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Имейте в виду правила П. Халмоша

Выдающийся американский математик П. Халмош написал много работ, адресованных широкой публике и посвященных технике научной работы. Одна из наиболее известных таких его статей - How to Write Mathematics - содержит много полезных рекомендаций. Вот некоторые из них.

Write Good English

...Good English style implies correct grammar, correct choice of words, correct punctuation, and, perhaps above all, common sense. There is a difference between "that" and "which", and "less" and "fewer" are not the same, and a good mathematical author must know such things. The reader may not be able to define the difference, but a hundred pages of colloquial misusage, or worse, has a cumulative abrasive effect that the author surely does not want to produce....

Honesty Is the Best Policy

The purpose of using good mathematical language is, of course, to make the understanding of the subject easy for the reader, and perhaps even pleasant. The style should be good not in the sense of flashy brilliance, but good in the sense of perfect unobtrusiveness. The purpose is to smooth the reader's way, anticipate his difficulties and to forestall them. Clarity is what's wanted, not pedantry; understanding, not fuss....

Down with the Irrelevant and the Trivial

...The first question is where the theorem should be stated, and my answer is: first. Don't ramble on in a leisurely way, not telling the reader where you are going, and then suddenly announce "Thus we have proved that...". Ideally the statement of a theorem is not only one sentence, but a short one at that....

The Editorial We Is Not All Bad

...Since the best expository style is the least obtrusive one, I tend nowadays to prefer neutral approach. That does not mean using "one" often, or ever; sentences like "one has thus proved that ..." are awful. It does mean the complete avoidance of first person pronouns in either singular or plural. "Since p, it follows that q." "This implies p." "An application of p to q yields r." Most (all ?) mathematical writing is (should be ?) factual; simple declarative sentences are the best for communicating facts.

A frequently effective and time-saving device is the use of the imperative. "To find p, multiply q by r." "Given p, put q equal to r." (Two digressions about "given". (1) Do not use it when it means nothing. Example: "For any given p there is a q." (2) Remember that it comes from an active verb and resist the temptation to leave it dangling. Example: Not "Given p, there is a q", but "Given p, find q".)

There is nothing wrong with the editorial "we", but if you like it, do not misuse it. Let "we" mean "the author and the reader" (or "the lecturer and the audience")....

Use Words Correctly

...in everyday English "any" is an ambiguous word; depending on context it may hint at an existential quantifier ("have you any wool?", "if anyone can do it, he can") or a universal one ("any number can play"). Conclusion: never use "any" in mathematical writing. Replace it by "each" or "every", or recast the whole sentence.... "Where" is usually a sign of a lazy afterthought that should have been thought through before. "If n is sufficiently large, then |a_n| < e, where e is a preassigned positive number"; both disease and cure are clear. "Equivalent" for theorems is logical nonsense.... As for "if ... then ... if ... then", that is just a frequent stylistic bobble committed by quick writers and rued by slow readers. "If p, then if q, then r." Logically all is well (pЮ(qЮ r)), but psychologically it is just another pebble to stumble over, unnecessarily. Usually all that is needed to avoid it is to recast the sentence, but no universally good recasting exists; what is best depends on what is important in the case at hand. It could be "If p and q, then r", or "In the presence of p, the hypothesis q implies the conclusion r", or many other versions.

Use Technical Terms Correctly

...I belong to the school that believes that functions and their values are sufficiently different that the distinction should be maintained.

"Sequence" means "function whose domain is the set of natural numbers." When an author writes "the union of a sequence of measurable sets is measurable" he is guiding the reader's attention to where it doesn't belong. The theorem has nothing to do with the firstness of the first set, the secondness of the second, and so on; the sequence is irrelevant. The correct statement is that "the union of a countable set of measurable sets is measurable" (or, if a different emphasis is wanted, "the union of a countably infinite set of measurable sets is measurable"). The theorem that "the limit of a sequence of measurable functions is measurable" is a very different thing; there "sequence" is correctly used.

I have systematically and always, in spoken word and written, use "contain" for О and "include" for М . I don't say that I have proved anything by this, but I can report that (a) it is very easy to get used to, (b) it does no harm whatever, and (c) I don't think that anybody ever noticed it.

Consistency, by the way, is a major virtue and its opposite is a cardinal sin in exposition....

Resist Symbols

...The best notation is no notation; whenever it is possible to avoid the use of complicated alphabetic apparatus, avoid it....

The rule of never leaving a free variable in a sentence, like many of the rules I am stating, is sometimes better to break than to obey. The sentence, after all, is an arbitrary unit, and if you want a free "f" dangling in one sentence so that you may refer to it in a later sentence in, say, the same paragraph, I don't think you should necessarily be drummed out of the regiment. The rule is essentially sound, just the same, and while it may be bent sometimes, it does not deserve to be shattered into smithereens....

Use Symbols Correctly

...How are we to read "О": as the verb phrase "is in" or as the preposition "in"? Is it correct to say: "For x О A, we have x О B", or "If x О A, then x О B"? I strongly prefer the latter (always read "О" as "is in") and I doubly deplore the former (both usages occur in the same sentence). It's easy to write and it's easy to read "For x in A, we have x О B"; all dissonance and all even momentary ambiguity is avoided. The same is true for "М" even though the verbal translation is longer, and even more true for "і". A sentence such as "Whenever a possible number is і, its square is і 9" is ugly.

Not only paragraphs, sentences, words, letters, and mathematical symbols, but even the innocent looking symbols of standard prose can be the source of blemishes and misunderstandings; I refer to punctuation marks. A couple of examples will suffice. First: an equation, or inequality, or inclusion, or any other mathematical clause is, in its informative content, equivalent to a clause in ordinary language, and, therefore, it demands just as much to be separated from its neighbors. In other words: punctuate symbolic sentences just as you would verbal ones. Second: don't overwork a small punctuation mark such as a period or a comma. They are easy for the reader to overlook, and the oversight causes backtracking, confusion, delay. Example: "Assume that a О X. X belongs to the class C, ...". The period between the two X's is overworked, and so is this one: "Assume that X vanishes. X belongs to the class C, ...". A good general rule is: never start a sentence with a symbol. If you insist on starting the sentence with the mention of the thing the symbol denotes, put the appropriate word in apposition, thus: "The set X belongs to the class C, ...".

The overworked period is no worse than the overworked comma. Not "For invertible X, X* also is invertible", but "For invertible X, the adjoint X* also is invertible". Similarly, not "Since p № 0, p О U", but "Since p № 0, it follows that p О U". Even the ordinary "If you don't like it, lump it" (or, rather, its mathematical relatives) is harder to digest than the stuffy-sounding "If you don't like it, then lump it"; I recommend "then" with "if" in all mathematical contexts. The presence of "then" can never confuse; its absence can....