[FOM] Platonism and Formalism

Torkel Franzen torkel at sm.luth.se
Tue Sep 30 07:34:12 EDT 2003

Harvey Friedman says:

 >Their best argument, or at least one of their very strong arguments, to
 >convince non Platonists, would be the startling consequences in V(9) of
 >their most powerful principles that they would claim are absolutely true,
 >and which cannot be proved without accepting their most powerful principles.

 >Yet even more startling consequences of a similar nature in V(9) would come
 >out of principles that they would claim are absolutely false. And of course
 >the Platonist isn't in a position to prove or refute these more startling
 >consequences in V(9).

I think Platonism in the sense of holding certain set-theoretical
principles to be descriptive of an objective realm of sets is not a
central issue here. The significance of the fact that set-theoretical
principles (whether compatible with AC or not) do settle questions in
concrete mathematics is enormous, but does not lie in supporting some
principles before others.  Rather, the significance is that such
principles do have consequences in concrete mathematics, and that even
if we do not regard it as meaningful to ask whether the principles are
true, we can meaningfully ask whether their consequences in concrete
mathematics are true (however exactly we define concrete mathematics),
and as long as we take questions in concrete mathematics, such as
those about V(9), to turn on mathematical facts, not only on taste and
aesthetics and consensus and free choice, we are faced with the
philosophical and foundational problem of how to understand this
connection between abstract principles and actual or hypothetical
computations, and how to justify a claim that some particular set of
principles yields true consequences in concrete mathematics. This
problem presupposes a form of Platonism with regard to concrete
mathematics, and a consistent non-Platonist will reject the idea of
there being "mathematical facts" about V(9) other than those that we
accept as such.

Torkel Franzen

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