Frege's Platonism and Platonism in Mathematics Today

在20世纪70年代，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声称涉及这些对象的数字表达式进行语义分析的问题，这是使形而上学和认识论问题更易于处理的重要步骤，从而使解决这些问题的系统性进展成为可能。这场“哥白尼革命”导致了一个非常富有成效的研究项目在哲学的算术，即弗雷格的逻辑主义，这反过来又激发了所谓的新弗雷格项目。最近，受语言学转向的启发，出现了一种从广义上的弗雷琴方法论的转变，而又出现了所谓的“建构性元哲学”的复兴。然而，本申请人确信，首先，当处理我们的算术知识和我们对基础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解决关于我们对数的知识和关于数的性质的基本问题的最有希望的方法之一;但其次，这种方法需要从一个重新设计的过程中汲取新的灵感和方向，评价弗雷格对数学本体论的介绍和论证的实际方法的关键但被忽视的细节。本研究项目的中心目的是进一步调查弗雷格的方法和意见，通过在他的主要工作，'模具Grundlagen之算术'和'模具Grundgesetze之算术'的核心主题的哲学数学。这些见解，然后将被用来提出新的挑战，数学的弗雷琴哲学，并进一步在算术和元本体论的哲学目前的辩论。研究项目分为三个阶段。第一阶段集中在弗雷格自己的概念的逻辑对象。这项研究是追求与马库斯Rossberg教授（UConn）合作，第二阶段发展弗雷格的标识数与逻辑对象，即与扩展的悖论。这项研究是追求与罗伊T库克教授（明尼苏达州）合作;和最后和第三阶段的目的是克服僵局的辩论中的“厚度”的数学对象内的新弗雷琴传统的算术哲学今天。该项目的核心目标是产生三个新的和原始的研究文章和一个编辑收集弗雷格。

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.