Is mathematics invented or discovered?

Mathematics starts with rules somebody writes down, and everything those rules force is discovered. Start from ab and allow two moves, putting an a in front and a b behind, or writing the string twice. Nobody wrote a rule about counting letters, yet every string you can reach has as many a's as b's, so aab is unreachable. That fact was fixed the moment the moves were written. Whitehead and Russell spent ten years finding out what their marks committed them to, because the answer was not up to them. In 1733 Giovanni Saccheri derived a geometry of curved space, called it repugnant and rejected it, and it held regardless. Nature's numbers are found the same way. Nobody chose the fine-structure constant, the strength of electromagnetism, any more than anybody chose the square root of two.

Also in Chapter 36: Why These Laws and Not Others?.