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

数学哲学中实在论与建构主义关系探讨

基本信息

批准号：
06610013
负责人：
KANEKO Hiroshi
金额：
$ 0.32万
依托单位：
Senshu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1994
资助国家：
日本
起止时间：
1994 至 1995
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-06610013/
关键词：
realism constructivism platonism intuitionism Frege philosophy of mathematics mathematical object abstract object 正準証明

项目摘要

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.

这项研究的焦点问题是数学是什么或者数学对象是什么。关于这个问题有很多哲学解释。鉴于数学哲学的最新发展，我特别关注这样的猜想：我们无法维持既定的实在论（柏拉图主义）和反实在论（唯名论）二分法。为了证实这个猜想，我首先分析了M.Detlefsen最近所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。人们还认为布劳威尔认识论总体上与逻辑相反。但在这项研究中，我展示了这种认识论可以对逻辑（包括经典逻辑）发挥补充作用的可能性，以解决例如穆勒的困境。其次，我研究了赖特-黑尔提出的新柏拉图主义，并将其与达米特的立场进行了比较。我这样指出，新柏拉图主义与所谓柏拉图主义有本质的不同，而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，例如柏拉图主义的建构主义，将会向我们开放。结果，我们将失去维持旧的数学哲学框架的理由。