Explanatory Abstractions

Explanatory Abstractions Abstract A number of philosophers have recently suggested that some abstract, plausibly non-causal and/or mathematical explanations explain in a way that is radically different from the way causal explanations explain. While causal explanations explain by providing information about causal dependence, some abstract explanations allegedly explain in a way tied to the independence of the explanandum from the micro-details or causal laws, for example. We oppose this recent trend to regard abstractions as explanatory in some sui generis way, and argue that a prominent account of causal explanation can be naturally extended to capture explanations that radically abstract away from micro-physical and causal-nomological details. To this end, we distinguish different senses in which an explanation can be more or less abstract, and analyse the connection between an explanation's abstractness and its causal power. According to our analysis, abstract explanations have much in common with counterfactual causal explanations. 1 Introduction 2 Three Dimensions of Abstraction 3 Explanatory Power and Independence 4 Abstraction in a Counterfactual Framework 5 Königsberg—Encore! 6 Conclusion 1 Introduction There is broad agreement that many explanations derive their explanatory power from information about dependence. Causal explanations involve causal dependence, and various kinds of non-causal explanations can, arguably, be similarly understood in terms of non-causal dependence of the explanandum on the explanans.1 Indeed, it may be that (almost) all explanations involve information about what the explanandum (causally or non-causally) depends on. This hypothesis, if tenable, could be part of a unified account of explanation, according to which explanations—causal or otherwise—explain by virtue of providing information about dependences. Against this hypothesis, consider explanatory abstractions. Explanations vary in their abstractness: some explanations appeal to relatively abstract features of reality, whereas others turn on much less abstract explanans. We value explanatory abstraction, when appropriate. It filters out details that are irrelevant to the explanation in question. But abstraction does not directly concern what the explanandum depends on. Rather, abstraction is about the independence of the explanandum from irrelevant details. It is tempting to think, along with various recent authors, that this information about independence is where the explanatoriness of many abstract explanations lies. This mode of explaining, associated with what is irrelevant to the explanandum, can seem sui generis, different in kind from explanations in terms of dependences. We will argue that explanatory abstractions do not point to a sui generis mode of explaining. There is no need to revise the idea that explanatory power is a matter of providing dependence information (and the more the better); no need to abandon the hope for a unified account of explanatory power. Here we stand against a clear trend that urges the opposite: that some abstract explanations do not fit an account of explanatory power that is focused on (non-)causal dependence.2 More specifically, we aim to undercut prominent arguments supporting the trend we oppose. We will present and argue for a unified view that extends the account of causal explanation developed by Woodward ([2003]) and Woodward and Hitchcock ([2003a], [2003b]) to allow it to capture influential exemplars of abstract explanations behind the anti-unificationist trend. Before we get to the nitty-gritty, let us illustrate explanatory abstraction and sketch some of the key points of the article. The explanation of why Mother fails to divide her twenty-three strawberries equally among her three children (without cutting any strawberries) turns on the fact that twenty-three is not evenly divisible by three.3 Citing the causal details or laws involved in Mother’s attempt or the physical constitution of the strawberries does not suffice to explain this fact. The explanation is independent of such details; the explanatory mathematical facts hold irrespective of contingent causal laws and details. So where does the explanatory power associated with mathematics come from? Mathematical facts do not seem to capture dependence relations between numbers in a way analogous to contingent causal and nomological connections. So perhaps the explanatory contribution of mathematics here cannot be captured in terms of dependences? Perhaps the explanatory contribution of mathematics is somehow sui generis, turning on the independence of mathematics from the details of the physical constitution and the causal laws? We will be concerned with explicating and rebutting this intuition. We see little reason to shift focus from dependence to independence in analysing abstract mathematical explanations of this kind. One easily overlooks a wealth of dependence information that we take for granted here. The explanation does not just tell us that Mother must fail as a matter of mathematical necessity. It also tells us what the failure depends on—the number of strawberries—and how. For example, if Mother had two fewer strawberries or one strawberry more, then she could have succeeded. This change-relating, what-if-things-had-been-different information is exactly the kind associated with explanatory dependence.4 We will argue that this information is doing all the explaining, and that this is captured by a natural extension of the ideology of causal explanation, according to which explanations function by providing what-if-things-had-been-different information. It has been broadly thought that this ideology is ill-fitting for capturing abstract non-causal and/or mathematical explanations, since the relevant modal and counterfactual notions only apply to causal connections.5 This, we will argue, is mistaken. The reasons given for leaving behind the counterfactual dependence ideology associated with causal explanations in connection with abstract explanations are faulty. And so are the arguments for sui generis, independence-based accounts of abstract explanations. The key to seeing the latter is to recognize, first of all, that practically all explanations have some degree of abstraction and provide some dependence information. What we need is a way of gauging the explanatory importance of dependence information, relative to information about independence. We will argue that the explanatory importance of dependence information is revealed by varying the information while keeping the degree of abstraction fixed. In the context of the strawberry case, consider removing the dependence information altogether, pretending that we can take away all dependence information based on background arithmetical knowledge, and only retain information about the explanandum’s independence from causal laws and features. If all we know is that, as a matter of mathematical necessity, Mother's attempt with twenty-three strawberries must fail, so that we are unable to answer any questions about possible changes that would render her success possible, then it is not clear that we have any explanation of Mother's failure. Consider, now, adding a little bit of information about dependence, pretending that we only get to know that Mother could have succeeded with twenty-four strawberries. Assume that this is all the information we have, besides knowing that twenty-three is not divisible by three. Here we may have an explanation, but it looks like a very thin one. Yet, in terms of independence from particular causal laws, processes, and micro-physical conditions, it is on a par with the general explanation that appeals to background arithmetical knowledge. If the original explanatory information explains by virtue of showing the explanandum to be independent of physical details and laws, then we would expect the modified explanation to explain to a commensurate degree. But we do not see this. When we vary the amount of information about dependence, we also seem to vary the degree of explanatoriness, in a way that suggests that it has little to do with independence information. To develop the line of thought sketched above we need to lay some conceptual groundwork. To this end, we distinguish three different dimensions of abstraction (Section 2). Then we show how these distinctions apply to prominent recent conceptions of explanatory abstraction (Section 3). Our key claim is that not all dimensions of abstraction are responsible for explanatoriness, even if good explanations are typically abstract along several dimensions. Against the backdrop of these conceptual distinctions, we will then present an alternative counterfactual account of the explanatoriness of abstract explanations in the spirit (but not to the letter) of the dependency-oriented account developed by Woodward ([2003]) and Woodward and Hitchcock ([2003a], [2003b]). In particular, we focus on how to select the right what-if-things-had-been-different counterfactuals (Section 4). The virtues of this unified account are illustrated by revisiting a graph-theoretic explanation of Königsberg’s bridges (Section 5), and further underlined in relation to other challenges faced by less unified views of explanation (Section 6). 2 Three Dimensions of Abstraction We will challenge the idea that explanatoriness turns on the independence of the explanation from particular physical laws, processes, or concrete physical structures. Nonetheless, we think that abstract explanations typically do exhibit this kind of independence. The task of this section is to distinguish three conceptually different ways in which explanations can be abstract. They will all involve independence in one sense or another. However, we will later argue that only one kind of independence carries information about explanatory dependence and, as a consequence, contributes to explanatory power. Let us recall some simple examples of abstract, plausibly non-causal mathematical explanations (what we have been calling ‘abstract explanations’) in order to bring out intuitions about different senses of explanatory abstraction. The following examples have featured prominently in recent philosophical analyses: (i) Economical bees build hexagonal honeycombs because this is the most resource-efficient way to divide a Euclidean plane into regions of equal area with least total perimeter (Lyon and Colyvan [2008]). (ii) Plateau’s laws hold for soap film geometry, because this geometry minimizes the system’s energy by minimizing the surface area (Lyon [2012]; Pincock, [2015]). (iii) Königsberg cannot be toured by crossing each and every bridge exactly once because of the relational configuration of the bridges (Pincock [2007]). (iv) The strawberry case discussed above. We will focus on (iii) and (iv). Abstract explanations hinge on (relatively) abstract features of reality that are non-causal (or perhaps causal in a very broad sense of ‘causal’).6 This abstractness is partly a matter of relative independence of the explanans from the actual physical structure of the entities involved: strawberries, bridges, soap films, honeycombs, and so on. This is the first intuitive aspect of explanatory abstraction that we wish to highlight. There is nothing special about the physical nature of strawberries, for example, that makes them indivisible in this way. The very same explanation explains why twenty-three marbles (say) cannot be evenly divided. The strawberry explanation abstracts away from whatever concrete features need to be in place in order for there to be a set of twenty-three ‘individuals’ that can be divided into mutually exclusive and collectively exhaustive subsets. A second, more radical aspect of abstraction pertains to the fact that the explanations above are independent from the actual laws of nature that underlie the physical processes presupposed by the why-question: the processes of group division, bridge crossing, or comb building, for example. There is nothing special, as far as these explanations are concerned, about the actual physics that underlies the solidity of honeycomb walls, strawberries, or bridges. The same explanations would work just the same in more exotic, ‘counter-legal’ possible worlds where the underlying physical laws are rather different (assuming the relevant why-questions still make sense). The explanation of Plateau’s laws, for example, transcends the nomologically possible causes or grounds of the area-minimizing action of soap bubbles. This explanation is modally robust with respect to variation in the underlying physical laws, as long as the laws still give rise to area-minimizing action for some surfaces. To this extent the explanation abstracts away from the actual nomological facts pertaining to the systems in question. So far, we have identified a conceptual distinction between two senses of abstraction. There is also a third aspect of abstraction in the above explanations that is easily missed. This has to do with the explanatory regularities involved, and the degree to which an explanatory generalization is independent from the actual value of an explanans variable. For example, the fact that twenty-three is not evenly divisible by three is an instance of the more general fact that no integer apart from whole multiples of three is evenly divisible by three. In so far as we judge that the explanatory work is really done by this more general arithmetical background knowledge, we can say that this explanation would answer the question of why Mother would have failed had she had twenty-two strawberries (or why she could have succeeded had she had twenty-one). The explanation turns on a regularity that is robust by virtue of allowing the same explanation to be given for a wide range of changes in the number of strawberries. In this third sense of abstraction, the wider this range is, the more abstract the explanation. These three intuitive aspects of explanatory abstraction all have something in common, namely, each involves a way that an explanation can be independent of some feature of the system in question. This independence is naturally thought of as follows: An explanation can cover a range of possible systems. Explanatory abstraction is a matter of independence of the explanation from the details of ‘realization’. In the above cases of abstract explanations, the extent of unnecessary details omitted is striking. Consider the Königsberg case, for instance. The explanation is independent of all the physical and geometrical features of the bridges and the crossing processes, and of the underlying nomological facts pertaining to the bridges and their possible crossings, only relying on a very high-level, global, structural feature associated with a wide-ranging explanatory regularity. (We will make this more precise later in the article.) The extent of such independence is most striking in the case of abstract explanations, but more or less any explanation actually exhibits the three dimensions of abstraction to some degree. Consider the mundane case of gravitational pendulums, for instance. Why is the period of a given (more-or-less) ideal pendulum two seconds? Explaining this causally, in terms of the pendulum's length and gravitational acceleration, relies on a law-like regularity that supports the same explanation in a range of possible cases in which these two parameters will vary (the third dimension of abstraction). Furthermore, the explanation can apply regardless of whether the underlying nomological facts are rooted in a classical Newtonian world with gravitational force acting at a distance, or whether the pendulum occupies a general relativistic world with gravity being a manifestation of curved spacetime (the second dimension of abstraction). Finally, the why-question presupposes that the pendulum cord is inextensible, but the explanation is independent from whatever microstructural facts underlie this property (the first dimension of abstraction). In light of the fact that more or less all explanations are abstract to some degree along all three dimensions, it is prima facie surprising that there would be two radically different sources of explanatory power in play: one for the cases that are commonly thought of as causal and a different source for cases of abstract explanation. We will argue that the difference between this mundane causal explanation and abstract explanation is one of degree, not of kind. The notion of explanatory abstraction as a matter of independence is clearly a modal notion: it concerns the range of possible systems covered by the explanation. Different aspects of explanatory abstraction concern different dimensions of modal variation with respect to these possible systems. We can think of possible systems that vary from the actual explanandum only in the (nomologically) possible physical structures of the entities involved; possible systems that vary in the microphysical or dynamical laws they obey; possible systems that vary from the actual explanandum by varying a variable that explicitly features in the explanatory regularity employed. Although modal variations along these different dimensions are often interlinked and metaphysically intertwined in complex ways, the dimensions themselves are conceptually independent.7 In the strawberry case where we considered the explanatory regularity with respect to a system of twenty-three versus twenty-five strawberries, one need not think about variation in the physical structures of these entities or the division processes. Similarly, in thinking about the explanation with respect to systems that vary in the physical structures of the entities to be divided, one need not think of variation in their number, or of counter-legal possibilities where the underlying physical laws are different. And vice versa. Often abstract non-causal and mathematical explanations—like the ones above—exhibit a striking level of abstraction in each of the three senses. We believe this has confounded some recent commentators. If one thinks, prima facie, that abstraction in a particular sense is the source of explanatoriness, it is all too easy to find corroborating evidence when so many good explanations are highly abstract in this sense. Now that we have conceptually separated the different dimensions of abstraction, however, we can ask if explanatory abstraction has something to do with explanatory power (as is commonly accepted), then exactly which dimension or dimensions of abstraction can contribute to it? In its full generality, we cannot hope to comprehensively answer this question in a single paper. There is, however, a narrower question that we can answer. It is commonly agreed that explanation (and the associated notion of explanatory power) has both worldly and pragmatic or communicative aspects. Here, we will largely set aside the communicative and pragmatic aspects. What is at stake in the debate at hand is whether to function as a source of explanatory power, abstract explanations require facts tied to independence, instead of (or in addition to) facts about dependence. In particular, is it the case that for the paradigmatic cases of abstract explanations (mentioned above), explanatory power is underpinned by worldly facts related to independence? Our first aim is to undermine prominent recent arguments that answer in the affirmative to the last question. Our second, positive aim is to show how these abstract explanations can be accommodated in more unified terms, in the spirit of the counterfactual account developed by Woodward and Hitchcock ([2003a], [2003b]) (henceforth, the W-H account). According to the W-H account, explanations provide a particular kind of information about dependence. This account creates an essential association between explanatoriness and our third dimension of abstraction. (Roughly, an explanatory generalization that has a broader range under which we can vary the values of the variables provides more explanatory information about the dependence of the explanandum on these variables than a corresponding explanatory generalization with a more restricted range of invariance.) This is in sharp contrast to recent claims according to which (some) abstract explanations explain either (i) by virtue of abstracting away from all the concrete physical features of the system in question (our first dimension above), or (ii) by virtue of abstracting away from the underlying laws of nature (our second dimension). The W-H account is well liked in connection with various causal explanations (even amongst the advocates of alternative analyses of abstract explanations), but there has been resistance to extending this counterfactual account to abstract explanations that are (plausibly) non-causal or mathematical. We will argue that this resistance is entirely unnecessary. In the spirit of the W-H account, issues concerning explanatory power and its relation to abstraction are tractable in very natural terms for abstract explanations too, after the key notions of this account are appropriately clarified and refined. Next, we will critically review some recent claims regarding explanatory abstraction, questioning the intuitions that support them. After this, we will discuss more systematically the third dimension of abstraction, which has a better claim for association with the source of explanatory power than either of the first or second dimensions. 3 Explanatory Power and Independence Various philosophers have recently argued (in different ways) for the disunity of explanatoriness by pointing to the sui generis role of abstraction as a source of explanatory power. We will focus on two authors who particularly clearly associate explanatory power with either the independence from concrete physical details (the first dimension), or the independence from the particular laws (the second dimension). Pincock ([2007], p. 257) brought the Königsberg bridge example into philosophical prominence by identifying it as an abstract explanation that ‘appeals primarily to the formal relational features of a physical system’. Relational features that are ‘formal’ are clearly meant to stand apart from causal relations. Furthermore, regarding the explanation of the impossibility of touring the town by crossing each and every bridge exactly once, Pincock ([2007], p. 259) writes: […] an explanation for this is that at least one vertex [in the formal graph structure instantiated by the bridges] has an odd valence. Whenever such a physical system has at least one bank or island with an odd number of bridges from it, there will be no path that crosses every bridge exactly once and that returns to the starting point. If the situation were slightly different [so that] the valence of the vertices were to be all even, then there would be a path of the desired kind. This is all surely correct. The question is how to capture this explanation in philosophical terms. Pincock ([2007], p. 260) intimates that the explanation critically turns on abstraction: The abstract explanation seems superior [to a microphysical explanation] because it gets at the root cause of why walking a certain path is impossible by focusing on the abstract structure of system. Even if the bridges were turned into gold, it would still have the structure of the same graph, and so the same abstract explanation would apply. By abstracting away from the microphysics, scientists can often give better explanations of the features of physical systems. The notion of ‘abstracting away from the microphysics’ is naturally construed as being along the first dimension identified above: a matter of relative independence of the explanans from the physical structure of the entities involved. As Pincock ([2011], p. 213) puts it: ‘the explanatory power [in this case] is tied to the simple way in which the model abstracts from the irrelevant details of the target system’. The intuition is that the explanation comes from stripping away as irrelevant all the physical details pertaining to the bridges’ make-up, length, location, angles, and so on, thereby highlighting what is relevant, namely, the formal ‘mathematical structure found in the target system itself’ (Pincock [2011], p. 213). We can see Pincock’s intuition, but we are unable to see a good reason for viewing this as a sui generis abstract explanation. The explanation is undeniably highly abstract and plausibly non-causal. But the notion that good explanations only provide relevant information, leaving out unnecessary details, is common ground with dependence accounts of explanation. According to dependence accounts one ought to provide suitable information about what the explanandum depends on, and one also ought not to claim (or imply) that the explanation depends on something that it does not. The leaving out of irrelevant detail is insufficient to motivate the view that abstract explanations are sui generis in the way Pincock regards them. How about the notion that the explanatorily relevant features of Königsberg are ‘formal-cum-mathematical’? Does this motivate a departure from familiar counterfactual accounts of explanation? We do not think so. We do not regard the explanans formal or mathematical in any sense that makes the application of counterfactuals difficult. The explanation involves applied mathematics; it is not an intra-mathematical explanation. There is a clear sense in which the explanans is just concerned with how many bridges there are to each of the islands. It is even unclear why the valence of an island—there being an even or odd number of bridges to it—is not a high-level physical feature of it. And the same is true for the yet more abstract feature of there being at least one odd island in the whole system of many islands. Furthermore, there seems to be a clear and straightforward sense in which the explanandum at stake depends on these, arguably high-level physical, features: the explanation tells us how the explanandum would change if there were a different number of bridges (as Pincock himself notes; see the first quotation above). Whether this explanatory dependence is causal or not is neither here nor there for the prospects of sticking to the core idea of the W-H account of explanation: explanatoriness is ultimately a matter of telling us what the explanandum depends on.8 When it comes to identifying the worldly source of explanatory power, we have not yet been given a reason to think that in the Königsberg case explanatoriness derives (even partly) from the independence of the explanandum from some worldly features. Let us now move on to consider Lange’s analysis of the explanatory power of the cited abstract explanations (as ‘distinctly mathematical’).9 Lange ([2013], pp. 486–8) argues that his analysis of these explanations reveals ‘a fundamental difference’ between causal explanation and abstract non-causal explanation, and its significance ‘lies in what it reveals about the kinds of scientific explanations there are’. Abstract explanations, Lange argues, explain not by supplying information about the world’s network of causal relations but by ‘showing how the fact to be explained was inevitable to a stronger degree than could result from the causal powers [actually] bestowed by the possession of various properties’. This is a nod towards Wesley Salmon’s ([1984], p. 293) ‘modal conception’ of scientific explanations, according to which such explanations ‘do their jobs by showing that what did happen had to happen’. As Lange ([2013], p. 505) puts it: [They explain] not by describing the world’s actual causal structure, but rather by showing how the explanandum arises from the framework that any possible causal structure must inhabit, where the ‘possible’ causal structures extend well beyond those that are logically consistent with all of the actual natural laws there happen to be. According to Lange, abstract explanations may happen to also provide modal information that a counterfactual account (in the spirit of the W-H account) would count as explanatory. But he explicitly rejects the idea that their explanatoriness in any way derives from such modal information, relating it instead to information about the independence of the explanandum from some contingent features of the system in question.10 In contrast to Pincock, the critical sense of independence for Lange is in the direction of the second dimension of abstraction identified above: abstraction away from the actual laws underlying the features of the system presupposed in the context of the relevant why-question. In relation to the Königsberg case, for instance, Lange ([2013], pp. 505–6) writes: [The] explanation of the repeated failure to cross the Königsberg bridges shows that it cannot be done (where this impossibility is stronger than physical impossibility) […] The explanans consists not only of various mathematically necessary facts, but also […] of various contingent facts presupposed by the why question that the explanandum answers, such as that the arrangement of bridges and islands is fixed. The distinctively mathematical explanation shows it to be necessary (in a way that no particular force law is) that, under these contingent conditions, the bridges are not crossed. Lange’s account thus emphasizes the way in which the actual physical laws do not come into play in explanations such as this. Undoubtedly, Lange is right to note that the sense of abstraction at play goes beyond the first dimension concerning the irrelevance of nomologically possible physical realizations of the contingent features presupposed in the context of the why-question. The same explanation would work, in the same way and to same extent, in a far removed possible world with alien properties and laws—as long as there is a system with traversible bridges for which the why-question makes sense. But having said that, why should we regard this sense of abstraction as a source of explanatoriness in connection with abstract explanations? The answer to this question is surprisingly difficult to find in (Lange [2013]).11 There is little to directly motivate this, beyond the commonplace idea that good explanations do not mention irrelevant details, which in this case include all causal laws. Yet again, the idea that good explanations only provide relevant information, leaving out unnecessary details, is common ground with any account according to which we ought to provide information about dependence, and we ought not to claim (or imply) that the explanandum depends on something that it does not. Thus the independence from the actual laws seems insufficient in itself to motivate the view that abstract mathematical explanations are sui generis in the sense of Lange. Both Lange and Pincock are contrasting their views to causal accounts of explanation. It is natural, then, to take them to provide a competing account of the worldly source of explanatory power. So far we have undermined broad motivations for thinking that abstract explanations require a source of explanatory power that is not centred on dependence. Our view is that two of the three dimensions of abstraction—even though undeniably exhibited by abstract explanations—do not actually provide a source of explanatoriness.12 The problem is that these dimensions of abstraction do not provide information about explanatory dependence in the W-H sense. We do not, unlike Lange and Pincock, take it to be the case that explanations that are highly abstract in the first two dimensions require an account of explanation where the source of explanatory power is not in the W-H spirit.13 In the rest of the article we will discuss this specific notion of explanatory dependence and how it connects to the third dimension of abstraction. But before we move on, let us address a natural worry. It may seem that dependence and independence are conceptually so closely connected that we cannot cleanly separate the two. In particular, any information about independence is also information about dependence: ‘E is independent from A’ means that ‘E does not depend on A’, and if we assume that explanandum E depends on something, then ‘E is independent from A’ entails that ‘E depends on something other than A’. Thus it is the case, the worry goes, that information about independence along every dimension of abstraction can provide information about dependence. In response, we simply note that it is not true that information about independence ipso facto provides information about dependence that is explanatory—that is, dependence information in the W-H sense. It is undeniably true that one provides what-if-things-had-been-different information, broadly speaking, by showing that even if the laws of nature were different, Mother could not divide twenty-three strawberries equally among three children. (What if things had been different with respect to laws of nature? Then Mother would have failed just the same!) But this is not the sort of what-if-things-had-been-different information that counts as explanatory in the counterfactual framework that we favour, according to which only what we call ‘change-relating’ counterfactual information counts as explanatory.14 To understand Mother's failure, we need to be able to say under what conditions Mother could have succeeded. We now move on to discuss this framework. 4 Abstraction in a Counterfactual Framework The starting point for the W-H account is that explanation ‘is a matter of exhibiting systematic patterns of counterfactual dependence’ (Woodward [2003], p. 192). To develop this into a theory of explanation, we need to be more precise about which patterns of counterfactual dependence matter for explanation. On the W-H account this is done by focusing on modal information that allows one to answer questions about how the explanandum would have been different under a special type of change in the explanans (Woodward [2003], p. 192). The changes in the explanans that are relevant are those that result from interventions on the explanans variable(s) (with respect to the explanandum variable). The notion of intervention is technical; it plays the role of ruling out changes in the explanans variable that are brought about by (i) changes in the explanandum variable, and (ii) changes to some other (‘common cause’) variable that changes the explanandum variable independently of the explanans variable. This approach has a number of attractive features. First, it is natural to take relations (whether these are nomological, causal, or mathematical) that are explanatory to be capable of providing some sort of modal information. Second, by requiring more specifically that the modal information relates to different possible states of the explanandum—how the actual explanandum would have been different had the explanans been different—the account captures the natural idea that explanatory information is information about worldly dependences. In the W-H account the critical modal notion of dependence gets cashed out in causal terms through the notion of a (testing) intervention. This allows one to rule out backtracking counterfactuals—for example, ‘had the period of the pendulum been different, then the length of the pendulum would have been different’—as providing the right dependences. The focus on what-if-things-had-been-different questions that concern changes in the target explanandum also provides a clear contrast to mere subsumption under modal regularities in the spirit of the deductive-nomological account of explanation. Third, this focus on interventions squares with much of our experimental practice. Fourth, the account is, as Woodward and Hitchcock ([2003a], [2003b]) discuss at length, very well suited for capturing not only what it takes to have an explanation, but also what makes an explanation better or worse. Having an explanation is a matter of having the right information about the dependences, and the more the better. The W-H account separates an explanans into two parts: a specification of an invariant explanatory generalization and a specification of the actual values of the variables in the explanatory generalization.15 For a simple illustration, let us look at the explanation of the period of a simple gravitational pendulum. Let us assume that we want to explain the fact that period T takes the value t1 (our explanandum, M) by using the simple pendulum law. In terms of the W-H account, we have explanatory generalization T≈2πlg. The explanans consists of this generalization and a specification of the actual values of l and g. In order to have an explanation according to the W-H account, the simple pendulum law has to give the (at least approximately) correct value of the period as t1 under an intervention that fixes the values of l and g to the actual values. Moreover, the simple pendulum law must capture (in at least one case) how the period would change under at least one and ideally more interventions that change the length or the gravitational acceleration. It thus captures the dependence of the explanandum on these variables. Now we are in a position to make more precise the contrast between the different dimensions of abstraction. Let us begin with the first dimension of abstraction. In the context of the simple pendulum explanation, we can ask whether explanandum T = t1 is independent of particular background conditions that are in some sense part of the system but not represented as variables in explanatory generalization T≈2πlg. We can think of the microphysical features that make the pendulum cord inextensible, for example, or the mass of the bob, and so on. The degree of independence of the explanation from such background conditions was the focus of the first dimension of abstraction. (In contrast to such ‘internal’ background conditions, there are other background conditions that are external to the system and clearly irrelevant, such as the bob being made in Japan, the exchange rate of the US dollar to the Malaysian ringgit, the colour of the shirt of the person setting the bob in motion, and so on) In the second dimension of abstraction, we considered the degree of independence of the explanation from the actual laws of nature. In the context of the pendulum example, we can ask whether we can vary the actual laws of nature and still expect the simple pendulum explanation to apply. We can alter many of the laws of, for example, electromagnetism without affecting the explanation, provided that those alterations do not run afoul of the assumptions presupposed in the context of the why-question, for example, the cord being nearly inextensible. As mentioned earlier, we can also alter many of the fundamental aspects of gravitational acceleration, such as whether its nomological basis is an action-at-a-distance effect or a manifestation of curved space-time. Finally, in the third dimension of abstraction, we considered the range of conditions directly relevant to changes in the variables in the explanatory generalization, T≈2πlg, such that the simple pendulum explanation of the period holds. For example, we could decrease g by moving the pendulum to a higher altitude or to the moon, and we could change the length of the pendulum; nonetheless, the explanation of the period would work in just the same way. On the W-H account, abstractness in this dimension is just a matter of the invariance of the explanatory generalization, ‘measured’ by the range of alternative values of variables l and g for which the generalization holds and is change relating. So far we have simply described the W-H account. Before we can apply these ideas to the paradigmatic examples of abstract explanations, we need to extend and refine some of them. In particular, the W-H account was developed as an account of causal explanation, while the abstract explanations we are interested in are plausibly non-causal. As we move beyond causal explanation, we can no longer appeal to the exact understanding of the relevant class of explanatory counterfactuals that Woodward and Hitchcock use. They delineate the class of relevant counterfactuals in terms of the causal notion of intervention: explanatory counterfactuals describe the effect of a testing intervention. These counterfactuals are underwritten by contingent explanatory generalizations, and the effects of surgical testing interventions are naturally understood as concerning causal dependences. In the non-causal cases, this notion of causal dependence becomes inapplicable or unnatural, because the notion of causal intervention is inapplicable, or because the counterfactuals are not underwritten by contingent laws of nature (but, rather, by logic or mathematics or metaphysical truths). Is it still possible in these cases to characterize explanatory dependences in counterfactual terms? The answer is yes; we have analogous explanatory dependences in the situations involving Mother's strawberries, Königsberg’s bridges, and so on.16 The challenge is how to select the right counterfactuals without assuming that causal notions are applicable. In responding to this challenge we note, first of all, that regardless of whether or not we can apply the idea of a causal intervention to the explanans variable, we can still have a good grasp of what it means to change the explanans—for example, to change the number of strawberries, or the bridge configuration in Königsberg. Furthermore, regardless of whether or not the explanatory generalization is a contingent law of nature or a stronger modal truth, we can reason about how the explanandum depends on these changes, and whether the dependence is non-symmetric in a way that supports explanatory what-if-things-had-been-different reasoning. For example, in the the strawberry example, we have a very good grasp of what changing the explanans variable amounts to. Indeed, changing the number of strawberries seems as straightforward an intervention as any. We also know how this changes the system’s divisibility by three, and consequently the possibility of Mother's success or failure. We thus grasp how Mother's success or failure explanatorily depends on the number of strawberries that she has (and the number of children). In order for the dependence to be regarded as explanatory, it needs to be appropriately directed. Intuitively, it is. As we indicated above, the notion of intervention plays the role of blocking changes to the explanans that go through changes to the explanandum (or through a common cause variable, and so on). Similarly, changes to the number of strawberries do not seem to go through changes to the system’s divisibility by three, or through changes to Mother's success or failure. After all, changing the system’s divisibility by three does not mandate any particular number of strawberries. However, changes to Mother's success or failure do seem to go through changes to her number of strawberries (or children). This intuition can be supported in a more precise manner. On the W-H account, a successful explanation makes use of an explanatory generalization for two main tasks (as outlined above): first, to show that setting the value of the explanans variable to its actual value fixes the value of the explanandum variable to its actual value; and, second, to show that changes to the value of the explanans variable that move it away from its actual value will, in some cases, change the value of the explanandum variable. Now, consider the intuition that changes to the system’s divisibility by three occur by means of changes in the number of strawberries, but not vice versa. This is underwritten by the fact the explanatory generalization regarding the number of strawberries and their divisibility by three supports the first task asymmetrically. Fixing the explanans variable to its actual value should fix the explanandum variable to its actual value, but fixing the system’s (in)divisibility by three does not fix the number of strawberries to any particular value. In contrast, the number of strawberries being twenty-three does fix the system’s (in)divisibility by three. This asymmetry underlies the intuition that changes to the number of strawberries do not come about via changes in the system’s divisibility by three.17 The asymmetry does not seem to be a causal matter, however, since the intervention on the number of strawberries can have an ‘effect’ (for example, rendering the set divisibility by three) that is rather more intimately related to it. To this extent Lange ([2013]) is absolutely right to stress the independence of this connection from any causal laws. Very similar considerations apply to the case of Königsberg’s bridges.18 In this case, it’s not clear whether the explanans variable—concerning the global configuration of the whole bridge system—can be changed in a way that necessarily satisfies our intuitions about what counts as a causal intervention. But be that as it may, we can surely have a good grasp of what it means to change the explanans variable (by strategically building and destroying bridges so as to achieve a particular overall configuration). And again, the explanatory generalization tells us how this changes the bridge system’s traversability, which depends on the configuration being in a certain way. (We will lay this out in more detail in the next section.) This dependence is again appropriately directed: while fixing the bridges in a particular way fixes the (non-)traversability of the bridge system, the reverse does not hold. We cannot fix the bridges to a particular configuration merely by ensuring that the bridge system is (say) non-traversable.19 There is more to be said about the nature of explanatory dependence in abstract explanations, of course. One can wonder about the metaphysics of the dependence relation, for example, as we have not said anything about it. Our hope is that one doesn’t have to get into the metaphysics of dependence to support the broader idea of there being such explanatory dependencies in the world, transcending those readily incorporated into the W-H account as it stands; it is enough to show how we can pick out the right counterfactuals. (This is in tune with Woodward’s own attitude to this issue.) One may also wonder what to make of the fact that in the examples at hand, the explanatory relation does not seem to connect two distinct events; it is not the case that ‘we change one thing and another change follows’.20 For instance, the event of destroying some of Königsberg’s bridges can ipso facto be the event of rendering the bridges Euler-tourable. In response, we would like to point out that leaving behind the idea that explanatory relations hold between events (or event-types) is critical to the W-H account; rather, explanatory relata are treated as variables (Woodward and Hitchcock [2003a], pp. 10–1). In causal cases, this amounts to a notion of explanatory causal relevance between such variables. Similarly, in the examples at hand it is natural think of explanatory variables as designating different aspects of the system in question. We can make sense of such variables standing in an explanatory dependence relation, even when changes in the values of the variables involve one and the same event. (See Saatsi [2016a] for related discussion of explanatory variables in connection with non-causal explanations from ‘geometry of motion’.) There is also more to be said about the directionality of explanation. Here we refer the reader to work done elsewhere that offer ways of developing further the ideas we have only briefly sketched here.21 But we hope to have said enough to show that the prospects for a more detailed account are no worse—and, in our opinion, are better—when extending a dependence account in the W-H spirit than when focusing on independence along the lines of Lange and Pincock.22 Our suggested diagnosis of the original W-H account is that it correctly identifies explanatory counterfactuals as those that are appropriately change-relating and directed. From now on, let us call the explanatory counterfactuals simply ‘change-relating counterfactuals’ (adapting Woodward’s use of the term), to keep in mind that they are a strict subset of all counterfactuals and a broader class than interventionist causal counterfactuals. Before we can move on to consider a detailed application, we need to consider one final aspect of extending the W-H account. Since the relevant counterfactuals are selected by focusing on changes to some object or system, the modal information that we are interested in is tied to the system or object in question. They are what Woodward ([2003], p. 281) calls ‘same object’ counterfactuals: Suppose that we wish to explain the behavior of some object or system o. As the standard view is usually understood, it claims that generalizations of form ‘All As are Bs’ are explanatory of the behavior of o if they support counterfactuals of the following form: […] If some object o*, different from o and that does not possess property A, were to be an A, then it would be a B […]. Call such counterfactuals ‘other object’ counterfactuals: they describe what the behavior of objects other than o would be under the counterfactual circumstances in which they are A. By contrast, according to the view I have been defending, to count as invariant and hence explanatory with respect to o, a generalization must support ‘same object’ counterfactuals that describe how the very object o would behave under an intervention. At first glance, this may seem to suggest that all of the explanations that we have in mind must be about specific, particular systems or objects in order for the distinction—central to the W-H account—between same object (SO) and other object (OO) counterfactuals to apply. Pincock ([2015]) takes this to rule out the application of the W-H account to abstract explanations, since some highly abstract explanations do not seem to provide information about any particular object at all (much less information about changes to any particular object). In our view, one should not think of the contrast between SO counterfactuals and OO counterfactuals in the same way as Pincock. First, note that even in the case of causal explanations, most scientific explanations do not have a particular, individual object as the explanatory target. The explanandum is typically generic (and general, even when couched in language such as ‘the period of a simple pendulum’). The importance of the distinction between SO and OO counterfactuals lies in the fact that it focuses our attention on the right kind of change-relating counterfactuals. When we make use of the simple pendulum law to explain the period of a simple pendulum, the right counterfactuals to have in mind are those that ask how the period of a generic kind of dynamical system—namely, a simple pendulum—is affected by changing its length, say. Counterfactuals concerning objects other than simple pendulums do not come into play, even when they are reasonable and well defined. (It may be true that had a hammer been suspended around an appropriate pivot with the head down, then we could have used the simple pendulum law to explain its period; but such OO counterfactuals are not required in order to explain the period of a simple pendulum.) Although the W-H account does not apply directly to the case of Königsberg’s bridges, the central notion of change-relating (SO) counterfactuals can be carried over to this case. This is the work of the next section. 5 Königsberg—Encore! We will now revisit the Königsberg case to illustrate the above counterfactual account and its virtues. So far, we have mainly criticized Pincock’s and Lange’s motivations for thinking that there is a sharp distinction between causal explanations and abstract non-causal explanations, maintaining that one can instead approach both types of explanations in the fundamentally same spirit. We now push for a stronger claim, arguing that the core claim of the counterfactual account (as we understand it)—that explanatoriness is associated exclusively with the third dimension of abstraction—can be tested against the alternative viewpoints. This will provide a clear reason to prefer the counterfactual account. Let us go back to the eighteenth-century Königsberg, and ask why it is impossible to make a round tour of Königsberg crossing each of its seven bridges exactly once (QK). An intuitive explanation-sketch response goes as follows: Clearly, each visit to a landmass (an island) requires the use of two bridges: one in and one out, otherwise one gets stuck. Therefore, in order for a network of bridges to allow for a round tour—to be ‘tourable’—each island must have a number of bridges that is some multiple of two. On the other hand, if there are one or more islands with an odd number of bridges, the system is not tourable. Eighteenth-century Königsberg had four such troublesome junctions, rendering a round tour of Königsberg impossible. Euler articulated a famed graph-theoretic explanation that makes the above sketch precise. This is standard material in graph-theory textbooks, illustrating the explanatory use of mathematical notions (such as ‘connected graph’, its ‘vertices’, and their ‘degrees’).23 But before we get to the graph-theoretic explanation in its full generality, it is worth attempting to answer QK without graph theory. In the counterfactual framework, explanations must involve an invariant, change-relating generalization that supports counterfactuals indicating an explanatory dependence of the explanandum on the explanans. A generalization that thus underwrites an answer to QK could be markedly less abstract and less general than the graph-theoretic explanation, without thereby being unexplanatory. As a matter of fact it is easy to find a simple invariant generalization that furnishes a non-mathematical answer to QK. Let us focus our attention on the kind of bridge system that connects exactly four islands, with at most two bridges between any two islands. Euler’s Königsberg is one of these systems (see Figure 1). Figure 1. View largeDownload slide Königsberg’s bridge system. Figure 1. View largeDownload slide Königsberg’s bridge system. There are 395 such bridge systems in total. We can classify them as follows: Call a bridge system ‘even’ if and only if each island has an even number of bridges. In this particular case, this means that each island must have two, four, or six bridges. If a bridge system is not even, call it ‘odd’. Considering the specific type of bridge system exhibited by Königsberg, we can ask why it is not tourable. In answering this question, we naturally look for an explanatory generalization capable of providing suitable modal information by supporting appropriate, change-relating counterfactuals. Focusing our attention, for now, only on the bridge systems consisting of four connected islands with at most two bridges between any two islands, a fitting generalization is not hard to find: of all these bridge systems, all and only the even ones are tourable.24 This is a true generalization about this set of 395 types of bridge system. It is also a generalization that can afford us a degree of explanatory purchase on QK. That is, by reference to this generalization, we can begin to answer QK simply by noting that Königsberg’s bridge system is not tourable because it is not even; it would be tourable if it were odd. Königsberg’s tourability—the feature that is our explanandum—depends on this high-level physical property of the system.25 More formally, we can define the following binary variables X and Y:   X=even/oddsystemx=1:evenx=0:odd  Y=(non-)tourablesystemx=1:tourablex=0:non-tourable The simple explanatory generalization at stake then states that for bridge systems of this kind—four islands, with a maximum of two bridges between any pair—it holds that   X=Y. This equation should be read left to right, as indicating an asymmetric dependence of Y-variable on the X-variable. The tourability (or otherwise) of a bridge system depends on it being even (or odd). Being even (or odd) does not depend on tourability; it only depends on the number of bridges. All in all, the explanation naturally fits the counterfactual framework presented above (Section 4). The explanatory generalization, while narrow, is explanatory nevertheless, even if minimally so. Undoubtedly, the explanation provided immediately raises further questions. Why exactly is this generalization true, for example? What if we start adding or subtracting islands? What if we relax the restriction that there are at most two bridges between any pair of islands? This clearly renders the explanation shallow and somewhat contrived, especially in comparison to a full-blown graph-theoretic account. But none of this diminishes the philosophical significance of this explanation. For however minimal and shallow the toy explanation is, we deem it to have some explanatory power nevertheless, and we maintain that this is due to the explanation providing modal information of the right sort.26 The shallow explanation above clearly has a degree of abstractness along the three dimensions introduced in Section 2. Indeed, the explanation is highly abstract with respect to the specific material realizations of the bridges (the first dimension) and also with respect to the underlying laws of physics (the second dimension). As a matter of fact, the explanation is as abstract along these dimensions as the full-blown graph-theoretic explanation! Yet the explanation is shallow. This clearly speaks against the idea that the (minimal) explanatoriness in question springs from abstraction along these lines, as Pincock and Lange would have it. With regard to the third dimension of abstractness, the explanatory generalization does allow us to answer some change-relating, what-if-things-had-been-different questions. We have counterfactual information of the right, SO kind. We can answer counterfactual questions about how the tourability of a generic kind of structural system—the bridge system—is changed by changes to the oddness or evenness of the system (at least as long as we stay within the constraints of four connected islands and a maximum of two bridges between any two islands). This is what makes it explanatory, even if shallowly so. Yet, its degree of abstractness along this dimension is very limited. Although the explanatory generalization never delivers the wrong answer, the generalization is simply silent on what happens in cases of more than four islands or more than two bridges between some islands. Thus, the explanation using the generalization breaks down in these cases.27 In order to understand precisely how the shallow explanation compares in its abstractness to the deeper graph-theoretic explanation, we need to pay attention to the fact that the variable X concerns a determinable property of the system: it being even or odd. This determinable property is determined by the bridges’ configuration. There are various ways for a bridge system to be even. One way is for each island to have (say) four bridges. This is still a determinable property of the system, determined by a specific way of having four bridges for each island (see Figure 2). Figure 2. View largeDownload slide Two determinate 4–4–4–4 configurations. Figure 2. View largeDownload slide Two determinate 4–4–4–4 configurations. Here is another way for a bridge system to be even: one of the islands has eight bridges, and the other three islands have two, two, and four bridges, respectively. While our generalization supported some explanatory what-if-things-had-been-different questions, it is simply silent on the system’s tourability if we change X from odd to even by allowing one island to have as many as eight bridges. Similarly, it is silent on what happens if we change X from even to odd by including seven bridges for some island. To answer what-if-things-had-been-different questions corresponding to these cases, we need a broader generalization, X = Y, that applies to systems as rich in bridges as these. Differences in the determinate configurations do not explicitly feature as a variable in explanatory generalization X = Y. Indeed, on the face of it, the explanatory generalization looks the same regardless of the range of determinate what-if-things-had-been-different questions it is taken to support. Differences in the range of determinate realizations of evenness and oddness do come into play through restrictions in application. When we consider changes to the variable (even, odd), these changes have to go through changes in the bridge system (there is no way to change the evenness of the system that does not go through changing the specific configurations of the bridges). If we are justified in taking generalization X = Y to apply to systems that have (say) a maximum of three bridges between any two islands, then our explanation covers a wider range of conditions directly relevant to changing the value of the explanans variable, X. In particular, we can also consider what-if-things-had-been-different situations with seven, eight, or nine bridges for some island(s). In this way, we can straightforwardly compare different Königsberg explanations with respect to their degree of abstraction in the third dimension. The explanation where we restrict the generalization to a maximum of three bridges between any two islands applies to all the ways of altering the variable in the explanatory generalization that has a two-bridge restriction—and then some! The full-blown graph-theoretic explanation is, of course, maximally abstract along this third dimension of abstractness. The explanatory generalization now covers any (connected) bridge system of arbitrarily many islands and bridges. The graph-theoretic explanation has considerable depth, in contrast to the shallow, non-mathematical explanation. This is due solely to increased abstraction along the third dimension. The explanatory dependence of the shallow explanation is subsumed under a more general explanatory dependence, enabling us to answer a much wider range of change-relating, what-if-things-had-been-different questions. The shallow explanation does not contain irrelevant information about the nature of the bridges, their material constitution, or their length, say, nor does it contain irrelevant information about the underlying nomological features, such that the increase in explanatory depth could be due to abstracting away from such information. Indeed, and as already noted, the explanans of these two explanations are already maximally abstract along the first and second dimensions, and the considerable increase in the explanatory power should be attributed solely to the third dimension of abstraction. 6 Conclusion We have argued that paradigmatic abstract (plausibly) non-causal explanations can be naturally accommodated within an account that associates explanatoriness with suitable information about dependence. This improves the prospects of subsuming many explanations under a unified counterfactual framework, opposing the current trend that emphasizes the explanatory value of abstraction as a sui generis source of explanatoriness. We found the motivations for this trend questionable (Section 3), leaving room for a more unified account. More unified theories are often better, ceteris paribus, but we are not just expressing this kind of a prima facie preference for unification. Rather, we motivated the unified account with the following considerations: First, we argued that the unified account better captures intuitions about ‘explanatory depth’. Part of the force behind the W-H framework comes from the fact that it provides a natural starting point for capturing intuitions about explanatory goodness (Woodward and Hitchcock [2003a], p. 3; Ylikoski and Kuorikoski [2010]). We argued, in the same spirit, that our viewpoint can be tested by varying dependence information—the hypothesized source of explanatoriness—keeping other things fixed (Section 5). We maintain that our intuitions about radically varying explanatory power naturally correspond to radically varying amounts of W-H dependence (but not independence) information, in a way that is difficult to accommodate from the alternative viewpoints. Second, the unified account has a further virtue worth flagging. This has to do with dissolving difficult questions facing the more disjunctive accounts that regard abstract explanations as fundamentally different from causal explanations in their source of explanatoriness. Noting that causal explanations also typically abstract away from a huge amount of physical detail raises questions about Pincock’s point of view, for example: What explains the distinct qualitative difference between causal explanations that exhibit a degree of abstraction along each of the three dimensions, on the one hand, and the sui generis abstract explanations, on the other? Is there a ‘threshold’ of abstraction above which the counterfactual conception fails, despite capturing abstract causal explanations so well? A similar question can be raised for Lange’s account. Many causal explanations also abstract away from a huge amount of underlying nomological detail. Thus, all causal explanations that incorporate a degree of abstraction along the second dimension of abstraction also show how the explanandum is, to a corresponding degree, ‘necessary to a stronger degree of necessity’ by virtue of showing the irrelevance of some of the actual laws involved. So why is it that such modal information about independence becomes explanatory in a sui generis way in connection with abstract explanations? Or is it the case that such modal information always contributes to the explanatory power, but it contributes in a different way when an explanation abstracts away from (almost) all nomological information, as in the case of ‘mathematical’ explanations?28 Why do abstract explanations explain so differently—in Salmon’s ‘modal’ mode—from causal explanations that incorporate a degree of similar abstraction? In our view, Pincock and Lange have not provided satisfactory answers to these questions. In particular, their arguments that abstract explanations do not work by providing information about causal dependence do not show that they work in a radically different way instead (as opposed to providing information about non-causal W-H dependence). The more unified account we advocate has the virtue of sidestepping these issues entirely. According to this account, paradigmatic abstract explanations, despite their non-causal character, are explanatory for the same reason as causal explanations. In both cases, the explanatory power springs from counterfactual information of the same sort, and the paradigmatic exemplars of abstract explanation need not be regarded as sui generis. Acknowledgements We are grateful for the comments, questions, and criticisms from the participants at BSPS15 and EPSA15. Thank you also to the participants of the Leeds philosophy of science reading group. The support of MOE Tier 1 grant RG162/14 is gratefully acknowledged. Footnotes 1 See, for example, (Woodward [2003]; Woodward and Hitchcock [2003a], [2003b]; Ylikoski and Kuorikoski [2010]; Strevens [2008]). Ruben ([1990]) is an early proponent. 2 See, for example, (Lange [2013]; Pincock [2007], [2015]—for arguments that none of the existing dependence accounts can capture highly abstract explanations; Batterman and Rice [2014]). 3 Cf. (Lange [2013]; Braine [1972]). 4 Most prominently in (Woodward [2003]). 5 There are exceptions to this. For example, (Bokulich [2008], [2012]; Rice [2015]; Reutlinger [2016]). We take ourselves to be in the same tradition as these accounts. However, we will differ in that we need to specify more precisely the kind of what-if-things-had-been-different information that counts as explanatory (which we do in Section 4). 6 The issue of exactly what kinds of dependences count as causal is convoluted and largely orthogonal to our interests in this article. What we want to understand here is how various abstract explanations explain. 7 The dimensions are metaphysically intertwined because, for example, various possible changes in the physical laws are bound to imply changes in the physical structures. 8 After heavily emphasizing the idiosyncratic features of abstract explanations, Pincock ([2015]) suggests that it may be possible to provide a unified account in terms of dependence. However, Pincock’s notion of dependence is explicitly not in the spirit of the W-H account. We will address Pincock’s objection to the W-H account in Section 4. 9 The term ‘distinctly mathematical’ is a term of art in (Lange [2013]). It is aiming to capture many of the cases that Pincock ([2015]) would call ‘highly abstract’. 10 In relation to the strawberry example, Lange ([2013], pp. 493–6) argues that the counterfactuals pertaining to the number of Mother’s strawberries provide causal information, noting, for example, that ‘clearly manipulation of the numbers of strawberries or children would bring about corresponding changes in the outcome of Mother's attempt’; yet he maintains that ‘this explanation is non-causal because it does not work by describing the outcome’s causes or, more broadly, the world’s network of causal relations’. 11 Lange’s motivations are best understood in the broader context of his related views on other kinds of non-causal explanations (Lange [2015]), and on the modal metaphysics of laws of nature in general (Lange [2009]). Like other philosophical views, Lange’s account of abstract explanations can gain indirect support from being a coherent part of a ‘bigger picture’. Here we do not wish to assess the pros and cons of this bigger picture; we will focus on Lange’s association of explanatoriness with the second dimension of abstraction, in connection with the paradigmatic examples of abstract explanations. 12 This is not to say that they can never be relevant to any aspect of good explanation. To only provide relevant information is a shared commitment of any account of explanation (cf. Strevens [2008] on causal explanation). This goes some way towards explaining the intuition that the first two dimensions provide a separate source of explanatory power. For example, if we assume that we have a mistaken belief that taking the island as a starting point is relevant to explaining the failure to complete a round tour of Königsberg, then it is enlightening to find out that the starting island is irrelevant. However, a dependence account can easily capture this intuition. The failure to complete a round tour of Königsberg does not depend on the starting island. A good explanation cannot cite misinformation about the dependences. Note that the intuition that this kind of information about independence is a source of explanatory power disappears once we are not focused on correcting a mistaken belief about dependences. We are not tempted to regard as explanatory the information that failure to complete the tour is independent of the fact that the bridges are made of stone and not wood. We never mistakenly believed that the failure to complete the tour depended on these features. 13 As we noted above, Lange ([2013]) explicitly considers and rejects this option. Pincock ([2015]) is sympathetic to the general idea of dependence accounts, but explicitly rejects W-H dependence as a viable candidate. 14 In the terminology of Woodward and Hitchcock ([2003a], [2003b]), an explanatory generalization must be invariant under at least one testing intervention. 15 See (Woodward [2003], p. 203) for a detailed account. 16 Woodward ([2003], p. 221) notes in passing that it seems natural to extend the account to non-causal explanations, but he does not do so. For more discussion on how to decouple the counterfactual aspect and the causal aspect of Woodward’s account, see (Saatsi and Pexton [2013]; Saatsi [2016b]; Rice [2015]). 17 The explanatory generalization regarding the number of strawberries and their (in)divisibility by three supports some counterfactuals running in the opposite direction, of course. For instance, it’s true that had the system’s (in)divisibility by three been otherwise, then the number of strawberries would have been different. So why is this counterfactual not explanatory of the number of strawberries? Answer: the explanatory generalization, with the supposed ‘explanans’ variable fixed, does not state the actual number of strawberries. 18 The full story here is somewhat more complicated since we are modelling the physical bridge system and there are considerations that arise from the applicability of the model. The importance of considerations of applicability for directionality of explanation is stressed by Jansson ([2015]). 19 The true counterfactual, ‘Had the traversability of the bridge system been different, then the system of bridges would have been different’, does not explain the actual bridge configuration for reasons analogous to those discussed in Footnote 17. 20 We thank an anonymous reviewer for raising this questions; (s)he put it in these terms. 21 See (Jansson [2015]) for a nomological dependence account and (Jansson [2016]) for a dependence account for metaphysical cases. There is also more to say about how we should understand optimality explanations—for example, the honeycomb case. We believe that a similar account to the one that we have given here holds for the honeycomb case, but that this has to be argued for on a case-by-case basis. The discussion of many of these cases is made more complicated by the existence of closely related causal explanations, as Lange ([2013], p. 500) notes for the honeycomb case. 22 In (Pincock [2015], p. 865) the directionality comes from an otherwise unanalysed relation of instantiation, and in (Lange [2013], p. 508) the directionality comes from an otherwise unanalysed notion of some features being constitutive of a system. In our view, while these are potential avenues for identifying the source of non-causal directionality, they are neither more plausible nor better understood than the alternative provided by change-relating counterfactuals. 23 It is unsurprising that many authors refer to this explanation as a paradigmatic case of a (distinctly) mathematical explanation of an empirical fact. In addition to the work already covered, see, for example, (Lyon [2012]). 24 This fact about these bridge systems can be established by a variety of means, for example, by attempting to draw every such system without lifting your pen, or by playing with a comprehensive collection of miniature models of such systems. Getting epistemic access, or representing this fact, need not involve mathematics (cf. Saatsi [2011]). 25 Why do we regard this as a (high-level) physical property? Because adding or subtracting a bridge between any two islands plainly makes for a physical difference, and we can get from an even configuration to an odd configuration simply by adding or subtracting a sufficient number of bridges. Furthermore, if we are dealing with a finite number of possible bridge systems (as above, focusing on 395 particular configurations), we can in principle do without the mathematical concepts of even and odd in delineating the two kinds of configurations, since the difference between even and odd is expressible logically, without mathematics. So, we see little reason to regard these as mathematical or purely formal properties of the bridges. Having said that, we should stress that we are not claiming that the Königsberg explanation is a causal one, and nothing in our analysis hangs on whether or not the explanation counts as mathematical. 26 Is minimal explanation of this sort ever completely satisfactory? That depends on the context of the why-question. Imagine Königsberg without Euler—or any other mathematician for that matter. The King of Königsberg, annoyed by his failure to do a round tour of the city, wishes to understand the situation. ‘This city must be made thus tourable, at whatever cost!’, he commands, putting his best people to work. ‘Well, actually, the less it costs the better—and we certainly cannot afford more than two bridges between any two islands!’. The King’s minions get to work, and painstakingly demonstrate, by non-mathematical means, that for any financially feasible set-up of bridges, it holds that it is tourable if and only if it is even. Equipped with this knowledge, they can explain the situation to the King: ‘Whatever you do with the bridges, make sure the whole system is even, and you will be able to tour it, since tourability depends on this feature alone (at least for the kinds of bridge systems we can afford)’. 27 Here, two differences between our third dimension of abstraction and (even the extended) notion of invariance in (Woodward and Hitchcock [2003b]) are important. First, we take the question to be whether the variations destroy the explanation (not merely the generalization). The generalization itself does not break down; it just does not apply. We take this focus on the whole explanation from Potochnik ([2010]) and Weslake ([2010]). Second, the range of variations are understood as the range of cases in which one of the variables in the explanans can be changed without destroying the explanation. While this is also the idea of the W-H account, in (Woodward and Hitchcock [2003b]) it is often natural to interpret the range of invariance as the range of changes in the variable values for which the generalizations continue to hold. 28 Almost all, since these explanations still contain information associated with the various contingent facts presupposed by the why question. References Batterman R., Rice C. [ 2014]: ‘Minimal Model Explanations’, Philosophy of Science , 81, pp. 349– 76. Google Scholar CrossRef Search ADS   Bokulich A. [ 2008]: ‘Can Classical Structures Explain Quantum Phenomena?’, British Journal for the Philosophy of Science , 59, pp. 217– 35. Google Scholar CrossRef Search ADS   Bokulich A. [ 2012]: ‘Distinguishing Explanatory from Non-explanatory Fictions’, Philosophy of Science , 79, pp. 725– 37. Google Scholar CrossRef Search ADS   Braine D. [ 1972]: ‘Varieties of Necessity’, Proceedings of the Aristotelian Society , 46, pp. 139– 70. Google Scholar CrossRef Search ADS   Jansson L. [ 2015]: ‘Explanatory Asymmetries: Laws of Nature Rehabilitated’, Journal of Philosophy , 111, pp. 577– 99. Google Scholar CrossRef Search ADS   Jansson L. [ 2016]: ‘Explanatory Asymmetries, Ground, and Ontological Dependence’, Erkenntnis , 82, pp. 17– 44. Google Scholar CrossRef Search ADS   Lange M. [ 2009]: Laws and Lawmakers: Science, Metaphysics, and the Laws of Nature , Oxford: Oxford University Press. Google Scholar CrossRef Search ADS   Lange M. [ 2013]: ‘What Makes a Scientific Explanation Distinctively Mathematical?’, British Journal for the Philosophy of Science , 64, pp. 485– 511. Google Scholar CrossRef Search ADS   Lange M. [ 2015]: ‘How to Explain the Lorentz Transformations’, in Mumford S., Tugby M. (eds), Metaphysics and Science , Oxford: Oxford University Press. Lyon A. [ 2012]: ‘Mathematical Explanations of Empirical Facts, and Mathematical Realism’, Austrasian Journal of Philosophy , 90, pp. 559– 78. Google Scholar CrossRef Search ADS   Lyon A., Colyvan M. [ 2008]: ‘The Explanatory Power of Phase Spaces’, Philosophia Mathematica , 16, pp. 227– 43. Google Scholar CrossRef Search ADS   Pincock C. [ 2007]: ‘A Role for Mathematics in the Physical Sciences’, Noûs , 42, pp. 253– 75. Google Scholar CrossRef Search ADS   Pincock C. [ 2011]: ‘On Batterman’s on the Explanatory Role of Mathematics in Empirical Science’, British Journal for the Philosophy of Science , 62, pp. 211– 17. Google Scholar CrossRef Search ADS   Pincock C. [ 2015]: ‘Abstract Explanations in Science’, British Journal for the Philosophy of Science , 66, pp. 857– 82. Google Scholar CrossRef Search ADS   Potochnik A. [ 2010]: ‘Levels of Explanation Reconceived’, Philosophy of Science , 77, pp. 59– 72. Google Scholar CrossRef Search ADS   Reutlinger A. [ 2016]: ‘Is There a Monist Theory of Causal and Non-causal Explanations?’, Philosophy of Science , 83, pp. 733– 45. Google Scholar CrossRef Search ADS   Rice C. [ 2015]: ‘Moving beyond Causes: Optimality Models and Scientific Explanation’, Noûs , 49, pp. 589– 615. Google Scholar CrossRef Search ADS   Ruben D.-H. [ 1990]: Explaining Explanation , London: Routledge. Google Scholar CrossRef Search ADS   Saatsi J. [ 2011]: ‘The Enhanced Indispensability Argument: Representational versus Explanatory Role of Mathematics in Science’, British Journal for the Philosophy of Science , 62, pp. 143– 54. Google Scholar CrossRef Search ADS   Saatsi J. [ 2016a]: ‘On Explanations from Geometry of Motion’, British Journal for the Philosophy of Science , in press, 10.1093/bjps/axw007. Saatsi J. [ 2016b]: ‘On Mathematics’ ‘Indispensable Explanatory Role’’, Mind , 125, pp. 1045– 70. Google Scholar CrossRef Search ADS   Saatsi J., Pexton M. [ 2013]: ‘Reassessing Woodward’s Account of Explanation: Regularities, Counterfactuals, and Noncausal Explanations’, Philosophy of Science , 80, pp. 613– 24. Google Scholar CrossRef Search ADS   Salmon W. C. [ 1984]: ‘Scientific Explanation: Three Basic Conceptions’, Proceedings of the Biennial Meeting of the Philosophy of Science Association , 1984, pp. 293– 305. Google Scholar CrossRef Search ADS   Strevens M. [ 2008]: Depth: An Account of Scientific Explanation , Cambridge, MA: Harvard University Press. Weslake B. [ 2010]: ‘Explanatory Depth’, Philosophy of Science , 77, pp. 273– 94. Google Scholar CrossRef Search ADS   Woodward J. [ 2003]: Making Things Happen: A Theory of Causal Explanation , Oxford: Oxford University Press. Woodward J., Hitchcock C. [ 2003a]: ‘Explanatory Generalizations, Part I: A Counterfactual Account’, Noûs , 37, pp. 1– 24. Google Scholar CrossRef Search ADS   Woodward J., Hitchcock C. [ 2003b]: ‘Explanatory Generalizations, Part II: Plumbing Explanatory Depth’, Noûs , 37, pp. 181– 99. Google Scholar CrossRef Search ADS   Ylikoski P., Kuorikoski J. [ 2010]: ‘Dissecting Explanatory Power’, Philosophical Studies , 148, pp. 201– 19. Google Scholar CrossRef Search ADS   © The Author 2017. Published by Oxford University Press on behalf of British Society for the Philosophy of Science. All rights reserved. For Permissions, please email: journals.permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The British Journal for the Philosophy of Science Oxford University Press

Explanatory Abstractions

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Oxford University Press
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© The Author 2017. Published by Oxford University Press on behalf of British Society for the Philosophy of Science. All rights reserved. For Permissions, please email: journals.permissions@oup.com
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0007-0882
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10.1093/bjps/axx016
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Abstract

Abstract A number of philosophers have recently suggested that some abstract, plausibly non-causal and/or mathematical explanations explain in a way that is radically different from the way causal explanations explain. While causal explanations explain by providing information about causal dependence, some abstract explanations allegedly explain in a way tied to the independence of the explanandum from the micro-details or causal laws, for example. We oppose this recent trend to regard abstractions as explanatory in some sui generis way, and argue that a prominent account of causal explanation can be naturally extended to capture explanations that radically abstract away from micro-physical and causal-nomological details. To this end, we distinguish different senses in which an explanation can be more or less abstract, and analyse the connection between an explanation's abstractness and its causal power. According to our analysis, abstract explanations have much in common with counterfactual causal explanations. 1 Introduction 2 Three Dimensions of Abstraction 3 Explanatory Power and Independence 4 Abstraction in a Counterfactual Framework 5 Königsberg—Encore! 6 Conclusion 1 Introduction There is broad agreement that many explanations derive their explanatory power from information about dependence. Causal explanations involve causal dependence, and various kinds of non-causal explanations can, arguably, be similarly understood in terms of non-causal dependence of the explanandum on the explanans.1 Indeed, it may be that (almost) all explanations involve information about what the explanandum (causally or non-causally) depends on. This hypothesis, if tenable, could be part of a unified account of explanation, according to which explanations—causal or otherwise—explain by virtue of providing information about dependences. Against this hypothesis, consider explanatory abstractions. Explanations vary in their abstractness: some explanations appeal to relatively abstract features of reality, whereas others turn on much less abstract explanans. We value explanatory abstraction, when appropriate. It filters out details that are irrelevant to the explanation in question. But abstraction does not directly concern what the explanandum depends on. Rather, abstraction is about the independence of the explanandum from irrelevant details. It is tempting to think, along with various recent authors, that this information about independence is where the explanatoriness of many abstract explanations lies. This mode of explaining, associated with what is irrelevant to the explanandum, can seem sui generis, different in kind from explanations in terms of dependences. We will argue that explanatory abstractions do not point to a sui generis mode of explaining. There is no need to revise the idea that explanatory power is a matter of providing dependence information (and the more the better); no need to abandon the hope for a unified account of explanatory power. Here we stand against a clear trend that urges the opposite: that some abstract explanations do not fit an account of explanatory power that is focused on (non-)causal dependence.2 More specifically, we aim to undercut prominent arguments supporting the trend we oppose. We will present and argue for a unified view that extends the account of causal explanation developed by Woodward ([2003]) and Woodward and Hitchcock ([2003a], [2003b]) to allow it to capture influential exemplars of abstract explanations behind the anti-unificationist trend. Before we get to the nitty-gritty, let us illustrate explanatory abstraction and sketch some of the key points of the article. The explanation of why Mother fails to divide her twenty-three strawberries equally among her three children (without cutting any strawberries) turns on the fact that twenty-three is not evenly divisible by three.3 Citing the causal details or laws involved in Mother’s attempt or the physical constitution of the strawberries does not suffice to explain this fact. The explanation is independent of such details; the explanatory mathematical facts hold irrespective of contingent causal laws and details. So where does the explanatory power associated with mathematics come from? Mathematical facts do not seem to capture dependence relations between numbers in a way analogous to contingent causal and nomological connections. So perhaps the explanatory contribution of mathematics here cannot be captured in terms of dependences? Perhaps the explanatory contribution of mathematics is somehow sui generis, turning on the independence of mathematics from the details of the physical constitution and the causal laws? We will be concerned with explicating and rebutting this intuition. We see little reason to shift focus from dependence to independence in analysing abstract mathematical explanations of this kind. One easily overlooks a wealth of dependence information that we take for granted here. The explanation does not just tell us that Mother must fail as a matter of mathematical necessity. It also tells us what the failure depends on—the number of strawberries—and how. For example, if Mother had two fewer strawberries or one strawberry more, then she could have succeeded. This change-relating, what-if-things-had-been-different information is exactly the kind associated with explanatory dependence.4 We will argue that this information is doing all the explaining, and that this is captured by a natural extension of the ideology of causal explanation, according to which explanations function by providing what-if-things-had-been-different information. It has been broadly thought that this ideology is ill-fitting for capturing abstract non-causal and/or mathematical explanations, since the relevant modal and counterfactual notions only apply to causal connections.5 This, we will argue, is mistaken. The reasons given for leaving behind the counterfactual dependence ideology associated with causal explanations in connection with abstract explanations are faulty. And so are the arguments for sui generis, independence-based accounts of abstract explanations. The key to seeing the latter is to recognize, first of all, that practically all explanations have some degree of abstraction and provide some dependence information. What we need is a way of gauging the explanatory importance of dependence information, relative to information about independence. We will argue that the explanatory importance of dependence information is revealed by varying the information while keeping the degree of abstraction fixed. In the context of the strawberry case, consider removing the dependence information altogether, pretending that we can take away all dependence information based on background arithmetical knowledge, and only retain information about the explanandum’s independence from causal laws and features. If all we know is that, as a matter of mathematical necessity, Mother's attempt with twenty-three strawberries must fail, so that we are unable to answer any questions about possible changes that would render her success possible, then it is not clear that we have any explanation of Mother's failure. Consider, now, adding a little bit of information about dependence, pretending that we only get to know that Mother could have succeeded with twenty-four strawberries. Assume that this is all the information we have, besides knowing that twenty-three is not divisible by three. Here we may have an explanation, but it looks like a very thin one. Yet, in terms of independence from particular causal laws, processes, and micro-physical conditions, it is on a par with the general explanation that appeals to background arithmetical knowledge. If the original explanatory information explains by virtue of showing the explanandum to be independent of physical details and laws, then we would expect the modified explanation to explain to a commensurate degree. But we do not see this. When we vary the amount of information about dependence, we also seem to vary the degree of explanatoriness, in a way that suggests that it has little to do with independence information. To develop the line of thought sketched above we need to lay some conceptual groundwork. To this end, we distinguish three different dimensions of abstraction (Section 2). Then we show how these distinctions apply to prominent recent conceptions of explanatory abstraction (Section 3). Our key claim is that not all dimensions of abstraction are responsible for explanatoriness, even if good explanations are typically abstract along several dimensions. Against the backdrop of these conceptual distinctions, we will then present an alternative counterfactual account of the explanatoriness of abstract explanations in the spirit (but not to the letter) of the dependency-oriented account developed by Woodward ([2003]) and Woodward and Hitchcock ([2003a], [2003b]). In particular, we focus on how to select the right what-if-things-had-been-different counterfactuals (Section 4). The virtues of this unified account are illustrated by revisiting a graph-theoretic explanation of Königsberg’s bridges (Section 5), and further underlined in relation to other challenges faced by less unified views of explanation (Section 6). 2 Three Dimensions of Abstraction We will challenge the idea that explanatoriness turns on the independence of the explanation from particular physical laws, processes, or concrete physical structures. Nonetheless, we think that abstract explanations typically do exhibit this kind of independence. The task of this section is to distinguish three conceptually different ways in which explanations can be abstract. They will all involve independence in one sense or another. However, we will later argue that only one kind of independence carries information about explanatory dependence and, as a consequence, contributes to explanatory power. Let us recall some simple examples of abstract, plausibly non-causal mathematical explanations (what we have been calling ‘abstract explanations’) in order to bring out intuitions about different senses of explanatory abstraction. The following examples have featured prominently in recent philosophical analyses: (i) Economical bees build hexagonal honeycombs because this is the most resource-efficient way to divide a Euclidean plane into regions of equal area with least total perimeter (Lyon and Colyvan [2008]). (ii) Plateau’s laws hold for soap film geometry, because this geometry minimizes the system’s energy by minimizing the surface area (Lyon [2012]; Pincock, [2015]). (iii) Königsberg cannot be toured by crossing each and every bridge exactly once because of the relational configuration of the bridges (Pincock [2007]). (iv) The strawberry case discussed above. We will focus on (iii) and (iv). Abstract explanations hinge on (relatively) abstract features of reality that are non-causal (or perhaps causal in a very broad sense of ‘causal’).6 This abstractness is partly a matter of relative independence of the explanans from the actual physical structure of the entities involved: strawberries, bridges, soap films, honeycombs, and so on. This is the first intuitive aspect of explanatory abstraction that we wish to highlight. There is nothing special about the physical nature of strawberries, for example, that makes them indivisible in this way. The very same explanation explains why twenty-three marbles (say) cannot be evenly divided. The strawberry explanation abstracts away from whatever concrete features need to be in place in order for there to be a set of twenty-three ‘individuals’ that can be divided into mutually exclusive and collectively exhaustive subsets. A second, more radical aspect of abstraction pertains to the fact that the explanations above are independent from the actual laws of nature that underlie the physical processes presupposed by the why-question: the processes of group division, bridge crossing, or comb building, for example. There is nothing special, as far as these explanations are concerned, about the actual physics that underlies the solidity of honeycomb walls, strawberries, or bridges. The same explanations would work just the same in more exotic, ‘counter-legal’ possible worlds where the underlying physical laws are rather different (assuming the relevant why-questions still make sense). The explanation of Plateau’s laws, for example, transcends the nomologically possible causes or grounds of the area-minimizing action of soap bubbles. This explanation is modally robust with respect to variation in the underlying physical laws, as long as the laws still give rise to area-minimizing action for some surfaces. To this extent the explanation abstracts away from the actual nomological facts pertaining to the systems in question. So far, we have identified a conceptual distinction between two senses of abstraction. There is also a third aspect of abstraction in the above explanations that is easily missed. This has to do with the explanatory regularities involved, and the degree to which an explanatory generalization is independent from the actual value of an explanans variable. For example, the fact that twenty-three is not evenly divisible by three is an instance of the more general fact that no integer apart from whole multiples of three is evenly divisible by three. In so far as we judge that the explanatory work is really done by this more general arithmetical background knowledge, we can say that this explanation would answer the question of why Mother would have failed had she had twenty-two strawberries (or why she could have succeeded had she had twenty-one). The explanation turns on a regularity that is robust by virtue of allowing the same explanation to be given for a wide range of changes in the number of strawberries. In this third sense of abstraction, the wider this range is, the more abstract the explanation. These three intuitive aspects of explanatory abstraction all have something in common, namely, each involves a way that an explanation can be independent of some feature of the system in question. This independence is naturally thought of as follows: An explanation can cover a range of possible systems. Explanatory abstraction is a matter of independence of the explanation from the details of ‘realization’. In the above cases of abstract explanations, the extent of unnecessary details omitted is striking. Consider the Königsberg case, for instance. The explanation is independent of all the physical and geometrical features of the bridges and the crossing processes, and of the underlying nomological facts pertaining to the bridges and their possible crossings, only relying on a very high-level, global, structural feature associated with a wide-ranging explanatory regularity. (We will make this more precise later in the article.) The extent of such independence is most striking in the case of abstract explanations, but more or less any explanation actually exhibits the three dimensions of abstraction to some degree. Consider the mundane case of gravitational pendulums, for instance. Why is the period of a given (more-or-less) ideal pendulum two seconds? Explaining this causally, in terms of the pendulum's length and gravitational acceleration, relies on a law-like regularity that supports the same explanation in a range of possible cases in which these two parameters will vary (the third dimension of abstraction). Furthermore, the explanation can apply regardless of whether the underlying nomological facts are rooted in a classical Newtonian world with gravitational force acting at a distance, or whether the pendulum occupies a general relativistic world with gravity being a manifestation of curved spacetime (the second dimension of abstraction). Finally, the why-question presupposes that the pendulum cord is inextensible, but the explanation is independent from whatever microstructural facts underlie this property (the first dimension of abstraction). In light of the fact that more or less all explanations are abstract to some degree along all three dimensions, it is prima facie surprising that there would be two radically different sources of explanatory power in play: one for the cases that are commonly thought of as causal and a different source for cases of abstract explanation. We will argue that the difference between this mundane causal explanation and abstract explanation is one of degree, not of kind. The notion of explanatory abstraction as a matter of independence is clearly a modal notion: it concerns the range of possible systems covered by the explanation. Different aspects of explanatory abstraction concern different dimensions of modal variation with respect to these possible systems. We can think of possible systems that vary from the actual explanandum only in the (nomologically) possible physical structures of the entities involved; possible systems that vary in the microphysical or dynamical laws they obey; possible systems that vary from the actual explanandum by varying a variable that explicitly features in the explanatory regularity employed. Although modal variations along these different dimensions are often interlinked and metaphysically intertwined in complex ways, the dimensions themselves are conceptually independent.7 In the strawberry case where we considered the explanatory regularity with respect to a system of twenty-three versus twenty-five strawberries, one need not think about variation in the physical structures of these entities or the division processes. Similarly, in thinking about the explanation with respect to systems that vary in the physical structures of the entities to be divided, one need not think of variation in their number, or of counter-legal possibilities where the underlying physical laws are different. And vice versa. Often abstract non-causal and mathematical explanations—like the ones above—exhibit a striking level of abstraction in each of the three senses. We believe this has confounded some recent commentators. If one thinks, prima facie, that abstraction in a particular sense is the source of explanatoriness, it is all too easy to find corroborating evidence when so many good explanations are highly abstract in this sense. Now that we have conceptually separated the different dimensions of abstraction, however, we can ask if explanatory abstraction has something to do with explanatory power (as is commonly accepted), then exactly which dimension or dimensions of abstraction can contribute to it? In its full generality, we cannot hope to comprehensively answer this question in a single paper. There is, however, a narrower question that we can answer. It is commonly agreed that explanation (and the associated notion of explanatory power) has both worldly and pragmatic or communicative aspects. Here, we will largely set aside the communicative and pragmatic aspects. What is at stake in the debate at hand is whether to function as a source of explanatory power, abstract explanations require facts tied to independence, instead of (or in addition to) facts about dependence. In particular, is it the case that for the paradigmatic cases of abstract explanations (mentioned above), explanatory power is underpinned by worldly facts related to independence? Our first aim is to undermine prominent recent arguments that answer in the affirmative to the last question. Our second, positive aim is to show how these abstract explanations can be accommodated in more unified terms, in the spirit of the counterfactual account developed by Woodward and Hitchcock ([2003a], [2003b]) (henceforth, the W-H account). According to the W-H account, explanations provide a particular kind of information about dependence. This account creates an essential association between explanatoriness and our third dimension of abstraction. (Roughly, an explanatory generalization that has a broader range under which we can vary the values of the variables provides more explanatory information about the dependence of the explanandum on these variables than a corresponding explanatory generalization with a more restricted range of invariance.) This is in sharp contrast to recent claims according to which (some) abstract explanations explain either (i) by virtue of abstracting away from all the concrete physical features of the system in question (our first dimension above), or (ii) by virtue of abstracting away from the underlying laws of nature (our second dimension). The W-H account is well liked in connection with various causal explanations (even amongst the advocates of alternative analyses of abstract explanations), but there has been resistance to extending this counterfactual account to abstract explanations that are (plausibly) non-causal or mathematical. We will argue that this resistance is entirely unnecessary. In the spirit of the W-H account, issues concerning explanatory power and its relation to abstraction are tractable in very natural terms for abstract explanations too, after the key notions of this account are appropriately clarified and refined. Next, we will critically review some recent claims regarding explanatory abstraction, questioning the intuitions that support them. After this, we will discuss more systematically the third dimension of abstraction, which has a better claim for association with the source of explanatory power than either of the first or second dimensions. 3 Explanatory Power and Independence Various philosophers have recently argued (in different ways) for the disunity of explanatoriness by pointing to the sui generis role of abstraction as a source of explanatory power. We will focus on two authors who particularly clearly associate explanatory power with either the independence from concrete physical details (the first dimension), or the independence from the particular laws (the second dimension). Pincock ([2007], p. 257) brought the Königsberg bridge example into philosophical prominence by identifying it as an abstract explanation that ‘appeals primarily to the formal relational features of a physical system’. Relational features that are ‘formal’ are clearly meant to stand apart from causal relations. Furthermore, regarding the explanation of the impossibility of touring the town by crossing each and every bridge exactly once, Pincock ([2007], p. 259) writes: […] an explanation for this is that at least one vertex [in the formal graph structure instantiated by the bridges] has an odd valence. Whenever such a physical system has at least one bank or island with an odd number of bridges from it, there will be no path that crosses every bridge exactly once and that returns to the starting point. If the situation were slightly different [so that] the valence of the vertices were to be all even, then there would be a path of the desired kind. This is all surely correct. The question is how to capture this explanation in philosophical terms. Pincock ([2007], p. 260) intimates that the explanation critically turns on abstraction: The abstract explanation seems superior [to a microphysical explanation] because it gets at the root cause of why walking a certain path is impossible by focusing on the abstract structure of system. Even if the bridges were turned into gold, it would still have the structure of the same graph, and so the same abstract explanation would apply. By abstracting away from the microphysics, scientists can often give better explanations of the features of physical systems. The notion of ‘abstracting away from the microphysics’ is naturally construed as being along the first dimension identified above: a matter of relative independence of the explanans from the physical structure of the entities involved. As Pincock ([2011], p. 213) puts it: ‘the explanatory power [in this case] is tied to the simple way in which the model abstracts from the irrelevant details of the target system’. The intuition is that the explanation comes from stripping away as irrelevant all the physical details pertaining to the bridges’ make-up, length, location, angles, and so on, thereby highlighting what is relevant, namely, the formal ‘mathematical structure found in the target system itself’ (Pincock [2011], p. 213). We can see Pincock’s intuition, but we are unable to see a good reason for viewing this as a sui generis abstract explanation. The explanation is undeniably highly abstract and plausibly non-causal. But the notion that good explanations only provide relevant information, leaving out unnecessary details, is common ground with dependence accounts of explanation. According to dependence accounts one ought to provide suitable information about what the explanandum depends on, and one also ought not to claim (or imply) that the explanation depends on something that it does not. The leaving out of irrelevant detail is insufficient to motivate the view that abstract explanations are sui generis in the way Pincock regards them. How about the notion that the explanatorily relevant features of Königsberg are ‘formal-cum-mathematical’? Does this motivate a departure from familiar counterfactual accounts of explanation? We do not think so. We do not regard the explanans formal or mathematical in any sense that makes the application of counterfactuals difficult. The explanation involves applied mathematics; it is not an intra-mathematical explanation. There is a clear sense in which the explanans is just concerned with how many bridges there are to each of the islands. It is even unclear why the valence of an island—there being an even or odd number of bridges to it—is not a high-level physical feature of it. And the same is true for the yet more abstract feature of there being at least one odd island in the whole system of many islands. Furthermore, there seems to be a clear and straightforward sense in which the explanandum at stake depends on these, arguably high-level physical, features: the explanation tells us how the explanandum would change if there were a different number of bridges (as Pincock himself notes; see the first quotation above). Whether this explanatory dependence is causal or not is neither here nor there for the prospects of sticking to the core idea of the W-H account of explanation: explanatoriness is ultimately a matter of telling us what the explanandum depends on.8 When it comes to identifying the worldly source of explanatory power, we have not yet been given a reason to think that in the Königsberg case explanatoriness derives (even partly) from the independence of the explanandum from some worldly features. Let us now move on to consider Lange’s analysis of the explanatory power of the cited abstract explanations (as ‘distinctly mathematical’).9 Lange ([2013], pp. 486–8) argues that his analysis of these explanations reveals ‘a fundamental difference’ between causal explanation and abstract non-causal explanation, and its significance ‘lies in what it reveals about the kinds of scientific explanations there are’. Abstract explanations, Lange argues, explain not by supplying information about the world’s network of causal relations but by ‘showing how the fact to be explained was inevitable to a stronger degree than could result from the causal powers [actually] bestowed by the possession of various properties’. This is a nod towards Wesley Salmon’s ([1984], p. 293) ‘modal conception’ of scientific explanations, according to which such explanations ‘do their jobs by showing that what did happen had to happen’. As Lange ([2013], p. 505) puts it: [They explain] not by describing the world’s actual causal structure, but rather by showing how the explanandum arises from the framework that any possible causal structure must inhabit, where the ‘possible’ causal structures extend well beyond those that are logically consistent with all of the actual natural laws there happen to be. According to Lange, abstract explanations may happen to also provide modal information that a counterfactual account (in the spirit of the W-H account) would count as explanatory. But he explicitly rejects the idea that their explanatoriness in any way derives from such modal information, relating it instead to information about the independence of the explanandum from some contingent features of the system in question.10 In contrast to Pincock, the critical sense of independence for Lange is in the direction of the second dimension of abstraction identified above: abstraction away from the actual laws underlying the features of the system presupposed in the context of the relevant why-question. In relation to the Königsberg case, for instance, Lange ([2013], pp. 505–6) writes: [The] explanation of the repeated failure to cross the Königsberg bridges shows that it cannot be done (where this impossibility is stronger than physical impossibility) […] The explanans consists not only of various mathematically necessary facts, but also […] of various contingent facts presupposed by the why question that the explanandum answers, such as that the arrangement of bridges and islands is fixed. The distinctively mathematical explanation shows it to be necessary (in a way that no particular force law is) that, under these contingent conditions, the bridges are not crossed. Lange’s account thus emphasizes the way in which the actual physical laws do not come into play in explanations such as this. Undoubtedly, Lange is right to note that the sense of abstraction at play goes beyond the first dimension concerning the irrelevance of nomologically possible physical realizations of the contingent features presupposed in the context of the why-question. The same explanation would work, in the same way and to same extent, in a far removed possible world with alien properties and laws—as long as there is a system with traversible bridges for which the why-question makes sense. But having said that, why should we regard this sense of abstraction as a source of explanatoriness in connection with abstract explanations? The answer to this question is surprisingly difficult to find in (Lange [2013]).11 There is little to directly motivate this, beyond the commonplace idea that good explanations do not mention irrelevant details, which in this case include all causal laws. Yet again, the idea that good explanations only provide relevant information, leaving out unnecessary details, is common ground with any account according to which we ought to provide information about dependence, and we ought not to claim (or imply) that the explanandum depends on something that it does not. Thus the independence from the actual laws seems insufficient in itself to motivate the view that abstract mathematical explanations are sui generis in the sense of Lange. Both Lange and Pincock are contrasting their views to causal accounts of explanation. It is natural, then, to take them to provide a competing account of the worldly source of explanatory power. So far we have undermined broad motivations for thinking that abstract explanations require a source of explanatory power that is not centred on dependence. Our view is that two of the three dimensions of abstraction—even though undeniably exhibited by abstract explanations—do not actually provide a source of explanatoriness.12 The problem is that these dimensions of abstraction do not provide information about explanatory dependence in the W-H sense. We do not, unlike Lange and Pincock, take it to be the case that explanations that are highly abstract in the first two dimensions require an account of explanation where the source of explanatory power is not in the W-H spirit.13 In the rest of the article we will discuss this specific notion of explanatory dependence and how it connects to the third dimension of abstraction. But before we move on, let us address a natural worry. It may seem that dependence and independence are conceptually so closely connected that we cannot cleanly separate the two. In particular, any information about independence is also information about dependence: ‘E is independent from A’ means that ‘E does not depend on A’, and if we assume that explanandum E depends on something, then ‘E is independent from A’ entails that ‘E depends on something other than A’. Thus it is the case, the worry goes, that information about independence along every dimension of abstraction can provide information about dependence. In response, we simply note that it is not true that information about independence ipso facto provides information about dependence that is explanatory—that is, dependence information in the W-H sense. It is undeniably true that one provides what-if-things-had-been-different information, broadly speaking, by showing that even if the laws of nature were different, Mother could not divide twenty-three strawberries equally among three children. (What if things had been different with respect to laws of nature? Then Mother would have failed just the same!) But this is not the sort of what-if-things-had-been-different information that counts as explanatory in the counterfactual framework that we favour, according to which only what we call ‘change-relating’ counterfactual information counts as explanatory.14 To understand Mother's failure, we need to be able to say under what conditions Mother could have succeeded. We now move on to discuss this framework. 4 Abstraction in a Counterfactual Framework The starting point for the W-H account is that explanation ‘is a matter of exhibiting systematic patterns of counterfactual dependence’ (Woodward [2003], p. 192). To develop this into a theory of explanation, we need to be more precise about which patterns of counterfactual dependence matter for explanation. On the W-H account this is done by focusing on modal information that allows one to answer questions about how the explanandum would have been different under a special type of change in the explanans (Woodward [2003], p. 192). The changes in the explanans that are relevant are those that result from interventions on the explanans variable(s) (with respect to the explanandum variable). The notion of intervention is technical; it plays the role of ruling out changes in the explanans variable that are brought about by (i) changes in the explanandum variable, and (ii) changes to some other (‘common cause’) variable that changes the explanandum variable independently of the explanans variable. This approach has a number of attractive features. First, it is natural to take relations (whether these are nomological, causal, or mathematical) that are explanatory to be capable of providing some sort of modal information. Second, by requiring more specifically that the modal information relates to different possible states of the explanandum—how the actual explanandum would have been different had the explanans been different—the account captures the natural idea that explanatory information is information about worldly dependences. In the W-H account the critical modal notion of dependence gets cashed out in causal terms through the notion of a (testing) intervention. This allows one to rule out backtracking counterfactuals—for example, ‘had the period of the pendulum been different, then the length of the pendulum would have been different’—as providing the right dependences. The focus on what-if-things-had-been-different questions that concern changes in the target explanandum also provides a clear contrast to mere subsumption under modal regularities in the spirit of the deductive-nomological account of explanation. Third, this focus on interventions squares with much of our experimental practice. Fourth, the account is, as Woodward and Hitchcock ([2003a], [2003b]) discuss at length, very well suited for capturing not only what it takes to have an explanation, but also what makes an explanation better or worse. Having an explanation is a matter of having the right information about the dependences, and the more the better. The W-H account separates an explanans into two parts: a specification of an invariant explanatory generalization and a specification of the actual values of the variables in the explanatory generalization.15 For a simple illustration, let us look at the explanation of the period of a simple gravitational pendulum. Let us assume that we want to explain the fact that period T takes the value t1 (our explanandum, M) by using the simple pendulum law. In terms of the W-H account, we have explanatory generalization T≈2πlg. The explanans consists of this generalization and a specification of the actual values of l and g. In order to have an explanation according to the W-H account, the simple pendulum law has to give the (at least approximately) correct value of the period as t1 under an intervention that fixes the values of l and g to the actual values. Moreover, the simple pendulum law must capture (in at least one case) how the period would change under at least one and ideally more interventions that change the length or the gravitational acceleration. It thus captures the dependence of the explanandum on these variables. Now we are in a position to make more precise the contrast between the different dimensions of abstraction. Let us begin with the first dimension of abstraction. In the context of the simple pendulum explanation, we can ask whether explanandum T = t1 is independent of particular background conditions that are in some sense part of the system but not represented as variables in explanatory generalization T≈2πlg. We can think of the microphysical features that make the pendulum cord inextensible, for example, or the mass of the bob, and so on. The degree of independence of the explanation from such background conditions was the focus of the first dimension of abstraction. (In contrast to such ‘internal’ background conditions, there are other background conditions that are external to the system and clearly irrelevant, such as the bob being made in Japan, the exchange rate of the US dollar to the Malaysian ringgit, the colour of the shirt of the person setting the bob in motion, and so on) In the second dimension of abstraction, we considered the degree of independence of the explanation from the actual laws of nature. In the context of the pendulum example, we can ask whether we can vary the actual laws of nature and still expect the simple pendulum explanation to apply. We can alter many of the laws of, for example, electromagnetism without affecting the explanation, provided that those alterations do not run afoul of the assumptions presupposed in the context of the why-question, for example, the cord being nearly inextensible. As mentioned earlier, we can also alter many of the fundamental aspects of gravitational acceleration, such as whether its nomological basis is an action-at-a-distance effect or a manifestation of curved space-time. Finally, in the third dimension of abstraction, we considered the range of conditions directly relevant to changes in the variables in the explanatory generalization, T≈2πlg, such that the simple pendulum explanation of the period holds. For example, we could decrease g by moving the pendulum to a higher altitude or to the moon, and we could change the length of the pendulum; nonetheless, the explanation of the period would work in just the same way. On the W-H account, abstractness in this dimension is just a matter of the invariance of the explanatory generalization, ‘measured’ by the range of alternative values of variables l and g for which the generalization holds and is change relating. So far we have simply described the W-H account. Before we can apply these ideas to the paradigmatic examples of abstract explanations, we need to extend and refine some of them. In particular, the W-H account was developed as an account of causal explanation, while the abstract explanations we are interested in are plausibly non-causal. As we move beyond causal explanation, we can no longer appeal to the exact understanding of the relevant class of explanatory counterfactuals that Woodward and Hitchcock use. They delineate the class of relevant counterfactuals in terms of the causal notion of intervention: explanatory counterfactuals describe the effect of a testing intervention. These counterfactuals are underwritten by contingent explanatory generalizations, and the effects of surgical testing interventions are naturally understood as concerning causal dependences. In the non-causal cases, this notion of causal dependence becomes inapplicable or unnatural, because the notion of causal intervention is inapplicable, or because the counterfactuals are not underwritten by contingent laws of nature (but, rather, by logic or mathematics or metaphysical truths). Is it still possible in these cases to characterize explanatory dependences in counterfactual terms? The answer is yes; we have analogous explanatory dependences in the situations involving Mother's strawberries, Königsberg’s bridges, and so on.16 The challenge is how to select the right counterfactuals without assuming that causal notions are applicable. In responding to this challenge we note, first of all, that regardless of whether or not we can apply the idea of a causal intervention to the explanans variable, we can still have a good grasp of what it means to change the explanans—for example, to change the number of strawberries, or the bridge configuration in Königsberg. Furthermore, regardless of whether or not the explanatory generalization is a contingent law of nature or a stronger modal truth, we can reason about how the explanandum depends on these changes, and whether the dependence is non-symmetric in a way that supports explanatory what-if-things-had-been-different reasoning. For example, in the the strawberry example, we have a very good grasp of what changing the explanans variable amounts to. Indeed, changing the number of strawberries seems as straightforward an intervention as any. We also know how this changes the system’s divisibility by three, and consequently the possibility of Mother's success or failure. We thus grasp how Mother's success or failure explanatorily depends on the number of strawberries that she has (and the number of children). In order for the dependence to be regarded as explanatory, it needs to be appropriately directed. Intuitively, it is. As we indicated above, the notion of intervention plays the role of blocking changes to the explanans that go through changes to the explanandum (or through a common cause variable, and so on). Similarly, changes to the number of strawberries do not seem to go through changes to the system’s divisibility by three, or through changes to Mother's success or failure. After all, changing the system’s divisibility by three does not mandate any particular number of strawberries. However, changes to Mother's success or failure do seem to go through changes to her number of strawberries (or children). This intuition can be supported in a more precise manner. On the W-H account, a successful explanation makes use of an explanatory generalization for two main tasks (as outlined above): first, to show that setting the value of the explanans variable to its actual value fixes the value of the explanandum variable to its actual value; and, second, to show that changes to the value of the explanans variable that move it away from its actual value will, in some cases, change the value of the explanandum variable. Now, consider the intuition that changes to the system’s divisibility by three occur by means of changes in the number of strawberries, but not vice versa. This is underwritten by the fact the explanatory generalization regarding the number of strawberries and their divisibility by three supports the first task asymmetrically. Fixing the explanans variable to its actual value should fix the explanandum variable to its actual value, but fixing the system’s (in)divisibility by three does not fix the number of strawberries to any particular value. In contrast, the number of strawberries being twenty-three does fix the system’s (in)divisibility by three. This asymmetry underlies the intuition that changes to the number of strawberries do not come about via changes in the system’s divisibility by three.17 The asymmetry does not seem to be a causal matter, however, since the intervention on the number of strawberries can have an ‘effect’ (for example, rendering the set divisibility by three) that is rather more intimately related to it. To this extent Lange ([2013]) is absolutely right to stress the independence of this connection from any causal laws. Very similar considerations apply to the case of Königsberg’s bridges.18 In this case, it’s not clear whether the explanans variable—concerning the global configuration of the whole bridge system—can be changed in a way that necessarily satisfies our intuitions about what counts as a causal intervention. But be that as it may, we can surely have a good grasp of what it means to change the explanans variable (by strategically building and destroying bridges so as to achieve a particular overall configuration). And again, the explanatory generalization tells us how this changes the bridge system’s traversability, which depends on the configuration being in a certain way. (We will lay this out in more detail in the next section.) This dependence is again appropriately directed: while fixing the bridges in a particular way fixes the (non-)traversability of the bridge system, the reverse does not hold. We cannot fix the bridges to a particular configuration merely by ensuring that the bridge system is (say) non-traversable.19 There is more to be said about the nature of explanatory dependence in abstract explanations, of course. One can wonder about the metaphysics of the dependence relation, for example, as we have not said anything about it. Our hope is that one doesn’t have to get into the metaphysics of dependence to support the broader idea of there being such explanatory dependencies in the world, transcending those readily incorporated into the W-H account as it stands; it is enough to show how we can pick out the right counterfactuals. (This is in tune with Woodward’s own attitude to this issue.) One may also wonder what to make of the fact that in the examples at hand, the explanatory relation does not seem to connect two distinct events; it is not the case that ‘we change one thing and another change follows’.20 For instance, the event of destroying some of Königsberg’s bridges can ipso facto be the event of rendering the bridges Euler-tourable. In response, we would like to point out that leaving behind the idea that explanatory relations hold between events (or event-types) is critical to the W-H account; rather, explanatory relata are treated as variables (Woodward and Hitchcock [2003a], pp. 10–1). In causal cases, this amounts to a notion of explanatory causal relevance between such variables. Similarly, in the examples at hand it is natural think of explanatory variables as designating different aspects of the system in question. We can make sense of such variables standing in an explanatory dependence relation, even when changes in the values of the variables involve one and the same event. (See Saatsi [2016a] for related discussion of explanatory variables in connection with non-causal explanations from ‘geometry of motion’.) There is also more to be said about the directionality of explanation. Here we refer the reader to work done elsewhere that offer ways of developing further the ideas we have only briefly sketched here.21 But we hope to have said enough to show that the prospects for a more detailed account are no worse—and, in our opinion, are better—when extending a dependence account in the W-H spirit than when focusing on independence along the lines of Lange and Pincock.22 Our suggested diagnosis of the original W-H account is that it correctly identifies explanatory counterfactuals as those that are appropriately change-relating and directed. From now on, let us call the explanatory counterfactuals simply ‘change-relating counterfactuals’ (adapting Woodward’s use of the term), to keep in mind that they are a strict subset of all counterfactuals and a broader class than interventionist causal counterfactuals. Before we can move on to consider a detailed application, we need to consider one final aspect of extending the W-H account. Since the relevant counterfactuals are selected by focusing on changes to some object or system, the modal information that we are interested in is tied to the system or object in question. They are what Woodward ([2003], p. 281) calls ‘same object’ counterfactuals: Suppose that we wish to explain the behavior of some object or system o. As the standard view is usually understood, it claims that generalizations of form ‘All As are Bs’ are explanatory of the behavior of o if they support counterfactuals of the following form: […] If some object o*, different from o and that does not possess property A, were to be an A, then it would be a B […]. Call such counterfactuals ‘other object’ counterfactuals: they describe what the behavior of objects other than o would be under the counterfactual circumstances in which they are A. By contrast, according to the view I have been defending, to count as invariant and hence explanatory with respect to o, a generalization must support ‘same object’ counterfactuals that describe how the very object o would behave under an intervention. At first glance, this may seem to suggest that all of the explanations that we have in mind must be about specific, particular systems or objects in order for the distinction—central to the W-H account—between same object (SO) and other object (OO) counterfactuals to apply. Pincock ([2015]) takes this to rule out the application of the W-H account to abstract explanations, since some highly abstract explanations do not seem to provide information about any particular object at all (much less information about changes to any particular object). In our view, one should not think of the contrast between SO counterfactuals and OO counterfactuals in the same way as Pincock. First, note that even in the case of causal explanations, most scientific explanations do not have a particular, individual object as the explanatory target. The explanandum is typically generic (and general, even when couched in language such as ‘the period of a simple pendulum’). The importance of the distinction between SO and OO counterfactuals lies in the fact that it focuses our attention on the right kind of change-relating counterfactuals. When we make use of the simple pendulum law to explain the period of a simple pendulum, the right counterfactuals to have in mind are those that ask how the period of a generic kind of dynamical system—namely, a simple pendulum—is affected by changing its length, say. Counterfactuals concerning objects other than simple pendulums do not come into play, even when they are reasonable and well defined. (It may be true that had a hammer been suspended around an appropriate pivot with the head down, then we could have used the simple pendulum law to explain its period; but such OO counterfactuals are not required in order to explain the period of a simple pendulum.) Although the W-H account does not apply directly to the case of Königsberg’s bridges, the central notion of change-relating (SO) counterfactuals can be carried over to this case. This is the work of the next section. 5 Königsberg—Encore! We will now revisit the Königsberg case to illustrate the above counterfactual account and its virtues. So far, we have mainly criticized Pincock’s and Lange’s motivations for thinking that there is a sharp distinction between causal explanations and abstract non-causal explanations, maintaining that one can instead approach both types of explanations in the fundamentally same spirit. We now push for a stronger claim, arguing that the core claim of the counterfactual account (as we understand it)—that explanatoriness is associated exclusively with the third dimension of abstraction—can be tested against the alternative viewpoints. This will provide a clear reason to prefer the counterfactual account. Let us go back to the eighteenth-century Königsberg, and ask why it is impossible to make a round tour of Königsberg crossing each of its seven bridges exactly once (QK). An intuitive explanation-sketch response goes as follows: Clearly, each visit to a landmass (an island) requires the use of two bridges: one in and one out, otherwise one gets stuck. Therefore, in order for a network of bridges to allow for a round tour—to be ‘tourable’—each island must have a number of bridges that is some multiple of two. On the other hand, if there are one or more islands with an odd number of bridges, the system is not tourable. Eighteenth-century Königsberg had four such troublesome junctions, rendering a round tour of Königsberg impossible. Euler articulated a famed graph-theoretic explanation that makes the above sketch precise. This is standard material in graph-theory textbooks, illustrating the explanatory use of mathematical notions (such as ‘connected graph’, its ‘vertices’, and their ‘degrees’).23 But before we get to the graph-theoretic explanation in its full generality, it is worth attempting to answer QK without graph theory. In the counterfactual framework, explanations must involve an invariant, change-relating generalization that supports counterfactuals indicating an explanatory dependence of the explanandum on the explanans. A generalization that thus underwrites an answer to QK could be markedly less abstract and less general than the graph-theoretic explanation, without thereby being unexplanatory. As a matter of fact it is easy to find a simple invariant generalization that furnishes a non-mathematical answer to QK. Let us focus our attention on the kind of bridge system that connects exactly four islands, with at most two bridges between any two islands. Euler’s Königsberg is one of these systems (see Figure 1). Figure 1. View largeDownload slide Königsberg’s bridge system. Figure 1. View largeDownload slide Königsberg’s bridge system. There are 395 such bridge systems in total. We can classify them as follows: Call a bridge system ‘even’ if and only if each island has an even number of bridges. In this particular case, this means that each island must have two, four, or six bridges. If a bridge system is not even, call it ‘odd’. Considering the specific type of bridge system exhibited by Königsberg, we can ask why it is not tourable. In answering this question, we naturally look for an explanatory generalization capable of providing suitable modal information by supporting appropriate, change-relating counterfactuals. Focusing our attention, for now, only on the bridge systems consisting of four connected islands with at most two bridges between any two islands, a fitting generalization is not hard to find: of all these bridge systems, all and only the even ones are tourable.24 This is a true generalization about this set of 395 types of bridge system. It is also a generalization that can afford us a degree of explanatory purchase on QK. That is, by reference to this generalization, we can begin to answer QK simply by noting that Königsberg’s bridge system is not tourable because it is not even; it would be tourable if it were odd. Königsberg’s tourability—the feature that is our explanandum—depends on this high-level physical property of the system.25 More formally, we can define the following binary variables X and Y:   X=even/oddsystemx=1:evenx=0:odd  Y=(non-)tourablesystemx=1:tourablex=0:non-tourable The simple explanatory generalization at stake then states that for bridge systems of this kind—four islands, with a maximum of two bridges between any pair—it holds that   X=Y. This equation should be read left to right, as indicating an asymmetric dependence of Y-variable on the X-variable. The tourability (or otherwise) of a bridge system depends on it being even (or odd). Being even (or odd) does not depend on tourability; it only depends on the number of bridges. All in all, the explanation naturally fits the counterfactual framework presented above (Section 4). The explanatory generalization, while narrow, is explanatory nevertheless, even if minimally so. Undoubtedly, the explanation provided immediately raises further questions. Why exactly is this generalization true, for example? What if we start adding or subtracting islands? What if we relax the restriction that there are at most two bridges between any pair of islands? This clearly renders the explanation shallow and somewhat contrived, especially in comparison to a full-blown graph-theoretic account. But none of this diminishes the philosophical significance of this explanation. For however minimal and shallow the toy explanation is, we deem it to have some explanatory power nevertheless, and we maintain that this is due to the explanation providing modal information of the right sort.26 The shallow explanation above clearly has a degree of abstractness along the three dimensions introduced in Section 2. Indeed, the explanation is highly abstract with respect to the specific material realizations of the bridges (the first dimension) and also with respect to the underlying laws of physics (the second dimension). As a matter of fact, the explanation is as abstract along these dimensions as the full-blown graph-theoretic explanation! Yet the explanation is shallow. This clearly speaks against the idea that the (minimal) explanatoriness in question springs from abstraction along these lines, as Pincock and Lange would have it. With regard to the third dimension of abstractness, the explanatory generalization does allow us to answer some change-relating, what-if-things-had-been-different questions. We have counterfactual information of the right, SO kind. We can answer counterfactual questions about how the tourability of a generic kind of structural system—the bridge system—is changed by changes to the oddness or evenness of the system (at least as long as we stay within the constraints of four connected islands and a maximum of two bridges between any two islands). This is what makes it explanatory, even if shallowly so. Yet, its degree of abstractness along this dimension is very limited. Although the explanatory generalization never delivers the wrong answer, the generalization is simply silent on what happens in cases of more than four islands or more than two bridges between some islands. Thus, the explanation using the generalization breaks down in these cases.27 In order to understand precisely how the shallow explanation compares in its abstractness to the deeper graph-theoretic explanation, we need to pay attention to the fact that the variable X concerns a determinable property of the system: it being even or odd. This determinable property is determined by the bridges’ configuration. There are various ways for a bridge system to be even. One way is for each island to have (say) four bridges. This is still a determinable property of the system, determined by a specific way of having four bridges for each island (see Figure 2). Figure 2. View largeDownload slide Two determinate 4–4–4–4 configurations. Figure 2. View largeDownload slide Two determinate 4–4–4–4 configurations. Here is another way for a bridge system to be even: one of the islands has eight bridges, and the other three islands have two, two, and four bridges, respectively. While our generalization supported some explanatory what-if-things-had-been-different questions, it is simply silent on the system’s tourability if we change X from odd to even by allowing one island to have as many as eight bridges. Similarly, it is silent on what happens if we change X from even to odd by including seven bridges for some island. To answer what-if-things-had-been-different questions corresponding to these cases, we need a broader generalization, X = Y, that applies to systems as rich in bridges as these. Differences in the determinate configurations do not explicitly feature as a variable in explanatory generalization X = Y. Indeed, on the face of it, the explanatory generalization looks the same regardless of the range of determinate what-if-things-had-been-different questions it is taken to support. Differences in the range of determinate realizations of evenness and oddness do come into play through restrictions in application. When we consider changes to the variable (even, odd), these changes have to go through changes in the bridge system (there is no way to change the evenness of the system that does not go through changing the specific configurations of the bridges). If we are justified in taking generalization X = Y to apply to systems that have (say) a maximum of three bridges between any two islands, then our explanation covers a wider range of conditions directly relevant to changing the value of the explanans variable, X. In particular, we can also consider what-if-things-had-been-different situations with seven, eight, or nine bridges for some island(s). In this way, we can straightforwardly compare different Königsberg explanations with respect to their degree of abstraction in the third dimension. The explanation where we restrict the generalization to a maximum of three bridges between any two islands applies to all the ways of altering the variable in the explanatory generalization that has a two-bridge restriction—and then some! The full-blown graph-theoretic explanation is, of course, maximally abstract along this third dimension of abstractness. The explanatory generalization now covers any (connected) bridge system of arbitrarily many islands and bridges. The graph-theoretic explanation has considerable depth, in contrast to the shallow, non-mathematical explanation. This is due solely to increased abstraction along the third dimension. The explanatory dependence of the shallow explanation is subsumed under a more general explanatory dependence, enabling us to answer a much wider range of change-relating, what-if-things-had-been-different questions. The shallow explanation does not contain irrelevant information about the nature of the bridges, their material constitution, or their length, say, nor does it contain irrelevant information about the underlying nomological features, such that the increase in explanatory depth could be due to abstracting away from such information. Indeed, and as already noted, the explanans of these two explanations are already maximally abstract along the first and second dimensions, and the considerable increase in the explanatory power should be attributed solely to the third dimension of abstraction. 6 Conclusion We have argued that paradigmatic abstract (plausibly) non-causal explanations can be naturally accommodated within an account that associates explanatoriness with suitable information about dependence. This improves the prospects of subsuming many explanations under a unified counterfactual framework, opposing the current trend that emphasizes the explanatory value of abstraction as a sui generis source of explanatoriness. We found the motivations for this trend questionable (Section 3), leaving room for a more unified account. More unified theories are often better, ceteris paribus, but we are not just expressing this kind of a prima facie preference for unification. Rather, we motivated the unified account with the following considerations: First, we argued that the unified account better captures intuitions about ‘explanatory depth’. Part of the force behind the W-H framework comes from the fact that it provides a natural starting point for capturing intuitions about explanatory goodness (Woodward and Hitchcock [2003a], p. 3; Ylikoski and Kuorikoski [2010]). We argued, in the same spirit, that our viewpoint can be tested by varying dependence information—the hypothesized source of explanatoriness—keeping other things fixed (Section 5). We maintain that our intuitions about radically varying explanatory power naturally correspond to radically varying amounts of W-H dependence (but not independence) information, in a way that is difficult to accommodate from the alternative viewpoints. Second, the unified account has a further virtue worth flagging. This has to do with dissolving difficult questions facing the more disjunctive accounts that regard abstract explanations as fundamentally different from causal explanations in their source of explanatoriness. Noting that causal explanations also typically abstract away from a huge amount of physical detail raises questions about Pincock’s point of view, for example: What explains the distinct qualitative difference between causal explanations that exhibit a degree of abstraction along each of the three dimensions, on the one hand, and the sui generis abstract explanations, on the other? Is there a ‘threshold’ of abstraction above which the counterfactual conception fails, despite capturing abstract causal explanations so well? A similar question can be raised for Lange’s account. Many causal explanations also abstract away from a huge amount of underlying nomological detail. Thus, all causal explanations that incorporate a degree of abstraction along the second dimension of abstraction also show how the explanandum is, to a corresponding degree, ‘necessary to a stronger degree of necessity’ by virtue of showing the irrelevance of some of the actual laws involved. So why is it that such modal information about independence becomes explanatory in a sui generis way in connection with abstract explanations? Or is it the case that such modal information always contributes to the explanatory power, but it contributes in a different way when an explanation abstracts away from (almost) all nomological information, as in the case of ‘mathematical’ explanations?28 Why do abstract explanations explain so differently—in Salmon’s ‘modal’ mode—from causal explanations that incorporate a degree of similar abstraction? In our view, Pincock and Lange have not provided satisfactory answers to these questions. In particular, their arguments that abstract explanations do not work by providing information about causal dependence do not show that they work in a radically different way instead (as opposed to providing information about non-causal W-H dependence). The more unified account we advocate has the virtue of sidestepping these issues entirely. According to this account, paradigmatic abstract explanations, despite their non-causal character, are explanatory for the same reason as causal explanations. In both cases, the explanatory power springs from counterfactual information of the same sort, and the paradigmatic exemplars of abstract explanation need not be regarded as sui generis. Acknowledgements We are grateful for the comments, questions, and criticisms from the participants at BSPS15 and EPSA15. Thank you also to the participants of the Leeds philosophy of science reading group. The support of MOE Tier 1 grant RG162/14 is gratefully acknowledged. Footnotes 1 See, for example, (Woodward [2003]; Woodward and Hitchcock [2003a], [2003b]; Ylikoski and Kuorikoski [2010]; Strevens [2008]). Ruben ([1990]) is an early proponent. 2 See, for example, (Lange [2013]; Pincock [2007], [2015]—for arguments that none of the existing dependence accounts can capture highly abstract explanations; Batterman and Rice [2014]). 3 Cf. (Lange [2013]; Braine [1972]). 4 Most prominently in (Woodward [2003]). 5 There are exceptions to this. For example, (Bokulich [2008], [2012]; Rice [2015]; Reutlinger [2016]). We take ourselves to be in the same tradition as these accounts. However, we will differ in that we need to specify more precisely the kind of what-if-things-had-been-different information that counts as explanatory (which we do in Section 4). 6 The issue of exactly what kinds of dependences count as causal is convoluted and largely orthogonal to our interests in this article. What we want to understand here is how various abstract explanations explain. 7 The dimensions are metaphysically intertwined because, for example, various possible changes in the physical laws are bound to imply changes in the physical structures. 8 After heavily emphasizing the idiosyncratic features of abstract explanations, Pincock ([2015]) suggests that it may be possible to provide a unified account in terms of dependence. However, Pincock’s notion of dependence is explicitly not in the spirit of the W-H account. We will address Pincock’s objection to the W-H account in Section 4. 9 The term ‘distinctly mathematical’ is a term of art in (Lange [2013]). It is aiming to capture many of the cases that Pincock ([2015]) would call ‘highly abstract’. 10 In relation to the strawberry example, Lange ([2013], pp. 493–6) argues that the counterfactuals pertaining to the number of Mother’s strawberries provide causal information, noting, for example, that ‘clearly manipulation of the numbers of strawberries or children would bring about corresponding changes in the outcome of Mother's attempt’; yet he maintains that ‘this explanation is non-causal because it does not work by describing the outcome’s causes or, more broadly, the world’s network of causal relations’. 11 Lange’s motivations are best understood in the broader context of his related views on other kinds of non-causal explanations (Lange [2015]), and on the modal metaphysics of laws of nature in general (Lange [2009]). Like other philosophical views, Lange’s account of abstract explanations can gain indirect support from being a coherent part of a ‘bigger picture’. Here we do not wish to assess the pros and cons of this bigger picture; we will focus on Lange’s association of explanatoriness with the second dimension of abstraction, in connection with the paradigmatic examples of abstract explanations. 12 This is not to say that they can never be relevant to any aspect of good explanation. To only provide relevant information is a shared commitment of any account of explanation (cf. Strevens [2008] on causal explanation). This goes some way towards explaining the intuition that the first two dimensions provide a separate source of explanatory power. For example, if we assume that we have a mistaken belief that taking the island as a starting point is relevant to explaining the failure to complete a round tour of Königsberg, then it is enlightening to find out that the starting island is irrelevant. However, a dependence account can easily capture this intuition. The failure to complete a round tour of Königsberg does not depend on the starting island. A good explanation cannot cite misinformation about the dependences. Note that the intuition that this kind of information about independence is a source of explanatory power disappears once we are not focused on correcting a mistaken belief about dependences. We are not tempted to regard as explanatory the information that failure to complete the tour is independent of the fact that the bridges are made of stone and not wood. We never mistakenly believed that the failure to complete the tour depended on these features. 13 As we noted above, Lange ([2013]) explicitly considers and rejects this option. Pincock ([2015]) is sympathetic to the general idea of dependence accounts, but explicitly rejects W-H dependence as a viable candidate. 14 In the terminology of Woodward and Hitchcock ([2003a], [2003b]), an explanatory generalization must be invariant under at least one testing intervention. 15 See (Woodward [2003], p. 203) for a detailed account. 16 Woodward ([2003], p. 221) notes in passing that it seems natural to extend the account to non-causal explanations, but he does not do so. For more discussion on how to decouple the counterfactual aspect and the causal aspect of Woodward’s account, see (Saatsi and Pexton [2013]; Saatsi [2016b]; Rice [2015]). 17 The explanatory generalization regarding the number of strawberries and their (in)divisibility by three supports some counterfactuals running in the opposite direction, of course. For instance, it’s true that had the system’s (in)divisibility by three been otherwise, then the number of strawberries would have been different. So why is this counterfactual not explanatory of the number of strawberries? Answer: the explanatory generalization, with the supposed ‘explanans’ variable fixed, does not state the actual number of strawberries. 18 The full story here is somewhat more complicated since we are modelling the physical bridge system and there are considerations that arise from the applicability of the model. The importance of considerations of applicability for directionality of explanation is stressed by Jansson ([2015]). 19 The true counterfactual, ‘Had the traversability of the bridge system been different, then the system of bridges would have been different’, does not explain the actual bridge configuration for reasons analogous to those discussed in Footnote 17. 20 We thank an anonymous reviewer for raising this questions; (s)he put it in these terms. 21 See (Jansson [2015]) for a nomological dependence account and (Jansson [2016]) for a dependence account for metaphysical cases. There is also more to say about how we should understand optimality explanations—for example, the honeycomb case. We believe that a similar account to the one that we have given here holds for the honeycomb case, but that this has to be argued for on a case-by-case basis. The discussion of many of these cases is made more complicated by the existence of closely related causal explanations, as Lange ([2013], p. 500) notes for the honeycomb case. 22 In (Pincock [2015], p. 865) the directionality comes from an otherwise unanalysed relation of instantiation, and in (Lange [2013], p. 508) the directionality comes from an otherwise unanalysed notion of some features being constitutive of a system. In our view, while these are potential avenues for identifying the source of non-causal directionality, they are neither more plausible nor better understood than the alternative provided by change-relating counterfactuals. 23 It is unsurprising that many authors refer to this explanation as a paradigmatic case of a (distinctly) mathematical explanation of an empirical fact. In addition to the work already covered, see, for example, (Lyon [2012]). 24 This fact about these bridge systems can be established by a variety of means, for example, by attempting to draw every such system without lifting your pen, or by playing with a comprehensive collection of miniature models of such systems. Getting epistemic access, or representing this fact, need not involve mathematics (cf. Saatsi [2011]). 25 Why do we regard this as a (high-level) physical property? Because adding or subtracting a bridge between any two islands plainly makes for a physical difference, and we can get from an even configuration to an odd configuration simply by adding or subtracting a sufficient number of bridges. Furthermore, if we are dealing with a finite number of possible bridge systems (as above, focusing on 395 particular configurations), we can in principle do without the mathematical concepts of even and odd in delineating the two kinds of configurations, since the difference between even and odd is expressible logically, without mathematics. So, we see little reason to regard these as mathematical or purely formal properties of the bridges. Having said that, we should stress that we are not claiming that the Königsberg explanation is a causal one, and nothing in our analysis hangs on whether or not the explanation counts as mathematical. 26 Is minimal explanation of this sort ever completely satisfactory? That depends on the context of the why-question. Imagine Königsberg without Euler—or any other mathematician for that matter. The King of Königsberg, annoyed by his failure to do a round tour of the city, wishes to understand the situation. ‘This city must be made thus tourable, at whatever cost!’, he commands, putting his best people to work. ‘Well, actually, the less it costs the better—and we certainly cannot afford more than two bridges between any two islands!’. The King’s minions get to work, and painstakingly demonstrate, by non-mathematical means, that for any financially feasible set-up of bridges, it holds that it is tourable if and only if it is even. Equipped with this knowledge, they can explain the situation to the King: ‘Whatever you do with the bridges, make sure the whole system is even, and you will be able to tour it, since tourability depends on this feature alone (at least for the kinds of bridge systems we can afford)’. 27 Here, two differences between our third dimension of abstraction and (even the extended) notion of invariance in (Woodward and Hitchcock [2003b]) are important. First, we take the question to be whether the variations destroy the explanation (not merely the generalization). The generalization itself does not break down; it just does not apply. We take this focus on the whole explanation from Potochnik ([2010]) and Weslake ([2010]). Second, the range of variations are understood as the range of cases in which one of the variables in the explanans can be changed without destroying the explanation. While this is also the idea of the W-H account, in (Woodward and Hitchcock [2003b]) it is often natural to interpret the range of invariance as the range of changes in the variable values for which the generalizations continue to hold. 28 Almost all, since these explanations still contain information associated with the various contingent facts presupposed by the why question. References Batterman R., Rice C. [ 2014]: ‘Minimal Model Explanations’, Philosophy of Science , 81, pp. 349– 76. Google Scholar CrossRef Search ADS   Bokulich A. [ 2008]: ‘Can Classical Structures Explain Quantum Phenomena?’, British Journal for the Philosophy of Science , 59, pp. 217– 35. Google Scholar CrossRef Search ADS   Bokulich A. [ 2012]: ‘Distinguishing Explanatory from Non-explanatory Fictions’, Philosophy of Science , 79, pp. 725– 37. Google Scholar CrossRef Search ADS   Braine D. [ 1972]: ‘Varieties of Necessity’, Proceedings of the Aristotelian Society , 46, pp. 139– 70. 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Google Scholar CrossRef Search ADS   © The Author 2017. Published by Oxford University Press on behalf of British Society for the Philosophy of Science. All rights reserved. For Permissions, please email: journals.permissions@oup.com

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Published: Nov 30, 2017

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