Which Mathematical Logic is the Logic of Mathematics?

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 independence-friendly logic which hence is not a new logic nor a further development of ordinary first-order logic but a richer version of it.