Frege's Platonism and Platonism in Mathematics Today

在20世纪70年代的《S》一书中，迈克尔·达米特描绘了哲学中的语言转向，他认为这是弗雷格讨论算术时的一个时刻，在这个时刻，一个认识论和形而上学的问题，即我们的知识和数字的性质的问题，被转化为一个需要对旨在指代这些对象的数字表达式进行语义分析的问题，这是赋予形而上学和认识论问题更容易处理的形状，从而使解决这些问题的系统进展成为可能的关键步骤。这场“哥白尼革命”导致了算术哲学中一个非常富有成效的研究项目，即弗雷格的逻辑主义，这反过来又启发了所谓的新弗利根项目。最近，在语言学转向的启发下，出现了一种宽泛的弗雷格方法论的转变，以及可以被称为“建设性形而上学”的复兴。然而，这位申请人坚信，首先，当处理我们的算术知识和我们对潜在数学对象的知识的基本问题时，弗雷格启发的方法论仍然提供了最有希望的方法之一来解决关于我们对数的知识和关于数字本质的基本问题；但其次，这种方法需要从对弗雷格引入和证明数学本体论的实际方法的关键但被忽视的细节的重新评估中获得新的灵感和方向。本研究项目的中心目的是进一步研究弗雷格的方法论和他在其主要著作《数学哲学的核心主题》和《数学哲学的核心主题》中所采用的观点。这些洞见将被用来为弗利葛式的数学哲学提出新的挑战，并进一步推动当前算术哲学和元本体论的辩论。研究项目分为三个阶段。第一阶段主要讨论弗雷格自己的逻辑客体概念。这项研究是与康涅狄格州大学的Marcus Rossberg教授合作进行的；第二阶段发展了Frege对数字与逻辑对象的识别中的一个悖论，即带有扩展。这项研究是与罗伊·T·库克教授(明尼苏达州)合作进行的；最后和第三阶段的目的是打破当今算术哲学中新弗莱格主义传统中关于数学对象“厚度”的争论陷入僵局。该项目的核心目标是制作三篇新的原创研究文章和一本关于弗雷格的编辑文集。

In the 1970's Michael Dummett portrayed the 'linguistic turn' in philosophy, which he identified with the moment in Frege's discussion of arithmetic at which an epistemological and metaphysical question, that is the question of our knowledge and of the nature of numbers, is transformed into one requiring a semantic analysis of numerical expressions purporting to refer to these objects, as the essential step in giving a more tractable shape to metaphysical and epistemological issues and so making possible systematic progress towards their resolution. This 'Copernican Revolution' has lead to a very fruitful research project in the philosophy of arithmetic, namely Frege's Logicism, which, in turn inspired the so-called Neo-Fregean project. More recently there has been a shift away from a broadly Fregean methodology inspired by the linguistic turn, and a resurgence of what may be called 'constructive metaphysics'. However, it is the conviction of this applicant that, first, when dealing with the fundamental questions of our knowledge of arithmetic and our knowledge of the underlying mathematical objects, a Frege-inspired methodology still offers one of the most promising approaches to resolving the fundamental questions about our knowledge of and about the nature of numbers; but secondly, that this approach needs to draw fresh inspiration and direction from a re-evaluation of crucial but neglected details of Frege's actual approach to the introduction and justification of a mathematical ontology.The central aim of this research project is to further investigate Frege's methodology and views as adopted in his main work, 'Die Grundlagen der Arithmetik' and 'Die Grundgesetze der Arithmetik' on core themes in the philosophy of mathematics. These insights will then be used to raise new challenges for a Fregean philosophy of mathematics and to further the current debate in the philosophy of arithmetic and meta-ontology.The research project is divided into three stages. The first stage focuses on Frege's own conception of logical objects. This research is pursuit in collaboration with Professor Marcus Rossberg (UConn); the second stage develops a paradox in Frege's identification of numbers with logical objects, that is, with extensions. This research is pursuit in collaboration with Professor Roy T Cook (Minnesota); and the last and third stage aims to overcome a stalemate reached on the debate of the 'thickness' of mathematical objects within the Neo-Fregean tradition in the philosophy of arithmetic today. The core aims of the project is to produce three new and original research articles and one edited collection on Frege.