[FOM] Is Godel's Theorem surprising?
laureano luna
laureanoluna at yahoo.es
Fri Dec 8 09:23:23 EST 2006
On Charlie Silver wrote:
>I'm just wondering what accounts for the shock so to
>speak of Godel's
>Theorem.
One possibility is that Gödel's theorem was surprising
because of the predominant philosophy of mathematics
at his time, due to the Vienna Circle.
Since most neo-empiricists admitted of no a priori
synthetic truths, non empirical truths were regarded
as arising from linguistic conventions (this was at
least Carnap's explicit program). But if we reach
mathematical truth by developping our implicit
linguistic conventions, we are most probably using a
finitely describable code, an implicit finite grammar.
So, a complete formal axiomatization of logic and
mathematics should be possible.
>I'm also wondering, though this is a separate point,
>whether today
>the theorem is not only not surprising, but perhaps
>even intuitively
>obvious.
As for this second question, I can only refer to my
own case; I would say we now know much better the
possibilities and the logic of diagonalization in a
broad sense. This helps me to see why logical and
mathematical truth is inexhaustible by means of finite
resources.
I should add that diagonalization is itself
surprising, so Gödel's theorem still retains a bit of
its mystery.
Since yours is a philosophical question, I dare
suggest that we should be able to account for the very
way human reason works in its deep level in order to
account for the ultimate root of Gödel's result.
Regards,
Laureano Luna Cabañero.
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