Part 1 Field's response. Part 2 Another anti-realist response - mathematics is really talk about possibilities involving concrete objects. Part 3 Maddy's realism - mathematical objects are like ordinary concrete objects and known by similar means. Part 4 Other approaches to realism. Part 5 Second-order logic - a new tool for logicists and other mathematical reductionists? Part 6 Structuralism - mathematics studies structures.
