Re: Platonism

From: Acid Pooh (poopdeville_at_gmail.com)
Date: 11/30/04 

Date: 30 Nov 2004 00:20:24 -0800

tchow@lsa.umich.edu wrote in message news:<41abb653$0$571$b45e6eb0@senator-bedfellow.mit.edu>...
> In article <39d6e584.0411291439.3bf05395@posting.google.com>,
> J.E. <troubled6man@yahoo.com> wrote:
> >tchow@lsa.umich.edu wrote in message
> >news:<41aa2a59$0$566$b45e6eb0@senator-bedfellow.mit.edu>...
> >> Actually having ZFC in mind when you say "exist" is very different
> >> from pulling out ZFC as a magic amulet when cornered into defending
> >> your beliefs.
> >
> >Apples and oranges. If the only time that you discuss "what you
> >really meant" is when cornered, then what you say then about your past
> >intent is the only information available to outsiders about your past
> >intent. We have introsepction about ourselves, but if we discuss the
> >intentions of others, we have to go by the best available evidence.
>
> This is an irrelevant philosophical tangent. Establishing the difference
> between "having ZFC in mind" and "pulling out ZFC when cornered" is easily
> done even in your verificationist framework. Simply ask, "Did you have ZFC
> in mind when you said, `exist'?" The answer will come back, "No." This is
> different from the question, "What ultimately justifies the validity and
> correctness of your mathematical discourse?" to which the reply will be
> something like, "Er...um...well, it can all be formalized in ZFC, can't it?
> Not that I know what the axioms of ZFC actually are."
>
> >That's an interesting viewpoint, full on ontological content. If
> >natural numbers are thought of as finite ordinals, then the modern
> >theorems about natural numbers are different than old platonic
> >theorems because they are about sets, specifically ordinals.
>
> If you pull this one on the typical mathematician, they'll probably react
> with some form of "structuralism," saying that the theorems are equivalent,
> and all we care about in mathematics is whether the theorems are equivalent
> (rather than "the same" in some sense that only logicians care about).

If he has some logical sophistication, he'll note that the set of
finite ordinals is a model for PA and mention that the theorems of PA
are theorems "about" any objects which satisfy the PA axioms. But
this might be that sense of "the same" only logicians care about. :-)

'cid 'ooh

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