One of the important discussions in the philosophy of mathematics, is that centered on Benacerraf’s Dilemma. Benacerraf’s dilemma challenges theorists to provide an epistemology and semantics for mathematics, based on their favourite ontology. This challenge is the point on which all philosophies of mathematics are judged, and clarifying how we might acquire mathematical knowledge is one of the main occupations of philosophers of mathematics. In this thesis I argue that this discussion has overlooked an important part of mathematics, namely mathematics (...) as it is exercised by ordinary people. I do so by looking at the different theories that have been put forward in the recent debate, and showing for each of these that they are unable to account for the mathematical practices of ordinary people. In order to show that these practices do need to be accounted for, I also argue that ordinary people are doing mathematics, i.e. that they engage in properly mathematical practices. Because these practices are properly mathematical, they should be accounted for by any philosophy of mathematics. The conclusion of my thesis, then, is that current theories fail to do something that they should do, while remaining neutral on how well they perform when it comes to accounting for the practices of professional mathematicians. (shrink)

In this article I offer an explicating interpretation of the procedure of content recarving as described by Frege in §64 of the Foundations of Arithmetic. I argue that the procedure of content recarving may be interpreted as an operation that while restricting the subject matter of a sentence, performs a generalization on what the sentence says about its subject matter. The characterization of the recarving operation is given in the setting of Yablo’s theory of subject matter and it is based (...) on the relation of determination between properties. The main advantage of the proposal is its generality, for it is applicable not just to the case of abstraction principles. (shrink)

I extend theorems due to Roy Cook on third- and higher-order versions of abstraction principles and discuss the philosophical importance of results of this type. Cook demonstrated that the satisfiability of certain higher-order analogues of Hume's Principle is independent of ZFC. I show that similar analogues of Boolos's new v and Cook's own ordinal abstraction principle soap are not satisfiable at all. I argue, however, that these results do not tell significantly against the second-order versions of these principles.