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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. Conclusion
I have surveyed a wide array of salient arguments for mathematical realism, and against
moral realism, and have argued that, on the face of it, each argument seems to have a
relevant analog of quite comparable strength. Obviously I have not considered every
argument for mathematical realism, or against moral realism, and I have not considered
any such argument in great detail. Nevertheless, the above survey is sufficient to lend
credibility to the bold thesis that moral realism is on “no worse footing” than
mathematical realism. Parallels between problems in the relevant areas seem to be much
more pervasive than is normally supposed. The burden is on the moral antirealist to show
that this appearance is misleading.
24
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