Research project
Frege tried to explain our knowledge of the natural numbers by reducing arithmetic to logic. This program, however, could not be carried out. The main reason was that the theory of classes that Frege used in defining the natural numbers turned out to be inconsistent.
For a long time, therefore, Frege´s philosophy of mathematics came to be regarded as hopelessly passé. Recently, however, the situation has changed, mainly due to the revision of Frege´s program by Crispin Wright and Bob Hale and logical investigations carried out by the late George Boolos, Richard Heck, and others. The project concerns the semantics, metaphysics and epistemology of mathematics. If we take mathematical statements at face value, they assert the existence of various kinds of mathematical objects such as numbers, sets, functions etc. This gives rise to different philosophical questions: Should we accept mathematics as being literally true and thereby commit ourselves to the existence of mathematical objects? Or should we rather think of mathematics as some kind of useful fiction? If there are mathematical objects, what is their nature and how can we gain knowledge about them?
The aim of the project is to subject the neo-Fregean program in the philosophy of mathematics to a critical examination. Many of the assumptions and presuppositions of this program can be questioned. What is the epistemic status of the higher-order logic that is assumed? Can all the principles and rules of inference of this logic be justified on the basis of conceptual connections? Or is it rather, as critics have claimed, that the Neo-Fregeans have provided substantial mathematical assumptions with an innocent-looking logical disguise? What is the status of Hume´s principle and similar so-called abstraction principles? The focus will be on questions concerning the limits of logic, the interpretation of higher order logic, and the status of abstraction principles. The importance of the project lies in the clarity that it can provide concerning philosophically important concepts and problems.
