Frege's Platonism and Platonism in Mathematics Today

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.

The core questions of this six-month research project are concerned with our knowledge of mathematical objects and the nature of such objects. These questions are of central importance in theoretical philosophy, in particular the philosophy of mathematics.

It is not difficult to explain the relevance and importance of these questions to the field of philosophy and mathematics to a non-academic audience and it is not too difficult to provide an initial idea of how such answers might be approached. An attempt in this direction was made by the applicant in a previously published paper 'What mathematical knowledge could not be' (available on my website:
http://web.mac.com/philipebert/Site/Online_Papers/Entries/2006/12/6_'What_Mathematical_Knowledge_could_not_be'.html) and in open lectures given by this author in Edinburgh and Dundee with the same title. Yet, to provide excellent research which is intended to make a contribution to the academic field, while at the same time directly impacting individuals or groups outside of academia is, given this area of research, a near impossible task. The reason for this disconnect is due both to the abstract nature of my research project as well as a lack of professional training in philosophy and mathematics by those outside academia so as to properly understand the answers offered. After all, the aim of my project is not to discuss a certain phenomena that an individual outside academia would have sufficient pre-theoretical understanding of, or intuitions about, in order to critically engage with the research material - such as some topics in ethics that aim to clarify our conception of justice or our notion of fairness. Instead, the core results will be concerned with a highly theoretical type of engagement in an area with no practical or ethical implications so that direct impact on individuals with no academic interest on such research questions is extremely unlikely.

Still, it is worth highlighting that the proposed research will be a contribution within a tradition of philosophy which has, over a large timescale, profoundly informed our society's modes of thinking and living. The ongoing cultural impact of this tradition does not of course depend directly on this six-month research project. However, it would be wrong to infer that because the cultural impact of a tradition of theoretical philosophy is not dependent upon some one piece of research within it, that piece of research plays no role in the impact of the whole to which it is a contribution.

A prime example of a research project that initially did not seem to impact on society at all was Frege's Logicism. During his lifetime, his work was not acknowledged outside academia and even within academia only very few paid his work much attention. Now, however, Frege is generally regarded as the inventor of modern logic who paved the way to extremely fruitful research in mathematics and the rise of computer science, artificial intelligence and linguistics. It would, of course, be preposterous to think that the work proposed here will have impact in a similar way, yet to clarify the views of a philosopher who had profound impact on our society, even in a way that might not be readily understandable to most of society, might itself be in the interest of society.