Philosophers identify Frege with the doctrine that arithmetic truths are “analytic.” But Frege's main goal, as many now realize, was mathematical: to show that the truths of arithmetic are provable, against many who denied this, each for a different reason. Mathematicians had by Frege's time shown that analysis could be grounded in what we would today call second-order arithmetic: arithmetic plus “logic” Frege's goal was to remove the dependence on arithmetic by beefing up the “logic.”

This observation explains what otherwise seems a puzzling digression in the Grundlagen. At Grundlagen §10, Frege suddenly considers the question of whether the laws of arithmetic might not be “inductive truths,” that is, established nondeductively by verifying particular cases. This question is not reducible to the question of whether mathematics is “analytic” in Kant's (or even Frege's) sense. Nor is Frege concerned here, as he is elsewhere, with the more specific claim of John Stuart Mill that arithmetic is empirical. Even if arithmetic is non-empirical, even if it is analytic, it might still be inductive – a possibility that few great philosophers had considered, but that could not escape a mathematician.

Frege's reply is: