Mathematics can be broadly characterized as the study of non-trivial apriori symbolically displayed axiomatic systems. Or more elaborately the study of non trivial apriori implicit or explicit axiomatic systems that are usually written in a symbolic inferential language.
If I want to put the above in an exact terms, then I'd say that mathematics is the study of non-trivial apriori justified sets of sentences that are interpretable in an axiomatic system.
In that sense what I've previously said to be an axiomatic system can be taken here to mean just a part of that system. An axiomatic system is composed to two parts the semantics and the syntactical side, it has axioms which are sentences that are considered as true by stipulation, and has inference rules by which we derive sentences from prior ones, the deductive closure of all these sentences constitute the axiomatic theory. The formal system is the syntactical part of an axiom system. (see Shoenfield) (1.1, 1.2)
However here what I mean by an axiomatic system is more appropriately understood as a "part of an axiomatic system". Now arithmetic is written in sentences with inference rules guiding formation of sentences from prior ones and with primary sentences that are considered as rules of arithmetic that are not derived by arithmetic inference rule from other sentences, although one doesn't see the logical background governing many of the processing in mathematics, and one doesn't see statements explicitly called "axioms", nor statements called "inference rules", however in action definitely they are all there, one can easily interpret arithmetic studied in our schools in an axiomatic system like Peano arithmetic, now the arithmetic we study is apriori justified, and it is non-trivial, so it satisfies all the conditions in this definition. Geometry I mean Euclid's is clearly an axiomatic system, and even Geometry preceding it, since ever, all is interpretable in an axiomatic system, and is non-trivial and apriori justified.
I personally don't know of a discipline belonging to mathematics that evades that definition.
One word needs to be said is that this definition is not a formalist stand point, nor belong to deductivism or the alike. To the formalist mathematics is just manipulation of string of symbols, there is no meaning (except of some basic entities like the naturals and logic) to the symbols, and virtually has no semantics in the true sense of the word, so they don't advocate something like "apriori" justification (which is related to semantics of an axiomatic system) or the alike (abstract, necessary, etc...). So definitely this definition incorporates a semantic side, which are semantics the truth of which is apriori justified, in other words to a formalist this account translates to saying that mathematics is applied pieces of formal systems to semantics with apriori justification, so he'll sum it up by saying that what is said here is that mathematics is non-trivial apriori applied formalism, and to the formalist this is not mathematics as he conceives, since to him he doesn't define mathematics to be about a certain kind of concepts or semnatics, he just think it is the formal side, i.e. the syntactical symbolic stream. The account here greatly differs, of course here when I said apriori then it can be apriori analytic which is logic in my understanding, or apriori synthetic which is the extra-logical parts of mathematics. That mathematics was not largely axiomatized until later in history is irrelevant to this definition, just because it wasn't axiomatized, it doesn't mean it is not using actually an axiomatic framework in implicit manner, and it doesn't mean that it cannot be interpretable in an axiomatic system, all mathematics before the 20th century are actually interpretable in an axiomatic system. Almost all of mathematical presentations can be easily axiomatized since they are written using string of symbols with clear mathematical operators that are deductively closed under mathematical inference rules, all of those can be added as primitives, and added on top of a logical frame, and the mathematical inference rules can be added as inference rules of the system, and as long as the system is deductively non-trivial and is apriori justified, then it is clearly mathematical, so the criterion of interpretability in an axiom system is straightforward for almost all of mathematically presented material. There might be some informal concepts the stance of which from axiomtizability is unknown, and that are considered as mathematical. I'd like to know of those really, since they'd constitute counter-examples to this broad characterization.
So in nutshell what I'm saying here is broadly speaking:
Mathematics= Interpretability in an axiomatic system + apriori justified semantics + Non-trivialism.
Are there counter-examples to this broad characterization?