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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 mathematical writing, the custom is to refer to anything in the remainder of the paper in the present tense. I don't believe there is any grammatical reason for it, and the custom may be different in other disciplines. – Peter Shor Jul 29 '16 at 12:10
  • You've already proven it, which led to writing the paper. "We prove the claim as follows" summarizes that the paper describes what you did to prove it. Since the proving has already been completed and you are only reporting it, the actual proving is past tense and demonstrating the proof is present tense; it's always the present for the reader at reading time ("prove" is used here to mean demonstrate). There is no need for future tense ("will prove"). Beyond that, it's a matter of style preferences, which aren't really English rules. – fixer1234 Jun 8 '17 at 1:01
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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.

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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)

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    Most of the "answer" is a rant about what the construction "will prove" should be called. There are only two sentences here that actually address the question. – Peter Shor Jun 6 '17 at 21:37
  • @PeterShor Is there any difference in scholarly writing (specifically in mathematical writing, about which I know naught) between the meanings of "In what follows, we prove..." and "In what follows, we will prove..."? I've always assumed that they mean the same thing, but that assumption is not based on anything but intuition! – P. E. Dant Jun 7 '17 at 5:27
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    @P.E. Dant: They mean the same thing. Several authorities suggest that the present and not the future should be used for this, but in actual fact people use both. I would strongly advise against using the future in the abstract. The abstract is not supposed to be an introduction, but is supposed to be an executive summary of the paper, and using the future doesn't make any grammatical sense in it. I checked the abstract of around 20 papers in Annals of Math, and not one of them used the future. – Peter Shor Jun 7 '17 at 10:58

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