Ontological Methodology and the Philosophy of Arithmetic: A Critique of Abstractionism

Abstract

My dissertation is a study of Bob Hale and Crispin Wright’s abstractionism, a realist philosophy of mathematics that originates from the philosophical and technical work of Gottlob Frege (1884-1925). More specifically, I am concerned with the structure and viability of their metaontology: the method by which Hale and Wright establish the existence of numbers as mind-independent objects. At the heart of their view is the claim that the truth of abstraction principles (a special type of implicit definition) and the Syntactic Priority Thesis (a special semantic principle) is enough to guarantee mathematical realism. Hence, they adopt the language-first approach, according to which facts about language can decide metaphysical questions about mathematicalia. The first chapter is introductory; it serves to situate the subject matter of this thesis and provide the necessary background information. The second chapter is historical. Therein I offer a novel interpretation of the language-first arguments for mathematical realism put forth by Frege. On this reading, his metaontology relies on the aboutness properties of arithmetical terms and sentences. This chapter serves to make explicit the mechanics of Frege’s metaphysical arguments, which have hitherto remained somewhat mysterious; and to place the metaontology of abstractionism in relief. The third chapter is critical. There I present Hale and Wright's methodology and level several criticisms against it: First, I demonstrate that their argument for the truth of a given abstraction principle is unsuccessful. Second, I show that the Syntactic Priority Thesis has plausible counterexamples. Thus, a different approach must be taken if abstractionism is to count as a promising species of mathematical realism. The fourth and final chapter is constructive. I lay the foundations of a new language-first metaontology for abstractionism that is inspired by the work of Stewart Shapiro, Øystein Linnebo, and Roy T. Cook. As a species of coherentist minimalism, this approach is committed to the following claim: if a formal theory of abstraction meets stringent coherence conditions (or is coherent-plus), then the entities it purports to be about exist as mind-independent abstract objects. Lastly, I show that a particularly important theory of abstraction that grounds arithmetic does, in fact, meet said coherence conditions.