Confusion about mathematical writing? 
Is it true that short forms like "haven't", "don't", "let's" should not be used in serious mathematical texts?

 A: There is nothing inherently wrong with using contractions in mathematical writing; it's a matter of style.
Using contractions tends to make writing more conversational, since people usually do use contractions when speaking with each other.  Conversely, avoiding contractions results in text that is more formal, perhaps even cold.  These distinctions are probably more obvious to native English speakers, and in some cases non-native authors may not recognize the effect of such subtle choices.
There is a spectrum of style between formal and informal, and every author is a bit different.  In general I think more authors prefer to write more formally, and thus tend to avoid contractions, but this may be shifting.
A: I personally avoid them, but I saw many published papers full of "don't"'s.
Much depends on the editors: once the editor told me that "do not have" is bad english (he/she was clearly british).
So, unless you are publishing on the Slang Math. Journal, you'd better use the long forms.
A: I do get annoyed when semi-literate editorial assistants insist on changing "don't" to "do not" and thereby change the rhythm of a sentence. Don't write what doesn't sound good to the ear; if the natural shape of a sentence seems to call for a contraction, then use it. And I say this as someone who edited a journal for twelve years, and having just sent another book off to press with oodles of contractions. 
If you find writing with zest and lively naturalness means using occasional contractions, do so (within reason). If a stodgy editor tells you to remove the "don't"s and "won't"s later, then you might have to sigh and conform. But the more people send in articles in a lively readable style the better, and the more chance of journals relaxing their sometimes over-conservative style guides. 
A: As long as you can clearly convey your message, it does not matter whether you use contractions or not.
When publishing an article, thesis or another formal document, keep in mind that a good number of your readers will not know the language to the level where these subtleties begin to elicit a change in their thought process while reading. That is, they will not perceive the "informality" of your writing.
In my opinion, when writing mathematics, what is most important is that the writer transmit their written message in a way that is as clear and concise as our elegant equations.
