request for ideas
From: Craig Feinstein (cafeinst_at_msn.com)
Date: 08/22/04
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Date: 22 Aug 2004 08:31:04 -0700
I am planning to write a paper which surveys mathematical results that
show that the old "axiom->proof->theorem" way of doing mathematics
does not always yield complete information about mathematics. The
prime example of this (which started it all) is Godel's Incompleteness
Theorem, but there has been a lot of work in this area since then.
For instance, Gregory Chaitin has an incompleteness theorem which
shows conclusively that a certain number which he calls Omega, which
is really the probability that a computer program halts (defined in a
way that makes sense), is a random number - which implies that there
is no finite axiom system that can yield all of the bits of Omega. He
concludes from all of his work that sometimes one has to simply
perform experiments in mathematics and form conclusions from the
experiments without being absolutely certain that the conclusions are
correct.
It is these types of very original ideas that I am looking for to put
in my paper, that there are some problems out there that are so
difficult for us to get a grip on that we might have to approach them
like a chemist approaches chemistry, never being 100% sure that his or
her theories are always correct.
Anyone who knows of results like these or has done work in this area
or has original ideas is welcome to respond to me on usenet or if you
want, you can email me directly.
Craig
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