[FOM] mathematics as formal
Eray Ozkural
examachine at gmail.com
Sat Mar 8 22:46:00 EST 2008
I think it may be a skewed picture of history to suggest that
formality in mathematics arose with the formalization of foundations.
While the axiomatic approach and proof-theory has been vastly
fruitful, with far-reaching applications, I think that in essence not
much has changed from conducting mathematics in an "informal" way
(except that better automation has been secured). Also, these
meta-theories were always present in some form, as explained in detail
in our moderator's excellent book "The Universal Computer", Although
the initial theories "proto logical languages" etc. were too simple to
formalize any substantial piece of mathematics, with the advent of
something as simple as first order we can begin to reason about such
distinctions as axioms and derivative theorems in everyday
mathematics, which is nice as it goes to show that we are being more
conscious of ourselves when we are making mathematics.
As Bhupinder would agree, the most interesting aspect of mathematical
discourse is that I can transfer an abstract idea from a mind to
another mind without any ambiguity. Such is the basic formal character
of mathematics, and I believe that no philosophical approach can
liberate mathematics from the necessity of formality: it is a
linguistic requirement. If it were not possible for me to disambiguate
mathematical statements with a little bit of the right kind of
mathematical context, then I would be at a loss, whenever I had to
read a piece of mathematics. A trivial fact, but it cannot be
neglected. Being form-al wasn't discovered, it was simply better
understood through meta-theories. (I always thought that was clear
from the start, though). Just my ideas regarding a consistent
understanding of mathematics no matter what foundation has been
chosen!
The next level, I think, is a more philosophically sound way of
establishing computationalism, but computationalism isn't favored much
among mathematicians so I'll let the programs speak for themselves
first. If computationalism is right we must be able to pinpoint
certain programs, or at least the possibility thereof. Thus, again, we
must not be dogmatic about our assumptions, we must have very good
reasons for accepting any of them (semantic reasons, that is).
Best,
--
Eray Ozkural, PhD candidate. Comp. Sci. Dept., Bilkent University, Ankara
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