![]() ![]() ![]() He argues roughly that PVW conditionals cannot be understood in terms of things as large as possible worlds that PVW conditionals are peculiar and so cannot be accommodated by general accounts of counterfactuals, thereby reflecting the piecemeal character of scientific practice and standing at odds with the one-size-fits-all approach of “analytic metaphysicians” and that PVW counterfactuals are not made true partly by natural laws. Wilson derives various broad philosophical morals from the scientific role played by the Principle of Virtual Work. The paper then sketches a more positive proposal for dealing with explanatory asymmetry in non-causal explanations. This paper argues that several recent accounts Explanation beyond causation: philosophical perspectives on non-causal explanations, Oxford University Press, Oxford, pp 117–140, 2018 Jansson and Saatsi in Br J Philos Sci, forthcoming Jansson in J Philos 112:7–599, 2015 Saatsi and Pexton in Philos Sci 80: 613–624, 2013 French and Saatsi, in: Reutlinger and Saatsi Explanation beyond causation: philosophical perspectives on non-causal explanations, Oxford University Press, Oxford, pp 185–205, 2018) fail to meet this challenge. Whether that further ingredient, even if applicable to causal explanation, can fit non-causal explanation is the challenge that explanatory asymmetry poses for counterfactual accounts of non-causal explanation. Therefore, something more than mere counterfactual dependence is needed to account for explanatory asymmetry. These attempts recognize that even when there is explanatory asymmetry, there may be symmetry in counterfactual dependence. This paper examines some recent attempts that use counterfactuals to understand the asymmetry of non-causal scientific explanations. The paper sketches an account of what makes a proof explanatory and uses that account to defend the morals drawn from the examples already given. These examples suggest several reasons why explaining and grounding tend to come apart, including that explanatory proofs need not exhibit purity, tend not to be brute force, and often unify separate cases by identifying common reasons behind them even when those cases have distinct grounds. The paper offers several examples from mathematical practice to illustrate these points. It argues that oftentimes, a proof specifying a mathematical fact’s grounds fails to explain why that fact obtains whereas any explanation of the fact does not specify its ground. The paper argues that a mathematical fact’s grounds do not, simply by virtue of grounding it, thereby explain why that fact obtains. This paper explores whether there is any relation between mathematical proofs that specify the grounds of the theorem being proved and mathematical proofs that explain why the theorem obtains. This paper offers an inductive account of why mathematical IBE tends to lead to mathematical truths. Both platonist and non-platonist interpretations of mathematics ought to accommodate explanation in mathematics and ought to recognize IBE in mathematics, though these interpretations disagree on the ontological commitments that mathematicians ought to have. IBE supplements other forms of non-deductive reasoning in mathematics, avoiding obstacles sometimes faced by enumerative induction or hypothetico-deductive reasoning. Analogously, when applied to mathematical evidence, IBE sometimes helps to justify mathematicians' in expanding the range of their ontological commitments. When applied to empirical evidence, IBE sometimes helps to justify the expansion of scientists’ ontological commitments. ![]() IBE operates in mathematics in the same way as IBE in science. This paper argues that in mathematical practice, conjectures are sometimes confirmed by “Inference to the Best Explanation” as applied to some mathematical evidence. ![]()
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