Symmetries, Structure and Determinables

Consider the so-called Standard Model, which has been the subject of much dis­cussion in the popular and philosophical literature, especially following the dis­covery of the Higgs boson.

Furthermore, the Standard Model is a gauge theory, represented by the group SU (3) x SU(2) x U(1) via which further relevant symmetries can be captured within the theory. What this means, broadly speaking, is that the Lagrangian of a system— which basically captures the dynamics—remains invariant under a group of transformations, where the ‘gauge' denotes certain redundant degrees of freedom of that Lagrangian. Thus, consider electrodynamics, for example, for which U(1) above is the relevant gauge symmetry group associated with the property of charge and the photon (a gauge boson) effectively drops out of this requirement that the theory be gauge invariant. Extending this requirement to the other forces, we obtain, for the weak nuclear force, the SU(2) symmetry group associated with isospin, a property of protons and neutrons, and for the strong nuclear force, SU(3) associated with the colour property of quarks.

That, crudely sketched, is the relevant exemplar. Now, it has been argued that the appropriate realist stance that should be adopted towards this exemplar is that of the structuralist, where the metaphysical notion of ‘object’ is at best set on a par with that of ‘relation’ (Ladyman and Ross 2007) or removed from the picture altogether in favour of a fundamental conception of ‘structure’ (French 2014). The obvious question that has been asked (repeatedly) is ‘What is that structure?’, or putting it more generally, ‘What is the world like, if it is structural?”. Again, one answer would be to write out the details of the Standard Model on a whiteboard and pointing, insist ‘It is like that!’. As before, this yields a thin sense of metaphysically informed understanding. An alternative is to attempt some form of bespoke account. Thus Eddington, before he went off the metaphysical deep end as it were, expressed such group theoretically described invariances in terms of ‘patterns of interweaving’, which at least is evocative if not perhaps very precise (and perhaps not really very bespoke, given the connotations associated with ‘weaving’!).

Instead we might appeal to certain devices in the metaphysical toolbox to help capture the nature of ‘structure’ in this context. So, consider the way in which the fundamental properties from ‘being a fermion’ to charge and spin ‘drop out’ of the above symmetries. This is a core feature of this structuralist view: rather than con­sidering the world as built from the bottom up, as it were, beginning with objects that have properties, between which there hold relations, which are expressed by laws, that are constrained, in some sense, by these symmetries, the structural realist inverts that order, and sees the relevant metaphysics as proceeding from the top down, so that we take the symmetries and laws as fundamental, and the properties to be derivative.

This has been extensively discussed of course (for an excellent overview of the various positions, issues and concerns that have been raised, see Wilson forth­coming) and the central idea is that determinables and determinates stand to one another in a certain specification relation, as the determinable ‘colour’ does to the determinate ‘red’, or the latter as determinable does to a particular shade of red, or as mass, qua determinable, does to a specific mass value. Part of the extensive discussion here has focussed on the nature of this relation but the crucial point is that it relates properties that are more or less specific, relative to one another; so, ‘red’ is more specific than ‘colour’ and a particular value of mass is more specific than ‘mass’. ‘Increased specificity’ is just one of the features of the determinable- determinate relationship that Wilson helpfully lists (ibid., pp. 8-9). Others that also motivate its deployment in this case include: ‘determinate incompatibility’, according to which if something has a certain determinate of a given determinable, then it cannot at the same time have a different determinate of that determinable (at least, not of the same or lower specificity); ‘determinate opposition’, according to which different determinates of the same determinable are not just incompatible but are relevant alternatives (so ‘red’ and ‘blue’ are determinates of ‘colour’ but ‘red’ and ‘square’ are not); ‘requisite determination', which requires that anything that has a given determinable, must have some determinate of it; and ‘asymmetric dependence’ which states that for any determinable of some determinate, anything that has that determinate must have that determinable, but something could have that determinable without having that determinate (so anything that is red must be coloured but something coloured, may not be red of course).

Now, the suggestion is that we can apply this metaphysical tool to the case of the symmetries that the structural realist takes to be a fundamental feature of the structure of the world, in the sense that we regard such symmetries as relational determinables generating determinable properties and associated determinate val­ues. Thus, consider the permutation group, mentioned above: this encodes a range of possible particle statistics, but in this world it appears that only two of those determinates are manifested, namely those corresponding to the kinds fermion and boson (yielding Fermi-Dirac and Bose-Einstein statistics and characterised by anti-symmetric and symmetric state functions, respectively). Likewise, the sym­metry of relativistic space-time, characterised by the Poincare group can be regarded as a determinable which also yields spin as a property-determinable, which in turn yields the property spin /2, associated with the electron for example, as a determinate. Again, it is through the determinable (with the emphasis on the - able) that the relevant possibilities are encoded (French 2014, p. 283). So, being a fermion, say, is more specific than being subject to permutation symmetry, so we have increased specificity; and being a fermion and being a boson are not just incompatible but are relevant alternatives; and anything that is subject to permu­tation symmetry must behave according to some particle statistics, whether fer­mionic, bosonic or, but not apparently in this world, parastatistical. Finally, of course, anything that is a boson is subject to permutation symmetry but something that is subject to the latter may not be a boson—it could be a fermion, for example.

Now applying this device to help flesh out the metaphysics of structure raises a number of issues. First of all, some have argued that increased specificity feature implies that determinates must be metaphysically prior to or more fundamental than determinables and if this were accepted, we could not take permutation symmetry to be prior to and more fundamental than bosonic or fermionic statistics.

Not only can we use metaphysics as a constructive tool, but we can also use it as a contrastive one. Thus to get a (hopefully) clearer picture of the view being presented, and of the way in which the determinable-determinate relation can help as a metaphysical tool in fleshing out that picture, let us compare it to Paul's recent development of a ‘one-category' ontology (2012; 2013).

She begins with the core question that obviously resonates with the structural realist: ‘What is the fundamental structure of the world?' (2012, p.

So, for the ‘building rule' she takes composition, on the grounds that we have a direct, intuitive grasp of proper parthood which forms the heart of the composition relation. Here immediately the likes of Ladyman might object that such intuitions, based as they are on naive view of ‘everyday' objects or, at best, classical mechanics, fail utterly when it comes to modern physics where the notion of being a part of is much slipperier and harder to grasp.

However, when it comes to the fundamental categories, Paul does draw on certain features of quantum physics to argue, first, that we should reject what she calls the ‘traditional spatiotemporal view' that runs throughout much of contem­porary metaphysics and which ‘...takes some or all of the fundamental constituents of the world to be spatiotemporal parts, i.e., chunks of spacetime, many of which are qualitatively rich, and the building relation to be spatiotemporal composition.' (ibid., pp. 233-234). And here she acknowledges that the fault of such a view is that it conflates the metaphysics of the everyday, or ‘manifest image' with that of ‘the real' (ibid., p. 239) Secondly, and more importantly for my purposes, she maintains that we can still retain ‘the world-building relation' but now applied to a different set of fundamental categories.

Thus Paul collapses the category of property into that of substance (2013). On her view the world is built from n-adic properties via property composition, which effects a kind of fusion or bundling (2012, p. 242) and since this ‘mereological bundle theory' does not require this fundamental category to include spatio-temporal properties, it can accommodate a much broader range of possi­bilities when it comes to the nature of the fundamental entities. And since ‘... every fundamental physical theory ever given, including all of those currently on offer, is or can be couched in terms of properties and relations, even if these properties and relations are extremely abstractly specified...' (ibid., p. 245), such theories will mesh with this particular metaphysics.

This is certainly an attractive metaphysics and it is for that reason that it acts as a useful contrastive tool. The contrast, of course, comes from Paul's reading of physical theories as couched in terms of properties and relations and the con­comitant insistence on a metaphysical ‘bundling' or property composition relation. Although the former is obviously true to a certain extent, this reading omits the crucial role of laws and symmetries. Indeed, if we take this role seriously in the context of the Standard Model say, then it would seem that although Paul has gone some way in the right direction by dropping spatio-temporal composition, she still retains an overall ‘bottom-up' approach. At the very least mereological bundle theory needs to be able to accommodate the relationship between symmetries and kinds, as in the case of permutation symmetry and the boson/fermion distinction, and properties, as in the case of Poincare symmetry and spin, say.

Paul herself explicitly invites the structural realist to adopt her mereological bundle theory, on the grounds that, ‘. structuralists can make good use of an n-adic property mereology, since they don't need substances or even monadic properties in order to construct the world.' (2012, p. 248; see also 2013 pp. 110­111). Indeed, she suggests, such a marriage would lead to a ‘super-sophisticated structuralism' (2013, p. 111) that avoids certain of the problems that its less sophisticated form is held to face.^[75] The idea then is that we take relations as constituting our fundamental base and then apply bundling as the appropriate building relation, thereby effectively constructing the structure of the world via fusion (Paul 2012 p. 245), with putative objects as ‘nodes' in this structure, or as Cassirer put it, as ‘intersections' of these relations.

Understood this way, mereological bundle theory would be in effect a further metaphysical tool that the structural realist could use (see French 2014, pp. 186-189). However the issue of how to accommodate symmetries remains. If they are viewed as merely ‘by-products' of laws, expressing certain features of the latter then with laws themselves expressing the relations that sit in the fundamental base, mereological bundle theory might seem the appropriate metaphysical device for accommodating the relevant structure construction. On this account, each such relation would exhibit a certain feature that when ‘fused' to create the network of relations that the laws of physics describe manifest the global features that we describe via symmetries. Of course, further work is required to ‘mesh' this meta­physics with the physics.

In particular it might be objected that in the practice of physics, symmetries act as constraints on laws, or as ‘meta-laws', which suggests more of a ‘top down' stance, in contrast with Paul's. Now of course, one could respond that this might be correct when it comes to the heuristic use of symmetries but that doesn't require that they be regarded as ‘standing above' laws, metaphysically speaking. A third way between these two extremes is to follow Cassirer and take symmetries, laws and measurement results as being on a par and together constituting the structure of the world (French 2014). This removes the necessity of adopting either a ‘bottom up' or ‘top down' stance towards symmetries but of course the relationship between them and the properties typically taken to be monadic must still be accommodated. In particular, although it might be regarded as merely ‘loose talk' to say that a property such as spin ‘drops out' of Poincare symmetry, the close relationship here on the physical side needs to be matched by a similar relationship on the metaphysical.

Consider again permutation symmetry and the distinction between bosons and fermions. One could begin with that distinction as fundamental, so that quantum entities possess ‘being a boson' or ‘being a fermion' as kind properties. Bundling such properties together yields the relevant feature of assemblies of such entities that is represented by either the bosonic or fermionic representation of the per­mutation group, respectively. And the fact that this group yields other representa­tions, corresponding to paraparticle statistics, for example, is primarily of mathematical rather than physical significance—unless of course, such statistics turns out to be physically realized (as was suggested for a short time in the case of quarks), in which case the relevant group representation would be applied (so this comes down to an issue in the applicability of mathematics), but metaphysically, of course, according to mereological bundle theory we would still begin with ‘being a paraparticle (of a certain order)' and build up from that.

The alternative is to begin with the symmetry itself as part of the fundamental base and take the ‘dropping out' of bosonic and fermionic statistics to express the relationship between this symmetry, as part of the fundamental structure of the world, and these kind properties. In terms of the mathematics this amounts to no more than the relationship between the group and its representations but this obviously needs to be matched on the metaphysical side. Here the determinable- determinate relationship seems to do the job, so that ‘being a boson' or ‘being a fermion' are simply determinate aspects of that partly determinable structure. Note the further contrast with mereological bundle theory: instead of a ‘building relation' we have something akin to a ‘manifestation relation' and instead of thinking of the structure as built up from certain parts (even if these are properties and relations rather than objects and substances), we are invited to think of it as given holisti­cally, as it were, and as manifesting certain determinate features.^[76]

Another way of seeing the contrast between this and Paul's approach is to consider the question of what should be our attitude towards the other possible properties, such as ‘being a paraparticle' for example. According to mereological bundle theory we begin our construction with the properties that we actually dis­cover in the world, such as ‘being a boson.' We represent those properties math­ematically via group theory and we find that such mathematics includes alternatives that do not appear to be realized—from this perspective these are just so much ‘surplus structure'. According to the alternative, this ‘surplus' represents certain possibilities which may or may not be actualized and the determinable nature of the structure flags the point that it encodes such possibilities. Thus rather than begin­ning with the actual, and building up from that, we begin with what is modally allowed and show how the actual world fits into that, as a determinate manifestation of that modally informed structure.

There is more to say here, of course, but this is perhaps enough to highlight the differences between these metaphysical tools.^[77]

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