Ingo Brigandt

We give an answer to the question as to whether quantifier elimination is possible in some infinite algebraic extensions of Qp (‘infinite p-adic fields’) using a natural language extension. The present paper deals with those infinite p-adic fields which admit only tamely ramified algebraic extensions (so-called tame fields). In the case of tame fields whose residue fields satisfy Kaplansky’s condition of having no extension of p-divisible degree quantifier elimination is possible when the language of valued fields is extended by the power predicates Pn introduced by Macintyre and, for the residue field, further predicates and constants. For tame infinite p-adic fields with algebraically closed residue fields an extension by Pn predicates is sufficient.

In this chapter I lay out a notion of philosophical naturalism that aligns with pragmatism. It is developed and illustrated by a presentation of my views on natural kinds and my theory of concepts. Both accounts reflect a methodological naturalism and are defended not by way of metaphysical considerations, but in terms of their philosophical fruitfulness. A core theme is that the epistemic interests of scientists have to be taken into account by any naturalistic philosophy of science in general, and any account of natural kinds and scientific concepts in particular. I conclude with general methodological remarks on how to develop and defend philosophical notions without using intuitions.