FOM: Re: Kline/Davis/Friedman
Dana_Scott@POP.CS.CMU.EDU
Dana_Scott at POP.CS.CMU.EDU
Sat Jan 3 11:27:32 EST 1998
Martin Davis said about Morris Kline "I believe that he never grasped
the concept of a formal logical system." Friedman agrees. Both Davis
and Friedman, however, give Kline good marks for many aspects of his
synoptic exposition of the history of mathematics (though, In my
opinion, he never achieves the depth of, say, A. Weil or J. Dieudonne'
in their historical writings). I wonder what they think of Kline's
later book, "Mathematics: The Loss of Certainty."
I would like to ask if the problem Prof. Kline had was not so much
with formal systems as with FORMAL LANGUAGES. Even before getting to
a concept of formal proof, mathematicians -- from my observation --
have a very hard time getting the idea of quantifiers and bound
variables. Leaving proof aside, the idea of first-order or
higher-order DEFINABILITY is hard to put over.
This is certainly why Go"del went to a finitely axiomatizable version
of set theory with classes in his monograph on the CH: he wanted to
avoid metamathematical discussions and making the formal language
explicit. And the move made it very hard for me to understand as a
student why V=L was consistent! But, this also is why I think set
theory had a lot of success: many constructions that logicians might
describe with formulae are replaced by set manipulations. They seem
"more concrete". Tarski, Halmos and others over many years tried --
without enough success, I think -- to replace logical languages with
"algebraic logic". And algebraists go to a lot of trouble to speak of
"semi-algebraic sets" rather than "first-order definability in
real-closed fields". Also many people love Church/Curry combinators
because they eliminate bound variables.
In connection with nonstandard analysis, it is particularly hard for
mathematicians to get clear about the idea of truth for higher-order
statements RELATIVE to a model (the "internal vs. external" conflict).
So I feel it is no wonder that Kline could not see why the proof of
the categoricalness of the axioms of complete ordered fields did not
outlaw nonstandard models. His brief description of NSA is not too
bad on pp 274-5 of "Mathematics: The Loss of Certainty." But he did
not seem to get the idea of the "transfer principle" and "conservative
extension".
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