Stating structural realism: mathematics‐first approaches to physics and metaphysics

I respond to the frequent objection that structural realism fails to sharply state an alternative to the standard predicate-logic, object / prop-erty / relation, way of doing metaphysics. The approach I propose is based on what I call a ‘math-ﬁrst’ approach to physical theories (close to the so-called ‘semantic view of theories’) where the content of a physical theory is to be understood primarily in terms of its mathematical structure and the representational relations it bears to physical systems, rather than as a collection of sentences that attempt to make true claims about those systems (a ‘language-ﬁrst’ approach). I argue that adopting the math-ﬁrst approach already amounts to a form of structural realism, and that the choice between epistemic and ontic versions of structural realism is then a choice between a language-ﬁrst and math-ﬁrst view of metaphysics ; I then explore the status of objects (and properties and relations) in fundamental and non-fundamental physics for both versions of math-ﬁrst structural realism. ordinary-language semantics: satisfaction.