From: gordon brown (gbrown@euclid.colorado.edu)
Date: Thu Mar 06 2003 - 17:29:54 EST
Jim,
If when you refer to acceptance of the starting assumptions, you mean
believing the truth of the axioms for a mathematical system, you are
accepting an antiquated view of what constitutes mathematics. The problem
of proving Euclid's Fifth Postulate changed our attitude toward
mathematics since it was shown that if Euclidean geometry is consistent,
so is hyperbolic geometry, and vice versa. We no longer presume to know
the meaning of the terms used in the axioms. They are regarded as
undefined regardless of what they may mean in other contexts, and the
truth of the axioms or of theorems derived from them does not depend on
the truth of the statement when the words are given some particular
meaning. For example, the truth of a theorem in Euclidean geometry does
not depend on our world being Euclidean.
Gordon Brown
Department of Mathematics
University of Colorado
Boulder, CO 80309-0395
On Thu, 6 Mar 2003, Jim Armstrong wrote:
> Again, the essence of all "proofs" is the sufficiency of persuasive.
> That intrinsically introduces a subjective element since our individual
> criteria for "sufficiently persuasive" vary all over the map. Even
> "rigorous" mathematical proofs are, for example, subject to one's
> acceptance of the starting assumptions, and in these days even the most
> basic of assumptions in mathematics are subject to review and even
> reframing. So it is not surprising that what may be sufficient to
> constitute convincing proof for one of us may not satisfy another who
> per force applies a different set of criteria.
>
> With that, I favor the flavor of the amended Burgy-statement. JimA
>
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