 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 not the result of empirical generalization based on an inductive process, and that arithmetic, like geometry, is a deductive science based on a priori and necessary truths, which are justified by the evidence of internal perception. (Stumpf, 2008, Bogen 5–4) Stumpf therefore agrees with Kant that axioms are necessary truths, but he denies that they are based on synthetic a priori judgments. Stumpf argues rather that Kant’s synthetic judgments presuppose a kind of “mental machinery” (Stumpf, 2008, Bogen 16–3) in order to account for their formation, as they involve numerous assumptions, notably the a priori forms of sensibility, intuition, and Denkformen, which Stumpf takes to be mere constructions. Stumpf criticizes Kant especially for the thesis according to which the origins of the necessity of our knowledge of axioms lie in intuition, in that knowledge of mathematics then becomes dependent upon experience. However, Stumpf argues that intuition understood in the Kantian sense is inductive and only leads to concepts and never to propositions. That is why Stumpf claims that the very idea of synthetic a priori judgments is a contradiction in terms. Axioms and propositions are analytic judgments constructed deductively from concepts. We can see that Stumpf’s fundamental argument against empiricism and transcendental criticism rests on the fundamental distinction between concept and proposition, which are of major philosophical significance in Stumpf’s subsequent work on the theory of knowledge, as we will later see. In his habilitation thesis, Stumpf is more concerned with delimiting the field of logic and mathematics from that of psychology. In this regard, Stumpf clearly separates the question of the origins of concepts, which is a psychological question, from the logico-mathematical domain to which propositions and axioms belong. For, as Stumpf argues in his posthumous work, one can agree with empiricism on the psychological origin of concepts while acknowledging that there is a priori knowledge which is independent from experience. (Stumpf, 1939–1940, p. 126) For Stumpf admits that the basic concepts of arithmetic (the concept of number) and geometry (the concept of space) have their origins in experience. In his Raumbuch, published three years after his habilitation thesis, Stumpf attempts to demonstrate the thesis that the concept of space has psychological origins. However, the position one takes on this issue is distinct from that which one adopts regarding the nature of propositions and necessary truths in the logico-mathematical domain. For, in this domain, one is solely concerned with axioms and propositions, which can be deductively inferred, and one also assumes the origins of the axioms and their justification as necessary truths. Stumpf argued in 1870 that these axioms are analytic a priori propositions and that arithmetic and geometry are deductive sciences.