Use of the word "shall" in mathematical writing Sometimes I really want to use the word "shall" in my writing. But I get unnerved because I got a feeling it's pretentious or outdated. Any opinions?
Is it a nice variation on "will"?
Other alternatives?
 A: This is an English issue, not limited to math writing. Briefly, "shall" is rather more common in the UK then the USA. In practice they are more or less interchangeable in the context you describe.
To my (North American) ear, "shall" sounds a little old-fashioned, but not pretentious or outdated.
A: It is nothing to do with being stuck up or pretentious. The rule in English is that 'shall' ought to be used with 'I' and 'we', and 'will' ought to be used with any other pronoun (you, they, he, she, it). If you wanted emphasis, this was reversed: 'I will do it', 'You shall go to the ball'. Although nowadays this is increasingly ignored and 'will' seems to be used with every pronoun.
Interestingly, the occasion when you have to use 'shall' is when in the form of a question, i.e.: 'Shall I wash the dishes?' and not 'Will I wash the dishes?' which has a different meaning.
A: One thing to keep in mind is that good mathematical writing is (in part) just good writing. Overusing one word (without a good reason) makes your writing less interesting (Knuth specifically talks about this http://jmlr.csail.mit.edu/reviewing-papers/knuth_mathematical_writing.pdf). So shall (or some other synonym) could be useful in writing as a nice change of pace to break up a bunch of "will"s and to keep your word choice more interesting.
A: While user50229 is right about the complex rule that emerged in the 18th and 19th centuries an older sense of < shall > versus < will > might be simpler to revive. The reason that the 19th-century reversal used to distinguish the two words was courtesy. Because < shall > implies a degree of compulsion or duty, the courtesy rule was to impose an obligation on yourself, but allow volition (< will >) for others. 
An easy way to see the simpler, older rule is to look at the two conditional forms < should > and < would >, which are still used fairly regularly in their proper sense, with < should > implying some kind of obligation and < would > implying some kind of volition. 
There's little harm in using both interchangeably in maths writing to give variety, as Gage says, but the essence of the real difference is that < shall > and < should > are somewhat stronger in feel. 
