Abstract
In this chapter I consider what recent work on background independent physics can do for structuralism, and what structuralism can do for background independent physics. I focus on the problems of time and observables in gravitational physics. The ‘frozen’ character of the observables of general relativity is usually considered to constitute a serious problem for the theory. I argue that by invoking correlations between physical quantities we can provide a natural explanation of the appearance of time and change in timeless structures. I argue that this response can resolve a problem with Max Tegmark’s ‘extreme structuralist’ position. I then consider what bearing the mathematical representation used (namely Rovelli’s framework of ‘partial’ and ‘complete’ observables) has on the debate over the nature of structure in discussions of structural realism (i.e. the question of how structures are to be conceived). I argue that it has both the resources to ground the notion of structure in physics and to answer the ‘no relations without relata’ objection.
This chapter was originally written for the FQXi’s Nature of Time essay competition (http://www.fqxi.org/community/essay/winners/2008.1#Rickles). I thank FQXi for permission to reproduce the essay (albeit in modified form) here.
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- 1.
As he puts it: ‘the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena’ [16, p. 230].
- 2.
Though not all structural realists would go this far. Some, for example, would prefer to say that structure is physical, and that there might be biological and social structures that are not necessarily mathematical. However, the recent trends are towards extending structural realism across the entire domain of science, including biology [5] and economics [12]. The distinction between ‘physical’ and ‘mathematical,’ if it can be established at all, is not quite as simple as it might seem prima facie—see [8] for a discussion of this difficulty in the context of string theory.
- 3.
For example, one can conceive of the laws being different, and indeed, as in David Lewis’ theory, the existence of a plurality of structures of the sort described can provide the machinery to ground such possibilities.
- 4.
Related to this is the problem of equivalent mathematical structures that correspond to distinct physical situations. In other words, one and the same structure can be taken to represent very different systems. I don’t see this to be as problematic as it is sometimes taken to be. If there are indeed differences in the physical systems, then though we can indeed, in many cases, represent them using the same mathematical structure (for example, the Navier-Stokes equations can be applied to all manner of prima facie very different systems), that does not thereby mean that the systems would not have some other structures more closely corresponding to them. Any physical difference would simply mean that there ought to be a structural difference too, so long as we use a fine enough resolution of the structure.
- 5.
To this we might add various geometric object fields representing the observed matter and radiation.
- 6.
- 7.
I restrict the discussion to classical systems in order to make the presentation easier to follow. For the technically savvy, one can transform to the quantum case, roughly, by thinking of the functional relation or correlation \(\mathcal{A}(\mathcal{B})\) as representing the expectation values of \(\mathcal{A} \) relative to the eigenvalues of \(\mathcal{B} \).
- 8.
It seems that Einstein might have been aware of this implication soon after completing his theory of general relativity, for he writes that ‘the connection between quantities in equations and measurable quantities is far more indirect than in the customary theories of old’ [4, p. 71].
- 9.
There are two types of constraint in general relativity : the Hamiltonian (or scalar) constraint and the momentum (or vector) constraint. These can be understood as encoding indeterminacy about ‘when and where’ some quantity is measured.
- 10.
There is a proof (for the case of closed vacuum solutions of general relativity) that there can be no local observables at all [15], where ‘local’ here means that the observable is constructed as a spatial integral of local functions of the initial data and their derivatives.
- 11.
We might also call them ‘Kretschmann observables’ since they stem from Kretschmann’s objection to general covariance later incorporated into Einstein’s own ‘point-coincidence’ argument.
- 12.
Once again, we find that Einstein was surprisingly modern-sounding on this point, writing that ‘the gravitational field at a certain location represents nothing “physically real”, but the gravitational field together with other data does’ [4, p. 71]. Likewise, the ‘other data’ will represent nothing without yet more data (such as the gravitational field). The correlations are the fundamental things.
- 13.
Here then we have a clear answer to Chakravartty’s question ‘in what sense are relations more fundamental?’ (Chapter 10, p. 199). The physical weight is carried by the complete observables (the relational structures), and the partial observables (relata) are certainly not fundamental in the sense that they are not even physical but artefacts of the gauge choice employed.
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Rickles, D.P. (2012). Time, Observables, and Structure. In: Landry, E., Rickles, D. (eds) Structural Realism. The Western Ontario Series in Philosophy of Science, vol 77. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2579-9_7
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