Reading [this question](https://mathoverflow.net/questions/54818/consistency-strength-needed-for-applied-mathematics), and the [Wikipedia page](http://en.wikipedia.org/wiki/Reverse_mathematics) 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!



Do we need more than the periods?
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.


