Another question about math is whether it is something intrinsic to nature which people have discovered, or a very useful tool which we have invented, and which we impose upon nature. I've always seen it as the latter, which diminishes, at least for me, the intrinsic interest of those other questions you mention.
Of course, I may have been entirely wrong this entire time. I have the impression that most contemporary mathematicians and physicians and zoologists and botanists would say that I'm wrong.
Nietzsche believed we invented math. See Menschliches Allzumenschliches, vol 1, section 1, "Von den ersten und letzten Dingen," paragraph 19, "Die Zahl." Mathematicians and physicists might find this passage interesting, among other reasons for the grasp of atomic theory which Nietzsche demonstrates in something he published in the late 1870's.But many years after I first read that, it suddenly struck me, like a hammer striking a gong, that everyone knows exactly what a circle is, although none of us has ever seen a perfect circle. This very simple fact, available to anyone who thinks about it for as long as a moment, seems to me to be a very strong argument in favor of Plato's forms, and in favor in math being something we discover as opposed to something we invent.
Nietzsche despised Plato more intensely than he did any other single human being. I went through a period of very intense admiration for Nietzsche (except for his sexism and enthusiasm for war, which I always rejected), and I adopted his contempt of Plato. But my gong-moment, my insight about circles, has forced me to reconsider Plato. And when you reconsider something as influential as Platonic philosophy, you necessarily re-consider many other things.
