Self-Evidence, Proof, and Disagreement

This chapter argues that our mathematical beliefs have no better claim to being a priori justified than our moral beliefs. In particular, they have no better claim to being self-evident, provable, plausible, “analytic,” or even initially credible than our moral beliefs, despite widespread allegations to the contrary. It considers the objection that pervasive and persistent moral disagreement betrays a lack of parity between the cases, and argues that there is no important sense in which there is more moral than mathematical disagreement, or in which moral disagreement is less tractable than mathematical disagreement. A common argument to the contrary simply confuses logic with mathematics (though can even make a parity argument in the case of metalogic). The chapters conclude with the suggestion that the extent of disagreement in an area, in any familiar sense, is of little epistemological consequence.

Realism, Ontology, and Objectivity

This chapter explicates the concept of realism, and distinguishes it from related concepts with which it is often conflated. It shows that, properly conceived, realism has no ontological implications, and that influential epistemological objections to moral and mathematical realism fallaciously assume otherwise. One upshot of the discussion is that it is no response to Paul Benacerraf’s epistemological challenge to claim that there are no special mathematical entities with which to “get in touch.” The chapter concludes with a distinction between realism and objectivity, a distinction which is central to Chapter 6. It uses the Parallel Postulate, understood as a claim of pure geometry, as a paradigm of a claim that fails to be objective, even if mathematical realism is true. Conversely, it explains how realism about claims of a kind may be false even though they are objective in a sense in which the Parallel Postulate is not.

Observation and Indispensability

This chapter argues that mathematical beliefs have no better claim to being empirically – i.e., a posteriori -- justified than our moral beliefs. It shows that Harman’s influential argument to the contrary is fallacious. It then formulates a better argument for a lack of parity between the cases, in terms of indispensability. It argues that, while the necessity of mathematics is no bar to developing a mathematics-free alternative to empirical science, the contents of our arithmetic beliefs, realistically and even objectively construed, do seem to be indispensable to metalogic. But this at most shows that a subset of our mathematical beliefs have better claim to being empirically justified. Surprisingly, however, the range of moral beliefs that we actually have may be so justified, in a more direct way. The chapter concludes with the prospect that there is no principled distinction between intuition and perception, and, hence, between a priori and a posteriori justification.

Introduction

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.

Conclusion

The Conclusion suggests a general partition of areas of philosophical interest into those which are more like mathematics and those which are more like morality. In the former category are questions of possibility, grounding, essence, logic, and mereology. In the latter are questions of epistemology, political philosophy, aesthetics, and prudential reasoning. The chapter argues that the former questions are like the question of whether the Parallel Postulate is true, qua a pure mathematical conjecture. By contrast, practical questions are immune to deflation in this way. The conclusion is that the objective questions in the neighborhood of questions of modal metaphysics, grounding, nature, and so forth are practical. Practical philosophy should, therefore, take center stage.

Realism, Objectivity, and Evaluation

This chapter discusses “realist pluralism” in mathematics and morality. It argues that, under the assumption of pluralism, factual questions get deflated while practical -- i.e., what-to-do -- questions do not. It then uses this contrast to formulate a radicalization of Moore’s Open Question Argument. Practical questions remain open even when the facts, including the evaluative facts, come cheaply. The chapter concludes that practical realism must be false, but practical questions are objective in a paradigmatic respect. Conversely, mathematical realism is true, but mathematical questions fail to be objective. An important upshot of the discussion is that the concepts of realism and objectivity, which are widely identified, are actually in tension.

Explaining our Reliability

This chapter discusses the Benacerraf–Field Challenge – i.e., the reliability challenge. It argues that neither Benacerraf’s formulation of the challenge, nor any simple variations on it, satisfies key constraints which have been placed on it. It then turns to more promising analyses, in terms of sensitivity and safety. The challenge to show that our beliefs are sensitive is widely supposed to admit of an evolutionary answer in the mathematical case, but not in the moral. The chapter argues that it does not, but that, even if it did, this is an inadequate formulation of the challenge. But understanding the reliability challenge as the challenge to show that our beliefs are safe is more promising. The chapter shows that whether this challenge is equally pressing in the moral and mathematical cases depends on whether “realist pluralism” is equally viable in the two areas.

Genealogical Debunking Arguments

This chapter examines Genealogical Debunking Arguments. It argues that they misunderstand the significance of explanatory indispensability. Debunkers observe that whether the proposition that P is implied by some explanation of our coming to believe that P is predictive of its having epistemically desirable qualities when the fact that P would be causally efficacious if it obtained. The problem is that these things are independent when the fact that P would be causally inert, as moral facts would be according to Genealogical Debunking Arguments. The chapter formulates a principle governing undermining defeat, Modal Security. This constitutes a criterion of adequacy for debunking arguments. It says that if such arguments undermine our beliefs, then they give us reason to doubt their safety or sensitivity. But they do not. The chapter concludes that the real problem to which Genealogical Debunking Arguments point is an application of the Benacerraf–Field Challenge. But this challenge has nothing to do with explanatory indispensability.