How far should our realism extend? For many years philosophers of mathematics and philosophers of ethics have worked independently to address the question of how best to understand the entities apparently referred to by mathematical and ethical talk. But the similarities between their endeavours are not often emphasised. This book provides that emphasis. In particular, it focuses on two types of argumentative strategies that have been deployed in both areas. Thefirst-debunking arguments-aims to put pressure on realism by emphasising the seeming redundancy of mathematical or moral entities when it comes to explaining our judgements. In the moral realm this challenge has been made by Gilbert Harman and Sharon Street; in the mathematical realm it is known as the'Benacerraf-Field' problem. The second strategy-indispensability arguments-aims to provide support for realism by emphasising the seeming intellectual indispensability of mathematical or moral entities, for example when constructing good explanatory theories. This strategy is associated with Quine and Putnam in mathematics and with Nicholas Sturgeon and David Enoch in ethics. Explanation in Ethics and Mathematics addresses these issues through an explicitly comparative methodologywhich we call the 'companions in illumination' approach. By considering how argumentative strategies in the philosophy of mathematics might apply to the philosophy of ethics, and vice versa, the papers collected here break new ground in both areas. For good measure, two further companions for illumination arealso broached: the philosophy of chance and the philosophy of religion. Collectively, these comparisons light up new questions, arguments, and problems of interest to scholars interested in realism in any area.

1 Neil Sinclair & Uri Leibowitz: Introduction: Explanation in Ethics and Mathematics; Part I: Evolutionary Debunking Arguments; 2 Justin Clarke-Doane: Debunking and Dispensability; 3 Folke Tersman: Explaining the Reliability of Moral Beliefs; 4 Toby Handfield: Genealogical Explanations of Chance and Morals; 5 Erik J. Wielenberg: Evolutionary Debunking Arguments in Religion and Morality; 6 Hallvard Lillehammer: 'An Assumption of Extreme Significance': Moore, Ross and Spencer on Ethics and Evolution; 7 Richard Joyce: Reply: Confessions of a Modest Debunker; Part II: Indispensability Arguments; 8 Alexander Miller: Moral Explanation for Moral Anti-Realism; 9 David Liggins: Grounding, Explanation, and Multiple Realization in Mathematics and Ethics; 10 Debbie Roberts: Explanatory Indispensability Arguments in Metaethics and Philosophy of Mathematics; 11 Mary Leng: Taking Morality Mathematically: Enoch's Indispensability Argument; 12 Alan Baker: Non-Optional Projects: Mathematical and Ethical; 13 David Enoch: Indispensability Arguments in Metaethics: Even Better than in Mathematics?