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Article 5985 of comp.ai.philosophy:
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>From: torkel@sics.se (Torkel Franzen)
Newsgroups: comp.ai.philosophy
Subject: Re: penrose
Message-ID: <1992May30.071423.5597@sics.se>
Date: 30 May 92 07:14:23 GMT
Article-I.D.: sics.1992May30.071423.5597
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In-Reply-To: costello@CS.Stanford.EDU's message of 29 May 92 22:47:04 GMT

In article <1992May29.224704.22798@CSD-NewsHost.Stanford.EDU> costello@CS.
Stanford.EDU (T Costello) writes:

  >I on the other hand was considering, not "true" theories but
  >consistent ones. It does not hold that adding your reflection principle 
  >to a  consistent theory results in a consistent theory.

  No, but why should consistency be of any interest here? The whole point of
reflection principles is, as Feferman puts it, that they "express a
certain trust" in the axioms of a theory, and in particular the extension
by reflection of a theory T that I described is epistemologically
worthless (for all we know) unless we accept all theorems of T as true.

  >I thought (and still think) that assumptions 
  >about truth of theories often muddy the water, as it is a purely semantic
  >issue, and not fully open to logical analysis.

  What does this mean? Let us be a bit more concrete and consider
Peano arithmetic and its extensions. From the point of view of
mathematical knowledge, consistency of these extensions is by no means
sufficient to give their theorems any interest. For example, if I have
proved in an extension P' of P that there are infinitely many twin
primes, I want to know whether this gives me any reason for believing
that the theorem is true. And that the theorem is true means only
this, that there are infinitely many twin primes. If the axioms I have
added to P are axioms that I know are true - e.g. if P' is an
extension by reflection of P - then I do know, on the basis of the
proof, that there are infinitely many twin primes. If I merely know
that P' is consistent, I cannot conclude from the proof that there are
infinitely many twin primes. What do you consider unclear about this, and
how would you look at mathematical knowledge without introducing "semantic
issues"?


