A Metaphysics for Mathematical and Structural Realism

The goal of this paper is to preserve realism in both ontology and truth for the philosophy of mathematics and science. It begins by arguing that scientific realism can only be attained given mathematical realism due to the indispensable nature of the latter to the prior. Ultimately, the paper argues for a position combining both Ontic Structural Realism and Ante Rem Structuralism, or what the author refers to as Strong Ontic Structural Realism, which has the potential to reconcile realism for both science and mathematics. The paper goes on to claims that this theory does not succumb to the same traditional epistemological problems, which have damaged the credibility of its predecessors.

Permutations and referential indeterminacy

This chapter begins with a definition of isomorphism, and then introduces the Push-Through Construction, which allows us to generate many distinct isomorphic copies of a model. Benacerraf and Putnam have used this construction to raise certain problems for realism and the determinacy of reference. These problems seem most threatening in the case of mathematics, where nothing like causation could help pin down reference. In this connection, we introduce Putnam’s famous just-more-theory manoeuvre for the first time, and a position which is vulnerable to it, namely moderate objects-platonism. We then evaluate various attempts to salvage determinacy of reference, including Shapiro’s ante rem structuralism, before outlining a supervaluationist semantics which allows for referential indeterminacy. The appendices to this chapter contain a proof that isomorphism implies elementary equivalence, and a discussion of recent work on eligibility.

An ‘i’ for an i, a Truth for a Truth†

Stewart Shapiro’s ante rem structuralism recognizes the structural or ‘algebraic’ aspects of mathematical practice while still offering a face-value semantics. Fictionalism, as a purely ‘algebraic’ approach, is held to be at a disadvantage, as compared with Shapiro’s structuralism, in not interpreting mathematics at face value. However, the face-value reading of mathematical singular terms has difficulty explaining how we can use such terms to pick out a unique referent in cases where the relevant mathematical structures admit non-trivial automorphisms. Shapiro offers a solution to this difficulty, but his solution, I argue, evens the score between Shapiro’s structuralism and fictionalism.

Realistic structuralism’s identity crisis: A hybrid solution.

Keränen (2001) raises an argument against realistic (ante rem) structuralism: where a mathematical structure has a non-trivial automorphism, distinct indiscernible positions within the structure cannot be shown to be non-identical using only the properties and relations of that structure. Ladyman (2005) responds by allowing our identity criterion to include ‘irreflexive two-place relations’. I note that this does not solve the problem for structures with indistinguishable positions, i.e. positions that have all the same properties as each other and exactly the same relations to all objects (including themselves). I conclude that realistic structuralists must compromise and treat some structures eliminativistically.Published in Analysis 66.3: 216–22.