I shall provide several non-standard theories for mathematics. Such theories aim to solve inherent problems of the failed project of providing a strong foundation for mathematics by means of standard mathematical theories. The non-standard theories are meant as formalizations of mathematics that provide a philosophical pragmatic foundation for mathematical practices and results. The underlying logic of the theories I shall investigate is a non-monotonic adaptive logic, respectively the rich non-transitive paraconsistent logic CLminus. Precisely the deviant properties of these logics, i.e. non-monotonicity and non-transitivity, make it possible to tackle the problems of standard existing theories. More particularly, some of the theories I shall investigate are inconsistent but not trivial and at the same provide an explication for the validity of all of the established results within existing mathematical theories. In that sense they form a flexible and pragmatic foundation for the entire (relevant) domain of mathematics.