An Epistemic Structuralist Account

ii This thesis aims to explain the nature and justification of mathematical knowledge using an epistemic version of mathematical structuralism, that is a hybrid of Aristotelian structuralism and Hellman’s modal structuralism. Structuralism, the theory that mathematical entities are recurring structures or patterns, has become an increasingly prominent theory of mathematical ontology in the later decades of the twentieth century. The epistemically driven version of structuralism that is advocated in this thesis takes structures to be primarily physical, rather than Platonically abstract entities. A fundamental benefit of epistemic structuralism is that this account, unlike other accounts, can be integrated into a naturalistic epistemology, as well as being congruent with mathematical practice. In justifying mathematical knowledge, two levels of abstraction are introduced. Abstraction by simplification is how we extract mathematical structures from our experience of the physical world. Then, abstraction by extension, simplification or recombination are used to acquire concepts of derivative mathematical structures. It is argued that mathematical theories, like all other formal systems, do not completely capture everything about those aspects of the world they describe. This is made evident by exploring the implications of Skolem’s paradox, Gödel’s second incompleteness theorem and other limitative results. It is argued that these results demonstrate the relativity and theory-dependence of mathematical truths, rather than posing a serious threat to moderate realism. Since mathematics studies structures that originate in the physical world, mathematical knowledge is not significantly distinct from other kinds of scientific knowledge. A consequence of this view about mathematical knowledge is that we can never have absolute certainty, even in mathematics. Even so, by refining and improving mathematical concepts, our knowledge of mathematics becomes increasingly powerful and accurate.