Structuralism and Symmetry

Poincare is generally taken to be the arch-epistemic structuralist about physics. In an oft quoted passage he claims that physical theories "teach us ...that there is such and such a relation between this thing and that", and that in naming these "things" we only name "images we substituted for the real objects which Nature will hide for ever from our eyes. The true relations between these real objects are the only reality we can attain ..." ([1905], p. 161). The crucial thing to note about this passage is that it clearly expresses an unbridgeable gap between epistemology and ontology: there are real objects external to our minds, but the only access we have to them comes in the form of relations between them. Since the relations express something true about the world we can gain restricted knowledge about it, but only about it's structure. The Kantian sentiment of this stance is unlikely to impress many realists, for it implies that our theories do not and cannot tell us anything about the nature of the fundamental elements of the worlds, only about their relational structure; we are forever prevented from gaining complete knowledge about the natures of objects.

A perhaps more satisfying structuralist position for the realist to adopt is an on-tological version of structuralism, according to which the Kantian notion of 'Nature hiding real objects from our eyes' is replaced by the notion that, 'fundamentally, there just are no objects' (see Ladyman [1998]). The true relations that we do have access to exhaust the ontology. The ontology is one of pure structure, and we gain knowledge of this structure by observing relations. But the relational structure does not supervene on a set of noumenal objects as per Poincare's epistemic perspective. Instead, we say that the structure is ontologically subsistent. It might be that the structure is ontologically 'prior' to or ontologically 'on all fours' with objects (relata), that are then characterized as intersections or nodes in the relational structure. Or it might be that objects simply drop out of the ontology altogether at a fundamental level. According to this stance our theories give us knowledge of reality. This is bound to not satisfy many philosophers, who will wish for a more robust characterization of this ontology of structure. This will generally amount to a desire for a reduction of structure to something else. Perhaps the only way to deal with these requests is to paraphrase Lewis ([1975], p. 85), who faces a similar problem with regard to his treatment of possible worlds: Structures are what they are and not some other thing! Fortunately, I think one can do better than this; it is symmetry that is the key to structure but the structuralism that follows is one of extreme deflation metaphysically speaking.

Let me quickly distinguish structuralist positions from relationalist positions; many philosophers have a strong tendency to conflate the two. Recall that relationalism was characterized by a material dependency thesis. The idea was that a certain structure can be given a relational description if it can be reduced to a family of relations on a primitive set of material objects. The ontological primacy is held by the set of material objects such that without them the relational structure would not exist at all. Hence, though one can be a realist about the structure, one must view it as being supervenient on (reducible to; dependent upon) some more primitive facts involving objects. Substantivalism also involves a dependency thesis and a set of primitive objects, though in this case the objects are spacetime points and it is their relations to one another that determine the overall structure. On the other hand, according to structuralism, the relational structure is not characterized by any dependency thesis at all. In fact, if anything, the dependency works the other way around (though I will argue against this below): objects are reduced to nodes in the relational structure. We might, if we wish to cling to the idea that structuralism expresses a form of relationalism, distinguish between two forms: reductive relationalism, corresponding to what I have been calling relationalism, and non-reductive relationalism, corresponding to what I have been calling structuralism. I believe that this is what Saunders [2003a] has in mind.

I have been suggesting that we should adopt a structuralist position with respect to the physics of general relativity and quantum gravity (and theories with gauge-type symmetries more generally) for two key reasons. The first is the un-derdetermination of metaphysics by physics—e.g. as brought out by French in the context of quantum statistics [1989].268 This was seen to be a fairly ubiquitous phenomena in mathematically dense physical theories, and we found counterparts in classical mechanics, electromagnetism, gauge theory, general relativity, and quantum gravity. The second reason concerns the role that symmetries play in each of theories just mentioned. Whether we use an interpretation that retains symmetries or not, it is evident that they should play a key role in determining the ontology. We use symmetries to get to the invariant structures and these structures comprise what is physically observable. The physically invariant structures of general relativity and quantum gravity are just the diffeomorphism invariant ones (orbit constant quantities, or quantities that commute with the first class constraints in the canonical, Hamiltonian case).269 The main interpretive debates that rage in these theories (and any other theories with symmetries on the state space) concern the status attributed to the symmetries: does one retain the symmetry or factor it out? does one quantize with the symmetries or factor them out? do the objects related by symmetries get eliminated if one factors the symmetry out? does one view the possibility set generated by the symmetry as representing physically possible worlds one-to-one, many-to-one, or not at all? I argued that in large part the issue of the reality or ontological nature of certain elements of the theory was secondary to one connected to modality and transworld identity. Once this is realised, there is

268 French [2001] also makes use of the 'covering space'/'intrinsic space' underdetermination in the context of spacetime theories (a favourite example of conventionalists).

269 Recall that the constraints encode in the (3+1)-dimensional, canonical context the 4-dimensional spacetime diffeo-morphism invariance of the covariant theory. The diffeomorphisms comprise the gauge freedom of general relativity.

seen to be underdetermination between incompatible stances concerning ontology virtually across the board as far as theories with symmetries are concerned. The underdetermination is mathematical too, and I showed that there can be no (internal) formal basis for choosing one of these stances over any other. This leaves us without an internally well motivated metaphysical, formal, or physical basis on which to base our interpretations.270 Hence my desire for a somewhat 'neutral' position between ontological deflationism (without objects) and inflationism (with objects—be they individuals or 'non-individuals').

In this book I have tried to bracket the issue of realism as much as possible; I took both relationalists and substantivalists to be espousing realist positions about spacetime, differing merely in how it is to be understood: spacetime points versus relational structure supervening upon material objects. Structuralism too is to be understood as a realist position differing from each of the other two positions in that the focus is shifted to the relational structure itself. All three types of position will generally agree that the structure in question is mind-independent, and that qualifies them all as espousing realism (however, minimally construed) in my book.271 However, just as there are many quite distinct views that sit under the banners of 'relationalism' and 'substantivalism', so there are many flavours of structuralism. In this section I outline three types of structuralism (about spacetime), and show how they connect up to the issues discussed in this book. I show that each is flawed in their own quite distinctive way, and proceed to diagnose the problems. I then outline the version I think has most chance of success. The version I present focuses on the nature of symmetry and observables and the precise way that these two notions interact in the context of the problems considered in the current and previous chapters.

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