Abstract

This work assesses Peirce's epistemology of mathematics. The central insight is that mathematical objects are represented by logical, diagrammatic constructions. But the constructions do not represent actualities. If they did, our knowledge of them would be contingent knowledge. But it is not. Although the same mathematical proposition might be true in one state of affairs and false in another, and accordingly, we might be cautious towards mathematical propositions true in every possible world, our knowledge when we have it is of necessary truths. This is because logical constructions represent ideal objects of our own invention. In effect, we know a lot about our own ideas or common suppositions in a possible world. Thus, when we know a proposition in mathematics, we have a greater degree of certainty compared with other sorts of claims that may be false in the same possible world in which they are true. The explanation for this is simply that we have nothing to compare a mathematical object with other than what we agree to talk about