A phenomenological account of mathematical modes of intuition

If mathematics has to provide some knowledge, this means, for Husserl, that it does not only conceive or mean possible (non-contradictory) contents but also give “objects” that make its statements true. Now, these objects must not necessarily be intuited in authentic presentations; they can also be intuited mediately through symbolic presentations. This, actually, is the case for most of mathematical objects, especially for the manifolds (Mannigfaltigkeiten) or structures that are investigated through the formal systems of contemporary arithmetic and geometry. Yet these formal systems are not just arbitrary rule-based games; they are related to the initial theories of numbers and of space by continuity relations that can be seen through some metamathematical considerations. And this is why they keep on providing knowledge on numerical and spatial manifolds.