The correct answer is: "why give a fuck?".

The only practical value of having an answer to this question is giving philosophy students something to write about.

Mathematics is a human activity, so whether you think it's "exploring" or "inventing" mathematics, you would agree that we aren't pursuing the exploration or invention in all possible directions.

So let's look at what we, the humans, actually do.

Any mathematical object ever defined is ultimately an abstraction and a model of something the humans actually experience in the physical world.

Number theory is abstract, numbers are not. You can count your sheep in a pasture; you can count your neighbor's. Natural numbers model the idea that neither of you has more than the other.

This is just a model, of course; you have a sheep named Dolly which is dear to you, and your neighbor has one named Sue which gives more wool than any sheep in the county.

So you really don't have the same thing, because these sheep are not interchangeable. But we have a concept — natural numbers — which models the idea of sameness if we, for a second, forget that.

It's an imperfect model; the concept of something that can grow without bound is an outcome of this imperfection. When you use numbers, you might as well assume that 10^(10^100) is the largest number there is. Numbers of that scale are so large, that all computers in the world taken together aren't enough to just store this number. But modeling the concept of "not knowing what the bound is" is modeling an experience we run into all the time. 640KB isn't enough for everyone.

We dispense with this abstraction with ease, though. Most calculations in practice are done with 64-bit "numbers". These numbers don't behave like the one in math do, but it's OK.

Even simpler, our clocks have just 12 numbers, even though the time doesn't stop running.

Euclidean geometry can be verified experimentally. Cut out a triangle out of paper; tear off the angles; put them together, and see that they make up a straight line.

It's an imperfect model, and you don't need Einstein or Lobachevsky to tell you that. Long-distance sailing made people run into this model breaking. The corners of a triangle you cut out of orange peel with a straight knife won't make a straight line.

But that only gives rise to other kinds of geometries being interesting to us.

People do math because it's beautiful, and something that doesn't connect to any human experience is decidedly not beautiful.

Take a deck playing cards, or just a dozen cards, and come up with a set of rules of how a pair of cards can be traded for a single one. You can make different rules depending on which of the two cards is on top; 2-for-1 is the only restriction.

We could come up with an immense number of complicated systems that way. But most of them would be so arbitrary, they won't connect to anything. So nobody cares about systems like that. Unless they come from something else and therefore have more than an arbitrary structure.

Like finite symmetry groups, e.g. rotations of a cube. You can model that with such card rules. Then it's interesting.

So the question of whether mathematical objects exist is beside the point.

The reality is that all mathematical objects that humans care to study are, ultimately, modeling an experience that does.

And this is consistent with mathematics truly being an art, not a science.

Because art that doesn't abstract human experience in some way is not something we care about. Strings of letters spit out by a random number generator aren't an exciting read.

That's what Vladimir Arnold means when he says, seemingly controversially, that mathematics is a branch of physics (in which experiments are cheap)[1].

It's that mathematics that doesn't connect to the world we live in is too boring to exist - or be invented and remembered.

[1] https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html