From: http://www.seop.leeds.ac.uk/archives/win1999/entries/mathphil-indis/

One of the most intriguing features of mathematics is its applicability
to empirical science. Every branch of science draws upon large
and often diverse portions of mathematics, from the use of Hilbert
spaces
in quantum mechanics to the use of differential geometry in general
relativity. It's not just the physical sciences that avail themselves
of the services of mathematics either. Biology, for instance, makes
extensive use of difference equations and statistics. The roles
mathematics plays in these theories is also varied. Not only
does mathematics
help with empirical predictions, it allows elegant and economical
statement of many theories. Indeed, so important is the language
of mathematics to science, that it is hard to imagine how theories
such as quantum mechanics and general relativity could even be
stated without employing a substantial amount of mathematics.

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.

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 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.

(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. 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.

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 (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.

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. 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.

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 (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.

It is not clear how damaging the above criticisms are to the
indispensability argument. 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 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]

Although the indispensability argument is to be found in many places in Quine's writings (including 1976; 1980a; 1980b; 1981a; 1981c), the locus classicus is Putnam's short monograph Philosophy of Logic (included as a chapter of the second edition of the third volume of his collected papers (Putnam, 1979b)). See also Putnam (1979a) and the introduction of Field (1989) which has an excellent outline of the argument. Colyvan (2000) is a sustained defence of the argument.

See Chihara (1973), and Field (1980; 1989) for attacks on the second premise and 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), Peressini (1997), Sober (1993) and Vineberg (1996) for attacks on the first premise. Colyvan (1998; 1999a; 2000), 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).

Azzouni, J., 1997, "Applied Mathematics, Existential Commitment
and the Quine-Putnam Indispensability Thesis", Philosophia
Mathematica (3) 5/3 (October): 193-209

Balaguer, M., 1996a, "Towards a Nominalization of Quantum
Mechanics", Mind 105/418 (April): 209-226

Balaguer, M., 1996b, "A Fictionalist Account of the Indispensable
Applications of Mathematics", Philosophical Studies 83/3 (September):
291-314

Balaguer, M., 1998, Platonism and Anti-Platonism in Mathematics,
New York: Oxford University Press

Benacerraf, P., 1983a, "What Numbers Could Not Be", reprinted
in Benacerraf and Putnam (1983), pp. 272-294

Benacerraf, P., 1983b, "Mathematical Truth", reprinted
in Benacerraf and Putnam (1983), pp. 403-420 and in Hart (1996),
pp. 14-30

Benacerraf, P. and Putnam, H. (eds.), 1983, Philosophy of Mathematics:
Selected Readings, 2nd edition, Cambridge: Cambridge University
Press

Burgess, J., 1983, "Why I Am Not a Nominalist", Notre
Dame Journal of Formal Logic 24/1 (January): 93-105

Burgess, J. and Rosen, G., 1997, A Subject with No Object: Strategies
for Nominalistic Interpretation of Mathematics, Oxford: Clarendon

Chihara, C., 1973, Ontology and the Vicious Circle Principle, Ithaca,
NY: Cornell University Press

Colyvan, M., 1998, "In Defence of Indispensability",
Philosophia Mathematica (3) 6/1 (February): 39-62

Colyvan, M., 1999a, "Contrastive Empiricism and Indispensability",
Erkenntnis 51/2-3 (September): 323-332

Colyvan, M., 1999b, "Confirmation Theory and Indispensability",
Philosophical Studies 96/1 (October): 1-19

Colyvan, M., 2000, The Indispensability of Mathematics, New York:
Oxford University Press

Feferman, S., 1993, "Why a Little Bit Goes a Long Way: Logical
Foundations of Scientifically Applicable Mathematics", Proceedings
of the Philosophy of Science Association 2: 442-455

Field, H.H., 1980, Science Without Numbers: A Defence of Nominalism,
Oxford: Blackwell

Field, H.H., 1989, Realism, Mathematics and Modality, Oxford: Blackwell

Hart, W.D. (ed.), 1996, The Philosophy of Mathematics, Oxford:
Oxford University Press

Hellman, G., 1999, "Some Ins and Outs of Indispensability:
A Modal-Structural Perspective", in A. Cantini, E. Casari
and P. Minari (eds.), Logic and Foundations of Mathematics, Dordrecht:
Kluwer, pp. 25-39

Irvine, A.D. (ed.), 1990, Physicalism in Mathematics, Dordrecht:
Kluwer

Kitcher, P., 1984, The Nature of Mathematical Knowledge, New York:
Oxford University Press

Maddy, P., 1990, "Physicalistic Platonism", in A.D. Irvine
(ed.), Physicalism in Mathematics, Dordrecht: Kluwer, pp. 259-289

Maddy, P., 1992, "Indispensability and Practice", Journal
of Philosophy 89/6 (June): 275-289

Maddy, P., 1995, "Naturalism and Ontology", Philosophia
Mathematica (3) 3/3 (September): 248-270

Maddy, P., 1997, Naturalism in Mathematics, Oxford: Clarendon Press

Maddy, P., 1998, "How to be a Naturalist about Mathematics",
in H.G. Dales and G. Oliveri (eds.), Truth in Mathematics, Oxford:
Clarendon, pp. 161-180

Malament, D., 1982, "Review of Field's Science Without Numbers",
Journal of Philosophy 79/9 (September): 523-534 and reprinted in
Resnik (1995b), pp. 75-86

Parsons, C., 1980, "Mathematical Intuition", Proceedings
of the Aristotelian Society 80 (1979-1980): 145-168 and reprinted
in Resnik (1995b), pp. 589-612 and in Hart (1996), pp. 95-113

Parsons, C., 1983, "Quine on the Philosophy of Mathematics",
in Mathematics in Philosophy: Selected Essays, Ithaca, NY: Cornell
University Press, pp. 176-205

Peressini, A., 1997, "Troubles with Indispensability: Applying
Pure Mathematics in Physical Theory", Philosophia Mathematica
(3) 5/3 (October): 210-227

Putnam, H., 1979a, "What is Mathematical Truth", in Mathematics
Matter and Method: Philosophical Papers Vol. 1, 2nd edition, Cambridge:
Cambridge University Press, pp. 60-78

Putnam, H., 1979b, "Philosophy of Logic", reprinted in
Mathematics Matter and Method: Philosophical Papers Vol. 1, 2nd
edition, Cambridge: Cambridge University Press, pp. 323-357

Quine, W.V., 1960, Word and Object, Cambridge, MA: Massachusetts
Institute of Technology Press

Quine, W.V., 1976, "Carnap and Logical Truth" reprinted
in The Ways of Paradox and Other Essays, revised edition, Cambridge,
MA: Harvard University Press, pp. 107-132 and in Benacerraf and
Putnam (1983), pp. 355-376

Quine, W.V., 1980a, "On What There Is", reprinted in
From a Logical Point of View, 2nd edition, Cambridge, MA: Harvard
University Press, pp. 1-19

Quine, W.V., 1980b, "Two Dogmas of Empiricism", reprinted
in From a Logical Point of View, 2nd edition, Cambridge, MA: Harvard
University Press, pp. 20-46 and in Hart (1996), pp. 31-51 (Page
references are to the first reprinting)

Quine, W.V., 1981a, "Things and Their Place in Theories",
in Theories and Things, Cambridge, MA: Harvard University Press,
pp. 1-23

Quine, W.V., 1981b, "Five Milestones of Empiricism",
in Theories and Things, Cambridge, MA: Harvard University Press,
pp. 67-72

Quine, W.V., 1981c, "Success and Limits of Mathematization",
in Theories and Things, Cambridge, MA: Harvard University Press,
pp. 148-155

Quine, W.V., 1984, "Review of Parsons', Mathematics in Philosophy",
Journal of Philosophy 81/12 (December): 783-794

Quine, W.V., 1986, "Reply to Charles Parsons", in L.
Hahn and P. Schilpp (eds.), The Philosophy of W.V. Quine, La Salle,
ILL: Open Court, pp. 396-403

Resnik, M.D., 1985, "How Nominalist is Hartry Field's Nominalism",
Philosophical Studies 47 (March): 163-181

Resnik, M.D., 1995a, "Scientific Vs Mathematical Realism:
The Indispensability Argument", Philosophia Mathematica (3)
3/2 (May): 166-174

Resnik, M.D. (ed.), 1995b, Mathematical Objects and Mathematical
Knowledge, Aldershot (UK): Dartmouth

Resnik, M.D., 1997, Mathematics as a Science of Patterns, Oxford:
Clarendon Press

Shapiro, S., 1983, "Conservativeness and Incompleteness",
Journal of Philosophy 80/9 (September): 521-531 and reprinted in
Resnik (1995b), pp. 87-97 and in Hart (1996), pp. 225-234

Sober, E., 1993, "Mathematics and Indispensability",
Philosophical Review 102/1 (January): 35-57

Urquhart, A., 1990, "The Logic of Physical Theory", in
A.D. Irvine (ed.), Physicalism in Mathematics, Dordrecht: Kluwer,
pp. 145-154

Vineberg, S., 1996, "Confirmation and the Indispensability
of Mathematics to Science" PSA 1996 (Philosophy of Science,
supplement to vol. 63), pp. 256-263