Are there synthetic a priori propositions ? The paradigmatic case of mathematics, from Kant to Frege and Peirce

Bernard Bolzano notoriously rejected Immanuel Kant’s claim that arithmetic and geometry were grounded on synthetic a priori judgements based on pure intuition. According to Bolzano, only analysis of concepts could ground the generality of mathematical statements and proofs; such a stand would later lead to logicism (from Frege to Carnap). Far from going in this way, however, Charles Sanders Peirce, who was one of the fathers of formal logic but also a great admirer of Kant, provided semiotic reasons to believe that diagrams do have a general meaning and that they can provide a knowledge which is both general and “ampliative”. Unlike mere logical analysis, diagrams (the analogon of Kant’s schemas) help to explore concepts by going somehow “outside of them” in such a way that new knowledge is gained. This provides new support to Kant’s notion of intuitive construction, which is supposed to be both deductive and inventive