Abstract
We address a recent proposal concerning ‘surplus structure’ due to Nguyen et al. (Br J Phi Sci, 2018). We argue that the sense of ‘surplus structure’ captured by their formal criterion is importantly different from—and in a sense, opposite to—another sense of ‘surplus structure’ used by philosophers. We argue that minimizing structure in one sense is generally incompatible with minimizing structure in the other sense. We then show how these distinctions bear on Nguyen et al.’s arguments about Yang-Mills theory and on the hole argument.
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Notes
For a discussion of the role of this theme in Earman’s thinking on the hole argument during the 1970s and 1980s, see Weatherall (2018c) and references therein.
Earman, for instance, proposed moving to Einstein algebras (Geroch 1972) as a suitably ‘relationist’ alternative to standard formulations of general relativity (Earman 1986a, b, 1989a, b; Rynasiewicz 1992; Bain 2003; Rosenstock et al. 2015). Similar issues are at stake when, for instance, Rovelli (2006, p. 31) argues that the manifold is ‘a gauge artifact’ in general relativity or Smolin (2000, p. 5) argues that there are no points in physical spacetime. We take these arguments to assume, often implicitly, that something like manifold substantivalism is the ‘default’ interpretation of the standard formalism, and to avoid that interpretation, one needs a formalism with a different, weaker metaphysics as its ‘default’ interpretation. (Other authors, such as Field (1984) and Friedman (1983), offer more direct arguments for positions similar to manifold substantivalism on the basis of the standard formalism of general relativity.)
Weatherall uses the expression ‘excess structure’; nothing turns on the difference between ‘excess’ and ‘surplus’ here.
Nguyen et al. (2018) focus on Yang-Mills theory, but if Weatherall’s argument fails there, it will fail in general relativity too; indeed, Weatherall (2016c, 2016a) has argued that, at least in this connection, Yang-Mills theory and general relativity are strongly analogous, and neither has surplus structure.
Apparently bolstering their case, Prop. 2 of Weatherall (2016c) is false as stated (Weatherall 2018a), leading to the surprising conclusion that, on Weatherall’s account, theories that are often taken to have ‘surplus structure’ actually have less structure than ones that are said not to have surplus structure (as opposed to being equivalent, as Weatherall originally claimed).
See, for instance, the Abstract and Introduction of their paper—and, indeed, the title, which makes sense only insofar as one might have initially thought surplus structure were superfluous. We emphasize this point because an anonymous referee suggests that Nguyen et al. (2018) may not have intended to reject the maxim that one should always minimize structure, but we think the plain meaning of their texts suggests otherwise.
To be sure, we are not in the business of policing language: Nguyen et al. are clear and precise about what they mean by ‘surplus structure’, and we think that, on their understanding of the expression, their argument is compelling and insightful. Our point, rather, is to clearly distinguish two different, nearly opposite, meanings of an expression both of which seem to be in use in the literature, and to emphasize that showing that surplus structure in one sense is not eliminable does not imply that surplus structure in other, very much distinct, senses is also not eliminable.
These ideas were introduced to philosophy of science by Weatherall (2015) as a means of comparing different physical theories, following a suggestion by Halvorson (2012)—though similar ideas have long been used in mathematics. A review of applications of this approach—termed ‘Theories as Categories of Models’ by Rosenstock (2018)—is given in Weatherall (2017). For background on category theory, see Mac Lane (1998); for a gentler introduction, see Leinster (2014).
Barrett (2018) and Rosenstock (2018) both give reasons why, in some applications, it is important to consider more than just isomorphisms; for present purposes, little turns on whether one considers categories with a broader notion of arrow, as long as one does so consistently across all theories under discussion.
We observe that there is an \(n-\)categorical perspective on this classification, where each of these three notions of ‘forgetting’ correspond to forgetting structure at different ‘levels’: forgetting properties means forgetting 0-structure; forgetting structure means forgetting 1-structure; forgetting stuff means forgetting 2-structure; and so on, where one extends these notions to a hierarchy of ‘essentially k-surjective’ functors between \(n-\)categories (Baez and Shulman 2010). This alternative perspective may make it seem as if all of these notions of ‘forgeting’ correspond to different kinds of ‘structure’ that may be forgotten. But what is important to emphasize is that, as we discuss below, it is 1-structure that most naturally corresponds to what is usually meant by the structure of a mathematical object or model of a physical theory.
We remark that, although this is fair to say, adopting this prescription for what should be meant by ‘surplus’ does not recover common claims that gauge theories exhibit ‘surplus’ anything—because, as Nguyen et al. (2018) go on to argue, what would putatively be ‘surplus’ in such cases is ineliminable.
One way of thinking about what we have done here is to choose a particular (partial) reference relation. We have not added any structure to V; we have just made a choice of mapping from V to the world.
Observe, however, that we do not have the same freedom for choosing ‘South’, once we have chosen ‘North’. In fact, a choice for ‘South’ is (essentially) fixed by our choice of ‘North’: ‘South’ must be represented by (some positive multiple of) \(-v\), since it is essential to ‘South’ that it be the opposite direction of ‘North’. We can drop the parentheticals if we adopt the convention, as we will in what follows, that ‘directions’ are all represented by vectors of the same length. But nothing that has been said thus far forces us to do this.
We acknowledge that the expression ‘as we usually understand them’ is doing a fair amount of work, here. In particular, we have fixed a meaning for ‘orthogonal’ in both the mathematical context and in the world, and we are insisting that whatever reference relations we adopt regarding which vectors represent which direction respect those meanings.
Again, we note that the sense of ‘requirement’ here turns on the prior assumption that any suitable reference relation for vectors in the present context should respect the plain meaning of terms such as ‘orthogonal’.
Assuming fixed conventions, shared across members of the comparison class, concerning what reference relations are acceptable.
This view is defended by Barrett (2014).
This intuition can be made precise in various contexts, including the first order case, where adding further relations to a theory (for instance) reduces the number of symmetries of its models. See, for instance, Barrett (2018).
It is perhaps worth noting that the same intuitions play out in standard discussions of classical space-time structure: Newtonian space-time has more structure than Galilean space-time, which has more structure than Leibnizian space-time; this is all reflected by the fact that Leibnizian space-time has more automorphisms than Galilean space-time, which has more automorphisms than Newtonian space-time. These relationships are described in somewhat more detail by Barrett (2015) and Weatherall (2016b) in a way that connects directly to how we discuss them here, though they were already recognized and well understood by, for instance, Stein (1967) and Earman (1989b).
This sort of ‘opposite direction’ functor can be complicated to define (it generally, as here, involves the axiom of choice), and it does not always exist—this is why we have limited attention to a highly simplified case, to avoid complicated constructions that obscure the basic conceptual point. In a sense, this is the core of Nguyen et al.’s argument, as we discuss in Sect. 5.
If the gas particles really did have colour, then one would want the models to be non-isomorphic. But then there would be no representational redundancy because one would think that there was in fact a correct colour attribution.
We remark that, although this is hardly a slight against Nguyen et al., it is not at all clear that the plausible positions in the non-Abelian case look very much like those in the U(1) case. (See, for instance, Healey (2007); Weatherall (2016a); Gilton (2018) for discussions of some of the ways in which non-Abelian Yang-Mills theory resists interpretations that seem natural in electromagnetism—among which is the fact that field strength [curvature] is not a gauge-invariant quantity in non-Abelian theories.)
In fact, they go somewhat further than this, and make a proposal concerning how to think of the spaces of possible field configurations over all manifolds M at once. They conclude that to treat this problem adequately, one should move from thinking about theories as categories of models to thinking of theories as functors—in this case, as a functor from a category of manifolds to a category of groupoids. This proposal has many virtues, but it does not bear directly on the issues we discuss here.
To unpack this equation: by \(d_a g\), we mean the pushforward map along g defined at each point, which, at each \(p\in M\), is a map from \(T_pM\) to \(T_{g(p)}U(1)\). Then \(g^{-1}\) is the pushforward along the translation on U(1) determined by the inverse of the group element g(p), yielding an element of the tangent space at the identity of U(1), i.e., an element of the Lie algebra of U(1) (which happens to be \({\mathbb {R}}\)). Thus, \(g^{-1}d_a g\) is a (closed) one-form on M.
In all of these categories, following Nguyen et al., we ‘fix’ M. For some purposes, one might wish to include diffeomorphisms acting on M among the morphisms of the categories, but nothing is lost for present purposes by neglecting them.
This is because there are (gauge-equivalent) objects \(A_a\) and \(A'_a\) of \({\mathcal {S}}_A\) that are mapped to the same object [A] of \({\mathcal {S}}_{[A]}\), but which have no arrows between them that could map to the identity on [A].
We remark that there is also a functor going in the opposite direction that (using Choice) chooses, from each equivalence class [A], a representative \(A_a\). It is interesting to note that this functor is full and faithful, because every arrow is mapped to an identity arrow, and no two objects have more than one arrow between them; but not essentially surjective, because each equivalence class is identified with a single representative. So this functor forgets forgets property—not structure or stuff. To see what is going on here, note that what this functor is doing is associating with each equivalence class a single, preferred representative. But from the point of view of \({\mathcal {S}}_A\), there are many other fields around that do not get mapped to, representing physical possibilities that are inequivalent to those in the image of the functor, but which do not correspond to any possibility represented in \({\mathcal {S}}_{[A]}\), according to that functor. We can think of the property that is forgotten as the property of being the (privileged) representative of an equivalence class (or the ‘one true gauge’).
One might even worry about the following proposal: suppose we have a theory, and we would like another theory with ‘less structure’. We could simply stipulate that every model of the theory is equivalent to itself in more ways, by introducing trivial maps. For instance, in a model of general relativity, consider the new metric automorphisms which are maps of the form \(g_{ab}\mapsto g_{ab} + n{\mathbf {0}}\), for all n. Suddenly metrics have a new automorphism group!
A principal G bundle, for some Lie group G, is a smooth surjective map \(P\xrightarrow {\pi } M\), where M and P are smooth manifolds with the following property: there is a smooth, free, fiber-preserving right action of G on P such that given any point \(p\in M\), there exists a neighborhood U of p and a diffeomorphism \(\zeta :U\times G\rightarrow \wp ^{-1}[U]\) such that for any \(q\in U\) and any \(g,g'\in G\), \(\zeta (q,g)g'=\zeta (q,gg')\). A (global) section of a principal bundle is a smooth map \(\sigma :M\rightarrow P\) satisfying \(\pi \circ \sigma =1_M\). A principal connection on \(\pi\) is a smooth Lie-algebra-valued one form \(\omega ^{{\mathfrak {A}}}{}_{\alpha }\) on P satisfying certain further conditions, including that \(\omega ^{{\mathfrak {A}}}{}_{\alpha }\) be surjective on the Lie algebra. (Here the lowered Greek index indicates action on tangent vectors to P and the raised capital fraktur index indicates membership in the Lie algebra of G, \({\mathfrak {g}}\). Since the Lie algebra of U(1) is \({\mathbb {R}}\), we drop the fraktur index when discussing principal connections on U(1) bundles.)
A vertical principal bundle automorphism on \(\pi\) is a diffeomorphism \(\Psi :P\rightarrow P\) such that (a) \(\pi \circ \Psi = \pi\) and (b) for any \(x\in P\) and \(g\in G\), \(\Psi (xg)=\Psi (x)g\). We remark that although these maps are automorphisms on the principal bundle, they are not necessarily automorphisms once one fixes a connection.
See also p. 2: ‘How can ‘redundancy’ be an essential feature of a theory?’. This formulation is more congenial to our perspective here.
Here M is a smooth, four-dimensional manifold, which we assume to be Hausdorff and paracompact; and \(g_{ab}\) is a smooth, Lorentz-signature metric \(g_{ab}\) defined on M. For further details on the mathematical background of general relativity, see Malament (2012) or Wald (1984). Our discussion in what follows depends only on the sorts of mathematical facts that are usually at issue in the literature on the hole argument.
Of course, some authors have taken the hole argument to do something like what is described here. But as Weatherall (2018b) argues, this is chimerical: to get to the conclusion that the hole argument generates empirically equivalent but non-isomorphic possibilities, one uses the identity map on the manifold to compare particular points on the manifold. Under such a comparison, the models are not equivalent, either representationally (because it does not give rise to an isomorphism) or observationally.
The argument in Weatherall (2018b) concerns just this issue—or rather, a possible misunderstanding concerning how to understand this ‘freedom’.
Rynasiewicz (1994) draws a very similar moral regarding the hole argument, relating it to Putnam’s famous permutation argument.
Or, we hasten to add, Einstein algebras, since the latter are, in a precise sense, equivalent to relativistic spacetimes (Rosenstock et al. 2015).
Note the echoes of the move from \({\mathcal {C}}_A\) to \({\mathcal {E}}_A\) as described in the previous section. Note that one cannot generally introduce a global coordinate system, in the sense of a smooth map from a generic four dimensional manifold to \({\mathbb {R}}^4\), but one can always assign unique labels to points, for instance by fixing some (non-smooth) map to \({\mathbb {R}}^4\).
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Acknowledgements
This material is partially based upon work produced for the project “New Directions in Philosophy of Cosmology”, funded by the John Templeton Foundation under Grant Number 61048. We are grateful to Thomas Barrett, James Ladyman, and Nic Teh for helpful discussions and suggestions as we prepared this paper.
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Bradley, C., Weatherall, J.O. On Representational Redundancy, Surplus Structure, and the Hole Argument. Found Phys 50, 270–293 (2020). https://doi.org/10.1007/s10701-020-00330-y
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DOI: https://doi.org/10.1007/s10701-020-00330-y