 12  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 |an | < s, where s 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 (p ^ (q ^ 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. 12 |
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