0
$\begingroup$

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!

Improve this question
$\endgroup$
8
  • $\begingroup$ Just an idle question... $\endgroup$
    – David Roberts ♦
    Commented Jul 2, 2011 at 1:40
  • 3
    $\begingroup$ The Wikipedia article says that it's not known whether $e$ is a period or not. I daresay $e$ is useful for applied mathematics... $\endgroup$
    – Qiaochu Yuan
    Commented Jul 2, 2011 at 1:43
  • 6
    $\begingroup$ 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-going-with-limerick response). $\endgroup$
    – Ramsey
    Commented Jul 2, 2011 at 2:16
  • $\begingroup$ Heh, alright, Qiaochu and Cam. Serves me right for being slack. The 'weak forms of pure mathematics' par of the question still stands, though. :) $\endgroup$
    – David Roberts ♦
    Commented Jul 2, 2011 at 2:46
  • 3
    $\begingroup$ I'm finding "applied mathematics" and "weak forms of pure mathematics" pretty vague or not terribly meaningful signifiers. $\endgroup$
    – Todd Trimble
    Commented Jul 2, 2011 at 11:43

1 Answer 1

Reset to default
3
$\begingroup$

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.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Thanks, Stefan. I can't believe I forgot that periods only form a ring, again. Please vote to close this question. $\endgroup$
    – David Roberts ♦
    Commented Jul 2, 2011 at 21:45

Not the answer you're looking for? Browse other questions tagged .