3.1.1. Modal Information

It is often said that nonfundamental explanations provide modal information of a kind that is absent from fundamental explanations. Views of this kind have been defended in one way or another by Garfinkel (Reference Garfinkel1981), Jackson and Pettit (Reference Jackson and Pettit1992), Wilson (Reference Wilson1994), and passages in Woodward (Reference Woodward2003). In order to evaluate this strategy, consider an example from Woodward (Reference Woodward2003, 231–33):

First, there is the following microscopic strategy for explaining the new pressure P′:

In short, the microscopic strategy consists in a DN or interventionist explanation employing fundamental laws. Second, there is the following macroscopic strategy for explaining the new pressure P′:

The macroscopic strategy can be summarized in the form of the ideal gas law $PV=nRT$. In short, the macroscopic strategy consists in a DN or interventionist explanation employing nonfundamental laws. Regarding the connection between these two explanations, Woodward (Reference Woodward2003, 232–33, 355–56) claims that the information provided by the macroscopic explanation is not “captured” or “represented” by the microscopic explanation, that the microscopic explanation is deficient in not providing information on the conditions under which P′ would have been different, that an explanation employing the ideal gas law does provide this information and so provides a better explanation of P′, and that while it is true that the microscopic explanation answers a wide range of w-questions, it does not answer the w-question demanded.

The basic doubt I will raise against these claims can be seen by noting that the macroscopic explanatory strategy is constructed precisely by appealing to the microscopic strategy over a range of possible initial conditions, which is to say that the modal information described by the macroscopic explanation has been exclusively procured from the microscopic explanation. This problem with Woodward's claims can be put more simply and generally as follows. The fundamental physical explanation for any event will employ laws of maximal generality in the sense of both scope and invariance. In particular, for any determinate w-question framed in terms of the variables employed by the fundamental physical explanation, the explanatory model will specify a determinate answer.Footnote ¹¹ If we assume a reasonable form of physicalism, then there are no questions that can be formulated in terms of any other variables that do not correspond to one of these questions. So there are no physically possible counterfactuals on which the fundamental physical explanation is silent. The fundamental physical explanation provides the resources to answer any possible w-question. In short, a reasonable physicalism entails that there is no missing modal information of the kind claimed.Footnote ¹²

A similar mistake is made by Jackson and Pettit (Reference Jackson and Pettit1992, 173–75), who likewise describe the difference between microscopic and macroscopic explanations in terms of the information the explanations provide. Jackson and Pettit present the idea in a slightly different way, focusing on information concerning the causes of a particular event and formulating the informational claim in terms of what it is possible for one to be ignorant about. In particular, Jackson and Pettit (Reference Jackson and Pettit1992, 177) argue that the information provided by a macrocausal explanation that is not provided by a microcausal explanation concerns counterfactuals of the form “if the actual history described by the microcausal explanation had not obtained, the explanandum would still have occurred.”Footnote ¹³ Of course, not all grounds for believing a counterfactual of this kind provide explanations. For example, knowledge of an unrelated backup cause for any event will ground such a counterfactual while making no contribution to the explanation of that event. To respond to this problem, Jackson and Pettit appeal to the Lewis (Reference Lewis1986) account of causal explanation, according to which an explanation must provide information about the actual causal history. This means that the difference between the acceptable macrocausal information and the unacceptable backup cause information must amount to a difference in the information provided about actual causal history, which Jackson and Pettit (Reference Jackson and Pettit1992, 178) say amounts to the question of whether “the counterfactual is grounded in the nature of the actual [causal] history.” But this amounts to the admission that the information provided by the macrocausal explanation is in fact already implicit in the microcausal explanation, which by hypothesis is a microphysical description of the entire actual causal history. So again, there is no missing modal information of the kind claimed.

Nevertheless, perhaps there is a way to make the distinction between microcausal and macrocausal explanations in a way that saves the idea that macrocausal explanations provide modal information that microcausal explanations do not. This would require microcausal explanations to take a different form than either DN or interventionist explanations since, as I have argued, these explanations implicitly contain all the modal information there is concerning the system in question. Since the idea here would be that the general modal information carried by laws in DN and interventionist explanations is absent from this variety of explanation, I will call these explanations singular microcausal and macrocausal explanations. The modal informational account of explanatory depth would then provide a sense in which singular macrocausal explanations are deeper than singular microcausal explanations in virtue of conveying modal information that singular microcausal explanations do not.Footnote ¹⁴ But this would be a Pyrrhic victory for the modal informational account of depth, for it would remain the case that fundamental DN and interventionist explanations are always deeper than their nonfundamental analogues, which is the result I have supposed an account of depth should license us to deny. The explanations provided by fundamental physics are simply not singular explanations in this sense.

3.1.2. Taxonomic Information

A second way in which informational accounts of explanatory depth have been formulated is in terms of what I will call taxonomic information, nonmodal information concerning the way in which explananda are described or individuated. For example, at one point Woodward (Reference Woodward2003, 232–33) suggests that the information missing from the microscopic explanation of the gas pressure concerns the relationship between microscopic and macroscopic variables. This is puzzling given Woodward's description of the microscopic explanation since the last sentence of that description shows how the macroscopic variable P′ is derived by aggregating the energy and momentum imparted by each molecule to the walls of the container. Nevertheless, suppose it were true that the microscopic explanation did not itself contain the information required to describe the explananda in macroscopic terms. In that case, the explanation should be rejected simply because it does not answer the original explanatory question. But this would again be a Pyrrhic victory for the informational account, for if we now suppose that the microscopic explanation were supplemented with the information required, we would have a fundamental explanation as deep as the nonfundamental explanation.

The basic problem here is that the taxonomic informational account of explanatory depth appeals to information that no reasonable account of explanation ought to count as explanatory. The debates concerning the relationship between the description of the world provided by fundamental physics and the descriptions given by the nonfundamental sciences are vexed and interesting, but they are orthogonal to the explanatory question under discussion. Indeed, the interventionist account of explanation itself connects explanation with the provision of modal information in the form of answers to specific w-questions, so this itself rules out nonmodal taxonomic information as beside the point from an explanatory point of view. Exactly the same point applies to the idea that nonfundamental sciences “capture patterns” that fundamental physics does not or provide explanations that are “simpler” or more “understandable.”Footnote ¹⁵ In each case, the relevant informational notion, being nonmodal, is simply explanatorily irrelevant.

3.1.3. Conclusion

I conclude that the difference in depth between microscopic and macroscopic explanations cannot be elaborated in terms of the idea that there is information included in macroscopic explanations that microscopic explanations leave out. Either the supposed information is modal, in which case it is not missing, or it is taxonomic, in which case it is explanatorily irrelevant. The informational account of explanatory depth cannot deliver autonomy.

3.2.1. Three Varieties of Explanatory Generality

As I noted earlier, both the amended DN and the interventionist account of explanatory depth focus on a feature of explanations in virtue of which they can be more or less general. While the amended DN view defines depth in terms of the range of possible systems to which an explanatory generalization potentially applies (scope), the interventionist view defines depth in terms of the range of counterfactual questions an explanatory generalization answers (invariance). As I have shown, both views entail fundamentalism since the fundamental laws are exactly those generalizations that are maximally deep in both these senses.

In my view, these views are correct to make a connection between generality and depth but have overlooked the dimension of generality required to secure autonomy. The key point to notice is that both views focus on varieties of generality possessed by the generalizations employed in explanations. The dimension of generality that has been overlooked, I suggest, is one possessed not by explanatory generalizations but by explanations per se. To see the distinction, consider again the microscopic and macroscopic explanations of the pressure in Woodward's example, focusing on a macroscopic explanation that appeals to the ideal gas law $PV=nRT$. The explanation shows how the new pressure P′ of the gas is a function of the temperature T′ and volume V′ of the gas. Since the ideal gas law holds only under a restricted range of microscopic conditions, the microscopic explanation is more general than the macroscopic explanation in terms of both scope and invariance: the situations in which the ideal gas law applies form a subset of those in which the fundamental laws apply, and the w-questions addressed by the ideal gas law form a subset of those addressed by the fundamental laws. However, there is a sense in which the ideal gas law explanation as a whole is more general than the microscopic explanation since the ideal gas law explanation as a whole applies to a wider range of physically possible systems than does the microscopic explanation, which by hypothesis applies to a single type of physically possible system. To adopt the terminology of Garfinkel (Reference Garfinkel1981), the microscopic explanation is in this sense hyperconcrete. Call the degree to which a whole explanation applies to a range of possible situations abstraction. My proposal, which I dub the abstractive account of explanatory depth, is that abstraction provides a theoretically important dimension of explanatory depth.

Note that I do not suggest that there exists a measure of abstraction. Since typically an explanation will be potentially satisfied by an infinite number of physical systems, the provision of a measure of the degree of abstraction of an explanation looks unforthcoming.Footnote ¹⁶ This in turn lends some credence to the intuition that the very idea of comparing explanatory depth across different explananda is incoherent.Footnote ¹⁷ However, a partial ordering of explanations in terms of abstraction is sufficient to secure autonomy. For notice that in Woodward's example, every case in which the microscopic explanation applies is a case in which the macroscopic explanation applies but not vice versa. It follows that the macroscopic explanation is more abstract, whether or not we can compare the abstraction of the macroscopic explanation with the abstraction of other unrelated explanations.

As noted earlier, the abstraction of an explanation is an explanatory virtue that must be traded off against the virtues of scope and invariance. I take it that it is a contingent matter that the world is structured so as to permit a gain in abstraction without significant cost in invariance. The evidence for the world being this way comes from the very existence of nonfundamental sciences and from the particular varieties of explanation I consider in what follows. At least part of the explanation for the world being this way is provided by fundamental physics itself, for example, by way of renormalization group methods (Batterman Reference Batterman2000b).Footnote ¹⁸ However, for my purposes it is important only that this trade-off exists.

3.2.2. Placing Abstraction

It is surprising that the abstractive account of explanatory depth has never been stated in a form free of further theoretical commitments. The basic idea behind the abstractive account of explanatory depth originates with Hilary Putnam (Reference Putnam, Capitan and Merrill1967, Reference Putnam1973, Reference Putnam1975). However Putnam claimed not merely that abstraction is a dimension of explanatory depth but that it dominates other dimensions in all contexts, for some explananda. I see no reason to think this is the case.Footnote ¹⁹

The idea also bears some resemblance to the unificationist account of explanation defended by Philip Kitcher (Reference Kitcher1981, Reference Kitcher, Kitcher and Salmon1989, Reference Kitcher1999), although the motivation for the requirement is different. In contrast with the unificationist criterion for explanatory depth, abstraction does not necessarily result in the minimization of “the number of types of facts that we have to accept as … brute” (Kitcher Reference Kitcher, Kitcher and Salmon1989, 432). Moreover the account is not subject to some important problems for Kitcher's account raised by Woodward and Hitchcock (Reference Woodward and Hitchcock2003b).Footnote ²⁰ The first problem turns on the example of the local temperature of microwave background radiation, which can be repeatedly derived from the assumption that it is homogenous throughout the universe. The problem is that the derivation does not provide answers to any w-questions and so fails to be explanatory. This is not an issue for the abstractive account of depth, which is perfectly compatible with the interventionist theory of explanation. The second problem turns on an example involving two neural mechanisms, N ₁ and N ₂, the first of which is actually pervasive and the second of which is rare, but both of which can be used to answer the same range of w-questions. The problem for Kitcher's account is that the unification afforded by the explanation of N ₁ is greater than that afforded by N ₂, which is in tension with the intuition that the two explanations are equally good. What this highlights is that, as with scope (see n. 7), abstraction should be taken to measure the number of possible situations to which an explanation applies, not the number of actual situations to which it applies. Setting aside worries concerning the comparability of explanations for different explananda, presumably N ₁ and N ₂ are equally abstract in this sense. So again, this is no problem for the abstractive account. The third problem is that it looks as if Kitcher's way of handling explanatory asymmetries rules out the possibility of saying that there can be two explanations of the same phenomena, one more unified and one less unified. This means that degree of unification cannot provide an account of explanatory depth capable of handling cases such as the gas example from Woodward. Again, it is clear that the abstractive account does not have this problem.

The account of explanatory depth most similar to the abstractive account I have outlined is provided by the kairetic theory of explanation due to Strevens (Reference Strevens2004, Reference Strevens2009). Strevens rightly emphasizes the generality of whole explanations, rightly criticizes the unificationist emphasis on actual generality rather than possible generality, and rightly notes some of the explanatory benefits of abstraction. However Strevens, following Putnam, develops his account by drawing a tight connection between abstraction and the notions of explanatory relevance and difference making. I have shown that the abstractive account of depth can be motivated without making this commitment.

3.2.3. Problems for Abstraction

I turn now to a problem concerning disjunctive predicates that arises in one way or another for any account of explanatory depth that ties depth with generality. The problem is that it seems possible to gain generality cheaply by employing disjunctive predicates in either the particular or the general components of an explanation. For instance, suppose we aim to explain why some given configuration of planets obtains. Consider the explanation with a particular component consisting in the disjunction of the initial configuration of the plants and the initial macroscopic configuration of a particular chamber of gas and a general component consisting in the disjunction of Newton's laws and the ideal gas law. Call this gerrymandered law L ₄. This explanation is more general than an explanation employing any of L ₁–L ₃ since it can explain not only the configuration of the planets but the macroscopic properties of gases. But no account of explanatory depth should say that it provides a deeper explanation in either context.

The solution to this problem is straightforward. The problem with the explanation employing L ₄ is that it contains redundant, nonexplanatory information in any particular context in which it is employed. This is in effect the position taken both by Strevens (Reference Strevens2004, 170–72), who describes the redundancy in terms of a loss of cohesion (which demands that a causal model contain as many causally active elements as possible), and by Woodward and Hitchcock (Reference Woodward and Hitchcock2003b, 190), who describe the redundancy in terms of a lack of increase in the interventionist w-questions addressed by the causal model.Footnote ²¹ Either way, the problem is that generality has been bought at the cost of explanatory redundancy. This in turn shows that abstraction is to be understood as not simply requiring maximal explanatory generality but requiring maximal explanatory generality consistent with explanatory nonredundancy.

A second problem may be thought to arise in connection with disjunctive predicates introduced into fundamental explanations.Footnote ²² Notice that the microscopic explanation for the gas pressure involves a particular component consisting in an exact description of the initial positions of all gas molecules and so is hyperconcrete and minimally abstract. However, the corresponding explanation involving a particular component consisting in the infinite disjunction of all physically possible initial positions consistent with the gas pressure is, it may be alleged, exactly as abstract as the macroscopic explanation employing the ideal gas law. In reply, I deny that the disjunctive microscopic explanation is as abstract as the macroscopic explanation. To see this, notice that the ideal gas law is independent of whether the underlying mechanics is Newtonian or quantum mechanical. This means that there are physically impossible systems to which the macroscopic explanation applies but to which the microscopic explanation does not. So the macroscopic explanation is more abstract than the disjunctive microscopic explanation. I do not deny that there exist macroscopic explanations that do not have this feature, in which case they will be explanatorily equivalent to disjunctive microphysical explanations. And of course, in this case we still prefer the macroscopic explanation to the gerrymandered microscopic explanation—but as I argued earlier, this preference is not grounded in a difference in explanatory virtue.Footnote ²³

3.2.4. The Virtues of Abstraction

Why believe that abstraction provides a genuine dimension of explanatory depth? My central claim, of course, is that abstraction provides the best explanation for the truth of autonomy. So far I have defended this claim by way of a single example, in which I showed that abstraction grounds the claim that an ideal gas law explanation is in one respect deeper than the corresponding fundamental explanation for the volume of a gas. It is important to see, however, that the example generalizes. The conditions under which fundamental and nonfundamental explanations will be related in this way are as follows:

• • Every possible situation in which the fundamental explanation applies is a case in which the nonfundamental explanation applies.

• • There are possible situations in which the nonfundamental explanation applies in which the fundamental explanation does not apply.

This will be the case whenever the nonfundamental explanation supervenes on and is multiply realizable by the fundamental explanation.Footnote ²⁴ So it is reasonable to believe that nonfundamental explanations are often autonomous to the extent that it is reasonable to believe that the variables employed in explanations provided by the nonfundamental sciences often supervene on, and are multiply realizable by, the variables employed in fundamental physics. In my view, this makes the autonomy of nonfundamental explanations with respect to fundamental explanations pervasive. Indeed, there is a continuum here corresponding to the traditional hierarchy of the sciences, with physics at the bottom and the social sciences at the top. From the perspective of the abstractive account of explanatory depth, the explanatory strategies of these sciences can be seen to involve different norms concerning the way invariance and abstraction are to be weighed against each other. An explanation in fundamental physics has the advantage of maximizing invariance, while explanations in the nonfundamental sciences have the advantage of variably increasing abstraction.Footnote ²⁵

In making this claim, I should emphasize that I am engaging in a certain idealization. I do not claim that any of the explanations currently provided by the fundamental and nonfundamental sciences are related in this way. Nor do I claim that at the limit of inquiry explanations will be related in this way. Rather, I am making a claim concerning what we might think of as ideally complete explanations in terms of the variables distinctive of a particular science.Footnote ²⁶ In physics, these ideally complete explanations will take the form described by Woodward in his example, and my claim is that abstraction provides a dimension according to which these explanations are inferior to certain nonfundamental explanations. Potochnik (Reference Potochnik2010) has argued convincingly that many, if not most, of the explanations actually provided by more fundamental sciences are not ideally complete in this sense and do not serve as subvenience bases for less fundamental explanations. In her example, the variables composing genetic explanations for the evolution of a trait typically do not provide subvenience bases for the variables composing corresponding phenotypic explanations. Instead of it being the case that one explanation is more abstract than the other, they are abstract in different ways—one abstracts away from phenotype, and one abstracts away from genotype. Both explanations, meanwhile, abstract away from a vast array of microphysical detail. My claim, then, is that abstraction provides the best explanation for the autonomy of nonfundamental explanations with respect to ideally complete fundamental explanations.

It would be nice to have some additional reasons to believe the abstractive account of explanatory depth, and indeed there are many to be found. As it turns out, a large array of explanatory practices and preferences manifested in the sciences can be explained and unified by the abstractive account of explanatory depth.

First, the abstractive account of depth tracks the attitudes of scientists. For instance, as documented by Weisberg (Reference Weisberg2004), Hoffmann (Reference Hoffmann1995, Reference Hoffmann1998) has claimed that more general, less accurate “qualitative models” in chemistry aid understanding in a way not captured by more precise computational models. The abstractive account provides an objective notion of explanatory virtue that justifies this claim, without requiring recourse to subjective notions of simplicity and understandability.Footnote ²⁷

Second, the abstractive account explains why indeterministic explanations can be preferable to deterministic explanations since a gain in abstraction can often be made at the price of predictive accuracy. As Glymour (Reference Glymour1980, 37) writes, it “is a striking feature of scientific reasoning that we are willing to sacrifice a bit of empirical accuracy for a gain in explanatory unification.”Footnote ²⁸

Third, the abstractive account explains why it is that idealizing models possess a distinctive kind of explanatory value since a gain in abstraction can often be made by omitting representational details, or by sacrificing representational accuracy, while only negligibly sacrificing invariance.Footnote ²⁹ Fourth, the abstractive account justifies the explanatory intuitions behind some classic problem cases for theories of causal explanation, in which deep explanations are provided precisely by abstracting away from causal details. I have in mind, in particular, equilibrium explanations and geometric explanations.

In equilibrium explanations, the final state of a system is explained by showing that it is an endpoint of many different initial configurations. Equilibrium explanation was initially brought to the attention of philosophers by Sober (Reference Sober1983), who took as an example Fisher's famous explanation for equilibrium sex ratios in biological populations (see Charnov Reference Charnov1982 for a survey). Other examples abound. The “asymptotics of the first kind” discussed in Batterman (Reference Batterman2000a) are equilibrium explanations. The explanatory advantage provided by the inflationary universe hypothesis over the standard big bang model in cosmology is the advantage of an equilibrium explanation over a nonequilibrium (Maudlin Reference Maudlin2007, sec. 7). And part of the role of the concept of equilibrium in evolutionary game theory is to provide equilibrium explanations in this sense (Skyrms Reference Skyrms and Hull1992, Reference Skyrms2000).

In geometric explanations, geometric facts combine with empirical facts to explain. Examples from folk physics include the inability to untie a knot explained by topological features of the knot (Kitcher Reference Kitcher, Kitcher and Salmon1989, 426) and the geometrical explanation for why a square peg cannot fit into a round hole (Putnam Reference Putnam1973, 295–97). A more sophisticated example along the same lines is Euler's explanation for the impossibility of walking certain kinds of paths over the bridges of Königsberg (Pincock Reference Pincock2007, 257–60). Examples from real physics include dimensional explanations (Lange Reference Lange2009) and explanations invoking space-time geometries (Nerlich Reference Nerlich1979). A particularly lovely example from biology is the explanation for allometric scaling laws due to West, Brown, and Enquist (Reference West, Brown and Enquist1997), which makes central use of the fractal properties of vascular systems.

What is the distinctive explanatory virtue of all of these explanations? In each case, the equilibrium or geometric explanation supervenes on and is multiply realizable by more fine-grained explanations and so is more abstract. According to the abstractive account of explanatory depth, it is the abstraction of equilibrium and geometric explanations that makes for their explanatory virtue—and the mistake of traditional causal theories of explanation such as Lewis's (Reference Lewis1986) is to suppose that explanations can never be (nonpragmatically) improved by omitting rather than incorporating causal details.

Note that many of the explanations I have described provide more examples of explanations that would remain explanatory even if the fundamental laws of nature were different, within a certain set of constraints, from what they actually are. To take just one example, just as the ideal gas law is independent of whether the underlying mechanics is Newtonian or quantum mechanical, the square peg explanation works not just in the nomologically possible worlds but in all possible worlds consistent with basic laws concerning rigid bodies.Footnote ³⁰ It is for this reason that I propose that abstraction is to be understood in terms of the set of logically possible systems compatible with a given explanation. An explanation is deeper in this sense, then, if it applies not merely to a wide range of other nomologically possible situations but to a wide range of logically possible situations.

Finally, we are often interested in explanatory questions that demand an abstracted answer. For instance, to take an example from Batterman (Reference Batterman2002a, 26), suppose we observe a stiff ribbon of steel being gradually loaded up until it buckles. We may be interested in the precise details of how this particular ribbon buckled, but more typically we are interested in general properties of steel with the same macroscopic characteristics and so seek an explanation that would have applied not just to this piece but to all pieces sharing those characteristics.