Philosophies of Mathematics:
I like to compare philosophy of mathematics today to philosophy of science in
the 30's and 40's. That subject was dominated by logical positivists: Rudolf
Carnap and his friends of the ``Wiener Kreis'' (Vienna Circle). As a result of
taking Bertrand Russell and Ludwig Wittgenstein too seriously, they believed
they knew the correct methodology for scientific work: (1) state the axioms;
(2) give correspondence rules between words and physical observables; (3)
derive the theory, as Euclid derived geometry, or Mach derived mechanics.
It was noticed after a while that what logical positivists said had little in
common with what scientists did or wanted to do. New ideas in philosophy of
science came from Karl Popper, Tom Kuhn, Imre Lakatos, Paul Feyerabend. These
subversives disagreed with each other. But they all thought philosophers of
science could think about what scientists actually do, not bring
presuppositions and instructions for scientists to ignore.
Philosophy of mathematics is overdue for its Popper, Kuhn, Lakatos, and
Feyerabend. It's overdue for analysis of what mathematicians actually do, and
the philosophical issues therein. In fact, this turn is taking place. Wittgenstein and Lakatos helped start it. In recent years Michael Polanyi, George Polya, Alfred Renyi, Leslie White, Ray Wilder, Greg Chaitin, Phil Davis, Paul Ernest, Nick Goodman,
Phil Kitcher, Penelope Maddy, Michael Resnik, Gian-Carlo Rota, Brian Rotman, Gabriel Stolzenberg, Robert Thomas, Tom Tymoczko, Jean Paul van Bendegem, and Hao Wang have participated.
Here are some ideas some of these people hold.