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

本研究的焦点问题是数学是关于什么的，或者说数学对象是什么。关于这个问题，有很多哲学上的说法。鉴于数学哲学的最新发展，我特别注意到这样一个猜想，即我们不能维持现实主义(柏拉图主义)和反实在论(唯名论)的既定二分法。为了证实这一猜想，我首先分析了M.Detlefsen最近描述的布劳威尔建构主义(直觉主义)。他将其描述为一种数学认识论。也有人认为布劳威尔的认识论在总体上是与逻辑对立的。但在这项研究中，我提出了这样一种可能性，即这种认识论可以对逻辑学(包括古典逻辑)起到补充作用，以解决例如密尔困境。其次，研究了莱特-黑尔提出的新柏拉图主义，并将其与达米特的观点进行了比较。通过这种方式，我指出了新柏拉图主义与所谓的柏拉图主义根本不同，相反，它应该被视为一种唯名论。如果这些分析是正确的，数学哲学中的一些新的立场组合，例如柏拉图建构主义，将向我们敞开大门。结果，我们将失去维持数学哲学旧框架的理由。

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.