Path: utzoo!attcan!uunet!lll-winken!lll-tis!helios.ee.lbl.gov!nosc!humu!uhccux!lee
From: lee@uhccux.uhcc.hawaii.edu (Greg Lee)
Newsgroups: comp.ai
Subject: Re: The Ignorant assumption
Message-ID: <2365@uhccux.uhcc.hawaii.edu>
Date: 9 Sep 88 14:27:25 GMT
References: <3546@s.cc.purdue.edu>
Organization: University of Hawaii
Lines: 50

From article <3546@s.cc.purdue.edu>, by afo@s.cc.purdue.edu (Neil Rhodes):
" ...
" I have a problem with Mr. Lee's reasoning in the above statement, and it
" seems to be the foundation of most of his recent arguments.
" 
" If a formal system were to contain "only rules of derivation," what would
" these rules act upon to form statements (theorems) about the system?
" Rules alone in a formal system give you nothing.  For this reason, you

They give you nothing but tautologies, at least.

" need a given set of statements (axioms) from which these rules can derive
" other statements (theorems).  Since these axioms are not derived and are
" necessary to the formal system, then you must "believe" them to be true
" while working within the system.

There are formalizations of logic that require axioms, but not all
do.  Gerhard Gentzen created systems that have no axioms.  For
instance:
	Suppose p (one can introduce provisional assumptions freely)
	Conclude p (one can repeat an assumption as a conclusion)
  So, p implies p (since p was concluded on the basis of the provisional
	  assumption p, one can derive the implication)

" Since many scientific statements are derived within formal systems, to
" believe these statements you must also believe other statements which
" cannot be proved.

Perhaps that's so.  My example does not concern "scientific statements".
I was reacting to a statement that "formal systems" require assumptions.
They don't -- maybe formalized scientific systems do, in a sense,
but even there assumptions can be treated as provisional rather than
as axioms.  This is not to disagree with what Neil Rhodes said just
above.

As you will observe, a Gentzen system does involve assumptions, but
no specific assumption is given as part of the system.  That is, there
are no axioms.

" If Mr. Lee still believes that science asks us to take nothing on
""faith," then I am curious to know what flaws he finds in *my*
" reasoning.

I find no flaws.  If you are to have faith in scientific conclusions,
you must have faith in scientific assumptions.  But why have faith in
anything?  Why does "science ask us" to do this?  If you have a need to
believe in things, other than tautologies, I think you ought not to lay
this at the door of science.  It's a personal problem, which I think you
should try to get over.

		Greg, lee@uhccux.uhcc.hawaii.edu