Huggett and Wüthrich (2013) are interested in the empirical coherence of quantum gravity approaches. A theory is defined as empirically incoherent “in case the truth of the theory undermines our empirical justification for believing it to be true.” (Huggett and Wüthrich 2013, p. 277). First, Huggett and Wüthrich (2013) make the assumption that empirical science ultimately rests on tracking pointer coincidences sorted by place and time stamps (known as ‘beables’):
A central concern of philosophy of science is understanding how the theoretical connects to the empirical, the nature and significance of ‘saving the phenomena’. ...At a high level of generality, however, presumably the link is established by observing (in some sense) a material ‘something’, in some determinate state or other, at some spatial location at some moment in time and connecting this occurrence to our theory, for instance by postulating, in our theory, entities which behave in ways that would explain our observation. This is crude, no doubt, but seems to capture quite generally the nexus between our theorising about the world and our experiencing it, from meter readings in the lab to observing distant galaxies with a radio telescope to the results of high energy collisions. (p. 276)
Then, the general concern of Huggett and Wüthrich towards quantum gravity is that (many) approaches to quantum gravity (at least at first sight) feature empirical coherence issues, as their actual structures are so distinct from familiar spacetime theories that they either already do not allow for local beables at the fundamental level or at least do not suggest a clear path for connecting the familiar local beables at the derivative level (those beables we are using and referring to in our phenomenal spacetime) to the local beables in the fundamental theory. The issue resembles the debate on 3N-dimensional (configuration) space vs. 3-dimensional (position) space in the philosophy of (non-relativistic) quantum mechanics, and the specific question therein of how one can make sense of the idea that 3N-dimensional space could ground a 3d-dimensional space if indeed more fundamental.Footnote 5 In both the QG and the QM case, the putative empirical coherence threat is thus rooted in the lack of fundamental beables and/or a lacking connection to observable beables.
In response to Huggett and Wüthrich’s concerns with empirical coherence in quantum gravity approaches I will work through the following points:
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Strictly speaking, empirical coherence can only occur in models of an empirically relevant physical theory but not in a physical theory as such. Moreover, empirical coherence worries would be abundant in modern physics if they were just about the lack of local beables at the fundamental level of the theories (Sect. 2.1).
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An interesting example for a spacetime theory whose models partly feature empirical coherence issues is given by general relativity (GR). Thus, empirical coherence issues at the model level even among spacetime theories are not special to quantum gravity but already occur in GR (Sect. 2.2), rendering their occurence in approaches to QG as less dramatic.
The misconceived empirical (in)coherence issue
Theoretical laws can only be tested through testing the empirical laws derived from them.Footnote 6 In this sense, it would be wrong to believe that an approach to quantum gravity (as a candidate for an empirical theory) itself can bring along an issue of empirical coherence; a genuine issue of empirical coherence could only arise if the laws of quantum gravity were not connected to empirical laws at all. This will however not be the case for any viable approach to quantum gravity, as it is required to reduce to GR or at least to a theory empirically equivalent to GR for the regime within which GR has been successfully tested. So, unlike claimed by Huggett and Wüthrich, it is not relevant in this context to ‘understand’ how spacetime emerges, that is say in the context of loop quantum gravity (LQG) how space-time arises from spin networks. It only matters that spacetime does emerge since otherwise the approach to quantum gravity is not worth considering in the first place. The idea of a theory of quantum gravity fully detached from current physics (that is a theory not reducing to any physical theory at all currently in use, or at least not even numerically agreeing with one of them in some regime) is far from any reality.
In other words, physical theories cannot be empirically incoherent—or else they are simply not physical but just mathematical structure. (If a theory was completely empirically incoherent in the first place, it would be disconnected to all currently known physical theories as they are after all linked to the empirical to some degree.) Still, one could argue that the relevant question is about whether currently known quantum gravity approaches are just mathematics or whether they can reduce to measurable statements. But this is to be decided by the common derivational work as usually done in the physics and not in the philosophy department.
Now, if one were to look for empirical coherence issues, one should rather expect them in particular models of theories not in (empirically relevant) theories as such. In fact, single models of more fundamental theories might not all be connected to models of their derivative theories nor to directly empirically relevant laws (see next section). (For saying that a more fundamental theory reduces a putatively less fundamental theory it is after all sufficient that all models of the less fundamental theory are reduced by models from the more fundamental theory—but not vice versa.) It is those kinds of models which we should count as ‘empirically incoherent’.Footnote 7\(^{,}\)Footnote 8 Thus, rather than talking about how a theory is empirically incoherent “... in case the truth of the theory undermines our empirical justification for believing it to be true ...” (Huggett and Wüthrich 2013, p. 277), we should be interested in the notion of a model being empirically incoherent in case the truth of the model undermines our empirical justification for believing it to be true.
It could be objected to my complaints above that Huggett and Wüthrich themselves already concede with respect to most approaches they consider that there is no empirical coherence issue at the theory level linked to them. And that the decisive question is thus rather whether the empirical coherence worry towards (some) approach to quantum gravity is a legitimate philosophical prima facie problem.Footnote 9 However, as I will argue in the following, accepting an empirical coherence issue for QG approaches as a relevant prima facie problem draws on an overly naive realist picture which ignores as to how much theories are generally in need of interpretation—and thus arguably hardly attractive.
In a first step, it is worth stressing that (i) a part of the empirical coherence issue (Huggett and Wüthrich 2013) have in mind cannot count as a merely epistemological one but that it presupposes some commitment to scientific realism. A pragmatic physicist clearly need not come up with this worry: Rather, once she has managed to mathematically reduce results from a quantum gravity approach to GR she will just go out—or let others go out—and measure quantum-gravitational corrections to GR quantities. It is in this sense that it is essentially a realist’s worry.
In a second step, it is important to (ii) acknowledge that the empirical coherence issue of Huggett and Wüthrich (2013) could have been posed in an analogous way in other cases just as well (in the context of thermodynamics for instance) but that it was not: After all, just as it is true that there might be no local beables (at least not in the standard sense) available at the fundamental level in quantum gravity, thermodynamics does not feature beables in a standard sense, either (it is primarily a theory about how different equilibrium states can be linked to one another). We can only ‘embed’ thermodynamic systems into spacetime regions in a highly crude manner:
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Time is only brought into the theory in a very minimal sense, namely as the posit that systems not in equilibrium approach equilibrium (while systems in equilibrium remain in equilibrium—see Brown and Uffink 2001). In particular, there is no sense of duration for time in thermodynamics.
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Space only features in the theory indirectly via notions such as pressure and volume.
In particular, as thermodynamics is duration-less it can strictly speaking not be linked to determinate spacetime regions (it is not all clear how long a process will take). Assignment to a determinate spacetime region only becomes possible from past experience (based on previous experience with specific systems, one learns that the approach to equilibrium occurs within a certain time-span), or from statistical mechanics (which allows for making more quantitative statements about the approach to equilibrium). Now, given that the (classical mechanistic) sense of beables as a precise marker of events in space and time is not available in thermodynamics either, the naive realist could equally stumble into an empirical incoherence issue when facing thermodynamics: after all, she could wonder how a theory of thermodynamics without duration could be empirically adequate if it was not straightforwardly relatable to our phenomenal spacetime.
That the naive realist as a matter of fact did not stumble here, might strike one as an important disanalogy between the case of thermodynamics, and the case of QG approaches. Well, that we are not aware of any bigger empirical coherence issue in the face of thermodynamics, lies in the fact that thermodynamics happened to be developped as a generalisation of how heat machines, engines, and all that work (that is, as a phenomenological theory). Only later thermodynamics got abstracted into a more universal framework (whose interpretation on its own then, as argued, is not immediate anymore). In other words, possible interpretations of thermodynamics and linkages to the world were part of its development from the very beginning. Then, given the contingency of such a generative story, a naive realist would have to recognise a prima facie empirical coherence worry towards thermodynamics—if not as one that actually unfolded in our world, then at least in some (nomically identical) world in which thermodynamics was arrived at via a less phenomenological route. Note, however, that if the theory of thermodynamics is not seen to have suffered from an empirical coherence issue in our world but only in some (nomically equivalent) other one, the naive realist should also expect that some specific approaches to QG she currently labels as prima facie empirically incoherent will most likely not suffer from a prima facie empirical coherence issue in some other (nomically identical) world; say, because in that world we had already adapted our way of talking about physics, and doing experiments beforehand in such a way that the interpretation of the structures of these approaches in question becomes immediate. Fair enough. But, as long as not just a prima facie worry contingent on quite specific albeit QG-independent background knowledge is meant by the empirical coherence issue in QG (but rather a general prima facie worry about the nature of specific QG approaches and their in-principle linkability to the world—as I will assume in the following), a naive realist will have to regard both thermodynamics and approaches to QG as suffering from a (prima facie) empirical coherence issue.Footnote 10
In sum then, the empirical coherence worry can thus be said to only arise with full force on us realists when we ignore the general knowledge on how much physical theories—such as thermodynamics and not just QM or QG—stand in need of interpretation anyway (that is, if we are straightforwardly naive realists). To stress: None of this is to say that I disagree with the answer of Huggett and Wüthrich (2013) to the worry, namely that it can be dismissed through acknowledging that physical salience can flow from the derivative level to the fundamental level as well. My point is rather that there is no need to present the (so-called) empirical coherence problem as a genuine worry which could perhaps not be overcome: it is simply a task to address (a task of interpretation) albeit none whose eventual success is seriously at issue.
More precisely, and to put a positive spin on things, Huggett and Wüthrich (2013) (and also Lam and Wüthrich 2018) should thus simply be understood as making explicit (in particular to the naive realist) the specific interpretational need(s) linked to different approaches to quantum gravity. Generally, empirical interpretation is loosely about how to link the theory to the world. Curiel (2009) for instance introduces the notion of concrete interpretation:
Concrete. The fixation of a semantics for the formalism, in the sense that the formalism under the semantics expresses the empirical knowledge the framework contains—for example, the fixation of a Tarskian family of models, or, less formally, the contents of a good, comprehensive text-book. (p. 46)
Every physical theory—at least with respect to some of its models—needs to have a concrete interpretation. Building on a further notion by Curiel (called ‘metalinguistic interpretation’), a theory can be said to stand in need of empirical interpretation in an interesting sense if it either (1) can only be empirically understood upon providing sophisticated qualifications as to in what sense its empirical attributions apply, or (2) requires extratheoretical import to allow for or make sense of empirical attributions. Quantum mechanics proper is a prime example for a theory in need of interpretation in an interesting sense—a naive concrete interpretation of the formalism alone leaves it open how determinate measurement results are obtained. In contrast to this, empirical coherence is about whether a theory/model can be interpreted at all.
The need of quantum gravity for an empirical interpretation in an interesting sense is two-fold then:
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Any theory of quantum gravity needs an interpretation qua being a quantum theory.Footnote 11
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Even when ignoring quantum mechanical superposition effects, a theory of quantum gravity needs an interpretation if it rests on spacetime-less structures which do not allow for straightforwardly connecting to our usual ‘beables’/familiar measurement procedures in \(4 = 3+1\) dimensional spacetime. For instance, borrowing an example from Huggett and Wüthrich (2013), target space with respect to which string theory is standardly formulated only superficially looks like a higher-dimensional generalisation of 4d dimensional (curved) spacetime. Provided their assessment is correct, string theory itself is only empirically understandable via approximation and limit procedures since we can only this way make contact to expressions associable to objects in 4d spacetime. A similar observation could be made with respect to the spin-network structures of LQG, causal sets in causal set theory, and so on (for details see the tour de force on what is left of spacetime in various approaches to quantum gravity by Huggett and Wüthrich 2013).
Proper issues of empirical incoherence
In the previous section, it was argued that the notion of empirical coherence in quantum gravity should be considered only at the level of single models but not at the level of theories, i.e. whole sets of models.
For example: Although LQG in so far as it is an approach to quantum gravity is empirically coherent, some LQG models will not lead to GR spacetime models (or any spacetime-like models) upon applying the appropriate reduction scheme (see Wüthrich 2006, p. 169). However, as I will explain in this section now, already our currently best spacetime theory—GR—features an analogous empirical coherence issue at the level of its models. (In other words, there is nothing special about the existence of an empirical coherence issue to quantum gravity qua being a spacetime theory either.)
Empirical access in GR ultimately rests on the availability of a coordination procedure, that is a protocol for how a relativistic observer can set up spacetime coordinate frames in order to sort and keep track of measurement results (one might very well in the end use a coordination scheme without an operational meaning for calculations etc. but this is a different matter). I will now argue that certain models do not even allow for such a coordination, and must thus count as empirically incoherent.Footnote 12\(^{\text {, }}\)Footnote 13
Now, the only fully operationalist protocol for setting up (local) coordinates in a general relativistic spacetimeFootnote 14 employs radar (light) signals: light signals and their echos are used to probe an observer’s environment relative to an (admittedly) arbitrary parameterisation of her own worldline \(C_0: t \mapsto \gamma (t)\) (which serves as a time standard). In particular, the observer does not need to have a standard clock at her disposal, that is a clock that would show a reading proportional to the proper time of the observer.Footnote 15 For a sent-off time \(t_1\) and return time \(t_2\) the following (local) radar coordinatesFootnote 16 are then assigned to an event P relative to the observer’s worldline parameterisation \(t \mapsto \gamma (t)\) (see also Fig. 1):Footnote 17\(^{,}\)Footnote 18
$$\begin{aligned} T&= \frac{1}{2} \cdot (t_1 + t_2)\\ R&= \frac{1}{2} \cdot (t_2 - t_1)\\ \end{aligned}$$
A direction with respect to which the signal is emitted needs to be determined relative to a (infinitesimally close-by) neighbouring worldline \(C'\). We can then say that two signals are emitted in the same direction if both cross this same neighbouring worldline.Footnote 19 How the direction is exactly picked, is not important for the following, however.Footnote 20
Provided that light signalling is available in the first place, radar coordinates are guaranteed to exist within a sufficiently small neighbourhood of any GR spacetime (see Proposition 1 in Perlick 2008). But unlike often assumed (say in the constructive axiomatic approach to GR of Ehlers et al. 2012), light signals simply do not necessarily belong to the idealised ontology of GR. After all, light is internally first of all described by the electrodynamics sector of GR, and can as such only be said to trace out null geodesics in an idealised fashion if what is known as the geodesic-optic limit holds (that is if light wave packets can indeed be shown to be idealised as rays tracing out null geodesics when taking a high frequency limit).Footnote 21
Following Asenjo and Hojman (2017), light signals will in fact not follow null geodesics in Gödel spacetime (neither in rotating spacetimes more generally) but move at varying speed.Footnote 22 As a consequence, at least in Gödel spacetime, radar coordinates cannot be set up since light signals will not generally move on null geodesics and can thus not be used as a fixed, coordinate-independent reference standard (thus, suffering from the same problems like timelike signals). However, given that radar coordinates (or radar-type coordinates) are the only notion of operational coordination available, sorting and recording events thus becomes—even under all forms of idealisation—impossible.Footnote 23
We do not need to go into detail here. What matters for us, is that we have found that the very basis for empirical access in GR—radar(-like) coordinates—cannot (not even in an idealised fashion) be set-up in certain spacetime models (first and foremost not in Gödel spacetime). In a lack of an operationalist sense of coordinates, pointer coincidences cannot be kept track of and in this sense there are effectively no local beables to work with. Models like Gödel spacetime are thus first of all empirically incoherent. This is not to exclude that there might be some operational procedure from outside of GR—say quantum information—for circumventing the sketched obstacles to setting up radar coordinates, and tracking point coincidences but, from a GR point of view, the mentioned spacetimes are just empirically incoherent.Footnote 24