Kant has earned the great compliment of having detractors who criticize him with great insight and ingenuity. German idealism, which dominated nineteenth-century philosophy, finds its footing by attacking Kant’s conception of things-in-themselves. Idealists such as Hegel argue that there is something deeply suspicious about these mysterious entities, which Kant claims are the source of our sensations while claiming we can have no direct knowledge of them. Idealism jettisons things-in-themselves and the whole noumenal realm, arguing instead that reality consists primarily of mental phenomena. Analytic philosophy, which is one of the leading schools of twentieth-century philosophy, also gets its start through an attack on Kant. The logician Gottlob Frege criticizes Kant for basing the analytic–synthetic distinction on the subject-predicate form of grammar, which is not a necessary feature of the logical structure of language or reality. Frege argues that we should base the analytic–synthetic distinction on whether we justify a given judgment by appealing to its logical form or to empirical investigation and that, according to this distinction, the category of the synthetic a priori becomes unnecessary. Kant is only able to argue that geometry, for instance, relies on synthetic a priori knowledge because he fails to distinguish between pure geometry—the stuff of mathematical axioms and proofs—and empirical geometry—the application of geometrical principles to science. Pure geometry is a priori, but it is also analytic, since it is justified according to logical principles alone. Empirical geometry is synthetic, but it is also a posteriori, since we only learn from experience what sort of geometry applies to the real world.