The wave function in quantum mechanics presents an interesting challenge to our understanding of the physical world. In this paper, I show that the wave function can be understood as four intrinsic relations on physical space. My account has three desirable features that the standard account lacks: it does not refer to any abstract mathematical objects, it is free from the usual arbitrary conventions, and it explains why the wave function has its gauge degrees of freedom, something that are usually (...) put into the theory by hand. Hence, this account has implications for debates in philosophy of mathematics and philosophy of science. First, by removing references to mathematical objects, it provides a framework for nominalizing quantum mechanics. Second, by excising superfluous structure such as overall phase, it reveals the intrinsic structure postulated by quantum mechanics. Moreover, it also removes a major obstacle to ``wave function realism.''. (shrink)

Relaxed realists hold that there are deep differences between moral truths and the truths studied by the empirical sciences, but they deny that these differences raise troubling metaphysical or epistemological questions about moral truths. On this view, although features such as causal inefficacy, perceptual inaccessibility, and failure to figure in the best explanations of our empirical beliefs would raise pressing skeptical concerns were they claimed to characterize some aspect of physical reality, the same is not true when it comes to (...) the moral domain. This chapter raises some doubts about this general picture of morality and some prominent ways of defending it. First, it takes up a comparison that is frequently invoked by relaxed realists, and one on which they often place a great deal of weight: a comparison between irreducibly normative properties and truths on the one hand, and mathematical properties and truths on the other. It argues that this comparison is much less favorable to the relaxed realist’s cause than is often thought. It then offers an extended critique of a particularly vigorous and sustained presentation of relaxed realism: that offered by Ronald Dworkin in Justice for Hedgehogs. (shrink)

This chapter accepts for the sake of argument Ronald Dworkin’s point that the only viable form of normative skepticism is internal, and develops an internal skeptical argument directed specifically at normative realism. There is a striking and puzzling coincidence between normative judgments that are true, and normative judgments that causal forces led us to believe—a practical/theoretical puzzle to which the constructivist view has a solution. Normative realists have no solution, but are driven to conclude that we are probably hopeless at (...) recognizing the independent normative truths they posit. Since this is an unacceptable conclusion, we must conclude that normative realism is false. Drawing on evolutionary considerations, it is explained why this internal skeptical argument does not carry over to our knowledge of objects in our manifest surroundings, and why the challenge does not depend on any assumption that the epistemology of the normative domain must be a causal one. (shrink)

The existence of fundamental moral disagreements is a central problem for moral realism and has often been contrasted with an alleged absence of disagreement in mathematics. However, mathematicians do in fact disagree on fundamental questions, for example on which set-theoretic axioms are true, and some philosophers have argued that this increases the plausibility of moral vis-à-vis mathematical realism. I argue that the analogy between mathematical and moral disagreement is not as straightforward as those arguments present it. In particular, I argue (...) that pluralist accounts of mathematics render fundamental mathematical disagreements compatible with mathematical realism in a way in which moral disagreements and moral realism are not. 11. (shrink)

In both the early modern period and in contemporary debates, philosophers have argued that there are analogies between mathematics and morality that imply that the ontology and epistemology of morality are crucially similar to the ontology and epistemology of mathematics. I describe arguments for the math‐moral analogy in four early modern philosophers (Locke, Cudworth, Clarke, and Balguy) and in three contemporary philosophers (Clarke‐Doane, Peacocke, and Roberts). I argue that these arguments fail to establish important ontological and epistemological similarities between morality (...) and mathematics. There are analogies between the two areas, but the disanalogies are more significant, undermining the attempt to confer on morality the same ontological and epistemological status that mathematics possesses. (shrink)

I this article, I introduce the notion of pluralism about an area, and use it to argue that the questions at the center of our normative lives are not settled by the facts -- even the normative facts. One upshot of the discussion is that the concepts of realism and objectivity, which are widely identified, are actually in tension. Another is that the concept of objectivity, not realism, should take center stage.

This paper seeks to question the position of ZF as the dominant system of set theory, and in particular to examine whether there is any philosophical justification for the axiom of foundation. After some historical observations regarding Poincare and Russell, and the notions of circularity and hierarchy, the iterative conception of set is argued to be a semi-constructvist hybrid without philosophical coherence. ZF cannot be justified as necessary to avoid paradoxes, as axiomatizing a coherent notion of set, nor on pragmatic (...) grounds, and should be replaced by a system allowing non-well-founded sets. (shrink)

Evolutionary debunking arguments are arguments that appeal to the evolutionary origins of evaluative beliefs to undermine their justification. This paper aims to clarify the premises and presuppositions of EDAs—a form of argument that is increasingly put to use in normative ethics. I argue that such arguments face serious obstacles. It is often overlooked, for example, that they presuppose the truth of metaethical objectivism. More importantly, even if objectivism is assumed, the use of EDAs in normative ethics is incompatible with a (...) parallel and more sweeping global evolutionary debunking argument that has been discussed in recent metaethics. After examining several ways of responding to this global debunking argument, I end by arguing that even if we could resist it, this would still not rehabilitate the current targeted use of EDAs in normative ethics given that, if EDAs work at all, they will in any case lead to a truly radical revision of our evaluative outlook. (shrink)

This paper takes up an important epistemological challenge to the naturalistic moral realist: that her metaphysical commitments are difficult to square with a plausible rationalist view about the epistemology of morality. The paper begins by clarifying and generalizing this challenge. It then illustrates how the generalized challenge can be answered by a form of naturalistic moral realism that I dub joint-carving moral realism. Both my framing of this challenge and my answer advertise the methodological significance of non-fundamental epistemological theorizing, which (...) defends and deploys epistemological claims without adverting to the most fundamental epistemological facts. (shrink)

This is a contribution to the symposium on Tim Scanlon’s Being Realistic about Reasons. We have two aims here: First, we ask for more details about Scanlon’s meta-metaphysical view, showing problems with salient clarifications. And second, we raise independent objections to the view – to its explanatory productivity, its distinctness, and the argumentative support it enjoys.

What should a Quinean naturalist say about moral and mathematical truth? If Quine’s naturalism is understood as the view that we should look to natural science as the ultimate ‘arbiter of truth’, this leads rather quickly to what Huw Price has called ‘placement problems’ of placing moral and mathematical truth in an empirical scientific world-view. Against this understanding of the demands of naturalism, I argue that a proper understanding of the reasons Quine gives for privileging ‘natural science’ as authoritative when (...) it comes to questions of truth and existence also apply to other stable and considered elements of our inherited world-view, including, arguably, our firmly held mathematical and moral beliefs. If so, then the ‘thin’ mathematical and moral realisms of Penelope Maddy and T. M. Scanlon, respectively, are vindicated. We do not need to shoehorn mathematical and moral truths into the pushings and pullings of our empirical scientific world-view; for the busy sailor adrift on Neurath’s boat, mathematical and moral truths already have their place. (shrink)

In this paper, I will attempt to develop and defend a common form of intuitive resistance to the companions in guilt argument. I will argue that one can reasonably believe there are promising solutions to the access problem for mathematical realism that don’t translate to moral realism. In particular, I will suggest that the structuralist project of accounting for mathematical knowledge in terms of some form of logical knowledge offers significant hope of success while no analogous approach offers such hope (...) for moral realism. (shrink)

At the beginning of Die Grundlagen der Arithmetik [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”. This, of course, is true, but thinkers differ on why it is that mathematicians prefer proof. And what of propositions for which no proof is possible? What of axioms? This talk explores various notions of self-evidence, and the role they play in various foundational systems, notably those of Frege and Zermelo. I argue that both (...) programs are undermined at a crucial point, namely when self-evidence is supported by holistic and even pragmatic considerations. (shrink)

The Evolution of Morality attempts to accomplish two tasks. The first is to clarify and provisionally advocate the thesis that human morality is a distinct adaptation wrought by biological natural selection. The second is to inquire whether this empirical thesis would, if true, have any metaethical implications.

I argue that certain species of belief, such as mathematical, logical, and normative beliefs, are insulated from a form of Harman-style debunking argument whereas moral beliefs, the primary target of such arguments, are not. Harman-style arguments have been misunderstood as attempts to directly undermine our moral beliefs. They are rather best given as burden-shifting arguments, concluding that we need additional reasons to maintain our moral beliefs. If we understand them this way, then we can see why moral beliefs are vulnerable (...) to such arguments while mathematical, logical, and normative beliefs are not—the very construction of Harman-style skeptical arguments requires the truth of significant fragments of our mathematical, logical, and normative beliefs, but requires no such thing of our moral beliefs. Given this property, Harman-style skeptical arguments against logical, mathematical, and normative beliefs are self-effacing; doubting these beliefs on the basis of such arguments results in the loss of our reasons for doubt. But we can cleanly doubt the truth of morality. (shrink)

Humans and other animals have an evolved ability to detect discrete magnitudes in their environment. Does this observation support evolutionary debunking arguments against mathematical realism, as has been recently argued by Clarke-Doane, or does it bolster mathematical realism, as authors such as Joyce and Sinnott-Armstrong have assumed? To find out, we need to pay closer attention to the features of evolved numerical cognition. I provide a detailed examination of the functional properties of evolved numerical cognition, and propose that they prima (...) facie favor a realist account of numbers. (shrink)

Anyone who has read Plato’s Republic knows it has a lot to say about mathematics. But why? I shall not be satisfied with the answer that the future rulers of the ideal city are to be educated in mathematics, so Plato is bound to give some space to the subject. I want to know why the rulers are to be educated in mathematics. More pointedly, why are they required to study so much mathematics, for so long?

Metaethical—or, more generally, metanormative— realism faces a serious epistemological challenge. Realists owe us—very roughly speaking—an account of how it is that we can have epistemic access to the normative truths about which they are realists. This much is, it seems, uncontroversial among metaethicists, myself included. But this is as far as the agreement goes, for it is not clear—nor uncontroversial—how best to understand the challenge, what the best realist way of coping with it is, and how successful this attempt is. (...) In this paper I try, first, to present the challenge in its strongest version, and second, to show how realists—indeed, robust realists—can cope with it. The strongest version of the challenge is, I argue, that of explaining the correlation between our normative beliefs and the independent normative truths. And I suggest an evolutionary explanation as a way of solving it. (shrink)

There is a long tradition comparing moral knowledge to mathematical knowledge. In this paper, I discuss apparent similarities and differences between knowledge in the two areas, realistically conceived. I argue that many of these are only apparent, while others are less philosophically significant than might be thought. The picture that emerges is surprising. There are definitely differences between epistemological arguments in the two areas. However, these differences, if anything, increase the plausibility of moral realism as compared to mathematical realism. It (...) is hard to see how one might argue, on epistemological grounds, for moral antirealism while maintaining commitment to mathematical realism. But it may be possible to do the opposite. (shrink)

In "Mathematical Truth", Paul Benacerraf articulated an epistemological problem for mathematical realism. His formulation of the problem relied on a causal theory of knowledge which is now widely rejected. But it is generally agreed that Benacerraf was onto a genuine problem for mathematical realism nevertheless. Hartry Field describes it as the problem of explaining the reliability of our mathematical beliefs, realistically construed. In this paper, I argue that the Benacerraf Problem cannot be made out. There simply is no intelligible problem (...) that satisfies all of the constraints which have been placed on the Benacerraf Problem. The point generalizes to all arguments with the structure of the Benacerraf Problem aimed at realism about a domain meeting certain conditions. Such arguments include so-called "Evolutionary Debunking Arguments" aimed at moral realism. I conclude with some suggestions about the relationship between the Benacerraf Problem and the Gettier Problem. (shrink)

It is commonly suggested that evolutionary considerations generate an epistemological challenge for moral realism. At first approximation, the challenge for the moral realist is to explain our having many true moral beliefs, given that those beliefs are the products of evolutionary forces that would be indifferent to the moral truth. An important question surrounding this challenge is the extent to which it generalizes. In particular, it is of interest whether the Evolutionary Challenge for moral realism is equally a challenge for (...) mathematical realism. It is widely thought not to be. In this paper, I argue that the Evolutionary Challenge for moral realism is equally a challenge for mathematical realism. Along the way, I substantially clarify the Evolutionary Challenge, discuss its relation to more familiar epistemological challenges, and broach the problem of moral disagreement. The paper should be of interest to ethicists because it places pressure on anyone who rejects moral realism on the basis of the Evolutionary Challenge to reject mathematical realism as well. And the paper should be of interest to philosophers of mathematics because it presents a new epistemological challenge for mathematical realism that bears, I argue, no simple relation to Paul Benacerraf's familiar challenge. (shrink)

We attempt to extend the nominalistic project initiated in Hartry Field's Science Without Numbers to modern physical theories based in differential geometry.

Modern empiricism has been conditioned in large part by two dogmas. One is a belief in some fundamental cleavage between truths which are analytic, or grounded in meanings independently of matters of fact, and truth which are synthetic, or grounded in fact. The other dogma is reductionism: the belief that each meaningful statement is equivalent to some logical construct upon terms which refer to immediate experience. Both dogmas, I shall argue, are ill founded. One effect of abandoning them is, as (...) we shall see, a blurring of the supposed boundary between speculative metaphysics and natural science. Another effect is a shift toward pragmatism. (shrink)

J. L. Mackie argued that if there were objective moral properties or facts, then the supervenience relation linking the nonmoral to the moral would be metaphysically queer. Moral realists reply that objective supervenience relations are ubiquitous according to contemporary versions of metaphysical naturalism and, hence, that there is nothing especially queer about moral supervenience. In this paper we revive Mackie's challenge to moral realism. We argue: (i) that objective supervenience relations of any kind, moral or otherwise, should be explainable rather (...) than sui generis; (ii) that this explanatory burden can be successfully met vis-à-vis the supervenience of the mental upon the physical, and in other related cases; and (iii) that the burden cannot be met for (putative) objective moral supervenience relations. (shrink)

There is a fairly widespread—and very infl uential—hope among philosophers interested in the status of normativity that the solution to our metaethical and, more generally, metanormative problems will emerge from the philosophy of action. In this essay, I will argue that these hopes are groundless. I will focus on the metanormative hope, but—as will become clear—showing that the solution to our metanormative problems will not come from what is constitutive of action will also devastate the hope of gaining significant insight (...) into first-order, normative truths by focusing on what is constitutive of action. (shrink)

The paper is a defense of the project of explaining the a priori via the notion of meaning or concept possession. It responds to certain objections that have been made to this project—in particular, that there can be no epistemically analytic sentences that are not also metaphysically analytic, and that the notion of implicit definition cannot explain a priori entitlement. The paper goes on to distinguish between two different ways in which facts about meaning might generate facts about entitlement—inferential and (...) constitutive. It concludes by outlining a theory of the latter. (shrink)

Contemporary realist theories of value claim to be compatible with natural science. In this paper, I call this claim into question by arguing that Darwinian considerations pose a dilemma for these theories. The main thrust of my argument is this. Evolutionary forces have played a tremendous role in shaping the content of human evaluative attitudes. The challenge for realist theories of value is to explain the relation between these evolutionary influences on our evaluative attitudes, on the one hand, and the (...) independent evaluative truths that realism posits, on the other. Realism, I argue, can give no satisfactory account of this relation. On the one hand, the realist may claim that there is no relation between evolutionary influences on our evaluative attitudes and independent evaluative truths. But this claim leads to the implausible skeptical result that most of our evaluative judgements are off track due to the distorting pressure of Darwinian forces. The realist’s other option is to claim that there is a relation between evolutionary influences and independent evaluative truths, namely that natural selection favored ancestors who were able to grasp those truths. But this account, I argue, is unacceptable on scientific grounds. Either way, then, realist theories of value prove unable to accommodate the fact that Darwinian forces have deeply influenced the content of human values. After responding to three objections, the third of which leads me to argue against a realist understanding of the disvalue of pain, I conclude by sketching how antirealism is able to sidestep the dilemma I have presented. Antirealist theories of value are able to offer an alternative account of the relation between evolutionary forces and evaluative facts —
an account that allows us to reconcile our understanding of evaluative truth with our understanding of the many non-rational causes that have played a role in shaping our evaluative judgements. (shrink)

Since Edmund L. Gettier reminded us recently of a certain important inadequacy of the traditional analysis of "S knows that p," several attempts have been made to correct that analysis. In this paper I shall offer still another analysis (or a sketch of an analysis) of "S knows that p," one which will avert Gettier's problem. My concern will be with knowledge of empirical propositions only, since I think that the traditional analysis is adequate for knowledge of nonempirical truths.

One of the most significant disputes in early modern philosophy was between the moral rationalists and the moral sentimentalists. The moral rationalists — such as Ralph Cudworth, Samuel Clarke and John Balguy — held that morality originated in reason alone. The moral sentimentalists — such as Anthony Ashley Cooper, the third Earl of Shaftesbury, Francis Hutcheson and David Hume — held that morality originated at least partly in sentiment. In addition to arguments, the rationalists and sentimentalists developed rich analogies. The (...) most significant analogy the rationalists developed was between morality and mathematics. The most significant analogy the sentimentalists developed was between morality and beauty. These two analogies illustrate well the main ideas, underlying insights, and accounts of moral phenomenology the two positions have to offer. An examination of the two analogies will thus serve as a useful introduction to the debate between moral rationalism and moral sentimentalism as a whole. (shrink)

This is a review of Christopher Peacocke's book The Realm of Reason (Oxford University Press, 2004).

In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic concepts and conventions of set theory.§0. The ordinal numbers were Georg Cantor's deepest contribution to mathematics. After the natural numbers 0, 1, …, n, … comes the first infinite ordinal number ω, followed by ω + 1, ω + 2, …, ω + ω, (...) … and so forth. ω is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {ν ∣ ν α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ω is identified with the first infinite cardinal ℵ0, similarly for the first uncountable ordinal number ω1 and the first uncountable cardinal number ℵ1, etc. We thus arrive at the following picture:The von Neumann hierarchy divides the class V of all sets into a hierarchy of sets Vα indexed by the ordinal numbers. The recursive definition reads: ;Vλ = ∪v<λVv for limit ordinals λ. We can represent this hierarchy by the following picture. (shrink)

This book does three main things. First, it defends mathematical platonism against the main objections to that view (most notably, the epistemological objection and the multiple-reductions objection). Second, it defends anti-platonism (in particular, fictionalism) against the main objections to that view (most notably, the Quine-Putnam indispensability objection and the objection from objectivity). Third, it argues that there is no fact of the matter whether abstract mathematical objects exist and, hence, no fact of the matter whether platonism or anti-platonism is true.

The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set-theoretic possibilities, a phenomenon that challenges the universe (...) view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for. (shrink)

According to a certain pluralist view in philosophy of mathematics, there are as many mathematical objects as there can coherently be. Recently, Justin Clarke-Doane has explored what consequences the analogous view on normative properties would have. What if there is a normative pluriverse? Here I address this same question. The challenge is best seen as a challenge to an important form of normative realism. I criticize the way Clarke-Doane presents the challenge. An improved challenge is presented, and the role of (...) pluralism in this challenge is assessed. (shrink)

Modal Security is an increasingly discussed proposed necessary condition on undermining defeat. Modal Security says, roughly, that if evidence undermines (rather than rebuts) one’s belief, then one gets reason to doubt the belief's safety or sensitivity. The primary interest of the principle is that it seems to entail that influential epistemological arguments, including Evolutionary Debunking Arguments against moral realism and the Benacerraf-Field Challenge for mathematical realism, are unsound. The purpose of this paper is to critically examine Modal Security in detail. (...) We develop and discuss what we take to be the strongest objections to the principle. One of the aims of the paper is to expose the weakness of these objections. Another is to reveal how the debate over Modal Security interacts with core problems in epistemology — including the generality problem, and the distinction between direct and indirect evidence. (shrink)