X-Message-Number: 10239 Date: Sat, 15 Aug 1998 09:36:26 -0400 From: Thomas Donaldson <> Subject: CryoNet #10229 - #10236 Hi everyone! Yet more on Goedel's Theorem etc. There is another approach entirely to the results Goedel found. The question it might raise in some minds is whether or not we want to use a formal system at all. We are after all living in the world without benefit of a formal system which explains the world to us. The whole idea of formal systems began with the Greeks, and has now been developed far enough that some people such as Goedel have come upon one of their central problems. People were able to invent and think things before the Greeks, and the Chinese (with no well developed notion of theory) went quite far -- for some time farther than any other civilization on Earth. I am not doubting Goedel's Theorem. I am raising a question about what it means. It's even likely that we can do mathematics, though it would not be the kind done with formal systems, axiomatically. Even the Pythagorean theorem, as a statement about how lengths and triangles behaved, was known to the Assyrians if not before them. And we might even avoid the problems of such systems by refusing to admit proofs by contradiction and insisting that the entities we discuss must always be constructible. Symbols are NOT the world. We do not live in symbols, we live in the world, and ultimately our language and words are defined not by other symbols and language but by their use in the world, sometimes by just pointing, sometimes other ways. And the entire idea of formal systems has a history, and may someday be abandoned for other means of doing math, or physics, or in any way dealing with the world. After all, it was experimental science, not mathematics, which brought the West first even and then far ahead of the Chinese. As someone who taught math at college level and beyond, I will add that axioms do provide a useful form of exposition. But for some time now that idea has begun to look shaky --- Goedel's Theorem providing an example. I doubt that axiomatic math and formal systems will disappear entirely, but they may well end up as useful tools for exposition, to which we do not and should not attach any special metaphysical significance. Or does anyone on Cryonet seriously believe that we've come to the acme of understanding, even of mathematics and how to do it? Best and long long life, Thomas Donaldson Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=10239