Abstract This book explores arguments for and against moral realism and mathematical realism, how they interact, and what they can tell us about areas of philosophical interest more generally. It argues that our mathematical beliefs have no better claim to being self-evident or provable than our moral beliefs. Nor do our mathematical beliefs have better claim to being empirically justified. It is also incorrect that reflection on the “genealogy” of our moral beliefs establishes a lack of parity between the cases. In general, if one is a moral anti-realist on the basis of epistemological considerations, then one ought to be a mathematical anti-realist too. And yet, the book argues that moral realism and mathematical realism do not stand or fall together – and for a surprising reason. Moral questions, insofar as they are practical, are objective in a sense in which mathematical questions are not, and the sense in which they are objective can only be explained by assuming practical anti-realism. It follows that the concepts of realism and objectivity, which have been widely identified, are actually in tension. The author concludes that the objective questions in the neighborhood of questions of logic, modality, grounding, nature, and more are practical questions as well. Practical philosophy should, therefore, take center stage. Keywords morality, mathematics, realism, anti-realism, objectivity, practical, self-evidence, proof, genealogy, Benacerraf, logic, Indispensability Argument, Open Question Argument, <p. i: half-title page, recto> Morality and Mathematics <p. ii: half-title page, verso: blank> <p. iii: title page, recto> Morality and Mathematics JUSTIN CLARKE-DOANE OXFORD UNIVERSITY PRESS <p. iv: title page, verso: imprint and copyright details to follow> <p. v: dedication page, recto> For Dad (1948–2017) <p. vi: dedication page, verso: blank> <p. vii: TOC page, recto> Contents Acknowledgments Introduction 0.1 Science and Value 0.2 The Status of the Question 0.3 Overview of the Book 1. Realism, Ontology, and Objectivity 1.1 Individuating Areas 1.2 Bare-Bones Realism 1.3 Faithfulness 1.4 Knowledge 1.5 Ontology 1.6 Objectivity 1.7 Conclusions 2. Self-Evidence, Proof, and Disagreement 2.1 A Priori Justification 2.2 Axioms and Proofs 2.3 Self-Evidence 2.4 Plausibility and Disagreement 2.5 Extent of Disagreement 2.6 Philosophical Corruption 2.7 The Concept of Set 2.8 Error Theory 2.9 Philosophy Everywhere 3. Observation and Indispensability 3.1 Indispensability 3.2 The Quine–Putnam Thesis 3.3 Harman’s Thesis 3.4 Instrumentalism and Modality 3.5 Indispensable Mathematics and Metalogic 3.6 Perception 3.7 Justification and Explanation 3.8 Recreational Mathematics 3.9 A Priori/A Posteriori Revisited 4. Genealogical Debunking Arguments 4.1 Harman’s Thesis as an Underminer 4.2 Three Constraints 4.3 Sensitivity 4.4 Safety 4.5 “Connection” 4.6 Modal Security 4.7 Conclusions 5. Explaining our Reliability 5.1 Justification and Reliability 5.2 Clarifying the Challenge 5.3 Causation, Explanation, and Connection 5.4 Counterfactual Dependence 5.5 Selection for Truth 5.6 The Mathematical Indifference of Evolution 5.7 The Inadequacy of the Sensitivity Challenge 5.8 Contingency 5.9 Pluralism and Safety 5.10 Conclusions 6. Realism, Objectivity, and Evaluation 6.1 Pluralism and Safety Again 6.2 Formulating Mathematical Pluralism 6.3 A Contrast 6.4 The Irrelevance of Semantics 6.5 Radicalizing the Open Question Argument 6.6 Objections and Replies 6.7 Realism, Objectivity, and Practical Safety Conclusion 7.1 Key Themes 7.2 Toward Practical Philosophy Bibliography Index <p. ix: Acknowledgments page, recto> CF Acknowledgments This manuscript has greatly benefited from the criticism of friends and colleagues. I received CF.P1 written comments on chapters 1-6 at a workshop on the manuscript hosted by Columbia University. Thanks to Wolfgang Mann for making that happen, and to Russ Shafer-Landau, Paul Boghossian, Mary Leng, Paul Horwich, Mark Balaguer, and Gideon Rosen for written comments on Chapters 1 to 6, respectively. Thanks to the excellent oral comments from participants, including, but not limited to, David Albert, Sean Carroll, Jessica Collins, Hartry Field, Joel David Hamkins, Tamar Lando, Jennifer McDonald, Michaela McSweeney, Thomas Nagel, Elliot Paul, Chris Peacocke, Lisa Warenski, and Crispin Wright. Thanks to Vera Flocke, David Kaspar, John Morrison, Olle Risberg, Katja Vogt, and an anonymous referee for meticulous comments on the entire document. Matthew Bedke, Colin Marshall, Conor Mayo-Wilson, and Michael Raven also offered penetrating commentary on a draft at an author-meets-critics symposium at the June 2019 Canadian Philosophical Association meeting. Thanks to Christopher Stephens for organizing that. Finally, I am grateful for the engaging discussion of chapters 3-6 which took place in Elizabeth Harman’s and Sarah McGrath’s Metaethics graduate seminar in Spring 2019. Jens Haas, Brian Leiter, Peter Momtchiloff, Achille Varzi, and Katja Vogt guided me through the process. I deeply appreciate their encouragement. Thanks to Dan Baras, John Bengson, Betsy Clarke, Sinan Dogramaci, Emily Fletcher, Martha Gibson, David Keyt, Patricia Kitcher, Philip Kitcher, Karen Lewis, John Mackay, Farid Masrour, Christia Mercer, James Messina, Michael Ridge, Chris Scambler, Anat Schechtman, Mark Schroeder, Alan Sidelle, Alex Silk, Jon Simon, Neil Sinclair, Rob Streiffer, Bill Talbott, Mike Titelbaum, Jack Woods, and Justin Zacek for helpful discussion. Thanks to Matti Eklund for invigorating exchanges. Chapter 6 is indebted to them and his work. Thanks to Rani Rachavelpula for compiling the index. Thanks, finally, to Jenn McDonald, for everything. <p.1: Introduction = 1st page main text, recto> Abstract I discuss specialization in philosophy, and the threat it poses to understanding “how things hang together.” I illustrate the problem using naturalism, a prominent view which combines realism about the sciences with anti-realism about value. Whether this view is tenable depends on whether one can be a mathematical realist and a moral anti-realist. But nobody knows whether one can, because metaethics and the philosophy of mathematics are mutually insulated research fields. I conclude that whether one can be a moral anti-realist and a mathematical realist, or whether metaethics and the philosophy of mathematics have anything else to teach us about how things “hang together,” requires bringing the areas into meaningful contact. Keywords specialization, philosophy of mathematics, metaethics, naturalism, realism, anti-realism <next fresh page> CI Introduction Philosophy is arguably the most inclusive discipline. Its subjects include everything from art to CI.P1 consciousness, from morality to mathematics. Moreover, it approaches its subjects from historical, conceptual, and scientific angles. Its breadth is not coincidental. Many take the defining aim of philosophy to be “to understand,” as Wilfrid Sellars puts it, “how things in the broadest possible sense of the term hang together in the broadest possible sense of the term” [2007/1962, 369]. CI.P2 Despite its breadth, philosophy has become highly specialized. Corresponding to each of its myriad subjects are subdisciplines. Corresponding to morality is the subdiscipline of ethics, and corresponding to mathematics is the subdiscipline of the philosophy of mathematics. These subdisciplines, in turn, branch into sub-subdisciplines, including normative ethics, applied ethics, metaethics, and moral psychology in the one case, and mathematical logic, set theory, mathematical epistemology, and mathematical ontology in the other. To each sub-subdiscipline corresponds an enormous, often difficult, literature. CI.P3 Specialization facilitates a kind of progress in philosophy. Problems are understood in depth, and theoretical options are developed with ever more sophistication. Such progress resembles Thomas Kuhn’s normal science (Kuhn [1962]), minus the agreed-upon paradigm. But it also threatens progress toward the defining aim to which Sellars alludes. There is little hope of understanding “how things hang together” absent serious engagement between philosophy’s diverse subfields—just as there is little hope of this absent serious engagement between philosophy and other fields. CI.S1 0.1 Science and Value Consider the question of realism. To what extent are the subjects of our thought and talk real? CI.P4 We all have a sense of it, prior to philosophical indoctrination. It is the question of whether the subjects of our thought and talk are “out there in the world” existing “independent of us.” We are inclined to be realists about some areas while being anti-realists about others. For example, a common naturalist position among philosophers and scientists combines realism about the sciences with anti-realism about value. Naturalists, in the relevant sense, believe in independent facts about gene expression, plate tectonics, and quantum mechanics but do not believe in independent facts about what is morally good for us, what we epistemically ought to believe, or how prudentially we should live. Sean Carroll summarizes the naturalist’s position on morality, in particular, as follows. CI.P5 There are not…moral truths…existing independently of human invention…but there are real human beings with complex sets of preferences. What we call “morality” is an outgrowth of the interplay of those preferences with the world around us, and in particular with other human beings. The project of moral philosophy is to make sense