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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.]

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    $\begingroup$ I'll stick with the classical $\{(x,y) : y \geq f(x)\}$. $\endgroup$ – Federico Poloni Sep 29 '15 at 10:01
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    $\begingroup$ @FedericoPoloni: however, this one is not proper :-) $\endgroup$ – M.G. Sep 29 '15 at 10:41
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    $\begingroup$ 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. $\endgroup$ – Olivier Sep 29 '15 at 11:31
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    $\begingroup$ 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 $\endgroup$ – user9072 Sep 29 '15 at 13:37
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    $\begingroup$ Isn't this the epitome of opinion-based questions? $\endgroup$ – Daniel Griscom Sep 29 '15 at 15:50

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I personally am fond of the epigraphs in Zettl, Anton. Sturm-liouville theory. No. 121. American Mathematical Soc., 2010. Here are a selected few:

(Chapter 10: Singular Self-Adjoint Operators)
One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it.
In H. Eves Mathematical Circles Revisted, Boston: Prindle, Weber and Schmidt, 1971

Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country
In H. Eves Mathematical Circles Revisted, Boston: Prindle, Weber and Schmidt, 1971

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(Chapter 11: Singular Indefinite problems)
Biographical history, as taught by public schools, is still largely a history of boneheads: ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals - the flotsam and jetsam of historical currents. The men who radically altered history, the great scientists and mathematicians are seldom mentioned, if at all.
Martin Gardner

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(Chapter 13: Two Intervals)
Each problem I solved became a rule which served afterwards to solve other problems.
René Descartes

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Go back to An-Fang, the Peace Square at An-Fang, the beginning place at An-Fang, where all things start (...) An-Fang was near a city, the only living city with a pre-atomic name (...) the headquarters of the people programmer was at An-Fang, and there the mistake happened : a ruby trembled. Two tourmaline nets failed to rectify the laser beam. A diamond noted the error. both the error and the correction went into the general computer. -Cordwainer Smith, The Dead Lady Of Clown Town, 1964

Epigraph to Jean-Yves Girard, "Locus Solum" http://iml.univ-mrs.fr/~girard/0.pdf

There was no turning back after I read that one.

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I'm fond of the one from Volume 1 of Zariski and Samuel's Commutative Algebra (Springer-Verlag, New York, 1958):

Le juge: Accusé, vous tâcherez d'être bref.

L'accusé: Je tâcherai d'être clair.

        —G. Courteline

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