waves & intuition

We talked about oceans today in Earth Science class and waves and all that stuff. But all the while my mind was on intuitionism...a philosophy of mathematics. this particular philosophy focuses on the constructive nature of mathematics. it remedies the antinomes of classical mathematics (e.g. Russell's paradox) by disallowing the use of infinite totalities taken as entirely given. the theory's rational for this is that one cannot ever fully construct (in any way) an infinite totality, e.g. the numbers 1, 2, 3, so on (since there's always something more to be constructed...i.e. the nature of being infinite). The paradoxes arise because of classical mathematics' taking an infinite totality as given absolutely and actually. This goes beyond what one's mind can "see" and so one cannot hope to be sure that one's constructions using these infinite totalities will be consistent whereas if one remains with what one can see clearly, i.e. the method of constructing the numbers 1, 2, 3, ..., 10, 11, 12, 13, ..., 100, 101, 102, and so on then one cannot ever arrive at a contradiction because what one sees clearly cannot admit of a contradiction. contradiction arises only when we try to say something about that which we cannot see clearly, e.g. the set of natural numbers taken as a completed whole, the set of all sets taken as a whole, etc. because we have no grasp of these things except in a very nebulous metaphysical way. Interestingly enough a lot of mathematics can be done this way, though it comes at the cost of a little more complexity. I find it highly interesting because it considers the only true mathematical meaning of a statement such as "M exists" (where M is some mathematical object such as a set or a number or a triangle) to be "M is constructible" (i.e. one can effect a construction in one's mind [possibly using paper and pencil as aid to this]). All other possible meanings are "metaphysical" and therefore not mathematical. This is good.

I don't know why i tried to say all that because the ideas are incredibly complex and I myself am only beginning to see them. Just thought i'd say something about what the hell i'm doing these days intellectual-wise.