The basic concepts of arithmetic (number) and geometry (space) have their origins in experience, but axioms and propositions in these domains are deductively inferred and justified as necessary truths
Knowledge claimed to be independent of experience, associated with the intuitionist school; Mill denies its existence as part of his radical empiricism.
axioms(Stumpf, 1891)
Propositions that we assume to be true and necessary, originating in the content of judgments.
necessary truths(Leibniz's argument for God's existence from eternal truths)
Truths that hold independently of whether any finite minds exist to think them
In the critical part of his work, Stumpf raises the problem of the origins of the laws and principles of logic and mathematics as follows: if these principles are inductive in nature, as Mill believes them to be, then they do not constitute necessary truths; if, on the contrary, they are necessary truths, then the question arises as to whether they are synthetic a priori judgments as Kant claims or analytic a priori propositions as Stumpf claims. Against Mill, Stumpf argues that the axioms are n