Poincaré on Understanding Mathematics

Poincaré holds up that the varieties of formal logical theories don’t express the essential proof-theoretical structure in order to understand mathematics. Intuition and æsthetic reasoning, the later depending from the criterion “harmony by a surprising order-character”, are other decisive proof-aspects. In what sense elements from Peirce’s semiotics and Goodman’s æsthetics contribute something to Poincaré’s aim of mathematical reasoning besides logical inference ?