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!