Abstract
This paper is a critical response to Andreas Bartels’ (Theoria 55, 7–19, 2006) sophisticated defense of a structural account of scientific representation. We show that, contrary to Bartels’ claim, homomorphism fails to account for the phenomenon of misrepresentation. Bartels claims that homomorphism is adequate in two respects. First, it is conceptually adequate, in the sense that it shows how representation differs from misrepresentation and non-representation. Second, if properly weakened, homomorphism is formally adequate to accommodate misrepresentation. We question both claims. First, we show that homomorphism is not the right condition to distinguish representation from misrepresentation and non-representation: a “representational mechanism” actually does all the work, and it is independent of homomorphism – as of any structural condition. Second, we test the claim of formal adequacy against three typical kinds of inaccurate representation in science which, by reference to a discussion of the notorious billiard ball model, we define as abstraction, pretence, and simulation. We first point out that Bartels equivocates between homomorphism and the stronger condition of epimorphism, and that the weakened form of homomorphism that Bartels puts forward is not a morphism at all. After providing a formal setting for abstraction, pretence and simulation, we show that for each morphism there is at least one form of inaccurate representation which is not accommodated. We conclude that Bartels’ theory – while logically laying down the weakest structural requirements – is nonetheless formally inadequate in its own terms. This should shed serious doubts on the plausibility of any structural account of representation more generally.
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Notes
We are using the term morphism to refer to any structural mapping regardless of the kind of transfer of structure from \(\mathbb {A}\) to \(\mathbb {B}\) that it implies. Therefore the term should not be understood as a synonym for homomorphism, which is at best the basic, or most elementary, form a morphism can take.
Jorge Luis Borges’ wonderful discussion of the one-to-one scale map is an exemplary parody of how a perfectly accurate representation is also perfectly useless (Borges 1954).
One of the referees points out that this is in fact an incredibly strong assumption. As he or she puts it: “A system of gas molecules is not a set of elements and a family of labelled relations, etc. It has no labelled relations because it contains no labels [...] The real world thing being represented is not a structure, whereas the author’s ‘target’ has to be a structure for the author’s discussion to make any sense at all”. We agree wholeheartedly with this referee. It is indeed the case that a real physical object, a system, or a phenomenon, can only be said to be a structure under a description. And it is clear (as one of us has often pointed out, in e.g. Suárez (2010, p.96)) that any structural description is necessarily vastly underdetermined: Every real object exemplifies multiple, perhaps an infinite number of, structures. This simple fact puts great pressure upon structuralist claims regarding ontology (to the extent that claims to the effect that the “world consists only of structure” or some such thing, are rendered vacuous or, worse, incoherent – as pointed out by e.g. van Fraassen (2006)). We ignore this issue because almost all the literature that we do address ignores it too, and also because it can only strengthen our critique of the homomorphism theory of representation. But it is worth pointing out with the referee that general widespread acquiescence with a false assumption does not make it any less false or unwarranted.
While misrepresentation as inaccuracy is taken into account in Cartwright (1983), Contessa (2011), Frigg (2006), Giere (1988), Pincock (2011), Teller (2001), Teller (2008), Suárez (2003), Suárez (2004), and van Fraassen (2008), misrepresentation as mistargetting is presented in Suárez (2003) and Suárez (2004).
In fact, Bartels’ attempt to accommodate the conceptual adequacy seems to resolve in a form of deflationary, or functional, account. Deflationary (Suárez 2004) or functional (Chakravartty 2010) approaches treat representation as a function of models which allows model users to gain information about the target at stake via the model. The ascription, or recognition, of the representational function of a model by a user is then essential to have representation. The crucial role played by the representational mechanism’s choice in Bartels’ homomorphism theory puts his theory very much in line with those accounts.
“A relational structure B is said to be a homomorphic image of A if there exists a homomorphism from A to B that is onto B (in symbols, B=h ∗(A)). (A function f maps A onto B [it should be A onto B] if for every b∈B there is an a∈A such that h(a)=b).” (Dunn and Hardegree 2001, 15). Read the bold character in the quote as our \(\mathbb {A}\) and \(\mathbb {B}\).
Consider two similar structures, \(\mathbb {A} = \langle A, ({R^{A}_{1}}, {R^{A}_{2}}) \rangle \) and \(\mathbb {A} = \langle B, ({R^{B}_{1}}, {R^{B}_{2}}) \rangle \), with \(A \in \mathbb {A} = \{a_{1}, a_{2}, a_{3}, a_{4}\}\), \(B \in \mathbb {B} = \{b_{1}, b_{2}, b_{3}\}\). The mapping \(f: A \rightarrow B\) is surjective, and the condition of completeness holds. Therefore, \(\mathbb {B}\) is a homomorphic image of \(\mathbb {A}\). To find a case where the conditions of completeness and the surjectivity of f (and \(\mathbb {A}\) and \(\mathbb {B}\) are similar structures) are satisfied, but \(\mathbb {B}\) is not faithful, we need a relation \({R^{B}_{j}} \in \mathbb {B}\) which has no counterpart \({R^{A}_{j}} \in \mathbb {A}\) and, at the same time, we need to assure that all the relations in \(\mathbb {A}\) have their counterparts in \(\mathbb {B}\). The function \(f: A \rightarrow B\) is surjective (and not injective) and ascribes to each argument the following images: f(a 1)=b 1,f(a 2)=b 2,f(a 3)=b 3,f(a 4)=b 3. Consider now the case that \(\mathbb {A}\) has the following family of relations: \({R^{A}_{1}} \subseteq A^{2} = \{(a_{1}, a_{2}), (a_{1}, a_{3})\}\) and \({R^{A}_{2}} \subseteq A^{2} = \{(a_{1}, a_{2}), (a_{3}, a_{4})\}\). As for \(\mathbb {B}\): \({R^{B}_{1}} \subseteq B^{2} = \{(b_{1}, b_{2}), (b_{1}, b_{3})\}\) and \({R^{B}_{2}} \subseteq B^{2} = \{(b_{2}, b_{1}), (b_{3}, b_{2})\}\). The relation \({R^{B}_{1}}\) in \(\mathbb {B}\) thus corresponds to both the relation \({R^{A}_{1}}\) and \({R^{A}_{2}}\) in \(\mathbb {A}\), while the relation \({R^{B}_{2}}\) has no counterpart in \(\mathbb {A}\). Therefore, \(\mathbb {B}\) is a homomorphic image of \(\mathbb {A}\) while faithfulness is violated.
The standard definition of epimorphism is “surjective homomorphism”. Therefore Rothmaler adds faithfulness as a further condition. As it will turn out in Section 4.2, this notion of epimorphism works fine also to distinguish the conditions for having epimorphism from those required for having a “homomorphic image”.
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Acknowledgments
We would like to thank two anonymous referees for helpful comments and suggestions. Mauricio Suárez acknowledges the support of the European Commission (Marie Curie grant FP7-PEOPLE-2012-IEF: project grant 329430) and of the Spanish Government (DGICT grant: FFI2014-57064-P), and helpful discussions at the Universities of London, Florence, and Complutense of Madrid. Francesca Pero acknowledges the support of the Department of Philosophy, University of Florence and of the Complutense University of Madrid, as well as the helpful exchanges at the SILFS conference held in Rome, June 2014, and at the Universities of Florence and Complutense of Madrid.
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Pero, F., Suárez, M. Varieties of misrepresentation and homomorphism. Euro Jnl Phil Sci 6, 71–90 (2016). https://doi.org/10.1007/s13194-015-0125-x
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DOI: https://doi.org/10.1007/s13194-015-0125-x