On the Uniqueness and Determinacy of the Standard Model of Arithmetic

This dissertation concerns the intended interpretation of mathematical theories, in light of model-theoretic underdetermination. When considering non-algebraic mathematics (i.e. those theories that we take as providing an axiomatisation of a particular mathematical structure or domain), a natural philosophical question comes to the fore: how is it that we manage to successfully discern the subject matter of these theories? Although on the face of it, this question may not appear worrying, it becomes far more pressing once we note that the first-order theories we accept as constituting current mathematical practice fail to uniquely determine their own intended interpretation(s) — even up to isomorphism. Quite generally, any mathematical theory purporting to describe a particular subject matter will have unintended, non-isomorphic models of its axioms. These unintended interpretations cannot be excluded on mathematical grounds, and satisfy not only the axioms of the theory but indeed the complete set of truths concerning the domain. As such, axiomatic theories cannot uniquely determine the content of mathematics. But this raises a worrying question; if our best mathematical theories cannot uniquely determine their mathematical content, what possibly could? In order to explore this problem, my dissertation consists of four chapters each addressing a different aspect of this problem. My first chapter situates the project with respect to the broader philosophical literature, introduces some basic procedures for the construction of non-standard models, and introduces the general challenge with which the dissertation is concerned. My second chapter explores three prominent anti-skeptical arguments, and shows that they each fail for deep structural reasons that cannot be easily fixed. I first show that existing arguments fail to accurately characterize problem at hand. I then demonstrate that even if these arguments were successful, one could still nonetheless construct a revenge argument which is not subject to them. I therefore conclude that the problem of unintended interpretations remains a significant problem for the philosophy of mathematics. In my third chapter, I draw attention to an important presupposition that has been widely overlooked. Namely, the presupposition of classical mathematics. In doing so, I utilize some theorems of intuitionistic mathematics to show that intuitionistic arithmetic and set theory do not face the same problems as their classical counterparts. I note that this can be seen as an independent argument for intuitionism. In my fourth chapter, I develop a novel proposal for a purely algebraic conception of mathematics which seeks to reframe the problem of unintended interpretations as a feature of modern axiomatic systems, rather than a bug. The dissertation concludes that there is significant value to be had both mathematically and philosophically from investigating unintended interpretations, and urges further research into the intersection of philosophy and model theory.