That's actually an example of what OP was talking about. You have defined + as the operator that mimics what piles of rocks do, and defined numbers as counting rocks.
That's only a tiny fraction of what math does. An interesting and useful one, and mathematicians have put a lot of work into studying basic arithmetic. They have expanded out into numerous other forms, some of which turn out to have correspondence to the real world like non-Euclidean geometry.
Others turn out to be completely abstract and are merely curiosities. There are an infinite number of them, each containing truths, almost all of them of no interest. Interest is defined by mathematicians, not physics. Even so it turns out to sometimes be useful, such as the beautiful theorems of prime numbers that drive Internet security centuries after they were invented.
That is what the OP means. You can make up any axioms you want and prove true theorems. But the hard part is convincing other mathematicians to care.
> But the hard part is convincing other mathematicians to care.
My point is that whether other mathematicians care or not is completely irrelevant and doesn't subtract from mathematics' power of predicting phenomena in the real world.
Each and every mathematical theory has to be consistent with basic rules of reality - if nothing else, symbolic manipulation relies on basic arithmetic and set theory. Without symbolic manipulation, you can't even express all those "abstract" mathematics - to say that "abstract" mathematics can not have correspondence to the real world is completely false, because of this basic connection.
Now that I think about it - claiming that "mathematics is a social consensus" is exactly what I'd expect from a mathematics professor - a person whose whole life is isolated from reality, limited to the rigid structure of academia, and whose whole existence depends on other people caring. I doubt there is a single (professional) engineer that would say something like that.
You can construct a formal system with any axioms you choose. It is not arbitrary that some of these systems turn out to be useful in modeling the world. But there are other systems that are only dry exercises in symbol manipulation that may be of little use to physics or engineering. Or of course they might end up being super important 100 years later. But in the meantime mathematicians might be interested in them anyway.
> I doubt there is a single (professional) engineer that would say something like that.
I suspect an engineering professor would be even more compelled to highlight how social consensus is the foundation of everything else that follows. You only have to read the surface layers of this or that performance debate to see how it is unfortunately the case. "Performance" is a fluid term that doesn't mean much of anything on its own -- and people arguing that they've juiced another drop of performance juice from this or that application's fruit are often just talking past each other in terms of priorities or perspective.
The symbolic manipulation at the heart of mathematics is a byproduct of language -- another system beholden to social consensus. It's inescapable.
Reading the discussion, I think you're both right. There is a common sense notion of mathematics that exists independent of what people want to believe or agree upon.
There are also ways to philosophize about the nature of things in order to frame math as a human construct. I think both views can be simultaneously correct
That's only a tiny fraction of what math does. An interesting and useful one, and mathematicians have put a lot of work into studying basic arithmetic. They have expanded out into numerous other forms, some of which turn out to have correspondence to the real world like non-Euclidean geometry.
Others turn out to be completely abstract and are merely curiosities. There are an infinite number of them, each containing truths, almost all of them of no interest. Interest is defined by mathematicians, not physics. Even so it turns out to sometimes be useful, such as the beautiful theorems of prime numbers that drive Internet security centuries after they were invented.
That is what the OP means. You can make up any axioms you want and prove true theorems. But the hard part is convincing other mathematicians to care.