ORI GIN AL ARTICLE
Evidential Holism and Indispensability Arguments
Joe Morrison
Received: 17 February 2010 / Accepted: 26 May 2011 / Published online: 22 July 2011
� Springer Science+Business Media B.V. 2011
Abstract The indispensability argument is a method for showing that abstract
mathematical objects exist (call this mathematical Platonism). Various versions of
this argument have been proposed (§1). Lately, commentators seem to have agreed
that a holistic indispensability argument (§2) will not work, and that an explanatory
indispensability argument is the best candidate. In this paper I argue that the
dominant reasons for rejecting the holistic indispensability argument are mistaken.
This is largely due to an overestimation of the consequences that follow from
evidential holism. Nevertheless, the holistic indispensability argument should be
rejected, but for a different reason (§3)—in order that an indispensability argument
relying on holism can work, it must invoke an unmotivated version of evidential
holism. Such an argument will be unsound. Correcting the argument with a proper
construal of evidential holism means that it can no longer deliver mathematical
Platonism as a conclusion: such an argument for Platonism will be invalid. I then
show how the reasons for rejecting the holistic indispensability argument impor-
tantly constrain what kind of account of explanation will be permissible in
explanatory versions (§4).
1
Mathematical Platonism is a realist ontological position: it asserts that mathematical
statements are true, that mathematical statements are committed to the existence of
mathematical objects (such as numbers, sets, functions etc.), and thus that these
mathematical objects exist (Maddy 1990). In addition to realism, Platonism insists
that these mathematical objects are abstract; the Platonist maintains that there exist
J. Morrison (&)
Department of Philosophy, University of Birmingham, Birmingham B15 2TT, UK
e-mail: [email protected]
123
Erkenn (2012) 76:263–278
DOI 10.1007/s10670-011-9300-4
non-spatiotemporal, causally inefficacious mathematical objects (Hale 1994, p. 299;
Linnebo 2006 p. 545).
A major contemporary source of motivation for Platonism is the indispensability
argument (Colyvan 2001, p. 6). The argument proceeds from the fact that we make
indispensable use of mathematics and mathematical claims in our scientific
theorising to the conclusion that we ought to be ontologically committed to the
existence of such things as numbers, sets, models and functions. In addition, there is
general consensus that if such mathematical objects exist at all, then they are
abstract.1 So the argument is controversial; it suggests that normal scientific
practices commit us to the existence of abstract objects. This is unpalatable to
contemporary metaphysical tastes on at least two counts: one might repudiate the
idea that scientists can furnish one’s ontology at all, and more specifically, one
might be horrified to discover that one’s ontology has been furnished with such
bizarre entities as abstracta. Since indispensability arguments explicitly involve the
claim that scientific theories can be a source of ontological commitments, the
argument does not attempt to convince either those who think that matters of
ontology are the proprietary domain of metaphysicians, or those who think that
ontology can’t be done. But for scientific realists, who maintain that science can
discover what exists, the indispensability argument does stand as a challenge, since
it suggests that scientists are committed to the existence of abstracta in the same
way as they are to atoms and genes.
It is inaccurate to say that there is such a thing as the indispensability argument
since there are many different forms. The characteristic that is common to all of
them is the dialectical procedure from the premise that science indispensably
depends on mathematics to the conclusion that mathematical objects exist. I will not
question the initial premise that scientific theorising indispensably depends on
mathematical claims. Showing that science could be done without mathematics is
one option available to those scientific realists who wish to avoid Platonism about
mathematical objects. Alternatively one could agree that science depends on
mathematical claims, but deny that those claims commit to mathematical abstracta;
giving an alternative semantics or a reason for understanding mathematical claims
non-literally will allow one to accept the indispensability of mathematical claims
while trying to avoid their ontological commitments. This paper will assume,
without argument, that mathematical claims are indispensable to scientific
theorising, and that the ontological commitments of sentences like:
1. the specific heat capacity of H2O is 4,200 J/kg/K
can be understood directly and literally from their paraphrases into canonical
notation, such as:
1 If most mathematical realists are Platonists, this is because most people agree that it ‘‘appears to make
no sense to ask where numbers or sets are located, or when they came into existence’’ (Hale 1994, p. 299).
One exception is Penelope Maddy, who attributes physical/causal properties to some mathematical
objects. Maddy maintains that sets can be perceivable (Maddy 1990, pp. 58–63). Michael Resnik (Resnik
1997, pp. 93–95) critically discusses her position.
264 J. Morrison
123
2. there exists an x such that x = specific heat capacity of H2O = the number of
Joules energy required to raise one kilogramme of water by one degree
Kelvin = 4,200
It may be that these two assumptions, along with the commitment to scientific
realism, are sufficient to secure mathematical Platonism. Let’s call such an
argument a Semantic Indispensability Argument (SIA) (see Liggins 2008 and
Wagner 1996). It says that we should accept the ontological commitments of our
best scientific theories, that our best scientific theories indispensably appeal to
mathematical claims, and that those mathematical claims are committed to the
existence of abstracta; hence we should accept the existence of abstracta. But our
scientific realist who has no taste for Platonic entities may not feel the pull of SIA.
Being as respectably-well-informed about actual scientific practices as she is, our
scientific realist points out that the overwhelming majority of scientific theories
involve explicit idealizations and simplifications: real-world complex systems are
reduced to partial models in which only one or two variables contribute any causal
influence. She notes that while scientists might indispensably depend upon
idealizations, neither they nor she are thereby committed to thinking that the world
is anything like the way it is represented in those models. Not all parts of a
scientist’s theory should be treated equally; the scientific realist looks to make some
sort of distinction between the parts of the theory which are ontologically-
committing conclusions, and which parts are merely instrumental devices for
getting those conclusions; a distinction between those claims which are tools and
those which are results.2 So, for example, she might try to argue that scientific
discussion of atoms and genes is talk about substantial, ontologically-committing
results, while talk of numbers is merely instrumental, the invocation of a tool. Our
scientific realist feels no need to provide any alternative or non-literal semantics for
the instrumental parts of a theory. Rather, she simply takes actual scientific practice
to licence a restricted notion of scientific realism: no scientist would think that we
should be wholesale scientific realists, ignoring the many differences between toolsand results. Our restricted scientific realist maintains that numbers are tools, and
that as such SIA does not commit her to mathematical Platonism. SIA will only
convert wholesale scientific realists into Platonists.
If this is right, then we can understand other varieties of indispensability
argument either as attempts to force the restricted realist to be a wholesale realist, or
as attempts to demonstrate that her restricted version of realism is actually
committed to numbers after all. Examples of the former are Holistic IndispensabilityArguments—these approaches have the consequence that the distinction between
2 Clearly, it’s not enough just to stipulate that some claims are ontologically committing while others are
only instrumental—appealing to scientific practice might allow us to invoke such a distinction (see
Maddy 1990, p. 280), but it doesn’t licence drawing this line arbitrarily. Such a distinction is a
consequence of prior philosophical theory; various arguments can be given for putting particular cases of
apparent ontological commitment into either the ‘tool’ or ‘result’ categories. For example, perhaps, of all
the propositions indispensable to science, the ontologically committing propositions are just those that
can act as explanations, or that identify causes, or that are falsifiable, etc. Regardless of how such a
distinction is drawn, the present point is that various types of indispensability argument attempt to either
remove it or redraw it. (Thanks to an anonymous referee for encouraging this point).
Evidential Holism and Indispensability Arguments 265
123
tools and results is not available to the scientific realist: the tools are actually a lot
more like results. Explanatory Indispensability Arguments work in the other way: a
distinction between tools and results might seem to be permissible, and perhaps it is
also acceptable that mathematical talk is more tool-like than result-like. Neverthe-
less, the argument maintains that a restricted scientific realist will have to be a
mathematical Platonist, since her ontologically privileged category of scientific
results is really no different from the instrumentally useful mathematical tools—
both are importantly instrumental for giving explanations of physical phenomena. In
§2 and §3 I explore holistic indispensability arguments. I return to explanatory
indispensability arguments in §4.
2
Many commentators maintain that W.V.O. Quine gave an indispensability argument
for mathematical Platonism that invokes evidential holism as a premise, but the role
which holism is supposed to play is not immediately clear. This is, in part, because it
is not entirely clear quite what Quine’s evidential holism amounts to. Recently, I
have identified three distinct evidentially holistic theses that are commonly treated
interchangeably (Morrison 2010)3:
Prediction Thesis (PT): Only whole theories imply observations.
Falsification Thesis (FT): Observations only falsify whole theories, not individual
parts thereof.
Confirmation Thesis (CT): Observations confirm entire theories, not individual
parts thereof.
These distinctions are useful as they enable us to identify candidate roles for
holism in indispensability arguments. For example, the falsification thesis has long
since been invoked in Quinean-style arguments against the possibility of a priori
knowledge. The reasoning is as follows: if the relata of falsification are observations
and theories, then any particular observation underdetermines which of a theory’s
constitutive claims is false. As such, it remains in principle possible to blame the
empirical failures of a theory on any of its a priori constituents (i.e. any
mathematical or logical parts). If a necessary condition for a proposition to be
known a priori is that it is empirically indefeasible, and if any propositions can in
principle be empirically defeated, then no propositions are known a priori.
Variations on such an argument are used to provide support for epistemic
naturalism. Epistemic naturalism tells us that only scientifically acceptable methods
of belief-formation are sufficient to form justified beliefs. So holism’s role in this
indispensability argument is to secure this premise: holism (FT) is taken to imply
epistemic naturalism. The argument then proceeds as follows: epistemic naturalism
tells us that scientific inquiry is the only source of knowledge or justified belief. The
indispensability thesis tells us that scientific inquiry cannot proceed without
existentially quantifying over mathematical objects. It follows that the existence of
3 Strictly speaking there are only two distinct theses here, since I show that PT and FT are contapositives.
266 J. Morrison
123
mathematical objects becomes a necessary condition for the possibility of any
beliefs being justified.
We should note that the pivotal premise of this holistic/naturalistic-indispensa-
bility argument is not evidential holism; instead, it is epistemic naturalism, which is
purported to be a consequence of holism (in the form of FT). So this indispensability
argument will only work if it is true that evidential holism entails that there is no
a priori knowledge. As such, the question as to whether holism entails mathematical
Platonism is dependent on whether holism entails the repudiation of apriority. This
issue has been much discussed. Some maintain that respectable analyses of a priori
knowledge should not make empirical indefeasibility a necessary condition (Rey
1998; Jenkins 2008). Others argue that the argument sketched above turns on a
pivotal and fallacious inference from the underdetermination of falsification to the
possibility of revising a mathematical or logical belief (Klee 1992; Morrison 2010;
Resnik and Orlandi 2003). But for our purposes here, it is sufficient to note that this
holistic/naturalistic indispensability argument need not convert our restricted
scientific realist into a wholesale scientific realist. She can agree that claims which
quantify over mathematical abstracta are a necessary condition for scientifically
justified beliefs, and she can agree that there are no other sources of epistemic
justification, all the while maintaining that none of this compels her think that
mathematical claims are anything more than (importantly) useful tools.
A more widely recognised role for evidential holism in indispensability
arguments invokes the confirmation thesis (CT). The first premise of the argument
is the indispensability thesis, which we saw above: scientific theorising indispens-
ably depends upon claims that existentially quantify over mathematical abstracta.
The next part of the argument invokes evidential holism. The idea is that when a
scientist carries out an experiment, the results confirm not only the experimental
hypotheses at stake but also all of the mathematical claims (along with all other
additional theoretical claims). As such, scientists have empirical confirmation of the
mathematical claims on which they (indispensably) depend. We can see this move
being made when Michael Resnik claims that indispensability arguments involve
the following premise:
Evidential holism: The evidence for a scientific theory bears directly upon its
theoretical apparatus as a whole and not upon its individual hypotheses.
(Resnik 1995, p. 166)
Similarly, Penelope Maddy claims that the argument involves the premise that
‘‘as holists, we take a scientific theory to be confirmed as a whole, the mathematical
along with the physical hypotheses’’ (Maddy 2005b, p. 444). Likewise, Elliott Sober
tells us that
[t]his indispensability argument for mathematical realism gives voice to an
attitude towards confirmation elaborated by Quine. Quine’s holism—his
interpretation of Duhem’s thesis—asserts that theories are confirmed only as
totalities. A theory makes contact with experience only as a whole, and so it
receives confirmation only as a whole. If mathematics is an inextricable part of
Evidential Holism and Indispensability Arguments 267
123
a physical theory, then the empirical success of the theory confirms the entire
theory—mathematics and all. (Sober 1993, p. 35)
The argument starts by claiming that scientists often must use mathematical
claims. By invoking holism, we can infer that we can have empirical support for
these mathematical claims. The last step of the argument is to show that this
empirical support means that a scientist cannot fail to be committed to the existence
of mathematical objects. Resnik goes on to describe how this inference might work:
[I]f mathematics is an indispensable component of science, then, by holism,
whatever evidence we have for science is also evidence for the mathematical
objects and mathematical principles it presupposes. So … the existence of
mathematical objects is as well grounded as that of the other entities posited
by science. (1995, p. 166, my emphasis)
Resnik’s description of the argument suggests that there would be something
remiss in a) supposing that an experiment could confirm two different claims, one
physical and one mathematical, to the same extent, and yet b) believing ourselves to
be committed only to the objects invoked by the physical claim and not to the entities
of the mathematical. This is consonant with Putnam’s account of indispensability
arguments, where he discusses the ‘‘intellectual dishonesty of denying the existence
of what one daily presupposes.’’ (Putnam 1971, p. 347).4 In effect, holism generates a
problem: how are we to account for the empirical successes of our mathematical
claims? And a constraint on a reply is that it seems that we should explain the
empirical successes of mathematics in the same way that we explain the empirical
successes of any other scientific hypothesis. If we explain the success of physical
hypotheses by thinking that the entities they discuss are real, then we should give a
similar explanation of the empirical successes of mathematics; namely that the
entities they discuss are real. As such, any scientific realist should properly be
wholesale, rather than restricted, about her realism.
3
It should come as no surprise that the premise expressing evidential holism has
come under fire. Penelope Maddy thinks that holism so expressed is empirically
false as it makes a mistaken claim about the nature of evidence in mathematics, and
that it misdescribes actual mathematical and scientific practices and the important
differences between them (Maddy 1992). Elliott Sober thinks that holism so
expressed is an incorrect theory of confirmation; he too suggests that it
mischaracterises actual scientific practice, and he raises several problems in
confirmation theory which he claims it cannot answer (Sober 1993). I disagree that
CT gives a mistaken account of actual scientific and mathematical practices. But my
4 I do not mean to imply that Hilary Putnam should be understood as having argued for the existence of
abstract mathematical objects. Commentators often refer to indispensability arguments as ‘Quine-Putnam
indispensability arguments’; David Liggins (2008) makes it perfectly clear that the elision is incorrect,
and that any overlap is illusory.
268 J. Morrison
123
assessment of the holistic indispensability argument has this much in common with
both Sober and Maddy: evidential holism, as it appears in that argument, is false.
Following the line taken in (Morrison 2010), I maintain that evidential holism
proper describes the holistic nature of the deductive relations that must hold
between observations and hypotheses. Morrison (2010) argues that Quine’s
evidential holism is properly understood as a combination of only the prediction
and falsification theses (PT and FT), and that CT is a consequence of the mistaken
assumption that PT and FT will yield holistic conclusions about inductive relations
such as confirmation and disconfirmation. In this section I will argue that the holistic
argument just sketched does fail, and that its failure is a direct consequence of its
invocation of CT. The real problem with the holistic indispensability argument
given above is that it depends on CT; that there are no good reasons for endorsing
CT, and that there are plenty of good reasons for rejecting it.
Sober’s objections depend on his giving a particular construal of how he thinks
that the holist must ‘distribute’ confirmation accruing to an entire theory among its
constitutive parts. He maintains that evidential holists are committed to CT, and that
if they want to distribute the confirmation of an entire theory amongst its individual
parts then they cannot do so differentially—every part of a theory is confirmed tothe same extent by the evidence. But since scientists do not (and indeed, should not)
think about evidence this way, CT must have something wrong. I think that CT is
consistent with differential assignments of evidential values among the parts of a
theory. We should not conflate:
Confirmation Thesis (CT): Observations confirm entire theories, not individual
parts thereof.
With:
No individual theoretical claims can be differentially supported.
The confirmation thesis is a claim about the relata of confirmation: observations
taken by themselves only support entire theories; taken individually, observations
don’t single out any particular propositions/hypotheses for support. But this is entirely
consistent with the possibility that observations, taken in conjunction with various
auxiliary assumptions about the experimental set-up and about the relative likelihoods
of various hypotheses, will be able to identify particular propositions as being better
supported than others. The Bayesian formula (plus some argument about the correct
measure of confirmation) is just one example of the kind of supplementary
presupposition which might enable an observation to be said to support one particular
theoretical claim more than another. That is to say that even if one held CT, one could
still endorse differential distribution of support. Sober seems to think that by
definition such a position would no longer be ‘holistic’ (Sober 2000, p. 268 footnote
28). But CT is a holistic position: it denies that observations taken by themselves
admit differential support for the constituent parts of a theory, but that is all it denies.
Maddy’s concerns about CT and mathematical practice are as follows. She
suggests that if CT is correct we should expect mathematicians to be looking to
developments in science to tell them which of their theories are confirmed: ‘‘[i]f this
is correct, set theorists should be eagerly awaiting the outcome of debate over
Evidential Holism and Indispensability Arguments 269
123
quantum gravity, preparing to tailor the practice of set theory to the nature of the
resulting applications of continuum mathematics. But this is not the case; set
theorists do not regularly keep an eye on developments in fundamental physics.’’
(Maddy 1992, p. 289). Since they do not, CT must get something wrong. I do not
dispute her assessment of the practice of mathematicians. I think that the
expectation is misplaced—CT does not entail that each individual part of a theory
is supported by observation when the theory-as-a-whole is supported by observa-
tion. If CT is consistent with a notion of differential support, as I’ve suggested
above, then we should have no reason to expect mathematicians to look to scientific
results to find support for their theories.
Maddy’s argument about CT and scientific practice is as follows. Quine’s
evidential holism tells us that ‘‘our statements about the external world face the
tribunal of sense experience not individually but only as a corporate body’’ (Quine
1951, p. 41). But ‘‘the actual practice of science presents a very different picture’’
(Maddy 1992, p. 280), in so far as scientists have withheld their assent to the
existence of some of the posits of well-confirmed theories until claims about those
particular objects have received ‘direct verification’. If practicing scientists do not
think about evidence holistically, if they operate as though not all parts of a theory
are equally confirmed, then ‘‘[i]f we remain true to our naturalistic principles, we
must allow a distinction to be drawn between parts of a theory that are true and parts
that are merely useful.’’ (ibid. p. 281). Maddy argues that evidential holism is
descriptively false: it mischaracterises the actual nature of the evidential relation,
describing scientists as acting as though all parts of a theory are equally confirmed,
without differential support, when in fact they do not. I do not dispute her assessment
of the practice of scientists.5 I think that the expectation is misplaced—CT does not
entail that each individual part of a theory is supported by observation when the
theory-as-a-whole is supported by observation. And if CT is consistent with a notion
of differential support, as I’ve suggested above, then we should have no reason to
expect scientists to think that all parts of their theories receive equal confirmation.
However, if CT can be compatible with a notion of differential support, then the
holistic indispensability argument will not work. The mere fact that a theory that
indispensably relies on mathematical claims is supported by observation is no
reason to think that the mathematical claims are thereby supported. I can only
speculate that the many commentators on holistic indispensability arguments have
assumed a stronger version of CT that rules out the possibility of differential
support, although it is hard to see why that should be the case. A proper diagnosis
would require an examination of the motivations behind CT to see if there are any
particular reasons for thinking that holism about confirmation should be understood
in the stronger form which rules out any notion of differential support. Such an
examination is undertaken in Morrison (2010); what is surprising is the scarcity of
any arguments motivating CT in either form.
5 Quine clearly does not place too much emphasis in what scientists think that they are up to when
espousing his holism: ‘‘the scientist thinks of his experiment as a test specifically of his new hypothesis,
but only because this was the sentence he was wondering about and is prepared to reject.’’ (1990a, p. 14).
So while he was concerned to give an accurate explanation of scientific practice, it was not necessary that
it be consonant with the descriptions that scientists might use to describe their own activities.
270 J. Morrison
123
The confirmation thesis (CT) is openly endorsed in many mainstream epistemo-
logical discussions, where it is rarely given any explicit defence or motivation, and
at the same time it is regarded as highly controversial in discussions of evidence in
the philosophy of science. This rather remarkable discrepancy can be explained. In
contrast, the prediction thesis (PT) seems to be straightforwardly motivated; indeed,
Sober says it is something like an ‘‘unexceptional observation’’ (Sober 1993, p. 35).
I have suggested that the prediction thesis is broadly equivalent to the claim that
scientific observations are theory-laden, and thus that whatever intuitive plausibility
attaches to the theory-ladenness of observation is thereby motivation for the
prediction thesis (Morrison 2010). I suspect that adherents of the confirmation thesis
suppose that there is an intimate link between the prediction thesis and the
confirmation thesis. This supposition makes it seem as though the credibility
associated with the prediction thesis, which seems to be well motivated, will transfer
to the confirmation thesis. The inferential link between CT and PT is at once primafacie plausible and fallacious—indeed, for that very reason it shows up in paradoxes
to do with confirmation.
A minimum condition for thinking that the prediction thesis implies the
confirmation thesis will involve accepting a principle of the hypothetico-deductive
(HD) theory of confirmation. A crude account of this HD principle maintains that
for a to confirm b it suffices for a to be deductively derivable from b. By endorsing
PT, the holist is committed to the idea that an entire theory predicts a testable
observation, which is to say that the testable observation is deductively derived from
an entire theory. As a result, if the holist additionally endorses this principle of
hypothetico-deductivism then they should also accept that the observation confirms
the entire theory (CT).
In order to make an inference from the prediction thesis to the confirmation
thesis, we need to have a principled reason for thinking that whatever predicts an
observation gets confirmed by it. This HD principle does precisely that: it equates
confirming instances with deductive consequences. So we can see that in order to
establish any sort of deductive link between holism about prediction and holism
about confirmation we will need to appeal to something which makes the same
connection between prediction and confirmation as this HD principle. It is important
to note that any such adherent of CT is thereby relying upon some extra-holistic
machinery, which is itself in need of independent motivation. The holist who
accepts the prediction thesis needs some reason to think that they should be
committed to this hypothetico-deductive principle of confirmation before endorsing
the confirmation thesis. What motivates endorsing the HD principle as an additional,
extra-holistic component?
Arguments for this HD principle stem from the view that scientific laws are
universal generalisations, and that inductive support for these laws can be garnered
from particular instances of these laws. So, for example, we think that seeing white
swans lends inductive support for the claim that all swans are white. Prima facie, the
general idea that ‘instances support their generalisations’ seems innocent enough.
Indeed, at first glance to anyone who is not familiar with the debates that go on in
the philosophy of science, the principle seems an intuitive and eminently plausible
theory of how confirmation relations might work. As such, it is no surprise that
Evidential Holism and Indispensability Arguments 271
123
someone might think that the intuitive appeal of PT is sufficient to motivate CT.
However, the formalisation that any observed deductive consequence of b thereby
confirms b quickly leads to problems. For example, all swans are white entails that
all swans are swans, but it sounds strange to think that an observation of a swan that
is a swan confirms or lends inductive support to the claim that all swans are white.
This issue alone is suggestive that the hypothetico-deductive principle as it is stated
above requires modification. Indeed, no contemporary account of hypothetico-
deductivism would maintain the HD principle stated above, simply because it is
such a naı̈ve position that it results in these unintuitive consequences. All candidate
examples of sophisticated hypothetico-deductive positions, from its founder, Carl
Hempel, onwards have tried to capture the intuition that ‘instances support their
generalisations’ while avoiding the formalisation that equates confirming instances
with deductive consequences precisely because of these problems. Moreover, these
are not the only problems for hypothetico-deductive accounts of evidence.
Once we see that CT is not a consequence of ‘unexceptional observations’ about
the nature of prediction, it becomes difficult to discern any other motivations to
accept it.6 It is for this reason, rather than any particular concerns about
mathematical or scientific practice, that we should reject the holistic indispensability
argument that depends on CT. Furthermore, while PT and FT are well motivated
and defensible statements of evidential holism, they cannot can be employed to
generate indispensability arguments which should compel a restricted scientific
realist to be a wholesale scientific realist. Consider, for example, the evidential
holist who accepts PT and FT while repudiating CT—we might think that his theory
of evidence is somewhat more like a holistic falsificationism. His account of
evidence says that only entire theories yield observational predictions, and if those
predictions are incorrect then the observations, taken alone, only falsify the entire
theory. Suppose that mathematical claims are necessary constitutive parts of the
theory. Suppose further that this holistic falsificationist is a scientific realist, and is
willing to accept those ontological commitments of his best as-yet-unfalsified
theory. Should he also be committed to the existence of mathematical abstracta?
His theory of evidence makes it look as though the mathematical claims are
evidentially on a par with the scientific claims. If so, we could ask, as we did above,
whether it is acceptable for him to (a) suppose that two different claims, one
physical and one mathematical, might be evidentially on a par, both being as-yet-
unfalsified, and yet (b) believe himself to be committed only to the objects invoked
by the physical claim and not to the entities of the mathematical? The relevant
consideration here seems to be whether he can insist on treating mathematical
statements as mere instrumental tools in his theory rather than substantive
ontologically-committing parts of his theory. How could this restricted scientific
realism be defended?
6 It might be that various Quinean doctrines are supposed to yield CT. Jerry Fodor and Ernie Lepore
suggested that something like CT is a consequence of Quine’s dissolution of the analytic/synthetic
distinction or his semantic holism, but this seems unlikely (Fodor and Lepore 1992). Samir Okasha (2000)
explains why this gets the direction of fit the wrong way round. In general, the Quinean doctrines which
could plausibly be employed to produce CT tend to be more controversial than CT itself, so such
arguments won’t be straightforwardly motivating to most.
272 J. Morrison
123
One strategy is to explore what would happen if a whole theory (scientific and
mathematical claims combined) made a false prediction. In this case, PT tells us that
the whole theory has been falsified. Does it follow that all of the parts of the theory
are epistemically equal? We get the result that the falsifying-observation alone
underdetermines which part of the theory is to blame, and that in principle any one
of the constituent claims of the theory could be responsible for generating the false
prediction. In this narrow sense, the parts of the theory are epistemically equal: they
are observationally on a par. But it doesn’t follow that they are epistemically on a
par more generally. That is, we shouldn’t conflate:
Falsification thesis: Observations only falsify theories, not individual parts
thereof.
With:
No individual theoretical claims can be falsified.
Individual theoretical claims cannot be falsified by failed predictions alone, as
failed predictions fail to determine which of our theory’s claims is at fault. But this
is consistent with the possibility that failed predictions in addition with some otherauxiliary might allow us to single out particular claims for refutation. It follows that
the falsification thesis (FT) does not entail that every claim in a theory must be
epistemically on a par.
We should separate FT from the claim that no individual claims can be falsified,
and recognize that FT is consistent with understanding failed predictions as
allowing for a notion of differential evidential consequences for theoretical claim.
Doing so enables the holistic falsificationist to treat some claims differently from
others: for example, he can consistently maintain that those mathematical parts of
his theory, which he thinks are purely instrumental, are in no danger of being
falsified. His holistic falsificationism can provide a license for treating strictly
mathematical claims differently from the strictly scientific. Quine has long since
advertised this consequence of (falsificationist) evidential holism: ‘‘We exempt
some members of [theory] S from this threat [of falsification] on determining that
the fateful implication still holds without their help. Any purely logical truth is thus
exempted, since it adds nothing to what S would logically imply anyway; and
sundry irrelevant sentences in S will be exempted as well.’’ (Quine 1990b, p. 11).
Again, this demonstrates that FT does not entail that every claim in a theory must be
epistemically on a par.
So evidential holism, in the form of PT and FT but not CT, does not generate
indispensability arguments that compel a restricted scientific realist to be a
wholesale scientific realist—quite to the contrary, it provides a principled method
for resisting going wholesale.
4
Holistic indispensability arguments are supposed to suggest that a distinction
between the tools of a theory and a theory’s results is not defensible, and that as
Evidential Holism and Indispensability Arguments 273
123
such scientific realists should be as committed to mathematical abstracta as they are
to genes and atoms. Many commentators have agreed that evidential holism is false,
largely because of the sorts of criticisms suggested by Maddy and Sober against CT,
and that as such the holistic indispensability arguments are unsound. I have
disagreed with their specific criticisms of CT, but for all that I agree that CT is
indefensible. In contrast, I maintain that evidential holism is true, in the form of PT
and FT. However, there is no valid indispensability argument from this form of
evidential holism to the conclusion that scientific realists should be as committed to
mathematical abstracta as they are to genes and atoms. Most contemporary
discussions of indispensability arguments no longer turn on issues to do with
holism. While this is the right result, I have given reasons above for thinking that it
has come about for the wrong reasons.
Current interests have turned to explanatory indispensability arguments (Baker
2005, 2009; Bangu 2008; Colyvan 2002; Leng 2005; Melia 2002). The idea is that
even if the scientific realist wishes to maintain that claims involving mathematical
objects are mere tools, if it can be shown that those tools are indispensable to
explaining physical phenomena, then she should grant them the same ontological
status as she does to other theoretical posits. After all, the reason she willingly
accepts ontological commitments to genes and atoms is principally that they are
indispensably instrumental in explanations of physical phenomena. Much of these
discussions concern whether there are any genuine mathematical explanations of
physical phenomena. I do not intend to contribute to that issue. Rather, I think the
preceding discussion about evidential holism might constrain what can be said about
explanation in these new indispensability arguments.
Hempel thought that explanation is a relation that is symmetrical with
confirmation. That is, e confirms h if and only if h, if true, would explain e. Such
a symmetry thesis is not without intuitive appeal: it seems to make sense that where
an observation of litmus paper turning blue is taken to confirm that the liquid is
alkali, the fact (if it is one) that the liquid is alkali explains why the litmus paper
turned blue (see Bird 2010a for discussion). There are also good reasons for thinking
that the biconditional does not hold without exception—most counterexamples put
pressure on the conditional that if e confirms h it follows that h, if true, would
explain e. So, for example, the observation of litmus paper turning blue might be
taken to confirm that the liquid will be dangerous if ingested in large quantities, but
if true, this claim does not seem to explain why the liquid turned the litmus blue.
The converse conditional, however, seems much more defensible: that a hypothesis,
if true, could explain some physical phenomena is generally taken to be a reason for
thinking that observing the physical phenomena adduces some degree of empirical
confirmation on the hypothesis.
It is this consequence that is significant. Many commentators have been
convinced that Maddy’s concerns about mathematical practice are sufficiently
problematic to repudiate holistic indispensability arguments, and in its stead they
have turned to examination of explanatory indispensability arguments. In these, the
live concern is whether there are any genuine mathematical explanations of physical
phenomena. Alan Baker has suggested two such physical explanada: the prime-
numbered periodic life cycles of North American cicadas, and the relative efficiency
274 J. Morrison
123
of using hexagonal structures in building hives (Baker 2005, 2009). For each
physical phenomenon he argues that the correct explanans is a mathematical claim.
If Baker is correct that the mathematical propositions are the best explanations of
the physical phenomena, and if the relationship between explanation and
confirmation just described is plausible, then we have a reason for thinking that
in each case the physical phenomenon confirms the mathematical claim. If this is the
case, Maddy’s concerns about mathematical practice should follow: mathematicians
should augment their normal standards and methods of evidence with the additional
consideration that if scientists successfully use some mathematical propositions in
explaining physical phenomena then those propositions are better confirmed than
their rivals. If we take Maddy’s objections about mathematical practice seriously
enough to repudiate holistic indispensability arguments, we should take care not to
offend against them with explanatory indispensability arguments. As such, those
who are adherents of explanatory indispensability arguments because of Maddy’s
criticisms of holistic indispensability arguments should reject the prima facieplausible link between explanation and confirmation.
For the most part, scientific realists will not want to reject this prima facieplausible link between explanation and confirmation, since it goes to the heart of
their primary motivation for scientific realism: inference to the best explanation
(IBE). Alexander Bird expresses this point succinctly:
The basic idea behind IBE is that if a putative hypothesis would explain some
evidence, then that evidence provides some degree of confirmation to that
hypothesis. Thus the fact that Einstein’s general theory of relativity could
explain the anomalous precession of the perihelion of Mercury was a reason to
think that Einstein’s theory is correct. In some cases, several competing
hypotheses each provide possible explanations of the evidence. IBE tells us
that, subject to various constraints, the evidence most strongly confirms that
hypothesis which best explains the evidence. (Bird 2010b, p. 11)
Since explanatory indispensability arguments are aimed at scientific realists, and
since scientific realists appeal to IBE, which in turn appeals to a direct link between
explanation and confirmation, explanatory indispensability arguments show that
mathematical claims are confirmed. Thus, Maddy’s objection from mathematical
practice is seen to reapply: if scientific evidence can confirm mathematical
statements, then we should expect mathematicians to be looking to developments in
science to tell them which of their theories are confirmed. Since they do not, the
argument must get something wrong.7
7 Maddy’s mathematical practice objection has little force if mathematicians either do look to
developments in science to see which of their theories is confirmed, or if they have good reasons for not
doing so. I have not argued that mathematicians do not take considerations of the applied parts of their
theories into account, or that mathematicians should look for empirical support for their theories. Rather,
the argument here proceeds by parallel steps with the case against holistic indispensability arguments: if
we do take Maddy’s concerns about mathematical practice seriously enough to repudiate holistic
indispensability arguments, then the fact that explanatory indispensability arguments offend against those
same concerns is equally problematic.
Evidential Holism and Indispensability Arguments 275
123
The wider point of this attack on the explanatory indispensability argument is as
follows. The epistemological premise in the indispensability arguments considered
here has changed from confirmational holism to IBE, but the objection from practice
remains: since actual mathematicians don’t look to science to discover which of
their theories is confirmed, why should a scientific realist be committed to
mathematical abstracta? What’s stopping a scientific realist from accepting IBE,
endorsing mathematical claims as best explanations, but denying that the
mathematical parts are confirmed? If it seems ad hoc or arbitrary to think that
best explanations provide confirmation in all cases except those involving
mathematical claims, then the objection gives a principled reason for doing so:
mathematicians don’t take scientific evidence as confirmation of mathematical
claims, so neither should scientific realists. The principled reason for denying a link
between explanation and confirmation in the case of mathematical explanations of
physical phenomena is that mathematicians don’t consider scientific applicability to
be a source of support for their theories.
Supposing that IBE is one of the methods of science, then we have uncovered a
methodological difference between mathematics and science: scientific theorising
involves inferring the confirmation of propositions from their status as best
explanations of physical phenomena, where mathematics involves no such
inference.8 The tension that is raised by the mathematical practice objection is as
follows: naturalistic philosophers of science, who think that the epistemology of
science should follow from the actual evidential standards employed by scientists,
should endorse inferential methods such as IBE. Consistent application of IBE,
following the explanatory indispensability argument, entails the confirmation of
mathematical claims, and commitment to the existence of mathematical objects.
Mathematical naturalists, who think that the epistemology of mathematics should
follow from the actual evidential standards employed in maths, should not consider
a mathematical theory confirmed as a result of its ability to explain physical
phenomena.
Considered like this, the issue is not so much of a debate about the metaphysics
of mathematics, nor is it a disagreement between whether one should inquire into
the metaphysical questions prior to or only after settling the methodological
questions.9 Rather, the mathematical practice objection stands as a disagreement
between competing methodologies: should the question ‘‘do mathematical objects
exist?’’ be answered by scientific or mathematical ways and means, given that they
disagree? The holistic indispensability argument can be understood as an attempt to
deny that there are entirely distinct evidential methodologies available for
mathematics and science—but I’ve shown how evidential holism (properly
construed) fails to support such a conclusion. The explanatory indispensability
8 Mathematicians might still employ IBE within mathematics for inferring the confirmation of
mathematical theories or propositions from their status as best explanations of mathematical phenomena.
Such applications of IBE wouldn’t be sufficient to compel scientific realists to accept the existence of
mathematical abstracta since in these inferences the explananda aren’t sufficiently connected to the
general motivations of scientific realists (see also Leng 2005; Bangu 2008; Baker 2009). (Thanks to an
anonymous referee for encouraging this point).9 Penelope Maddy represents the debate this way in her 2005a (see p. 358)
276 J. Morrison
123
argument invokes a different evidential norm that’s found in science: IBE, which
appeals to a link between explanation and evidence, but which conflicts with the
evidential norms found in mathematics. In this sense, explanatory indispensability
arguments make the same mistake as holistic ones: they are insufficiently attentive
to the different standards of evidence at work in mathematics and in science.
Acknowledgments Thanks to Jacob Busch for his feedback on earlier drafts of this work and ongoing
encouragement. I’m indebted to the anonymous reviewers for this journal for having provided such
useful, clear and incisive comments. I’m also grateful to the editors of this journal for having been so
responsive with keeping communications clear, timely and relevant. Many thanks to my colleagues and
the audience at the University of Birmingham, where this work was presented to a research seminar, May
24th 2010. Further thanks for discussion and comments go to Darragh Byrne, Sean Cordell, Paul
Faulkner, Chris Hookway, Gerry Hough, Mary Leng, David Liggins, Arash Pessian, Joe Melia, Bob
Plant, Duncan Pritchard, Kirk Surgener, David Walker, Nick Wiltsher and Rich Woodward.
References
Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena? Mind, 114,
223–238.
Baker, A. (2009). Mathematical explanation in science. British Journal for the Philosophy of Science, 60,
611–633.
Bangu, S. (2008). Inference to the best explanation and mathematical realism. Synthese, 160, 13–20.
Bird, A. (2010a). Inductive knowledge. In S. Bernecker & D. Pritchard (Eds.), The Routledge companionto epistemology. London: Routledge.
Bird, A. (2010b). The epistemology of science – a bird’s-eye view. Synthese, 175, 5–16.
Colyvan, M. (2001). The indispensability of mathematics. Oxford: Oxford University Press.
Colyvan, M. (2002). Mathematics and aesthetic considerations in science. Mind, 111, 69–74.
Fodor, J., & Lepore, E. (1992). Holism: A shopper’s guide. Oxford: Blackwell Publishers.
Hale, B. (1994). Is platonism epistemologically bankrupt? Philosophical Review, 103(2), 299–325.
Jenkins, C. (2008). A priori knowledge: debates and developments. Philosophy Compass, 3(3), 436–450.
Klee, R. (1992). In defense of the Quine-Duhem thesis: a reply to Greenwood. Philosophy of Science, 59,
487–491.
Leng, M. (2005). Mathematical explanation. In C. Cellucci & D. Gillies (Eds.), Mathematical reasoning,heuristics and the development of mathematics (pp. 167–189). London: King’s College Publications.
Liggins, D. (2008). Quine, Putnam, and the ‘Quine-Putnam’ indispensability argument. Erkenntnis, 68(1),
113–127.
Linnebo, Ø. (2006). Epistemological challenges to mathematical Platonism. Philosophical Studies, 129,
545–574.
Maddy, P. (1990). Realism in mathematics. Oxford: Oxford University Press.
Maddy, P. (1992). Indispensability and practice. Journal of Philosophy, 89, 275–289.
Maddy, P. (2005a). Mathematical existence. The Bulletin of Symbolic Logic, 11(3), 351–376.
Maddy, P. (2005b). Three forms of naturalism. In S. Shapiro (Ed.), Oxford handbook of philosophy ofmathematics and logic (pp. 437–459). Oxford: Oxford University Press.
Melia, J. (2002). Response to Colyvan. Mind, 111, 75–79.
Morrison, J. (2010). Just how controversial is evidential holism? Synthese, 173(3), 335–352.
Okasha, S. (2000). Holism about meaning and about evidence: in defence of W. V. Quine. Erkenntnis, 52,
39–61.
Putnam, H. (1971). Philosophy of logic. New York: Harper.
Quine, W. V. (1951). Two Dogmas of empiricism. The Philosophical Review, 60, 20–43.
Quine, W. V. O. (1990a). Pursuit of truth. Cambridge, Mass: MIT Press.
Quine, W. V. O. (1990b). Three indeterminacies. In Barrett & R. Gibson (Eds.), Perspectives on Quine(pp. 1–16). Oxford: Blackwell Publishers.
Resnik, M. D. (1995). Scientific vs. Mathematical realism: The indispensability argument. PhilosophiaMathematica, 3(2), 166–174.
Evidential Holism and Indispensability Arguments 277
123
Resnik, M. D. & Orlandi, N. (2003). Holistic realism: a response to Katz on holism and intuition.
Philosophical Forum, 34 (3 & 4), 301–316.
Rey, G. (1998). A naturalistic a priori. Philosophical Studies, 92, 25–43.
Sober, E. (1993). Mathematics and indispensability. Philosophical Review, 102(1), 35–57.
Sober, E. (2000). Quine: I—Elliott Sober, Quine’s Two Dogmas. Proceedings of the Aristotelian Society,
74 (Supplementary Volume), 237–280.
Wagner, S. (1996). Prospects for platonism. In Morton & S. Stich (Eds.), Benacerraf and his critics (pp.
73–99). Oxford: Blackwell.
278 J. Morrison
123