mangoe@umcp-cs.UUCP (Charley Wingate) (10/09/85)
In article <574@spar.UUCP> ellis@spar.UUCP (Michael Ellis) writes: > On the contrary, purely rational systems are true independently of > any observation whatsoever. > Why is 1+1=2? Why does A&~B <=> ~(~A|B)? Why is the sum of the angles > in any Euclidean triangle invariably 180 degrees? > Not because they are observable facts, although the utility of such > totally certain facts derives from their applicability to physical > laws. Sure, 1 rock plus 1 rock equals two rocks. But 1 cloud plus 1 > cloud often equals 1 cloud. And 1 rabbit plus 1 rabbit may equal > thousands of rabbits. Nonetheless, we do not consider that such > observable facts contradict mathematics. > In fact, we usually assert that math and logic are true in any > conceivable universe. Science, whose central axiom is empirical > induction, IS a religion from the purely rational viewpoint, for which > mathematics and logic represent the highest level of certainty. If you want to persist in this folly, the least you can do is establish several flavors of truth, because there is a quite obvious qualitative difference between the following three statements: "I am typing this in on my computer." "Gravitation and the curvature of space are related in such-and-such way." "One plus one equals two." In particular, the first two possess a relationship with Reality that the third lacks. You cannot test mathematics against reality (although you can certainly argue that some part of mathematics and some aspect of reality are not related-- but then you're talking science and not math). The fact that Euclidean geometry does not describe nature does not make it "false". Take a look again at that last sentence. Doesn't it look a little rediculous? The reason for this is that in math we talk about the "truth" of math in isolation with a different, more appropriate word. Rather than saying it is true, we say that it is consistent, preferring to reserve truth to refer to relationships to the world (even statements about consistency within mathematics too, but then the "world" is obviously restricted). And as can be easily illustrated by some elementary paradoxes, one can easily make rational arguments for statements that are actually false, even though the arguments appear to be completely consistent. Charley Wingate