Logic and Modern Philosophy of Mathematics

Especially since World War II, new philosophies have emerged separate from logicism, formalism, and intuitionism. One kind is called naturalism, in which mathematical objects are said to be accessible to ordinary sense perception. Forms of Platonism are advocated, partly inspired by the enthusiasm of Gödel. Disaffection from concerns with mathematical truth has stimulated conventionalism; Henri Poincaré was an eminent early advocate. Versions of constructivist mathematics have been developed, broadly following the same prohibitions as Brouwer's but avoiding his peculiar philosophy; Erret Bishop was a notable figure. Structuralism emphasizes the ubiquity of structures, perhaps to excess; mathematics has them rather than is them.

Most of this philosophy is by and for philosophers; the mathematical content is usually limited to set theory and/or arithmetic, even among empiricists who claim to attend to mathematical practice. In particular, little is said about the creation and development of mathematical theories in the first place. Here the eminent Hungarian Georg Polya was an important contributor, with books such as Mathematics and Plausible Reasoning (1954) discussing themes such as proofs themselves modifying old theorems and motivating new ones. A rather neglected area is the philosophy of mechanics and (classical) mathematical physics, despite the attraction of physical interpretation as well as rigor and proof.

By contrast, quantum mechanics has gained much attention, partly in connection with the philosophy of probability, which has a rather separate history. What kind of knowledge is expressed by the probability that the next throw of the die will be 5 is 1/6 ? Influential answers include the logical (it is a deduction from axioms), propensity (it is an assertion about the die and also its environment), subjective or Bayesian (it is a rational belief, drawing upon prior performance), and frequency (it is the limiting case of evidence drawn from long runs of throws).