## 1. Spelling Out the Quine-Putnam Indispensability Argument



The Quine-Putnam indispensability argument has attracted a great deal
of attention, in part because many see it as the best argument for
mathematical realism (or platonism). Thus anti-realists about
mathematical entities (or nominalists) need to identify where the
Quine-Putnam argument goes wrong. Many platonists, on the other hand,
rely very heavily on this argument to justify their belief in
mathematical entities. The argument places nominalists who wish to be
realist about other theoretical entities of science (quarks,
electrons, black holes and such) in a particularly difficult position.
For typically they accept something quite like the Quine-Putnam
 argument[2])
 as justification for realism about quarks and black holes. (This is
what Quine (1980b, p. 45) calls holding a "double
standard" with regard to ontology.) 



For future reference, we'll state the Quine-Putnam
indispensability argument in the following explicit form:



> 
> (P1)
>  We ought to have ontological commitment to all and only the entities
> that are indispensable to our best scientific theories.
> 
> 
> (P2) Mathematical entities are indispensable to our best scientific
> theories.
> 
> 
> 
> (C) We ought to have ontological commitment to mathematical
> entities.
> 
> 
> 



Thus formulated, the argument is valid. This forces the focus onto the
two premises. In particular, a couple of important questions naturally
arise. The first concerns how we are to understand the claim that
mathematics is indispensable. We address this in the next section. The
second question concerns the first premise. It is nowhere near as
self-evident as the second and it clearly needs some defense.
We'll discuss its defense in the following section. We'll
then present some of the more important objections to the argument,
before considering the Quine-Putnam argument's role in the
larger scheme of things -- where it stands in relation to other
influential arguments for and against mathematical realism. 


## 2. What is it to be Indispensable?



The question of how we should understand
'indispensability' in the present context is crucial to
the Quine-Putnam argument, and yet it has received surprisingly little
attention. Quine actually speaks in terms of the entities quantified
over in the canonical form of our best scientific theories rather than
indispensability. Still, the debate continues in terms of
indispensability, so we would be well served to clarify this term.




The first thing to note is that 'dispensability' is not
the same as 'eliminability'. If this were not so,
*every* entity would be dispensable (due to a theorem of
 Craig).[3]
 What we require for an entity to be 'dispensable' is for
it to be eliminable *and* that the theory resulting from the
entity's elimination be an attractive theory. (Perhaps, even
stronger, we require that the resulting theory be *more*
attractive than the original.) We will need to spell out what counts
as an attractive theory but for this we can appeal to the standard
desiderata for good scientific theories: empirical success;
unificatory power; simplicity; explanatory power; fertility and so on.
Of course there will be debate over what desiderata are appropriate
and over their relative weightings, but such issues need to be
addressed and resolved independently of issues of indispensability.
(See Burgess (1983) and Colyvan (1999) for more on these issues.)



These issues naturally prompt the question of *how much*
mathematics is indispensable (and hence how much mathematics carries
ontological commitment). It seems that the indispensability argument
only justifies belief in enough mathematics to serve the needs of
science. Thus we find Putnam speaking of "the set theoretic
'needs' of physics" (Putnam 1979b, p. 346) and Quine
claiming that the higher reaches of set theory are "mathematical
recreation ... without ontological rights" (Quine 1986, p. 400)
since they do not find physical applications. One could take a less
restrictive line and claim that the higher reaches of set theory,
although without physical applications, do carry ontological
commitment by virtue of the fact that they have applications *in
other parts of mathematics.* So long as the chain of applications
eventually "bottoms out" in physical science, we could
rightfully claim that the whole chain carries ontological commitment.
Quine himself justifies some transfinite set theory along these lines
(Quine 1984, p. 788), but he sees no reason to go beyond the
constructible sets (Quine 1986, p. 400). His reasons for this
restriction, however, have little to do with the indispensability
argument and so supporters of this argument need not side with Quine
on this issue.


## 3. Naturalism and Holism



Although both premises of the Quine-Putnam indispensability argument
have been questioned, it's the first premise that is most
obviously in need of support. This support comes from the doctrines of
naturalism and holism. 



Following Quine, naturalism is usually taken to be the philosophical
doctrine that there is no first philosophy and that the philosophical
enterprise is continuous with the scientific enterprise (Quine 1981b).
By this Quine means that philosophy is neither prior to nor privileged
over science. What is more, science, thus construed (i.e. with
philosophy as a continuous part) is taken to be the complete story of
the world. This doctrine arises out of a deep respect for scientific
methodology and an acknowledgment of the undeniable success of this
methodology as a way of answering fundamental questions about all
nature of things. As Quine suggests, its source lies in
"unregenerate realism, the robust state of mind of the natural
scientist who has never felt any qualms beyond the negotiable
uncertainties internal to science" (Quine 1981b, p. 72). For the
metaphysician this means looking to our best scientific theories to
determine what exists, or, perhaps more accurately, what we ought to
believe to exist. In short, naturalism rules out unscientific ways of
determining what exists. For example, naturalism rules out believing
in the transmigration of souls for mystical reasons. Naturalism would
not, however, rule out the transmigration of souls if our best
scientific theories were to require the truth of this
 doctrine.[4]




Naturalism, then, gives us a reason for believing in the entities in
our best scientific theories and no other entities. Depending on
exactly how you conceive of naturalism, it may or may not tell you
whether to believe in *all* the entities of your best
scientific theories. We take it that naturalism does give us
*some* reason to believe in all such entities, but that this is
defeasible. This is where holism comes to the fore: in particular,
confirmational holism.



Confirmational holism is the view that theories are confirmed or
disconfirmed as wholes (Quine 1980b, p. 41). So, if a theory is
*confirmed* by empirical findings, the *whole* theory is
confirmed. In particular, whatever mathematics is made use of in the
theory is also confirmed (Quine 1976, pp. 120-122). Furthermore,
it is the same evidence that is appealed to in justifying belief in
the mathematical components of the theory that is appealed to in
justifying the empirical portion of the theory (if indeed the
empirical can be separated from the mathematical at all). Naturalism
and holism taken together then justify
 P1.
 Roughly, naturalism gives us the "only" and holism gives
us the "all" in P1.



It is worth noting that in Quine's writings there are at least
two holist themes. The first is the confirmational holism discussed
above (often called the Quine-Duhem thesis). The other is semantic
holism which is the view that the unit of meaning is not the single
sentence, but systems of sentences (and in some extreme cases the
whole of language). This latter holism is closely related to
Quine's well-known denial of the analytic-synthetic distinction
(Quine 1980b) and his equally famous indeterminacy of translation
thesis (Quine 1960). Although for Quine, semantic holism and
confirmational holism are closely related, there is good reason to
distinguish them, since the former is generally thought to be highly
controversial while the latter is considered relatively
uncontroversial.



Why this is important to the present debate is that Quine explicitly
invokes the controversial semantic holism in support of the
indispensability argument (Quine 1980b, pp. 45-46). Most
commentators, however, are of the view that only confirmational holism
is required to make the indispensability argument fly (see, for
example, Colyvan (1998a); Field (1989, pp. 14-20); Hellman
(1999); Resnik (1995a; 1997); Maddy (1992)) and my presentation here
follows that accepted wisdom. It should be kept in mind, however, that
while the argument, thus construed, is Quinean in flavor it is not,
strictly speaking, Quine's argument.


## 4. Objections



There have been many objections to the indispensability argument,
including Charles Parsons' (1980) concern that the obviousness
of basic mathematical statements is left unaccounted for by the
Quinean picture and Philip Kitcher's (1984, pp. 104-105)
worry that the indispensability argument doesn't explain
*why* mathematics is indispensable to science. The objections
that have received the most attention, however, are those due to
Hartry Field, Penelope Maddy and Elliott Sober. In particular,
Field's nominalisation program has dominated recent discussions
of the ontology of mathematics. 



Field (2016) presents a case for denying the second premise of the
Quine-Putnam argument. That is, he suggests that despite appearances
mathematics is not indispensable to science. There are two parts to
Field's project. The first is to argue that mathematical
theories don't have to be true to be useful in applications,
they need merely to be *conservative.* (This is, roughly, that
if a mathematical theory is added to a nominalist scientific theory,
no nominalist consequences follow that wouldn't follow from the
nominalist scientific theory alone.) This explains why mathematics
*can* be used in science but it does not explain why it
*is* used. The latter is due to the fact that mathematics makes
calculation and statement of various theories much simpler. Thus, for
Field, the utility of mathematics is merely pragmatic --
mathematics is not indispensable after all.



The second part of Field's program is to demonstrate that our
best scientific theories can be suitably nominalised. That is, he
attempts to show that we could do without quantification over
mathematical entities and that what we would be left with would be
reasonably attractive theories. To this end he is content to
nominalise a large fragment of Newtonian gravitational theory.
Although this is a far cry from showing that *all* our current
best scientific theories can be nominalised, it is certainly not
trivial. The hope is that once one sees how the elimination of
reference to mathematical entities can be achieved for a typical
physical theory, it will seem plausible that the project could be
completed for the rest of
 science.[5]




There has been a great deal of debate over the likelihood of the
success of Field's program but few have doubted its
significance. Recently, however, Penelope Maddy, has pointed out that
if
 P1
 is false, Field's project may turn out to be irrelevant to the
realism/anti-realism debate in mathematics.



Maddy presents some serious objections to the first premise of the
indispensability argument (Maddy 1992; 1995; 1997). In particular, she
suggests that we ought not have ontological commitment to *all*
the entities indispensable to our best scientific theories. Her
objections draw attention to problems of reconciling naturalism with
confirmational holism. In particular, she points out how a holistic
view of scientific theories has problems explaining the legitimacy of
certain aspects of scientific and mathematical practices. Practices
which, presumably, ought to be legitimate given the high regard for
scientific practice that naturalism recommends. It is important to
appreciate that her objections, for the most part, are concerned with
methodological consequences of accepting the Quinean doctrines of
naturalism and holism -- the doctrines used to support the first
premise. The first premise is thus called into question by undermining
its support.



Maddy's first objection to the indispensability argument is that
the actual attitudes of working scientists towards the components of
well-confirmed theories vary from belief, through tolerance, to
outright rejection (Maddy 1992, p. 280). The point is that naturalism
counsels us to respect the methods of working scientists, and yet
holism is apparently telling us that working scientists ought not have
such differential support to the entities in their theories. Maddy
suggests that we should side with naturalism and not holism here. Thus
we should endorse the attitudes of working scientists who apparently
do not believe in *all* the entities posited by our best
theories. We should thus reject
 P1.



The next problem follows from the first. Once one rejects the picture
of scientific theories as homogeneous units, the question arises
whether the mathematical portions of theories fall within the true
elements of the confirmed theories or within the idealized elements.
Maddy suggests the latter. Her reason for this is that scientists
themselves do not seem to take the indispensable application of a
mathematical theory to be an indication of the truth of the
mathematics in question. For example, the false assumption that water
is infinitely deep is often invoked in the analysis of water waves, or
the assumption that matter is continuous is commonly made in fluid
dynamics (Maddy 1992, pp. 281-282). Such cases indicate that
scientists will invoke whatever mathematics is required to get the job
done, without regard to the truth of the mathematical theory in
question (Maddy 1995, p. 255). Again it seems that confirmational
holism is in conflict with actual scientific practice, and hence with
naturalism. And again Maddy sides with naturalism. (See also Parsons
(1983) for some related worries about Quinean holism.) The point here
is that if naturalism counsels us to side with the attitudes of
working scientists on such matters, then it seems that we ought not
take the indispensability of some mathematical theory in a physical
application as an indication of the truth of the mathematical theory.
Furthermore, since we have no reason to believe that the mathematical
theory in question is true, we have no reason to believe that the
entities posited by the (mathematical) theory are real. So once again
we ought to reject
 P1.



Maddy's third objection is that it is hard to make sense of what
working mathematicians are doing when they try to settle independent
questions. These are questions, that are independent of the standard
axioms of set theory -- the ZFC
 axioms.[6]
 In order to settle some of these questions, new axiom candidates have
been proposed to supplement ZFC, and arguments have been advanced in
support of these candidates. The problem is that the arguments
advanced seem to have nothing to do with applications in physical
science: they are typically intra-mathematical arguments. According to
indispensability theory, however, the new axioms should be assessed on
how well they cohere with our current best scientific theories. That
is, set theorists should be assessing the new axiom candidates with
one eye on the latest developments in physics. Given that set
theorists do not do this, confirmational holism again seems to be
advocating a revision of standard mathematical practice, and this too,
claims Maddy, is at odds with naturalism (Maddy 1992, pp.
286-289).



Although Maddy does not formulate this objection in a way that
directly conflicts with
 P1
 it certainly illustrates a tension between naturalism and
confirmational
 holism.[7]
 And since both these are required to support P1, the objection
indirectly casts doubt on P1. Maddy, however, endorses naturalism and
so takes the objection to demonstrate that confirmational holism is
false. We'll leave the discussion of the impact the rejection of
confirmational holism would have on the indispensability argument
until after we outline Sober's objection, because Sober arrives
at much the same conclusion.



Elliott Sober's objection is closely related to Maddy's
second and third objections. Sober (1993) takes issue with the claim
that mathematical theories share the empirical support accrued by our
best scientific theories. In essence, he argues that mathematical
theories are not being tested in the same way as the clearly empirical
theories of science. He points out that hypotheses are confirmed
relative to competing hypotheses. Thus if mathematics is confirmed
along with our best empirical hypotheses (as indispensability theory
claims), there must be mathematics-free competitors. But Sober points
out that *all* scientific theories employ a common mathematical
core. Thus, since there are no competing hypotheses, it is a mistake
to think that mathematics receives confirmational support from
empirical evidence in the way other scientific hypotheses do.



This in itself does not constitute an objection to
 P1
 of the indispensability argument, as Sober is quick to point out
(Sober 1993, p. 53), although it does constitute an objection to
Quine's overall view that mathematics is part of empirical
science. As with Maddy's third objection, it gives us some cause
to reject confirmational holism. The impact of these objections on P1
depends on how crucial you think confirmational holism is to that
premise. Certainly much of the intuitive appeal of P1 is eroded if
confirmational holism is rejected. In any case, to subscribe to the
conclusion of the indispensability argument in the face of
Sober's or Maddy's objections is to hold the position that
it's permissible at least to have ontological commitment to
entities that receive no empirical support. This, if not outright
untenable, is certainly not in the spirit of the original Quine-Putnam
argument.


## 5. Explanatory Versions of the Argument



The arguments against holism from Maddy and Sober resulted in a
reevaluation of the indispensability argument. If, contra Quine,
scientists do not accept all the entities of our best scientific
theories, where does this leave us? We need criteria for when to treat
posits realistically. Here is where the debate over the
indispensability argument took an interesting turn. Scientific
realists, at least, accept those posits of our best scientific
theories that contribute to scientific explanations. According to this
line of thought, we ought to believe in electrons, say, not because
they are indispensable to our best scientific theories but because
they are indispensable in a very specific way: they are
*explanatorily* indispensable. If mathematics could be shown to
contribute to scientific explanations in this way, mathematical
realism would again be on par with scientific realism. Indeed, this is
the focus of most of the contemporary discussion on the
indispensability argument. The central question is: does mathematics
contribute to scientific explanations and if so, does it do it in the
right kind of way.



One example of how mathematics might be thought to be explanatory is
found in the periodic cicada case (Yoshimura 1997 and Baker 2005).
North American Magicicadas are found to have life cycles of 13 or 17
years. It is proposed by some biologists that there is an evolutionary
advantage in having such prime-numbered life cycles. Prime-numbered
life cycles mean that the Magicicadas avoid competition, potential
predators, and hybridisation. The idea is quite simple: because prime
numbers have no non-trivial factors, there are very few other life
cycles that can be synchronised with a prime-numbered life cycle. The
Magicicadas thus have an effective avoidance strategy that, under
certain conditions, will be selected for. While the explanation being
advanced involves biology (e.g. evolutionary theory, theories of
competition and predation), a crucial part of the explanation comes
from number theory, namely, the fundamental fact about prime numbers.
Baker (2005) argues that this is a genuinely mathematical explanation
of a biological fact. There are other examples of alleged mathematical
explanations in the literature but this remains the most widely
discussed and is something of a poster child for mathematical
explanation.



Questions about this case focus on whether the mathematics is really
contributing to the explanation (or whether it is merely standing in
for the biological facts and it is these that really do the
explaining), whether the alleged explanation is an explanation at all,
and whether the mathematics in question is involved in the explanation
in the right kind of way. Finally, it is worth mentioning that
although the recent interest in mathematical explanation arose out of
debates over the indispensability argument, the status of mathematical
explanations in the empirical sciences has also attracted interest in
its own right. Moreover, such explanations (sometimes called
"extra-mathematical explanations") lead one very naturally
to think about explanations of mathematical facts by appeal to further
mathematical facts (sometimes called "intra-mathematical
explanation"). These two kinds of mathematical explanation are
related, of course. If, for example, some theorem of mathematics has
its explanation rest in an explanatory proof, then any applications of
that theorem in the empirical realm would give rise to a prima facie
case that the full explanation of the empirical phenomenon in question
involves the intra-mathematical explanation of the theorem. For these
and other reasons, both kinds of mathematical explanation have
attracted a great deal of interest from philosophers of mathematics
and philosophers of science in recent years.


## 6. Conclusion



It is not clear how damaging the above criticisms are to the
indispensability argument and whether the explanatory version of the
argument survives. Indeed, the debate is very much alive, with many
recent articles devoted to the topic. (See bibliography notes below.)
Closely related to this debate is the question of whether there are
any other decent arguments for platonism. If, as some believe, the
indispensability argument is the *only* argument for platonism
worthy of consideration, then if it fails, platonism in the philosophy
of mathematics seems bankrupt. Of relevance then is the status of
other arguments for and against mathematical realism. In any case, it
is worth noting that the indispensability argument is one of a small
number of arguments that have dominated discussions of the ontology of
mathematics. It is therefore important that this argument not be
viewed in isolation. 



The two most important arguments *against* mathematical realism
are the epistemological problem for platonism -- how do we come
by knowledge of causally inert mathematical entities? (Benacerraf
1983b) -- and the indeterminacy problem for the reduction of
numbers to sets -- if numbers are sets, which sets are they
(Benacerraf 1983a)? Apart from the indispensability argument, the
other major argument *for* mathematical realism appeals to a
desire for a uniform semantics for *all* discourse:
mathematical and non-mathematical alike (Benacerraf 1983b).
Mathematical realism, of course, meets this challenge easily, since it
explains the truth of mathematical statements in exactly the same way
as in other
 domains.[8]
 It is not so clear, however, how nominalism can provide a uniform
semantics.



Finally, it is worth stressing that even if the indispensability
argument *is* the only good argument for platonism, the failure
of this argument does not necessarily authorize nominalism, for the
latter too may be without support. It does seem fair to say, however,
that if the objections to the indispensability argument are sustained
then one of the most important arguments for platonism is undermined.
This would leave platonism on rather shaky ground.