Abstract
I have two central aims in this paper, both relatively modest. The first is to present some ways of thinking about structuralism about scientific representation COMP: Please set ‘Chapter 7’ before the title and make the digit 7 be prefixed in all section heads in this chapter.. After distinguishing two distinct structuralist theses about representation – “vehicle structuralism” and “content structuralism” – from one another and from various other structuralist theses about the sciences (Section 7.2), I will separate three different non-formal concepts of structure (Section 7.3.1), discuss their relationship to the familiar formal concepts (Section 7.3.2), and consider two different ways of explicating vehicle structuralism (Section 7.4). I will then go on (and this is the second aim) to present a line of argument for the conclusion that structural realists of a certain familiar sort should reject both vehicle structuralism and content structuralism, and, relatedly, the semantic view of theory structure (Sections 7.5 and 7.6). Appallingly, this conclusion may not be at odds with the commitments of any particular philosopher or group of philosophers, but I think it will be useful to spell out the connections explicitly nonetheless.
Some of the points made in this paper were first presented as part of a talk given at PSA 2004, but did not make it into the paper which appeared in the proceedings (Thomson-Jones 2006). Thanks to Craig Callender, Anjan Chakravartty, James Ladyman, and Bas van Fraassen for helpful correspondence and discussion.
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Notes
- 1.
It is at least suggested by passages in French and Ladyman (1999), Brading and Landry (2006), and van Fraassen (2006a,b) and (2008, ch. 11), for example – which is not to say that all these authors would embrace the view, or that any of them would embrace it exactly as I have formulated it here. See also the last paragraph of this section for an important qualification.
- 2.
- 3.
These are only partial characterisations of the two varieties of structural realism; in particular, no claim has been made about the knowability of the structure of the unobservable world. Note also that there is a harmless redundancy involved in this characterisation of metaphysical structural realism: given MS, a commitment to LS is unavoidable.
- 4.
I will, for the most part, leave MS aside.
- 5.
Note that this way of putting things assumes nothing about the relationship between models and theories.
- 6.
See n. 23 for the one point at which it makes a difference.
- 7.
Though I will not be presenting an overall assessment of VS or CS in this paper, I offer arguments against VS in “Thomson-Jones” (in preparation a,b). I do not and would not argue that no scientific representation is representation by means of structure, however, nor (of course) that no scientific representation is of structure.
- 8.
It does not matter for present purposes whether structures of this second sort are in fact universals, or even whether there are any universals; I mention the issue only as a way of characterising the notion of structure I have in mind.
- 9.
If we so wish, this schema can be treated as providing a reductive analysis of the ‘X has S’ locution when that locution is employed in the sense I am hereby specifying (so that the reductive analysis is correct by stipulation, so to speak). This means that we need not worry about the ontological status of structures as entities in themselves, or of such things as “arrangements of properties and relations”; talk of structures can be treated simply as indirect talk of parts, properties, and relations, and so as no more mysterious than talk of those things (however mysterious that may or may not be).
- 10.
We might also want to allow schemas of the form ‘X has S iff there is a partition of X into at least nparts such that…’; this would amount to allowing structures which are instantiated by an object purely in virtue of the properties and relations of just some of its parts.
- 11.
Whether one should think that there is, or might be, a structured object which instantiates only one concrete structure will depend on one’s views about the metaphysics of properties, amongst other things. But even if there are such objects, they are clearly atypical.
- 12.
Incidentally, the present approach to thinking about structures can easily accommodate the notion of a substructure: we can simply say that S is a substructure of S′ iff necessarily, for all X, if X has S′, then X has S. (So S2 and S3 are both substructures of S1, for example.)
- 13.
Note that I am not thinking of the relation in question as defined exhaustively by its extension on the set {a, b, c, d, e}; if it were, the mention of transitivity would be redundant.
- 14.
The examples I have given are ones in which either every mention of a property or relation is specific (S1, S2, and S3), or none are (S4). I will take it that those are the cases of interest, and leave aside the category of “mixed” cases.
Note also that I am using the labels ‘abstract’ and ‘concrete’ differently than (Redhead, 2001, pp. 74–75) and (Votsis, 2003, p. 881, and 2005, p. 1363); see Section 7.3.2 for more, and especially n. 17.
- 15.
The substructure relation can also be defined for abstract structures in just the same way as for concrete structures – see n. 12.
- 16.
The ‘Pi’ is often absent. Sometimes this is because properties-in-extension are treated as unary relations-in-extension, so that the ‘Rjn’ is read as covering them; and sometimes it is because the focus is exclusively on relations-in-extension, and so on a formal analogue of what I have called purely relational structures. I have nonetheless included the ‘Pi’ for greater ease of comparison to the informal notions laid out in the last subsection. The relevant adjustments are easily made when we want to focus on the special case of purely relational structures. Similarly, we might in some contexts want to consider tuples also containing distinguished members of U, or functions-in-extension, or both.
It is worth noting that a structure of this sort is also a structure in one established logical sense – namely, it is the sort of thing which provides an interpretation for a set of sentences in a first-order language. This is no coincidence: see n. 26.
- 17.
Votsis uses the labels ‘concrete’ and ‘abstract’ rather than ‘tuple’ and ‘isoclass’; I have not followed his usage for obvious reasons.
Incidentally, (Votsis, 2005, pp. 1362–1363) seems to attribute both the tuple and the isoclass notions to Redhead (2001), but it is not clear to me that Redhead means to be employing either of them. Redhead gives as one example of what he calls a concrete structure “a pile of bricks, timbers and slates, which are then ‘fitted together’ to make a house” (2001, p. 74), and makes no mention of ordered tuples, so his notion of a concrete structure does not seem to be Votsis’s (i.e., the tuple notion). Redhead then says that we can think of what he calls an abstract structure as either an isomorphism class of the sorts of things he calls concrete structures, or “in an ante rem Platonistic sense as the second-order Form which is shared by all the concrete relational structures in a given isomorphism class…” (ibid. p. 75). The second notion is clearly not equivalent to Votsis’s notion of an abstract structure (which is the isoclass notion), and if I am right that Redhead’s notion of a concrete structure is not the tuple notion, then the notion of an isomorphism class of (his) concrete structures is not Votsis’s notion of an abstract structure, either.
- 18.
In that approach, the relations-in-extension are partial relations – relations-in-extension which may be defined only on a proper subset of U – but that difference will not bear on the present discussion.
- 19.
Of course, it may also be true that two or more properties or relations can share their extension simpliciter – perhaps even necessarily – but the claim I need here is a much less controversial one.
- 20.
This example suggests that the “underdetermination” of concrete structure by structures-as-tuples will be a widespread phenomenon even if our ontology of properties is a sparse one, or (alternatively) we are restricting attention to a limited range of natural properties. On a more promiscuous ontology of properties, or given a less restricted field of attention, such underdetermination will be ubiquitous.
- 21.
Or between structures of either formal kind and structured objects, for that matter. (To see that there is no one-to-one correspondence between isomorphism classes of tuples and abstract structures, incidentally, consider the fact that the extension of a relation on a restricted domain does not, in general, determine its logical properties.)
- 22.
The isoclass notion does allow one to say “Same structure!” when pointing at the two desks, of course; but that notion does not enable us to capture the sense in which there is a structure shared by the two desks which is shared only by objects which have parts instantiating a certain pattern of relations of aboveness to one another. If, on the other hand, we are interested in focussing on a more abstract sort of structure the desks have, one which is not tied to the possession of any specific properties or relations, then the informal notion of abstract structure will serve our purposes just as well as the isoclass notion.
- 23.
See the next section, when it will become clearer what I mean by talk of singling out arrangements of properties and relations and employing them in representation.
Incidentally, this is the one point at which the strength of my argument might be thought to depend on the fact that I have adopted an unqualified formulation of VS. It will seem to many less implausible that all scientific representation by means of models and theories involves the employment of formal structures than that all scientific representation does so, and not only because the former is a logically weaker thesis: the weaker claim may, in fact, have a fairly high degree of plausibility for adherents of at least some central varieties of the semantic view. Leaving aside the big question of whether the semantic view gets things right, however, we can say that even the qualified version of VS – the weaker claim – seems (pre-theoretically) more plausible when construed in terms of the informal notions of structure than the formal, even if the plausibility gap is smaller than the gap between the informal and formal construals of the unrestricted version of VS. I would thus maintain that this argument for using the informal notions goes through either way.
- 24.
This is quite compatible with the claim that it might be helpful, for certain purposes, to use formal structures in the philosophical representation of the workings of scientific representation. (French and Ladyman, 1999, p. 107) and French (2010) emphasise that various of their claims are to be taken in this latter spirit. It seems to me, nonetheless, that if we want to understand how scientific representation works, then sooner or later we will need to know what sorts of thing scientists in fact employ when representing the world.
- 25.
Especially given the points made in n. 9.
- 26.
Attachment to the < U, Pi, Rjn > notion of structure may come from a commitment to a certain programme in the semantics of natural language (or at least the language of science), or, relatedly, from linking a notion of model appropriate to the philosophy of science to the notion of model found in the standard Tarskian semantics of first-order languages. I do not wish to presuppose the former commitment, however, and have argued that the latter tendency is ill-advised (Thomson-Jones (2006)). Note, too, that although my emphasis here is on making room for non-formal notions of structure, we can also challenge the assumption that a formal approach should rely primarily on set-theoretic tools; see (Thomson-Jones, 2006, esp. p. 534) and Landry (2007).
- 27.
Note that no claim is made here about when an act counts as an act of representation.
- 28.
I am not assuming here that the act in question counts as an act of scientific representation; the point here is only to get some purchase on the notion of x’s carrying all the content conveyed by an act of representation, whatever the do problem of saying what it is for a representational act to count as an act of scientific representation is one I have purposefully left aside.
- 29.
It is easier to construct artificial cases which fit the picture, especially if we are not concerned with limiting ourselves to scientific representation: see the “British are coming” case discussed at the beginning of Section 5, below.
- 30.
See Watson (1969), pp. 45, 67, 97, 108, and 113.
- 31.
I am leaving aside the sugar and phosphate molecules.
- 32.
I will not attempt to settle here the question of whether a representational act which centres on the tin-plate model but specifies the four kinds can be fitted to the DVS picture. There are at least some complications there, however. In particular, no such representational act could be taken to be of the especially simple sort just described, in which it is a straightforward matter of attributing the displayed structure to the target system, because the tin-plate model does not instantiate any structure which involves the property of being, say, a guanine base (which is just to say that no part of the tin-plate model is a guanine base). Relatedly, if no representational acts of the more limited sort described in the text took place, the proponent of DVS has more work to do to make her view seem plausible.
- 33.
We will consider the consequences of taking a different view of mathematics – the structuralist view – in a few moments.
- 34.
Note that here I have considered a representational act in which we do no more than represent time as having a certain abstract structure. If more is being said when we say that the parts are times, or moments, and that the relation is (say) the earlier-than relation, then there will be complications involved in fitting a representational act which conveys that additional content to the DVS picture, complications of the same sort as the ones discussed in n. 32.
- 35.
Bear in mind that the point here is not to decide whether it is accurate, in the final analysis, to claim that any or all scientific representation works this way; I am merely trying to fill out the structuralist picture of scientific representation.
Incidentally, the points made in this paragraph apply in equal measure to the second explication of VS, to which we are about to turn.
- 36.
- 37.
I am using ‘single out’ and ‘pick out’ interchangeably; unmodified, both phrases are intended to leave open the means, so to speak, and that is how I have been using them thus far.
- 38.
For one thing, note that simple vehicle structuralism has no difficulty in accommodating the mathematical structuralist’s understanding of the real line/time case.
- 39.
By saying that simple vehicle structuralism is in danger of seeming trivial, I do not mean that it begins to look like an analytic or logical truth. I mean rather that it begins to look too weak and uncontroversial to be staking out a specifically “structuralist” approach to scientific representation.
- 40.
This is especially plausible if we count as a partitioning the trivial partitioning of a thing into its sole improper part.
- 41.
If the parenthetical ‘virtually all, or most’ is needed in this argument, then although simple vehicle structuralism might still be true, it will only apply for the very general reasons just presented in virtually all or most cases, and thus will only be nearly trivial.
- 42.
We have already seen (in Section 7.3.2) that there is a formal notion of abstract structure in the literature, often traced back to Russell (e.g., Votsis (2003), p. 881); and some who employ the tuple notion of structure sometimes focus exclusively on relations. The proposals I am considering here are distinct, of course, because they centre on informal notions of structure; but the parallels are clear.
A third option, incidentally, is to impose both constraints, but the points I am about to make will extend to that option automatically.
- 43.
- 44.
I do not mean to insinuate that it will be impossible to provide such accounts; my point is just that there are questions that need to be answered on this sort of approach, and it is not obvious how those questions are to be answered.
- 45.
In other words, although VS may entail CS as stated, given the notions of structure in play, strengthening CS to yield a sufficiently substantive thesis will undermine the entailment.
Note also that a defender of CS who responds to this triviality problem by restricting attention to abstract structures will have to claim that Bohr acts attribute no specific properties or relations to parts of the hydrogen atom, whereas one who responds by restricting attention to purely relational structures will have to claim that Bohr acts attribute no intrinsic properties.
- 46.
See Goodman (1976). On p. 5, for example, Goodman makes the slightly qualified claim that “almost anything may stand for almost anything else,” and follows up with the claim that a particular sort of standing-for, denotation, is “the core of representation.” (Of course, even the correspondingly qualified claim that almost anything can represent almost anything else does not follow from these two claims, but Goodman has been taken to assert that, too.)
- 47.
We could also call them “pragmatic” considerations, provided we do not allow the term to trigger unwanted associations.
- 48.
I do not mean to suggest that Goodman failed to take account of these points – he clearly did not.
- 49.
The emphasis here is on the word ‘special’ – see the next paragraph.
- 50.
At worst, there is some redundancy involved in declaring a commitment to CS, LS, and SS: if our representations tell us only about structure (CS), there is little point in saying that we can know only what they tell us about structure (LS). Note, however – and this is to anticipate the point that the ESRist should reject CS – that this combination of views is clearly not what the ESRist has in mind, as she clearly means to be describing a restriction on what we should believe, from amongst all the things our scientific representations say.
- 51.
Of course, this point, and the points which follow, extend to any metaphysical structural realist who might wish to draw support for his position from the argument we have been considering.
- 52.
For my own view about the best way of understanding the seminal variants due to Patrick Suppes and Bas van Fraassen, see Thomson-Jones (2006). Despite the fact that the work of Ronald Giere and Frederick Suppe is often mentioned in the same breath as that of Suppes and van Fraassen, their accounts of theory structure diverge in a number of significant (and different) ways from the Suppes-van Fraassen approach. (See, e.g., Giere (1988); Suppe (1989)). Particularly relevant here is the fact that models, on Giere’s account, are not mathematical structures; see Thomson-Jones (2010).
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Thomson-Jones, M. (2010). Structuralism About Scientific Representation. In: Bokulich, A., Bokulich, P. (eds) Scientific Structuralism. Boston Studies in the Philosophy and History of Science, vol 281. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9597-8_7
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