I'll use here some personal terminologies, I hope it won't be disturbing.

We may divide truths into two kinds: receptive and consequential.
With the first kind I mean truths that are external to our makings and procedures, that
impose itself on us, that our main stance from it is that of "receiving it", like for example that the earth orbits the sun, that the sun is bigger than earth, like water is composed of hydrogen and oxygen, etc.. those are truths that impinge itself on us and we have no choice but to accept them as such, and to discover them we need to use apparatus related to detecting them, primarily being our five senses, devices, etc..We can call such truths as reality kind of truths.

The other kind of truth is what I label as "consequential", here this kind of truth has a human role
in delivering it, for example we begin with an arbitrary set of rules, and see were those rules lead us to. A simple example is to have a rule that: R is transitive (if x R y R z then x R z), and another rule stating that there are objects x,y,z,u such that x R y R z R u R x, then from these two rules we'll have the following 12 consequences:

x R z, x R u, x R x, y R u , y R x, y R y, z R x, z R y, z R z, u R y, u R z, u R u.

The fellowship of those consequences from the two input rules is a kind of TRUTH. This is a consequential fact. In the sense that we are obligated to hold that those outcomes are the results of applications of the input rules. The main difference between the consequential and the receptive kind, is that with the consequential kind we have control over the conditions from which we harvested that truth, while with the receptive kind we don't control the medium from which that truth sprang. Now consequential truth might lead to outcomes that might oppose receptive truths (reality truths), for example: if we input the rule that all Males have XX chromosomes, the rule that Napoleon is a male, then the outcome is Napoleon has XX chromosomes. Here this outcome is against reality, but as a consequence of the aforementioned rules, the consequence itself is TRUE, i.e. it is TRUE that if we hold the first inputs to be true then it follows that the outcome is true.

Emmanuel Kant once spoke about what he termed as ANALYTIC truths, like in the sentence "All bald men are bald". which is a kind of affirmation consequential rule. Here the consequential rule is about "are", it affirms the acquisition of a predicate before it, so the general rule is ALL objects that possess predicate phi ARE in possession of predicate phi. Another way is to break it downs to rules related to conjunction and implication: that is [A & B -> A] , [ A & B -> B]. This is just a rule following scenario, so what he called as "analytic" fact, is actually nothing but a kind of consequential truth.

Once Hume had said that mathematics is absolute because it deals with those kinds of analytic truths, so 2+2=4, comes from the roles assigned to the symbols 2, +, =. The complete story is if we hold the rules of the game of arithmetic, then we'll have 2+2=4 being an outcome (a consequence) of this game. This is absolute, because the rules are put in absolute manner. And it is a kind of TRUTH.

Here I'm stating a similar stance that is: Mathematics is the study of consequential truth.

So mathematics is about studying rule following games. I call them games because the choice of the primary rules is IMMATERIAL, we can even call them ARBITRARY, the most important is to harvest consequential truths in those games. The reality of the games, i.e. the stance of its primary rules and consequential outcomes from reality, is not relevant to mathematics itself, it is however relevant to its application, but not to mathematics per se.

As such mathematics need not essentially have an application. However, that doesn't meant that it ought not to. On the contrary SOME parts of mathematics do have many applications, and those parts have their primary rules related in some sense to reality, that's why it had applications, and thus were significant in increasing our knowledge about reality, some parts of reality are rule driven, so examining rule driven scenarios can be related to reality. Applied mathematics is the part of mathematics whose primary rules can be said to have some connection to reality and so can be labeled as quasi-empirical or the alike.

Of course it would be understandable that only beneficial rule following games would survive, and be studied more. However, that is not truly the primary job of the mathematician, the mathematician is concerned with consequential truths in rule following games, she\he has nothing to do with justification of the primary rules of those games from the perspective of match-ability with reality.

So what I'm advocating here is a line in philosophy of mathematics that can be termed as: consequential-ism. So mathematicians negotiate themselves with scenarios fruitful in consequential thought and study the consequential outcomes of those scenarios.

So mathematics per se is not necessarily empirical or quasi-empirical, although some parts of mathematics can be so.

There is the point of view that even if mathematics is not about reality-matching rule following games, yet it is about some kind of platonic realm matching. In other words mathematicians are somehow perceiving a kind of reality that is not the concrete physical reality, and that mathematical truths can be seen as a kind of receptive truth about that realm. This is what is generally labeled as mathematical Platonism.

The main problem with this viewpoint is that it is to strong, for although a working mathematician when he brings about some new mathematical rules or define new objects, he's indeed working within such context, that it appears as if they he is "discovering" those, but at the and of the day, no such a claim would be granted or even posed. What is granted is the consequential facts of his work, and it is that what counts. While its stance from such realm or even from reality is not even posed.

This philosophical line doesn't assert any of the known prior logical schools in philosophy of mathematics, so it is not to be confused with logicism, nor with formalism (which is the nearest to it), nor with nominalism, or structuralism, nor with fictionalism.

Here I'm not asserting that the rule following games must be purely logical (has no extra-logical primitive concepts), or that mathematics is nothing but a string manipulation rules of meaningless (empty) symbols. I'm simply saying that the rule following game can be of ANY motivation even empirically motivated, but if the sole study of it was as a rule following device, and the study was just for the consequential load of it, then it is a mathematical study. It can be a meaningful game, where meaning itself is stipulated through rules, so it would be a meaning rule manipulating system, it can also be an empty string of symbols rule manipulating system. It doesn't matter how much meaning are there and their stance from reality or even from a platonic realm, or from apriori concepts, what counts is the consequential bearings of those games.

So mathematics is the study of consequential truths in rule following games.

Of course this study does include heuristics involved in bringing about such games, and bringing about useful definitions and proofs in those games, for although those heuristics themselves are not decided by the primary rules themselves, yet they are finally about consequential bearing within those scenarios. Those heuristics can virtually have imports from any realm of thought actually, logical, apriori, structural, platonic, pragmatic, empirical, formal, etc.. so there is no limit to it, and it constitutes the mathematical ingenuity no doubt. But mathematics cannot be defined after those heuristics, since such a thing is clearly un-definable. That ingenuity can be understood as the tool that leads us to bring about mathematical systems, rather than it itself being identified with mathematics. Mathematics is the consequential bearings that this ingenuity finds in what it creates of rule following games. So philosophy of mathematical practice is not identical with that of mathematics.

What are the drawbacks of that personal line of thought of mine about philosophy of mathematics.