Final Remarks
We have argued that mathematics does not rely on any absolute notion of truth. In contrast, by studying different structures truth should be relativized to such structures while the structures themselves are used as tools in applications (outside and within Mathematics).
Thus, the question of the truth of an axiom (like the Parallel Postulate or CH) could either only be asked with respect to a chosen structure or would have to be abandoned in favour of a decision concerning the usefulness of it (and the structure in which it is fulfilled) and/or its alternatives.The view presented here is very much in line with Bernays’s analysis of the existence of ideal objects in terms of “bezogene Existenz” (Bernays 1950).[182] In the very same way, as Bernays let existence refer to a structure, we would like to let truth refer to a structure.
The existence of different structures leads naturally to “different truths” in these ones. But this does not imply, by no means, that Mathematics would lose its objectivity; as existence and truth, it also refers to structures, and all we have to concede is a certain form of pluralism in Mathematics.
A question which remains concerns the very existence of a structure and its internal configuration. We will discuss this question in more detail elsewhere, but remark here only that the structures considered in our examples are concrete structures. In contrast to such structures Bourbaki, for instance, considers abstract structures which are supposed to encompass abstract properties of several, quite different, concrete structures.[183] Their ontological status differs substantially from that of the structures we have considered here, but for both sorts of structures we can say with Bour- baki (1950, p. 231) that:
mathematics appears thus as a storehouse of abstract forms—the mathematical structures.
And Bourbaki (1950, p. 227):
The “structures” are tools for the mathematician.