Revisiting “The New 4CT Problem”

According to what one might label the traditional view of proof in mathematics, proofs have the following characteristics. They are knowable a priori, the knowledge they provide is certain, rather than merely probable, they are surveyable, and, because of these other features, a mathematical proof is convincing to one who understands it. Opponents of this view typically drew their motivation not from the study of mathematics, but rather from a more general antipathy to apriority in epistemology and necessity in metaphysics (Mill, Putnam and Quine all spring to mind here). Tymoczko (1979) provides a different sort of challenge, taking as its cue a development within mathematics itself, rather than being based on philosophical scruples over apriority or necessity.