Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
mareg_at_mimosa.csv.warwick.ac.uk
Date: 01/29/05
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Date: Sat, 29 Jan 2005 18:38:03 +0000 (UTC)
In article <41fbd499$0$557$b45e6eb0@senator-bedfellow.mit.edu>,
tchow@lsa.umich.edu writes:
>In article <ctg43l01cjo@drn.newsguy.com>,
>Daryl McCullough <stevendaryl3016@yahoo.com> wrote:
>>I don't understand why not. What is there to mathematical
>>platonism other than commitment to the existence of sets?
>
>First of all, I'll say that personally I lean towards platonism. I didn't
>always used to be this way; many years ago I was much more influenced by
>formalism, but have gradually changed my point of view. So *my* commitment
>to the existence of sets is tinged with platonism.
>
>That aside, I would answer your question by quoting something you said later
>in the same article:
>
>>I think it's a little hazy in what sense sets exist or don't
>>exist. We can certainly talk about them meaningfully *as* if
>>they exist, but that doesn't require anything any more than
>>some kind of coherence of our story about them.
>
>As you say, committing to the existence of sets is something of a hazy
>commitment until the nature of this "existence" is fleshed out a bit.
>I can commit to the existence of sets, yet flesh out the details in
>different ways. I might adopt a fictionalist posture, somewhat like
>what you describe here. That is, I regard my statements as meaningful
>*according to a certain story*. Oliver Twist exists according to a
>certain story; this is a meaningful statement and not just a syntactic
>entity to be manipulated according to syntactic rules, yet Oliver Twist
>is of course a fictional character.
>
>A platonist doesn't regard sets as "useful fictions" but as real, and
>hence is likely to say that meaningful statements about sets are either
>true or false, whereas a fictionalist is usually happy to say that some
>statements are intelligible ("Oliver Twist's left thumbprint was a whorl")
>but have indeterminate truth value.
Hmmm! Well I think of myself as a platonist, but I don't consider that
it is meaningful to say that the axiom of choice, which is a meaningful
statement about sets, is either true or false in any absolute sense.
So perhaps I am being inconsistent? I find it strange that there are
some mathematicians who do claim to believe that ACC is true or false,
although they do not generally expect ever to find out which!
Derek Holt.
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