Abstract
Linnebo and Pettigrew (Philos Q 64:267–283, 2014) have recently developed a version of non-eliminative mathematical structuralism based on Fregean abstraction principles. They recognize that this version of structuralism is vulnerable to the well-known problem of non-rigid structures. This paper offers a solution to the problem for this version of structuralism. The solution involves expanding the languages used to describe mathematical structures. We then argue that this solution is philosophically acceptable to those who endorse mathematical structuralism based on Fregean abstraction principles.
Highlights
Mathematical structuralists who think that abstract structures exist face a problem
A non-trivial automorphism is an isomorphism from a structure to itself that is not the identity mapping; under the automorphism, there is an object in the structure that is not mapped to itself
This paper offers a solution to the problem of non-rigid structures for a version of non-eliminative structuralism based on Fregean abstraction principles
Summary
Structuralists argue that mathematics is not primarily about mathematical objects, like numbers, points, and sets. Mathematics is more concerned with the structures that these objects form, like the field of real numbers, the Euclidean plane, and the cumulative hierarchy of sets, as well as the mathematical theories that describe these structures. Eliminative structuralists, on the other hand, deny the existence of unique abstract structures, arguing that claims about the real number structure should be understood in terms of quantification over all complete ordered fields. On the fine-grained classification, ante rem and in re structuralism are both versions of non-eliminative structuralism, as both views hold that there exists some (abstract) entity that is, e.g., the structure of the reals. The difference between ante rem and in re non-eliminative structuralism concerns the relationship between an abstract structure and the systems exhibiting that structure. Though the problem of non-rigid structures threatens both ante rem and in re structuralism, this paper addresses the problem only for a specific version of in re structuralism According to this version of in re structuralism, which we call abstraction-based structuralism, abstract structures are obtained through a process of Fregean abstraction.
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