[FOM] Platonism and Formalism

[Haim Gaifman]

Sandy Hodges wrote:

>
>  If I confine myself to asserting:
> S |- F
> when I know that F is provable in system S, and to asserting:
> ~ ( S |- F )
> when I know that F is not so provable, and if I demur when some
> Platonist proposes
> ( \/ S, F ) ( ( S |- F ) V ~ ( S |- F ) )
> as a metamathematical axiom, then I fail to see that I have done
> anything that makes no sense.   Have I?
>
> It is less clear, but I think I could even describe myself as behaving
> in this way, were I indeed a formalist, without saying anything that
> makes no sense.
>
> If I could behave as a formalist, and describe what it means to behave
> as a formalist, without making any utterances that make no sense, then
> in what sense does formalism make no sense?
>

This is essentially the position Abraham Robinson took in his  1964 paper
``Formalism 1964'' in {\it Logic Methodology and
Philosophy of Science}, Y. Bar-Hillel ed., North-Holland 1965, pp.
228-246.
It is a coherent description of a "formalist's behavior". But perhaps we
should
distinguished between describing one's *behavior* and giving an *account*
of a certain view. For example, we believe that the system we employ is
consistent. Can a formalist give grounds for this belief, over and above
the brute fact that so far no contradiction has been discovered? Note that
even
the disjunction "S is consistent V S is inconsistent" involves an
application of
the excluded middle in an infinite domain, and it cannot be reduced to
something more palatable to a finitist (along Hilbert's lines).

Haim Gaifman