Frege's Platonism and Platonism in Mathematics Today

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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.