The fallacy of concept swapping (i.e. replacement of a synthetic by an analytic judgment) which is responsible for the logicist position on the epistemological status of the axioms of geometry also underlies the empiricist position. The fallacy is shared by famous scientists (e.g. Schröder, Ostwald, and Mach), and it has pushed several high-calibre mathematicians (Gauss, Lobachevsky, Riemann, and Helmholtz) into empiricism, and another (Poincaré) into conventionalism. They all resisted Kant's solution—the idea of synthetic a priori judgments.