The logical empiricism of the Vienna Circle presented itself as the reconciliation of a thoroughgoing empiricism as regards substantive knowledge with the certainty and necessity of mathematics. According to empiricism, sense experience is the only source of substantive knowledge of facts, but sense experience might always have been different from what it actually was. Moreover, anything we extrapolate from what we experience may be falsified by further experiences. Knowledge grounded in sense experience is thus knowledge of contingencies, and except perhaps for knowledge of what is immediately experienced (“Red here now”), it is revisable on the basis of further experiences. In contrast, knowledge of mathematics appears to be unrevisably certain substantive knowledge of necessary truths. The contingencies of our sense experience thus seem to afford no basis for knowledge of mathematics. In a word, empiricism must be false, because mathematics is substantive a priori knowledge.

The logical empiricist approach to this dilemma is to deny that knowledge of mathematics is substantive knowledge. The apriority and necessity of mathematics is the apriority and necessity of logic. Logic is in its turn grounded in tacit conventions for the use of certain symbols, symbols that do not themselves stand for anything, like the signs for negation and conjunction. As Hans Hahn put it: