In my view, mathematics is definitely *not* the study of thinking. There are huge areas in cognitive psychology for that and mathematics exists irrespective of one's level of thinking (yes, my philosophy is showing). I lean toward the view that math is the study of structure, patterns, and form that arise as a consequence of rule sets. The things that Warren listed that *seem* formless are not actually. Chaos has form; fluid turbulence has structure. Even truly random events & sets have structure that can be studied mathematically. Kerry On Mon, Jul 6, 2015 at 9:30 AM, Warren D Smith <warren.wds@gmail.com> wrote:
I still like mine better than everybody else's: it is the study of thinking & how to be a better thinker.
Is it the study of structure, patterns & form? No, because it also encompasses the study of phenomena that seem structureless and formless. How about chaos, fluid turbulence, and random number generators? Are they to be excluded from mathematics? Is it algebraic & symbolic manipulations? No, that would exclude much ancient greek geometry-based mathematics. Is it the study of "numbers" and their descendants? How about set theory, topology, and logic as counterexamples? Is it physics? Well, plenty of maths has nothing to do with physics, in fact sometimes intentionally so, such as the study of laws intentionally designed to be wrong-physics. Is it the study of tautologies and obviously-true statements? Dubious, counterexamples have already been posted. Is it the study of rule-sets we make up (Vi Hart)? Not a bad try, that one. I think all those definitions were trying to be my definition but falling a little short, except for maybe Vi Hart's definition which arguably is equivalent to mine.
So there :)