# Most intriguing mathematical epigraphs

Good epigraphs may attract more readers. Sometimes it is necessary. Usually epigraphs are interesting but not intriguing.

To pick up an epigraph is some kind of nearly mathematical problem: it should be unexpectedly relevant to the content.

What successful solutions are known for you? What epigraphs attracted your attention?

Please post only epigraphs because quotes were collected in Famous mathematical quotes.

There are certain common Privileges of a Writer, the Benefit whereof, I hope, there will be no Reason to doubt; Particularly, that where I am not understood, it shall be concluded, that something very useful and profound is coucht underneath. (JONATHAN SWIFT, Tale of a Tub, Preface 1704)

[Taken from Knuth, D. E. The art of computer programming. Volume 3: Sorting and searching.]

• I'll stick with the classical $\{(x,y) : y \geq f(x)\}$. – Federico Poloni Sep 29 '15 at 10:01
• @FedericoPoloni: however, this one is not proper :-) – M.G. Sep 29 '15 at 10:41
• When I defended my PhD, a renown mathematician and authority in my field showed up. This surprised me slightly because, though I of course knew her, I had never interacted with her in any way. After the defense, she came to see me and asked if we could talk about "some of the beautiful ideas in the manuscript", which surprised me much more. Turns out she wanted to talk about the epigraph. – Olivier Sep 29 '15 at 11:31
• I think it would be helpful if you made clear in the question what exactly you are looking for/what an epigraph is. Some answers seem just like quotes, and we had such a question already mathoverflow.net/questions/7155/famous-mathematical-quotes – user9072 Sep 29 '15 at 13:37
• Isn't this the epitome of opinion-based questions? – Daniel Griscom Sep 29 '15 at 15:50

The most interesting epigraphs I have seen in mathematical books are in:

1. Bender and Orszag, Advanced mathematical methods for scientists and engineers. I. Asymptotic methods and perturbation theory. Every chapter is decorated by an epigraph from Sherlock Holmes. For example:

The triumphant vindication of bold theories - are these not the pride and justification of our life's work? (Conan Doyle, The Valley of Fear)

Just feel like re-reading Sherlock Holmes:-)

1. Reed and Simon, Methods of Mathematical physics, especially volume 1. For example, the chapter on Unbounded Operators has this:

I tell them that if they will occupy themselves with the study of mathematics, they will find that it is the best remedy against the lusts of the flesh. (Th. Mann, Magic Mountain).

But my favorite one is the following, from Kirillov, What's a number?:

Examiner: What is a multiple root of a polynomial?

Student: Well, this is when we plug a number to a polynomial and obtain zero; plug it again and obtain zero again... And this happens $$k$$ times. But on the $$(k+1)$$-st time we do not obtain zero.

Cannot help citing one more. Brocker, Lander, Differentiable germs and catastrophes:

• The last one is really fantastic! – Asvin Sep 30 '15 at 7:59
• IIRC the first part of the last one is an ancient Chinese proverb, but the last two lines are due to René Thom. – Robert Israel Oct 1 '15 at 15:31
• @Robert Israel: you are right this is how the authors refer to it. I just wanted to make my answer shorter, and keep the essence. I suspect that the whole thing is invented by Broker and Lander:-) But I like it. – Alexandre Eremenko Oct 1 '15 at 20:21
• @RobertIsrael YYRC, 朱泙漫學屠龍於支離益，單千金之家，三年技成，而無所用其巧。(Zhu Ping-man learned how to slaughter the dragon from Zhi-li Yi, expending (in doing so) all his wealth of a thousand ounces of silver. In three years he became perfect in the art, but he never exercised his skill.) - from Zhuangzi, chapter 列御寇 (Lie Yu-kou) – მამუკა ჯიბლაძე Oct 2 '15 at 4:43
• @მამუკა ჯიბლაძე: Great. So the reference by Broker and lander are genuine, and the end of the story is indeed due to Thom:-) – Alexandre Eremenko Oct 2 '15 at 12:02

Chapter 0 of Abstract and Concrete Categories - The Joy of Cats by Adamek, Herrlich and Strecker opens with

There’s a tiresome young man in Bay Shore.
When his fiancée cried, ‘I adore
The beautiful sea’,
He replied, ‘I agree,
It’s pretty, but what is it for?’

Morris Bishop


No category theorist needs any explanation why they chose this.

By the way this book also has absolutely marvelous illustrations by Marcel Erné, both inside text and also used as kind of epigraphs for chapters.

(I just can't stop)

Mathematics is the art of giving the same name to different things.

—Henri Poincaré, in Science and Méthode, 1908. English translation.

• In contrast/reference to: "Poetry is the art of giving different names to the same thing". – Kevin Casto Sep 30 '15 at 4:59

A famous quote by John von Neumann:

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.

• It is an epigraph where? – მამუკა ჯიბლაძე Sep 30 '15 at 4:24
• In "Analysis and Probability: Wavelets, Signals, Fractals" by Palle E. T. Jorgensen, Springer-Verlag, New York, 2006, p. vii. – José Hdz. Stgo. Sep 30 '15 at 6:38

My immediate reaction to this question was - the epigraph used for "Methods of Homological Algebra" by Gelfand and Manin

I find this a golden standard for good epigraphs - intriguing, cool, and truly holding the quintessence of the subject.

this seems intriguing enough:

Development of mathematics resembles a fast revolution of a wheel: sprinkles of water are flying in all directions. Fashion – it is the stream that leaves the main trajectory in the tangential direction. These streams of epigone works attract most attention, and they constitute the main mass, but they inevitably disappear after a while because they parted with the wheel. To remain on the wheel, one must apply the effort in the direction perpendicular to the main stream.

Though this has not really much to do with mathematics, you might be intrigued by the epigraph of Compact complex surfaces by Barth, Peters, Van de Ven (+ Hulek for the 2nd edition):

Par une belle matinée du mois de mai, une élégante amazone parcourait, sur une superbe jument alezane, les allées fleuries du Bois de Boulogne. (A. Camus, La peste)

(By a beautiful morning of May, an elegant horsewoman rode a superb sorrel mare along the flowery avenues of the Bois de Boulogne -- A. Camus, The Plague).

In the book, a secondary character is trying to write a novel, but he is stuck with the first sentence, which he tries to improve again and again -- the epigraph is one of these attempts. Of course it is supposed to give some indication on the difficulty of writing such a book...

• As an aside, this reminds me of "If on a winter's night a traveler", one of my all-time favorite books. – Simon Rose Sep 30 '15 at 9:20

The preface to the 2nd edition of John McCleary's A User's Guide to Spectral Sequences has the following epigraph:

"For I know my transgressions, …" (Psalm 51)

See p. 185ff for transgressions in spectral sequences. And note the title "Sins of Omission" for Part III of the book.

"Algebraic Number Theory" by Cassels and Froehlich was (so Birch told me once) one of the first mathematics books typeset by Academic Press. According to Birch, when he received the first offprints of his article there was "about one error per line, on average", due perhaps to the combination of the way books were printed in the 1960s (lumps of metal with symbols embossed on them, I guess) and the large collection of perhaps unfamiliar symbols used in a mathematics textbook (which presumably had to be physically made to print the book). This might go some way towards explaining why in the original preface the editors write

The editors must emphasize, however, that neither the lecturers nor the note-takers have any responsibility for any inaccuracies which may remain : they are an act of God.

In the early 2010s the book was reprinted by the London Mathematical Society and I was asked to collect up any known errata, which were inserted at the beginning of the book. So I got to choose my own epigraph, and as some sort of an attempted response to the original preface I went for

Consider what God has done: Who can straighten what he has made crooked?

[Ecclesiastes 7:13]

Reports of other typos/errors still come trickling in, so in particular I don't think anyone has succeeded just yet.

Here are some mathematical and quasi-mathematical head-of-chapter epigraphs that I used in my book The Big Questions:

To create a healthy philosophy, you should renounce metaphysics but be a good mathematician. --- Bertrand Russell
They are ill discoverers who think there is no land, when they can see nothing but sea. --- Sir Francis Bacon
The question of the actual or potential reach of the human mind when it comes to proving theorems in arithmetic is not like the question how high it is possible for humans to jump, or how many hot dogs a human can eat in five minutes, or how many decimals of $$\pi$$ it is possible for a human to memorize, or how far into space humanity can travel. It is more like the question of how many hot dogs a human can eat in five minutes without making a totally disgusting spectacle of himself, a question that will be answered differently at different times, in different societies, by different people. --- Torkel Franzen
If it is a Miracle, any sort of evidence will answer, but if it is a Fact, proof is necessary. --- Mark Twain
God exists since mathematics is consistent, and the Devil exists since we cannot prove it. --- André Weil
God wrote the Universe in the language of mathematics. --- Galileo Galilei
If a religion'' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one. --- John Barrow
The point of philosophy is to start with something so simple as not to seem worth stating and to end with something so paradoxical that no one will believe it. --- Bertrand Russell
It is hard enough to remember my opinions without also remembering my reasons for them. --- Friedrich Nietzsche
• Regarding John Barrow's one... I am thinking isn't it ironic that Godel's first completeness theorem is provable! – JP McCarthy Oct 1 '15 at 6:31
• The quote from Galilei is often presented in this way, but the original version of Galilei has actually a slightly different feel (and is much more poetic). – Olivier Oct 2 '15 at 8:11
• @Olivier I guess it is this one - La filosofia è scritta in questo grandissimo libro, che continuamente ci sta aperto innanzi agli occhi (io dico l'Universo), ma non si può intendere, se prima non il sapere a intender la lingua, e conoscer i caratteri ne quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi ed altre figure geometriche, senza i quali mezzi è impossibile intenderne umanamente parola; senza questi è un aggirarsi vanamente per un oscuro labirinto. – მამუკა ჯიბლაძე Jul 27 '17 at 4:39
• @ მამუკა ჯიბლაძე Yes, precisely. I find it interesting that, contrary to the common apocrypha, the actual quote makes the role and even form of mathematics precise and that, far from being theistic, it also makes it clear that it is because we are human that mathematics is required. – Olivier Jul 27 '17 at 5:27

Lévy [...] once remarked to me that reading other mathematicians’ research gave him actual physical pain.

A quotation from J.L. Doob, on Paul Lévy, used as an epigraph by Tom Leinster in A survey of definitions of n-category. Lightly amusing on first reading; more seriously so when one recalls it while going through that pain; most serious, and less amusing, when one tries to write with one’s readers’ pain in mind.

The two volumes Computational Commutative Algebra by Martin Kreuzer and Lorenzo Robbiano (Springer, 2000 and 2005) contain lots of epigraphs. Some of them are rather weird.

Here are some examples from Volume 1:

Dura Lex, sed Lex. (Ancient Latin Proverb)

(Section on term orderings)

Divide et impera. (Philip of Macedonia)

(Section on the division algorithm)

Eliminate, eliminate, eliminate. Eliminate the eliminators of elimination theory. (Shreeram S. Abhyankar)

(Section on, well, elimination)

No keyboard present. Hit F1 to continue. (DOS Error Message)

(Appendix on CoCoA)

• Can you give the most interesting examples? – Alexey Ustinov Oct 1 '15 at 5:23

One of my personal favourites, which describes a lot of what goes on in maths:

In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics.

-- Hermann Weyl

I think that many a MO-er will find very intriguing the epigraph at the beginning of chapter one of the book A=B by Marko Petkovšek, the late Herbert Wilf, and Doron Zeilberger:

The ultimate goal of mathematics is to eliminate any need for intelligent thought. — Alfred N. Whitehead

By the way, when I shared it in the thread on "Famous mathematical quotes", my answer received several comments on the accuracy and source of the quote; if you want to take a look at them, follow the link below:

https://mathoverflow.net/a/8238/1593

From Streater & Wightman, "PCT, Spin and Statistics, and All That" (Ch. 2):

In the thirties, under the demoralizing influence of quantum-theoretic perturbation theory, the mathematics required of a theoretical physicist was reduced to a rudimentary knowledge of the Latin and Greek alphabets. - R. Jost

Montesinos's Classical Tessellations and Three-Manifolds has an epigraph from Don Quixote in every chapter which is relevant to the content:

Preface: "You have said more than you know, Sancho", said Don Quixote, "for there are some who tire themselves out learning and proving things which, once learnt and proved, do not concern either the understanding or the memory a jot."

Chapter 1 ($$S^1$$- bundles over surfaces): Chapter LXXVII. In which a Thousand Trifles are recounted, as nonsensical as they are necessary to the True Understanding of this great History.

Chapter 2 (manifolds of tessellations on the Euclidean plane): Now, overlooking the courtyard of our prison were the windows of the house of a rich and important Moor, which, as is usual in Moorish houses, were more like loopholes than windows, and even so were covered by thick and close lattices.

Appendix A (orbifolds): Chapter LXVIII. In which we are told who the Knight of the Mirrors and his Squire were, and given some account of them.

Chapter 3 (manifolds of spherical tessellations): Chapter CII. What happened to Sancho Panza on the Rounds of his Isle.

Chapter 4 (Seifert manifolds): "Take care, your worship," said Sancho; "those things over there are not giants but windmills, and what seem to be their arms are the sails, which are whirled round in the wind and make the millstone turn."

Chapter 5 (manifolds of hyperbolic tessellations): Chapter CXXIII. Which follows the one hundred twenty-second and deals with matters indispensable for the clear understanding of this history.

Appendix B (the hyperbolic plane): As all human things, especially the lives of men, are transitory, being ever on the decline from their beginnings till they reach their final end, and as Don Quixote has no privilege from Heaven exempting him from the common fate, his dissolution and end came when he least expected it.

If we read Frege's Grundlagen der Arithmetik from the end, the following note in the appendix would be the most courageous of epigraphs all.

A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press.

I am charmed by the epigraphs in Herrlich and Strecker's book on Category Theory, where the first substantive chapter (titled "Foundations") opens with these lines from Dr. Seuss's On Beyond Zebra:

Because finally he said:

"This is really great stuff!

"And I guess the old alphabet

"ISN'T enough!"

and the Appendix (also titled "Foundations") closes with these lines from the same source:

So you see!

There's no end

To the things you might know,

Depending how far beyond Zebra you go!

Since both Knuth and Sherlock Holmes have already been mentioned here, I'd like to add this one:

You will, I am sure, agree with me $\dots$ that if page 534 finds us only in the second chapter, the length of the first one must have been really intolerable. -- SHERLOCK HOLMES, in The Valley of Fear (1888)

[D.E.Knuth, The Art of Computer Programming, Vol. 1, at the end of chapter 2 (page 463 in my edition)]

The fact that it occurs at the end rather than the beginning of the chapter it references, makes it a kind of dual of an epigraph.

The distinction between "epigraph" and mere "quote" is not entirely clear, since the latter may be used as the former. For example, I used Millay's "Euclid alone has looked on Beauty bare ..." as an epigraph to the Introduction of my Friendly Intro to Number Theory book, but of course, it did not originate as an epigraph. Similarly, I used "It is the star to every wandering bark, whose worth's unknown, although his height be taken" as an epigraph for my PhD thesis, and Marc Hindry and I had fun with such epigraphs for each chapter of our joint book.

Another one - not so much intriguing maybe but charming imo -

« La mer était étale mais le reflux commençait à se faire sentir. »

Hugo, Les travailleurs de la mer

("Étale cohomology" by J. S. Milne)

Flajolet-Sedgewick's Analytic Combinatorics is very rich in epigraphs, the most interesting being Painlevé's famous

Entre deux vérités du domaine réel, le chemin le plus facile et le plus court passe bien souvent par le domaine complexe.

("The shortest and easiest path between two truths of the real domain most often passes through the complex domain"), quite appropriately introducing the chapter on "Complex Analysis, Rational and Meromorphic Asymptotics".

Other pretty intriguing epigraphs I know of are:

When Gauss says he proved something, it is very probable... when Cauchy says it, you can bet equally well pro or contra, but when Dirichlet says it, it is certain. I prefer to leave myself out of this Delikatessen.

from Jacobi's letter to von Humboldt, at the beginning of the chapter on (of course) Dirichlet's theorem on arithmetic progressions in Pollack's Not Always Buried Deep, and

George Pólya once had a young mathematician confide to him that he was working on the great Riemann Hypothesis. "I think about it every day when I wake up in the morning", he said. Pólya sent him a reprint of a faulty proof that had once been submitted by a mathematician who was convinced he'd solved it, together with a note: "If you want to climb the Matterhorn you might first wish to go to Zermatt where those who have tried are buried".

by Lars Hörmander, in the chapter "Failed Attempts at Proof" of Borwein-Choi-Rooney-Weirathmueller's The Riemann Hypothesis.

A round man cannot be expected to fit in a square hole right away. He must have time to modify his shape. (Mark Twain)

This epigraph is used in Bredon's book Topology in Geometry. It is a very nice illustration of the idea of homotopy theory. Also nice is the epigraph used in the preface:

The golden age of mathematics - that was not the age of Euclid, it is ours. (Keyser, 1916)

From the book Inexhaustibility by Torkel Franzen, which is about, let's say, natural numbers!

A Druidic myth relates how Lucanor, coming upon the other gods as they sat at the banquet table, found them drinking mead in grand style, to the effect that several were drunk, while others remained inexplicably sober; could some be slyly swilling down more than their share?

The disparity led to bickering, and it seemed that a serious quarrel was brewing. Lucanor bade the group to serenity, stating that the controversy no doubt could be settled without recourse either to blows or to bitterness. Then and there Lucanor formulated the concept of numbers and enumeration, which heretofore had not existed. The gods henceforth could tally with precision the number of horns each had consumed and, by this novel method, assure general equity and, further, explain why some were drunk and others not. "The answer, once the new method is mastered, becomes simple!" explained Lucanor. "It is that the drunken gods have taken a greater number of horns than the sober gods, and the mystery is resolved." For this, the invention of mathematics, Lucanor was given great honour.

It occurred to me only now: it is absolutely unthinkable not to mention John Horton Conway in this context. His epigraphs I find unsurpassed. To every chapter in "On Numbers and Games", and as well in "Winning Ways for your Mathematical Plays" by Berlekamp, him and Guy.

I am actually tempted to reproduce all epigraphs in "On Numbers and Games". I'll try hard to omit at least some.

ZEROTH PART (ON NUMBERS . . .)

A Hair, they say, divides the False and True-
Yes; and a single Alif were the clue,
Could you but find it—to the Treasure-house,
And peradventure to The Master too!

The Rubaiyat of Omar Khayyam

CHAPTER 0 (All Numbers Great and Small)

Whatever is not forbidden, is permitted.
J. C. F. von Schiller, Wallensteins Lager

CHAPTER 1 (The Class No is a Field)

Ah! why, ye Gods, should two and two make four?
Alexander Pope, "The Dunciad"

CHAPTER 2 (The Real and Ordinal Numbers)

Don't let us make imaginary evils, when you know we have so many real ones to encounter.
Oliver Goldsmith, "The Good-Natured Man"

CHAPTER 3 (The Structure of the General Number)

We admit, in Geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth in the largest heads.
Voltaire, Article "Infinity", in A Philosophical Dictionary, Boston 1881

CHAPTER 4 (Algebra and Analysis of Numbers)

Now as to what pertains to these Surd numbers (which, as it were by way of reproach and calumny, having no merit of their own, are also styled Irrational, Irregular, and Inexplicable) they are by many denied to be numbers properly speaking, and are wont to be banished from Arithmetic to another Science (which yet is no science) viz., algebra.
Isaac Barrow, "Mathematical Lectures", 1734

CHAPTER 7 (Playing Several Games at Once)

For when the One Great Scorer comes
to write against your name,
He marks—not that you won or lost—
but how you played the game.

Grantland Rice,
Alumnus Football

And of course,

CHAPTER 8 (Some Games are Already Numbers)

"Reeling and Writhing, of course, to begin with," the Mock Turtle replied; '''And then the different branches of Arithmetic— Ambition, Distraction, Uglification, and Derision."
Lewis Carroll, "Alice in Wonderland".

OK I think I am still able to stop. I believe anybody who knows the book will agree with me - all these epigraphs fit the contents perfectly, and those I have omitted are as brilliant.

I was tempted to use this one for my PhD thesis (in the end I didn't, out of fear it might sound pretentious):

There is nothing so practical as a good theory (Ludwig Boltzmann)

I like how it reconciles the concepts of theory and practice. All too often the former is dismissed as "yes, that's what would happen in theory; but in practice..."

• In theory, there is no difference between theory and practice – but, in practice, there is. – Gerry Myerson Jul 26 '17 at 23:23
• Depends on what you mean by practice – timur Jul 31 '18 at 3:59

Qimh Xantcha, Binomial Rings: Axiomatisation, Transfer, and Classification, arXiv:1104.1931v3 begins with an epigraph so apt that I had to google it to make sure it was not invented:

At the age of twenty-one he wrote a treatise upon the Binomial Theorem, which has had a European vogue.

Sherlock Holmes’s description of Professor Moriarty;

Arthur Conan Doyle, The Final Problem

(The later version v4 omits this epigraph, along with some gorgeous illustrations of combinatorial principles in Example 3. Perhaps the referees were unwilling to let so much beauty go undisturbed...)

I like the epigraph to the book of "Thom Spectra, Orientability and Cobordism" by Yuli Rudyak:

First, tell what you are going to talk about,
then tell this, and then
tell what you have talked about.

(from "Manuals of a senior country priest for beginners")

Another nice epigraph is in the book "Discovering Modern Set Theory" of Just and Weese. This book contains a chapter entitled "How to read this book" which starts with the epigraph taken from a book of Saharon Shelah:

So we shall now explain how to read the book. The right way is to put it on your desk in the day, below your pillow at night, devoting yourself to the reading, and solving the exercises till you know it by heart. Unfortunately, I suspect the reader is looking for advice how not to read, i.e. what to skip, and even better, how to read only some isolated highlights.

Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. (John von Neumann, 1951)

Knuth, D. E. The art of computer programming. Volume 2. Ch. 3 Random numbers.

Sсhülег:
Kann Euch nicht eben ganz verstehen.

Mephistopheles:
Das wird nächstens schon besser gehen,
Wenn Ihr lernt alles reduzieren
Und gehörig klassifizieren

(Goethe, Faust)

Manin Ju. I. Theory of commutative formal groups over fields of finite characteristic. - Uspehi Mat. Nauk 18 1963 no. 6 (114), 3–90.

Bayard Taylor’s English translation of the latter (Faust, Part I, Scene IV):

Student:
I cannot understand you quite.

Mephistopheles:
Your mind will shortly be set aright,
When you have learned, all things reducing,
To classify them for your using.

Dennis Gaitsgory begins his "Outline of the proof of the geometric Langlands conjecture for $GL_2$" (available here) with the following interesting quote in German:

“In jedem Minus steckt ein Plus. Vielleicht habe ich so etwas gesagt, aber man braucht das doch nicht allzu wörtlich zu nehmen.” - Robert Musil, Der Mann ohne Eigenschaften.

• Can you explain what is hidden behind this epigraph? – Alexey Ustinov Oct 3 '15 at 2:07
• "There's a plus in every minus. Maybe I did say something like that but one doesn't need to take it so literally." – Kristal Cantwell Oct 3 '15 at 16:57