Some mathematical proofs explain why the theorems they prove hold. This paper identifies several challenges for any counterfactual account of explanation in mathematics (that is, any account according to which an explanatory proof reveals how the explanandum would have been different, had facts in the explanans been different). The paper presumes that countermathematicals can be nontrivial. It argues that nevertheless, a counterfactual account portrays explanatory power as too easy to achieve, does not capture explanatory asymmetry, and fails to specify why (...) certain proofs are explanatory and others are not. Greater informativeness about counterfactual dependence can even yield less explanatory power. (shrink)

In the spirit of explanatory pluralism, this chapter argues that causal and noncausal explanations of a phenomenon are compatible, each being useful for bringing out different sorts of insights. After reviewing a model-based account of scientific explanation, which can accommodate causal and noncausal explanations alike, an important core conception of noncausal explanation is identified. This noncausal form of model-based explanation is illustrated using the example of how Earth scientists in a subfield known as aeolian geomorphology are explaining the formation of (...) regularlyspaced sand ripples. The chapter concludes that even when it comes to everyday "medium-sized dry goods" such as sand ripples, where there is a complete causal story to be told, one can find examples of noncausal scientific explanations. (shrink)

Recent years have seen increasing interest in interventionist analyses of metaphysical explanation. One area where interventionism traditionally shines, is in providing an account of explanatory depth; the sense in which explanation comes in degrees. However, the literature on metaphysical explanation has left the notion of depth almost entirely unexplored. In this paper I shall attempt to rectify this oversight by motivating an interventionist analysis of metaphysical explanatory depth (MED), in terms of the range of interventions under which a metaphysically explanatory (...) generalization remains invariant. After elucidating the notion through a toy-example, I demonstrate the important work which MED can perform in characterizing debate within contemporary metaphysics. Focusing upon rival approaches to explaining the identity and distinctness of concrete objects, I argue that the progress achieved in this debate can be characterized in terms of increasing explanatory depth. Having made an initial case for the utility of MED, I then turn this analysis to the metaphysics of explanation itself. By adopting an interventionist framework with respect to MED, I will show that we can assess the depth of competing theories of explanation. This application has two interesting results: first, it suggests that an interventionist analysis of explanation provides deeper explanations of the connection between explanans and explanandum than rival accounts; and second, it suggests that explanations provided by interventionism become deeper still, if one accepts that this methodology ranges over metaphysical, as well as causal, instances. (shrink)

The goal of this paper is to show how scientific explanation functions in the context of idealized models. It argues that the aspect of explanation most urgently requiring investigation is the nature of the connection between global theories and explanatory local models. This aspect is neglected in traditional accounts of explanation. The paper examines causal, minimal model, and structural accounts of model-based explanation. It argues that they too fail to offer an account of the connection with global theory that can (...) justify the explanatory power of an idealized local model, and consequently these accounts are unable effectively to distinguish explanatory from non-explanatory models. On the account proposed here, scientific explanation requires theoretical integration between the local model described in the explanation and a global theory with independent explanatory power. (shrink)

The goal of this paper is to show how scientific explanation functions in the context of idealized models. It argues that the aspect of explanation most urgently requiring investigation is the nature of the connection between global theories and explanatory local models. This aspect is neglected in traditional accounts of explanation. The paper examines causal, minimal model, and structural accounts of model-based explanation. It argues that they too fail to offer an account of the connection with global theory that can (...) justify the explanatory power of an idealized local model, and consequently these accounts are unable effectively to distinguish explanatory from non-explanatory models. On the account proposed here, scientific explanation requires theoretical integration between the local model described in the explanation and a global theory with independent explanatory power. (shrink)

The problem of explanatory asymmetry remains a serious challenge for non-causal accounts of explanation. This paper proposes a novel solution, and it does so by appealing to the theoretical context in which an explanation is offered. The paper develops the problem of explanatory asymmetry for non-causal dependency accounts of explanation, focusing specifically on Alexander Reutlinger’s Counterfactual Theory of Explanation and recent work by Marc Lange and Lina Jansson. It defends the idea that nomological possibility with respect to a global theory (...) is the right constraint on explanation, and it shows how this breaks the apparent symmetry in counterfactual dependence that is the source of the problem. This solution appeals to theoretical context, and the paper develops and defends the version of contextualism in explanation that is required. (shrink)

One can find in the literature two sets of views concerning the relationship between understanding and explanation: that one understands only if 1) one has knowledge of causes and 2) that knowledge is provided by an explanation. Taken together, these tenets characterize what I call the narrow knowledge account of understanding. While the first tenet has recently come under severe attack, the second has been more resistant to change. I argue that we have good reasons to reject it on the (...) basis of theoretical models that provide how-possibly explanations. These models, while they do not explain in the strict sense, afford understanding. In response, I propose an alternative epistemology of understanding, broad KAU, that takes cases of theoretical modelling into account. (shrink)

This paper analyses the anti-reductionist argument from renormalisation group explanations of universality, and shows how it can be rebutted if one assumes that the explanation in question is captured by the counterfactual dependence account of explanation.

Contribution to a review symposium on Marc Lange's Because without cause: Non-causal explanation in science and mathematics. Oxford: Oxford University Press, 2017.

We explore the prospects of a monist account of explanation for both non-causal explanations in science and pure mathematics. Our starting point is the counterfactual theory of explanation for explanations in science, as advocated in the recent literature on explanation. We argue that, despite the obvious differences between mathematical and scientific explanation, the CTE can be extended to cover both non-causal explanations in science and mathematical explanations. In particular, a successful application of the CTE to mathematical explanations requires us to (...) rely on counterpossibles. We conclude that the CTE is a promising candidate for a monist account of explanation in both science and mathematics. (shrink)

The goal of this paper is to develop a counterfactual theory of explanation. The CTE provides a monist framework for causal and non-causal explanations, according to which both causal and non-causal explanations are explanatory by virtue of revealing counterfactual dependencies between the explanandum and the explanans. I argue that the CTE is applicable to two paradigmatic examples of non-causal explanations: Euler’s explanation and renormalization group explanations of universality.

In the recent philosophy of explanation, a growing attention to and discussion of non-causal explanations has emerged, as there seem to be compelling examples of non-causal explanations in the sciences, in pure mathematics, and in metaphysics. I defend the claim that the counterfactual theory of explanation captures the explanatory character of both non-causal scientific and metaphysical explanations. According to the CTE, scientific and metaphysical explanations are explanatory by virtue of revealing counterfactual dependencies between the explanandum and the explanans. I support (...) this claim by illustrating that CTE is applicable to Euler’s explanation and Loewer’s explanation. (shrink)

In this paper, I aim to provide access to the current debate on non-causal explanations in philosophy of science. I will first present examples of non-causal explanations in the sciences. Then, I will outline three alternative approaches to non-causal explanations – that is, causal reductionism, pluralism, and monism – and, corresponding to these three approaches, different strategies for distinguishing between causal and non-causal explanation. Finally, I will raise questions for future research on non-causal explanations.

Recent literature on non-causal explanation raises the question as to whether explanatory monism, the thesis that all explanations submit to the same analysis, is true. The leading monist proposal holds that all explanations support change-relating counterfactuals. We provide several objections to this monist position. 1Introduction2Change-Relating Monism's Three Problems3Dependency and Monism: Unhappy Together4Another Challenge: Counterfactual Incidentalism4.1High-grade necessity4.2Unity in diversity5Conclusion.

I examine explanations’ realist commitments in relation to dynamical systems theory. First I rebut an ‘explanatory indispensability argument’ for mathematical realism from the explanatory power of phase spaces (Lyon and Colyvan 2007). Then I critically consider a possible way of strengthening the indispensability argument by reference to attractors in dynamical systems theory. The take-home message is that understanding of the modal character of explanations (in dynamical systems theory) can undermine platonist arguments from explanatory indispensability.

So-called ‘distinctively mathematical explanations’ (DMEs) are said to explain physical phenomena, not in terms of contingent causal laws, but rather in terms of mathematical necessities that constrain the physical system in question. Lange argues that the existence of four or more equilibrium positions of any double pendulum has a DME. Here we refute both Lange’s claim itself and a strengthened and extended version of the claim that would pertain to any n-tuple pendulum system on the ground that such explanations are (...) actually causal explanations in disguise and their associated modal conditionals are not general enough to explain the said features of such dynamical systems. We argue and show that if circumscribing the antecedent for a necessarily true conditional in such explanations involves making a causal analysis of the problem, then the resulting explanation is not distinctively mathematical or non-causal. Our argument generalises to other dynamical systems that may have purported DMEs analogous to the one proposed by Lange, and even to some other counterfactual accounts of non-causal explanation given by Reutlinger and Rice. (shrink)

A number of philosophers have recently suggested that some abstract, plausibly non-causal and/or mathematical, explanations explain in a way that is radically dif- ferent from the way causal explanation explain. Namely, while causal explanations explain by providing information about causal dependence, allegedly some abstract explanations explain in a way tied to the independence of the explanandum from the microdetails, 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 ac- count of causal explanation can be naturally extended to capture explanations that radically abstract away from microphysical and causal-nomological details. To this end, we distinguish di erent senses in which an explanation can be more or less abstract, and analyse the connection between explanations’ abstractness and their explanatory power. According to our analysis abstract explanations have much in common with counterfactual causal explanations. (shrink)

Current analytic metaphysics has been claimed to be, at best, out of touch with modern physics, at worst, actually in conflict with the latter The continuum companion to the philosophy of science, Continuum, London, 2011; Ladyman and Ross Every thing must go: metaphysics naturalized, Oxford University Press, Oxford, 2007). While agreeing with some of these claims, it has been suggested that metaphysics may still be of service by providing a kind of ‘toolbox’ of devices that philosophers of science can access (...) in order to help provide an interpretation of theories in fundamental physics Metaphysics in contemporary physics, Rodopi, Amsterdam, 2015). In this context it has been argued that ‘standard’ forms of dispositionalism simply cannot be sustained in the context of modern physics but that certain ‘non-standard’ views may provide the resources to help explicate the sense in which physics may be regarded as ‘modally informed’. Here that suggestion will be further extended in order to consider the implications both with regard to the overall relevance of metaphysics given advances in science and for the prospects of a naturalised metaphysics more generally. In particular, this paper will focus on three concerns: that the particular tools identified are not, in fact, ‘scientifically disinterested’ and thus that the distinction between ‘naturalised’ and ‘non-naturalised’ metaphysics is at best vague or poorly drawn; that the usefulness of such tools depends on their being shaped to fit the relevant physics and thus the latter ‘guts’ metaphysics; that if metaphysics does prove to be useful in this sense then we have no reason to scorn non-naturalised metaphysics to begin with. (shrink)

Recent years have seen growing interest in modifying interventionist accounts of causal explanation in order to characterise noncausal explanation. However, one surprising element of such accounts is that they have typically jettisoned the core feature of interventionism: interventions. Indeed, the prevailing opinion within the philosophy of science literature suggests that interventions exclusively demarcate causal relationships. This position is so prevalent that, until now, no one has even thought to name it. We call it “intervention puritanism” (I-puritanism, for short). In this (...) paper, we mount the first sustained defence of the idea that there are distinctively noncausal explanations which can be characterized in terms of possible interventions; a position we call “intervention liberalism” (I-liberalism, for short). While many have followed James Woodward (2003) in committing to I-puritanism, we trace support for I-liberalism back to the work of Jaegwon Kim (1974). Furthermore, we analyse two recent sources of scepticism regarding I-liberalism: debate surrounding mechanistic constitution; and attempts to provide a monistic account of explanation. We show that neither literature provides compelling reasons for adopting I-puritanism. Finally, we present a novel taxonomy of available positions upon the role of possible interventions in explanation: weak causal imperialism; strong causal imperialism; monist intervention puritanism; pluralist intervention puritanism; monist intervention liberalism; and finally, the specific position defended in this paper, pluralist intervention liberalism. (shrink)

In the recent literature on causal and non-causal scientific explanations, there is an intuitive assumption according to which an explanation is non-causal by virtue of being abstract. In this context, to be ‘abstract’ means that the explanans in question leaves out many or almost all causal microphysical details of the target system. After motivating this assumption, we argue that the abstractness assumption, in placing the abstract and the causal character of an explanation in tension, is misguided in ways that are (...) independent of which view of causation or causal explanation one takes to be most accurate. On major accounts of causation, as well as on major accounts of causal explanation, the abstractness of an explanation is not sufficient for it being non-causal. That is, explanations are not non-causal by dint of being abstract. (shrink)

Opinionated state of the art paper on scientific explanation. Analysis and discussion of the most relevant models and theories in the contemporary literature, namely, the deductive-nomological model, the models of inductive-statistical and statistical relevance, the pragmatic theory of why questions, the unifying theory of standard arguments, and the causal/non-causal counterfactual theory.