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. Mathematics is human. It's part of and fits into human culture.
Mathematical knowledge is fallible.
Like science, mathematics can advance by making mistakes and then correcting and recorrecting them. There are different versions of proof or rigor, depending on time, place, and other things. The use of computers in proofs is a nontraditional version of rigor. Empirical evidence, numerical experimentation, probabilistic proof all help us decide what to believe in mathematics. Aristotelian logic isn't necessarily always the best way of deciding.