evidential holism and indispensability arguments.pdf

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]

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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

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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

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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


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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

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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


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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

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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


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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

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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


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.


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

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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.

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