Mathematical Contingentism
Platonists and nominalists disagree about whether mathematical objects
exist. But they almost uniformly agree about one thing: whatever the
status of the existence of mathematical objects, that status is modally
necessary. Two notable dissenters from this orthodoxy are Hartry Field,
who defends contingent nominalism, and Mark Colyvan, who defends
contingent Platonism. The source of their dissent is their view that the
indispensability argument provides our justification for believing in the
existence, or not, of mathematical objects. This paper considers whether
commitment to the indispensability argument gives one grounds to be a
contingentist about mathematical objects.
1. Introduction
Platonists believe that there exist mathematical objects, and that these mathematical
objects are abstracta. The reasons why they think such objects exist vary. But some
Platonists, including Colyvan and Resnik and indeed some nominalists, such as Field,
justify their claims about the existence, or lack thereof, of mathematical objects by
appeal to what has become known as the indispensability argument. Though found in
different forms, the indispensability argument appeals to a broadly Quinean
naturalistic account that tells us to which entities we should be ontologically
committed. The idea is familiar. A theory, suitably regimented into first order
classical logic, is committed to all and only the entities to which the bound variables
of the theory must be capable of referring in order that the claims made in the theory
are true.1 We should accept our best theories. Thus we should be committed to all and
only the entities over which our best theories quantify.2
Thus the indispensability argument moves from the Quinean claim that we
1
Quine (1953) 114—115.
How sparse the corresponding ontology is to which we will be committed depends on whether our
best theories are restricted to scientific theories, or includes theories more broadly which might include
various “folk” theories.
2
1
should be committed to all and only the entities quantified over by our best suitably
regimented theories, via the claim that mathematical objects are quantified over by
our best suitably regimented scientific theories, to the conclusion that we should be
committed to the existence of mathematical objects.3 Those who endorse the
indispensability argument hold that the reason we should be committed to the
existence of mathematical objects (or not) is their indispensability (or lack thereof), to
our best theories. Indeed, both Colyvan and Field think that the indispensability
argument is the most powerful, and in Colyvan’s case, the only, argument that can
give us reason to believe, or not, in mathematical objects.4
It is in virtue of their commitment to the indispensability argument that Colyvan
and Field explicitly embrace contingentism about the existence of mathematical
objects. Colyvan is a contingent Platonist, and Field a contingent nominalist. The
implication is that anyone who justifies their belief in the existence, or lack thereof, of
mathematical objects by appeal to the indispensability argument ought also to be a
contingentist.5
The first section of this paper sets out the indispensability argument and
explores some ways in which we might draw from it a conclusion that has modal
force.
In section 2 I explore why one might think that commitment to the
indispensability argument means being a contingentist about mathematical objects.
Section 3 then considers an epistemic objection according to which contingentism is
in tension with the idea that indispensability is a good guide to ontology. I suggest
that whether this objection succeeds depends on how one understands the evidential
connection between indispensability and ontology, but that on many plausible ways of
3
This argument has come under attack. The claim that we should be committed to all those entities
quantified over by our best theories—a claim that owes its origins to a kind of confirmational holism—
has been attacked by, among others Maddy (1992: 280) and Azzouni (1997).
4
Colyvan (1998a); Field (1980).
5
Since the purpose of this paper is to consider the relationship between the indispensability argument
and contingentism, throughout I assume that contingentism is a meaningful doctrine in at least some
minimal sense. If necessitarianism is true, then many of the statements the contingentist makes will
turn out to be necessarily false, and thus, according to some accounts of content, will be contentless or
nonsensical. Even the necessitarian that has this view of content, however, must surely give some
(perhaps small) credence to contingentism being true, and if she does that she must concede that the
doctrine is meaningful in some good sense (perhaps she will here appeal to some sort of
hyperintensional content). For the purposes of this paper that is all that is required. Another possibility,
raised by a referee, is that necessitarians might think that many of the statements expressed by
contingentists are meaningless in the more straightforward sense that they are ungrammatical, and that
the mistake of the contingentist is to continually commit grammatical solecisms. Since the claims all
appear to be perfectly grammatical I am not sure on what basis that claim would be made, but at any
rate, this paper assumes that contingentist statements are grammatical and therefore meaningful in this
sense.
2
construing that relationship, the epistemic objection has force. Contingentism does
not completely undermine the indispensability argument, but it is in tension it, and the
contingentist does need to provide a plausible account of the evidential role of
indispensability. Moreover, in section 4 I suggest that any plausible contingentist
semantics that makes sense of the apparent necessity of mathematical statements will
be incompatible with the indispensability argument as it is presented by
contingentists. Section 4 then proceeds to consider a further argument for
contingentism that proceeds from the idea that because the conclusion to the
indispensability argument is an a posteriori claim this gives us at least prima facie
reason to suppose that it is also a contingent claim. I argue, instead, that this is reason
to think that mathematical statements are not a posteriori. In light of this, section 5
offers an amended version of the indispensability argument that yields an a priori
necessitarian
conclusion.
Ultimately,
I conclude,
considerations
based
on
indispensability do not cut in favour of contingentism rather than necessitarianism.
2. The Indispensability Argument and Modality
As it stands, the indispensability argument is a simple argument whose conclusion is
silent about many aspects of the nature of mathematical objects. Accepting its
conclusion is not thereby to accept Platonism.6 To yield the conclusion that Platonism
is true we need an additional premise according to which if there are any
mathematical objects, then those objects are abstracta. That needs to be an
independently motivated claim. For various reasons, most philosophers of maths
accept something like that claim.7 In this paper I follow suit. Hence, relevantly
amended, in its simplest form the indispensability argument is as follows:8
1. We should be committed to all and only the entities quantified over by our best
suitably regimented theories.
2. Mathematical objects are quantified over by our best suitably regimented
theories.
6
In this context I take Platonism to be the conjunction of two doctrines: mathematical realism, namely
the doctrine that mathematical objects exist, and a claim about the nature of those objects, namely that
they are abstracta.
7
Though there are notable exceptions, such as Maddy (1990).
8
This version of the indispensability is a somewhat amended version of what Colyvan refers to as the
Quine-Putnam indispensability argument. See Colyvan (2008, 2001: 11).
3
3. Necessarily, if O is a mathematical object then O is an abstract object.
4. Conclusion: Therefore we should be committed to the existence of abstract
mathematical objects.
To assess this argument we need to know what makes a theory the best theory. But
before we turn to this question, it is noteworthy that so far the so-called
indispensability argument makes no mention of indispensability. Instead it uses the
locution of being quantified over by our best suitably regimented theories. The notion
of indispensability owes its origins to Quine, who in essence thinks that entities are
indispensable to theory T, just in case T, suitably regimented, quantifies over those
entities. Suitable regimentation, for Quine, is regimentation into first-order logic; the
idea being that this canonical way to regiment a theory will reveal the ontological
commitments of that theory.
Subsequent defenders of the indispensability argument, like Colyvan, reject this
account of indispensability on the grounds that a theory, construed as a set of
sentences, can be regimented into canonical notation in numerous ways, each with
very different ontological commitments. For Colyvan, an entity is dispensable to a
theory T if there exists a modification of T, resulting in a second theory with exactly
the same observational consequences as T, in which the entity in question is neither
mentioned nor predicted, and the second theory is preferable to the first (Colyvan
1999). The second theory is preferable to the first if it is more theoretically virtuous
than the first. Henceforth I refer to this modification of T as a nominalised version of
T.
While little is said about what makes a theory the best theory, in this paper I
assume, for present purposes, that locutions such as “T is the best theory of domain D
in w” have implicit temporal indexes, so that they ought be read as “T is the best
theory of domain D in w at t”. For now, let us suppose that this means something like:
given all of the empirical data available to agents in w at t, T is the most empirically
adequate theory at t, and T is sufficiently empirically adequate that given the evidence
available at t, agents are rational to conclude that T is true, or approximately true. But
now suppose there are empirically equivalent competitor theories of D, as for instance
in the case of competitor quantum mechanical theories. Then it seems plausible to
suppose that the best theory is the one that is the most theoretically virtuous. If the
theories are equally virtuous, then I assume that there is no unique best theory (at t)
4
and that there are equal best theories. In general where equal best theories have
different posits, it will be unclear which posits agents ought to believe to exist, and
likely the scientific realist needs to say more about this issue. This may not, however,
worry the mathematical Platonist unduly. Premise (1) of the argument below states
that we should be committed to all and only the entities that are indispensable to our
best theories. If there are equal best quantum mechanical theories it might be unclear
just which entities we ought to be committed to, but if all of the theories quantify over
mathematical objects then Platonism is vindicated regardless.
The real question is how to make sense of premise (1) in the light of the fact that
for any theory T that posits the existence of mathematical objects, there exists an
empirically equivalent nominalised version of T, T* that does not. For we know that
if the posits of T are dispensable, then T is not the best theory, since T* is
theoretically more virtuous. Thus claiming that we should be committed to the
indispensable posits of our best theory is potentially misleading, since it suggests that
there could be something that counts as a best or equal best theory, whose posits are
dispensable. Instead, we should really read (1) as claiming that where we have
empirically equivalent theories T and T* and where T* is a nominalised version of T,
we should be committed to the posits of T either where T is more theoretically
virtuous than T*, (and hence where T is straightforwardly the unique best theory of
the pair) or, if they are equally theoretically virtuous and hence equally “best”
theories, we should be committed to the indispensable posits of T.9 That is, we should
be committed to the posits of T even where T and T* are equally good theories, since
the nominalised theory T* is not preferable to T. In that case it is not that we are
committed to the indispensable posits of the unique best theory, but rather, that we are
committed to the posits of one of the equally best theories where those posits are
indispensable to that theory. This way of understanding the relationship between
indispensability and theoretical virtue will be revisited later in the paper; for now I
presuppose the account that is offered by the proponent of the indispensability
argument. Henceforth then, use of the phrase “the best theory” should be read as “the
best, or equal best theory”. Use of the phrase “the unique best theory” will be used
where the intention is to restrict the class of theories to the unique best.
9
The nominalist might dispute this latter half of the disjunct on the grounds that it is difficult to see
why to one ought to be committed to the existence of the posits of a non-nominalised theory if a
nominalised version of the theory is equally virtuous. I return to this issue later in the paper.
5
Notably then, what it is for a theory to be the best theory is defined in terms of
what evidence is available to agents at a time, and that makes good sense since the
most recent versions of the indispensability argument yield conclusions not about to
what entities theories are committed (as originally found in Quine) but rather, to what
entities agents ought to be committed. The thought is that agents ought to accept their
best theories as true or approximately true, and on that basis should believe that the
entities quantified over in those theories exist. That is, agents’ ontological
commitments, at any time, ought to match their views about which scientific theories
are true or approximately true, and hence which scientific theories are the best at that
time.
Subsequently the notion of the best theory simpliciter will be introduced, but for
now this temporally indexed notion will suffice to get a sense of how the argument is
supposed to proceed. Then substituting in this new understanding of indispensability
gives us the amended indispensability argument:
The Amended Indispensability Argument (AIS):
1. We should be committed to all and only the entities that are indispensable to our
best theories.
2. Mathematical objects are indispensable to our best theories.
3. Necessarily, if O is a mathematical object then O is an abstract object.
4. Conclusion: Therefore we should be committed to the existence of abstract
mathematical objects.
The modal force of premise 3 is supported by the fact that if they succeed, the
arguments for the conclusion that mathematical objects are abstract show that
mathematical objects could not be anything other than abstracta, not merely that in
fact, if there are any mathematical objects, they are abstracta. Henceforth I assume
that (3) is true, and therefore it will figure as a suppressed premise in all the
arguments that follow.
The conclusion to the AIS still does not tell us anything about the modal status
we should accord the existence of mathematical objects. We could extract a modal
conclusion from the AIS if modal claims were part of the relevant best theories.
6
Suppose, for instance, it was part of our best scientific theory about the laws of nature
that those laws hold of necessity. And suppose that mathematical objects are
indispensable to this theory. Then we should conclude that mathematical objects exist
and necessarily so.
Even if we include not only scientific theories but also folk theories in those we
are allowed to consider, it seems unlikely that modal claims will themselves be part of
our best theories.10 While scientists typically think that their claims about our world
are contingently true, it is not clear that this is ever, let alone often, part of any
particular scientific theory. Likewise, while the folk probably think that it is
contingent that tables exist, it is not obvious that this is any part of folk theory about
tables or macro objects more generally.
Since philosophers of maths want to draw modal conclusions there is an
obvious strategy to adopt according to which we should accord to entities that are
indispensable to our best theories whatever modal status is explicitly afforded them
by the theory in which they figure, if there is any, or is tacitly afforded them by the
entailments of the modal commitments of that theory if there are any. Otherwise we
should consider, on a case-by-case basis, independent reasons to hold that the entities
that are indispensable to our best theory have a particular modal status. If the reasons
in favour of one modal status outweigh those in favour of the other, then we should
add an additional premise to the indispensability argument that stipulates that the
entities in question, if they exist, have that modal status. Call this the Independent
Reasons Strategy:
Independent Reasons Strategy: (i) If it is part of best theory T that the entity E
over which T quantifies has a particular modal status, then that is the status we
should afford E. (ii) Otherwise, if T has certain modal commitments that entail
that E has a certain modal status, then this is the modal status we should afford
E. (iii) Otherwise, afford E the modal status for which independent reasons
provide the strongest support.
As we will see, proponents of the indispensability argument generally employ the
10
At least, this is a plausible claim about the way scientist’s themselves view their first-order claims.
Within the philosophy of science first-order scientific claims are sometimes viewed as disguised
counterfactual conditionals. Even if first-order claims are disguised conditionals though, that claim is
not itself part of scientific theory. So I think we should still see the modal status of scientific claims not
as falling out of just the scientific claims, but additional meta-claims about how we should interpret
those scientific claims.
7
independent reasons strategy. There are two ways we might tease out this strategy,
depending on which entities we focus on. Suppose we focus on the non-mathematical
entities quantified over by our best theory, and ask ourselves whether we have
independent reasons to suppose that those entities exist contingently. With some
notable exceptions11 most philosophers think that our best scientific theories are, if
true, contingently true. So it would be natural to suppose that the entities quantified
over by those theories exist contingently. Viewed from this perspective, it seems
natural to construct what I will call the first version of the modalised indispensability
argument, or V1 for short. As I show later, both Colyvan and Field use an argument
somewhat like this to argue for contingentism. A more sophisticated improved
version of the argument is considered subsequently, but this simple version is useful
as a starting point, and as a counterpoint to V2, below.
First version of the modalised indispensability Argument (V1):
1. We should be committed to all and only the entities that are indispensable to our
best theories.
2. We have independent reason to believe that our best theories are contingently
true.
3. If our best theories are contingently true, then we have reason to believe that
the entities that are indispensable to those theories exist contingently.
4. Mathematical objects are indispensable to our best theories.
5. Conclusion: Therefore we should be committed to the contingent existence of
mathematical objects.
I return to consider V1 shortly. First, let us introduce a second version of the
modalised indispensability argument. This version concentrates not on independent
reasons we have to suppose that the best theories themselves have a particular
modality, but rather, on independent reasons we have to suppose that the existence of
mathematical objects has a particular modality. Arguably, we have reason to think
that whether mathematical objects exist or not, they do so of necessity. Notice that
11
Swoyer 1982; Bird 2005; Shoemaker 1998.
8
this is not the much stronger claim, made by Hale and Wright, that it is a conceptual
necessity that numbers exist.12 It is the much weaker claim that mathematical objects
either exist or fail to exist of necessity. If we accept this claim we get the following
version of the modalised indispensability argument, V2 for short.
Second version of the modalised indispensability argument (V2):
1. We should be committed to all and only the entities that are indispensable to
our best theories.
2. Mathematical objects are indispensable to our best theories.
3. If mathematical objects exist, then they exist necessarily.
4. Conclusion: Therefore we should be committed to the necessary existence of
mathematical objects.
The interesting question then becomes, why accept V1 rather than V2?
2.1 From Indispensability to Contingentism
In broad brushstrokes, Field and Colyvan are contingentists about the existence of
mathematical objects because they endorse something like V1. I say they endorse
something like V1, because V1 is unsound. Even if our best theories are contingently
true, that is not reason to think that all of the entities quantified over by that theory
exist contingently. It is consistent with the contingency of our best theories that
mathematical objects exist of necessity.
Indeed we would have grounds to suppose that mathematical objects exist of
necessity if we had reason to think that for every world w, mathematical objects are
indispensable to some best theory of w. But to many this seems unlikely. Colyvan
and Field, for instance, think there are worlds where mathematical objects are
dispensable to the best theories of that world. Field’s attempts to nominalise scientific
theories are plausibly thought to show that such strategies succeed for Newtonian
physics. Thus a world in which Newtonian physics is the best theory is likely one in
12
Hale (1987); Wright (1983); (Hale and Wright 1992).
9
which mathematical objects are dispensable to the best theory of that world.13
The conclusion to the indispensability argument tells us what sorts of entities
we ought to be ontologically committed to: that is, which entities we ought to believe
exist. I ought to be ontologically committed, at t, to all and only the entities that are
indispensable to the best theories of my world at t. As we see below, however, in
order to render V1 sound we need to be able to talk about an agent’s ontological
commitments with respect to counterfactual worlds. To do so, we need to return to the
idea of a theory T being the best theory of a domain in a world simpliciter. Here the
thought is that T is the best theory of a domain in world w simpliciter iff given that an
agent knows all of the empirical facts about w, the agent thereby knows that T is
empirically adequate, and that any competitor theory that is empirically equivalent to
T is not more theoretically virtuous than T.
Let us call any world in which mathematical objects are indispensable to the
best theories of that world a mathematically indispensable world, and any world in
which mathematical objects are dispensable to the best theories of that world a
mathematically dispensable world. The key contingentist intuition is that in a
mathematically indispensable world I have reason to believe that mathematical
objects exist. In a mathematically dispensable world I have reason to believe that
mathematical objects do not exist. Since I have good reason to suppose that some
worlds
are
mathematically dispensable
worlds
and
others
mathematically
indispensable worlds, I have reason to believe that the modal status of the existence of
mathematical objects is contingent. Here is the appropriately amended version of V1.
Second version of the modalised indispensability Argument (V1*):
13
In fact, Field’s motivation for endorsing contingent nominalism goes beyond this reasoning. Field
holds that a mathematical theory M is conservative with respect to a nominalised scientific theory S iff
any nominalistic claim that is a consequence of M and S is also a consequence of S. Conservativeness
entails consistency, where for Field, consistency is defined in terms of a primitive modal notion of
possibility, such that a consistent theory is one that is (primitively) possibly true. So if mathematics is
conservative, then it is consistent. But if it is consistent, in this sense, then the various mathematical
axioms are possibly true. On at least one reading of Field’s notion of primitive possibility, it follows
that if these axioms are possibly true they are therefore true in some world. In the world where the
axioms are true, mathematics is true, and hence contingently true. Thus in our world, assuming that
mathematical theory is conservative but false, it is contingently false. (Field 1980). Field might resist
this conclusion, however, by denying that what is possibly true, in his sense, is thereby true in some
possible world. Perhaps the sphere of possible worlds includes only the metaphysically possible worlds
and not the merely logically possible worlds (in Field’s sense). Then there need be no world in which
the relevant axioms are true, and contingentism is not entailed by the view.
10
1. Relative to any world w I have reason to be committed to the existence of all
and only the entities that are indispensable to the best theory of w.
2. We have independent reason to believe that different theories will be the best
ones at different worlds.
3. We have independent reason to believe that for some world, w*, mathematical
objects will not be indispensable to the best theory of that world.
4. Therefore we have reason to believe that mathematical objects fail to exist in
w*.
5. We have independent reason to believe that for some world, w’, mathematical
objects will be indispensable to the best theory of that world.
6. Therefore we have reason to believe that mathematical objects exist in w’.
7. Conclusion: Therefore we should be committed to the contingent existence of
mathematical objects.14
3. Tensions between Indispensability and Contingentism
Let us consider the argument from indispensability to contingentism more carefully.
Indispensability is supposed to be an epistemic guide to when we should believe that
mathematical objects exist. If there existed either a constitutive or an appropriate
causal connection between the existence of mathematical objects and their being
indispensable to our best theories, then indispensability would be an infallible guide
14
One does not find the argument put this explicitly in Colyvan or Field, though this is clearly what
each has in mind. In effect Colyvan notes that the discovery that mathematical objects exist actually is
an empirical discovery about the quantifications of our best scientific theories, conceding that if our
world had been a Newtonian one then Platonism would have been falsified. He writes “It is generally
believed that empirical science provides us with propositions that are a posteriori, contingent and
revisable in the light of empirical evidence. Mathematical propositions, on the other hand, are generally
believed to be a priori, necessary and unrevisable in the light of empirical evidence. But
indispensability theory tells us that mathematical knowledge is in the same epistemic boat as empirical
knowledge. Mathematical statements such as are known to be true by the role they play in our best
scientific theories—in other words a posteriori…..Suppose that Hartry Field has completed the
nominalisation of Newtonian mechanics but that he and his successors repeatedly fail to nominalise
general relativity. Let's also suppose that this failure gives us good reason to believe that general
relativity cannot be nominalised. From this we conclude that mathematical entities are indispensable to
general relativity, but not to Newtonian mechanics. In this setting, then, can we imagine an experiment
to test the hypothesis that there are natural numbers?” (p 120-121). In essence Colyvan claims here that
the discovery that we are in a Newtonian world would be the discovery that actually there are no
numbers. In personal communication Colyvan notes that he has never explicitly set out the argument
for contingentism via the indispensability argument, but that the argument he and Field have in mind is
indeed that expressed by V1*. He suggests that where he comes closest to enunciating this argument is
in Colyvan (1998 pp115-119); Colyvan (2000 pp 87-91 and 91-92) and Colyvan (2007 pp 109-122).
11
to the existence of such objects. But neither of these is the relationship that defenders
of the indispensability argument intend to invoke between indispensability and
ontology. One might endorse a sort of minimalist Platonism that takes very seriously
the Quinean dictum that to be is to be the value of a bound variable, thus supposing
that all it is to be an abstracta is to be indispensable to some best theory and to fail to
be concrete. Then there would be a constitutive link between being indispensable and
existing. I set aside this minimalist Platonism, however, since Colyvan, and to a lesser
extent Field, have a less minimal conception of mathematical abstracta in mind.15
Defenders of contingentism need to say something about the link between
indispensability and ontology. The most general concern is why X’s being an
indispensable posit of one’s best theory is evidence that X’s exist, given that there is
an empirically equivalent nominalised version of that theory that does not posit Xs.
Moreover, one might worry that mathematical contingentism is an uneasy partner for
a defender of the indispensability argument. For one might wonder whether if
mathematical objects exist contingently, and if there is no constitutive or causal
connection between their existence and their being indispensable to our best theories,
then for every mathematically indispensable world, there is a physically
indistinguishable world in which mathematical objects fail to exist. Call the following
claim the matching claim: for any world, w, in which mathematical objects exist and
are indispensable to the best theory of w, there exists a physically indistinguishable
world, w*, in which mathematical objects fail to exist.
One might find the matching claim very plausible, since the fact that
mathematical objects are indispensable to the best theory of w does not entail, cause,
or constitute, their existence at w: it is just evidence that they exist. Thus one might
expect there to be a world just like w, but in which mathematical objects fail to exist.
But if the matching claim is true, one might further wonder whether indispensability
is really a good guide to ontology. The thought is this.
Suppose logical space can be exhaustively divided into four disjoint sets of
worlds. Two of these sets contain, between them, all of the mathematically
indispensable worlds. Remember, an object’s being indispensable to the best theory T
of a domain in a world does not entail that that object exists in w, since T’s being the
best theory in w does not entail its being true. Then let us suppose that one of these
15
For more discussion of this see Author (2009)
12
sets contains all of the mathematically indispensable worlds containing mathematical
objects, and the other set contains all of the mathematically indispensable worlds that
lack mathematical objects (since we know that some such worlds exist). Now suppose
there is a one to one correspondence between the members of the sets such that each
world in one set is mapped to a physically identical world in the other set. The other
two sets contain all of the mathematically dispensable worlds. One set contains all of
the dispensable worlds containing mathematical objects, the other set contains all of
the dispensable worlds that lack mathematical objects. Now suppose there is a one to
one correspondence between the members of these two sets such that each world in
one set is mapped to a physically identical world in the other set.
If this is a way that logical space can be partitioned, then by some sort of
principle of indifference it seems plausible that no agent should give any more than
50% credence to there existing mathematical objects in her world, regardless of
whether it is a mathematically dispensable or indispensable world. Call this the
epistemic objection. If the epistemic objection succeeds, it undermines premise 1 of
V1*, namely that relative to any world, w, an agent has reason to be committed to the
existence of all and only the entities that are indispensable to the best theory of w. For
it undermines any reason I have to think that indispensability is a guide to ontology.
The defender of mathematical contingentism via the indispensability argument
needs to resist the epistemic objection. She need not reject the matching claim, but
she does need to give us reason to suppose that logical space cannot be partitioned in
the way described. To put it intuitively, she needs to show that amongst the set of
worlds in which mathematical objects are indispensable, more of those worlds are
ones in which mathematical objects exist than fail to exist. Or to put it another way,
she needs to show that where there are competitor best theories T and T*, and where
T* is a nominalised version of T and where E’s are indispensable to T, T is more
likely to be true than is T* and hence it is more likely than not that E’s exist. If that
were not so, then indispensability would not be a good guide to ontology. Given that
she rejects causal and constitutive links between indispensability and ontology and
instead claims that the link is evidential, there appear to be two broad options for
explaining why indispensability is a guide to ontology, and hence why the epistemic
objection fails. Each approach links indispensability to some other feature, and
claims that that feature grounds the evidential relationship between ontology and
indispensability.
13
The first approach links indispensability to theoretical virtues. The argument
might proceed as follows:
Argument from Theoretical Virtues
1. Suppose T is the best theory of domain D, and T* is a nominalised version of T.
2. T and T* are empirically equivalent (this falls our of what it means for T* to be
a nominalisation of T).
3. If the objects quantified over by T are indispensable, then T is more
theoretically virtuous than T*.
4. The objects quantified over by T are indispensable.
5. So T is more theoretically virtuous than T*.
6. The theoretical virtues are a guide to truth.
7. So T is more likely to be true than T*.
8. Therefore we should be committed to the existence of the objects quantified
over by T.
This is a version of an indispensability argument that furnishes a Platonist conclusion,
but which also explicates the evidential link between indispensability and ontology in
a way that circumvents the epistemic objection.
As it stands, the problems lie with premises (3) and (6). On Colyvan’s definition
of indispensability, (3) is false. The Xs are dispensable to T iff there is a nominalised
version of T, T*, and T* is more theoretically virtuous than T. That means that the Xs
are indispensable to T iff there exists a nominalised version of T, T*, and T* is not
more theoretically virtuous than T. Thus there is no guarantee that mathematical
objects being indispensable to T entails that T is more theoretically virtuous than T*.
We could choose to redefine indispensability so that the X’s are indispensable
to T iff T is more theoretically virtuous than T*. This change of definition sets the bar
higher in terms of when the Xs will turn out to be indispensable to T. If nominalising
T renders T* more ontologically parsimonious, but also less simple, it might turn out
that overall the virtues come out the same for both theories. But that will not vindicate
the indispensability of the Xs as it would have under Colyvan’s preferred definition.
Nevertheless, let us henceforth redefine indispensability so that the Xs are
indispensable to T iff the nominalised version of T, T*, is less virtuous than T. This
guarantees the truth of (3) in the argument above.
14
The problem for premise (6) is its ambiguity. On one reading, the theoretical
virtues are a guide to truth in every world. That is not to say that the virtues are an
infallible guide to truth in every world. Rather, it is say that (at the very least) in any
world w in which there are competitor theories T and T* that are empirically
equivalent, T’s being more theoretically virtuous than T* is a reason to think that T is
more likely to be true than is T*. On a different reading, the theoretical virtues are, as
a matter of fact in the actual world, a good guide to truth. Thus in the actual world, A,
in which there are competitor theories T and T* that are empirically equivalent, T’s
being more theoretically virtuous than T* is a reason to think that T is more likely to
be true than T*.
The first reading is true only if for any arbitrary theory T and its nominalised
counterpart T*, and for any set of possible worlds W in which either T or T* is the
best theory of each of world in W, if T is more theoretically virtuous than T* then it is
true in more of the worlds in W than is T*. This is consistent with the truth of the
matching claim. But it is not consistent with the partitioning of logical space required
by the epistemic objection. Instead, more of the worlds amongst the mathematically
indispensable worlds will be ones in which there exist mathematical objects than in
which there do not. This is because the indispensability of the Xs to T is constitutively
linked to T’s being more theoretically virtuous than nominalised theory T*, and in
every world, a theory’s being more virtuous than a competitor theory is linked to its
being more likely to be true.
This gets the right result for the contingentist. But it relies on a controversial
interpretation of premise (6) according to which across the space of worlds, virtue is
correlated with truth. To be sure, if possibility space were like this it would explain
why virtue is correlated with truth actually. But given that the virtues are generally
held to be actual epistemic guides to truth, and again, not causally or constitutively
connected to them, it is hard to see why the space of worlds ought be like this. Why
think that the more virtuous of a pair of empirically equivalent theories will be true in
a greater proportion of the worlds in which that pair of theories are rivals? Arguably,
at best we have reason to think that the virtues are a guide to truth in our world. We
have no reason to suppose that virtue is correlated with truth in such a manner across
other worlds.
So let us suppose we read premise (6) as the claim that in the actual world, A, in
which there are competitor theories T and T* that are empirically equivalent, T’s
15
being more theoretically virtuous than T* is a reason to think that T is more likely
than T* to be true. This too is consistent with the matching claim. It is also consistent
with the mapping function posited by the epistemic objection. Suppose that actually,
mathematical objects are indispensable and we are therefore rationally obliged to hold
that they exist. On this view of the link between the virtues, indispensability, and
ontology, we know that there are worlds physically indistinguishable to ours in which
there exist no mathematical objects: that follows from the truth of the matching claim.
But once we come to see that we only have reason to think that the virtues are a guide
to truth in the actual world, we are left with no reason to hold that counterfactual
worlds in which T is more virtuous than T* are likely to be ones in which T is true. So
we have no reason to suppose that in counterfactual worlds, mathematical objects’
being indispensable to the best theories of those worlds is any grounds to conclude
that mathematical objects exist in those worlds. So we cannot rule out that logical
space can be partitioned in just the way that the epistemic objection suggests. Once
we admit that, it seems we have undermined our reason for thinking that we really are
in a world in which the virtues are a good guide to truth, since we know that there are
just as many worlds physically and evidentially like ours in which the virtues appear
to be good guides to truth, and in which mathematical objects are indispensable to the
best theory of the world, but in which mathematical objects fail to exist, as there are
worlds in which such objects do exist.
On the second interpretation of (6), the epistemic worry about indispensability
re-emerges. On the first reading it does not, but that is a controversial reading that
takes us well beyond anything that is supported by noting that amongst empirically
testable theories so far in our world, those that have been virtuous have been more
likely to be true or approximately true.
There is a second option available for linking indispensability to ontology that
does not require redefining the notion of indispensability. Instead, it might be that the
existence of mathematical objects at a world is not independent of the contingent
features of that world in virtue of which mathematical objects are indispensable to the
best theory of that world.16 Here is one way of thinking about this proposal. Perhaps
the existence of mathematical objects in a world is metaphysically grounded in the
particular physical structure of that world. Thus, for instance, perhaps actual
16
I thank the referee for pointing this out.
16
mathematical objects are grounded in the general relativistic space-time structure of
our world. Then one can understand the relationship between indispensability and
ontology as being mediated by metaphysical grounding. Indispensability is a good
guide to ontology because X’s being indispensable to the best scientific theory of
domain D (in w) is evidence that features of domain D ground the Xs.
This raises two possibilities. The first is that grounding relations like this are
necessary. For these purposes I will say that grounding relations are necessary iff for
any two worlds w and w* alike with respect to their distribution of fundamental
properties, those worlds are alike with respect to their grounding relations. I will say
that grounding relations are contingent iff for some pair of worlds w and w* alike with
respect to their distribution of fundamental properties, w and w* are not alike with
respect to their grounding relations. If grounding relations are contingent, then the
matching claim is true. There are pairs of worlds that are physically indistinguishable
but unalike with respect to the existence of mathematical objects. These are pairs of
worlds with the same distribution of fundamental properties, but different grounding
relations and hence different grounded objects. Suppose that in w the distribution of
fundamental properties P plus grounding relation R grounds the existence of
mathematical objects M. Consider the set S of worlds with the same distribution of
fundamental properties as w. The worlds in that set that contain grounding relation R
contain mathematical objects, and those without R lack mathematical objects. The
worlds in S are physically indistinguishable, so whichever theories are the best
theories of one of the worlds in S will be the best theories for all of the worlds in S.
Consider a particular theory T, that is the best theory of some domain D in each of the
worlds in S, and suppose that mathematical objects are indispensable to T. Being
indispensable to T is only good evidence for the existence of mathematical objects if
more of the worlds in S have mathematical objects than lack them. Otherwise
mathematical objects would be indispensable to the best theories of all of the worlds
in S, but in most of those worlds no mathematical objects exist. Then the epistemic
objection, or a very close cousin, would have force.
But if grounding relations are contingent, there is no particular reason to
suppose that grounding relation R is present in more than 50% of the worlds in S, and
hence no reason to suppose that mathematical objects exist in more than 50% of the
mathematically indispensable worlds. If we had an account that linked
indispensability to the presence of grounding relation R, then we might be able to
17
explain why R is more prevalent among the worlds in S, and thus why more of those
worlds have mathematical objects than lack them. For if grounding relations are
contingent, there is no sense in which it is the mere having of a particular physical
structure in a world necessitates the existence of mathematical objects in that world.
A further feature is necessary: a particular contingent grounding relation. But we have
no reason to think that where X’s are indispensable to theory T in w, it is more likely
than not that a particular grounding relation R obtains in w. So we cannot rule out that
logical space can be divided up in the way suggested by the epistemic objection.
If grounding relations are necessary, on the other hand, then the matching claim
is false: any pair of physically indistinguishable worlds will be alike with respect to
the existence, or lack thereof, of mathematical objects. Thus the epistemic objection
will fail. More of the mathematically indispensable worlds will be ones with
mathematical objects than without them. So if the mathematical contingentist is to
avoid epistemic problems, this is certainly one way to do so. There are two things to
note about the approach. First, it requires commitment to a fairly meaty view in the
metaphysics of grounding and dependence that goes well beyond a kind of Quinean
Platonism. Second, one might wonder how the existence of abstract objects could be
grounded in the physical structure of the world. It is relatively clear how, for instance,
chairs and tables are grounded in the existence of certain arrangements of simple
objects plus the composition relation. Not so in this case.
So there are ways for the contingentist Platonist who endorses the
indispensability argument to avoid the epistemic objection. But one of these requires
that there is a very strong connection between theoretical virtues and truth across
worlds, and the other requires that abstract objects are grounded in physical structures
of the world, and that these structures necessitate the existence of those objects via
necessitating the existence of certain grounding relations in worlds with that structure.
Other approaches to linking indispensability to ontology that do not wheel out some
strong connections between indispensability, some relevant second feature, and
ontology, cannot adequately respond to the epistemic objection. Given this, the
combination of the indispensability argument and contingentism is at least in some
tension. It is worth considering, therefore, whether given that one is antecedently
committed to holding that indispensability is a good guide to the existence or not of
mathematical objects, one can secure a more modally robust necessitarian conclusion
via the indispensability argument. I consider this issue in section 5. First, however, I
18
examine another argument that attempts to link the indispensability argument to
contingentism.
4. Mathematical Objects and Mathematical Statements
Unsurprisingly, there are strong links between the truth and modal status of
mathematical statements, and the existence and modal status of mathematical objects.
It is a widely accepted semantic claim that mathematical statements are true only if
there exist mathematical objects. This claim is in part motivated by the idea that we
should provide similar truth conditions for similar sentences. Since sentences that
quantify over mathematical objects are similar to sentences in our scientific theories
that quantify over scientific objects, we should interpret mathematical language on a
par with scientific language. We should read its quantificational claims to imply a
literal commitment to the entities quantified over.1718 Thus mathematical statements
are true only if the entities that those statements quantify over—mathematical
objects—exist. As we have seen, by far the most common view is that if mathematical
objects exist they are abstracta. This yields the following widely accepted semantic
claim:
(1) Mathematical statements are true only if there exist abstract mathematical
objects.
17
Benacerraf (1973) p 404.
One might think that talk of truth conditions is unnecessary, and we need only appeal to the Tschema. If “there exist infinitely many prime numbers” is true just in case there exist infinitely many
prime numbers, then we have good reason to think that there are infinitely many prime numbers. More
generally we have reason to think that mathematical statements are true only if mathematical objects
exist. Indeed, minimalists like Schiffer argue for the existence of mathematical objects on precisely
these grounds (Schiffer 1996). Minimalists about truth, then, will have no need (and indeed no desire!)
to talk about truth conditions to motivate the claim that mathematical statements are true only if
mathematical objects exist. Those who are not minimalists, however, may well think that the Tschema and the truth of some mathematical statement such as that just mentioned does not give one
reason to think that there exist abstract objects of the kind posited by Platonists. Consider: it is
consistent with the T-schema that the truth conditions for mathematical statements are various facts
about what mathematicians do in the mathematical discourse, or as some might say, in the
mathematical fiction rather than in virtue of the existence of abstract objects. Given these truth
conditions, the sentence “there exist infinitely many prime numbers” is true, and it is true iff there exist
infinitely many prime numbers. But it turns out that what it is for there to exist infinitely many prime
numbers turns out to be a bunch of facts about the fiction of maths rather than something about abstract
objects. So maximalists about truth might accept the T-schema but think that the truth of mathematical
statements that quantify over mathematical objects does not thereby commit them to Platonism as we
know it.
18
19
There are two other frequently accepted claims that reflect the near orthodoxy that
mathematical statements are paradigm instances of the necessary a priori. Namely:
(2) Mathematical statements are either necessarily true or necessarily false.
(3) Mathematical statements are true a priori.
One cannot endorse both (1) and (2) and contingentism about mathematical objects. If
mathematical objects exist in some worlds and not others, then given (1),
mathematical statements will be true in some worlds and not in others and hence will
be contingent. Necessitarians see the obvious truth of (1) and (2) as one reason to
reject contingentism. Contingentists like Colyvan and Field, who accept (1), are
therefore left rejecting (2). Colyvan offers an account of why (2) appears to be true,
and a way of mitigating the implausibility of (2) being false. Two questions naturally
present themselves: first, is Colyvan’s account of the apparent truth of (2) successful
and second, since it is agreed that (2) appears to be true, is there an independent
motivation for rejecting (2)? If the answer to the first question is flawed then we have
no explanation for the appearance of necessity and additional weight is placed on any
argument purporting to offer independent grounds for rejecting (2). I consider each
question in turn.
4.1 The appearance of necessity
Given the intuitive appeal of (2), the claim that mathematical statements are either
necessarily true or necessarily false and (3), the claim that mathematical statements
are true a priori, we are owed some story about why we have been so misled about
these statements. The story Colyvan offers19 borrows from the nominalist framework,
in distinguishing what is true within, or according to, some theory, and what is true
simpliciter. We come to know of the existence of mathematical objects a posteriori
through investigation of the quantifications of our best theory. But this is consistent
with statements within mathematical theory being what we might think of as pseudo a
priori, in the sense that understanding mathematical theory is sufficient to believe
such statements to be true or false. Relative to mathematical theory mathematical
statements are a priori because no empirical investigation is needed to determine
19
Colyvan (2001).
20
whether, according to mathematical theory, some statement will be true or false. But
mathematical statements are not a priori simpliciter, since whether such statements
are true depends on whether mathematical objects exist, and that can only be
determined a posteriori by examination of the (indispensable) ontological
commitments of our best theories. What we can know a priori is a conditional claim:
if mathematical objects exist, then certain mathematical statements—the ones we can
come to believe a priori from examination of mathematical theory—are true. Claims
made from within mathematical theory appear to be straightforwardly a priori
because from within the context of that theory there is a presumption that the
antecedent of the conditional is true. In a sense then, Colyvan’s approach mirrors
some of the nominalistic paraphrase strategies of old20 in that it appeals to
conditionals such as “if there were numbers then….” where the ellipses are filled in
by claims that are true within mathematical theory. But instead of reading the
conditional as a paraphrase that is supposed to capture what we mean by
mathematical statements, the strategy preserves a Platonist semantics for
mathematical statements and adopts the conditional claim as a way of explaining the
appearance of a priority.
Colyvan has more to say about the apparent a priori nature of mathematical
knowledge than he does about the apparently necessitarian modal status of its
statements. But we can see how an analogous move could be made. Mathematical
statements are not necessarily true. But there is a necessarily true conditional claim,
namely, that if there are any mathematical objects, then the same set of mathematical
statements will be true in all the worlds in which those objects exist. Specifically, if
there is a set of worlds W in which there exist mathematical objects, then for any
world w in W, and for any mathematical statement S that is true in w, S is true in every
world in W.
On at least some standard semantics this means that statements like 2 + 2 = 5
are necessarily false. They are straightforwardly false in worlds with mathematical
objects, and they are also false in worlds in which there are no mathematical objects,
on the assumption that sentences with singular terms that lack a referent are false.21
20
For instance Putnam (1967), and Hellman (1989).
One might deny this, and instead hold that such sentences lack a truth-value. This would render the
semantics somewhat more disjunctive, and on those grounds one might prefer the more unified
semantics suggested above. However, if one were committed to holding that sentences with singular
terms that fail to refer lack a truth value one could explain the appearance of necessity by noting that
21
21
Moreover, 2 + 2 = 4 is true in all worlds with mathematical objects. Indeed, the only
worlds in which it is false are worlds without mathematical objects, not worlds where
2 + 2 equals something other than 4. We could express this by saying that
mathematical statements are mathematically necessary, where a statement S is
mathematically necessary iff it is true in every mathematically possible world.
Intuitively, the mathematically possible worlds are the worlds in which there exist
mathematical objects—the worlds in W. Then the contingentist could embrace
something like Field’s fictionalism about the mathematically impossible worlds—the
worlds that lack mathematical objects. In those worlds, strictly speaking all
mathematical statements that take the form of positive existential claims are false.22
But some mathematical statements are true in the fiction, or in the story of maths,
(such as 2 + 2 = 4) and others are not true in the story of maths (such as 2 + 2 = 5).
Indeed, the contingentist has a nice way of specifying the semantics of the “in the
story of maths” operator. She can say that for any mathematical statement S, S is true
in the story of maths iff S is true in every mathematically possible world. Then the
appearance of necessity is explained by the fact that for every mathematical statement
S, (a) S is either true in every mathematically possible world or false in every
mathematically possible world, and (b) S is either true in the story of maths in every
mathematically impossible world or false in the story of maths in every
mathematically impossible world. Since if S is true in every mathematically possible
world it follows that it is true in the story of maths, we know that for any
mathematical statement, S, if S is true in every mathematically possible world then it
is true in the story of maths in every mathematically impossible world and mutatis
mutandis for a mathematical statement that is false in every mathematically possible
world. This explains the appearance of necessity.23
sentences like 2+2=5 are false in all the mathematically possible worlds, and lack a truth value
otherwise, and that that is sufficient to explain why the sentence seems to be necessarily false.
22
Negative existentials like “there is no largest prime number” come out true, but true because there
are no numbers and hence no prime numbers, not because there are prime numbers, but no largest
prime.
23
There are a couple of other options that the contingentist might consider. The first is to say that
mathematical statements are necessarily true, even though there are worlds in which there are no
mathematical objects. The idea is that our mathematical terms refer even in worlds that lack the
relevant entities, because our terms can refer to abstracta in any world. Then in a world without
numbers the statement “2+2=4” comes out true and necessarily so, because in every world in which
there are 2s, the statement is true. A second option is to accept that there is a mere appearance of
necessity, and to suggest that this is due to the necessity of some other claim. In particular, the
contingentist might suggest that all mathematical statements ought to be read as hidden conditionals of
the form: If there exist mathematical objects, then S. Such conditionals are necessarily true. The source
22
But if the set of mathematically possible worlds are to play the role just
assigned them, the same set of mathematical objects must exist in all those worlds.
Otherwise there will be some mathematically possible worlds at which a particular
mathematical statement S is true, and others at which S is false. Indeed, the
contingentist should want to say that something like what Balaguer calls full-blooded
Platonism24 is true at all the mathematically possible worlds, namely, that all possible
mathematical objects exist at those worlds. Since the mathematically possible worlds
represent the totality of worlds at which there exist mathematical objects, it is not very
helpful to talk of “all possible mathematical objects”. Instead, let us think of a
maximal set of mathematical objects as, roughly, the set of mathematical objects we
would need in order for any possible scientific theory to come out as true. Think of
each scientific theory T that quantifies over mathematical objects as having associated
with it a set S of mathematical objects, namely the set of mathematical objects
quantified over by that theory. For every possible scientific theory, there is an
associated set S of mathematical objects. Intuitively, the maximal set MS of
mathematical objects is the union of every set S1…Sn associated with a possible
scientific theory.25
We want every mathematically possible world to contain the maximal set of
mathematical objects. Suppose this were not so.
Then there is a mathematical
statement, S1, such that there is a mathematically possible world in which the
mathematical objects that are the truth-makers for S1 exist, and hence in which S1 is
true, and another mathematically possible world in which S1 is false because the
mathematical objects that are truth-makers for S1 do not exist. In such a world we
would have to say that S1 is true in the story of maths, though it is false simpliciter.
Moreover, we could no longer cash out the idea of “in the story of maths’ by appeal to
which statements are true in all of the mathematically possible worlds, since at least
some statements (like S1) will not be true at all such worlds. But this makes the
semantics of mathematical discourse massively disjunctive even within worlds. In any
of the apparent necessity of mathematical statements lies in us mistakenly forgetting that mathematical
statements are really conditionalised in this way, and hence mistakenly supposing that the consequents
of such conditionals are necessary when in fact they are contingent.
24
Balaguer (1998).
25
I appeal to all possible scientific theories that quantify over mathematical objects, rather than all
possible theories that quantify indispensably over mathematical objects because, first, if contingentism
is true then being indispensable is no guide to whether a mathematical object exists or not, and second,
we want MS to include all possible mathematical objects and there is no guarantee that restricting the
domain in this way would ensure that outcome.
23
particular mathematically possible world, some mathematical statements will be
straightforwardly true or false in virtue of the existence of the relevant mathematical
objects as truth-makers, while other statements will be false, but will be true or false
in the story of mathematics. This is not only pretty unpromising as a semantics of
mathematical statements, but it renders untenable the account of the appearance of
necessity offered earlier.
But then we have another argument against the link between the
indispensability argument and contingentism. For it is difficult to see how the
indispensability argument could furnish reasons to believe that the maximal set of
mathematical objects exist in all of the mathematically possible words. Since different
sets of mathematical objects will be indispensable to the best theories in different
worlds, we should be committed to the existence of different sets of mathematical
objects in different worlds. That requires us to come up with a radically different, and
very messy, semantics for mathematical statements, something that Quinean
naturalists like Colyvan and Field have eschewed. This means that the contingentist
who embraces contingentism in virtue of being committed to the indispensability
argument does not have a good account of why mathematical statements appear to be
necessary. This, in turn, puts additional pressure on the contingentist since in the
absence of a good independent reason to reject necessitarianism and in the further
absence of an explanation of the appearance of necessity, she should surely accept
that mathematical statements are necessary.
4.2 A posteriori necessities
Colyvan does tentatively gesture towards an independent, or semi-independent,
reason to deny the truth of (2). The suggestion is that the truth of (2) is intimately
linked to the truth of (3), and since the indispensability argument renders (3) false, it
implies that (2) is also false. Here is how the argument might proceed. Since it is an a
posteriori matter whether or not mathematical objects are indispensable to our best
theories, the indispensability argument yields an a posteriori conclusion regarding
whether mathematical objects exist. If mathematical statements are true only if there
exist abstract mathematical objects, then mathematical statements will be true a
posteriori. Hence those who endorse the indispensability argument reject (3). But
there is at least a prima facie relationship between a priority and necessity. Unless
24
mathematical statements turn out to be instances of a posteriori necessities, the fact
that they are a posteriori suggests that they will be contingent rather than necessary.
Colyvan and I both hold that the only kind of a posteriori necessity is what I
call post-Kripkean a posteriori necessities. These use a reference fixing description or
a name to rigidify on some actual referent of the term: they combine an a priori claim
about a term being a rigid designator, with a posteriori empirical findings about what
it is, actually, that the term refers to. Given this understanding of a posteriori
necessities, neither Colyvan not I find it plausible that mathematical statements will
be a posteriori necessary. We think that “water is H20” is a posteriori necessary
because the semantics of “water” is that “water is the actual watery stuff and
necessarily so”. Since H20 turns out to be the actual watery stuff, water is necessarily
H20. Watery stuff—stuff that plays the water role—in counterfactual worlds that is
not H20 is therefore not water, but schwater. By analogy, the claim that mathematical
statements are a posteriori necessary requires that we think that there could be
counterfactual abstract objects that play the mathematical objet role—ie the
equivalent of counterfactual watery stuff—but where these objects are not
mathematical objects but schmathematical objects.26 Since this seems crazy, we reject
the idea that mathematical statements function like this.
Not everyone accepts this as an account of the necessary a posteriori. One
might think that there are brute a posteriori necessities, or that the relevant necessities
are not the result of combining an a priori semantic claim with an empirical
discovery, but result from discovering the genuine essential nature of some kind of
object. The claim that mathematical statements are brute a posteriori necessities
strikes me as unappealing, and I am not convinced by the idea that we discover,
empirically, the essential nature of mathematical objects and hence discover some
necessary a posteriori truths. What sort of discovery would that be? Surely it is not
the discovery that it is the essential nature of mathematical objects that they are
abstracta, for not only is that not an empirical discovery, but if we were to discover
that mathematical objects are abstract objects and necessarily so the way we discover
that water is H20 and necessarily so, this would not be the discovery that
mathematical objects exist necessarily nor that mathematical statements are a
26
Mathematical statements then turn out to be necessarily true only because the worlds in which two
lots of the entities that play the schtwo role are not schfour, are worlds where strictly speaking there are
no mathematical objects, not worlds where 2+2 fails to equal four.
25
posteriori necessary. Nevertheless, I have not shown that either of these latter
suggestions could not be right, and therefore have not shown that the contingentist
cannot find any way to understand a posteriori necessities that will do the work she
wants. So though I am sceptical, this is a live option and I leave it open to the
contingentist to show how this will work if that is her preferred option.27
The indispensability argument yields the conclusion that mathematical
statements are a posteriori. Those who embrace the indispensability argument but
find the claim that mathematical statements are a posteriori necessities implausible
therefore have a prima facie reason to think that mathematical objects exist
contingently. The question is whether the necessitarian can reconstrue the
indispensability argument so that it yields an a priori rather than an a posteriori
conclusion.
5. An a priori indispensability argument?
Mathematical statements seem to be modally necessarily. Absent some independent
reason to endorse contingentism, the presence of this robust intuition is prima facie
reason to think that mathematical statements are necessary, particularly in the light of
the failure of contingentist strategies that explain away these intuitions.
In conjunction with plausible semantic claims, if mathematical statements are
modally necessary this entails that mathematical objects exist, or fail to, of necessity.
This is precisely the additional premise we find in V2. So it is tempting to suppose
that once we accept this premise, we can reconstrue the indispensability argument as
an a priori argument for the necessary existence of mathematical objects. Since I
earlier conceded that mathematical statements are not necessary a posteriori, if there
27
One last possibility, suggested by a referee, is that at least some mathematical statements might be a
posteriori necessities because they are theorems that are necessary, but that are humanly unprovable
and therefore not a priori. The right thing to say here, it seems, is that although such theorems are in
fact known a posteriori, they are still a priori because they are knowable a priori. Plausibly, x is
knowable a priori iff an ideal reasoner with infinite cognitive resources can know x a priori. Then just
as I can come to know that 2 +2 =4 a posteriori (by having someone tell me) so too these theorems are
known by humans a posteriori. But they are a priori truths, or, if you prefer, they are necessary a
priori theorems because they are knowable a priori even if no human can know them in this way.
Nevertheless, even if one held that such theorems are an example of a posteriori necessary truths,
appeal to such necessities will not provide a general characterisation of the necessary a posteriori
status of mathematical statements, since a good many mathematical statements are humanly provable.
So I do not think that the contingentist can in general appeal to this sense of the necessary a posteriori.
26
is to be a version of the indispensability argument that yields a necessitarian
conclusion, it had better be an argument that yields an a priori conclusion. To that
end, one might think that in order to obtain a necessitarian conclusion from the
indispensability argument all that matters is that there is a world in which
mathematical objects are indispensable to one’s best theories. It is irrelevant which
world that is. So we do not need to know any empirical facts about the commitments
of our own best theories. We merely need to consider, a priori, a range of possibly
best theories. If mathematical objects are indispensable to one of those theories, then
we have reason to suppose that mathematical objects exist in the world in which that
theory is the best theory. Since we know a priori that mathematical objects exist of
necessity if they exist at all, then we can conclude a priori that mathematical objects
exist of necessity. The a priori version of the argument might proceed as follows:
A priori version of the modalised indispensability argument (V3):
1. If mathematical objects exist, then they exist necessarily.
2. I should take the ontology of any world w to include all and only the entities
that are indispensable to the best theory of w.
3. There is a world, w1 at which mathematical objects are indispensable to the best
theory of w1
4. Therefore we should be committed to the existence of mathematical objects at
w1 .
5. If mathematical objects exist at w1, then they exist at every world.
6. Conclusion: Therefore we should be committed to the necessary existence of
abstract mathematical objects.
The problem with V3 is that an exactly analogous argument will yield the conclusion
that mathematical objects are impossible. We know a priori that there is a world
where mathematical objects are dispensable to the best theories in that world. If we
should be committed to all and only the entities quantified over by our best theories,
then being dispensable is a reason to think that mathematical objects fail to exist in
that world. Then there is an argument exactly like (V3) except that we replace w1 with
a world, w2, in which mathematical objects are dispensable to the best theories in w2,
to yield the conclusion that necessarily, mathematical objects fail to exist.
27
One response is to hold that there is a dilemma for those employing the
indispensability argument. Either contingentism is true or necessitarianism is true. If
contingentism is true, then on most ways of understanding the link between
indispensability and ontology, indispensability turns out to be no guide to the
existence of mathematical objects. At the very least, a lot of work remains in
explicating a plausible connection between the two, and, moreover, a plausible
connection that preserves a plausible mathematical semantics. But if V3 is the best
version of an indispensability argument that yields a necessitarian conclusion, then
since V3 is utterly equivocal about whether mathematical objects necessarily exist or
necessarily fail to, the indispensability argument turns out to be useless for the
necessitarian. Either way, the indispensability argument has significant problems.
I am sympathetic to this conclusion. Still, it would be nice to try and find a
way not to fall on one or other horn. To that end, there are two important things to
consider. First, suppose we are interested in whether some entity E, posited by a
theory T, exists in the actual world w, where we suppose that if E exists, it does so
contingently. Suppose E is indispensable to theory T, which is the best theory of its
particular domain, but that there are other theories (that are not rivals to T) that are the
best theories of their domain, relative to which E is dispensable. This is often true
(perhaps beliefs turn out to be dispensable to the best economic theory, but not to the
best psychological theories) and we conclude from this that entities like E are
indispensable in w, despite there being some best theories relative to which they are
dispensable. That is because the relevant sets of theories are not rivals, but are
theories of different parts or aspects of w. If E is indispensable to our best theories of
some parts or aspects of w, then the fact that E is dispensable to the best theories of
some other parts of aspects of w is irrelevant. We should be committed to all of those
entities that are indispensable to the complete best theory of w. So long as theory T is
part of that complete best theory, then E is indispensable to the complete theory and
we should be committed to the existence of E’s in w.
What is the appropriate domain when asking questions about the existence of
mathematical objects? So far we have implicitly assumed that we ask that question
separately at each world, and thus take each world as the appropriate domain. If a
necessitarian discovers that the best complete theory of a certain world w dispenses
with mathematical objects, then she has reason to conclude that mathematical objects
fail to exist of necessity. When she discovers that the best complete theory of a
28
different world, w*, cannot dispense with mathematical objects, then she has found
reason to think that mathematical objects exist and necessarily so. Hence we find
arguments like that of V3 utterly unhelpful.
If one is a necessitarian, however, it is not clear that one ought take each world
as occupying a different domain and ask, for each of those domains, whether
mathematical objects are dispensable to the complete theory of that domain. Claims
that are logically necessary (whether true of false) are claims about the structure of
logical space. The necessitarian asks whether logical space is constrained to include
only worlds containing mathematical objects, or only worlds that fail to contain those
objects. Epistemologically, the necessitarian nominalist might try to ague that
mathematical objects are logically or conceptually incoherent, while the necessitarian
Platonist might try to argue that mathematical objects are conceptually or logically
necessary. A necessitarian who embraces some kind of indispensability argument,
however, can be thought of as suggesting an entirely different a priori route.
Arguably, at least, such a necessitarian ought to think that what is relevant to
determining whether necessitarian Platonism or nominalism is true, is whether
mathematical objects are indispensable to characterising the totality of possible
worlds in logical space.
By analogy, the standard scientific realist assumes that if the best theory of
domain D of w posits quarks, then we should conclude that quarks exist in w, even if
the best theory of domain D* in w does not posit quarks. Or to put it another way, we
attempt to determine the total ontology of w, by considering the total best theory of w,
where the total best theory of w might include a long conjunction of theories of
various domains of w, or (in some worlds) one overarching theory about w. The
necessitarian presupposes that her claims are claims about the totality of possible
worlds. So it is plausible to suppose that she consider whether mathematical objects
are indispensable to the best total theory of what we might call the pluriverse, where
we can understand the pluriverse either as the domain of all worlds, or, as Sider
does28, as a single abstract entity that represents the totality of possible worlds.
Roughly, the best total theory of the pluriverse specifies, for each world w in the
pluriverse, the best total theory of w. If mathematical objects are indispensable to the
best total theory of the pluriverse, then they are indispensable simpliciter. Since it is
28
Sider (2002).
29
plausible that mathematical objects are indispensable to the total best theory of
various worlds in the pluriverse (quite likely our own among them) it follows that
they are indispensable to the total best theory of the pluriverse. Thus it follows that
mathematical objects are indispensable.
Clearly the fact that mathematical objects are indispensable to the best total
theory of the pluriverse does not entail that they exist necessarily: it is plausible that
donkeys are likewise indispensable. So if we approached this argument, which is set
out below as V4, with no antecedent commitment regarding the modal status of
mathematical objects, we could not extract a necessitarian conclusion from the
argument. V4 is not intended as an argument designed to show antecedently
committed contingentists that they ought to be necessitarians. Rather, our project was
to formulate an a priori indispensability argument that the necessitarian could use to
argue for the necessary existence of mathematical objects. This is what V4 seeks to
accomplish.
New a priori version of the modalised indispensability argument (V4):
1. Mathematical objects are either necessary or impossible (necessitarian
presupposition).
2. Within a domain D, we should be committed to the existence of the objects that
are indispensable to the best complete theory of D. (Indispensability claim).
3. Logical necessitarians about some entity E should take the domain in assessing
the dispensability or not of E to be the pluriverse.
4. Within the domain of the pluriverse, we should be committed to the existence
of objects that are indispensable to the best total theory of the pluriverse (from 2).
5. Mathematical objects are indispensable to the best total theory of the pluriverse.
6. Therefore we should be committed to the existence of mathematical objects.
7. If mathematical objects exist, they do so of necessity (from 1)
8. Therefore mathematical objects exist necessarily.
Notice that this argument does not have V3’s defect: there is no argument of an
analogous form that shows that mathematical objects fail to exist of necessity. V4 is
certainly not unassailable by necessitarian nominalists who might take issue with the
claim that the relevant domain in which to assess whether or not mathematical objects
30
are indispensable is the pluriverse. This is not the place to defend the argument, and
indeed, that is not my aim. The premises are, I think, plausible, though clearly they
are not indubitable. The purpose of V4 is to show that the necessitarian can offer a
plausible indispensability argument for the necessary existence of mathematical
objects, and therefore that insofar as indispensability arguments are of use in
determining mathematical ontology, they are at least as much, and probably more, use
to the necessitarian as to the contingentist. This is very much in opposition to the
orthodoxy, which take s it that indispensability arguments support contingentism.
6. Conclusion
The indispensability argument in itself gives us no reason to prefer contingentism to
necessitarianism. Moreover, the contingentist has substantial further work to do in
providing a plausible link between indispensability and ontology that does not
ultimately render indispensability a poor epistemic guide to the existence of
mathematical objects. This is no small task, and the accounts that might be amenable
to a neo-Quinean do not look destined to succeed.
At the heart of these worries about the use of indispensability arguments in the
philosophy of mathematics lies a mismatch between the Quinean naturalistic project
in which the indispensability argument is grounded, and the post-Quinean project of
taking modality seriously. Whether Quine is best thought of as an actualist or a modal
error theorist, it is clear that he did not think there was a metaphysically robust sense
of possibility and necessity, and that there really are ways things could be, and could
not be. He did not think that the modal facts about a world in any sense outstrip the
facts about the distribution of properties in the world. Given this assumption, it makes
sense to suppose that the indispensability of quantification over some entity in one’s
best theory entails ontological commitment to that entity. But once we allow that
there is a robust sense of modality, as many contingentists and necessitarians do, then
we have dispensed with a crucial aspect of the Quinean apparatus, and it is much less
clear why we should think that indispensability is a guide to ontological commitment.
For if modal facts outstrip the empirical facts about a world, it seems dubious to
suppose that any examination of the quantifications of our best empirical theories will
give us reason to be committed to the existence, or lack thereof, of abstract objects. If
there are genuine modal facts about the necessity, impossibility or contingency of
31
abstract objects, then these facts will not only be inaccessible through a posteriori
investigation of our best scientific theories, but whatever those modal facts are, they
facts, they will leave untouched the empirical facts about the world. Once we admit
there are such modal facts, we should be wary of drawing any ontological conclusions
about the existence of abstract objects based on what is ultimately empirical evidence.
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