ABSTRACT
The foundations of arithmetic and number relying on temporal intuition, and not requiring an axiomatic formulation, while the foundations of geometry, the paradigm case of an axiomatic deductive science, relied upon a theory of spatial intuition and the possibility of constructions performed in, or guaranteed by, the pure form of outer intuition. While abstractionism is no doubt a metaphysically important contribution to the existence of fundamental mathematical domains and their epistemology there are considerable difficulties facing the idea, not least technically with respect to applying the idea to set theory itself, for there the obvious abstraction principle is just the notorious Basic Law V. The second difficulty is metaphysical in kind and is that to do with the stipulative character of abstraction principles and how that is to be consistent with the avowed platonism of abstractionism, that a stipulation, in effect an implicit definition, is sufficient to characterise a truth about a pre-existent domain.
