Introduction

Epistemic structural realism (ESR) claims that we can know about the unobservable because we can know about its structure. Nevertheless, construed as an epistemic thesis, ESR has remained largely silent on the ontological status of the structure it so heavily relies on. In this paper, I argue that the challenge of describing the ontology of these structures is a serious one that demands a serious response. I proceed by recalling the so-called ‘Newman Problem’ to support the view that ESR’s favoured set-theoretic and Ramsey-sentence account of structure is untenable. I then present the Newman Problem as symptomatic of another threat to ESR which I call the ontological problem—that ESR has yet to give a satisfactory account of what kind of ‘thing’ structure is. With the problem stated, I present two alternative interpretations of structural realism, ontic structural realism (OSR) and the argument from mathematical representation (AMR). I argue that these, both with kinks that still need working out, do a better job at coming clean on the ontological status of structure than ESR does. In section 1, a classical statement of ESR is given, along with its history and motivation. Section 2 examines the preferred set-theoretic and Ramsey-sentence account of structure employed by ESR. I then turn my attention to the Newman Problem in section 3 and present it as symptomatic of the ontological problem that has hitherto lay dormant in structural realism. Finally, in section 4 the alternative interpretations of structural realism are presented as potential candidates for a way forward against the ontological problem. Their main difficulties are also briefly discussed.

1. Epistemic Structural Realism

ESR emerges as an attempt to reconcile the history of science with the appeal of scientific realism. In the case of the former, antirealists have pointed out that many historically successful scientific theories, once thought to be approximately true, have since been proven otherwise. The now-canonical example offered is the shift from Fresnel’s theory of light as constituted by vibrations in a luminiferous aether to Maxwell’s electromagnetic theory (Worrall, 1989; Poincaré, 1905). The historical record of once-successful, now-abandoned theories should offer the realist no reason to be optimistic about the prospects of our best contemporary theories. This has come to be known as the view from the pessimistic meta-induction. Worrall’s account of ESR attempts to give credence to the strength of the meta-induction whilst upholding scientific realism on the parallel strength of the so-called no-miracles argument—the view that only scientific realism ‘does not make the success of science a miracle’ (Putnam, 1975, p. 73). The position which offers the ‘best of both worlds’ according to Worrall is ESR. Here the continuity between a theory and its discarded predecessor is the retention of ‘structure’. Returning to the example, Worrall says:

‘…it seems right to say that Fresnel completely misidentified the nature of light, but nonetheless it is no miracle that his theory enjoyed the empirical predictive success that it did; it is no miracle because Fresnel’s theory, as science later saw it, attributed to light the right structure.’ (p. 117).

Worrall’s thesis then is a sort of compromise between realism and the pessimistic meta-induction that states plainly: if we want to be realists, we can only be realists about the structure of our best theories.

2. Structure

What exactly is structure then? According to Worrall (1989) and Poincaré (1905), it is best understood as a set of equations that encode the structure of a theory’s target domain (Frigg and Votsis, 2011). It seems clear to Worrall that the history of science affords many examples whereby structure, understood as the mathematical formulation of a scientific theory, is retained across radical theory change. In these cases, the ‘common pattern’ is for ‘…the old equations [to] reappear as limiting cases of the new’ (Worrall, p. 120, emphasis from source). For example, Newton’s equations describing motion are shown to be limiting cases of Einstein’s equations for general relativity.
Most modern proponents of ESR now endorse a set-theoretic definition of structure1 that goes as follows: a structure 𝑆 is determined by a domain of objects 𝐷, and relations 𝑅 on that domain. It is commonly presented as an ordered pair 𝑆= . The salient point here is that relations on 𝑆 are purely extensional—that is to say the extension of a relation is just the ordered n-tuples it ranges over, or as Frigg and Votsis (2011) call them, ‘featureless dummies’ with ‘virtually no intensional interpretation’. (p. 229). By defining structures in this extensional sense, their epistemological import is immediately restricted to their logico-mathematical properties. For example, given the relation 𝑅={(𝑎,𝑏),(𝑏,𝑎)}, we are free to make the claim that 𝑅 is a symmetric relation. What we are not free to do is make an intensional claim such as ‘𝑎 is bigger 𝑏’ or ‘𝑎 loves 𝑏’ etc.
The set-theoretic definition gives the ESRist a way to talk about structure in isolation, but what of structure embedded within scientific theories? The favoured approach here is the so-called Ramsey-sentence, introduced by Maxwell (1970). The Ramsey-sentence has also been endorsed by Worrall (2007) as the appropriate method of interpreting scientific theories in such a way that elucidates their structural elements. Let 𝑇 be a scientific theory to-be-Ramsified. We want to begin by partitioning all predicates within 𝑇 into observational and theoretical classes respectively. Allocation should be relatively straight-forward,2 things like ‘reading on a mercury thermometer’ or ‘click on a Geiger counter’ fall into the former while ‘is negatively charged’ or ‘is a hydrogen atom’ fall into the latter. Denote theoretical predicates with 𝑃′𝑠 and observational predicates with 𝑄′𝑠. We can then express 𝑇 in the following way: 𝑇(𝑃1,…,𝑃𝑛,𝑄1,…,𝑄𝑚). Ramsifying 𝑇 then has us replacing all theoretical terms with variables and existentially quantifying over them. The resulting Ramsey-sentence for 𝑇 is the following:
∃𝑋1,…,∃𝑋𝑛𝑇(𝑋1,…,𝑋𝑛,𝑄1,…,𝑄𝑚). Read naturally, the Ramsey-sentence says that there exists some theoretical relations such that 𝑇 holds with respect to those relations and observable ones (Frigg and Votsis, 2011). By way of example from Maxwell, let 𝑇=∀𝑥[(𝑃1𝑥&𝑃2𝑥)→∃𝑦(𝑄𝑦)] where ‘𝑃1𝑥’ means ‘𝑥 is a Radium atom’, ‘𝑃2𝑥’ means ‘𝑥 radioactively decays’ and ‘𝑄𝑦’ is an observational predicate meaning ‘𝑥 is a click in an appropriately located Geiger counter’. 𝑇-Ramsified gives us the following: ∃𝑋1&∃𝑋2𝑇[(𝑋1𝑥&𝑋2𝑥)→∃𝑦(𝑄𝑦)]. Supporters of the Ramsey-sentence approach within ESR argue that the Ramsey-sentence of a theory captures its ‘full cognitive content’ (Worrall, 2007, p. 147)—that is to say, a Ramsified theory captures both the observational elements of some theory along with its structural elements. The take-away here is the connection between the set-theoretic and Ramsey-sentence approach. Both approaches retain a commitment to a purely extensional interpretation of structure: the former by construing structural elements as ‘featureless dummies’ and the latter by maintaining that ‘theoretical predicates…should not be given an intensional interpretation’ (Ainsworth, 2009, p. 139).

3. The Newman Problem and the Ontological Problem

The extensional interpretation of structure gets ESR into some serious trouble. The most famous and damaging of which has become known as the Newman Problem, or ‘Newman’s objection’ after a criticism first put to the early Russellian (1927) form of ESR by Max Newman (1928). The problem has been taken up more recently by Demopoulos and Friedman (1985) and Ketland (2004). Put simply, the objection goes like this: consider any structure that organises a certain number of objects. Take any other collection of objects of equal number, now this collection may too be organised as to have the same structure. More formally:

for any structure 𝑆 and any collection of objects 𝐶, if the domain of 𝑆 has the same cardinality as 𝐶, the objects within 𝐶 can be organised as to have the same structure as 𝑆.

The problem emerges because structure has been defined purely extensionally—as we have already seen, in a structure 𝑆=, relations 𝑅 do not instantiate any physical or material elements. We are not granted useful information that would be available had the definition of structure been intensional rather than extensional. For example, suppose 𝑅 was the ‘heavier than’ relation over some arbitrary domain. In this case, we would be furnished with the information that the objects in our domain are of unequal weight (Ainsworth, 2009). The implication of the Newman Problem then is this: to talk about any particular relation as having ‘encoded’ the structure of a theory’s target domain becomes trivial and unduly specific (Ainsworth; Frigg and Votsis, 2011). More seriously, it has the effect of undercutting the ESR project by showing its version of structure to be weak and without much in the way of epistemic import. The Newman Problem has equally unwelcome implications for the Ramsey-sentence approach to structure, as shown by Demopoulos and Friedman. For the purposes of this paper, Newman’s original objection is sufficient.3 Various attempts have been made to salvage ESR from the Newman Problem,4 but for now we turn to another challenge for ESR that the aforementioned problem is symptomatic of: structural realism still needs to come clean on the ontological status of structure itself.
Presentations of structural realism in the literature thus far have been couched in largely epistemic terms—the principal claim of course being that structure is all we can know about the world. In that literature, less attention has been paid to what structure is beyond the popular set-theoretic and Ramsey-sentence approaches I have already covered. According to Frigg and Nguyen (2017), the reason behind this is one of job description: ‘…philosophers of science need not resolve this issue and can pass off the burden of explanation to philosophers of mathematics.’ (p. 67). The problem with this attitude is that thus far, for philosophers of science rather than mathematicians, it has been largely unproductive. In fact, if we take the Newman Problem to be symptomatic of an untenable approach to what structure is, the attitude has been wholly detrimental. Structural realists instead stand only to benefit from coming clean on what kind of a ‘thing’ structure is—a consequence of which is avoiding or assuaging the Newman Problem by clarifying the intentional/extensional distinction. From now on I will refer to the ontological problem for structural realism as the challenge to describe the kind of ‘thing’ structure is. So far, I’ve discussed the most popular account of structure in terms of set-theoretical entities and Ramsey-sentences. The Newman Problem has shown these to expose a soft underbelly. I have argued that weakness comes from a hitherto neglected ontological problem. I now want to turn to two alternative formulations of structural realism that do try to come clean on the ontological status of structure.

4.1 Ontic Structural Realism

Ontological, or ontic structural realism moves away from ESR by taking structure to be ontologically basic. Rather than claiming that structure is all we can know about the world; structure is all there is to it. In its most defensible form, largely due to French and Ladyman (1998), eliminative ontic structural realism (EOSR) construes relations as the basic units of reality. According to this view, relations need not have relata between which they hold—the objects that would ordinarily be related in some way or another are dropped from the ontological framework. Furthermore, EOSR abandons the purely extensional view of structure held by ESR. Rather, relations may be intensional and have some natural or material content to them. For example, the relation of ‘being larger than’ is a legitimate and desirable relation between objects in some domain, according to EOSR (Frigg and Votsis, 2011). Putting criticism aside for a moment, it is worth considering how EOSR tries to provide an answer to the ontological problem. The reasoning is twofold:

(i) There is an obvious and important shift in attitude that moves one from ESR to EOSR. In its crudest formulation, the former suggests that all we can know about the world is its structure while the latter argues that structure is all there is to that world. Put in this way, EOSR doubles as both an ontological and epistemic thesis—if it is true that all there is to the world is structure, it seems reasonable to suggest that structure is what we can know about that world. The important move here is to take on the additional burden of ontology in the approach.
(ii) EOSR makes explicit use of intensional instantiation and in doing so, escapes the Newman Problem (Frigg and Votsis 2011; French and Ladyman, 1998). Recall that the Newman Problem undercuts ESR by showing that the extensional reading of structure makes the set-theoretic/Ramsey-sentence approach essentially useless for ‘encoding’ the structure of a scientific theory. What EOSR does well then is construe structural relations as substantive and intensional. Viewed in this way, relations like ‘heavier than’ are acceptable. If we accept this version of structure, EOSR says that we can encode the structure of a scientific theory in some substantive, ontologically meaningful way that avoids the Newman Problem.

EOSR is a promising but problematic interpretation of structural realism. Of its kinks to be worked out, there is one that prominently sticks-out: there cannot be relations without relata (Busch, 2003; Psillos, 2001). This criticism construes EOSR’s ontological framework as incoherent since ‘…the very idea of structure presupposes some elements that go together…A relation might take anything as its relata, but it always takes something’ (Busch, p. 213, emphasis from source). Those defending EOSR have responded in largely two ways. The first response argues that the ‘no relations without relata’ criticism is based on an extensional reading of structure. If so, critics are correct to say EOSR would be incoherent since even simple relations, say 𝑅={(𝑎,𝑏),(𝑏,𝑎)}, relate elements of the domain 𝐷={𝑎,𝑏} even if these are just featureless dummies. EOSR, however, supports an intensional interpretation of relations. Accordingly, relations like ‘heavier than’ are taken to be substantive and ontologically meaningful in some sort of fundamental way, thus dispensing of the need for relata between them. Since this paper is not devoted to a discussion of EOSR, I leave the issue here. However, it may be important to note just how dramatically the notion of structure as being ‘ontologically basic’ will depart from most philosopher’s intuitions.
Those intuitions will be tested again by EOSR’s second response to the ‘no relations without relata’ criticism. The rejoinder, due to Ladyman and Ross (2007), says that relata does exist between relations but that this relata is also structure. If we look at some relation and the objects it relates, we will find that these objects themselves are structures. This response then suggests ‘The world is structure all the way down’ (Frigg and Votsis, 2011, p. 263). It is worth highlighting again how dramatic a departure this view is from the prevailing intuition towards the subject. In order to defend against the ‘no relations without relata’ criticism, supporters of EOSR must make some bold ontological claims about the world. It is difficult to see how the ontological framework EOSR supports would be digestible or attractive to philosophers of science. Nevertheless, I’ve presented it here as an impressive and largely promising attempt to answer the ontological problem in structural realism. Further work on its theoretical difficulties is sure to win EOSR more supporters.

4.2 The Argument from Mathematical Representation

Another interpretation of structural realism that proffers an answer to the ontological problem is the argument from mathematical representation (AMR). A classical statement of the position is due to van Fraassen (1997) who writes: ‘to present a scientific theory is, in the first instance, to present a family of models—that is, mathematical structures offered for the representation of the theory's subject matter’ (p. 526). The principal claim of the view then is that scientific theories are to be thought of as mathematical structures whereby the property of ‘being a structure’ here refers to ‘being a collection of models’. The AMR then gives a clear-cut response to the ontological problem: structures are models. Some important qualifications are in order:

(i) Though now construed as a collection of models, the set-theoretic definition of structure is retained (Frigg and Nguyen, 2017).
(ii) If structures are to be thought of as models, they have the additional burden of representing. A model, if it is to be such, must represent its target system in some way. According to the AMR, a model represents its target system by being isomorphic to it (van Fraassen, 1997; Frigg and Nguyen, 2017; Frigg and Votsis, 2011). Two structures 𝑆1= and 𝑆2= are isomorphic iff there is a bijective mapping 𝑓: 𝐷1→𝐷2 such that 𝑓 preserves the framework of relations in such a way that: for all relations 𝑟1∈𝑅1 and 𝑟2∈𝑅2, the elements 𝑎1,…,𝑎𝑛 of 𝐷1 satisfy the relation 𝑟1 iff the corresponding elements 𝑏1=𝑓(𝑎1),…,𝑏𝑛=𝑓(𝑎𝑛) in 𝐷2 satisfy 𝑟2, where 𝑟1 is the relation in 𝑅1 corresponding to 𝑟2 in the relation 𝑅2 (Frigg and Nguyen; Frigg and Votsis).
(iii) A collection of models is sufficient to adequately represent all parts of a scientific theory.

With these qualifications in place, the AMR offers another interpretation of structure that comes clean on the ontological problem: a structure is a collection of models that represent a scientific theory. Similarly, to EOSR, there are clearly some difficulties to be figured out.
First, supposing we accept that a structure is to be thought of as a collection of models; what exactly is the ontological status of those models? Throughout this paper I’ve persistently demanded that structure be accounted for in some ontologically meaningful sense i.e. one that would let us understand what kind of ‘thing’ it is. Nevertheless, the same demand can equally be made to the abstractness and diversity of scientific models. In fact, it has already been made by Frigg and Nguyen (2017) who argue that the ontological problem rears its ugly head in the area of model-representation:

‘What kinds of objects are models? Are they structures in the sense of set theory, fictional entities, descriptions, equations or yet something else? Or are there no models at all?’ (p. 54).

The same question has been posed by Suárez (2003) who asks what exactly the ontological link is between different kinds of models, say ‘…a toy model of a bridge; an engineer’s plan for a bridge…the billiard-ball model of gases…and the quantum state diffusion equation for a particle subject to a localization measurement’ (p. 225). Even if structural realism tries to defer its ontological commitments to the field of model-representation, it is unclear whether the latter itself has an adequate answer to the ontological problem.
It is worth noting another difficulty. Even if we permit that model-representation can give an answer to the ontological problem, it faces a further question: what version of scientific representation is the right one? We have already said that models are only models insofar as they represent their target systems. Though there is a vibrant and developing literature that discusses theories of scientific representation,5 there is little consensus on what theory of representation is best. The AMR relies on one version of representation, namely isomorphism between a model and its target system. Whilst we do not have space to discuss its merits or lack thereof, the isomorphic account of representation has been taken to task in Frigg (2006) and particularly Suárez (2003). The above criticisms show the AMR as being in danger of trying to take on too much. First, it must seek to solve the ontological problem as it occurs in model-representation; then it must take on the challenge of finding a satisfactory version of scientific representation. It is difficult to see how this load will not be too much. In spite of this, I am optimistic about the AMR’s chances as a solution to the ontological problem in structural realism. That optimism is owed chiefly to the quality and productivity of research in this area.

Conclusion

In this paper, I have argued that structural realism has yet to provide an adequate ontological account of structure. I did so by first revisiting the classical statement of ESR from Worrall (1989) and then presenting the two most popular formulations of structure in terms of set-theoretic entities and Ramsey-sentences. I discussed the Newman Problem and suggested it was symptomatic of the hitherto dormant ontological problem for ESR. Finally, I put forward two alternative interpretations of structural realism, EOSR and the AMR, that do try to answer the ontological problem. Both interpretations have some difficulties to be worked out and this paper has remained neutral on whether those difficulties can be given satisfactory solutions. Instead, I have referred to them with the hope that they encourage discussion and refinement of both EOSR and the AMR as potential solutions to the ontological problem. EOSR has been criticised on the basis of having an incoherent ontology. I’ve presented the main responses to this charge on behalf of EOSR but concluded by suggesting the more difficult task may be convincing philosophers of science to adopt the radical ontological framework that EOSR requires. For the AMR, resolving some of its issues seems a tall order, particularly because it shares its main difficulties with the field of model-representation. These are not simple problems: to understand what models are and to find a suitable theory of representation. Though challenging, these problems are not unsolvable, and model-representation is a productive research area within philosophy of science that promises to yield potential solutions. With luck and some necessary zeal, these solutions, along with solutions to the problems of EOSR, will provide a path forward on the ontological problem for structural realism.

Acknowledgments

I am grateful to Paul Daniell for his feedback on early versions of this paper.

Competing Interests

The author has no competing interests to declare

Notes

1. Most, though not all. The set-theory homogeny has been criticised by Landry (2007) as being overly restrictive.
2. Straight-forward only in this setting. I ignore the difficult debate on the observational/theoretical distinction. See Maxwell (1962) for a detailed discussion.
3. Readers interested in how the Newman Problem trivialises the Ramsey-sentence approach should be directed to Demopoulos and Friedman (1985) and Ketland (2004).
4. For critical discussion of these attempts, see Ainsworth (2009) and Frigg and Votsis (2011).
5. See Frigg and Nguyen (2017) for a survey of them all.

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