
@John, is this really an obstruction? – Mariano Suárez-Alvarez Oct 18 '10 at 18:50

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.

alt text

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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.

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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

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SPLAG?${}{}{}{}$ – Gerry Myerson Feb 3 '11 at 22:33
/Sphere Packing, Lattices, and Groups/, I believe. – Alison Miller Feb 4 '11 at 19:49
Theorem 100 reminds me of the second edition of Serre's "Cohomologie galoisienne" (1964, LNM 5) which contains a page of errata numbered E-1. The last line of that page is: "Page E-1: supprimer la dernière ligne." – Laurent Moret-Bailly Apr 6 '11 at 9:07

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.

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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.
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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:

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As an aside, I discovered by experience that searching for "Bing too old to spank" is NOT a good way to find a source for this quote. – Dave Futer Jul 5 '11 at 14:24

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.

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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."

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Perhaps only Dickens scholars would laugh out loud at the algebraic geometer's use of "elephants" for "elements", apparently a reference to a remark of Mr. Micawber in David Copperfield, upon entering from outdoors, where he felt he had been buffeted by "the elephants, uh... I mean elements". This presumably is also due to Miles Reid. – roy smith Jan 17 '11 at 17:08

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.

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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).

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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.

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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."

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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!

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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)."


"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....

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TMI, Will Turner. – Todd Trimble Mar 25 '14 at 12:19

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.

alt text

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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".

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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!?

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

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

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

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