Investigation into relations between realism and constructivism in the philosophy of mathematics

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The focal question of this research is what is mathematics about or what is it that there are mathematical objects. There are many philosophical accounts about this question. In view of recent developments of philosophy of mathematics, I gave a special attention to the conjecture that we could not sustain the established dichotomy of realism (platonism) and anti-realism (nominalism) . In oder to confirm this conjecture, I first analyzed the Brouwerian constructivism (intuitionism) as M.Detlefsen recently characterized. He characterized it as a kind of mathematical epistemology. It is also thought that Brouwerian epistemology is opposed to logic in general. But in this research I showed the possibility that such epistemology could play a complementary role to logics (including classical logic) in order to resolve, for example, Mill's Dilemma. Second, I researched Neo-platonism which Wright-Hale have proposed and I compared it with Dummett's position. In the way of that, I indicated that neo-platonism was fundamentally different from the so-called platonism and rather it should be viewed as a kind of nominalism. It these analyzes are right, some new conbinations of positions in philosophy of mathematics, for example platonistic constructivism, will be open to us. As a result we will lose reasons to sustain the old framework of philosophy of mathematics.