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C5: Methodological and Philosophical Issues in Physics This paper evaluates how the symmetry problem that has recently beenraised [2] for Friedmann-Lemaˆıtre-Robertson-Walker spacetimes in generalrelativity (GR) and according to which spacetime structuralism seems com-mitted to the claim that the universe has no spatial extension—it consistsof one lonely point—is transposed into the causal sets approach to quantumgravity. It is shown that the introduction of an irreflexive relation does not,in general, resolve the difficulty. Even though the problem resurfaces incausal set theory, however, there is reason to believe that—unlike in GR—itdoes not affect physically relevant models of the theory.
Spacetime structuralism rejects an ontological commitment to individualspopulating the fundamental level in favour of a reading of spacetime asa structure, i.e. as an ordered pair O, R consisting of a set of concretephysical relations R which take elements of the set O as their relata. Whatmatters to the structuralist is that elements of O do not possess intrinsicproperties, but assume their identity only by virtue of their position in thestructural complex O, R .
It has been shown [2] that the important family of cosmological mod- els in general relativity (GR) called Friedmann-Lemaˆıtre-Robertson-Walker(FLRW) spacetimes challenge this structuralist interpretation of spacetime.
An important feature of these spacetimes is that they encode the demand issued by the Cosmological Principle that there be no privileged spatial loca-tion in the universe. In other words, there cannot be any physical propertyhad by any point in the universe that is not also had by all other pointsat the same cosmological time. If a point cannot differ in its properties,including its relational properties, from any other point in space, then thePrinciple of the Identity of Indiscernibles (PII) appears to demand that theybe identified. PII holds that for any two objects, if they share all—possiblyrelational—properties, then they are identical. If sound, this argument es-tablishes that the spacetime structuralist is committed to the absurd claimthat FLRW spacetimes consist of only one point for any given value of cos-mological time—a pointlike universe! It has been suggested, most vigorously in [1], that the introduction of ir- reflexive relations resolves the difficulty insofar as these relations render therelata weakly discernible, i.e. the relata are numerically distinct by virtueof them exemplifying an irreflexive relation. The symmetry problem disap-pears since for an irreflexive relation to be exemplified at all, there must betwo numerically distinct objects. There is the general worry with this reso-lution, of course, that the assumption of there being an irreflexive relationexemplified in the physical structure at stake means, eo ipso, that there aretwo numerically distinct objects exemplifying the relation. If the point ofthis resolution was that numerical plurality was to be derived, rather thanstipulated, then it seems to fail.
However, GR is not the final word on gravity. Rather, it will be re- placed by a quantum theory of gravity. In the absence of a complete andempirically adequate quantum theory of gravity, we may nevertheless gleansome lessons from an approach to quantum gravity which is notable for be-ing conceptually particularly clean and simple: the causal sets approach. Itpostulates that the fundamental structure is a causal set, i.e. an orderedpair C, ≺ consisting of a set C of elementary and intrinsically featurelessevents and a relation defined on C, physically interpreted as causal—andhence temporal—and denoted by the infix ≺, which is irreflexive, antisym-metric, and transitive. It is further assumed that the fundamental structureis discrete. It is natural to interpret this theory structurally.
The only physically admissible relation is irreflexive. Ignoring the general worry above, the structuralist may therefore hope that she has been handedthe necessary tools to deal with the symmetry problem. Upon inspection,however, one quickly notes that the irreflexivity of the one and only physicalrelation only affords a partial resolution of the problem. Suppose the causalset exhibits a high degree of synchronic symmetry in that every element ina given ‘horizontal’ generation Gi exemplifies exactly the same relational properties as any element of the same generation, as depicted in Figure1, where points represent the elements of C and the arrows between themthe relation ≺.
There is no problem of diachronic plurality, since ≺ is antisymmetric. But just as forthe FLRW spacetimes, it seems as if on apurely structuralist reading, and using PII, for all the generations Gi, allthe points in it ought to be identified.
It is worth noting that in slightly less symmetric cases, such as the one of a ‘growing’ spacetime in Figure 2, not all points within a generation oughtto be identified—even though there are subsets of points in each generationthat share a relational profile. This may serve as an indication that thestrict symmetry is perhaps more fragile that in the classical case.
There is an important sense, however, in which the problem is not as grave as in GR. How generic are these highly symmetric causal sets amongall the dynamically possible ones? Even though this is hard to tackle an-alytically, one naturally conjectures that the fraction of highly symmetriccausal sets quickly becomes very small as the cardinality of C increases.
Analogously, the symmetric spacetimes ` zero in the space of all admissible spacetime models. To take this as groundfor dismissing the difficulty, however, was particularly unpalatable since theFLRW models are of such great theoretical importance [2].
In causal set theory, there is no reason to think that these highly sym- metric possibilities are of particular physical relevance. In fact, there aregrounds for thinking that they are not: even FLRW and similarly sym-metric spacetimes of GR generically emerge from causal sets not exhibitingthese perfect symmetries. That this is so is suggested by the fact that afaithful embedding of a causal set into a manifold can straightforwardly beobtained by randomly sprinkling points (at an appropriate density) into anFLRW spacetime and asserting causal relations between any pair of themjust in case they are timelike related. This will generate, ex constructione, a causal set that is faithfully embeddable into the FLRW spacetime. Therandom sprinkling, importantly, is required to guarantee the local Lorentzsymmetry, as a sprinkling onto a regular lattice would not leave the averagenumber of points in a volume invariant under Lorentz transformations.
The fact that causal sets thus necessitate a certain irregularity gives the structuralist reassurance that the symmetry problem vanishes in causal settheory. This, in turn, indicates that the problem may disappear altogetherin a full theory of quantum gravity.
[1] F.A. Muller, Phil. Sci. 78 (2011): forthcoming.
uthrich, Phil. Sci. 76 (2009): 1039-1051.

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