Parts of Structures

We contribute to the ongoing discussion on mathematical structuralism by focusing on a question that has so far been neglected: when is a structure part of another structure? This paper is a first step towards answering the question. We will show that a certain conception of structures, abstractionism about structures, yields a natural definition of the parthood relation between structures. This answer has many interesting consequences; however, it conflicts with some standard mereological principles. We argue that the tension between abstractionism about structure and classical mereology is an interesting result and conclude that the mereology of abstract structures is a subject that deserves further exploration. We also point out some connections between our discussion of the mereology of structures and recent work on non-well-founded mereologies.

Automated summary

This paper contributes to the discussion on mathematical structuralism by addressing the neglected question of when a structure is part of another structure. By defining structures through abstractionism, the paper provides an answer to this question and explores its interesting consequences. However, the proposed answer conflicts with some fundamental principles in mereology. The tension between abstractionism and classical mereology in the context of abstract structures suggests further exploration in this subject, which is also connected to recent work on non-well-founded mereologies.