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Justin Clarke-Doane Draft: 1/5/08 Moral Realism and Mathematical Realism1 Ethics and mathematics are normally treated independently in philosophical discussions. When comparisons are drawn between problems in the two areas, those comparisons tend to be highly local, concerning just one or two issues. Nevertheless, certain metaethicists have made bold claims to the effect that moral realism is on “no worse footing” than mathematical realism -- i.e. that one cannot reasonably reject moral realism without also rejecting mathematical realism.2 If such bold claims were even prima facie plausible, then realists in metaethics would have a powerful weapon at their disposal. In the absence of any remotely systematic survey of the relevant arguments, however, the prima facie plausibility of such claims cannot be usefully judged. There is no way to guess whether the few local parallels that have been observed are symptomatic of pervasive ones. What is needed is a general overview of the relevant dialectical landscape – one which serves to suggest the likely extent of commonality between arguments in ethics and arguments in the philosophy of mathematics. In this survey, I offer such an overview. I consider a wide array of arguments for mathematical realism, and against moral realism, and indiacte analogs to each. I argue that, while nothing definitive can be said at this point, the aforementioned bold claims do have significant prima facie plausibility. In particular, parallels between arguments in metaethics and arguments in the philosophy of mathematics seem to be much more systematic than is commonly supposed. 0. Contents The structure of the paper is as follows. First, I propose a definition of realism about an arbitrary discourse which I take to best capture the notion of realism as it is invoked by philosophers in ethics and philosophers in the philosophy of mathematics. I next consider parallels between viable realist and antirealist positions in relevant areas. Thereafter, I consider arguments for antirealism in ethics, and for realism in the philosophy of mathematics, and discuss the relative plausibility of analogous arguments in the philosophy of mathematics and ethics, respectively. I conclude with some morals. 1. Realism about a Discourse 2. Realist and Antirealist Positions in Ethics and the Philosophy of Mathematics 3. Realist and Antirealist Arguments in Ethics and the Philosophy of Mathematics A. Semantic Arguments -The Ordinary Language Argument -The Argument from Irrelevance -The Argument from Disagreement B. Epistemological Arguments 1 Thanks to Sharon Street, Tom Nagel, Derek Parfit, Hartry Field, Stephen Schiffer, Dale Jamieson, Kit Fine, Matt Evans, and members of the 2007 NYU Thesis Prep seminar for invaluable comments. 2 See, for example, Dworkin [1996], Shafer-Landau [2003], Putnam [2004], or Parfit [2006]. 1 -The Indispensability Argument -The Argument from “Intuition” -The Argument from Epistemic Inaccessibility C. Metaphysical Arguments -The Argument from Naturalism -The Open Question Argument -The Queerness Argument -Queerness Again 4. Conclusion 1. Realism about a Discourse Philosophers use the term “realism” in a bewildering variety of ways. Proposing a criterion of realism about an arbitrary discourse, D, must, therefore, be at least to some extent a matter of stipulation. That said, I believe that all and only the following three claims would be accepted by the majority of self-proclaimed realists with respect to a relevant discourse and, in particular, by the majority of realists with respect to ethical or mathematical discourses. (1) Sentences of D are truth-apt. (2) Some (atomic) sentences of D are true. (3) The truth-values of sentences of D are relevantly independent of anyone’s beliefs. Each of (1) – (3) is somewhat vague, though familiar. (1) is usually taken to entail that sentences of D have “truth-conditions” – i.e. conditions whose obtaining is necessary and sufficient for those sentences’ being true. It is supposed to be inconsistent with the proposition that sentences of D merely serve to express emotions or imperatives, or as meaningless pieces in a game, for example. Some philosophers have apparently wished to deny that (1) is inconsistent with such things,3 but, arguably, the debate between realists and antirealists only makes sense if we suppose that it is.4 I, thus, assume so much for the purposes of this essay. (2) is straightforward, given (1). It means that some sentences of D have truth-conditions which obtain. As before, some philosophers have apparently wished to deny that sentences can be true only if they have truth-conditions which obtain, but for the purposes of this essay, I must, again, set such philosophers aside. Finally, (3) is supposed to at least rule out the possibility that sentences of D are only true because some person or group believes them to be. It is supposed to entail the counterfactual claim that truth-values of sentences of D would remain largely intact even 3 With respect to ethical discourse, see, for example, Blackburn [1998], or Gibbard [2003]. With respect to mathematical discourse, see perhaps [Carnap 1950] or [Hilbert 1935]. Given the way I understand (1), such views with respect to ethical discourse become straightforward examples of what I below term “expressivism”, and such views with respect to mathematical discourse become straightforward varieties of what I below term “formalism”. 4 And philosophers such as Blackburn and Gibbard are often, themselves, quick to point this out. In this capacity, see also Dworkin [1996] and Fine [2001]. 2 if our beliefs were to diverge radically from what they are in fact. Perhaps some truthvalues of some sentences of D vary with the beliefs of individuals in an uninteresting sense – so that, for instance, the truth of “John is doing something evil right now” could vary with John’s enjoyment of his occurrent belief that everyone should be slaughtered mercilessly. But exactly how best to precisify (3) is, again, not a matter that will be relevant to my purposes here. One more clarificatory point about (1) – (3) is in order. I will be assuming throughout that one who grants (1) with respect to a discourse thereby also grants that the relevant sentences are to be semantically interpreted at face-value -- as being about the likes of numbers, spaces, points, etc. in the case of mathematical discourse, and about the likes of people, actions, events, etc. in the case of ethical discourse.5 Since the existence of people, actions, events, etc. is not in dispute in the context of realism-antirealism debates in ethics, there are virtually no philosophers who grant (1) with respect to ethical discourse but deny that the relevant sentences should be semantically interpreted at facevalue. On the other hand, the existence of numbers, spaces, points, etc. is in dispute in the context of realist-antirealist debates in the philosophy of mathematics, so some philosophers have at once granted (1) with respect to mathematical discourse, and denied that the relevant sentences are to be semantically interpreted at face-value.6 It is an interesting question whether the availability of denying the face-value constraint in the philosophy of mathematics in particular affords antirealism with respect to mathematical discourse significant additional plausibility. It certainly doesn’t afford it less plausibility, however, so I will not be taking up the issue here. 2. Realist and Antirealist Positions in Ethics and the Philosophy of Mathematics The last paragraph notwithstanding, viable realist and antirealist positions in ethics and the philosophy of mathematics are highly parallel. Realist positions in each domain are generated by taking different stands on the epistemic and metaphysical nature of the relevant body of truths. Realists about either discourse may, first, hold different positions on the question of whether the relevant truths are knowable a priori or (merely) a posteriori, on the basis of observation. Those of a naturalist bent will tend to regard the relevant body of truths as (merely) knowable a posteriori, perhaps via straightforward applications of enumerative induction, or perhaps via a substantive inference to the best explanation.7 Realists about 5 Note that this is not to assume anything about the nature of the properties predicated of these things in such sentences (which would be a contentious assumption in the case of ethics). This is just to rule out views as antirealist according to which such sentences are relevantly like “the average NYU professor is 5’8””, which, most would hold, is not about an individual, the average NYU professor, at all, but rather makes a general claim about a group of individuals. 6 See, for example, [Hellman 1989]. 7 With respect to mathematics, see, for instance, [Kitcher 1984], [Quine 1951], [Putnam 1971], or [Maddy 1990]. And, with respect to ethics, see, for instance, [Boyd 1988], [Brink 1989] or [Sturgeon 1988]. 3 either discourse that are not committed to naturalism, on the other hand, will tend to regard the relevant body of truths as knowable a priori somehow.8 Similarly, realists about ethics or mathematics may hold different positions on the question of whether the entities that are peculiar to the relevant discourses are reducible. Thus, mathematical realists may hold that numbers, points, groups, etc. are identical with prima facie distinct entities, and moral realists may say something similar about goodness, badness, etc.9 Realists of a reductionist persuasion may, moreover, hold different positions on the question of whether the relevant entities are reducible to natural ones. Thus, some reductive realists in the philosophy of mathematics may contend that mathematical objects are, in fact, just natural attributes or components of the spatiotemporal world, while others may contend that such objects are identical with prima facie distinct abstracta, such as (pure) sets or categories.10 Reductive realists in ethics may, correspondingly, contend that properties like goodness are just some natural sociological or psychological ones, while others may contend that they are, for instance, properties of a divine being.11 Alternatively, mathematical and ethical realists may hold that the entities peculiar to their subject matters are sui generis.12 Finally, realists about each domain may hold different positions on the modal status of the truths that they countenance. Though in each case realists are apt to regard the relevant truths as necessary, certain varieties of naturalism about them straightforwardly suggest that they are contingent. For example, if mathematical objects are just natural attributes of the contingent world, then one might think that facts about them are likewise contingent.13 And if moral properties are just natural psychological or sociological ones, then one might think something similar of them. Antirealist positions in ethics and the philosophy of mathematics are generated by denying different components of (1) – (3) with respect to the relevant discourse. Most straightforwardly, antirealists in ethics and the philosophy of mathematics may deny (1) with respect to the relevant discourse – they may deny that ethical or mathematical 8 With respect to mathematics, see, for instance, [Bealer 1982], [Katz 1998], or [Wright and Hale 2003]. And, with respect to ethics, see, for instance, Nagel [1986], Parfit [2006], Shafer-Landau [2003], or [Audi 2004]. 9 Arguments for “mathematical reductionism” include [Bealer 1982], Bigelow [1988], Landry [1999], [Maddy 1990], [Steinhart 2002], and [Clarke-Doane forthcoming]. Arguments for “moral reductionism” include [Boyd 1988], [Brink 1989], and [Jackson and Pettit 1995]. 10 Of the arguments for mathematical reductionism just listed, arguments for non-naturalist varieties include [Landry 1999], [Steinhart 2002], and [Bealer 1982]. [Bigelow 1988] and [Maddy 1990] outline naturalist varieties of reductionism. ([Clarke-Doane forthcoming] is an argument for reductionism in general.) 11 All of the arguments for moral reductionism just mentioned are arguments for naturalist varieties. [Moore 1903] famously challenges arguments for religious varieties of reductionism as well. 12 [McLarty 1993] and [Resnik 1997] are arguments for the view that mathematical objects could be sui generis. [Moore 1903] and perhaps [Sturgeon 2003] are arguments for the analogous view with respect to moral properties. 13 [Russell and Whitehead 1910, 1912, 1913] is widely thought to entail that mathematical theorems are contingently true, if true at all, in virtue of the contingent character of the Axiom of Infinity – the thesis that there are an infinite number of individuals in the universe. Quine, on the other hand, explicitly demurs from trafficking in modal terms throughout his corpus. Finally, [Field 1993] includes a sophisticated argument that mathematical objects do not exist of conceptual necessity if they exist at all. 4 sentences are truth-apt. Thus, certain expressivists in ethics contend that rather than having truth-conditions, moral sentences (merely) serve to express feelings, preferences, or plans, or to issue commands.14 And certain formalists in the philosophy of mathematics contend that rather than having truth-conditions, mathematical sentences serve a role much like that of chess pieces – as meaningless elements to be manipulated in the “game” of mathematics.15 The negation of (1) trivially entails the negation of (2). But one may, nonetheless, grant (1), yet deny (2). The basic position generated by such a combination of views is that sentences of the relevant discourse have truth-conditions, but the truth-conditions which they possess are never satisfied. Thus, error-theorists in ethics hold that “Killing is wrong” is truth-apt, but false.16 And fictionalists in the philosophy of mathematics say something analogous of “there is a perfect number greater than 19”.17 Such a position is normally motivated by skepticism about the entities that would be peculiar to the relevant discourses – moral properties and mathematical objects, respectively.18 Finally, one may deny the last clause of the above definition of realism, along with (2), but not (1). In every case that I am aware of, however, a philosopher denying (3) with respect to ethical or mathematical discourse will grant (2) in addition to (1) – the picture being that the truth-conditions of the relevant sentences are more easy to come by than it seemed (given that they depend somehow on our beliefs), so we should not doubt (2) with respect that discourse after all. Philosophers who embrace (1) and deny (3) with respect to mathematical discourse are known as intuitionists,19 and philosophers who embrace (1) and deny (3) with respect to ethical discourse are known as constructivists.20 In ethics, constructivists may contend that the truth-conditions of “Lying is wrong” are something like that one’s society disapproves or lying, or that rational agents could not will that lying was the universal law.21 And in the philosophy of mathematics, intuitionists may contend that the truth conditions of “for any number, n, there is a prime number, m, such that n < m” are that for any mentally constructible object, n, one could construct another number object, m, such that n < m, and m is prime. The viable realist and antirealist positions in ethics and mathematics are, thus, highly analogous. Even so, it could be that there are arguments for the disjunction of the relevant realist positions in ethics that do not have comparably strong analogs in ethics, or that there are arguments for the disjunction of the relevant antirealist positions in 14 See, for example, [Ayer 1952], [Carnap 1937], or [Hare 1952]. Again, [Blackburn 1998] and [Gibbard 1990] also qualify as examples of expressivism given the way that I’ve understood (1) with respect to ethical discourse. 15 See, for example, [Hilbert 1935], [von Neumann 1931], or [Curry 1958]. 16 See especially [Mackie 1977]. 17 See especially [Field 1980]. 18 Strictly speaking, one could, of course, be skeptical of mathematical properties in addition to mathematical objects – presumably, those properties that would only be exemplified by mathematical objects. 19 See, for example, [Brouwer 1952], [Heyting 1956], or [Dummett 1977]. 20 See, for example, [Korsgaard 1996], [Rawls 1999], [Harman 1996], or [Scanlon 1998]. 21 Note that both subjectivism and relativism reduce to forms of constructivism on my understanding of it. 5 ethics that do not have comparably strong analogs in the philosophy of mathematics. In what follows, I investigate the prima facie likelihood of this possibility. 3. Realist and Antirealist Arguments in Ethics and the Philosophy of Mathematics The arguments for mathematical realism, and for moral antirealism, that I shall consider may be segregated, roughly, into three groups: (A) Semantic, (B) Epistemological, and (C) Metaphysical, arguments. Arguments of type (A) concern the nature of ethical or mathematical discourse. Arguments of type (B) concern how we might know or be justified in believing ethical or mathematical truths. And arguments of type (C) concern the nature of the sorts of things that are presupposed by ethical or mathematical discourse. I will proceed by considering relevant arguments for mathematical realism, and against moral realism, together, beginning with all such arguments that are of type (A).22 A. Semantic Arguments There are three broadly semantic arguments that will be relevant here. The first is a longstanding argument for mathematical realism. I shall call it the “Ordinary Language Argument” (OLA), and it may be formulated as follows: The Ordinary Language Argument (4) We ascribe truth to (atomic) mathematical sentences. (5) We purport to bear propositional attitudes to what they express. (6) The best explanation of (5) and (6) entails mathematical realism. (7) So, by inference to the best explanation, mathematical realism is true. Whether (4) – (7) is a sound argument is clearly a contestable matter. Though (4) and (5) seem undeniable, antirealists of all stripes may vigorously challenge (6). Certain formalists may contend that (4) and (5) are better explained by a view entailing none of (1) – (3), on the grounds that the function of (pure) mathematical discourse is just the manipulation of characters according to certain rules. Fictionalists, in turn, may contend that (4) and (5) at best support the thought that it is as if (3) were true with respect to mathematical discourse in some relevant sense – in that mathematical sentences that we call “true” are conservative over nominalistic theories, for instance.23 Finally, intuitionists may point out that, as it stands at least, (OLA) seems to suggest nothing straightforward about whether the truth-values of mathematical sentences are independent of people’s beliefs – though intuitionists may still face a special case of (OLA) which does seem to do this (namely, an argument from our ascription of truth to certain mathematical sentences, such as undecidables, whose truth-conditions could apparently not be a function of our beliefs in any way). Does there exist an analogous argument to (OLA) for moral realism? Obviously there does; it runs as follows, (OLA)’: 22 I attach no special importance to these groupings. Surely a case could be made for putting some of the arguments that I consider under different headings. The segregation of arguments into groups is merely intended to ease their exposition. 23 For such an account of the sense in which it is as if mathematical sentences were true, see [Field 1980]. 6 (4)’ We ascribe truth to (atomic) moral sentences. (5)’ We purport to bear propositional attitudes to what they express. (6)’ The best explanation of (5)’ and (6)’ entails moral realism. (7)’ So, by inference to the best explanation, moral realism is true. Now, just like (4) – (7), (4)’ – (7)’ is, of course, contestable. But (4)’ and (5)’ seem just as undeniable as do (4) and (5). We just as surely ascribe truth to moral sentences as we do to mathematical ones (“We hold these truths to be self-evident…”), and we just as surely purport to bear propositional attitudes to what they express as we do to what mathematical ones express (“I believe that killing anyone is wrong.”). So, just as mathematical antirealists wishing to attack (OLA) must attack (6), moral antirealists wishing to attack (OLA)’ will have to attack (6)’. But what could be the peculiar sort of reasons for rejecting (6)’ as opposed to (6)? Expressivists may deny that (4)’ and (5)’ are best explained by a view that entails any of (1) – (3) in the criterion of realism with respect to ethical discourse, contending that the function of ethical discourse is the conveying of emotions or commands or whatever. But this, of course, is just the sort of consideration that formalists lobby against (6) – and, on the face of it, at least, it is not significantly more plausible when coming out of the moral antirealist’s mouth. Error-theorists, in turn, may contend that (4)’ and (5)’ at best motivate the thought that it is as if moral sentences were true in some relevant respect – that moral sentences that we call “true” contribute to social cooperation when accepted, for instance.24 And, yet, this objection, as well, is precisely parallel to the fictionalist’s objection with respect to (4) and (5), and there seems, again, to be no prima facie reason to find it especially plausible with respect to ethical discourse. Finally, constructivists may contend that (AOL)’ seems to suggest nothing straightforward about whether the truth-values of ethical sentences are independent of people’s beliefs – though, like intuitionists, constructivists may still face a special case of (AOL) which does seem to do this (namely, an argument from our ascription of truth to certain moral sentences, like certain counterfactuals, whose truth-conditions could apparently not be a function of our beliefs in any way).25 But this, as well, is just the sort of argument that we saw the intuitionist making against (AOL), and it certainly looks, on the face of it, at least, roughly as strong in the moral case as it does in the mathematical. Of course, it remains possible that, upon close inspection, the relevant lines of counterargument would show themselves to be much more persuasive in the case of ethics. But prima facie this does not seem to be so. The Argument from Irrelevance The expressivist’s response to (6)’ above naturally suggests a positive, broadly semantic, argument against moral realism.26 Arguably, there is a sense in which questions of 24 For such an account of the sense in which it is as if moral sentences were true, see [Mackie 1977]. I have in mind here counterfactuals like “if there were no humans around, it would still be the case that killing is wrong.” 26 For something along the lines of the argument that follows, see especially [Korsgaard 1996]. 25 7 ontology just seem beside the point of questions of ethics. In deciding what is right and wrong, one might argue, we do not, at a first approximation, check to see what kinds of things are out there in the world. We appeal to our feelings, abilities, wishes, or intentions. On this basis, one might contend that moral realism just gets the semantics of moral discourse wrong – at a minimum, moral sentences are not true relevantly independent of human minds at all. More explicitly, one might formulate a kind of “Argument from Irrelevance” as follows (AFI): (8) Moral realism entails that something is good or bad (right or wrong, etc.) just in case it instantiates a relevantly mind-independent property.27 (9) But whether people call moral sentences “true” does not seem to depend on whether they take any relevantly mind-independent properties to be instantiated. (10) The best explanation for (9) entails that moral sentences are not true relevantly independent of human minds at all (contrary to moral realism). (11) Hence, by inference to the best explanation, moral realism is false. Premise (8) is thought to be a straightforward consequence of the fact that moral realism entails that there are moral truths (interpreted at face-value) that obtain independent of human minds. If any sentence of the form “x is right (wrong, good, bad, etc.)” is true independent of human minds (when interpreted at face-value), then there is supposed to be, in some sense, a property of rightness (wrongness, goodness, badness, etc.) that is instanced in the world (namely by x) independent of human minds. Premise (9) is a consequence of the expressivist’s objection to (6)’. Thus, the expressivist may defend (9) on the grounds that we actually seem to call moral sentences “true” just in case we approve of certain states of affairs, or would wish for certain states of affairs to obtain were we ideally rational, for instance. The constructivist, on the other hand, may defend (9) on a prima facie different ground – namely, that we seem to call moral sentences “true” just in case we could will a certain rule to be a universal law, for example. (10) may be defended with reference to a kind of principle of charity. If members to a discourse appear to systematically predicate “true” of sentences depending on (relevantly) mind-dependent matters, then we should not, other things being equal, take the sentences so predicated to be mind-independently true. Finally, (11) follows by inference to the best explanation. (AFI) is hardly less challengeable than (OLA). Even supposing that (8) is uncontestable, (9) and (10) in combination look straightforwardly question-begging against the realist. For surely the realist either denies the data encoded in (9), or denies that it’s best explained by the negation of realism, contrary to (10) (perhaps because she thinks that we take our wishes, limitations of our will, or whatever, to be guides to the mindindependent moral truth). And, yet, if either (9) or (10) is implausible, then (in the absence of other considerations) so is (11) -- the conclusion that moral realism is false. By “mind-independent property” I mean a property that does not depend on human minds, in any interesting sense, for its instantiation. 27 8 However strong (AFI) might be, is there not an analogous argument of comparable strength against mathematical realism? It appears that there is.28 Just as it is arguable that questions of ontology seem beside the point of ethics, so too is it arguable that they seem beside the point of (pure) mathematics. In deciding what mathematical propositions to believe, one might contend that mathematicians do not seem to decide what things are out there in the world (as, say, physicists do with respect to physical propositions). They seem to decide what follows from, or is intuitive given, certain axioms. Whether there happens to be a mind-independent mathematical world conforming to the relevant axioms themselves just seems irrelevant. In particular, a mathematical antirealist might argue as follows (AFI)’: (8)’ Mathematical realism entails that 2 + 2 = 4, 7 < 9, etc. just in case certain mind-independent objects stand in certain relations.29 (9)’ But whether people call mathematical sentences “true” does not seem to depend on whether they take any mind-independent objects to stand in such relations. (10)’ The best explanation for (9)’ entails that mathematical sentences are not true relevantly independent of human minds at all (contrary to mathematical realism). (11)’ Hence, by inference to the best explanation, mathematical realism is false. Premise (8)’ is thought to be a straightforward consequence of the fact that mathematical realism entails that there are mathematical truths (interpreted at face-value) that obtain independent of human minds. If any sentence of the form “x + y = z”, say, is true independent of human minds (when interpreted at face-value), then there are supposed to be some numbers that stand in the plus relation to one another.30 Premise (9)’ is a consequence of the formalist’s objection to (6). Thus, the formalist may defend (9)’ on the grounds that we seem to call mathematical sentences “true” just in case we can “prove” them (generate them from certain statements according to certain rules), for instance, while the intuitionist, on the other hand, may defend (9)’ on a prima facie 28 For something like the argument that follows, see [Yablo 2000] or [Rayo unpublished]. By “mind-independent objects” I mean objects whose existence and nature do not depend, in any interesting sense, on human minds. 30 There does seem to me to be one interesting disanalogy between (8) and (8)’ which I shall not pursue (because it is complex and, if anything, lends more credibility to (AFI)’ relative to (AFI)). That is that the moral antirealist does not, in stating (8), strictly speaking, want to commit the moral realist to the existence of any additional things – goodness, badness, etc. (that is, surely the moral realist can be a nominalist about universals). On the other hand, the mathematical antirealist, in stating (8)’ does intend to, strictly speaking, commit the mathematical realist to the existence of additional things -- numbers, points, tensors, etc. (the mathematical realist believes, for instance, that among the furniture of reality is literally a perfect number greater than 19). But it is not obvious, to me, at least, that any nominalistically palatable paraphrase of (8) will make (9) look as plausible as (8) itself does. This is because (8) makes it sound like the moral realist is committed to our deciding whether things are right or wrong depending on whether we actually think that certain additional things exist. And surely this is not so. For, again, unlike the mathematical realist, the moral realist need not, strictly speaking, think that any additional things do exist -- in Quinean terms, she need not allow additional values of variables into her domain of quantification -- so she certainly need not think that we decide what is right or wrong by checking to see whether any do. 29 9 different ground – namely, that we call mathematical theorems “true” just in case we can successfully accomplish certain mental constructions. Finally, (10)’ is just a restatement of (10), and (11)’ follows from (10)’ by inference to the best explanation. Is there anything especially plausible (AFI) as opposed to (AFI)’? Certainly premise (8)’ is no less plausible than (8). Nor is it apparent how (9)’ and (10)’ could be significantly less plausible than (9) and (10). The mathematical realist may, of course, challenge (9)’ by contesting the data that it encodes, or can challenge (10)’ by granting that data, but denying that it is best explained by the negation of mathematical realism (perhaps we take marks on paper to be guides to the way the mind-independent mathematical world is). But we have already seen that the moral realist can respond to (9) and (10) in a completely analogous fashion – and, on the face of it, it seems, with comparable plausibility. And, yet, if (8)’ and (9)’ seem comparably strong to (8) and (9), then, since (10) and (10)’ depend for their plausibility only on (8) and (9), and (8)’ and (9)’, respectively, (AFI), as well, seems comparably strong to (AFI)’. The Argument from Disagreement Perhaps it is not terribly surprising that (OLA) and (AFI) seem, on the face of it, to have relevant analogs of comparable strength. But there is still one, broadly semantic, argument against moral realism that needs to be considered. It is the longstanding argument against moral realism known as the “Argument from Disagreement” (AFD), and may be formulated as follows:31 (12) There is pervasive disagreement among people on many moral issues. (13) The best explanation for this disagreement entails that moral realism is false. (14) Hence, by inference to the best explanation, moral realism is false. Premise (12) is a prima facie plausible empirical claim about people’s actual moral beliefs. Premise (13) is a less prima facie plausible claim about what accounts for (12). Certainly the best explanation of pervasive disagreement with respect to discourses generally does not entail that realism with respect to those discourses is false. There is pervasive disagreement concerning matters of cosmology, for instance, yet (virtually) no one would contend that the best explanation for this disagreement is that realism about cosmological discourse is false. However, moral antirealists have suggested that there are certain peculiar features to moral disagreements that suggest that they, in particular, are best explained by the supposition that realism with respect to moral discourse is false. Most notably, Mackie suggests that what distinguishes moral disagreements from cosmological ones, say, is that the latter result from inferences based on inadequate empirical evidence, whereas it is totally implausible that the former do.32 That is, it does not seem that moral disputes 31 32 See especially [Mackie 1977]. See Ibid. 10 could, in general, be settled by simply attending more responsibly to the available empirical data, whereas it does seem that cosmological ones could be settled this way. Set the question of exactly how strong (AFD) is aside. Is there not a prima facie comparably strong analogous argument against mathematical realism? It appears that there is. It runs as follows (AFD)’: (12)’ There is pervasive disagreement among people on many mathematical issues. (13)’ The best explanation for this disagreement entails that mathematical realism is false. (14)’ Hence, by inference to the best explanation, mathematical realism is false. (12)’ is not widely recognized by philosophers outside of the philosophy of mathematics. But among philosophers in the philosophy of mathematics (12)’ is an uncontroversial datum of some interest.33 It can be motivated with reference to “undecidable” sentences in mathematics, such as the Continuum Hypothesis, the Suslin Hypothesis, the Diamond Principle, the Kurepa Hypothesis, Martin’s Axiom, or the Axiom of Constructability.34 “Undecidable” sentences are sentences that are demonstrably not “decided” by the axioms of our mathematical theories – in the sense that neither they nor their negations can be proved on the basis of those axioms.35 Nonetheless, many mathematicians have positive views about the truth-values of many such sentences, and there is widespread disagreement among such mathematicians as to which such views are correct.36 Premise (13)’, as well, seems just as plausible as does (13), for it derives motivation from just the same considerations. In particular, (13)’ can be defended on the grounds that disputes over the truth-values of such sentences as the Continuum Hypothesis do not seem to arise because any of the members to the disputes are drawing inferences on the basis of inadequate empirical evidence. The truth-values of such sentences appear to be divorced from empirical evidence in just the way that truth-values of moral hypotheses do – and in just the way that the truth-values of, say, sentences about cosmology do not. One might think that mathematicians disagree over the truth-values of fewer hypotheses than do parties to moral disputes, but this seems to be confused. Presumably if anything is relevant here, it is not how many hypotheses people actually spend their time arguing about, but, rather, how many hypotheses people hold different beliefs on (or would argue about if they were raised). And while the number of mathematical hypotheses that 33 See, for example, [Feferman, Friedman, Maddy, and Steel 2000]. Note that while the last two undecidables mentioned have “axiom” in their names, they are not actually axioms of any currently accepted foundational theory (they are potential axioms). 35 Strictly: neither undecidables, nor their negations, can be proved on the basis of established axioms, given that those axioms are consistent. 36 With respect the Continuum Hypothesis, for example, see quotes from Cantor in [Dauben 1979] for a famous statement of the positive view, and [Godel 1940] for a famous statement of the negative. For an overview of some contemporary work in the relevant area, see [Woodin 2005]. (The Continuum Hypothesis is the conjecture that if A is an uncountable subset of the real numbers, R, then there exists a bijection between A and R.) 34 11 mathematicians spend their time arguing about is less than the number of moral hypotheses that people spend their time arguing about, obviously, for any mathematical hypothesis that is in dispute, disputing parties will generally hold different beliefs as to the truth-value of everything that is a consequence of it and established axioms, but isn’t a consequence of established axioms alone. And, given that there are an infinite number of such consequences, it’s hard to see how there could be “more” moral hypotheses that are in dispute.37 It might be countered that even if this is so, still there is marked uniformity of agreement as to the truth-values of a huge “core” of mathematical hypotheses. In particular, there is virtual consensus as to the truth of all theorems of our standard theories, such as Peano Arithmetic. So, maybe while the number of mathematical sentences that people disagree on is not less than the number of moral sentences that they disagree on, still the number of mathematical sentences that people do agree on is greater than the number of moral sentences that they agree on. And maybe this could still be thought to somehow cast peculiar doubt on (13)’. But even this much more limited suggestion seems mistaken. While there is, indeed, virtual consensus as to the truth of a huge “core” of mathematical hypotheses (such as the standard theorems of Peano Arithmetic), so too is there virtual consensus as to the truth of a huge “core” of moral ones. Just consider any instance of the open sentence “other things being equal, one ought not do x to someone for fun if she would prefer not to have x done to herself.”38 37 It is also a fairly trivial to establish the existence of infinitely-many substantial undecidable mathematical sentences in other ways. For example, in showing that the negation of the Continuum Hypothesis is consistent with both Zermelo-Fraenkel set-theory, and even Zermelo-Fraenkel set-theory supplemented with the Axiom of Choice, Cohen shows that the size of the continuum can consistently be supposed to be just about any transfinite cardinal at all. That is, there are an infinite number of claims as to the specific size of the continuum whose truth-value could be contested. See Cohen [1966]. 38 In conversation I have sometimes heard it claimed that there is still an apparent disanalogy here in that one can consistently deny such “core” moral principles, while one cannot even consistently deny such “core” mathematical ones (“one can consistently be a moral skeptic, but not a mathematical one”.) But this claim is almost certainly rooted in confusion of mathematical truths with corresponding logical ones. “1 + 1 = 2” (taken at face-value) makes existential commitments to numbers, and surely it is not logically inconsistent to deny that among the things that exist are numbers (or, as Kant pointed out, anything else for that matter). Indeed, this is just what fictionalists deny, and surely their position is not logically inconsistent. What it is logically inconsistent to deny is the logical truth that if there is exactly one F that is G and exactly one H that is G, and no F is an H, then there are exactly two things which are either F or H and also G (where the numerical quantifiers here are definable in terms of ordinary quantifiers plus identity) [Field 1989a]. But this truth (which does not even entail the existence of numbers) is not at issue here. All of this is, of course, to set aside the fact that it is unclear how the aforementioned claim, even if it were true, would be relevant to the relative plausibility of (13) and (13)’ in the first place. I’ll have a bit more to say on the relation between mathematics and logic below (in the section entitled “The Argument from Inaccessibility”). 12 On the face of it, then, (AFD), as well, seems not to afford a significant disanalogy between the case for moral realism and the case for mathematical realism. I now turn to (B) Epistemological Arguments. B. Epistemological Arguments The Indispensability Argument If there is anything like a standard story about why there is better reason to be a mathematical realist than there is to be a moral one, it is perhaps the story that Gilbert Harman briefly suggests at the end of “Ethics and Observation” [Harman 1977]. Roughly, he there endorses what has come to be known as the Quine-Putnam Indispensability Argument for mathematical realism, and suggests that there is no remotely plausible analogous argument for moral realism. The Quine-Putnam Indispensability Argument (IA) for mathematical realism runs as follows:39 (15) We are justified in believing the theories which figure into the best overall explanation of our observations. (16) Mathematical theories figure into the best overall explanation of our observations. (17) Hence, we are justified in believing mathematical theories. The first premise in the Quine-Putnam Indispensability Argument is a statement of the comparatively uncontroversial direction of the biconditional endorsed by Quinean empiricists -- that we are justified in believing a theory if and only if it figures into the best explanation of our observations. The intuitive motivation for the second premise is that mathematical theories figure into the working formulations of our (obviously) empirical scientific ones, and that the latter together constitute the best overall explanation of our observations. The conclusion follows by modus ponens. Mathematical antirealists of all stripes may contest (16). Formalists may deny it on the grounds that mathematical “theories” are not really theories, capable of being believed, at all. Fictionalists may deny it on the grounds that mathematics-free analogs to our going empirical theories can be constructed, and are better than the former in virtue of avoiding commitment to facts that are in dispute. And intuitionists may point out that, as it stands, (IA) seems to suggest nothing at all as to whether the truth-values of mathematical sentences would be independent of people’s beliefs.40 Whatever the prospects for these lines of response, anyone familiar with recent realist literature in metaethics will recognize that Harman’s suggestion that there is no remotely plausible analog to (IA) in metaethics is now surely false. We may formulate such an analog as follows (IA)’: 39 See especially [Quine 1951] and [Putnam 1971]. Of course, there may be a further argument that mathematical propositions whose truth would be relevantly independent of anyone’s beliefs, in particular, figure into the best overall explanation of our observations. 40 13 (15)’ We are justified in believing the theories which figure into the best overall explanation of our observations. (16)’ Moral theories figure into the best overall explanation of our observations. (17)’ Hence, we are justified in believing moral theories. Now, (15)’ is just a restatement of (15). And many moral realists have now offered what are, at the very least, remotely plausible defenses of (16)’.41 Some have noted, for instance, that a standard explanation offered by historians for the rise of abolitionism in the New World is that the form of slavery practiced there, chattel slavery, was especially bad. And it is natural to explain other historical facts, like Hitler’s slaughter of six million Jews, with reference to like facts – such as that Hitler was morally depraved.42 (17)’, in turn, follows by modus ponens. Of course, just as mathematical antirealists may attack (16), moral antirealists may attack (16)’. Expressivists, like formalists, may deny it on the grounds that our moral “theories” are not really theories, capable of being believed, at all. Error-theorists, in turn, like fictionalists, may deny it on the grounds that ethics-free analogs to our going empirical theories can be constructed, and are better than the former in virtue of avoiding commitment to the facts that are in dispute. And constructivists, like intuitionists, may point out that, as it stands, (IA)’ seems to suggest nothing at all as to whether the truthvalues of moral sentences would be independent of people’s beliefs, and, hence, fails to establish moral realism.43 But are these lines of response peculiarly plausible in the moral case? There does not seem to be any prima facie reason to find the expressivist’s ground for rejecting (16)’ more plausible than the formalist’s ground for rejecting (16), or to find the constructivist’s ground for rejecting that premise more plausible than the intuitionist’s. But perhaps there is more prima facie plausibility to the error-theorist’s thought that ethics-free analogs to our going empirical theories can be constructed, and that they’d be better than the originals, than there is to the fictionalist’s thought that mathematics-free analogs to our going empirical theories can be constructed, and that they’d be better than the originals. However, the pertinent point here is that the relative plausibility of the error-theorist’s thought is no longer altogether incommensurate with that of the fictionalist’s, as it likely was back when Harman published “Ethics and Observation”. This is because a tremendous amount of promising work has actually now been done to establish the relevant thought in the mathematical case, whereas virtually no such work has been done to establish the analogous thought in the moral one. Philosophers such as Field and Balaguer have now constructed what appear to be mathematics-free analogs to the going 41 For a foundational such defense, see [Sturgeon 1988]. See Ibid. 43 As with (IA), there may still be a further argument that moral propositions whose truth would be relevantly independent of anyone’s beliefs, in particular, figure into the best overall explanation of our observations. 42 14 formulations of a number of our best physical theories, and that philosophers cannot do something analogous for our physical theories quite generally is just not clear. 44 Of course, mathematical realists may still respond to these advances by denying that such analogs are really better than the originals (perhaps on the grounds that we should take our cue from the scientists themselves, or perhaps on the grounds that mathematical science is “simpler” or more aesthetically appealing in some relevant sense). But the important point is just that there seems to be no prima facie reason to assume that there is not a comparably strong analogous response open to moral realists with respect to ethicsfree analogs to the going formulations of our historical, sociological, psychological, etc. theories. Even if there is a bit more plausibility to (AI) than (AI)’, there no longer seems to be the kind of substantial disanalogy between these two arguments that is of interest to us here. The Argument from “Intuition” The Indispensability Argument for mathematical realism is apt to seem somewhat beside the point to many philosophers of mathematics in any case. Many philosophers are apt to think that the prima facie source of our justification for our mathematical beliefs is not observational in the first place – it is a priori.45 Whether or not we are justified in believing that 2 + 2 = 4 on the basis of observation we are, it seems, at least defeasibly, justified in believing it on the basis of something like its intuitive obviousness. And perhaps our justified belief in mathematical propositions quite generally can be explained in terms of our theorizing about, and drawing inferences from, such intuitively obvious propositions as 2 + 2 = 4. In particular, perhaps the following Argument from Intuition is sound (AI): (18) We are, at least defeasibly, justified in believing intuitively obvious propositions, and any propositions that “follow” from these via obviously valid inference rules (deduction, induction, inference to the best explanation, etc.).46 (19) Our mathematical theories are exhaustively composed of propositions that are either intuitively obvious or “follow” from intuitively obvious propositions via obviously valid inference rules. (20) Hence, we are, at least defeasibly, justified in believing our mathematical theories. Premise (18) is just a statement of a commonly-held element to rationalism in epistemology -- the view that we are non-experientially justified in believing some propositions. Premise (19) is a prima facie plausible thesis about the actual contents of our mathematical theories. With respect to Peano Arithmetic, for instance, the thought 44 See [Field 1980] and [Balaguer 1996]. For an especially elegant and thorough treatment of the relevant technical issues, see [Burgess and Rosen 1997]. 45 See, for example, [Parsons 1979], [Bealer 1982], and [Bonjour 1998]. 46 I put “follow” in scare quotes to distance myself from any substantive position on the relationship between deductive inference and other sorts of inference – maybe, as Hume would have it, it is only deductive conclusions that, strictly speaking, follow from premises. 15 might be that the relevant axioms are intuitively obvious themselves, or that they are part of the best explanation of intuitively obvious theorems (that 1 < 2, that 5 + 5 = 10, etc.). The conclusion, (20), follows by modus ponens. Whether (AI) is sound is, of course, contestable. Formalists may challenge (19) on the grounds that our mathematical “theories” do not, in fact, consist of propositions at all. Fictionalists may challenge (18), on the basis of a thorough-going empiricism in epistemology.47 And intuitionists may note that (AI) seems to leave open whether the truth-conditions of mathematical propositions are independent of everyone’s beliefs. But is there an analogous argument to (AI) for moral realism, (AI)’? It seems clear that there is.48 It may be formulated as follows: (18)’ We are, at least defeasibly, justified in believing intuitively obvious propositions, and any propositions that “follow” from these via obviously valid inference rules (deduction, induction, inference to the best explanation, etc.). (19)’ Our moral theories are exhaustively composed of propositions that are either intuitively obvious or “follow” from intuitively obvious propositions via obviously valid inference rules. (20)’ Hence, we are, at least defeasibly, justified in believing our moral theories. Premise (18)’ is just a restatement of (18). Premise (19)’ is a prima facie plausible thesis about the actual contents of our moral theories. Plausibly, certain general moral principles are either intuitively obvious (e.g. that other things being equal, one ought not cause needless harm) or are inductively derived from, or postulated to best explain, particular moral propositions which are (e.g. that Hitler’s slaughter of six million Jews was wrong). The conclusion, (20)’, follows by modus ponens. Just like (AI), (AI)’ is, of course, contestable. Expressivists may challenge (19)’ on the grounds that our moral “theories” do not, in fact, consist of propositions at all. Errortheorists may contest (18)’, on the basis of a thorough-going empiricism in epistemology.49 And constructivists may note that (AI)’ seems to leave open whether the truth of moral sentences is independent of everyone’s beliefs. Now, each of these responses to (AI)’ is precisely analogous to a response to (AI). The expressivist’s response corresponds to the formalist’s; the error-theorist’s response corresponds to the fictionalist’s; and the constructivist’s response corresponds to the intuitionist’s. Moreover, on the face of it, at least, there is nothing peculiarly plausible about the relevant lines of counterargument in the moral case as opposed to the mathematical one. It would seem, then, prima facie that the Argument from Intuition to mathematical realism, too, fails to afford a significantly disanalogy between the cases for mathematical realism and moral realism. 47 See, for example, [Field 1980]. See, for instance, [Schaffer-Landau 2003] or [Audi 2004]. 49 See, for example, [Mackie 1977]. 48 16 The Argument from Epistemic Inaccessibility But even ethicists who grant the soundness of (AI)’ may still argue that there is, after all, a relevant defeater. In particular, they may contend that that realists cannot afford a remotely plausible account of how it is that our moral “intuitions” would track the moral truth, and that, if this is right, then our justification for believing moral propositions is defeated.50 That is, moral antirealists may argue as follows (AEI): (21) If moral realism is true, then moral truths are true relevantly independent of human minds. (22) There is no plausible account of how our moral intuitions would reliably track the relevantly mind-independent moral truth. (23) If there is not a plausible account of how our moral intuitions would reliably track the relevantly mind-independent moral truth, then we should not believe that there is such a truth. (24) Hence, we should not believe moral realism. Whether (AEI) is sound is far from obvious. First, moral realists may challenge (23) as being too restrictive. They may point out that the question of how our moral intuitions would reliably track the mind-independent moral truth is just a special case of the question of how our intuitions generally would track the mind-independent truth. And while there may be a straightforward (presumably causal) story to tell with respect to some of these intuitions, it is far from obvious that there would be such a story to tell with respect to all of our other intuitions which we take to be truth-tracking.51 Moreover, realists may attack (22) as well. Moral naturalists, in particular, may point out that, on their view, the question of how our intuitions would track the mind-independent moral truth is just a special case of the question of how our intuitions would track mindindependent natural truths. And, yet, here, especially, there seems to be at least a prima facie case to be made for some sort of (causal) story. Whatever the prospects for the moral realist here may be, there certainly seems to be a comparably strong analogous argument to (AEI) against mathematical realism. Indeed, it is essentially the widely-hailed reliability argument for mathematical antirealism due to Field [1989a], taking his inspiration from Benacerraf [1973], and may be formulated as follows (AEI)’: (21)’ If mathematical realism is true, then mathematical truths are true relevantly independent of human minds. (22)’ There is no plausible account of how our mathematical intuitions would reliably track the relevantly mind-independent mathematical truth. (23)’ If there is not a plausible account of how our mathematical intuitions would reliably track the relevantly mind-independent mathematical truth, then we should not believe that there is such a truth. (24)’ Hence, we should not believe mathematical realism. 50 51 See, for instance, [Street 2006]. Consider, for example, modal intuitions, or our intuition that a believer doesn’t know in Gettier cases. 17 Now, just as (AEI) is capable of being challenged, so, too, is (AEI)’. Like moral realists, mathematical realists may challenge (23)’ as being too restrictive. And, like moral realists, mathematical realists may reject (22)’ as well. Mathematical naturalists, in particular, may point out that, on their view, the question of how our mathematical intuitions would track mind-independent mathematical facts just reduces to the one of how our intuitions would track mind-independent natural facts. And, here, again, there is at least prima facie plausibility to a sort of (causal) story. But is there anything that seems peculiarly plausible about the relevant lines of response in the mathematical case as opposed to the moral one? The mathematical realist’s objection to (23)’ is just a restatement of the moral realist’s objection to (23). I have sometimes heard it claimed in conversation that the mathematical realist’s objection to (22)’ is stronger than the moral realist’s objection to (22) on the grounds that mathematics is “reducible” to logic – the assumption being that a plausible story could be told about our knowledge of logic.52 But, first, however plausible it might be that mathematical sentences can be semantically reinterpreted as logical ones, that mathematics taken at face-value is a branch of logic (in any relevant sense of “logic”) is extraordinarily doubtful.53 Most straightforwardly, perhaps, mathematical sentences make existential commitments (e.g. that there is prime number greater than 25), while logical ones do not.54 Second, even if mathematics taken at face-value were just logic going by another name, still logic, itself, lacks a remotely satisfactory epistemology. Once the idea of logical truths being “true by convention” (as the logical empiricists were apt to put it) is given up, logical truths come to be seen as just another branch of truths which are true in virtue of the way the mind-independent world is. As such, it is as mysterious as ever how we could know such truths – even if knowing them involved reflecting on the meanings of our words, or the rules that govern our language and reasoning. After all, it could be that 52 I have also sometimes heard it claimed that it is more plausible that there is some sort of naturalistic explanation of our knowledge of mathematical facts because we wouldn’t know that we shouldn’t, for instance, let other animals eat most of our food if we didn’t know that any quantity minus some other (positive, finite) quantity is less than the original. But, first, this particular sort of worry seems, again, to mix up logical truths with mathematical ones in the style of fn. 33. No mathematics, in the relevant sense of “mathematics”, has evidently been invoked in this example – i.e. there has been no evident quantification over the likes of numbers here. But, second, even with respect to contexts where we do explicitly invoke mathematics, Russell [1936] long ago pointed out that for purposes of counting, mathematical truth, and thus knowledge, seems beside the point. And, more recently, drawing insight from the problem of multiple reductions in the foundations of mathematics (to be discussed briefly below under the heading “The Open Question Argument”), and considerations like those garnered in the “Argument from Irrelevance”, philosophers have noted that a similar point seems to hold with respect to real-world mathematical purposes very generally. In this capacity, see especially Benacerraf [1965], [Field 1989c], and [Edidin 1995]. 53 Of course I recognize that there have been, and even still are a few, philosophers of mathematics – namely, “logicists” – that hold, or at least seem to hold, that mathematics (taken at face-value), or certain fragments of it, is just logic in disguise. But such philosophers of mathematics are rare, and tend to hold highly nonstandard semantical and logical views. For a contemporary defense of one version of logicism which rests on such semantical and logical views, see [Wright and Hale 2003]. 54 Field is wont to stress this point in his [1989b]. 18 some of our conventionally stipulated definitions are simply not satisfied, or even satisfiable. If this were so, then knowing that logical terms mean what they do wouldn’t seem to entail knowing the relevant logical truths.55 It appears, then, on the face of it, at least, that (AEI) as well fails to afford a significant disanalogy between the cases for moral realism and mathematical realism. C. Metaphysical Arguments I have now surveyed an array of salient semantic and epistemic arguments for mathematical realism, and against moral realism, and argued that, on the face of it, each of them seems to have comparably strong analog of the relevant sort. In this section, I will consider what I take to be the salient, broadly metaphysical, arguments for mathematical realism or against moral realism, beginning with what I shall call “The Argument from Naturalism” (AFN). The Argument from Naturalism Next to Harman’s indispensability considerations, considerations of naturalism probably afford the closest thing to “the standard story” about why one should be more confident in mathematical realism than she should be in moral realism. Moral antirealists argue that moral realism would commit us to entities which would be somehow unnatural.56 That is, they argue as follows (AFN): (25) Naturalism is true. (26) Moral properties, which moral realism presupposes, would not be natural. (27) Hence, moral realism is false. (AFN) is structurally straightforward, though obscure in content. Naturalism is certainly the view that everything is natural, but what it is for something to be natural is very unclear. Whatever naturalism is, however, (25) claims it to be true, and (26) tells us that whatever things naturalism entails the nonexistence of, moral properties are among them. (27) then follows by modus ponens. It is obvious that there is an analogous argument to (AFN) against mathematical realism. It runs as follows (AFN)’: (25)’ Naturalism is true. (26)’ Mathematical objects, which mathematical realism presupposes, would not be natural. 55 Note that the most serious problem for the epistemology of logic, just like the epistemology of ethics or math, is not that of explaining why we have the beliefs that we do per se, but rather that of explaining why the beliefs that we do have tend to track the mind-independent truth. It might be that our having the logical beliefs that we do have is explicable from an evolutionary perspective, for instance. But that doesn’t mean that our having mind-independently true beliefs is. 56 The disanalogy, briefly discussed in footnote 30, above, between the cases against moral realism and mathematical realism will arise again pervasively throughout this section. But, as before, I cannot take the problem up here (recall that, if anything, it casts peculiar doubt on moral antirealism, as opposed to mathematical antirealism). 19 (27)’ Hence, mathematical realism is false. (25)’ is just a restatement of (25). (26)’ is just (26) with “moral” replaced by “mathematical”, and with “properties” replaced by “objects”.57 Like, (27), (27)’ then follows by modus ponens. Neither (AFN) nor (AFN)’ is stronger than its second premise, and there are two established lines of argument for that premise in the case of (AFN). I will now consider those lines of argument, and argue that they have relevant analogs, in turn. The Open Question Argument The first established line of argument for (26) takes its inspiration from Moore’s Open Question Argument [Moore 1903]. In its most plausible form, it proceeds as follows (OQA):58 (a) Competent speakers of our language can coherently doubt that any X is good (bad) for any naturalistic property-name, ‘X’. (b) The best explanation for (a) is that the good (bad) is not identical with any naturalistic property. (c) Hence, by inference to the best explanation, the good (bad) is not natural. It is obvious that (OQA) can be challenged by the naturalist, even setting aside worries about the obscurity of naturalism. Whether or not she grants (a), she can challenge (b) on numerous grounds. First, it’s not clear why the thought that the good is not reducible to any naturalistic property for which there is a preexisting name in some naturalistic vocabulary, does not afford an equally good explanation of (a). And, second, if the kind of doubt mentioned in (a) is merely the kind of doubt exemplified when we “imagine” it failing to be the case that water is H20, then surely the best explanation of (a) is not, at least obviously, that the good is not identical with any naturalistic property (contrary to (b)). Does there not appear to be an analogous, and comparably strong, argument to (26) for (26)’? It appears that there does – it is, in essentials, very similar to Paul Benacerraf’s widely-hailed argument from multiple reductions [Benacerraf 1965]:59 (a)’ Competent speakers of our language can coherently doubt that any X is the number 2 (3, 4…) for any naturalistic expression, ‘X’.60 In fact, (8)’ could be formed by, in addition to replacing “moral” with “mathematical”, just adding “objects” to the list – since, again, one could object to peculiarly mathematical properties as well. 58 See [Ball 1988] for a formulation of (OQA) roughly along these lines. 59 Actually, Benacerraf’s argument is somewhat stronger than this analog to Moore’s. Benacerraf doesn’t require of a successful reduction of the numbers that we can’t imagine it being false, but merely that there be some principled reason to prefer it to others. And he argues, plausibly, that even this condition is not met. 60 Of course, the argument doesn’t just work for numbers if it works. Choose your favorite mathematical object, and an analog to (a)’ should run equally smoothly with respect to it. 57 20 (b)’ The best explanation for (a)’ is that the number 2 (3, 4…) is not identical with any naturalistic one. (c)’ Hence, by inference to the best explanation, the number 2 (3, 4…) is not natural. It seems beyond serious controversy that (a)’ – (c)’ is at least as strong as (a) – (c). Obviously (a)’ is true in some sense of “doubt”. Indeed, it’s not even clear that competent speakers of our language can help but doubt that the number 2 is identical with some object nameable in naturalistic vocabulary. Does it even make sense to wonder whether the number 2 is identical with the Statue of Liberty, say?61 The closest thing to remotely fathomable identities of mathematical objects which could potentially qualify as naturalistic in the relevant sense are those which identify the numbers with cardinality or ordinality attributes, or with classes of entities exemplifying those attributes. And, yet, it is highly contentious that attributes or classes, qua objects in their own rights, are naturalistic entities in the first place. Of course, (b)’ can be challenged on just the grounds that (b) can be. But it doesn’t seem that it can be challenged on any grounds which (b) can’t be challenged on. So, (AFN) and (AFN)’, too, seem prima facie to be highly analogous. The Argument from Queerness The other standard argument for (26) takes its motivation from Mackie’s famous claim that moral properties would be “queer”, in that they’d be intrinsically motivating and reason-giving [Mackie 1977]. The argument proceeds as follows (AFQ): (d) Moral properties, which moral realism presupposes, would be intrinsically motivating and reason-giving. (e) Intrinsically motivating and reason-giving properties would not be natural. (f) Hence, moral properties, which moral realism presupposes, would not be natural. (AFQ) is clearly challengeable. Moral realists of all stripes may challenge (d), on the basis of thought experiments. It is certainly not obvious that there is anything incoherent in the thought of an ideally rational being that is neither motivated, nor has any reason, to do what is right. And, with respect to (e), moral realists may argue by analogy. Arguably uncontroversially natural properties like pain are intrinsically motivating and reason-giving. And if they are, then a property’s having the relevant features is not inconsistent with its being natural after all. There is a sense in which there does not seem to be any remotely plausible analog to (AFQ) for (26)’. In particular, there does not seem to be any plausibility to the thought that mathematical objects (or properties) would be intrinsically motivating. But what is relevant as far as (26)’ is concerned is not whether mathematical objects would be intrinsically motivating per se, but rather whether they’d have any peculiar features that 61 Frege famously claimed that it did in his [1884]. More recently, philosophers have been skeptical that such identities are even well-formed. See [Benacerraf 1965] or [Shapiro 1997]. 21 would qualify them as non-natural. And it is surely arguable that mathematical objects would have such features. In particular, one may argue as follows (AFQ)’: (d)’ Mathematical objects, which mathematical realism presupposes, would be aspatiotemporal, necessarily existing, and causally inert. (e)’ A-spatiotemporal, necessarily existing, and causally inert, objects would not be natural. (f)’ Hence, mathematical objects would not be natural. One common line of argument for (d)’ proceeds from the premise that mathematical truths, such as that there exists a perfect number greater than 19, are necessarily true.62 For if they are so true then it is true in every world that, for instance, there exists a perfect number greater than 19. But then, it is argued, it must be the case that such a number exists in every world – it must be the case that it exists of necessity. But nothing that exists of necessity could be identical with any spatiotemporally extended object (or anything ontologically parasitic on such an object), since there are worlds in which there are no such objects. Nor could it be that something that exists of necessity has any causal influence – for the world’s being any possible way is consistent with its existence. So mathematical objects would be necessarily existing, a-spatiotemporal, and causally inert. Much as moral realists can challenge (AFQ), mathematical realists can challenge (AFQ)’. Property-theoretic and impure set-theoretic naturalists may deny that mathematical objects would be necessarily existing after all, and hence also that they would be aspatiotemporal or causally inert as (d)’ entails.63 And even those who grant, (d)’ may still deny that mathematical objects would be unnatural, presumably on the grounds that they share more in common with uncontroversially natural objects than they seemed to. Whatever the subtleties of these issues, the relevant point is just that (AFQ)’ seems, on the face of it, to be at least as strong as (AFQ) -- indeed, both (d)’ and (e)’ are relatively uncontroversial, while (d) and (e) are anything but this. Queerness Again So, prima facie, for each of the two standard arguments for (26), (OQA) and (AFQ), there are comparably strong analogs for (26)’. But, since (25)’ is just a restatement of (25), and (27)’ follows by inference to the best explanation just like (27), (AFN)’ appears to be comparably strong to (AFN).64 62 See, for instance, [Shapiro 2000]. See, for instance, [Bigelow 1988] or [Maddy 1990] 64 I have heard the question raised in conversation whether there isn’t also an argument from the aspatiotemporal, necessarily-existing, and causally inert, character of moral properties to (8), in addition to one from their intrinsically motivating and reason-giving character (while there is not an argument from the intrinsically motivating and reason-giving character of mathematical objects to (8)’ in addition to one from their necessary existence, a-spatiotemporality, and causal inertness). The answer, I think, is that there would only be such an argument if Platonism about universals were true – i.e. if properties quite generally existed as objects in their own rights, of necessity, outside spacetime, and in a causally inert capacity. For, suppose that Platonism about universals is false. Then either property-talk is to be understood on the Quienean model as convenient shorthand for talk about propertied particulars, or it is to be understood as talk about a realm of non-Platonic entities. In the first case, “moral properties” are not really to be thought 63 22 Before concluding this section, however, I want to say one more word about “Arguments from Queerness” as they’re presented in the literature. It is not always clear that such arguments are intended to rely on (25), i.e. the assumption that naturalism is true. Sometimes moral antirealists seem to argue as follows: (28) Moral properties, which moral realism presupposes, would be relevantly unlike anything in the world. (29) It is not the case that anything relevantly unlike anything in the world exists. (30) Hence, moral realism is false. Prima facie, one could grant the soundness of this argument while still denying naturalism. One might maintain that a minimum condition that a postulate must meet is that it not be unlike uncontroversially extant things in certain relevant respects. Whether it needs to be natural per se can be left undecided. The obvious difficulty with such an argument, however, is that of specifying the relevant respect in which postulates must not diverge from uncontroversially existing things. If it is claimed that they must not diverge in being intrinsically motivating or reason-giving, then we will presumably need a reason for thinking so. And if the above argument is not going to rely on (25) – i.e. if it is not going to rely on the truth of naturalism -- then this reason cannot just reduce to the one that intrinsically motivating and reason-giving things would be unnatural.65 Of course, a moral antirealist could, I suppose, simply contend that there is something primitively objectionable about the supposition that there are intrinsically motivating or reason-giving entities in the world. But, so too, of course, could the mathematical antirealist contend that there is something primitively objectionable about the supposition that there are a-spatiotemporal, necessarily-existing, causally inert, ones.66 However of as additional elements to the world at all. There are things which are right and wrong, much as there are things which are white and black, but there is not, in addition, the properties of rightness and wrongness. In particular, there are not the properties of rightness and wrongness qua necessarily existing, aspatiotemporal, and causally inert objects. Alternatively, suppose that property-talk should be understood as talk about a non-Platonic realm of objects. Then those objects must at least fail to be “queer” in one of the relevant ways – they must either fail to be necessarily existing, a-spatiotemporal, or causally inert, insofar as they fail to be Platonic. But, then, here again there is no objection to their postulation that is analogous to the relevant one against the postulation of mathematical objects. Given that Platonism about universals is an extraordinarily contentious doctrine, the aforementioned worry does not, then, seem to carry much weight. 65 One could also offer an Open Question Argument against moral realism that did not rely on naturalism. Perhaps one allows that certain (maybe divine) non-natural properties exist, but denies that other nonnatural ones do. Or perhaps one requires of any postulate that it must be referred to in some informative identity statement – that is, roughly, that it not be sui generis (this seems, more or less, to be Benacerraf’s position a la mathematical objects in his [1965]). In the first case, one can argue that competent speakers of our language can coherently doubt that goodness is identical with any of the relevant non-natural properties, just as they can the natural ones. And in the second, one can argue, as Moore himself seemed to in his [1903], that such speakers can coherently doubt that goodness is identical to anything independently characterizable. 66 Indeed, this seems to be Quine’s and Goodman’s exact position a la mathematical objects in their [1947]. 23 plausible either of these claims might be, it is hard to see how one of them could be much more plausible than the other. 4. 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