The subject of this thesis is a position in the philosophy of mathematics - defended by Bob Hale and Crispin Wright - known variously as neo-Fregeanism, neo-Logicism or abstractionism, and which claims that knowledge of mathematical objects can be based on principles - known as abstraction principles - which are in important respects like definitions of mathematical language.
In the thesis, I make a distinction between two ways in which the abstraction programme might be carried out. These are the standardly defended static view, according to which abstraction principles can used to discover previously unrecognised objects lying within some fixed domain of quantification. The second is an expansionist view, according to which abstraction principles allow one to introduce new quantificational vocabulary, and thus expand ones domain of quantification to one which contains referents of mathematical terms.
There are then two main aims. The first is to examine the static position, so as to identify the components of that view which make it committed to a standard domain, and to argue against the view. My main argument against the view concerns what has become known as the bad company problem. I argue that there is an epistemological component to the bad company problem which can not be avoided by the static abstractionist.
The second aim of the thesis is to argue for and defend the expansionist view. In particular, I will claim that the expansionist view avoids the bad company problem, and that the expansionist view allows for an abstractionist foundation for set theory - an aim which (or so I will argue) has so far eluded the static view.
