Tips For Authors

If you write clearly, then your readers may understand your mathematics and conclude that it isn't profound. Worse, a referee may find your errors. Here are some tips for avoiding these awful possibilities.

1. Never explain why you need all those weird conditions, or what they mean. For example, simply begin your paper with two pages of notations and conditions without explaining that they mean that the varieties you are considering have zero-dimensional boundary. In fact, never explain what you are doing, or why you are doing it. The best-written paper is one in which the reader will not discover what you have proved until he has read the whole paper, if then.
2. Refer to another obscure paper for all the basic (nonstandard) definitions you use, or never explain them at all. This almost guarantees that no one will understand what you are talking about (and makes it easier to use the next tip).
3. When having difficulties proving a theorem, try the method of "variation of definition"---this involves implicitly using more that one definition for a term in the course of a single proof.
4. Use c, a, b respectively to denote elements of sets A, B, C .
5. When using a result in a proof, don't state the result or give a reference. In fact, try to conceal that you are even making use of a nontrivial result.
6. If, in a moment of weakness, you do refer to a paper or book for a result, never say where in the paper or book the result can be found. In addition to making it difficult for the reader to find the result, this makes it almost impossible for anyone to prove that the result isn't actually there. Alternatively, instead of referring to the correct paper for a result, refer to an earlier paper, which contains only a weaker result.
7. Especially in long articles or books, number your theorems, propositions, corollaries, definitions, remarks, etc. separately. That way, no reader will have the patience to track them down.
8. Write A==>B==>C==>D when you mean (A==>B)==>(C==>D), or (A==>(B==>C))==>D, or .... Also, always muddle your quantifiers.
9. Begin and end sentences with symbols wherever possible. Since periods are almost invisible (and may be mistaken for a mathematical symbol), most readers won't even notice that you've started a new sentence. Also, where possible, attach superscripts signalling footnotes to mathematical symbols rather than words.
10. Write "so that" when you mean "such that" and "which" when you mean "that". Always prefer the ambiguous expression to the unambiguous and the imprecise to the precise. It is the readers task to determine what you mean; it is not yours to express it.

Notes

... she writes on deeply technical matters in clear English without jargon. This does not inspire confidence. Obscurity, besides obscuring incomplete thought, often suggests that the thought was quite deep.

---J.K. Galbraith.

Those who know that they are profound strive for clarity. Those who would like to seem profound strive for obscurity.

---Nietzsche.

Regarding (1),

A recent (published) paper had near the beginning the passage `The object of this paper is to prove (something very important).' It transpired with great difficulty, and not till near the end, that the `object' was an unachieved one.

---Littlewood's Miscellany, p57.

An example of (3),

Crumbs are better than nothing; nothing is better than steak; therefore crumbs are better than steak.

Regarding (4),

It is said of Jordan's writings that if he had 4 things on the same footing (as a,b,c,d ) they would appear as a, M ₃ ', &epsilon ₂ , &Pi _1,2 .

---Littlewood's Miscellany, p60.

Regarding (5),

It is well known that to describe a result as well-known without giving either the proof or a reference is neither pleasing nor helpful to the reader.

J.W.P. Hirschfeld, Bull. Amer. Math. Soc. 27 (1992), p331.

Regarding (7),

One practical criticism applies to this book as well as a large part of contemporary mathematical production: the various statements are called by different names, such as Lemma, Theorem, Proposition, Corollary; the first three are numbered independently of each other, while the numbers assigned to corollaries are functions of several variables; in addition, numbered formulae have their own separate numeration. The strain placed on the reader by this partial ordering is obvious, but apparently readers seek vengeance on other readers when they turn into authors.

---I. Barsotti, MR 23#A2419.

Regarding (8),

In testifying before the House Appropriations Committee, the AMS president Felix Browder said: This poses difficult mathematical problems since all data sets do not have similar characteristics...

The QNS (quantifier negation syndrome) strikes again (Notices AMS, October 2000, p1041).
Compare
   Not all boys like mathematics.
with
   All boys do not like mathematics.

Regarding (10),

In standard English so that is used to denote result or logical consequence, and sometimes equals "in order that". The frequent confusion in the U.S. of so that with such that probably results from the corruption of the language from German.
We now take x to be a positive real number so that the square root of x is again real.[Correct]
We now take x to be a positive real number so that x^2+3 is greater than 4.[Incorrect]
We now take x to be a positive real number such that x^2+3 is greater than 4.[Correct]
We now take x to be a positive real number so that x^2+3 is greater than 3.[Correct, when "positive" is taken to mean "greater than zero".]

...careless use of "that" and "which" blurs the distinction between hypotheses and remarks, use of dangling participles leads to quantifiers out of order, and sloppy use of negatives and quantifiers can leave the reader totally confused.
         Anthony Knapp, Notices AMS 47, no. 11 (December 2000), p. 1356.

That is the defining, or restrictive pronoun, which the nondefining, or nonrestrictive.

The lawn mower that is broken is in the garage. (Tells which one.)
The lawn mower, which is broken, is in the garage. (Adds a fact about the only mower in question.)

Strunk and White, 3rd Edn, p59.

For an example of imprecise writing, see your (US) tax form, which says "combine" lines x and y when it means add lines x and y.

Compare:

A graph is r -regular if r edges e have origin( e)=v for all vertices v .

A graph is r -regular if each vertex is the origin of exactly r edges.

An example of a construction to avoid:

Two women were in a SoHo boutique when they were approached by an eager salesman.
"Ladies," he said. "If there's anything you need, I'm Nick."
"And if we don't need anything, who are you then?" one of the "ladies" asked.

New York Times (Metropolitan Diary), July 19, 1998.

For more tips, see: Reuben Hersh, How to do and write math research, Math. Intelligencer, 19 no2, 1997, p59.