1. Introduction
Since the mid-2000s, a consensus (or the closest one gets to a consensus in philosophy) has emerged in the literature on the scientific explanations, according to which there are causal and noncausal scientific explanations. I call this claim the ‘liberal consensus’. The liberal consensus has two sources. First, it rests on well-known examples of causal explanations in the natural and social sciences, including detailed mechanistic explanations, especially in the life sciences (Bechtel and Richardson Reference Bechtel and Richardson1993; Machamer, Darden, and Craver Reference Machamer, Darden and Craver2000), and less detailed ‘higher-level’ or ‘macro’ causal explanations (Cartwright Reference Cartwright1989; Woodward Reference Woodward2003). Second, the liberal consensus also gains support from compelling examples of noncausal explanations. Such examples include different kinds of ‘purely’ or ‘distinctively’ mathematical explanations such as graph-theoretic (Pincock Reference Pincock2012; Lange Reference Lange2013b), topological (Huneman Reference Huneman2010; Lange Reference Lange2013b), geometric (Lange Reference Lange2013b), and statistical explanations (Lipton Reference Lipton2004; Lange Reference Lange2013a). Other kinds of noncausal explanations, especially in physics, include explanations based on symmetry principles and conservation laws (Lange Reference Lange2011), kinematics (Saatsi, Reference Saatsi2016), renormalization group (RG) theory (Batterman Reference Batterman2000; Reutlinger Reference Reutlinger2014), dimensional analysis (Lange Reference Lange2009), and intertheoretic relations (Batterman Reference Batterman2002; Weatherall Reference Weatherall2011).
The liberal consensus is not an innocent assumption, because the currently dominating accounts of scientific explanation are causal accounts. According to causal accounts, there is a tight conceptual connection between explaining and identifying or representing causes (see, among many others, Salmon Reference Salmon1984; Cartwright Reference Cartwright1989; Machamer et al. Reference Machamer, Darden and Craver2000; Woodward Reference Woodward2003; Strevens Reference Strevens2008). The common core of seminal causal accounts of explanation can be expressed as follows: to explain some phenomenon P is just to identify the (type or token level) causes of P. The liberal consensus is a direct challenge to causal accounts, because causal accounts of explanation—prima facie—cannot accommodate noncausal explanations; hence, causal accounts do not provide a general account of all scientific explanations (as already noted by van Fraassen Reference van Fraassen1980, 123; Achinstein Reference Achinstein1983, 230–43; Lipton Reference Lipton2004, 32).Footnote 1
This dialectic situation leaves us with the task to come up with a theoretical response to the liberal consensus. In this paper, I will defend one possible (and particularly attractive) strategy for dealing with the liberal consensus: monism—more precisely, I will defend one specific monist approach to explanation, a counterfactual theory of explanation. I take monism to be the view that there is one single philosophical account capturing both causal and noncausal explanations. A monist holds that causal and noncausal explanations share a feature that makes them explanatory. Hempel’s covering law account is an instructive historical example for illustrating monism. Hempel argued that causal and noncausal explanations are explanatory by virtue of having one single feature in common: nomic expectability (Reference Hempel1965, 352). In the case of causal explanations, one expects the explanandum to occur on the basis of causal covering laws (laws of succession) and initial conditions; in the noncausal case, one’s expectations are based on noncausal covering laws (laws of coexistence) and initial conditions. However, owing to well-known problems of the covering law account (Salmon Reference Salmon1989, 46–50), Hempel’s monism is not a viable option for dealing with the liberal consensus.
My goal in this paper is to explore a monist account that does not collapse into Hempel’s untenable version of monism. I claim that a counterfactual theory of explanation is a promising candidate for playing this role (building on and elaborating recent work by Frisch Reference Frisch1998; Bokulich Reference Bokulich2008; Reutlinger Reference Reutlinger2013; Saatsi and Pexton Reference Saatsi and Pexton2013). However, I do not want to argue for full-fledged monism, that is, the claim that the counterfactual theory captures all kinds of noncausal explanations (including the ones listed above). My goal in this paper is more modest: I will argue that the counterfactual theory can be successfully applied to two paradigmatic examples of noncausal explanations, which I take to be representative of a larger class of noncausal explanations: Euler’s explanation and the RG explanation of universality.Footnote 2
The plan of the paper is as follows: In section 2, I introduce the counterfactual theory. In section 3, I argue that the counterfactual theory can be successfully applied to Euler’s explanation and RG explanations of universality.
2. The Counterfactual Theory
Is there a monist alternative to Hempel’s troubled monism? It is fruitful to take a suggestion of Peter Lipton’s as a stepping-stone for developing such a monist account. Having presented several examples of noncausal explanations, Lipton outlines a monist strategy for dealing with (what I call) the liberal consensus: “One reaction to this would be to attempt to expand the notion of causation to some broader notion of ‘determination’ that would encompass the non-causal cases” (Reference Lipton2004, 32). However, Lipton is skeptical as to whether one can prevent such a “broader notion of determination” from collapsing into Hempelian monism: “This approach has merit, but it will be difficult to come up with such a notion that we understand even as well as causation, without falling into the relation of deductive determination, which will expose the model to many of the objections to the deductive-nomological model” (32). I think that Lipton was too hasty in dismissing the merit of explicating “some broader notion of ‘determination’ that would encompass the non-causal cases,” because such a philosophical project need not necessarily rely on the covering law account and a “relation of deductive determination.”
Following Lipton’s original suggestion, my claim is that Lipton’s envisioned broader notion of determination is the notion of the counterfactual dependence (of the explanandum on the explanans), as captured by counterfactual theories of explanation (CTE). Perhaps the most influential version of the CTE is James Woodward’s:Footnote 3 “An explanation ought to be such that it enables us to see what sort of difference it would have made for the explanandum if the factors cited in the explanans had been different in various possible ways” (Reference Woodward2003, 11). And later, “Explanation is a matter of exhibiting systematic patterns of counterfactual dependence” (191). The CTE is appealing from a monist perspective for two reasons: First, Woodward’s (Reference Woodward2003, secs. 5.3 and 5.8) CTE avoids the notorious problems of the covering law account (see below for a qualification of this claim). Second, although Woodward’s version of the CTE—and the underlying interventionist theory of causation—is mainly intended to fit causal explanations, the core idea of the CTE provides a natural way for specifying Lipton’s “broader notion of determination.” As Woodward suggests himself (but does not elaborate), “The common element in many forms of explanation, both causal and non-causal, is that they must answer what-if-things-had-been-different questions” (221). Answering “what-if-things-had-been-different questions” amounts to revealing (or exhibiting)—in Woodward’s words—what sort of difference it would have made for the explanandum if the factors cited in the explanans had been different in various possible ways. The monist proposal, according to the CTE, is that causal and noncausal explanations are explanatory by virtue of exhibiting how the explanandum counterfactually depends on the explanans (13). Or, to put it in Lipton’s terms, the notion of counterfactual dependence is the broader notion of determination that one “expands” from causal explanations such that it encompasses the noncausal explanations. This CTE-based monism has been explored by Frisch (Reference Frisch1998), Bokulich (Reference Bokulich2008), Reutlinger (Reference Reutlinger2013), Saatsi and Pexton (Reference Saatsi and Pexton2013), and Saatsi (Reference Saatsi2016), among others. My goal is to further elaborate and advance the CTE and to apply it to two novel examples of noncausal explanations that proponents of the CTE have not yet addressed.
I will start with reconstructing the CTE in a way that emphasizes the “common element” (Woodward Reference Woodward2003, 221) of causal and noncausal explanations—this common element, I conjecture, is not essentially tied to an interventionist approach to causation (I will return to this issue below). In this reconstruction I largely follow Woodward’s (Reference Woodward2003, 203) and Woodward and Hitchcock’s (Reference Woodward and Hitchcock2003, 6, 18) exposition of the CTE. As a first step, the structure of an explanation has two parts: (1) a statement E about the explanandum phenomenon, and (2) an explanans consisting of generalizations G1, …, Gm and auxiliary statements S1, …, Sn. Auxiliary statements often are statements about initial or boundary conditions specifying the state of the explanandum system (as Woodward and Hitchcock highlight, those statements typically assert that variables take a certain value). But the auxiliary statements may also comprise other kinds of statements useful for explanations (e.g., Nagelian bridge laws, symmetry assumptions, limit theorems, and other modeling assumptions). According to the CTE, the relationship between the explanans and the explanandum is explanatory iff the following conditions are all satisfied:
1. Veridicality condition: G1, …, Gm, S1, …, Sn, and E are (approximately) true.
2. Implication condition: G1, …, Gm and S1, …, Sn logically entail E or a conditional probability P(E|S1, …, Sn)—where the conditional probability need not be ‘high’ in contrast to Hempel’s covering law account.
3. Dependency condition: G1, …, Gm support at least one counterfactual of the following form: had S1, …, Sn been different than they actually are (in at least one way deemed possible in the light of the generalizations), then E or the conditional probability of E would have been different as well.Footnote 4
The CTE provides a monist framework for causal and noncausal explanations—both kinds of explanation are explanatory because they reveal counterfactual dependencies between the explanandum and the explanans.
Let me add two further remarks in order to sharpen the CTE.
First, one may distinguish between causal and noncausal explanations within the CTE framework as follows: noncausal explanations are explanatory by virtue of exhibiting noncausal counterfactual dependencies; causal explanations are explanatory by virtue of exhibiting causal counterfactual dependencies. Although I consider the causal/noncausal distinction as a primitive for present concerns, I have argued for a positive account of how one might draw the distinction (Farr and Reutlinger Reference Farr and Reutlinger2013; Reutlinger Reference Reutlinger2013). Following Bertrand Russell (Reference Russell1912–13) and present-day neo-Russellians, I propose that causal relations are characterized by criteria such as asymmetry, time asymmetry, distinctness of the relata, metaphysical contingency, and so on. These ‘Russellian’ criteria may also be useful for identifying what makes an explanation causal or, respectively, noncausal in the CTE framework: causal explanations reveal counterfactual dependencies instantiating all of the Russellian criteria, while noncausal explanations exhibit counterfactual dependencies not instantiating all of the Russellian criteria. However, the success of the CTE does not depend on my particular proposal for drawing the causal/noncausal line.
Second, the dependency condition can, to a certain extent, be disentangled from an interventionist (or, more generally, causal) interpretation. In the context of causal explanations, Woodward interprets the dependency condition of the CTE in terms of interventionist counterfactuals whose antecedents state, “if there were a possible intervention on the initial or boundary conditions.” Woodward’s critics have recently argued that interventionist counterfactuals are inherently problematic and ultimately dispensable for understanding causation and causal explanation (see Strevens Reference Strevens2007, Reference Strevens2008; Reutlinger Reference Reutlinger2013). Since I want to sidestep this debate, I simply do not assume that the counterfactuals mentioned in the dependency condition need to be understood as interventionist counterfactuals. Woodward himself voices another prima facie convincing reason for not requiring that all explanatory counterfactuals have the form of interventionist counterfactuals: “When a theory or derivation answers a what-if-things-had-been-different question but we cannot interpret this as an answer to a question about what would happen under an intervention, we may have a non-causal explanation of some sort” (Reference Woodward2003, 221). Based on these two reasons, I assume, in the dependency condition, that the CTE should generally rely on noninterventionist counterfactuals of the form “if S1, …, Sn had been different than they actually are (in at least one way deemed possible in the light of the generalizations), then E or the conditional probability of E would have been different as well.”Footnote 5 There is a positive analogy to Woodward’s causal CTE regarding the existentially quantified form of the counterfactuals: the qualification ‘in at least one way deemed possible in the light of the generalizations’ is analogous to the interventionist requirement that there be some possible intervention on the antecedent variable that leads to a change in the consequent variable; it is not required that all possible interventions have such an effect.
I anticipate a potential worry at this point. One may wonder whether the noncausal version of the CTE avoids the problems of the covering law account. I assume here that the causal version of the CTE successfully solves these problems. I can only sketch how the noncausal version of the CTE responds to these problems. (a) To avoid counterexamples such as the birth control pills scenario (Salmon Reference Salmon1989, 50), the noncausal CTE distinguishes between explanatorily relevant and irrelevant factors as follows: a factor is relevant if the explanandum counterfactually depends on it (dependency condition); otherwise, it is an irrelevant factor. (b) To deal with the syphilis-paresis scenario (49), the noncausal CTE allows for low-probability explanations (implication condition). Finally, having the notorious flagpole-shadow scenario in mind (47), one may wonder whether the CTE accounts for the explanatory asymmetry in the case of noncausal explanations. This is one of the deepest puzzles of the current philosophy of explanation—not merely affecting the CTE—and it is an open research question as to how one can capture the explanatory asymmetry in the noncausal cases. It is even possible that noncausal explanations do not generally display such an asymmetry. This complex question will have to be addressed in another paper (for one recent attempt to make progress on the issue, see Lange Reference Lange2011).
3. Applying the Counterfactual Theory
I will now argue for the claim that the CTE applies to two paradigmatic examples of noncausal explanations. I will apply the CTE first to Euler’s explanation (sec. 3.1) and then to RG explanations of universality (sec. 3.2).
3.1. Euler’s Explanation
Let me start with an intuitively simple but powerful noncausal explanation (see Pincock Reference Pincock2012, 51–53; Lange Reference Lange2013b, 489). In 1736, Königsberg had four parts of town and seven bridges connecting these parts. Interestingly, no one ever succeeded in the attempt to cross all of the bridges exactly once. This surprising fact calls for an explanation. The mathematician Leonhard Euler provided an explanation. Euler’s explanation starts with representing relevant aspects of Königsberg’s geography with a graph. A simplified geographical map of Königsberg in 1736 represents only the four parts of town (the two islands A and B, and the two riverbanks C and D) and the seven bridges (part A is connected to five bridges, and parts B, C, and D are each connected to three bridges). This simplified geography of Königsberg can also be represented by a graph, in which the nodes represent the parts of town A–D and the edges represent the bridges.
Given this graph-theoretical representation, Euler defines an Euler path as a path through a graph G that includes each edge in G exactly once. Euler uses the notion of an Euler path to reformulate the explanandum in terms of the question, why has everyone failed to traverse Königsberg on an Euler path? His answer to this why question has two components:
1. Euler’s theorem: there is an Euler path through a graph G iff G is an Eulerian graph. Euler proved that a graph G is Eularian iff (i) all the nodes in G are connected to an even number of edges, or (ii) exactly two nodes in G (one of which we take as our starting point) are connected to an odd number of edges (Pincock Reference Pincock2012, 51).
2. Contingent fact: The actual bridges and parts of Königsberg are not isomorphic to an Eulerian graph, because conditions i and ii in the definition of an Eulerian graph are not satisfied: no part of town (corresponding to the nodes) is connected to an even number of bridges (corresponding to the edges), violating condition i; and more than two parts of town (corresponding to the nodes) are connected to an odd number of bridges (corresponding to the edges), violating condition ii. Königsberg could have been isomorphic to an Eulerian graph in 1736, but as a matter of contingent fact it was not.
Therefore, Euler concludes, there is no Euler path through the actual Königsberg. This explains why nobody ever succeeded in crossing all of the bridges of Königsberg exactly once.
Does the CTE capture Euler’s explanation? Euler’s explanation has the structure demanded by the CTE: the explanans consists of Euler’s theorem (a mathematical and intuitively noncausal generalization) and the statement that all parts are actually connected to an odd number of bridges. The explanandum phenomenon is that everyone has failed to cross the city on an Euler path. Moreover, all three conditions that the CTE imposes on the relation between explanans and explanandum are satisfied:
1. The veridicality condition holds because (a) Euler’s theorem, (b) the statement about the ‘contingent fact’ that each part of Königsberg is actually connected to an odd number of bridges, and (c) the explanandum statement are all true.
2. The implication condition is met, since Euler’s theorem and the statement about the ‘contingent fact’ entail the explanandum statement.
3. The dependency condition is satisfied, because Euler’s theorem supports the counterfactual “if all parts of Königsberg were connected to an even number of bridges, or if exactly two parts of town were connected to an odd number of bridges, then people would not have failed to cross all of the bridges exactly once.”
Therefore, I conclude that the CTE applies to Euler’s explanation.
3.2. Renormalization Group Explanations
So-called RG explanations constitute another, technically more sophisticated, kind of noncausal explanation (see Batterman Reference Batterman2000, Reference Batterman2002).Footnote 6 RG explanations are intended to provide understanding of why microscopically different physical systems display the same macrobehavior when undergoing phase transitions. For instance, near the critical temperature, the phenomenology of transitions of a fluid from a liquid to a vaporous phase, or of a metal from a magnetic to a demagnetized phase, is (in some respects) the same, although liquids and metals are significantly different on the microlevel. This ‘sameness’ or—to use a more technical term—‘universality’ of the macrobehavior is characterized by a critical exponent that takes the same value for microscopically very different systems (Batterman Reference Batterman2000, 125–26). How do physicists explain the remarkable fact that there is universal macrobehavior?
It is useful to understand the workings of RG explanations as consisting of three key explanatory elements: (1) Hamiltonians, (2) RG transformations, and (3) the flow of Hamiltonians. There is a fourth element—the laws of statistical mechanics, including the partition function—which I will leave in the background, for sake of brevity (Norton Reference Norton2012, 227; see also Wilson Reference Wilson1983). The exposition of these elements will be nontechnical because the paper is concerned with a nontechnical question (see Batterman Reference Batterman2000, 137–44; for a more detailed exposition, see Fisher Reference Fisher1982, Reference Fisher1998; Wilson Reference Wilson1983).
1. Hamiltonians: The Hamiltonian is a function characterizing, among other things, the energy of the interactions between the components of the system. One characteristic of the Hamiltonian of a physical system undergoing phase transition (say, a heating pot of water undergoing a transition from a liquid to a gaseous phase) is that each component of such a system does not merely interact with its nearby neighbors but is also correlated with distant components; in fact, the correlation length diverges (and becomes infinite). Adopting Batterman’s terminology, I call this complicated Hamiltonian of a system undergoing a phase transition the “initial” or “original” Hamiltonian.
2. Renormalization group transformations: Keeping track of the interactions and correlations between all the components of a system undergoing a phase transition is—given the large number of components and the diverging correlation length—practically impossible. So-called RG transformations deal with this intractability by redefining the characteristic length, at which the interactions among the components of the system at issue are described. Repeatedly applying RG transformations amounts to a redescription of the system, say, fluid F, on larger and larger length scales (while preserving the mathematical form of the original Hamiltonian). The transformed Hamiltonian describes a system (and the interactions between its components) with fewer degrees of freedom than the original Hamiltonian. In sum, the RG transformation is a mathematically sophisticated coarse-graining procedure eliminating microdetails that are irrelevant for the explanation of universality.
3. The flow of Hamiltonians: Suppose we start with the original Hamiltonian H of a fluid F undergoing a phase transition. Then, one repeatedly applies the RG transformation and obtains other more ‘coarse-grained’ Hamiltonians. Interestingly, these different Hamiltonians “flow” into a fixed point in the space of possible Hamiltonians, which describes a specific behavior characterized by a critical exponent (Batterman Reference Batterman2000, 143). Now suppose there is another fluid F* and its behavior (during phase transition) is described by the initial Hamiltonian H*. Repeatedly applying the RG transformation to H* generates other, more ‘coarse-grained’ Hamiltonians. If the Hamiltonians representing fluid F* and fluid F turn out to “flow” to the same fixed point, then their behavior, when undergoing phase transition, is characterized by the same critical exponent (Fisher Reference Fisher1982, 85; Batterman Reference Batterman2000, 143).
In sum, the three elements of an RG explanation allow us to determine whether systems with different original Hamiltonians belong to the same “universality class” and are characterized by the same critical exponent (Fisher Reference Fisher1982, 87). Two systems belong to the same universality class if reiterating RG transformations reveals that both systems “flow” to the same fixed point.
Now the decisive question is whether the CTE applies to RG explanations. First, RG explanations exhibit the required structure. The explanandum phenomenon is the occurrence of universal macrobehavior. The explanans of an RG explanation consists of the system-specific Hamiltonians describing the energy state of the physical systems in question—and, strictly speaking, the laws of statistical mechanics (the fourth element in the background); RG transformations and the flow of Hamiltonians are central auxiliary assumptions in the explanans (see sec. 2).
Second, the conditions that the CTE requires to hold are also satisfied:
1. The veridicality condition is satisfied, because the explanandum statement (that there is universal behavior) and the explanans can—at least for present purposes—be considered as being (approximately) true. Owing to space limits, I cannot discuss the role of idealizations (especially, limit theorems) in RG explanations posing a potential threat to the truth of the explanans. However, there are interpretations of the idealizations in question that are consistent with the veridicality condition (see Strevens Reference Strevens2008; Norton Reference Norton2012).
2. The implication condition holds, since the RG explanans entails that many physical systems with different original Hamiltonians display the same macrobehavior.
3. The dependency condition is met, because the RG explanans supports some counterfactuals of the form “There is a physically possible Hamiltonian H* such that if (1) a physical system had the original Hamiltonian H* (instead of its actual original Hamiltonian H), (2) H* were subject to repeated RG transformations, and (3) we determined the resulting flow of the Hamiltonians to a fixed point, then a system with original H* would be in a different universality class than a system with original Hamiltonian H.” Let me elaborate why I believe that some counterfactuals of this form are true in the light of RG theory. To avoid misunderstandings, a counterfactual of this form is not true for every physically possible (original) Hamiltonian, because the main accomplishment of RG explanations is to show that many systems with different original Hamiltonians belong to the same universality class. However, the dependency condition of the CTE does not require that the explanandum depend on all possible changes in the initial conditions. Instead, the condition merely requires that the explanandum counterfactually depend on some possible changes in the explanans. The latter claim receives support from RG theory, which (also) shows that and why some systems with different original Hamiltonians do not exhibit the same macrobehavior and in fact belong to different universality classes (Wilson Reference Wilson1983). As Batterman (Reference Batterman2000, 127) points out, RG explanations reveal that belonging to a particular universality class depends on features such as the symmetry properties of the order parameter and the spatial dimensionality of the physical system. Hence, if systems with H* and H—figuring in the counterfactual above—differ with respect to those features, then the counterfactual at issue seems to be true, according to RG theory.
Therefore, the CTE successfully captures RG explanations.
4. Conclusion
A ‘liberal consensus’ has emerged in the recent philosophy of scientific explanation: there are causal and noncausal explanations. In order to deal with the liberal consensus, I have argued for the counterfactual theory of explanation (the CTE). According to the CTE, causal and noncausal explanations are explanatory by virtue of revealing counterfactual dependencies between the explanandum and the explanans. I have argued that the CTE is applicable to two paradigms of noncausal explanations: Euler’s explanation and RG explanations of universality. For this reason, I believe that the CTE is a promising monist approach that deserves more attention and discussion.
