One central aim of science is to provide explanations of natural phenomena. What role(s) does mathematics play in achieving this aim? How does mathematics contribute to the explanatory power of science? Rules to Infinity defends the thesis, common though perhaps inchoate among many members of the Vienna Circle, that mathematics contributes to the explanatory power of science by expressing conceptual rules, rules which allow the transformation of empirical descriptions. Mathematics should not be thought of as describing, in any substantive sense, (...) an abstract realm of eternal mathematical objects, as traditional platonists have thought. In Rules to Infinity, this view, which I call mathematical normativism, is updated with contemporary philosophical tools, and it is argued that normativism is compatible with mainstream semantic theory. This allows the normativist to accept that there are mathematical truths, while resisting the platonistic idea that there exist abstract mathematical objects that explain such truths. Furthermore, Rules to Infinity defends a particular account of the distinction between scientific explanations that are in some sense distinctively mathematical – those that explain natural phenomena in some uniquely mathematical way – and those that are only standardly mathematical, and it lays out desiderata for any account of this distinction. Normativism is compared with other prominent views in the philosophy of mathematics such as neo-Fregeanism, fictionalism, conventionalism, and structuralism. Rules to Infinity serves as an entry point into debates at the forefront of philosophy of science and mathematics, and it defends novel positions in these debates. (shrink)

A general trend in recent philosophical and empirical work aims to undermine various traditional claims regarding the distinctive nature of self-knowledge. So far, however, this work has not seriously threatened the Cartesian claim that (at least some) self-knowledge is immune to the sort of sceptical problem that seems to afflict our knowledge of the external world. In this paper I carry this trend further by arguing that the Cartesian claim is false. This is done by showing that a familiar sceptical (...) argument that targets my knowledge of the external world can be adapted to target my belief that I exist, along with any of my self-knowledge that I know entails my own existence. Thus, my self-knowledge and my knowledge of the external world are subject to the same sort of sceptical problem. (shrink)