"As far as mathematics, Conceptualism states that we invent math, not discover it- it is our own modelling of reality in a way that we can intellectually understand. Math seems to fit nature so well (as per a Neo-Platonist claim) because we have designed it to do so." - Justin EllisI've read about a school of thought called Formalism that sounds similar to Conceptualism. The Formalists believed that arithmetic could be built up from logic and set theory into a formal system. Their goal was to prove this system was consistent and complete. They also believed mathematics was a kind of intellectual game, a pure invention that didn't exist before humans hit the scene and doesn't exist without us.
The opposing view was held by the Platonists, who said that theorems of mathematics are truths that exist independently of humans and are being discovered by us.
Now it's clear that mathematics was invented. We have systems of geometry that start out with fundamental axioms that seem self-evident to us based on our concepts of space. Axioms of arithmetic are also self-evident but the concepts here concern numbers, counting, and induction. The invention really stops at this point. What's involved henceforth is deriving theorems from the axioms, and new theorems from existing ones, and while this certainly involves ingenuity of the highest calibre, this is in retrospect merely following the rules of the game.
Up to this point it doesn't sound like the Platonists have a leg to stand on. But the game gets very interesting, in fact so interesting that it defies all expectations you had at the start. You could never have guessed, given Peano's five rather boring axioms, that irrational numbers exist, that pi appears again and again in the sum of so many different infinite series, or that there's an infinitude of primes. But all of these can be proved from this simple starting point.
It would still be considered no more than a fascinating game except now we see that nature is following rules, and they are mathematical. So when Justin says "Math seems to fit nature so well because we have designed it to do so.", the answer is yes, but no one could have anticipated the spectacular degree to which this is so. For example, the ideas that guided the foundations of mathematics could never have predicted that gravity behaves according to a mathematical law (an inverse square law), and further, no mathematical inventions need be conjured up on the spur of the moment to explain it. So I partly disagree with the Conceptualists position. Math is indeed an invention, but it is such a superb invention that its results seem like, or may very well be, truths that have been waiting for an eternity to be discovered.
The analog to Justin's statement is that "Quality seems to fit nature so well because Pirsig has designed it to do so." It could be suggested that Quality, like mathematics, may be just an invention that works at explaining reality, without being real itself. Well, Pythagoras was so enchanted by numbers that he thought they were the primal stuff of the universe. If you feel the same way about Quality, you could say the same of it. Pirsig himself recently argued that Quality is reality because of the harmony it produces. Presumably he means by this that it solves or dissolves the SOM platypuses and provides guidelines for solving practical moral problems. I'm not convinced, but you have to be the judge of this yourself.