This is my long-promised (as of 5 minutes ago) boring thread.
It seems pretty obvious numbers are real; after all, if I have an apple, and you give me another apple, I now have two apples, right?
Well, hang on; not so fast. The above thinking was exactly how I felt about it until like a day ago when I decided to argue the other side for fun.
The only way you have "2 apples" is if they are both "an apple." But what makes it an apple? Are they exactly the same? No; they've got enough things in common, though, for our brains to be satisfied thinking of them as the same thing, and therefore saying there are now 2 of the same thing.
In short, I'm now leaning toward numbers arising from categorization, rather than being something fundamental. There are no two completely identical things in the universe. Even electrons, which are so "the same" to our brains that we think of them as identical, are different from each other in terms of location and time and energy. There is no fundamental "2," because there is no completely 100% identical pair of anything. Just close enough for our brains to say, "Okay, that's the same."