### Conclusion

We have argued that mathematical statements are *a posteriori synthetic* statements because they are ultimately based on *empirical* facts or on *empirical* hypotheses. In some cases, mathematical statements can be verified by direct inspection of an empirical model of the relevant mathematical structure. In other cases, they must be inferred from the axioms of the relevant axiomatic theories, on the assumption that these axioms are *consistent*. Yet, the consistency of these axioms is always an *empirical* assumption: it may admit of empirical *verification* (by exhibiting an empirical model for these axioms), or at least it will admit of potential empirical *falsification* (by deriving contradictory implications from them).

On the other hand, mathematical statements are a posteriori synthetic statements of a very special sort, because they do not depend on the *contingent* features of the empirical world, but rather are logically *necessary* properties of the mathematical structures realized or potentially realized in the empirical world (as shown by the invariance of these properties under isomorphism). Moreover, even though they are synthetic statements, they resemble analytic statements in being logical consequences of the *axiomatic definition* of the mathematical structure they are dealing with. For this reason, we have proposed the term *structure-analytic* statements to describe them.

Their logical status as structure-analytic statements gives them a logical position *intermediate* between truly analytic statements and ordinary empirical statements. This explains the nontrivial and nontautological character of many important mathematical theorems, which often gives them the quality of *a priori* quite unexpected “brute facts.” This is an aspect of mathematics very hard to explain on the logical positivist assumption that mathematical statements are *truly* analytic.

We have also discussed some of the philosophic and mathematical problems posed by various limitational theorems. We have argued that for Peano arithmetic the danger of inconsistency can be minimized (though it cannot be fully eliminated), and the problem of noncategoricity can be fully overcome, by stating it in the form of a quantifier-free recursive theory.

On the other hand, in the case of set theory, we have argued that the Skolem paradox shows that we are logically free to *reject* the existence of absolutely nondenumerable sets, yet that, both on intuitive and on pragmatic grounds, it is preferable to *admit* their existence, as most set theorists do.

Finally, we have found that we are logically free to opt *either* for dualism (or for pluralism) *or* for monism in set theory. If we opted for the former position, this would mean only that the theory of finite sets can be extended to infinite sets in two (or more) different but equally admissible ways-just as other mathematical theories can often be generalized in more ways than one. (Moreover, if infinite sets have the nature of ideal elements, as Hilbert has suggested, it cannot really surprise us if we find ourselves to be logically free to invent two or more different but equally consistent stories about them.) Indeed, we have argued, again both on intuitive and on pragmatic grounds, that it seems preferable to admit a need for *both* Cantorian and non-Cantorian set theories.