# [FOM] Extended Formalization Thesis

A. Mani a_mani_sc_gs at yahoo.co.in
Thu Jan 10 19:42:04 EST 2008

```I think the formalization thesis may or may not be tautological. From the
point of view of expressive power. Richer languages will tend to accommodate
more scope for vague and fuzzy reasoning and at the same time provide greater
scope for understanding. Mathematics in the 'potentially formal' sense does
use a richer language ... that develops its meaning by contexts. Ultimately
any 'more formal approach' is the result of a dialectics between "reducing
the scope for vague and rough reasoning" and "improving comprehensibility and
clarity". (I am following Sazonov's notion of 'potentially formal')

Given a finite sequence of apparently clear instructions in the natural
language then it is not possible to expect different respondants to reason in
the same logical way. This holds even when 'the clear instructions' are
easily expressible in PC. This is because people reason logically
(classically) and not in the same way. All this is based on empirical
studies... of course these studies involve the individual researchers bias.

When it comes to "potentially formal" Mathematics, we have greater degree of
consensus on "intended interpretation". There are few empirical results that
prove the formalization thesis. The main problem of the formalization thesis
is this psychologism part.

The boundary between "Strictly formal" and "potentially formal" Mathematics is
also vague. It is obviously easier to stick to different versions of "formal"
Mathematics. Then I think the following "Restricted or Extended Formalization
Thesis" is much better (I will improve it later):

"Given a formal logic K in a language L in which we have a notion
of "potentially vague and rough expression and inferences", we can always
formulate a logic S in a language J (that is close to L), that captures every
proper theorem of K (by way of translations)  and is
a) minimal in that the language J is a minimal modification of L and S is
minimal among such other logics
b) avoids the potentially vague and rough inferences "

Best

A. Mani

--
A. Mani
Member, Cal. Math. Soc
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