Hofweber and Lange argue that the fragmentalist interpretation of special relativity is in tension with the proper explanation of why the Lorentz transformations hold (2017: 871). On the standard interpretation, the Lorentz transformations of the different frame descriptions obtain because of the way in which the same underlying world is coordinatized in the different frames of reference. Hofweber and Lange argue that this is the right explanation of why the Lorentz transformations hold and that the fragmentalist has no room for this explanation.
One preliminary point. I do not agree with the theoretical commitments attributed to the fragmentalist interpretation in their discussion. Hofweber and Lange assume that fragmentalism ‘takes frame-dependent facts to be fundamental rather than derivative from frame-invariant facts such as facts about the spacetime intervals between various events’ (2017: 874). But nothing that we have seen so far commits the fragmentalist to this. For one, Fine himself is not committed to the idea that something is in reality the case if and only if it is not grounded by anything, so two matters may both be real even when the one grounds the other (2001: 27). Fine’s notion of ‘reality’ is independent from ‘grounding’. Although grounding may be a good defeasible guide to what is real, there is no reason why there could not in principle be some fundamental facts grounding certain other fundamental facts (presumably, with the relevant grounding facts themselves being fundamental as well).
In other words, there is nothing in the fragmentalist interpretation that is incompatible with postulating a Minkowskian spacetime next to the various fragments and taking the distributions of the variant properties to be grounded in the spacetime intervals of the Minkowskian spacetime. Such a view does not disagree with the standard Minkowskian interpretation about the grounds or the explanations of the variant matters. The Minkowskian takes them to be explanations of mere appearances, whereas the fragmentalist takes them to be explanations of genuinely obtaining yet grounded facts. Such a view would only disagree with the Minkowskian view about the reality of the variant matters.
One might wonder whether there is still a substantive difference between the Minkowskian view and the current way of developing the fragmentalist account. What it is the theoretical upshot of the variant matters being ‘real’ in Fine’s sense? The importance is that of marking a metaphysical realism about those variant matters. The relevant question is whether realism or antirealism is true about the frame-relative facts, that is, whether consideration of the special theory of relativity removes all frame-relative facts from one’s metaphysical conception of reality: the Minkowskian answers yes, the fragmentalist answers no. There is of course a further question whether realism is best understood using Fine’s primitive notion of reality (see Horwich 2007; to which Fine 2007 responds; see also Lipman 2018).
As mentioned briefly in the introduction, alternative formulations of fragmentalism, given by Lipman (2015), Loss (2017) and Simon (forthcoming), do not invoke a primitive notion of ‘reality’ but rather only work with the notion of coherence, which they prefer to call ‘co-obtainment’. Two facts may obtain without co-obtaining and this is what the ‘fragmentation’ consists in. On these views, the debate with the Minkowskian is not whether the frame-relative facts are fundamental or real facts, as on Fine’s approach, rather, the debate is about whether they are facts at all. The fragmentalist believes that it can be a genuine fact that a piece of paper is square and a genuine fact that it is oblong, whereas the Minkowskian needs to say that the piece of paper does not have such a property but only seems to, given a certain frame-relative description. If grounding is assumed to be factive, the Minkowskian cannot be claiming that the frame-relative facts are themselves grounded in the spacetime interval (i.e. they cannot claim that it is the paper’s being square that is grounded in the interval), as this would imply that there are genuine frame-relative facts (i.e. that the paper is genuinely square and hence that there is the purely spatial property of being square). If there are such genuine frame-relative facts, this raises the question whether we should privilege one frame or accept the mutually incompatible facts of multiple frames. Neither of these options are typically taken to be live options for the Minkowskian view. Instead, the Minkowskian must be saying that, strictly speaking, it is only the mere appearance of frame-relative facts that is grounded in spacetime intervals. Though the Minkowskian and the current type of fragmentalist disagree about the exact nature of the explananda, they agree about the standard direction of explanation, both taking the spacetime interval to ground the variant matters (or, according to the Minkowskian, the appearances thereof).
The complaint that the fragmentalist interpretation conflicts with the right explanations of the Lorentz transformations is therefore somewhat off the mark: the objection targets something that is not really an essential commitment of the view, namely a claim about the direction of explanation. The fragmentalist can in principle adopt whatever direction of explanation the Minkowskian appeals to.
Having said this, Hofweber and Lange’s objection is interesting and it is worth considering the sort of fragmentalist view it does target. Let us therefore imagine a fragmentalist who believes: (1) that the variant properties are real, and (2) that the spacetime intervals consist in constant ratios of variant properties, and (3) that the variant properties—the different spatial and temporal distances in different frames—ground such ratios. The invariant intervals are the unified appearances of a fundamentally fragmented world on this way of developing the fragmentalist interpretation. The Lorentz transformations of the coordinate frames underwrite the law-governed regularity between the distributions of fundamental yet variant properties in the different fragments. It is this sort of view that Hofweber and Lange criticize and it is a grounding story that might indeed appeal to the fragmentalist. For the remainder of this section, the fragmentalist interpretation under discussion will be the one just outlined.
Both the Minkowskian and the fragmentalist agree that the Lorentz transformations hold and that the spacetime interval is invariant. Hofweber and Lange claim that the invariance of the interval is generally assumed to explain the Lorentz transformations (2017: 876). The assumed explanatory priority of the spacetime interval, they argue, is naturally explained by the standard interpretation of the special theory of relativity:
Why does the spacetime interval’s invariance count as explanatorily prior to various other facts, such as the transformation laws? After all, the transformation laws suffice to entail the spacetime interval’s invariance. Why does science take the direction of explanation as running from the spacetime interval’s invariance to the transformation laws rather than, say, in the reverse direction? Because the spacetime interval, as a frame-invariant fact, is the reality, whereas the facts related by the coordinate transformations are frame-dependent facts and hence are appearances of that reality. How things are explains how things appear from a given perspective. Therefore, the law that a certain quantity is invariant takes explanatory priority over the laws specifying how various frame-dependent quantities transform (2017: 876).
Starting from the Lorentz transformations plus supplementary assumptions, we can derive the invariance of the spacetime interval. In the other direction, so starting from the invariance of the spacetime interval plus supplementary assumptions, we can derive that the Lorentz transformations hold. About all of this, the Minkowskian and fragmentalist can agree.
Nevertheless, it is standardly assumed that the explanatory direction runs from the invariance of the spacetime interval to the Lorentz transformations. Hofweber and Lange suggest in the cited passage that this is because the spacetime interval is real and the variant properties mere appearances. The fragmentalist interpretation under discussion, which takes the variant properties to be real and the grounds for the spacetime interval, cannot offer the standard explanation. The standard interpretation underwrites the standardly assumed explanatory priority of the spacetime interval, whereas the fragmentalist conflicts with it.
It is unclear what the exact dialectical significance is of this point. We can discern two distinct criticisms. First there is the claim that the invariance of the interval is widely assumed to have explanatory priority within the scientific community. This should be distinguished from the claim that the invariance of the interval indeed has explanatory priority. The first is a sociological claim about, roughly, the assumptions at play in the scientific community; the second is a claim about which fact indeed takes explanatory priority. Let us consider these points in turn.
First there is the claim that the invariance of the spacetime interval is generally assumed to take explanatory priority. Hofweber and Lange offer various quotes (by Eddington, Mermin, Einstein and Holton) which express adherence either to the standard interpretation or to the basic assumption that different perspectives always present us with mere appearances of an underlying unified reality (2017: 874). This assumption—of taking reality to be unified behind the mere perspectival appearances—they claim is ‘alien to the spirit behind fragmentalism’ (2017: 877). Let us grant that this is so. What to make of this? One might here raise the methodological principle that in any disagreement between the views of scientists and metaphysicians, we ought to favour the views of the scientists, given that science is on a much firmer epistemological footing. Though we should all adhere to this methodological principle, it has to be wielded with care. The scientist’s views are authoritative because the scientific method has proven to be the best means of furthering knowledge. The epistemological deference to science is based on the methodological superiority of the scientific method, and hence the principle of ‘favoring the scientist’s views’ should clearly be restricted to only those matters that can be decided on the basis of those scientific methods—that is, to the empirical claims. The scientific method only provides the scientist with an authority over empirical matters. The fragmentalist does not question the special theory of relativity, the transformation laws, or any of the empirical evidence that supports them. There is no disagreement about the science itself. The debate is about the proper metaphysical interpretation of the relevant science: about how best to conceive of the world so as to accommodate the scientific facts. This is not decided by the standard empirical methods (if only we could), but on the basis of simplicity, plausibility, the ability to integrate with further theories, the ability to save the appearances, and so on. These methods are not a firmer footing than those of the metaphysician, in fact they arguably are amongst the methods of the metaphysician. There is no good basis for deferring to the views of scientists on these abstract and non-empirical matters.
Against this response, one might argue that we ought to adopt a more far-reaching naturalism on which we philosophers ought to adopt whatever views are congenial to current scientific practice, even on matters that lie beyond empirical matters. Physicists explain variant matters on the basis of invariant matters, and would it not obstruct science if fragmentalism demands a change in the current scientific practice?Footnote 5 Ultimately, current practice ought to conform to whatever turns out to be the most reasonable overall view of the world. The sensible fragmentalist metaphysician is not currently suggesting to overthrow current scientific practice, she is just showing that there is a coherent alternative to the standard interpretation and exploring its advantages. If the case for the standard interpretation remains stronger than the fragmentalist alternative, then this is reason enough to dismiss the fragmentalist interpretation. If there turn out to be major advantages to the fragmentalist interpretation over the standard interpretation, then it also would not obstruct science to change current practice, on the contrary. Either way, appeal to scientific practice at the current stage of investigation strikes me as a red herring.
There is another way of understanding the complaint that fragmentalism goes against general assumptions of the scientific community. The point may simply be that fragmentalism is a revisionary view and that this counts in favour of the standard interpretation, as the standard view has earned its credits and demonstrated it can provide an understanding of matters. Fragmentalism is indeed a revisionary view and this is indeed a comparative cost. This goes without saying. Further work will clarify how revisionary it is and whether advantages offset it. There are also distinctions to be made. The fragmentalist approach sketched earlier, which maintains explanations in terms of a fundamental Minkowskian spacetime, is less revisionary than the fragmentalist view currently under discussion, which also reverses the order of explanation. This may be a point in favour of the weaker approach, but not nearly a decisive point.
Let us turn instead to the endorsement of the standard explanatory direction. Hofweber and Lange claim that the standard assumption is right and that the invariance of the interval has indeed explanatory priority.Footnote 6 They write that the interval must be what explains Lorentz transformations ‘because the spacetime interval, as a frame-invariant fact, is the reality, whereas the facts related by the coordinate transformations are frame-dependent facts and hence are appearances of that reality’ (2017: 876). This is not an argument but the direct denial of fragmentalism, which just is the view that the frame-relative facts are real and not the mere appearances of frame-invariant facts.
Hofweber and Lange argue that the fragmentalist needs to postulate ‘brute’ connections across the fragments without giving any reason to expect such brute connections, and that this is a substantive cost of the view (2017: 878). The relevant question here, I take it, is this: why are there only ever fragments related to each other in accord with the Lorentz transformations, and no other fragments? The answer is that there are the Lorentz ‘laws’ relating the fragments. There is no further explanation of why these laws hold. The Minkowskian, in contrast, explains the Lorentz transformations, according to this objection.
That the lawlike-connections underwriting the Lorentz transformations take explanatory priority over the spacetime interval (and assumed variance of the variant properties) does not mean that there cannot be another explanation of the Lorentz transformations. One might offer an account along general Humean lines, for example, or one might endorse a so-called dynamical explanation of the Lorentz transformations, according to which transformations are taken to be grounded in the various forces acting on rods and clocks (Bell 1976/1987; Brown and Pooley 2006; and Brown 2005). Such explanations are not incompatible with fragmentalism (or the theory of special relativity, more generally).
If such explanations are not to be had, however, then there will indeed be a brute constraint on the distribution of variant properties, but if this is how things turn out, this type of brute constraint arises in some form for any view that does justice to the special theory of relativity. The constraint that the fragmentalist needs to impose on the fragments is effectively that light must have the constant velocity c, regardless of the state of motion of the emitting body, in any fragment. This is just the postulate of the invariant velocity of light. The Minkowskian captures this invariance of the speed of light in inertial reference frames in terms of the geometric structure imposed on spacetime (together with the way this is projected onto inertial frames). But that this constraint is incorporated as a geometric law of spacetime does not mean that it is somehow less brute. Why does spacetime have the Minkowski structure as opposed to a geometric structure that allows objects to move faster than the speed of light? The Minkowskian answer is that it just does. This is entirely fair. The fragmentalist’s postulation of the invariance of the velocity of light as a basic physical law is however no less fair.
One might think that the worry concerns not the brute imposition of the invariance of the speed of light, but rather the fact that the Minkowskian can explain why events have the variant properties that are attributed to them in a given frame. The Minkowskian explains variant properties in terms of the intervals of those events. The fragmentalist view under discussion takes the instantiations of the variant properties to be basic, and not to be the mere appearance of underlying properties. But that is only half of the story. This type of fragmentalism was characterized by the further idea that we can explain why the events have the spacetime intervals in terms of the variant properties of those events, in contrast to the Minkowskian view, on which the distribution of spacetime intervals are taken to be basic. Where two views differ in what grounds what, bruteness will emerge in different places. In such cases, our evaluation goes astray if we merely point out that a certain matter is brute on one view and not brute on the other.
Hofweber and Lange do not succeed in raising any serious problem for the fragmentalist interpretation of special relativity. It remains a live option, worthy of further investigation.