[FOM] The Gold Standard
Nik Weaver
nweaver at math.wustl.edu
Wed Feb 22 03:04:29 EST 2006
Harvey Friedman wrote (quoting me):
> > So I ask: is there any non-platonist justification of the
> > assertion that Z_2 has a natural model?
>
> Platonist: Is that your definition of Platonism? Then tautologically,
> the answer may well be no. But so what? Why aren't you a Platonist?
That isn't my definition of platonism. My definition is: belief
that mathematical "set" talk literally refers to a special kind
of abstract object. I'm surprised you haven't picked this up yet.
So what? You wrote "The claims that predicativity has some special
place in the robust hierarchy of logical strengths ranging from
EFA through j:V into V are unjustified." If you're now admitting
that impredicative systems can only be justified on platonistic
grounds, then predicativism has a very special place: it is the
most you can believe without being a platonist.
If you're not admitting this, please specify an impredicative
system and explain how it can be justified on non-platonistic
grounds.
Why I'm not a platonist: I've answered this several times
already; see
http://www.cs.nyu.edu/pipermail/fom/2006-January/009520.html
http://www.cs.nyu.edu/pipermail/fom/2006-February/009818.html
or better yet read Hartley Slater's analysis in
http://www.cs.nyu.edu/pipermail/fom/2006-February/009887.html
> Finitist: We know that PA does not have a model. Is there any
> non-platonist justification of the assertion that PA is consistent?
Yes. In order to believe that PA has a model we need to believe
in structures of type omega. For example, I have suggested a
structure involving marks on paper. There is no need to posit
the existence of natural numbers as some special kind of abstract
entities. You can doubt that omega structures exist but this is
not a doubt about platonism.
Nik
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