Kant's question “How are synthetic judgments a priori possible?” precipitated the Critique of Pure Reason. Question and answer notwithstanding, Mill and others persisted in doubting that such judgments were possible at all. At length some of Kant's own clearest purported instances, drawn from arithmetic, were sweepingly disqualified (or so it seemed; but see §11) by Frege's reduction of arithmetic to logic. Attention was thus forced upon the less tendentious and indeed logically prior question, “How is logical certainty possible?” It was largely this latter question that precipitated the form of empiricism which we associate with between-war Vienna – a movement which began with Wittgenstein's Tractatus and reached its maturity in the work of Carnap.

Mill's position on the second question had been that logic and mathematics were based on empirical generalizations, despite their superficial appearance to the contrary. This doctrine may well have been felt to do less than justice to the palpable surface differences between the deductive sciences of logic and mathematics, on the one hand, and the empirical sciences ordinarily so-called on the other. Worse, the doctrine derogated from the certainty of logic and mathematics; but Mill may not have been one to be excessively disturbed by such a consequence. Perhaps classical mathematics did lie closer to experience then than now; at any rate the infinitistic reaches of set theory, which are so fraught with speculation and so remote from any possible experience, were unexplored in his day.