[FOM] Truth and Consistency
4mjmu at rogers.com
Mon Jun 2 16:52:32 EDT 2003
----- Original Message -----
From: "Neil Tennant" <neilt at mercutio.cohums.ohio-state.edu>
To: "Lucas Wiman" <lrwiman at ilstu.edu>
Cc: "FOM math list" <fom at cs.nyu.edu>
Sent: June 2, 2003 10:18 AM
Subject: Re: [FOM] Truth and Consistency
> On Sun, 1 Jun 2003, Lucas Wiman wrote:
> > While I agree that [the fact that ZFC has been amply tested by
> > "mathematical experience"---Parsons] is a good reason to keep using ZFC,
> > I don't think that this tells us that ZFC is consistent. This tells us
> > that if ZFC has any inconsistencies, then they are probably extremely
> > non-obvious. They might have been missed by mathematicians and set
> > theorists for so long because they're extremely weird or long or
> > something like that. This is thus a contingent fact about human
> > psychology.
> Surely it would not be a contingent fact about human psychology if it
> turned out to be the case that the shortest proof of an inconsistency in
> ZF is extremely long? Wouldn't it be a mathematical fact?
Lucas Wiman wrote:
> > I am uncertain whether
> [Dummett's criticism of Field's inductivist justification of the
> consistency of ZF]
> > is correct; we have no non-inductive
> > verification of the facts of physics, yet they are generally considered
> > inductively verified. If we take Popper's falsification view, then
> > clearly it's quite reasonable to conjecture that ZF is consistent since
> > so little disproof exists of it, though accepting it as a verified fact
> > may be too strong.
Neil Tennant responded:
> Did Popper ever intend his falsificationist view of empirical theorizing
> to apply also to such claims as that ZF is consistent?
I don't know that Popper himself had an opinion, but the Popperian Imre
Lakatos did feel that mathematical reasoning proceeded by a process of
"proofs and refutations", which is taken to be analogous to the process of
"conjectures and refutations" outlined by Popper for the physical sciences.
He outlines this view in the book by the same name and elsewhere.
On a somewhat different topic (related to the Witgenstein and contraction
thread of April/May), it occurs to me that Lakatos was one writer who felt
that a scientific theory might contain contradictions without this impairing
its efficacy qua scientific theory. In fact, I believe he claims that it is
standardly the case for a theory to contain contradictions.
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