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I often read in published mathematics papers
We prove the claim as follows.
I am wondering why mathematicians use "prove" instead of "will prove"? (I sometimes read papers using "will prove but they seems less popular.)
Is this because everything is written in a paper, so there is no "future" involved?
In scholarly writing, this is standard form. I have always assumed that the present tense indicates that the proof is to be demonstrated in the present, which is to say: as we read it from this point forward. Linguists may well employ a specific term for this use of the present tense, such as immediate present.
As pointed out in another answer, English verbs do not inflect to express action in the future (as many other languages do) and English thus lacks a true future tense. Nevertheless, we must, can, and do talk about actions in the future, e.g. by combining the auxiliary verbs "will" and "shall" with the unmarked infinitive; and it is common (and useful) to refer to such usage colloquially as the future tense.
This ersatz future tense is occasionally seen in an introductory paragraph, of the form:
In what follows, we will prove...
In an abstract, the present is most commonly used.
Note that there are many manuals of style which address this and other aspects of writing within a given discipline.
A useful summary is presented in answer to the question "In what tense (present/past) should papers be written?" at our sister site.
There is no future tense in English. Both we prove and we will prove can refer to future time. Using the simple present we prove (a) gives the sentence a sense of immediacy and (b) is the strongest way we have in English to talk about the certainty of something happening in the future. For these two reasons, some scientific papers and many other contexts may prefer the simple present.
Calling will prove the "future tense" is a misnomer and arbitrary. We have over a dozen ways to talk about future time and none of these are "the future tense," because English does not have a special way of inflecting verbs to mean the future. You could just as well say be going to is the "future tense." Will prove is a modal use, one of whose functions is to talk about future time. But that is not its only function and it is only one of many ways to talk about future time.
You may wish to read The usage of the present simple for the future actions and other related questions. I also highly suggest the following:
What do you mean there's no future tense? (Polysyllabic)
English Has No Future Tense (The Ecphorizer)
Why does English not have a future tense? (Quora)
Future tense (Wikipedia)
