Indeterminacy and the law of the excluded middle

Abstract

This thesis is an investigation into indeterminacy in the foundations of mathematics and its possible consequences for the applicability of the law of the excluded middle (LEM). It characterises different ways in which the natural numbers as well as the sets may be understood to be indeterminate, and asks in what sense this would cease to support applicability of LEM to reasoning with them. The first part of the thesis reviews the indeterminacy phenomena on which the argument is based and argues for a distinction between two notions of indeterminacy: a) indeterminacy as applied to domains and b) indefiniteness as applied to concepts. It then addresses possible attempts to secure determinacy in both cases. The second part of the thesis discusses the advantages that an argument from indeterminacy has over traditional intuitionistic arguments against LEM, and it provides the framework in which conditions for the applicability of LEM can be explicated in the setting of indeterminacy. The final part of the thesis then applies these findings to concrete cases of indeterminacy. With respect to indeterminacy of domains, I note some problems for establishing a rejection of LEM based on the indeterminacy of the height of the set theoretic hierarchy. I show that a coherent argument can be made for the rejection of LEM based on the indeterminacy of its width, and assess its philosophical commitments. A final chapter addresses the notion of indefiniteness of our concepts of set and number and asks how this might affect the applicability of LEM.