X-Message-Number: 10259 Subject: Can we live without formal systems? Date: Mon, 17 Aug 1998 12:01:16 -0400 From: "Perry E. Metzger" <> > From: Thomas Donaldson <> > > 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. Formal systems don't exist to explain things. They exist to check things. Mathematics is a rigorous discipline. Formal systems are a way of making sure there is a mechanical way of checking your work, so that no two people can come to different conclusions. Godel didn't get rid of the value of formal systems -- he just showed that they have certain inherent limits. If you want to do math, you need formal systems. > 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. Then you would shatter many of the most valuable math proofs of all time. How, for instance, would you do any number theory this way? You would have eliminated the prime number theorem right off the bat. I doubt much else in number theory would stand, either. Perry Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=10259