[FOM] Re: Shapiro on natural and formal languages

[Vladimir Sazonov]

JoeShipman at aol.com wrote:

I claim that the reasoning involved in these [visual] proofs is valid 
and rigorous and something important is lost when they are converted 
into sententially formalized arguments.  That they (and, in our 
experience, all such visual arguments) CAN be so converted is an 
empirical and highly nontrivial sociological and epistemological fact 
about mathematics.

My comment:

These proofs are rigorous only to that degree to which they are 
formalizable - does not matter which way. We can try to formalize
such proofs very close to how they appeared visually, not necessary
in FOL. Moreover, we could have a metatheorem on a translation from
this formalization to FOL or ZFC. This may be a positive outcome,
and I see nothing so unusual here.

On the other hand, the discrepancy between intuition (via visual
images or some other way) and ANY kind of formalization, even that
looking very close to the intuition is evidently unavoidable.


Vladimir Sazonov