Talk: Algebraic Levels in Mathematical Structuralism

This was a talk at the Oslo Logic Seminar. Slides here.

This paper addresses the issue of the kinds of mathematical structure that exist, and the theories we use to talk about them. Often when philosophers discuss different varieties of structure and theories, they focus on the following distinction—particular structures and non-algebraic theories are those that are instantiated only by pairwise isomorphic structures (e.g. the axioms for second-order Dedekind-Peano arithmetic and the natural number structure), and general structures and algebraic theories are those that can have non-pairwise-isomorphic exemplars (e.g. the axioms for groups and the group structure).

In this paper I want to point to some results from model theory that show that the distinction between particular and general structures and non-algebraic and algebraic theories requires further refinement. General structures and algebraic theories can be subdivided into a further two philosophically salient different kinds, that I shall call LEGO-like and non-LEGO-like theories/structures. Roughly speaking a theory or structure is LEGO-like iff it is algebraic but any model/instantiation of a certain cardinality can be built from an initial template using a well-defined and determinate construction process. I’ll argue that the notions of model-theoretic geometry and strongly minimal set provide a nice way of capturing this idea. In this way, model theory provides philosophy with a rich variety of examples that have been neglected in virtue of our focus on the `canonical’ cases of arithmetic and basic abstract algebra.

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