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Reading this question, and the Wikipedia page on reverse mathematics, I wonder whether one needs more than the subfield $\mathcal{P} \subset \mathbb{C}$ of periods for applied mathematics, or indeed weak forms of pure mathematics.

Edit: This was the sort of question one poses to friends over a coffee, and be quickly reminded that again, somehow, one forgot that periods only form a ring. With some time passed, I think I shall vote to close it and leave it as a warning to others: this is not a good MO question!

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closed as not a real question by David Roberts, Qiaochu Yuan, Aaron Meyerowitz, Tom Church, Todd Trimble Jul 3 '11 at 3:38

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

Just an idle question... – David Roberts Jul 2 '11 at 1:40
The Wikipedia article says that it's not known whether $e$ is a period or not. I daresay $e$ is useful for applied mathematics... – Qiaochu Yuan Jul 2 '11 at 1:43
There once was a field named $\mathbb{Q}$, whose completions had among them "you". Your extension is $\mathbb{C}$, which has periods you see, but is algebraically closed too! (I guess "you" are $\mathbb{R}$ in this off-the-cuff-perhaps-not-even-worthy-of-a-comment-but-somehow-it's-what-I'm-goin‌​g-with-limerick response). – Ramsey Jul 2 '11 at 2:16
Heh, alright, Qiaochu and Cam. Serves me right for being slack. The 'weak forms of pure mathematics' par of the question still stands, though. :) – David Roberts Jul 2 '11 at 2:46
I'm finding "applied mathematics" and "weak forms of pure mathematics" pretty vague or not terribly meaningful signifiers. – Todd Trimble Jul 2 '11 at 11:43
up vote 3 down vote accepted

If my knowledge is sufficiently up to date, it is not known whether the periods form a field at all. So I would say yes! We need more than the periods. Even applied mathematics benefits from the stuctural simplicity of certain objects. And as Qiaochu pointed out, it is not known whether $e$ is a period.

In general I would say that having a field like $\mathbb C$ that is relatively easy to deal with is rather benificial compared to having your theory based on something that is technically difficult to handle such as the periods.

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Thanks, Stefan. I can't believe I forgot that periods only form a ring, again. Please vote to close this question. – David Roberts Jul 2 '11 at 21:45

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