## Abstract

The aim of the paper is to assess the relative merits of two formal representations of structure, namely, set theory and category theory. The purpose is to articulate ontic structural realism (OSR). In turn, this will facilitate a discussion on the strengths and weaknesses of both concepts, and will lead to a proposal for a pragmatics-based approach to the question of the choice of an appropriate framework. First, we present a case study from contemporary science—a comparison of the formulation of quantum mechanics in a language of Hilbert spaces and abstract \(C^\star \)-algebras. It is then shown how the method of structural representation can be determined based on the pragmatics of goal-oriented research, not a dogmatic choice. We investigate a hypothesis stating that use of the interplay between the powers of *abstraction* and *detail* of different representational methods results in adopting a pluralistic, as opposed to standard, unificatory, perspective on the role of structural representation in OSR.

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## Notes

It is worth noting that (Ladyman & Ross, 2007) acknowledged the possibility of using category theory as a valid approach in developing some of the details of the semantic view quite early: ”... the details of the semantic view are developed (and we think that lots of formal and informal approaches may be useful, perhaps, for example, using category theory rather than set theory for some purposes) ...” (Ladyman & Ross, 2007, p. 118). This remark was, however, not further elaborated on.

The relationship between pluralism and scientific realism, which is usually associated with monism, is further discussed in Sect. 5 of this paper.

This comprehensive and detailed case study is the product of many inspiring discussions with Michał Białończyk. I greatly appreciate his insights, that have given me an opportunity to dive into the mind of a working physicist, as well as his help in polishing up the formal aspects of the analysis.

Technically speaking, the state is modelled by a ray, which is an equivalence class of vectors, where two given vectors

*v*and*w*are in the same equivalence class, if and only if \(v = \lambda w\) for some non-zero complex number \(\lambda \). It reflects the fact that, according to quantum mechanics, vectors that are proportional to each other give the same physical predictions. For finite dimensional Hilbert spaces, which are of fundamental importance in quantum information, rays constitute the so-called complex projective space.It is important to emphasize, that the described formalism is not only applicable in easy, finite-dimensional cases, but it extends to a more general setup as well. The general noncommutative \(C^{\star }\)-algebra, relevant for the formulation of quantum mechanics, is then the algebra of bounded operators on some Hilbert space; the usual states can be extracted with help of Gelfand–Neimark–Segal construction (Arveson, 1981). In classical case, one takes the algebra of continuous, complex-valued functions on a locally compact Hausdorff space; then, due to the Riesz–Markov–Kakutani representation theorem, the states correspond to the probability measures defined in the space (Rudin, 1976). One should also keep in mind that the approach discussed above, although very advanced, is far from complete; for example, there are many subtleties connected to the unbounded operators. Meanwhile, many other routes are opened, for example, considering relations with the formulations using Jordan (nonassociative) algebras.

The close relation between Jordan operator algebras and \(C^\star \)-algebras provides the connection to the quantum-mechanical Hilbert space formalism, thus resulting in a novel axiomatic approach to general quantum mechanics.

The principles state: (i) no superluminal information transmission between systems by measurement on one of them, (ii) no broadcasting of the information contained in an unknown state, (iii) no unconditionally secure bit commitment.

Or, at least, there is no way to know that.

The problem of theory equivalence is strongly associated with the problem of structural representation of scientific theories. Different standards of equivalence say different things about which features of our theories are significant or contentful. For example, if one assumes the set-theoretic model isomorphism criterion (assigned to the standard semantic view), then one is also engaging with the idea of a theory expressed by its class of models. However, if one accepts the categorical equivalence criterion, then one is committing to the idea that a theory is given by its category of models. This is why it is so crucial to employ standards providing intuitive and desirable verdicts in particular research setups. For further discussions on the subject of theory (in)equivalence, especially the comparison between the model isomorphism and the categorical equivalence criteria, see (Barrett, 2019; Halvorson, 2012, 2013; Hudetz, 2019a, b; North, 2009; Weatherall, 2016, 2017.)

I thank Somayeh Towhidi for this observation. I also feel the difference in justification between such standard pragmatically motivated features, like popularity or historical priority of a given theory, and more ’antirational’ ones, e.g. fundamentalist Bourbaki-inspired claims about the

*proper*structure, but here I will not explore this issue further.Although Chang’s argumentation concerns scientific realism in general, I feel his ideas can be adapted to cover structuralist positions as well.

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## Acknowledgements

The author would like to thank the anonymous reviewers and the journal editor for the insightful comments concerning the paper. Special thanks to Michał Białończyk and Adrian Stencel for their expertise and assistance throughout all aspects of the study. The work was supported by the National Science Centre (Poland) grants 2018/29/N/HS1/02833 and 2020/36/T/HS1/00316.

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Proszewska, A.M. Goals shape means: a pluralist response to the problem of formal representation in ontic structural realism.
*Synthese* **200**, 245 (2022). https://doi.org/10.1007/s11229-022-03706-x

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DOI: https://doi.org/10.1007/s11229-022-03706-x