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From: ydobyns@tucson.princeton.edu (York H. Dobyns)
Subject: Re: Review of Shadows of the Mind
Message-ID: <1995Apr4.192829.27402@Princeton.EDU>
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Date: Tue, 4 Apr 1995 19:28:29 GMT
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In article <3lnp4b$mrb@mp.cs.niu.edu> rickert@cs.niu.edu (Neil Rickert) writes:
[and appears to contradict himself in so doing...]
>Firstly, I am not skeptical about our ability to know mathematical
>truths.  My skepticism is about the idea that there are such things
>as mathematical truths outside of axiom systems.
[...]
>I agree with what Boolos and Jeffrey say.  Here is the important
>part:
>
>>                                       hence for any
>>     interpretation of the language of the theory there will be
>>     truths in that interpretation which are not theorems of the
>>     theory.
>
>Notice that they do not speak of "mathematical truth".  They speak
>only of "truth in an interpretation."

This very phrasing seems to imply that there is a concept of truth
involved that is not merely provability under a given axiom system.
What can it even mean to say that something is true but not provable
under a given axiom system, if truth consists solely in derivability
under some set of axioms? If you admit, with Boolos and Jeffrey, that
there are statements that are true *of* an axiom system (plus
interpretation) other than the interpretations of those theorems
provable *in* the axiom system, you have just admitted a criterion of
"mathematical truth", whether you are willing to call it that or not.

-- 
York Dobyns		ydobyns@phoenix.princeton.edu
Honest skeptics must be willing to question *their own* beliefs, as
well as those of people with whom they disagree.
