Date: Apr 19 1997 4:50PM
"If I make false assumptions, then no matter how rational..."
Jeff, when I made my statement (a blatant ripoff of Hegel's quote, who unfortunately himself did not live up to it) about the rational being the real and the real being the rational, I did not define "rational" as simply "the valid". What is rational is both valid and sound. What is sound does not make false assumptions by definition.
In the case of Euclidean geometry, this is the most maligned case of mathematics failing to be an expression of reality yet being entirely rational. Euclidean geometry makes no claim upon any universe besides that consisting of plane surfaces. It is wholly consistent with reality as it claims to express it. It is no fault of the Euclidean system that for millenium after him, people assumed that the interior angles of a triangle under all conditions were one hundred eighty degrees. This is not rational. It is irrational based on a very false assumption about the universe and that being that triangles were always one hundred eighty degrees in all conditions. Namely, it assumes the univerese is a plane. This contingent proposition has no necessity about it, and currently such thinking is understood to be false.
So, Jeff, don't misunderstand "valid" for "rational".
valid that which follows from premisses irregardless of the truth of the premisses.
Socrates is a fish.
All fish are red.
Socrates is red.
sound that which is valid and for which the premisses are true.
I am a man.
All men are mortal.
I am mortal.
What I mean by the rational is also the sound, and what is most sound is that which is necessary. This is why I said that what is necessarily rational, or necessarily coherent, also corresponds to the real. However, that which is contingently coherent needs some other test in order for us to judge its correspondence. In the case of Euclidean geometry, we need only see empirically whether the world really is a plane or whether it exhibits other qualities, and that judges the coherence of Euclidean geometry as a system expressing reality. In other words, the ultimate coherence of a contingent proposition to reality is in some test of its correspondence as expressed by the nature of the proposition. What the ultimate coherence of a necessary proposition is is simply the proposition itself because that's what the nature of the proposition expresses.
Plato knew that mathematics in terms of reality was hypothetical in nature because it began with postulates it did not examine as to whether they were true or not. Plato, in his Republic listed mathematics along with science as hypothetical disciplines. He said that the only difference between philosophy and the hypothetical arts was that philosophy began with true premisses, and this is what he called dialectical. That Euclid's geometry was not taken to be an absolute expression of reality was known to some of the ancients. Why it's taken to be such a defeat for "rational is the real" today is generally based on a confusion between validity and rationality.