[FOM] Platonism and Formalism (answers to Sandy Hodges and Haim Gaifman)
Vladimir Sazonov
V.Sazonov at csc.liv.ac.uk
Wed Sep 10 14:55:51 EDT 2003
Sandy Hodges wrote:
> S.H. 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,
How can you ever "know" the latter? I see the only precise way to
assert this in the form
"It is proved in a metatheory MT that `~ ( S |- F )'",
and this metatheory is "respectable" in a sense (which should be
further explained or well-known).
Expressing thoughts in the above way makes mutual understanding
almost impossible. I do not insist to be absolutely pedantic,
but, at least, avoiding deliberately unclear phrasing is necessary.
and if I demur when some
> Platonist proposes
> ( \/ S, F ) ( ( S |- F ) V ~ ( S |- F ) )
> as a metamathematical axiom,
I guess the symbol \/ means forall S,F. Of course, this
metamathematical axiom is not related specifically with
Platonism at all. It is just the law of excluded middle of
the underlying logic. Any formalist (rejecting Platonist's
views) can consider ANY formal system (also any metatheory)
without any doubts on the law of excluded middle or any other
logical lows, just because they are both formal and intuitively
reasonable. Formalists do not reject any formal theories,
whether they are about infinite sets, large cardinals, etc.,
as well as any formal logic (intuitionistic, modal, etc.),
as soon as there is any reasonable intuition behind the
formalism considered.
Haim Gaifman wrote:
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?
According to a formalist position, how I understand it, the ground
for such a belief (of a formalist!) is some intuition on imaginary
"model" for the system considered. Of course, the intuition
is not absolutely reliable argument and, hence, is not an
absolute guarantee - just a good enough reason to believe.
(Such an intuition is, of course, different from Platonistic
one, as it has NO pretension on any absolute value.)
There is always a possibility for doubts, as well.
"The brute fact that so far no contradiction has been discovered"
in a formalism has, of course much less (almost zero) value.
We could just try in a wrong direction. Then, suddenly, some new
Bertrand Russell having a clear intuition (about a gap in the
intuition, if any, concerning a system considered) will come
with a new paradox (contradiction). But if we really have a good
enough intuition behind a system then there is a hope that this
will not happen. But only a hope. No guarantee!
I believe that formalist position should not be reduced to a
caricature, even if this view has been popular by some reason.
I also believe that this caricature is just a pure creature of
Platonists and has NO real prototype. People could express
in some situations extremists views like David Hilbert did
about the meaning of geometrical axioms. But this is the ordinary
way to stress on something important, to attract attention,
to present something most brightly, or the like, during polemics.
Was Hilbert a "fanatic formalist"? Did he ever consider seriously
any concrete formal system without paying attention to its
intuitive meaning?
Vladimir Sazonov
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