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Date: Thu, 27 Oct 94 09:47:43 +0100
From: dahn@mathematik.hu-berlin.de (Dahn)
Message-Id: <9410270847.AA01413@mathematik.hu-berlin.de>
To: qed@mcs.anl.gov
Subject: Semantics
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jmc:
> However, there is more to it for us than a philosophical controversy
> in the foundations of mathematics. If QED is to succeed, then informal
> mathematical arguments have to be translatable into QED. My opinion
> is that many of these arguments when formalized in logic will actually
> concern the relation of formulas to models, i.e. wil involve truth as
> a concept.
Agreed, but: Consider just the simple case of integers and take Hilbert's 10th problem into account.
>From Matijasevitch/Robinson we know, that there is such a simple thing like a Diophantine equation (polynomial equation with integral coefficients,+,*) the solvability of which in the integers can be neither proved nor refuted in ZFC (for other systems other equations serve the same purpose). So, if there are "the unique" integers (as most mathematicians believe though nobody has ever seen this infinite set) - what's the truth?
Everything we can! prove in mathematics depends only on the assumptions we used in the specific proof and on the logical calculus which is used. If the calculus is correct, the theorem will hold in any model satisfying the assumptions. If you are only interested in it's truth in "the unique" integers - well, that's a personal matter outside mathematics.
Ingo Dahn