Yes, mathematics exists – as a human cultural practice.
I think, however, that Antoine is really looking at the questions “Do mathematical objects (such as numbers) exist?” And people disagree about that.
But we might just as well ask:
- does existence exist?
- is truth true?
My point, of course, is that people disagree over the meaning of “exist” and over the meaning of “true”.
So Antoine discusses two possibilities – mathematical platonism, and formalism. But that’s too narrow a choice. I’m a fictionalist, so I am neither a platonist nor a formalist.
The mathematician Kronecker famously said “God gave us the natural numbers; all else is the work of man.” And that’s clearly a rejection of platonism. Personally, I think Kronecker gave God too much credit. The natural numbers are also human inventions.
Concepts such as “exist” and “true” emerge from cultural practices. And that’s why it is hard to pin them down.
When I am doing mathematics, then numbers exist. I can look at an equation, and talk about the existence of solutions (possibly unknown) for that equation. And when I do that, I am taking for granted that numbers exist. But when I am not actually doing mathematics, but instead trying to explain how it works, then numbers do not exist. They are useful fictions.
If I am reading a Sherlock Holmes story, then I have entered the fictional world of Sherlock Holmes. And, inside that world, Sherlock Holmes exists. There would be no fun in reading a Sherlock Holmes story if, every moment, I was questioning whether Holmes exists. But once I leave that fictional domain, no Sherlock Holmes does not exist.
For me, it is much the same with numbers and other mathematical entities.
As to why mathematics is so useful. I commented on that in another thread:
Science is systematic, and mathematics is about systematicity.