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.
$ above is called the "form" defined after R. /
Now the idea is that For any theory T if T can be interpreted in a fragment T* of ZFC+forms such that all "objects" of T are interpreted as "FORMS" in T*, then T is a mathematical theory.
Example: Peano Arithmetic "PA" is interpretable in in a fragment of ZFC +forms where all objects of PA (i.e. the natural numbers) are interpretable as forms defined after equivalence relation bijection in the following manner:
# is a natural number <-> (Exist x. for all y. y ~ # <-> y bijective to x)
The general idea is that mathematics is nothing but
"Discourse about possible form".
So ZFC+forms supplies the necessary extension of logic that covers a wide piece of Ontology, and thus provides a consistent discourse about its objects. ZFC+forms is viewed here as an extension of logic, and thus being essentially a kind of powerful logic, so it belongs more to arena of logic than to mathematics.
"Discourse" here follows the line of defining it as:
"Interpretability in a consistent extension of logic".
So here ZFC+forms is the necessary extension of logic that provides the basis for 'discourse'.
So Mathematics here is viewed as discourse about form, so any theory interpretable in ZFC+forms where all objects of that theory are definable as forms in ZFC+forms, then this theory is considered to provide a discourse about form, and thus being a mathematical theory.
Now according to the above understanding of what constitutes a mathematical theory, it is obvious that consistency plays a pivotal rule, it signifies that the mathematical theory in question is Possibly true, it doesn't prove that the relevant theory is THE truth about forms it negotiates. Being true requires more than just being possible, it requires that the theory fulfills some of the basic expectations about simple mathematical facts, and how that is done is actually the subject of many different disciplines like constructivists, predicativists, etc... so it is a debatable issue.
However if we provide a consistent discourse about form, then this signifies possible truth about that form which is what makes this theory mathematical or not.
Now the question about what is the 'nature' of those forms is not really mathematical, it is philosophical.
So roughly speaking this account view mathematics as being about "Logics of Form".
But is it the case that mathematics should engage itself with any possible form? The answer is of course no since that is not possible. So definitely mathematical discourse would be about "interesting" kinds of form. This almost grow in a recursive manner, starting from obviously interesting kinds of form like "number" ,"point" "line" etc.. then discourse about those yields problems the solutions of which inherits the interest, and those solutions lead to interesting problems the solutions of which are interesting etc.. This Tree of interest is what guides research about form.
Of course other kinds of discourse might be interesting. Actually any discourse other than the trivial discourse of proving everything is somehow interesting. However Consistent discourse represents the strongest and most interesting kind of those, and it might be possible that para-consistent (inconsistency tolerant) discourses prove to be dispensable with by the use of consistent discourse, however if that was not the case then para-consistent discourse about form would constitute a mathematical concept as well since it would be a kind of 'non trivial discourse about possible form'.
This of course gives a logicist flavor to mathematics albeit not completely so, since ideation about "form" is not solely a logical issue, it does have its innate nature that separates it from the logical background in which it is implemented.
Now all mathematical disciplines we know of do follow this line of thought: Arithmetic, Algebra, Geometry, Analysis, Group theory, topology, Category Theory, Graph Theory, Number theory, etc... All do clearly follow the above line of thought.
On the other hand: Set theory, Mereology, Recursion Theory, Modal theory, and Proof Theory and of course the logical systems, all belong to the logical background apparatus upon the shoulders of which mathematics stands.