FOM: Standards of mathematical rigour (and paradigms of Thomas Kuhn)

[Vladimir Sazonov]

This is reply to a posting of Charles Silver from 20 Oct 1998.

Thank you very much for recalling me the ideas of Thomas Kuhn on 
Structure of Scientific Revolutions. Really, about 15 or more 
years ago I participated in some meetings in Novosibirsk where 
this was actively discussed. Probably we can also consider any 
standard of mathematical rigour as a of paradigm.  Anyway, these 
standards are changed not so often. Only during a crisis 
mathematicians feel some discomfort. 

It seems that at present practically everybody agreed with the 
framework based on set theory so that no problems with the 
rigour arise in everyday mathematical practice.  Those who 
knows formal rules of predicate calculus may have somewhat 
different, detailed standards. But this is a paradise only for 
those who do not try to look outside this framework or who is 
irrelevant to length of (imaginary?) proofs and to uncontrolled 
using implicit abbreviating mechanisms in the proofs. All 
abbreviations are allowed?  What does it mean "all"? (Is not 
this "all" something contradictory?) If not all then which ones?  
What is the role of different kinds of abbreviations? If we will 
not decide this (possibly for each concrete formal system 
differently), what is then a *genuine* rigorous mathematical 
proof?. 


Vladimir Sazonov