Which Mathematical Logic is the Logic of Mathematics? Jaakko Hintikka

Abstract.

The main tool of the arithmetization and logization of analysis
in the history of nineteenth century mathematics was an informal logic of
quantifiers in the guise of the “epsilon–delta” technique. Mathematicians
slowly worked out the problems encountered in using it, but logicians from
Frege on did not understand it let alone formalize it, and instead used
an unnecessarily poor logic of quantifiers, viz. the traditional, first-order
logic. This logic does not e.g. allow the definition and study of mathematicians’
uniformity concepts important in analysis. Mathematicians’
stronger logic was rediscovered around 1990 as the form of independencefriendly
logic which hence is not a new logic nor a further development
of ordinary first-order logic but a richer version of it.