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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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closed as off topic by Loop Space, Felipe Voloch, Kevin Buzzard, Mark Sapir, quid Dec 25 '11 at 19:17

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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
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Maybe I should expand the question to include colorful language cut from serious mathematics papers :) –  John Stillwell Apr 23 '10 at 5:18
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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
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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
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@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

108 Answers 108

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.

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Heh. I am sure this was discovered by coincidence and kept by design. –  Yemon Choi Apr 25 '10 at 0:49
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Greg, congratulations on the great answer badge! –  John Stillwell Sep 17 '10 at 18:06
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Greg, I've just had a look at Hubbard's Teichmüller Theory, and a wonderful book it is. But, alas, I think your memory has deceived you, because his abbreviation for the category of Banach analytic manifolds (page 165) is in fact BanMan. –  John Stillwell Mar 28 '11 at 23:31
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Hmm, since my observation came from a course with Prof. Hubbard using a preprint of the book, I guess he changed it before publication. Thats a little disappointing. –  Greg Muller Mar 29 '11 at 16:27
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It has belatedly struck me that there should really be a contravariant equivalence of categories between Ban(Ana)Man and some category of algebraic objects, which could be abbreviated to ERIC. –  Yemon Choi Aug 23 '11 at 0:56

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]

Andreas Blass

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+1 A very nice alternative for using "it's easy to show", "trivial", "as one easily checks" etc. –  Johannes Hahn Apr 25 '10 at 11:24

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"

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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"

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Dear Charles: Ann. of Math. 69, 1959, pages 247-252. –  Georges Elencwajg Apr 23 '10 at 15:27
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This was a letter to the editor, not a math paper. –  KConrad Apr 23 '10 at 15:36
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Wow, I'll remember that one for some time. –  Pete L. Clark Apr 23 '10 at 20:12

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

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Or that they have dedicated some other book to themselves (perhaps secretly?) –  Will Sawin Jan 9 '12 at 23:07

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.

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I was a TA on a course taught by Sarason himself following this book. I had two students "solve" that Exercise during one of my office hours. –  Alfonso Gracia-Saz May 6 '10 at 5:02
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This reminds me of an exercise (C.67) from the chapter about linear transformations in Peter Hackman's Linear Algebra textbook Kossan (mai.liu.se/~pehac/titta/kossaboken). It's in Swedish, but here's an attempt to translate it: "Seize the ends of a pointer between your extended arms, and turn yourself an angle of $v$ radians about your own vertical axis. What have you proved then? Try the same maneuver with the pointer in your right hand, aligned with your straigh arm. Show this to someone who has never studied Linear Algebra. Interpret the result." –  Hans Lundmark Jul 1 '10 at 8:25
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Just a little correction: It's Exercise VIII.8.3, not VII.8.3. –  Irene Adler Aug 18 '10 at 8:48

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 fomulae." 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.

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Did you use (expletive deleted) for "damn", or was it rather more colorful? –  Harry Gindi Apr 23 '10 at 22:16
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Harry, it's here books.google.com/… and the text itself has (expletive deleted) –  Charles Siegel Apr 24 '10 at 0:58
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Yes, I disagree with Adams about double cosets. Then again the colourful stories about him that came out after he died seemed to me to indicate that I disagreed with him about a number of things (for example the merits of attacking people with axes) –  Kevin Buzzard Apr 24 '10 at 8:13
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@Kevin: Hmm, I don't know which side of the axe issue you stand on. You know what -- it'll come up eventually. Why don't you surprise me? –  Pete L. Clark Apr 24 '10 at 15:23
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I hope this is not seen as mean-spirited, but some years ago, once when I mentioned Adams over coffee (probably in the context of his Lie Groups book) someone asked if I'd heard the joke about the "unstable Adams spectral sequence". (It tickled my fancy; everyone else went back to talking about traffic or football.) –  Yemon Choi Apr 25 '10 at 0:54

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.

Michael Barr

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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.

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isn't there some humorous intention in the choice of the name "pointless topology" ? –  Pietro Majer Jan 17 '11 at 23:54
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+1 for Pooh, but: It should be remarked that Woozle-Hunting is a rather poor proof technique, given that it involves going in circles for a Long Time, and ends without capturing any Woozles at all. In fact, a proof by Woozle-hunt (that actually proved something) would be a remarkable achievement. –  Ketil Tveiten Feb 4 '11 at 9:27

A paper of David Bachman-Cooper-White describes a proof that a hyperbolic 3-manifold containing large embedded balls has large Heegaard genus. As they say at the end of the introduction,

a proper subset of the authors wish to subtitle this paper “Big balls imply big genus”

whch is indeed the best way to memorize the result.

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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."

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Yes, although is good to remember that this is an unpublished manuscript, and that Conway "has never approved of this exposition, which he regards as full of fluff." I think this paper would benefit itself immensely if 20 or so pages were left out. –  Andres Caicedo May 16 '10 at 14:39

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."

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For what it's worth, Timothy Chao had a piece "You could have invented spectral sequences" in the Notices (ams.org/notices/200601/fea-chow.pdf) that claims the name "spectral" is from some sort of analogy with eigenvalues. –  Michael Lugo Nov 5 '10 at 21:10
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I thought it was Timothy Chow? –  Todd Trimble Oct 22 '11 at 21:13

According to http://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.

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I wonder if this was a joke or if it was required by the BoP, just as some funding agencies require a similar note. –  Mariano Suárez-Alvarez Oct 21 '10 at 13:46

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.

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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:

American Mathematical Monthly volume 77 (1970) p. 78

American Mathematical Monthly volume 77 (1970) p. 78

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I couldn't possibly know, but my suspicion would be that a copyeditor bowdlerized the article without the author's knowledge or permission, and that the author, upon finding out, complained strongly enough for the magazine to give in and publish the correction. –  Ilmari Karonen Aug 29 '11 at 13:52

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.

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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!

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Truesdell is also the author of the single best Math Review ever: "In this paper are presented incorrect solutions to trivial problems. The basic error, however, is not new." –  Allen Knutson Apr 25 '10 at 16:05
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If you have access to MathSciNet, here's the review: ams.org/mathscinet-getitem?mr=39515 –  Jonas Meyer Apr 30 '10 at 6:47
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To be nitpicky, the quotation is not quite right. The exact words are "This paper, whose intent is stated in its title, gives wrong solutions to trivial problems. The basic error, however, is not new: [...]." –  Hans Lundmark Jul 1 '10 at 8:06
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I believe that Bass gets credit for a book review with the line- "this book fills a much needed gap in the literature". –  aginensky Jan 12 '11 at 2:52
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@aginesky: Actually the "much needed gap" is due to my colleague Lee Neuwirth. He put it in a review that the wrote either as a grad student or recent post-doc. Ralph Fox (his advisor) read it and roared with laughter. It was excised from the published version, but quickly made the rounds. –  Victor Miller Jan 16 '11 at 16:03

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.

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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)."

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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.

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"A differential geometer whose work often uses the simplifications obtained by considering the complex domain explained to me that the additional structure of complex manifolds makes them more interesting, just as two sexes are more interesting than one, but various aspects of this argument are open to debate." Volume 5, pg. 394, 3rd edition. –  Pait Jul 5 '11 at 16:44

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.

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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.

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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.

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Works in my building. –  Louigi Addario-Berry Sep 15 '10 at 18:46
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But then going home transposes the matrix. –  Gerry Myerson Sep 16 '10 at 0:41
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Unless there are two elevators in the building... –  Sándor Kovács Oct 21 '10 at 13:47

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 adviced not to commit seppuku instantly if he feels he does not quite understand 2. of chapter 1.

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1  
:D :D quote two made me laugh out loud. –  David Roberts Mar 11 '11 at 0:58

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'.

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Kato lectures like this too. His lectures (at least in the 1990s) often used to start with various bits of philosophy of this nature. I remember vividly his explaining at the IAS that the reason Bloch and Beilinson constructed the right zeta elements in K_2 was that they had very large mouths and loved their wives (and then a long explanation of why these things were relevant, which unfortunately this margin won't contain). It wouldn't surprise me if these comments ended up in print at some point---that's Kato. –  Kevin Buzzard Apr 24 '10 at 8:16
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Kato teaches like this as well. I remember him teaching theta functions, circa 2004, and coming through as a member of some strange cult (to me at least). Lots of mysticism, lots of references to the occult and kabbala and how the theta function is part of some spiritual realm, and the search for the "true theta function". –  Daniel Moskovich Apr 29 '10 at 4:43
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That is on the poster for the log-conf in Bordeaux in June. There's a picture to go with it, see math.u-bordeaux1.fr/Log_Conf_2010 –  Laurent Berger Apr 30 '10 at 15:40

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.

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@Mariano I've read the original, which is an outstanding book BTW, but the colourfulness is pretty much (90% at least) Iohara's in my opinion. Or perhaps, it sounds better in Japanese (the translator isn't making stuff up, but he certainly makes it sound more colourful than it was). Contrary to what my appearance might suggest, I speak and read Japanese fluently. –  Daniel Moskovich Apr 29 '10 at 7:19

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
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voting down per your wishes... –  S. Carnahan Jun 8 '10 at 0:45
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Now we just have to get your comment to 27 upvotes as well. –  Steven Gubkin Jun 8 '10 at 13:05
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+1 but not voted up! –  Abhishek Parab Jun 8 '10 at 13:27
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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

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."

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How a television, two typewriters, a ring and several pieces of cheese got into Chapter 8, I'll never know. Sorry, just found myself channelling Groucho Marx for a minute there. –  Gerry Myerson Aug 4 '11 at 5:48

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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