FOM: Standards of mathematical rigour (and paradigms of Thomas Kuhn)
Vladimir Sazonov
sazonov at logic.botik.ru
Sun Oct 25 09:18:23 EST 1998
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
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