&#62; <i>mathematical structures are not invented, they are discovered</i><p>This is a doorway to a pretty abstract philosophical discussion, not a statement of fact. I don't think that math is actually <i>real</i> in the strictest sense, so I'm on the fence about invent vs discover. While I think that mathematical constructions are idealizations similar to Platonic Ideals, I don't believe like Plato did that there's a perfect Plato-space where these idealizations live.<p>When an artist arranges common-place paints (or sounds or dance moves or whatever) into a novel combination, he is said to have created rather than discovered a new work of art. Is there a good reason why this should not be true when a mathematician takes common-place axioms and makes a new arrangement of them that validates as a proof?