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)

Are the special sciences autonomous from physics? Those who say they are need to explain how dependent special science properties could feature in irreducible causal explanations, but that’s no easy task. The demands of a broadly physicalist worldview require that such properties are not only dependent on the physical, but also physically realized. Realized properties are derivative, so it’s natural to suppose that they have derivative causal powers. Correspondingly, philosophical orthodoxy has it that if we want special science properties to (...) bestow genuinely new causal powers, we must reject physical realization and embrace a form of emergentism, in which such properties arise from the physical by mysterious brute determination. In this paper, I argue that contrary to this orthodoxy, there are physically realized properties that bestow new causal powers in relation to their realizers. The key to my proposal is to reject causal-functional accounts of realization and embrace a broader account that allows for the realization of shapes and patterns. Unlike functional properties, such properties are defined by qualitative, non-causal specifications, so realizing them does not consist in bestowing causal powers. This, I argue, allows for causal novelty of the strongest kind. I argue that the molecular geometry of H2O—a qualitative, multiply realizable property—plays an irreducible role in explaining its dipole moment, and thereby bestows novel powers. On my proposal, special science properties can have the kind of causal novelty traditionally associated with strong emergence, without any of the mystery. (shrink)

Turing patterns are a class of minimal mathematical models that have been used to discover and conceptualize certain abstract features of early biological development. This paper examines a range of these minimal models in order to articulate and elaborate a philosophical analysis of their epistemic uses. It is argued that minimal mathematical models aid in structuring the epistemic practices of biology by providing precise descriptions of the quantitative relations between various features of the complex systems, generating novel predictions that can (...) be compared with experimental data, promoting theory exploration, and acting as constitutive parts of empirically adequate explanations of naturally occurring phenomena, such as biological pattern formation. Focusing on the roles that minimal model explanations play in science motivates the adoption of a broader diachronic view of scientific explanation. (shrink)

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)

1000 time-travelers travel back in time, each with the intention of killing their own infant-self. If there is no branching time, then on pain of bringing about a logical contradiction, all must fail. But this seems inexplicable: what is to ensure that the time-travelers are stopped? For a time, this inexplicability objection was thought to provide evidence that there is something incoherent about the possibility of backwards time travel in a universe without branching time. There is now near-consensus, however, that (...) the objection has no bite: there is nothing inexplicable about the mass failure. Lewis, Sider and Ismael independently argue that since it is built into the description of the class of cases considered that the time-travelers must fail – and so we consider only unsuccessful attempts – there is no mystery. Smith argues that the absence of possible worlds at which auto-infanticide is committed suffices as a complete explanation for the failures. And Baron and Colyvan maintain that available causal and logical explanations jointly account for everything that needs accounting for. I argue that these are wrong. There is remaining, problematic inexplicability. For backwards time travel not to lead to logical contradiction, something would need to do logic’s bidding, after all. (shrink)

This article discusses the possibility of some outlandish tropes from time travel fiction, such as people reversing in age as they time travel or the universe being destroyed because a time traveler kills their ancestor. First, I discuss what type of possibility we might have in mind, detailing ‘close possibility’ as one such candidate. Secondly, I argue that—with only little exception—these more outlandish tropes fail to be closely possible. Thirdly, I discuss whether these outlandish tropes may nevertheless be more broadly (...) possible (e.g., metaphysically or logically possible), arguing that whether they are or not depends upon your favored metaphysics of the laws of nature. (shrink)

One of the oldest platitudes about beauty is that it is pleasant to perceive or experience. In this article, I take this platitude at face value and try to explain why experiences of beauty are seemingly always accompanied by pleasure. Unlike explanations that have been offered in the past, the explanation proposed is designed to suit a “realist” view on which beauty is an irreducibly evaluative property, that is, a value. In a nutshell, the explanation is that experiences of beauty (...) are experiences in which it appears that something is beautiful, and that such experiences are identical to experiences of aesthetic pleasure. (shrink)

Tim travels back in time and tries to kill his grandfather before his father was born. Tim fails. But why? Lewis's response was to cite "coincidences": Tim is the unlucky subject of gun jammings, banana peels, sudden changes of heart, and so on. A number of challenges have been raised against Lewis's response. The latest of these focuses on explanation. This paper diagnoses the source of this new disgruntlement and offers an alternative explanation for Tim's failure, one that Lewis would (...) not have liked. The explanation is an obvious one but controversial, so it is defended against all the objections that can be mustered. (shrink)

It has seemed, to many, that there is an important connection between the ways in which some theoretical posits explain our observations, and our reasons for being ontologically committed to those posits. One way to spell out this connection is in terms of what has become known as the explanatory criterion of ontological commitment. This is, roughly, the view that we ought to posit only those entities that are indispensable to our best explanations. Our primary aim is to argue that (...) the moral nonnaturalist places herself in an invidious position if she simply accepts that the nonnatural moral facts that she posits are not explanatory. Instead, we offer the nonnaturalist an alternative strategy for dealing with moral explanations. The strategy is to retain the explanatory criterion of ontological commitment and maintain that moral properties are, in fact, explanatory. The explanations they provide are not causal explanations; instead, moral properties make a non-causal difference to the physical facts. (shrink)

Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The final view (...) appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)

We argue that explanations appealing to logical impossibilities are genuine explanations. Our defense is based on a certain picture of impossibility. Namely, that there are impossibilities and that the impossibilities have structure. Assuming this broad picture of impossibility we defend the genuineness of explanations that appeal to logical impossibilities against three objections. First, that such explanations are at odds with the perceived conceptual connection between explanation and counterfactual dependence. Second, that there are no genuinely contrastive why-questions that involve logical impossibilities (...) and, third, that explanations appealing to logical impossibilities rule nothing out. (shrink)

This paper introduces a new kind of explanation that we describe as ‘purely theoretical’. We first present an example, E, of what we take to be a case of purely theoretical explanation. We then show that the explanation we have in mind does not fit neatly into any of the existing categories of explanation. We take this to give us prima facie motivation for thinking that purely theoretical explanation is a distinctive kind of explanation. We then argue that it can (...) earn its keep via application to two existing literatures: the literature on how we explain the truth of true negative existential propositions and the literature on how we explain the truth of true propositions about the past. We reply to some possible concerns regarding the introduction of purely theoretical explanations. We conclude that there is nothing obviously wrong with them and explore the ramifications for particular debates in metaphysics. (shrink)