[FOM] Platonism and Formalism (answers to Sandy Hodges and Haim Gaifman)

[Vladimir Sazonov]

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