FOM: second-order logic is biased
Stephen G Simpson
simpson at math.psu.edu
Wed Mar 24 18:38:34 EST 1999
My posting of 24 Mar 1999 17:55:24 (the previous posting, the one
about Platonist/realist assumptions implicit in the idea of absolute
standard semantics) was titled "second-order logic is a myth". But I
should have titled it "second-order logic is biased".
Randall Holmes 24 Mar 1999 13:48:53 writes:
> ... mathematics (even arithmetic) cannot be completely captured by
> any formal system. This is a well-known fact.
I'm not so sure that's a well-known fact. Isn't it more of a
well-known philosophical assumption? Extreme formalists might want to
argue that arithmetic is *nothing but* some particular formal system.
Thus your `well-known fact' appears to be a prejudgement against the
extreme formalists.
Tarski's theorem on undefinability of arithmetical truth may be viewed
as bolstering your `well-known fact'. But Tarski's theorem itself
assumes (on Platonist/realist grounds?) that there exists a truth
predicate for all arithmetical sentences. And, in order to conclude
that `arithmetic cannot be completely captured by any formal system',
you probably want to identify `arithmetic' with this truth predicate.
Don't these assumptions amount to a prejudgement against the extreme
formalists?
Could this be another instance of the built-in bias of `second-order
logic with standard semantics' against formalism and in favor of
realism/Platonism? See also my posting of 24 Mar 1999 17:55:24.
Once again, I am speaking as neither a formalist nor a
realist/Platonist. I only want all sides to get a fair hearing.
-- Steve
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