lets define a symbol as a well defined character.
A symbol would be referential if it there is a function assigned on it to a particular meaning.
A symbol is called empty if it is not associated with any meaning.
A symbol can be any written character, as far as its well defined! It can be a letter, a written shape, anything that one can write or draw if defined in a reproducible manner. And even it can be anything that one can "imagine" provided is a well defined piece of imagination. It can even be a well defined hypothetical concrete object (an object that exists in space and time obeying concreteness criteria of not being able to exist in whole in distinct places at the same time)
A "string of symbols" is any well specified collection of symbols, the string building operator can be simply concatenation, or it can be interlocking, branching, or any graphical setting. A string of symbols is well defined only if it is stipulated to be the result of some well defined combination of well defined string building operators that of course produce outputs in a controlled manner.
Now a symbolic inference rule is a relation on strings of symbols that couples a set of strings of symbols to specific set of strings of symbols.
Usually what is used is "functional inferential rules", i.e. having a single Output per a set of Inputs.
A symbolic language is any well defined set of well defined strings of symbols and a well defined set of inference rules on them.
A stipulation rule is a symbolic inference rule that enforce a certain string of symbols to be an output, without requiring specific inputs. This can be re-phrased as the stipulation rule being a rule with an empty set of inputs.
An analytic system is any set of strings of symbols written in some symbolic language, that are closed under some inference rules of that language!
A proof is a finite sequence of strings of symbols in which every string is the output of an inference rule whose inputs are items in the string or otherwise otherwise empty.
A string of symbols proved in an analytic system is the last string in a proof carried in that system. This string of symbols is called a "theorem" of that system.
A string of symbols t is provable from a set A of strings in system S, if and only if there is a proof carried in Swhere t is the last string in that proof and A is a set of sentences occurring earlier in the proof.
An analytic truth is an assignment from a triplet of a set A of strings and a string t and an analytic system S, to an element of a certain set of symbols, the latter stands as the set of truth values (usually dichotomous), that is determined by provability of t from A in S in a definable manner, like by stating if it is provable then T, if not then F.
Example:
⟨{Extensionality,Empty},∃!x(empty(x)),ZF⟩→T
An analytic system is consistent if no string of symbols belonging to it raise a conflicting truth assignment, i.e. assigned more than one truth value!
Now an analytic system is trivial if it is inconsistent.
An analytic system is POSSIBLY non-trivial if there is no proof of it being inconsistent.
An analytic truth is possibly non trivial if the analytic system in its antecedent is possibly non-trivial.
Mathematics is the study of analytic truth.
This means that it is the study of all kinds of analytic truths, trivial or not, but especially non-trivial of course.
It is to be noted that this doesn't at all mean that mathematics is a pure syntactical discipline. No! if the symbols in a system are referential in a well defined manner, i.e. each symbol is assigned a specific meaning by a well defined assignment function. Then the analytic truth traceable in that system would be also about meaning of theorems that is traceable to the meaning associated with their premises! In other words that referential kind of system would provide a kind of concept analysis!
Also its to be noted that this definition is much more general than just logical deductivity. Here it is about inferential thought about any written character, so it can be about letters, numbers, shapes, graphs, hypothetical concrete objects etc... Deduction as a term is usually reserved to the closure of sentences under logical or extended logical rules. But here the meaning goes way beyond that to involve any kind of string manipulation rules, actually any kind of graphics manipulation rules, as long as they are well defined, moreover any kind of an concrete object manipulation rules as far as it is well defined! However, I do believe that the expanded meaning adopted here CAN be reducible to the formal extended-logical deductivism.
So mathematics is Not restricted to the study of analytic complexity (system) of "abstract" independent objects.
Nor is restricted to the study of analytic complexity of apriori principles, since aposteriori motivated analytic complexities can be equally interesting.
Nor is restricted to the study of what is thought to be applicable in natural sciences, since studying analytic complexities bearing no close relationship to anything about natural science is well known. The idea that it must be necessarily directly or indirectly linked to improving knowledge about nature is not always very clear! Examples are the study of large cardinal axioms, NF, etc..
So mathematics is essentially the study of analytic complexities, and whether those can play a rule in natural science or some apriori logical or transcendental philosophical conception, etc.., it might, but it need not be restricted to those. Its not clear how it should be restricted to any of those. While its connection to analytic complexities is very clear.
The correct philosophy of mathematics is nearer to FORMALISM, or more appropriately what I term here as: ANALYTICISM. (I think Hume said something similar to that also). But this need not be connective to Logicism, since it might not be about logic.
However, mathematics "as PRACTICED", is something else, there is no doubt that there is a kind of naturalist look behind the development of axioms and definitions involved with mathematical systems, like in arithmetic and geometry, and also there is a kind of pragmatic motivation like in defining the integers, imaginary numbers, calculating areas . On the other hand Platonism is ongoing heuristic no doubt. Also some apriori logical thought is there underlying the development of many logical systems and even behind the standard model of arithmetic. So it couples all of the above. However what is shared in all of them is the analyticism! Although Analyticism itself is most of the times (with the exception of logic) not the motivation for the axioms, nor for the definitions; yet analyticism is the main interest in studying mathematical theories!
I think the true content of mathematics, i.e. its subject matter, doesn't lie in the kind of restriction on what motivates its axioms or definitions, since those belong to various arenas of thought. I rather think that it is related to providing a discourse where analytic complexities are examined and broken down! That is, it's about analytic truth about any concept or even going down to purely formal non-referential syntactical systems. As far as whether a system is mathematical or not, they all lie on a par.
It doesn't make difference to mathematics whether the axioms comes from apriori conceptualization or aposteriori grounds or are about abstract or concrete objects or whether they must be referential or not.
What makes a difference is that if they can produce non trivial analytic complexity whose truth is not easy to settle.
Mathematics is a kind of a complex rule following game.
Mathematics is the study of entities the investigation of which involves settling the truth about complex analytic issues.