Add a new primitive to the language of ZFC, this primitive is the binary relation "exemplifies" denoted by the infix dyadic symbol ~
R is a form defining relation iff R is an equivalence relation & For all x. ~x=0 -> For all s. Exist y. y R x & s in TC(y)
TC(y) refers to the 'transitive closure of y" defined in the usual manner as the minimal transitive superset of y.
To the axioms of ZFC add the following axiom scheme:
Forms: if R is a binary relation symbol, then
[R is a form defining relation -> For all x. Exist! $: For all y (y ~ $ <-> y R x)]
is an axiom.
Each $ here is said to be a form defined after R and x. /
Now the idea is that For any theory T if T can be interpreted in ZFC+forms in a manner such that each object in T is interpreted as some form in ZFC+forms and all primitive relations of T are interpreted by defined relations in ZFC+forms; then T is a mathematical Theory.
Example: Peano Arithmetic "PA" is interpretable in in ZFC+forms by an interpretation where all objects of PA are interpreted as forms each defined after equivalence relation bijection and some set in the following manner:
Define (natural number): # is a natural number iff (Exist x. x is finite & for all y. y ~ # <-> y bijective to x)
It is just straightforwards to interpret all primitive predicates of PA by 'defined' relations in ZFC+forms.
So PA is a mathematical Theory.
This is what I paraphrase as "PA supplying a consistent discourse about form". What I mean by "consistent discourse" is being interpretable in a consistent extension of logic, here ZFC+forms is the consistent extension of logic and since PA is interpretable in it then it provides a consistent discourse of its objects and since its objects are interpretable as forms in ZFC+forms then PA is said to provide a consistent discourse about form. Any theory that is so interpretable in any consistent theory that can define forms is accordingly said to be a theory that provides a consistent discourse about form and thus it is "MATHEMATICAL".
So roughly speaking this account views mathematics as being about "Logic of Form".
Para-consistent (inconsistency tolerant) discourse might be interesting if proves to be indispensable by the use of consistent discourse, and thus theories supplying para-consistent discourse about form would be designated also as MATHEMATICAL. Actually I tend to think that any discourse about form other than the trivial discourse of proving everything if proves indispensable by use of other kinds of discourse, then it would be mathematical
So Mathematics can be characterized as:
Non trivial discourse about form.
The difference between mathematics and science is that the later is about establishing the TURE discourse about the objects it negotiates, while mathematics is about supplying any NON TRIVIAL discourse about the objects it negotiates (i.e. forms) whether that discourse is what occurs in the real world (i.e. True) or whether it doesn't (fantasic). Mathematics supplies a non trivial discourse of form, so it supplies a language about form, and as said above more appropriately put as supplying: Logic of form, whether that logic is consistency based or paraconsistent. The truth of that, i.e. reality matching of the discourse about those forms is something else, this really belongs to a kind of physics rather than to mathematics. The job of mathematics is to supply the necessary language that enables us to speak about forms, not to validate its truth.
Anyhow that was my own personal opinion about what constitutes mathematics.