In philosophy "form" is a universal, however here the term "form" is used to designate a universal that is exemplified by all objects bearing some kind of isomorphic relation between them provided that the collection of all exemplifying objects do have all objects included in its transitive closure. I paraphrase that as: a universal that involves the whole universe. By contrast some forms as used in philosophy involves only a particular sector of Ontology, like for example "cat" which can only be exemplified by animals and so it is a restrictive kind of "form". Here any speech about forms will be meaning the non privet kinds of forms, i.e those that involves the whole universe after some isomorphic relation as mentioned above.
Mathematics is "discourse about form" with this it is meant any theory that can be interpreted in the set hierarchy having all its objects being interpreted as forms in the set hierarchy. So for example PA is a piece of mathematics since it can be interpreted in the set hierarchy with an interpretation in which all its "objects" are interpreted as "forms" defined after "bijection" relation in the Fregean manner. So it is a case of discourse about form, thus mathematical!
So here there is a line of separation between what is foundational and what is mathematical, the set\class hierarchy is foundational i.e. it belong "essentially" to logic! it is a sort of extended logic, although it definitely use some mathematics to empower it and actually it needs a mathematician to work it out, yet this doesn't make out of it mathematical, the piece of mathematics used in those foundational theories is just an application of mathematics to another field much as mathematics are used in physics. So what I'm saying here is that a theory like ZFC is not "Essentially" about mathematics, it is not even a piece of mathematics, it is a LOGICAL theory.
So Set theory is a kind of LOGIC. However one can easily see that such form of logic can only be handled by mathematicians really, but still that doesn't make out of it a piece of mathematics as mentioned above.
Mathematics is the study of "form" as mentioned above, it is "implemented" in the set\class hierarchy which provides a discourse about forms whether simple or structural. All known branches of mathematics: Arithmetic, Analysis, Geometry, Algebra, Number theory, Group theory, Topology, Graph theory, etc... all can be seen as discourse about form, since all its objects can be interpreted in the set hierarchy as forms.
Anyhow it is reasonable for branches of mathematics to be developed along some Foundation back-grounding in logic, and then the mathematical forms be implemented on that background logic, this can be seen clearly with topology which starts from set theory and then go higher to deal with forms like continuity and connectedness. However it can be seen to be essentially about the higher concepts it tries to manipulate, the back-grounding in sets is just the logical part of it, since what it tries to manipulate is a sort of "form", then topology is essentially mathematical.
Also I wanted to raise the issue that "any" consistent theory is speaking about a model that is "possible" to exist! So if we secure a consistent discourse about form then, we are speaking about forms that might possibly exist. And that's all what mathematics needs to bring about. Whether those forms really exist or not? this is not the discipline of mathematics. So consistency yields "possible" existence, and that's all what mathematics should yield, i.e. forms that could possibly exist.
How those forms are known to us? the answer is through their exemplification as part of the discourse of consistent theories about form. Whether they are platonic in the sense of being in no place no time, etc.., that is not relevant, we come to know about them by their exemplifications which are indeed not so abstract and can be grasped by our intellect. How can such an abstract notion be exemplified by such concrete objects, that's not the job of mathematics to explain.