One way of thinking of a structure is as a certain sort of set. So when we discuss the properties of structures satisfying the axioms of set theory, we seem already to be presupposing the notion of set. This is a version of an objection that is sometimes called Poincar ´ e’s petitio because Poincar ´ e (1906) advanced it against an attempt that had been made to use mathematical induction in the course of a justification of the axioms of arithmetic. In its crudest form this objection is easily evaded if we are sufficiently clear about what we are doing. There is no direct circularity if we presuppose sets in our study of sets (or induction in our study of induction) since the first occur- rence of the word is in the metalanguage, the second in the object language. Nevertheless, even if this is all that needs to be said to answer Poincar ´ e’s ob- jection in the general case, matters are not so straightforward in the case of a theory that claims to be foundational. If we embed mathematics in set the- ory and treat set theory implicationally, then mathematics — all mathematics — asserts only conditional truths about structures of a certain sort. But our metalinguistic study of set-theoretic structures is plainly recognizable as a spe- cies of mathematics. So we have no reason not to suppose that here too the correct interpretation of our results is only conditional. At no point, then, will mathematics assert anything unconditionally, and any application of any part whatever of mathematics that depends on the unconditional existence of mathematical objects will be vitiated.