Surplus Semantic Universalism And Minimal Structuralism

Following Ismael and Van Fraassen ([2003], p. 371), let us distinguish between 'elements' and 'structure'—where the latter is defined by a set of relations on the elements, as I defined in §1.2. If we use a structure to represent a physical system or an aspect of a physical system (i.e., if the structure is a mathematical model), then any surplus will, if it manifests itself at all, manifest itself through a many-to-one relationship between model and system. One and the same physical state of affairs being multiply represented is the hallmark of surplus structure. We have seen such structure in many different theories now and a (possible) many-to-one relationship is seen to occur when the structure admits a certain type of symmetry. The kinds of symmetry I have been concerned with in this book are such that they generate no change in the observable (in the technical sense) state of the system under consideration, i.e. global or gauge symmetries. That is to say, they do not alter the structure when they act on the elements—here, of course, I mean 'qualitative structure' and the observables provide the key to enable access to it. This picks out a class of symmetries that preserve all of the qualitative aspects of some model or world.

Of course, not all symmetries are like those mentioned above; most of those that occur in physics (in computations, and so on), change some qualitative features of the world. The world is changed but the system, taken intrinsically (i.e. without reference to anything external to the system), isn't. In their discussion of the connections between symmetries and Noether's theorems, Brading & Brown illustrate this difference with a simple example, as follows:

One way of getting our hands on the empirical significance of a symmetry is through 'Galilean ship' type experiments. Here, we take an effectively isolated subsystem of the universe, transform it (in the case of Galileo's ship we go from the ship being at rest to the ship being in uniform motion), and observe that the two states of the subsystem are empirically indistinguishable except in relation to (parts of) the rest of the universe. Thus, in the case of Galileo's ship, no experiments carried out inside the cabin of the ship, and without reference to anything outside the ship, enable us to tell whether the ship is at rest or moving uniformly The two states of motion are empirically indistinguishable except by looking out of the porthole. ...we apply the symmetry transformation to an effectively isolated subsystem of the universe, yielding two empirically distinct scenarios across which the internal evolutions of the subsystem are empirically indistinguishable. ...However, in the case of global gauge symmetry, this approach doesn't work. Gauge transformations have no analogue to the 'Galilean ship', they have no active interpretation. While it is true that global gauge symmetry does not have the same direct empirical significance ...this does not imply that global gauge symmetry is without empirical content. The very fact that a global gauge transformation does not lead to empirically distinct predictions is itselfnon-trivial. In other words, the freedom in our descriptions is no 'mere' mathematical freedom—it is a consequence of a physically significant structural feature of the theory. ([Brading and Brown, 2003], pp. 98-9)

Let us distinguish between these two types of symmetry, calling the former variety empirical symmetries and the latter variety gauge-type symmetries.288 Thus, the gauge-type symmetries preserve all qualitative structure, where 'qualitative structure' is determined by the (in principle, empirically accessible) properties and relations of a theory. The structuralism I have proffered is not general enough to include empirical symmetries; if it did include these the implication would be that there is but one unchanging structure in the world, for there would be a single element phase space representing it (after identifications have been made)! But it is the latter gauge-type symmetries that play a central role in the theories we are interested in, and these theories are the best ones we have. More to the point, within these theories, it is symmetry that plays a key role in determining what the theories are about, and therefore what the ontology of the theory is or could be. Where the symmetries of the first type are concerned, specifying an ontology (or a large part of it) will involve a significant amount of background ontology such as metrics and connections forming part of a fixed background spacetime. Without this background structure to hand, the analysis breaks down and so does the notion of particle constructed along group-theoretic lines (see the end of §9.1).

Recall the definition of a symmetry I gave in §1.2 as an automorphism of a structure, where a structure is a set of individuals with a set of properties and relations defined over them. Thus, we can view a symmetry as a permutation on the domain that has the effect of shuffling the properties and relations over the individuals without altering the structure: all relations and properties are preserved by the permutation. Now, clearly in the examples we have been considering some properties and relations will be altered by the permutation; if we take our individuals to possess haecceities then different properties will be assigned to the individuals in the two cases. If we fix our attention only on qualitative properties and relations then we get the desired result: invariant structure. The fact that the individuals are related in this way by a permutation implies that they are

288 Compare this with Giulini ([2003], p. 32) who distinguishes in a similar way, though he restricts the use of the term "symmetry" to what I have called empirical symmetries and calls my gauge-type symmetries "gauge redundancies". Similarly, Ismael & van Fraassen refer to the gauge-type symmetries as "beacons of redundancy" ([2003], p. 391).

qualitatively indistinguishable, for were they distinguishable the structure would certainly not be invariant: one could call upon some property or relation to discern the structures. The individuals thus related must therefore play the same roles in the structure. The fact that many non-trivial structures of modern physics behave in just this way is quite remarkable. I would suggest that this is one possibility for the "physically significant structural feature of the theory" that Brading and Brown mentioned; it is the (qualitative) indistinguishability of the symmetry operands that is responsible. This can be understood in terms of the observables simply as an indifference to how the individuals are distributed in the relational structure.

At least this is one story; but we must be careful about stripping metaphysical conclusions from the symmetries vis-à-vis the status of the individuals (this includes whether or not they exist): any such conclusion will most likely have involved putting a certain amount of metaphysics in by hand. For example, if we are anti-haecceitists then the idea permuting individuals to produce distinct possibilities will not make much sense. However, this hasn't stopped many authors 'pulling metaphysical rabbits out of physical hats' (to use French's phase)! One such conclusion that has been drawn from similar arguments and reasoning is semantic universalism—van Fraassen's phase for the Leibnizian view that one can describe the world using only general, purely descriptive terms (i.e. without haec-ceities); or, in his own words, that "all factual description can be completely given in entirely general propositions" ([1991], p. 465).289 Before we see how semantic universalism has been argued for, by Stachel and van Fraassen, let us get a better handle on what it says.

Here's how Cover and O'Leary-Hawthorne express the content of the semantic universalist account—or "generalist picture", as they call it:

A general proposition is ...any proposition that is not singular, containing no individuals or haecceities; whatever determines the individual(s) that a general proposition is about, it does so indirectly via qualitative properties and relations the individual(s) happens to bear. General propositions can in a suitably rich language be expressed by sentences void of any directly referential devices such as proper names or indexicals; they correspond to sentences constructed solely from quantifiers, variables, qualitative predicates not expressing haecceities, and logical operators. ([Cover and O'Leary-Hawthorne, 1996], p. 4)

Now, it should be clear that if semantic universalism is true, then we lose out on transworld identity and de re modal discourse (cf. ibid., p. 5). The reason is clear enough: there are not the conceptual resources to say of some particular thing in world w (with the array of properties P') that it is the same as the thing in world w* with those same properties (or some different array of properties Qj). In adopting this view, one can resolve the hole argument problem and those like it issuing from

289 van Fraassen understands a proposition as a set of possible worlds (the 'truth value' of the proposition). If a proposition is "general" then one can permute the individuals in the worlds salva veritae (ibid., p. 469). See Teller [1998] for an excellent presentation and critique of van Fraassen's notion of semantic universalism. I expound elements of Teller's account in which follows.

symmetries. Again, the reason is clear: those problems stem from a consideration of permuting individuals in a way that respects the overall qualitative structure (i.e. the qualitative properties and relations). Obviously, if one grounds all (relevant) truths in qualitative properties and relations alone (ruling out haecceities) then one avoids these problems at a stroke. Such a move is essentially made by all sophisticated substantivalists.290 The structuralist positions outlined above follow a similar path. Thus, the notion of individuals being swapped around to give new possibilities simply does not make sense in the semantic universalist's account (cf. Cover and O'Leary-Hawthorne [1996], pp. 13-4). Of course, there are different ways of implementing this position. Stachel, for example, argues as follows:

the points of the manifold are ...individuated entirely by the relational structure specified by some solution to the generally-covariant field equations. Remarkably enough, the elementary particles are similarly individuated by their position in a relational structure. Each particular kind of elementary particle ... may be characterized in a way that is independent of the relational structure in which its exemplars are imbricated: by their mass, spin, charge, half-life, etc.; but a particular elementary particle can only be individuated its role in such a structure. The reason for this is ...the requirement that all relations between N of these particles be invariant under the permutation group acting on these particles. But the elementary particles and the points of space-time are the basic building blocks of our current model of the universe. ...If individuality has been lost at the level of depth to which we have currently penetrated in our physical theories, it is hard to believe that it will re-emerge if we succeed in penetrating to a deeper level in our understanding of nature. This suggests that we impose the following generalized permutation principle as a requirement on any candidate for a future (more) fundamental theory: Whatever the nature of the basic elements out of which it is constructed, the theory should be invariant under all permutations of these basic elements. ([Stachel, 2002], pp. 29-30)

Thus, Stachel is convinced that individuality (i.e. primitive or intrinsic individuality) is ruled against by the diffeomorphism and permutation invariance of general relativity and quantum mechanics. Rather, individuality is reduced to the role played by the particular parts of a relational structure, and permutations map parts to parts in such a way as to 'swap' roles. Indeed, properly understood, the idea of permutations of the parts simply do not make sense since the individuality of a 'part' is determined dynamically by the role it plays in the structure. Stachel further believes that this will be a feature of the future theory of quantum gravity, whatever it may be. Indeed, Stachel draws rather strong conclusions about the form such a theory will take, essentially ruling out loop quantum gravity and string theory at a stroke on pain of violating his generalized principle of permutation invariance (both retain a degree of background dependence) ([2002], pp. 30-1).

290 I would place Saunders and Stachel similarly. For example, the latter speaks of a "principle of general permutation invariance" ([2002], p. 15) in a way that suggests he has the generalist picture in mind.

van Fraassen draws superficially similar conclusions in the context of quantum statistics and the question of particle identity, but draws from them deeper anti-metaphysical morals. He argues that there is an equivalence between quantum field theory (without particles; with occupation numbers) and the 'many-particle' picture:

All models of (elementary [first quantized], non-relativistic291) quantum field theory can be represented by (i.e. are isomorphic to) ...Fock space model constructions. ...Since the latter are clearly carried out within a 'labelled particle' theory, we have a certain kind of demonstrated equivalence of the particle—and the particleless—picture. ([van Fraassen, 1991], p. 448)

He generalizes this argument using the concept of semantic universalism, as we have seen, defined by him as the thesis that all factual descriptions can be completely given by general propositions. In terms of possible worlds the view corresponds to the claim that a permutation of individuals at a world does not affect the truth values of propositions (defined as a set of such worlds). His idea is to show that whenever semantic universalism is satisfied by some theory there is an equivalence between interpretations ("pictures") with and without individuals; moreover, whenever this is the case, we can be sure that physics cannot decide which package to choose.

Now, van Fraassen's argument takes place within a highly idealized 'possible worlds'-type model M = (D, C, R). The models comprise a domain of individuals D and an array of "cells" C that are taken to represent qualitative differences— these are analogous to the distinct gauge orbits of a theory with gauge symmetry (redundancy). Thus, if two distinct individuals occupy distinct cells they will differ with respect to at least one qualitative property (each cell represents a maximally consistent set of properties). A possible world corresponds to a distribution of individuals across the cells. The standard modalities are defined by an access relation R between worlds (necessity: satisfied at all R-accessible worlds; possibility (contingency): satisfied at some R-accessible worlds). A model is then a collection of such worlds sharing the same domain, array, and access relation. If a model is closed under permutations operating on the domain (i.e. shuffling individuals about across the cells) then it is said to be "full". Regarding the relationship between worlds with and without individuals, van Fraassen writes:

The models required by semantic universalism are exactly those which can be described equally on either view. So far we have described them in terms of individuals. But each world—a mapping of individuals into cells of a logical space—can be characterized simply as a set of occupation numbers for the cells. Closure under permutation of the access relation R entails that the R-modalities operate on fully general propositions without losing the generalities. Therefore every significant proposition can be restated entirely in terms of occupation numbers. ([van Fraassen, 1991], p. 475)

291 The qualification van Fraassen gives here is crucial since the shift to relativistic quantum field theory renders the 'particle picture' much more dubious; indeed, in the curved space quantum field theories the notion of particle doesn't seem to make any objective, invariant sense at all (see [Wald, 1994]).

van Fraassen claims that the occupation number representation allows one to "abstract" a model without individuals as follows: "A world is a mapping of cells into natural numbers" (ibid., p. 476). This simply tells us how many instances of some properties there are. As he notes, there are many 'individual-full' models corresponding to a single 'individual-free' model; however, the equivalence class of individual-full models (i.e. the unique full model closed under permutations) does correspond to a single individual-free model. Thus, van Fraassen's strategy is to assume haecceitism in the initial characterization of the models—indeed, the definition of a world is tantamount to an endorsement of haecceitism. This haecceitism produces an inflated possibility set consisting of qualitatively indistinguishable worlds (i.e. differing solely in terms of how haecceities are distributed). The connection to my original definition of symmetry is obvious, as is the similarity to the kinds of symmetry arguments we have looked at in this book. However, van Fraassen's next move (to an occupation representation) is in effect to eliminate the haecceitism of the model by imposing a principle of permutation invariance on them. The principle of permutation invariance is taken to correspond to a principle of generality; that is, generality (of propositions) is given precisely by invariance of truth-value under permutations. Of course, this is just the notion of fullness of models defined above. Semantic universalism is then cashed out as the thesis that all models are of this kind. van Fraassen argues that there is an equivalence between these full models and the abstracted models; this is the result mentioned above: for models related by a permutation of individuals, we can mod out by the permutation (i.e. take the isomorphism class) to get a single abstract model. In taking the equivalence, of course, we are ignoring the underlying identity of the individuals. This leads van Fraassen to anti-haecceitism: the abstract model (without individuals) is equivalent to a unique full model (with individuals), so the notion of 'individual' on which haecceitism rests is not doing any work (ibid., pp. 475-6).

This is, clearly, rather similar to what I myself have been arguing for, and something very like it formed my criticism of French and Ladyman's ontic structuralism. What are the differences? For one, I grounded my discussion more concretely in the geometric spaces used to actually describe theories; but, more importantly, I don't see that there is equivalence in the sense in which van Fraassen means. Simply by attending to the nature of the geometric spaces we can see that there is a definite inequivalence between the space corresponding to that with symmetries (extended: with symmetric possibilities and individuals) and that without (reduced: without symmetric possibilities or individuals).292 Thus, the possibility structures of these interpretations are radically different; modal talk available to the latter is simply not available in the former. Teller makes a similar point with regard to van Fraassen's claim of equivalence between labelled particle [Hn] and occupation number [H+ ] pictures of quantum field theory293:

292 Another point to bear in mind is that the Fock space description allows for the construction of separable state spaces for quantum field theories. The many-particle formalism (with labels) has a state space of the form H = (£^=1 ^ H (where the Hi are individual separable Hilbert spaces); a basis in this space is given by an infinite (uncountable) sequence \n\,..., n(). Perhaps this is another way to break the deadlock? Possibly, but it isn't clear to me that this demonstrates a mathematical inequivalence. For a nice discussion, see Streater & Wightman [1964].

293 See also French ([1998], p. 107) and Butterfield ([1993] §5 and §6).

I agree that the "pictures" are empirically equivalent, in the sense that all the facts that actually arise can, one way or another, be described in either framework. But it seems to me misleading to parley the empirical equivalence into equivalence of pictures across the board. In cutting down from Hn to H+ we have lost expressive power. In Hn but not in H+ we can describe cases that never occur. This fact, in turn, shows that although the cases that do occur can be described in either picture, there is an important sense in which the descriptions are not equivalent. One picture—the one using labels—describes cases that do occur in terms of a conceptual framework that facilitates saying things that cannot be said with the resources of the other picture. ([Teller, 1998], p. 131)

This corresponds to my points made within the context of the geometric spaces in the previous chapter. van Fraassen in effect uses the satisfaction of semantic universalism by certain physical theories to ontological effect. I agree that quantum statistics satisfies the principle of semantic universalism in this sense: facts about particle role do not figure in quantum statistics. This is the content—on one understanding—of permutation invariance. I argued that much the same holds for gauge invariance and diffeomorphism invariance. The observables of the theories I have considered can naturally be seen as implementing van Fraassen's idea but I don't wish to be led from this into constructive empiricism: I believe there's more to the world (that we are entitled to believe) than the empirical structure (given the empirically significant propositions). But I don't derive this from physics. Rather, all I wanted to show was that an underdetermination even more pernicious than that charted by French and Ladyman pervades our best theories. There are a number of ways to understand it; in brief: the natural space for their structuralism is the reduced space but (1) this space is (non-modally) equivalent to the extended space and (2) the reduced space allows all of the options, in particular, it is isomorphic to those spaces that result from gauge fixing most naturally aligned with robust object views. This underdetermination thus swallows up (eliminative) structural realism too. For this reason I do not wish to be lead into their brand of elimina-tivism. I am nonetheless a realist about our theories; certainly not an entity realist; but not a structural realist of the French & Ladyman stripe either. I believe in the invariant structure revealed by the symmetries of our theories, but I don't see how one can say that this structure is all there is: underdetermination forbids it.

I see two main reasons to avoid the kind of view expressed by van Fraassen in the context of the arguments considered in this book294—the second is, I think, stronger than the first. Firstly, it seems that, inasmuch as the position is invoked to resolve a specific type of problem in the philosophy of physics, it is liable to seem a trifle ad hoc. Certainly, if this avoidance is the only thing one bases the position on, then the charge of ad hocness is most likely vindicated. However, the general metaphysical package that the position is taken to go along with, namely anti-haecceitism, is itself rather well motivated.295 If one happens to endorse some

294 As a matter of fact, my metaphysical intuitions draw me towards the generalist picture and the anti-haecceitistic position it intuitively recommends. However, what I hope to have shown is simply that neither the generalist picture nor anti-haecceitism are uniquely supported by the physics in question.

295 See, for example, Lewis [1983a; 1986a].

modal metaphysics that respects generalism, then one might have a satisfying response. However, this metaphysics will hardly be the output of the symmetry arguments. This brings me to the second, and more serious point and this is simply that the generalist picture is not incompatible with individuals nor with haecceitism per se. As regards the coexistence of individuals and generalism, I find Saunders' arguments convincing: one can stick to the idea that a purely descriptive lexicon is adequate and yet retain a notion of object, even absolutely indiscernible ones. But Saunders, of course, grants the anti-haecceitism: de re modal claims for such individuals simply don't make any sense (unless, perhaps, one incorporates counterpart theory)—ditto for Hoefer's primitive identity denying substantivalist.

van Fraassen argues for semantic universalism in a rather curious way; he begins by assuming haecceitism in his semantic models (toy possible worlds models) and then imposes a principle of permutation invariance on these models, concluding that the latter principle enforces anti-haecceitism. It doesn't: I have argued at many places in this book that the principles of invariance can be accommodated with and without reduction, symmetries, objects, and haecceities.296 In other words, the principle of permutation invariance that van Fraassen's views as granting anti-haecceitism is perfectly compatible with individuals; one simply understands it as a principle of indifference concerning the structure: the (qualitative) structure is indifferent to which individuals play which roles. Thus, van Fraassen might be right that his principle of permutation invariance erases any reference to particular individuals, but that does not imply that one can or should dispense with individuals. Indeed, one can, but one need not; whatever option one chooses here is underdetermined by the physics, by the symmetries.

The observables of the theories we have considered do not distinguish between role swaps concerning individuals related by the relevant symmetries. There are a number of options, as we have seen. We can say that the objects are simply individuated by their role in the structure, as delineated by the observables (via gauge invariance). We can say that, although the swaps are not empirically significant, they none the less occur as distinct states and so at least have some conceptual significance (e.g. the individuals have haecceities). Or we can do away with the objects entirely, in which case the symmetries have a very different meaning.297

All of the above options have their plus points. None of them are dictated by the physics. We therefore need to be honest about what our theories (of the kind I have considered) can tell us about the world. In particular, they cannot tell us whether objects of the specified kind are individuals or non-individuals e.g. (without haecceities) and they cannot tell us if they even exist or not. Where does this leave the aspiring realist (which I claim to be myself)? I don't think panic stations need ensue for the simple reason that the symmetries, although clouding the question of objects, nonetheless provide us with definite physical

296 We have also seen how Brighouse's position allows for the irrelevance of haecceitism in considerations of physics without thereby ruling it out in principle.

297 Clearly they cannot be seen any longer as operating on objects. Indeed, I think that the question of how to understand (conceptually) symmetries is one of the most pressing problems facing advocates of the French and Ladyman ontic (elimi-nativist) position. This is made all the more pressing since invariance under symmetries is utilized to 'get to the structure'. Perhaps, however, the notion of using individuals as a 'heuristic crutch' can accommodate this (see, for example, French [1999])—still, the details need spelling out more than they have been thus far.

structure that is in principle measurable, invariant, and objective.298 I espouse a view that takes this structure as an aspect of the world—treating spacetime and matter as simply different aspects of the world, a la Stein, thanks to the ubiquitous underdetermination—but stays silent on the extent of this structure, on the issue of whether this structure is sufficient to 'cover' what there is (call this position 'minimal ontological structuralism'299). I for one think that there is a whole lot more, but honesty constrains these other aspects to the box marked 'metaphysics'; beyond the structure revealed by symmetries the thread holding together physics and metaphysics snaps. Decisions concerning the latter then have to be guided by something else.300

Thus what we are left with is a rather deflationary stance according to which our best theories of physics cannot furnish us with any information beyond what is contained in the structure of observables, but the structure of the observables themselves forbids reading entity realist or structural realist positions from this with good conscience. Perhaps the debate has run out of steam? I hope not and, in fact, I think that Eddington's structuralist position fits this state of affairs quite well.

Eddington eschews the question of priority when it comes to relations and relata (cf. French [2003], p. 233):

The relations unite the relata; the relata are the meeting points of the relations. The one is unthinkable apart from the other. I do not think that a more general starting-point of structure could be conceived. ([Eddington, 1928], pp. 230-1)

Thus, writes French, "Eddington did not regard the structure as ontologically prior ...[rather,] both relata and relations, structure and entity are ontologically entwined in that each is necessary to make sense of the other" (ibid., p. 257). Now, French draws attention to a criticism of Braithwaite to the effect that the relata must be originally (i.e. intrinsically) distinguishable in order to be distinguishable within the relational structure; that is, they must have existence independently of the relational structure they find themselves enmeshed within. Of course, as French notes, Braithwaite simply misses the point: the properties conferred by the relational structure are all there is to the relata. But this does not imply any priority of the relational structure over the relata, "the two came as a package" ([2003], p. 235). Of course, Braithwaite can be seen as employing the 'no relations without relata' objection to the ontic structuralist realist approach.301 Eddington answers

298 Not all of this structure is necessarily measurable in practice by us. Thus we are not restricted to the realm of the 'observable' in van Fraassen's sense. We nonetheless have 'access' to the structure through the observables (in the gauge theoretic sense).

299 This is not to be confused with van Fraassen's notion of "moderate structuralism" ([2006] 278) for that position requires an underlying substance that is the "bearer" of structure.

300 Thus, here I diverge from constructive empiricists who take empirical adequacy of theories as paramount and regard any claims that go beyond the empirical realm as out of bounds as far as belief is concerned (see van Fraassen [1980] 12). Indeed, my choice of geometric space was made to keep all of the options open where extra-empirical matters are concerned. I have no desire to wave "good-bye to metaphysics" (van Fraassen [1991], p. 480), and do not see that the underdetermination warrants it. But I also diverge from ontic structural realists who cut out any 'extra-structural' elements from their ontology as a matter of general principle.

301 Of course, the objections of Unruh and Kuchar to the (timeless) correlation responses to the problem of time are in the same vein. The idea there was that one could not have a correlation without things that are correlated. Of course, French it in a different, and I think correct, way to the ontic brigade. The latter simply eliminate relata and argue that it is the relations that have ontological clout, not the relata. They argue for this using the underdetermination the afflicts the metaphysical positions based on objects. However, I have shown that the elimination of objects that the ontic structuralists employ is likewise underdetermined by the physics. Thus, Eddington's response is seen to offer the better stance;302 what's more, as I showed in the previous chapters, it has the resources to deal with the problems of space and time of classical and quantum gravity when aligned with correlation-type approaches to the observables. Before we wrap things up, I first wish to further separate the position I have outlined from van Fraassen's position, thus making my own position more transparent.

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