Abstract
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.
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Notes
See Linnebo, 2017, 166-7, for a comparison between the conception of structures that we analyse in this paper (abstractionism about structures) and the view that structures are structured universals.
See Linnebo and Pettigrew (2014: 274) on how to formulate Frege’s abstraction consistently, avoiding the Burali-Forti paradox.
Suppose we take two structures S and T, such that S is a (induced) subgraph of T. Moreover, suppose that S is isomorphic to S′ and T is isomorphic to T’. It’s easy to see that S′ is not necessarily a (induced) subgraph of T’. Therefore, the relation of being an induced subgraph is not a congruence with respect to isomorphism.
This definition of embedding is equivalent to the standard one (S is embeddable in S′ if there is an injective function f : D → D′such that for all x…, xn ∈ D and all Ri: \({R}_i\left({x}_1,\dots, {x}_n\right)\leftrightarrow {R}_i^{\prime}\left(f{x}_1,\dots, f{x}_n\right)\Big).\) To see that embedding is a congruence with respect to isomorphism (i.e. that S′ ≃ S ↪ T ≃ T′ entails S′ ↪ T′) simply note that if f is an isomorphism from S′ to S, f′ is an embedding from S to T and f′′is an isomorphism from T to T′, then f′′ ∘ f′ ∘ f is an embedding from S′ to T′.
Thanks to Luca San Mauro for suggesting this example to us.
x ∘ y=df ∃ z(z ⊑ x ∧ z ⊑ y)
Indeed, given any two parts S1, S2 of an omega-sequence S, the structures of S1 and S2 can differ only because of their length. Nothing else can make them different. Therefore, they necessarily have a “segment” in common.
In presence of antisymmetry, strong supplementation entails weak supplementation, so the failure of weak supplementation entails the failure of strong supplementation. However, given that in the present context antisymmetry fails, we are not entitled to infer the failure of strong supplementation from the failure of weak supplementation.
It might be worth noticing that mereology for structures based on PS validates some composition principles. The disjoint union of two graphs G1, G2 is the graph G1∪∗G2 obtained by taking two isomorphic copies of the two graphs, \({G}_1^{\ast },{G}_2^{\ast }\), whose domains are disjoint, letting the domain of G1∪∗G2 be the union of the domains of \({G}_1^{\ast }\)and \({G}_2^{\ast }\), and the binary relation of G1∪∗G2 be the union of the binary relations of \({G}_1^{\ast }\)and \({G}_2^{\ast }\). The structure of G1∪∗G2 contains as parts the parts of the structure of G1 and the parts of the structure of G2, because any graph embeddable in either G1 or G2is embeddable in G1∪∗G2. In this sense, [G1∪∗G2] can be considered an upper bound of the pair {[G1], [G2]}, even though this terminology is strictly speaking improper since the notion of upper bound is standardly defined with relation to a partial order and the relation of parthood based on PS is not a partial order. Note that every induced subgraph of G1∪∗G2is the (disjoint) union of a subgraph of G1 and a subgraph of G2, hence for any graph G, [G] overlaps [G1∪∗G2] if and only if [G] overlaps either [G1] or [G2].
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Acknowledgements
Matteo Plebani acknowledges that the research activity that led to the realization of this paper was carried out within the Department of Excellence Project of the Department of Philosophy and Education Sciences of the University of Turin (ex L. 232/2016). Other relevant projects: Proyecto (FFI2017-82534-P) from FEDER/Ministerio de Ciencia, Innovación y Universidades-Agencia Estatal de Investigación and PID2020-115482GB-I00 from Ministerio de Ciencia e Innovación/Agencia nacional de Investigación. Michele Lubrano's research was supported by the project SPRJ_PRIN_2017_19_01 “From Models to Decisions”. Many thanks are due to Luca San Mauro, Øystein Linnebo and Leon Horsten.
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Plebani, M., Lubrano, M. Parts of Structures. Philosophia 50, 1277–1285 (2022). https://doi.org/10.1007/s11406-021-00453-0
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DOI: https://doi.org/10.1007/s11406-021-00453-0