[FOM] What produces certainty in mathematical proofs?

[Alex Blum]

Feng Ye wrote:

>On what ground can we decide certainty, or evaluate a judgment regarding certainty? What proofs or evidences can one
>offer for a judgment regarding certainty?
>  
>
Speculation: We know(from Saul Kripke) that identity/non-identiy 
statements(whose refering expressions refer "rigidly" such as constants 
or variables and some but not all descriptive phrases) are true/false 
iff necessarily true or false. Hence one instance will 
confirm/discomfirm the identity/non-identity. Thus one confirming 
instance of "water=H2O" to use his classic identiy statement should 
confirm it as a certainty. There are other innumerable examples of this 
kind.   
Can this underlying logic of identity be extended to a wider class of cases?

Alex Blum