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From the rather remarkable but seemingly uncontroversial fact that
mathematics is indispensable to science, some philosophers have drawn
serious metaphysical conclusions. In particular, Quine (1976;
1980a; 1980b; 1981a; 1981c) and Putnam (1979a; 1979b) have
argued that the indispensability of mathematics to empirical science
gives us good reason to believe in the existence of mathematical entities.
According to this line of argument, reference
to (or quantification over) mathematical entities such as sets,
numbers, functions and such is indispensable to our best scientific
theories, and so we ought to be committed to the existence of these
mathematical entities. To do otherwise is to be guilty of what Putnam
has called "intellectual dishonesty" (Putnam 1979b, p. 347).
Moreover, mathematical entities are seen to be on an epistemic par
with the other theoretical entities of science, since belief in the
existence of the former is justified by the same evidence that
confirms the theory as a whole (and hence belief in the latter).
This argument is known as the Quine-Putnam indispensability argument for
mathematical realism. There are other indispensability
arguments,^{[1]} but
this one is by far the most influential, and so in what follows
I’ll concentrate on it.

- Spelling Out the Quine-Putnam Indispensability Argument
- What is it to be Indispensable?
- Naturalism and Holism
- Objections
- Conclusion
- Bibliography
- Other Internet Resources
- Related Entries

For future reference I’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.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. I 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. I’ll discuss its defense in the following section. I’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.(P2) Mathematical entities are indispensable to our best scientific theories.

(C) We ought to have ontological commitment to mathematical entities.

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 (1999b) 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.

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. I 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, as
Putnam (1979a) has stressed, 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 (1998); Field (1989, pp. 14-20); Hellman (199?); 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.

Field (1980) 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.
I’ll leave the discussion of the impact the rejection of
confirmational holism would have on the indispensability argument
until after I 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.

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 is that it is desirable to provide 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.^{[9]}

See Chihara (1973), and Field (1980; 1989) for attacks on the second premise and Colyvan (1999b; 2001), Maddy (1990), Malament (1982), Resnik (1985), Shapiro (1983) and Urquhart (1990) for criticisms of Field’s program. For a fairly comprehensive look at nominalist strategies in the philosophy of mathematics (including a good discussion of Field’s program), see Burgess and Rosen (1997), while Feferman (1993) questions the amount of mathematics required for empirical science. See Azzouni (1997), Balaguer (1996b; 1998), Maddy (1992; 1995; 1997), Melia (2000), Peressini (1997), Sober (1993) and Vineberg (1996) for attacks on the first premise. Colyvan (1998; 1999a; 2001; 2002), Hellman (1999) and Resnik (1995a; 1997) reply to some of these objections.

For variants of the Quinean indispensability argument see Maddy (1992) and Resnik (1995a).

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*First published: December 21, 1998*

*Content last modified: July 5, 2001*