[FOM] Extended Formalization Thesis

[A. Mani]

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