OP-BJPS120056 255..278
Indispensability and Explanation
Sorin Bangu
ABSTRACT
The question as to whether there are mathematical explanations of physical phenomena
has recently received a great deal of attention in the literature. The answer is potentially
relevant for the ontology of mathematics; if affirmative, it would support a new version of
the indispensability argument for mathematical realism. In this article, I first review
critically a few examples of such explanations and advance a general analysis of the
desiderata to be satisfied by them. Second, in an attempt to strengthen the realist position,
I propose a new type of example, drawing on probabilistic considerations.
1 Introduction
2 Mathematical Explanations
2.1 ‘Simplicity’
3 An Average Story: The Banana Game
3.1 Some clarifications
3.2 Hopes and troubles for the nominalist
3.3 New hopes?
3.4 New troubles
4 Conclusion
1 Introduction
The indispensability argument (IA) for mathematical realism was advanced by
Quine and Putnam many years ago, but has been recently revived in a more
specific form, which can be called ‘explanationist’. This version was first inti-
mated, curiously, by the arch anti-realist Hartry Field ([1989], pp. 15–7); the
underlying idea is to combine the powerful and more general scientific realist
argument from ‘inference to the best explanation’ (IBE) with the view that
mathematics is indispensable to science. Field, however, has never fully
Brit. J. Phil. Sci. 64 (2013), 255–277
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articulated an argument to this effect; the insight has been developed by Alan
Baker in two recent articles ([2005], [2009]
1
).
But why do realists need a new version of the traditional IA? A good way to
understand their motivation is to review, briefly, the traditional IA as well as
some of the criticisms levelled against it.
Mark Colyvan ([2001], [2008]) presents the argument as follows:
(i) Mathematical entities are indispensable to our best scientific theories.
(ii) We ought to have ontological commitment to all and only the entities
that are indispensable to our best scientific theories.
2
From (i) and (ii), the conclusion is drawn that we ought to have ontological
commitment to mathematical entities. More precisely, a scientific realist must
be a mathematical realist: she must take numbers to be part of her ontology,
along with electrons and genes. Thus, in light of IA, a scientific realist who
doesn’t acknowledge herself as a mathematical realist is operating with a
‘double standard’ with regard to ontology (Quine [1980], p. 45), and thus is
guilty of ‘intellectual dishonesty’ (Putnam [1979], p. 347).
Its validity being beyond question, the criticisms of the argument focused on
the credibility of the premises. Notably, Field ([1980]) tried to show that (i) is
false.
3
Penelope Maddy ([1997]) doubted the ‘all’ part of premise (ii), i.e.
confirmational holism.
4
More recently, a novel criticism against IA was
launched by Joseph Melia ([2000], [2002]). Melia, a scientific realist, accepts
(i) and attacks the ‘all’ part of (ii) as well, but from a somewhat different angle.
He argues that one is not intellectually dishonest when refraining from making
ontological commitments to everything that our theories quantify existentially
over.
5
Mere indispensability of a posit is not enough, according to Melia; a
posit must be indispensable ‘in the right kind of way’, as Baker ([2009], p. 613)
put it, and one specific role a posit must necessarily play is in formulating
explanations.
Accordingly, the question Melia raises is to what extent mathematical posits
(can) fulfil this role. In essence, he proposes that to grant ontological rights to
mathematical objects only when positing them ‘results in an increase in the
1
In this latter article, Baker calls it ‘the Enhanced Indispensability Argument’. Steiner ([1978a],
[1978b]) is perhaps the first philosopher to reflect on this issue systematically. (See Baker [2009]
and Leng ([2005], [2010]) for recent reactions to his work.)
2
This premise incorporates Quine’s well-known criterion for ontological commitment: ‘a theory
is committed to those and only those entities to which the bound variables of the theory must be
capable of referring in order that the affirmations made in the theory be true’ ([1948], p. 33). For
a recent discussion of this criterion, see Azzouni ([1998], [2004]).
3
Many remain unconvinced, for a variety of reasons. As far as I can tell, Malament’s ([1982])
objections still stand.
4
Sober ([1993]) objects to confirmational holism too, on the basis of his views about confirmation
as being contrastive.
5
Melia’s strategy is called ‘weaselling’; see (Melia [2000]). (Colyvan [2010]) is a recent criticism of
this strategy.
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same kind of utility as that provided by the postulation of theoretical entities’
([2002], p. 75), where, as noted, explanation is a primary example of such
utility. Now, one reading of this proposal is that it advances a criterion for
ontological commitment linking existence with causal explanation—precisely
because theoretical entities are typically taken to play a causal role in explan-
ations.
6
Thus, the criterion for ontological commitment would change into ‘to
be real is to fall within the range of an existential quantifier and to have a
causal explanatory role’. But, for a naturalist–indispensabilist–realist, this
appeal to causality marks an unwelcome return to metaphysics (or ‘first-
philosophy’)
7
, hence it is a prima facie dubious move; on the other hand, if
the causal requirement is dropped from it, the criterion can be accepted by the
indispensabilist–realist. So, I submit, it is this construal of Melia’s challenge
that should be dealt with and this is something that realists attempted to do, as
we’ll see below.
Returning to the main argument, recall that at the very beginning I attrib-
uted the proto-explanationist version of IA to Hartry Field. More concretely,
the line of thinking he envisages is as follows: Suppose we hold a belief about a
physical phenomenon, and we present the best explanation of that phenom-
enon. Furthermore, suppose that a certain claim K is part of this explanation,
and no explanation of the phenomenon is possible without holding claim K.
Field notes:
[i]f a belief [K] plays an ineliminable role in explanations of our
observations, then other things being equal we should believe it,
regardless of whether that belief is itself observational, and regardless
of whether the entities it is about are observable. ([1989], p. 15).
The relevance of this idea for the issue of mathematical realism is clear. If a
physical phenomenon is best explained by making several assumptions,
and at least one is an ineliminable mathematical claim K, then IBE entitles
us to believe that K is true, and that the mathematical posits making it true
exist.
Yet, as I pointed out previously, this insight had not been worked into a full
explanationist–indispensabilist argument until very recently, when Baker
6
To use a well-known example, the neutrino was postulated (by W. Pauli) to explain the missing
amount of energy in a beta decay. This explanation can be said to be ‘causal’, in so far as it
appeals to the ‘causal power’ of the particle (its mass–energy).
7
As Quine points out, ‘the notion of cause is out of place in modern physics [ . . . ] Clearly the term
plays no role at the austere levels of the subject’ ([1974], p. 6) and ‘[s]cience at its most austere
bypasses the notion [of cause]’ ([1990], p. 76). By overlooking this aspect of the indispensabilist
position, Field’s (and others’) considerations from the role of causality in the decision to believe
in the existence of mathematical entities are beside the point. (See Field ([1980], p. 43), ([1989],
pp. 18–20), etc.). Burgess and Rosen ([1997]) also highlight the difficulties to pin down the
distinction between concrete and abstract, when this distinction is spelled out as the distinction
between causally efficacious versus causally inert.
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did it. He maintains that we ought to believe in the existence of mathematical
objects because
(i) we ought to believe in the existence of any entity that plays an indispens-
able explanatory role in our best scientific theories, and
(ii) mathematical entities do play such a role. (Baker [2009]).
This version of the argument is of course valid, yet two aspects of it require
clarification.
First, how do mathematical entities enter the picture? The answer an
explanationist–indispensabilist gives is that they are the truthmakers of the
true mathematical statements appearing among the explanans. Belief in the
existence of the mathematical objects is required by assuming the connection
between what Shapiro ([1997], [2000]) calls ‘realism in truth-value’ and ‘real-
ism in ontology’. Shapiro notes that the preliminary conclusion for which the
indispensabilist argues is that we ‘must accept mathematics as true’ ([2000],
p. 216). An explanationist can make this more precise: we must accept as true
the mathematical components of the explanans of the best explanation.
Accepting mathematics as true amounts to realism in truth value and, as
Shapiro puts it,
we get to realism in ontology by insisting that the mathematics be taken
at face value, just as we take physics at face value. Mathematical
assertions refer to (and have variables ranging over) entities like real
numbers, geometric points, and sets. Some of these assertions are literally
true. So numbers, points, and sets exist. ([2000], p. 216)
8
The second important aspect to be clarified is what sense of explanation is
used in this debate. Generally speaking, it is assumed that (i) the explanations
have the form of an argument in which the explanandum is the conclusion to
be derived from the premises (the explanans), and (ii) the explanans of good
explanations have to be true. Claim (ii) is potentially contentious because not
everyone agrees that these mathematical statements have to be true to serve
their role (those arguing along these lines typically endorse a form of fiction-
alism; see Leng ([2005], [2010])). However, in what follows, I assume—
together with other indispensabilist–explanationists—that we should require
the truth of the explanans (hence of their mathematical components) in a good
explanation.
9
Another typical realist assumption is also made here, namely,
that the simpler and more unifying an explanation is, the better. These points
8
Note, however, that this last inference is controversial. Helman ([1989]) and Chihara ([1990]),
for instance, reject it. They develop philosophies of mathematics which construe the truth of the
mathematical statements as not requiring the existence of mathematical ‘objects’. See Shapiro
([1997]) for criticisms of this strategy.
9
I suspect that the fictionalist conflates the explanatory role of the mathematical posits with their
representational (modelling) role. Baker ([2009], pp. 625–7) offers a more elaborate defense of
the conflation charge, and I’d direct the interested reader to his arguments.
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should be kept in mind, as they will be relevant for assessing the new example
of an explanation I propose later on.
A look at the recent literature reveals that both realists and nominalists
agree on where the disagreement between the two metaphysical camps
lies
10
: it is premise (ii). Both parties accept that mere indispensability might
not be enough. The question agreed by both sides to be crucial is whether
convincing examples can be found, in which, as Melia insisted, the best ex-
planation of a physical phenomenon features mathematics in an essential way.
Thus, the realists have set out to meet Melia’s challenge, as clarified previ-
ously. Such examples have been proposed, and extensively discussed recently,
especially by Colyvan ([2001], [2008]).
11
Against this background, let me finally state my goals in this article. I intro-
duce and discuss a number of examples, and I advance an analysis of the
desiderata that these explanations must satisfy to be able to support a realist
ontology. After I show why a new example is needed, I present such an ex-
ample and argue that it satisfies the constraints I outline in the first part of the
article.
2 Mathematical Explanations
A recent collection of interesting mathematical explanations is in Colyvan
([2001]). They are drawn from a wide range of scientific fields, from meteor-
ology to special relativity and dynamic non-linear systems theory. Consider
one from meteorology: given a certain moment of time, we want to explain
why are there two antipodal points, P0 and P1, on the earth’s surface with the
same temperature and barometric pressure ([2001], p. 49)? As Colyvan argues,
it is the Borsuk–Ulam theorem in algebraic topology that constitutes an es-
sential part of an explanation as to why such points exist. Another more recent
example appeals to the explanatory power of a mathematical construct called
the phase space. On Colyvan and Lyon’s account (which develops a sugges-
tion of Malament [1982]), the introduction of this notion (together with the
concept of a Poincare map) has a crucial role in explaining why ‘high energy
Henon–Heiles systems exhibit chaotic and predictable motion and why low
energy ones exhibit regular and predictable motion’ (Colyvan and Lyon
10
(Baker [2005]) is an intervention in a debate in Mind between Colyvan ([2002]), on the realist
side, and Melia ([2002]), on the anti-realist (or ‘nominalist’) side. Meanwhile, a growing number
of other authors joined the discussion–Azzouni ([2004]), Leng ([2005], [2010]), Pincock ([2007]),
Saatsi ([2007]), Mancosu ([2008]), Bangu ([2008]), Daly and Langford ([2009]), Batterman
([2010]), Colyvan and Bueno ([2011]), and so on–with divided sympathies. I don’t have space
here to get into any of the details of these positions, but the interested reader should be warned
that there are subtle differences even among the advocates of the same orientation, either realist
or antirealist.
11
(Balaguer [1998]) contains interesting examples too. Baker’s cicada example, in (Baker [2005]) is
also central in this context, and I’ll comment on it separately later on.
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[2008], p. 56). Colyvan also discusses Minkowski’s geometrical explanation of
certain relativistic effects (such as Lorentz contraction) and points out that it is
‘not obvious how such explanations could be obtained otherwise’—that is,
other than by introducing mathematical assumptions ([2001], p. 50).
Some of the examples mentioned involve explanatory power indirectly, by
drawing on the well-known connection between unification and scientific
explanation (Friedman [1974], Kitcher [1981], Morrison [2000]). More specif-
ically, the employment of certain mathematical concepts and structures often
results in unifying a scientific theory, and hence contribute to enhancing its
explanatory power. Arguing along these lines, Colyvan points out that it is
precisely this feature that is ‘hard to reproduce without mathematical entities’
([2001], p. 89).
12
2.1 ‘Simplicity’
Although I generally agree with all this, let’s observe that these examples come
from scientific fields characterized by a high degree of mathematical complex-
ity (applied algebraic topology, chaos theory, and so on). ‘Complexity’ is of
course a vague notion, but it is unquestionable that these examples don’t
feature anything properly called elementary, familiar, or simple. Note, further,
that by its very nature the challenge posed to the nominalist seems to require
an appeal to this complexity. The examples presented to the nominalist must
be non-elementary—hard—precisely because it must be hard to see how the
explanatory power of the mathematized theories can be reproduced without
mathematics. The idea is that the nominalist must be overwhelmed by the
complexity of the example and declare that nominalization manoeuvres
(such as the one described in Footnote 12) aren’t in sight, and thus mathem-
atics is indispensable to formulating the explanation. By presenting complex
examples, the realist wins. But this victory is not as crushing as it could be,
were the examples simpler. Faced with a very complex example, the nominalist
might admit defeat; yet, he is entitled to claim that this defeat is only tempor-
ary. He replies that nominalization is still possible, but just difficult to achieve
right away, as the theory is so complex—call this ‘the complexity excuse’ for
failing to nominalize.
12
In the current jargon, to purge a statement of its mathematical constituents is to ‘nominalize’
it (Field [1980]). The statement ‘there are two bananas on the table’ seems to make reference
to a mathematical object, the number two. Indeed, one way to reformulate it is as ‘the num-
ber of bananas on the table is two’. However, we can show that there is yet another way to
reformulate it in first order logic dispensing with reference to numbers altogether:
9x9yðFx ^ Fy ^ x 6¼ y ^8zðFz � ðz ¼ x _ z ¼ yÞÞÞ, where F is the concept ‘banana on the
table’. The nominalist project is to treat every single scientific theory (and thus every scientific
explanation) in this spirit.
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Moreover, the complexity of an example, by making nominalization hard to
achieve, generates two unwelcome effects. For one thing, it doesn’t allow us to
see how mathematics is actually explanatory; given the entanglement of highly
sophisticated mathematical and physical assumptions, the role of mathematics
becomes rather difficult to discern. Furthermore, because the nominalization
is not available, a comparison of the two versions of the explanations (the
mathematized and the nominalized one) becomes impossible. This seems to
me a disadvantage for the realist: in so far as a nominalized version of the
explanation is not entirely explicit, she can’t credibly argue that the mathe-
matized version of the explanation is better (the best).
13
These points should reveal the virtues of some simpler examples. While it
would indeed be much riskier for the realist to test whether the nominalist can
nominalize a simple example; such an example naturally increases the nom-
inalist’s chances to succeed in dealing with it. Hence, a more promising ap-
proach for the realist would be to challenge the nominalists by presenting
them with examples as elementary as possible, in which the appeal to some
basic mathematical vocabulary significantly enhances our explanatory re-
sources. An elementary example resisting nominalization will still amount
to a plausibility argument for realism; although nothing can guarantee that
all nominalization attempts will fail, such an example is more convincing in so
far as the complexity excuse will no longer be available to the nominalist.
To sum up, this desideratum—call it ‘Simplicity’—is reasonable because it
makes the challenge to present a successful nominalization more pressing for the
nominalist. The failure to succeed in nominalizing a simple example is surely
more telling in favour of realism than the failure to nominalize a highly complex
one. In addition, because it is easier to carry out a successful nominalization,
then, when this is done, a comparison becomes possible between the explanatory
virtues of the mathematized and the nominalized versions of an explanation.
Before I discuss an example that satisfies the simplicity desideratum (by
Baker [2005]), let me sketch out two other desiderata that should regulate
the use of the IBE strategy when employed to support mathematical realism.
If we recall Field’s idea from the Introduction, we note that a realist eager to
use it faces the following dichotomy. Either
(i) the explanandum/conclusion contains ineliminable (i.e. non-
nominalizable) mathematical vocabulary, or
(ii) the mathematics is eliminable.
If (i), the realist’s task becomes extremely difficult. Roughly speaking, the
problem is this: The realist has to take the explanandum to be true (otherwise
13
Not a serious disadvantage though; I agree with one of the referees for this article, who pointed
out that until an attractive nominalization is available, the realist wins.
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why bother explaining it?) and, in this case, the mathematical part of the
explanandum (which, again, can’t be eliminated) has to be true as well. But,
if this is so, she just assumes realism before arguing for it, so she would beg the
question against the nominalist. Consequently, the second desideratum is that
the explanandum be nominalizable (‘nominalize’ for short).
14
Branch (ii) is more promising for the realist. In this case, the mathematical
component of the explanandum is there but only superficially, as it can be
eliminated via a nominalization procedure. So, by taking the explanandum to
be true, the realist doesn’t beg the question anymore. But now the nominalist
needs to be provided with a further reason to see how the mathematical ex-
planans (which must occur as part of the premises to talk about mathematical
explanation in the first place) can have any explanatory relevance for an ex-
planandum that is in fact non-mathematical, that is, free of any traces of
mathematical vocabulary. Thus, the realist needs a further argument to
show that (at least one of) the explanans she uses to derive the explanandum
contains indispensable mathematical components. If they (these premises–
explanans) can be nominalized too, then what we get in the end is a nomina-
lizable explanandum (the conclusion) and nominalizable explanans (the
premises). Thus, to the nominalist’s satisfaction, it turns out that mathematics
was there only to capture (represent, describe, model, and so on) some essen-
tially non-mathematical content (the premises–explanans and the conclusion–
explanandum) in a more elegant fashion.
So, the challenge to the indispensabilist realist is to show that the mathem-
atics in the explanans is indispensable, given that the conclusion is nominaliz-
able. (I’ll abbreviate this desideratum as ‘indispensability’.) That the
mathematics in the explanans is not eliminable needs to be shown each time
one proposes a mathematical explanation, as a matter of principle. However,
whether this can be done in each case can’t be decided in advance, as it de-
pends on the specific form of the explanans.
Importantly, note that ‘indispensability’ is only a sufficient condition, not a
necessary one. It has to be balanced against the fourth (and essential) desid-
eratum—namely, that the mathematics involved in the premises be genuinely
explanatory (desideratum ‘explanation’ for short). I’ll discuss this constraint
14
This is essentially the point I made in my ([2008]). The problem is even more pressing for the
mathematical explanations of mathematical statements. Such explanations can’t count as sup-
porting mathematical realism, as Leng correctly notes:
one might wonder why it is mathematical explanations of physical phenomena that
get priority. For if there are [. . .] some genuine mathematical explanations [of
mathematical facts] then these explanations must also have true explanans. The
reason that this argument can’t be so is that, in the context of an argument for
realism about mathematics, it is question begging. For we also assume here that
genuine explanations must have a true explanandum, and when the explanandum
is mathematical, its truth will also be in question. ([2005], p. 174)
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later, in the context of the new example; however, before we move on, let me
clarify why ‘indispensability’ is not necessary. Indispensability realists say that
mathematics is indispensable in the sense that it is needed in achieving certain
goals. In this case, the goal is maximal explanatory power (other goals, such as
descriptive accuracy are often mentioned). So, even if a nominalist manages to
eliminate the mathematics from the explanans of a scientific explanation, his
case against the realist is not yet completed. He still has to show that the nom-
inalized version of the explanation is better—more attractive—than the math-
ematized version. So, if desideratum ‘explanation’ is satisfied and the
mathematics in the explanans is explanatory, but by nominalizing the explan-
ans the theory loses explanatory power, then achieving a nominalization of the
explanans is a hollow victory. The realist is entitled to claim that the mathe-
matized version of the explanation is superior, hence she is in the possession of
‘the best explanation’.
Back to the discussion of the examples in the literature, Baker ([2005]) de-
serves special mention here. His article argues that mathematical assumptions
feature essentially in the explanation of the fact that the life cycle of North
American magi-cicadas is a prime number. From the perspective I’ve articu-
lated here, what is important about this example (its intrinsic merits aside) is
that its complexity is rather minimal. (In essence, it is two lemmas drawn from
number theory that constitute the gist of the explanation. See Baker [2005],
p. 232)). Although desideratum ‘simplicity’ is met, the cicada example is not
without problems, as I argued in my previous article ([2008]) (for the larger
context of this argument, see (Mancosu [2008], Section 3])). The key difficulty
is that this example seems to fail to satisfy desideratum ‘nominalize’. Tellingly,
Baker ([2009], p. 619) revisits this issue and discusses the nature of the explan-
andum—‘the cicadas’ life-cycle is a prime number’—in the end acknowledging
that it can’t be nominalized! However, even assuming that this difficulty with
the cicada example gets sorted out eventually, I should emphasize that a new
example featuring the same low degree of complexity is necessary.
15
This is so
not only because the cicada example has already been attacked on other
grounds as well,
16
but also because the plausibility of the explanationist
15
To clarify: (Baker [2009]) also contains an argument to the effect that despite this feature of the
example, the charge of circularity I raised in my ([2008]), can be avoided. Therefore, until some
more careful assessments of Baker’s response are available, I should say that what I count
against the cicada example here is only the suspicions that it might fail to satisfy constraint
‘nominalize’. So, while I personally am not convinced of Baker’s response to my criticism, I’m
ready to admit that it might work. Yet, even if it does, a new example, free of such suspicions,
should be welcome by the realists.
16
See also (Leng [2005]) and (Saatsi [2007]). Daly and Langford ([2009]) point out that the fourth
desideratum (to use the terminology introduced here) is not satisfied. (Baker [2009]) addresses
these criticisms, but (Rizza [2011]) raises a new one.
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version of the indispensability argument is better supported by expanding the
number and variety of cognate examples.
17
In what follows, I propose such a new argument example.
18
After I describe it,
I point out the serious difficulties faced by the nominalist when attempting to
offer a non-mathematical treatment of it. I shall pay special attention to showing
that all four desiderata mentioned previously are satisfied in a very natural way.
3 An Average Story: The Banana Game
Consider the following game, played by two people, call them A and B. They
start by making two large crates, labelled ‘X ’ and ‘Y ’. Crate X contains two
identical smaller crates, labelled x1 and x2, each containing bananas.
Similarly, crate Y contains two smaller crates, labelled y1 and y2, each con-
taining bananas as well. A possible distribution of them would be: five ba-
nanas in x1, seven in x2, one in y1, and nine in y2. Such a distribution will be
described as [(5, 7); (1, 9)]. Of course, nominalists can’t express this informa-
tion in this way, as they lack numbers.
19
But they are able to do pairings
(one-to-one correspondences) and apply predicates such as ‘_ contains more
bananas than _’ or ‘_ contains as many bananas as _’ correctly when presented
with any two crates. (They just take all bananas out from the two crates they
want to compare and then form the pairs). Thus, the nominalists have no
difficulty noticing that, for example, crate x1 contains more bananas than
crate y1 and fewer bananas than y2, or that y2 contains more bananas than
any other small crate. They also have access to the fact about the two large
crates that ‘crate X contains more bananas than crate Y in total’. (Obviously,
they would have noticed this if they had availed themselves of numbers and,
by counting the bananas, they had gotten twelve bananas for crate X and ten
for Y.)
The rules of the game (call it ‘Game’) are as follows:
(a) The players know what is in each large crate, namely, that X and Y
contain small crates; they also know what is in each small crate.
(b) One player (the first player, say A) is asked to choose either crate X or
crate Y. The other player (B) will be left with the other large crate.
17
Baker himself invites such developments: ‘it is clearly less than ideal to rest the argument for the
existence of abstract mathematical objects on a single case study from science. Thus one line of
further inquiry on the platonist side is to look for more good examples of mathematical explan-
ation in science’ [2009, p. 631].
18
Its conceptual background is in economics. Similar games are discussed in von Neumann and
Morgenstern’s Theory of Games and Economic Behavior ([2007]; first published in 1944).
19
As a referee pointed out, can’t the nominalists express the content (for example, 5 bananas in x1,
7 in x2) in the familiar way using first-order logic and identity, as explained in Footnote 12? They
surely can; but they can’t use the signs ‘5’ or ‘7’ to express this content in the same way the realist
uses them, as standing for some objects.
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(c) Once A decides on a large crate, the choice of a small crate inside it is
purely chancy. However, the probabilities of choosing x1 or x2 are
equal, and the same goes for y1 and y2.
(d) Once a small crate is picked, A collects all the bananas in it; B does
the same.
(e) The goal of the game is to collect the most bananas. So, player
A should make the initial choice of the large crate with this goal in
mind.
(f) The choice of the large crates X or Y is made only once, at the begin-
ning of the game. If A has chosen to withdraw bananas from large
crate X (say), then he’ll keep doing this until the end of the game.
(g) After a small crate is emptied, it is refilled with as many bananas as
there were in it initially.
More concretely, the ‘Game’ is played as follows: Let us say that player
A goes for crate X, so B is left with crate Y. Now, let’s say A happens to
pick small crate x1 from large crate X, then he will collect all bananas in it.
B, the second player, is left with large crate Y, and he randomly picks a
small crate inside it. Let’s say he picks y1, then he collects what’s in it, and
the first run ends. Now crates x1 and y1 will be refilled with the same
number of bananas they contained initially, and a new turn begins, with
the first player picking from crate X. After many turns—say, a whole day of
playing—the two players’ piles are compared by using the predicate ‘_
contains more bananas than _’. Let’s also assume that, quirky characters
as they are, the two players enjoy the Game and play it for weeks. At the
end of one day, after many turns, the two players compare their piles, and
the winner is recorded. The next day, they play again, with player A making
his/her choice of a large crate, followed by many turns. The winner is
recorded again, and so on.
What is the outcome of the Game? Before start of play, the two large crates
don’t seem very different. The total amount of bananas each of them contains
is not markedly disproportionate (twelve versus ten), so the thought that each
large crate will win some runs would sound reasonable; in other words, it
might seem that any large crate can be chosen and only luck will decide.
Yet, as the game unfolds, and the days pass, the players observe an interesting
regularity, or pattern: there is a noticeable discrepancy in how many times one
and the same large crate wins, when compared with the other. Thus, one of the
crates (as it happens, crate X) wins overwhelmingly often (and, naturally, the
other one, crate Y, loses almost all the time).
Consider, furthermore, a new game (call it Game*) within the following
set-up. There are 0 bananas in x1, 16 in x2, 12 in y1, 12 in y2, and 6 in y3. The
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rules are the same as for the first Game, except for rule (c) that fixes the
probabilities:
Pr
� pick x1ð Þ ¼
1
4
Pr
� pick x2ð Þ ¼
3
4
Pr
� pick y1ð Þ ¼
1
6
Pr
� pick y2ð Þ ¼
2
6
Pr
� pick y3ð Þ ¼
3
6
Note that although the nominalist can’t use numbers, specific probabilities
(such as 1
2
or 3
4
) are still accessible to him. They can be constructed as follows:
Assume there is some randomization device yielding equi-probable outcomes.
For the first Game, a fair coin will do. A convention may be set up: if the coin
lands heads-up, then the player who collects from crate X will collect from x1;
if it lands tails-up, he’ll collect bananas from x2. A similar convention would
work just as well for crate Y and player B. Same reasoning can be used for
Game*. To understand the probability of 1
4
(say) nominalistically, we intro-
duce something like a physically symmetric four-sided die, whose faces (call
them s1, s2, s3, s4) are equi-probable. Thus, we convene that x2 would be
selected when the four-sided die lands any of faces s1 to s3 up, and we select
x1 when the four-sided die lands face s4 up.
On playing Game*, the players notice, again, that one of the crates wins
most of the time (crate X). Again, the wins are not evenly distributed, as one
might think on a superficial examination of the set-up: one crate, X, is (almost)
always the winner. In fact, this result is puzzling when compared with what
happened in the Game: while the winning crate in Game contained more
bananas than the losing crate (twelve vesus ten), in Game*, the winning
crate contains fewer bananas than the losing one (sixteen versus thirty).
Now, the players notice that the two games resemble one another in this
interesting respect: in both games, one crate wins overwhelmingly often. Is this a
mere accident, they wonder, or a fact that can be explained? Call the italicized
sentence ‘the explanandum’. Thus, the task set for the nominalist and the
realist is to identify what can account for it, or in other words, to explain
what makes the two games alike in this regard, i.e. unidirectional. This is, I
claim, something that the realist can handle relatively easily, whereas the
nominalist faces serious difficulties.
Let’s begin with the realist. She will proceed by introducing the elementary
mathematical notion of an expectation value, that is, a number associated
with picks from each large crate. More precisely, if we symbolize the
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expectation value associated with picks from X and Y by [X] and [Y], respect-
ively, in Game we have
20
:
X½ �¼ Pr pick x1ð Þ� # of bananas in x1ð ÞþPr pick x2ð Þ� # of bananas in x2ð Þ
Y½ �¼ Pr pick y1ð Þ� # of bananas in y1ð ÞþPr pick y2ð Þ� # of bananas in y2ð Þ
Because all probabilities are equal to 1
2
, given the numbers of bananas in each
small crate, we calculate [X]¼6 and [Y]¼5.
Moreover, the realist observes that there is a way to know what happens in
the long run. The player who selected crate X will gather approximately
([X]�n) bananas after n runs, and the one who chose Y will collect ([Y]�n)
bananas. This claim is justified by applying the well-known mathematical
theorem called the ‘(weak) law of large numbers’ or wLLN for short.
Consider V1, V2, . . . , Vn, a sequence of independent and identically distributed
random variables with finite expected values �¼ [V1]¼ [V2]¼ . . .¼ [Vn] and
finite variance. Let Vn ¼
1
n
(V1 + V2 + V3 + . . . + Vn) be the arithmetic mean (or
‘sample average’) of these variables. The (weak) law of large numbers states
that the arithmetic mean converges in probability to the expected value. (More
precisely, for any positive number ", P(jVn–�j<") !1, as n approaches in-
finity.) In essence, the theorem says that as n increases, the sample average Vn
gets closer and closer to the expected value, �.
The application of this theorem to our game is immediate. Consider crate X.
At every turn, a pick from crate X can yield either small crate x1 (i.e. five
bananas) or x2 (i.e. seven bananas), each of them with probability
1
2
. Thus, a
pick from crate X can be regarded as yielding values for a random variable (i.e.
two values, either five or seven) with equal probability 1
2
. Let A1 be this random
variable, associated with the first pick from X. Because A1 takes numerical
values (either five or seven) with probability 1
2
, the expected value is [A1]¼6.
The same holds for the second, third, etc. pick from X, so let us call these
random variables A2, A3, . . . , An. The expected values are [A2]¼ [A3]¼ . . .¼6.
In this case then, �¼6. Now, consider the arithmetic mean An ¼
1
n
(A1 + A2 + A3 + . . . + An). We know from wLLN that for sufficiently large n
(that is, after many turns), we have that An gets arbitrarily close to �. Or,
equivalently, the quantity of interest (i.e. the quantity of bananas in the first
player’s pile) is (A1 + A2 + A3 + . . . + An), and it approaches n�. More con-
cretely, after (say) n¼10,000 turns, the first player’s pile will contain approxi-
mately 60,000 bananas, whereas the second player’s will contain
20
How does a realist know that such a function exist? As it turns out, the existence of a function [�]
having the needed mathematical properties can be proved axiomatically (from three conditions
that hold for our games). A proof of the existence of a utility function (what function [�] actually
is) was given by von Neumann and Morgenstern); for a more modern presentation, see theorem
8.4 in Fishburn ([1970], pp. 112–5). I thank Professor Teddy Seidenfeld for drawing my atten-
tion to this literature.
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approximately 50,000. The difference in the amount of collected bananas in-
creases with n and, given that [X] > [Y], even if some statistical anomalies
occur now and then, it is clear that the pile associated with X will be recognized
as amassing more bananas than the one corresponding to Y (almost) all the
time.
The crucial point is that the result in the first Game (one crate winning
almost always) tends to occur because of an inequality of expectation
values: the value corresponding to crate X is higher than the one correspond-
ing to Y. Essentially, the same reasoning can be transferred to the other game,
Game*. In this game, we have that the expectation value associated with one
crate (X) is higher than the expectation value of the other (Y): compare
[X*]¼12 with [Y*]¼9. Hence, if one wants to know what is common to
both games, and thus what accounts for the explanandum, the realist offers
this: in both games, we have an inequality of expectation values. A common
feature of the games was identified, and this is what explains why the two
games evolve the same way in the long run. This feature has been shown, in a
rigorous fashion, to be responsible for the observed unidirectionality, that is,
the explanandum. This explanation is given in terms of a simple mathematical
notion (‘expectation value’), so we are entitled to count this explanation as a
mathematical one.
To ensure that this example complies with ‘nominalize’, the issue in need of
clarification at this point is whether the explanandum is an unquestionable
physical fact, expressible in nominalistic language. The method to deal with
this problem has been presented already, and now I’ll add a few, hopefully
useful, details. Let’s focus on Game. As we saw, using the pairing method, the
players notice that the pile collected from crate X contains more bananas than
the pile originating in crate Y. They record the result of the first day of playing:
next to crate X, they scratch a mark on the ground—say g. As the Game
progresses, after many days, the marks recording the wins for the two crates
accumulate (say the win mark for Y is f). Now, after many rounds (days),
they compare the two collections of physical marks using, again, the nomina-
listically acceptable procedure of pairing them. They note that there are over-
whelmingly more gs than fs, and this gives the outcome of Game: one crate
(X) wins. The players note a striking imbalance, as the wins don’t switch from
one crate to the other: the vast majority of wins corresponds to only one crate.
In essence, the same nominalistically acceptable procedures are used for
Game*. As we saw, the selection of small crates is made by tossing a fair
four-sided device and a six-sided one. In this game, the players also note a
conspicuous imbalance. Again, this game goes in one direction too: the wins
don’t shift from time to time, but one crate collects them all (almost). The
outcome of Game* can be expressed in nominalistic terms, like the outcome of
Game. Given that the outcome of Game* is the same as the outcome of Game
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(both are clearly unidirectional one-crate winners), the players wonder if this is
a mere accident or can it be explained? But, recall that this is the explanandum:
in both games, one crate almost certainly wins. This is a fact accessible to a
nominalist, insofar as the outcome of each game is, as we saw, nominalistically
expressible.
3.1 Some clarifications
Two contentious aspects of this example must be discussed before we evaluate
the nominalist resources to deal with it. First, one might be bothered by the
fact that the outcome of the games is not guaranteed to obtain; as said, crate X
wins ‘most of the time’ and not ‘all the time’ (‘overwhelmingly often’ and not
‘always’). Indeed, it’s possible that crate Y wins more rounds, and thus bal-
ances out the wins of crate X; hence, the players won’t be stricken by the
imbalance and won’t take the outcome of each of the games to be ‘one
crate wins overwhelmingly often’. Thus, at this point, an important empirical
assumption under which this example works needs to be made fully explicit:
namely, that the games are played in a world in which the (weak) law of large
numbers holds. The example stands or falls with this assumption; if the wins
are more evenly distributed, then such an example is out of the question, as the
explanandum is not true. But, as far as we know, this assumption is true: we do
live in such a world, so the outcomes and the explanandum, as described here,
do obtain. Virtually all physical processes we have ever observed strongly
corroborate this assumption.
21
This is precisely why the condition imposed
on these games is that they are played for a large number of rounds.
The second problematic issue is whether the nominalist can accept this
example in the first place because it is couched in terms of ‘games’ and these
are presumably abstract objects. This is a fair point, but one should keep in
mind that we talk of games here in purely operational terms, as a succession of
procedures and manipulations of otherwise unremarkable physical objects
(crates, bananas, scratches on the ground, and so on). Moreover, one
should also realize that if such over-stringent nominalist constraints are
imposed, virtually every example is blocked, including the cicadas one (on
the face of things, it requires at least quantification over species, which are,
one might argue, abstract entities too). Here, although quantification over
games might be necessary, if we want to express the explanandum in
first-order logic, the operational way in which it is meant to be understood
doesn’t existentially commit us to anything beyond mere physical objects.
21
Coin tossing is perhaps the simplest example of wLLN holding. In fact, an approximately equal
number of heads and tails begin to appear after a rather small number of tosses (hundreds, or
even tens).
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Essentially, in this example, unlike the cicadas one and others, no mathemat-
ical object or property appears in the formulation of the explanandum.
22
3.2 Hopes and troubles for the nominalist
So, what are the nominalist resources to deal with this example? If the nom-
inalist neglects utterly irrelevant features of the crates (such as them having
different colours, different textures, and so on), he will notice that the follow-
ing conditions hold true of Game:
(a) There are more bananas in one crate (the one that eventually wins)
than in the other one.
(b) There are as many small crates in one large crate as there are in the
other one.
(c) The probabilities of selecting small crates from the large crates are
equal.
Note that each of these three conditions is nominalistically acceptable. The
nominalist’s hope is that the conjunction of these conditions reveals what is
distinctive about Game. So, he advances the following ‘qualitative’ explan-
ation (call it Q) for the outcome of Game:
Assume (a), (b), and (c). Therefore, one crate (X) wins overwhelmingly
often.
But there is a problem. How does the nominalist show that this conditional
statement (‘If (a), (b), (c), then one crate wins overwhelmingly often’) holds?
Or, equivalently, that these three conditions are sufficient to derive the con-
sequent? More precisely, the question is why does he list only these three
conditions? Why not add a fourth one, for instance, that ‘one large crate
must not contain a small crate that contains fewer bananas than any other
small crate’? This condition holds true of the set-up of Game, and might be
relevant for the outcome. Why isn’t such a condition explanatory, like the
previous three?
Thus, there is apparently no principled reason to exclude a fourth, fifth, and
so on, condition from formulating an explanation. A correct and complete
formulation of Q (hence a rigorous proof of it) seems to be beyond the
22
This point has been added in response to a referee’s comment. Another referee pointed out that
an additional worry could be that talk of ‘manipulations’ and ‘procedures’ is also talk of
abstract objects, the reason being that we deal with types of procedures and types of manipu-
lations. The referee, however, also suggested that my reply to the first referee’s worry–roughly,
that we’re going to have to talk about some such abstract objects in presenting any examples
(and unless the nominalist can construe such talk in some nominalistically acceptable way or
other, then he is unable to describe empirical phenomena, let alone explaining them, nomina-
listically)–applies to this new worry as well. I agree.
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nominalist’s conceptual resources. Hence, he has no right to claim that the
antecedents of Q are the explanans for the explanandum in question (the
outcome of Game).
Yet things turn out to be even worse for the nominalist. In addition to not
being able to find out what the antecedents–explanans in Q should be, he faces
another difficulty. Suppose that we accept, for the sake of the argument, that
an incomplete list of these antecedents–explanans in Q has been compiled.
Hence, such an incomplete Q will count as a (quasi-)explanation of the result
of Game. It is now crucial to note that this (quasi-)explanation necessarily fails
as a possible (quasi-)explanation of the outcome of Game*. This is so because
even if the list of the antecedents–explanans can be eventually completed, none
of the explanans on the list so far (that is, conditions (a), (b), and (c)) holds for
Game*! In Game*, there are fewer bananas in the large crate that will even-
tually win (X), there are fewer small crates in the large crate that will eventu-
ally win (X), and the probabilities are not equal. It is thus important to stress
that even if the nominalist comes up with a (quasi-)explanation for the out-
come of Game* (call it Q*),
23
the nominalist has now two distinct (quasi-)
explanation: a (quasi-)explanation, Q, for the outcome of Game and another
(quasi-)explanation, Q*, for the outcome of Game*. And Q* is necessarily
different from Q regardless of how (or whether) the nominalist manages to
complete the list of antecedents–explanans. (It is different because Q and Q*
don’t—and can’t!—share some of the antecedent–explanans.)
3.3 New hopes?
Another route the nominalist can take is to devise an operational procedure by
which he can find out that one crate (X) will almost always be the winner in
Game. The procedure, carried out in nominalistic terms, is as follows. For
Game, he begins by simply listing a possible distribution of outcomes for the
withdrawals of bananas from X and Y.
Table 1 should be interpreted as follows. In round one, player A, who takes
out bananas from crate X, happens to pick small crate x1 and thus he collects
five bananas. Player B happens to pick y1 and thus collects only one banana.
In round two, A picks x1 and collects five bananas once again; this time, the
coin lands such that B picks y2 and collects nine bananas. In round three, A
adds seven bananas to his pile, whereas B adds only one, so on and so forth.
Two remarks about Table 1 are in order. Firstly (and obviously), it is just
illustrative; this is only one of the many possible ways in which Game might
unfold. The amount of bananas each player collects in each round depends on
23
Q* might be a conditional like this: ‘If (a*), (b*), and (c*), then one crate (X) is the winner’,
where (a*)¼there are fewer bananas in total in crate X, (b*)¼ there are fewer small crates in
crate X, and (c*)¼the probabilities are not equal.
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which small crate they collect from, which is in turn determined probabilis-
tically by how the coin lands. Secondly, and essentially, in so far as the prob-
abilities are fixed, we know for sure that no matter how the rounds go, in the
long run, there will be an approximately equal number of rounds when each of
these values (5 and 7, and 1 and 9, respectively) will appear in the two rows.
Thus, what the nominalist can do is just rearrange the values he tabulated
above (Table 2).
The next step then consists in dividing Table 2 in cells of four values, the
upper ones consisting in a (5, 7) pair, and the lower ones being a (1, 9) pair (the
first such ‘cell’ is indicated by the bold fonts). Given the probabilities, this
division in cells should be exhaustive. Now, just by simply applying the predi-
cate ‘_contains more than_’ to the two upper values and the two lower values
in each repeating cell, the nominalist can infer that A, the player who chose X,
will have more bananas in his crate in the end, as each cell indicates this.
(Using numbers: the advantage of A increases with two bananas, twelve versus
ten, with each cell.)
I’ll call such a procedure a ‘re-arrangement’. It is available to the nominalist
regardless of what the values in the cells are, and what the probabilities are.
There is no doubt that a rearrangement allows the nominalist to predict the
result of Game. But one might want to maintain that he can also explain why
Game almost certainly ends in this way. If I happen to expect that some wins
will go to one crate, some to the other, once you show me how the rearrange-
ment works, I’ll understand that only one will win. So, the existence of a
Table 1. Possible distribution
Round 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Player A
Crate X 5 5 7 7 7 7 5 5 7 5 5 5 7 7 7 5 7 5
Player B
Crate Y 1 9 1 9 9 1 9 9 1 1 9 1 9 1 9 1 1 9
Table 2. Rearrangement
Round 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Player A
Crate X 5 7 5 7 5 7 5 7 5 7 5 7 5 7 5 7 5 7
Player B
Crate Y 1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9
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rearrangement seems to show that a nominalistic explanation of the result in
Game is possible. But recall that the task set for the nominalist was to find a
nominalistic explanation of the result that one crate wins overwhelmingly
often in both games (recall that this is our explanandum). So, the next step
is to devise a rearrangement for Game*. To be sure, such a procedure can be
found, but I will not present it in detail.
24
Hence, the question now is what is the nominalistic account of the
explanandum? It must be this: the nominalist explains why is it the case that
‘in both games, one crate wins overwhelmingly often’ by pointing out that this
can be expressed as the conjunction of ‘in Game, one crate wins overwhelm-
ingly often’ and ‘in Game*, one crate wins overwhelmingly often’, and by
offering nominalistic explanations in the form of a rearrangement for each
conjunct.
3.4 New troubles
We can now draw the contrast between the nominalist and the realist situation
more forcefully. An analogy with the common-cause type of explanation
might be illuminating here.
25
(I stress that this is only an analogy, as nothing
has been said or implied about these explanations as being ‘causal’—on the
contrary.) Suppose two friends, Joe and Moe, arrive separately at the bus
station at 3 p.m. Also, suppose that they are followed by two detectives who
scrupulously record every move they make. The first detective has gathered a
lot of information, and is about to write down an account as to why Joe and
Moe arrived at the bus station at 3 p.m. The second detective followed Joe and
Moe closely too and gathered the same information as the first detective, but
this second detective, benefiting from some listening devices, also intercepted a
phone conversation between Joe and Moe the day before, in which they ac-
tually agreed to arrange their schedules for the next day such that they meet at
3 p.m. at the bus station.
Now consider the event ‘Joe and Moe both arrived at the bus station at
3 p.m.’. If asked to account for this event, the first detective explains it by
giving the full details of Joe’s trip from his place to the bus station. He will also
provide the details of Moe’s trip from work to the bus station. It is no surprise
for him that Joe went to the bus station (say, he knows he goes to visit his
mother); he can even explain why he was there at 3 p.m. instead of 4 p.m. (the
bus taking him to his mother leaves at 3:05 p.m., and so on). He has a similar
story about Moe as well (say, he goes to visit his father, and the bus taking him
there leaves at 3:07 p.m.). Knowing all these details, the detective actually
24
The reader can try to identify it herself. Hint: the repeating cell has the length of twelve boxes.
25
Reflections on (Owens [1992]) have been helpful in designing it.
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expects that the two will bump into each other at 3 p.m. at the town’s small bus
station. In brief, he has an explanation of the meeting event. If asked about the
meeting event, he describes it as a conjunction (of ‘Joe arrived at the bus
station at 3 p.m.’ and ‘Moe arrived at the bus station at 3 p.m.’), and he has
an explanation of each conjunct. Yet, it is intuitively clear that the second
detective has a better explanation of the event, in so far as he was able to
identify an additional relevant element—the phone call. Although for the first
detective, who missed it, the meeting is still explained, for the second, the
meeting event is better explained.
To see how this analogy is relevant for the banana game, we should begin by
observing that the realist was able to identify the factor responsible for the
unidirectional tendency observed in Game, expressed as an abstract mathem-
atical (structural) feature of the game: Game is such that one crate has a higher
expected value than the other. Moreover, the same factor is responsible for the
tendency exhibited in Game*. Hence, the realist is in the possession of the
‘common factor’ that accounts for the fact that in both games, one crate wins
overwhelmingly often (the explanandum). For the realist, the mathematical
apparatus of expectation functions enabled him to isolate this common uni-
fying element, just as the listening devices helped the second detective intercept
the phone call.
Does the nominalist have the resources to identify such a common unifying
factor? The discussion of the tentative nominalist explanations Q and Q*
made it clear that they are of no help in identifying such a common factor.
Hence, the only hope is that the explanations in terms of rearrangements
might yield such an element. A nominalist might claim that a rearrangement
like the one in Table 2 explains the result in Game. Because the same proced-
ure would explain the result in Game*, the nominalist too seems to be in the
possession of a common element responsible for the result in each game. Yet,
the realist is entitled to ask the nominalist to specify why the rearrangement
used in Game is the same as the rearrangement used in Game*. The question is
pertinent because, after all, one who sets up Tables 1 and 2 for Game doesn’t
set up identical tables for Game*. So, in what respect are the two rearrange-
ments the same? What do the two rearrangements have in common?
Thus, the realist presses the nominalist to acknowledge that what the two
procedures have in common is a certain structure, which in this case is a
mathematical one; more precisely, to acknowledge that it is no accident that
these procedures are available for the two games, and that they work for both.
The realist has what I take to be an excellent answer to the question above: the
games share the same abstract mathematical structure, they both instantiate
an inequality of expectation values. The realist’s challenge to the nominalist
can also be poignantly expressed in counterfactual terms: it is easy to see that if
one changed the mathematical structural relations (more precisely: the
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expectation value relations) among the crates, neither of the two procedures
would work anymore! It is this abstract mathematical factor that ‘lies under-
neath’ the efficacy of the two rearrangements. In so far as the nominalist can’t
recognize this factor, he must accept that the realist explanation is superior.
4 Conclusion
The bananas example fares well with respect to all four desiderata. Consider
desideratum ‘simplicity’ first: The example is simpler than other examples, and
it is at least as simple as Baker’s cicadas. But, unlike the cicada example, this
new example does not raise any suspicions with regard to the second desider-
atum ‘nominalize’, as no mathematical concept enters the formulation of the
explanandum. With regard to the third desideratum (‘indispensability’), we
must ask whether the mathematics of expectation values used here is indis-
pensable. Recall that the nominalist can come up with an account of the
explanandum, so the mathematics seems dispensable. Yet, when the fourth
desideratum (‘explanation’) is taken into account, realism remains the more
attractive position. As we saw, the nominalist’s conjunctive explanation of the
result scores lower than the realist’s precisely on the explanatory power scale.
Hence, I believe it is fair to say that the realist’s explanation is the best ex-
planation available.
Finally, let me stress that I offer this new type of example in a constructive
spirit. I propose it not so much as a replacement of the ones discussed so far in
the literature, but rather as providing necessary additional ammunition for the
realist camp.
Acknowledgements
I presented versions of this article in various places (Leeds, Seattle,
Amsterdam, and Cambridge), and I benefited from suggestions and criticisms
from the following philosophers, whom I wish to thank: Margaret Morrison,
Bob Batterman, Mark Colyvan, Jim Brown, Steven French, Alan Baker, Juha
Saatsi, Jacob Busch, Arthur Fine, Bill Talbott, Andrea Woody, Alison Willie,
Emily Grosholz, Michael Potter, Luca Incurvati, Alex Manafu, Kevin
Brosnan, Alex Broadbent, Paul Dicken, Teddy Seidenfeld, and Hugh
Chandler. The anonymous referees for this journal have been very helpful
and deserve thanks as well. Needless to say, I bear all responsibility for the
final version of the article.
Department of Philosophy
University of Bergen, Norway
Sorin.Bangu@fof.uib.no
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