Mathematics is essentially a form of abstraction, focusing on the attributes of spatiality (although temporality can be included), primarily position, dimension, and shape. We quantify objects by being able to count their differing positions in space, even if they are otherwise the same. We measure by defining a dimensive unit, and counting how many are in a particular object (linear, planar, or cubic). The ability to "mathematicize" experience is undoubtedly inborn, but there is a real empirical element insofar as we abstract from experience. It has elements of conjecture, as, for example, in the postulates, as well as elements of "self- evidency", that is, things that should be apparent to all with the ability to reason. But it is not really a matter of faith, insofar as the reasoning is rigorous, and time and again the applicability to real situations has been demonstrated.
The existence of things like irrational numbers has been a conundrum to philosophers since at least Pythagoras. If we assume that the universe is supposed to be completely rationalizable, then it is an offense that such things exist, and must reflect badly either on the material universe or on human reason. I think a sounder view is that such things reveal mysteries deep in the heart of the universe, and show that abstraction from Being is just that, only an abstraction, a model of greater or lesser accuracy, but not encompassing the fullness of Reality........ |
