In October of 1930 at a conference in Koenigsberg, three papers on the foundations of mathematics were debated. This is also the conference at which Goedel showed, first, that FOL formulas are provable iff true in every model (the completeness proof) and, second, that not all materially true theorems in a consistent axiomatic system are provable in that system (the first incompleteness theorem). At the conference, Carnap presented for the Frege/Russell camp of "logicism," Heyting for the camp of "constructivism/intuitionism," and von Neumann for the Hilbert camp of "formalism." Waismann was added at the last minute to represent Wittgenstein, whose views were highly respected but little understood.
W. consorted with many among the greats of his time. He worked regularly with Frege, Russell, Keynes, Schlick, and Ramsey. Turing attended W.'s lectures on mathematics. Everyone agreed that W. had much to say on the foundations of mathematics, but almost no one was quite sure what he was saying---a situation as frustrating for W. himself as for his interlocuters. In retrospect, it's clearer that Wittgenstein articluted an algorithmic view of mathematics remarkably similar in spirit and detail to Church's lambda calculus. W.'s views are also close to those of Skolem and Kronecker. The talk introduces W.'s work on the foundations of mathematics and tries to situate these views within the foundations of mathematics.
Caveat lector: "foundations of mathematics" is a euphemism for "philosophy of mathematics."