Erkenntnis

, Volume 74, Issue 1, pp 101–114 | Cite as

Magicicada, Mathematical Explanation and Mathematical Realism

Original Article

Abstract

Baker (2005) claims to provide an example of mathematical explanation of an empirical phenomenon which leads to ontological commitment to mathematical objects. This is meant to show that the positing of mathematical entities is necessary for satisfactory scientific explanations and thus that the application of mathematics to science can be used, at least in some cases, to support mathematical realism. In this paper I show that the example of explanation Baker considers can actually be given without postulating mathematical objects and thus cannot be used by the mathematical realist. I also show that, despite this, mathematics keeps playing an important methodological role in the explanation and does not reduce to a merely computational or descriptive framework.

1 Mathematical Explanation and Ontological Commitment

During the last decades philosophers of mathematics have been paying an increasing attention to the problem of applicability, especially with the objective of defending a form of realism about mathematical entities. This has been attempted by means of what have come to be known in the literature as indispensability arguments (see for instance Colyvan (2001) Putnam (1971), Quine (1981), Resnik (1995, 1997)).

These arguments are based on the striking fact that mathematics plays a major role in the formulation and articulation of many scientific theories. Such state of affairs seems to suggest that postulating abstract mathematical entities is necessary for the development of many among the sciences. If this conclusion is accepted, then true scientific theories involving mathematical postulations commit us to the existence of mathematical entities. In this sense, the role of mathematics within science, and thus its applicability, can be used to support mathematical realism.

The influential work of Hartry Field (especially Field (1980), (1984), but also see in this connection Burgess (1985, 1991, 1992)) has however shown that the mere pervasiveness of mathematics within science is neither necessary nor sufficient to support mathematical realism through indispensability. This is because there are scientific theories whose ordinary development makes use of a large amount of mathematics but which can be formulated without employing any mathematical references in such a way that their empirical consequences remain the same as those following from their mathematical presentation (Field (1980) shows this to be the case for the classical theory of the gravitational field). Although it is uncertain how far Field’s results extend,1 it is clear that they pose problems for the mathematical realist looking at the applicability of mathematics as a source of support for her view.

In particular, the realist needs to find applications of mathematics in which the reference to mathematical entities or the assumption that they exist is essential to scientific theorizing: roughly, examples are needed which show that a scientific theory would produce a less adequate picture of the world if its mathematical component were to be detached from it.2 During the last years some philosophers have attempted to establish this conclusion by means of suitable examples (see in particular Colyvan (2001), Baker (2005, 2009) and Lyon and Colyvan (2008)). The aim of these examples has been to show that mathematics plays a genuinely explanatory role in science. In other words, postulating mathematical entities makes it possible to produce scientific explanations which would not be otherwise available. In this context, references to mathematical entities may be compared to references to unobservable empirical entities: their presence can increase the explanatory power of a theory. If it is accepted that the assumption of unobservable empirical entities is ontologically committing because it increases explanatory power, the same conclusion must be drawn for mathematical entities. This type of argument is developed in Baker (2005), which contains an example of what Baker takes to be a genuine mathematical explanation of an empirical phenomenon, i.e. one where the existence of mathematical entities is required to explain the phenomenon in question. This is supposed to lead to ontological commitment to the relevant entities3 and thus to provide support for mathematical realism. The interest of Baker’s example lies in its elementary character and in the fact that a mathematical property (that of being a prime number) plays a central role in the relevant explanation. My plan in the next sections is to carefully look at the example, paying more attention to its mathematical development than it is done in Baker (2005), in order to assess whether it can really be used to draw Baker’s conclusions in favour of ontological commitment to mathematical entities.

My conclusion (in Sect. 4) will be in the negative: mathematical realism is not supported by Baker’s example. The reason is that his number-theoretical explanation can be formulated in purely non-numerical terms. This possibility sheds light on the nature of mathematical explanation in science, which I will discuss in Sects. 57.

2 Baker’s Example

The example of mathematical explanation described in Baker (2005) is taken from biology. The explanandum is the prime-numbered length of the life-cycle of certain species of North-American periodical cicada, belonging to the genusMagicicada.

Baker refers to two studies providing an explanation for this phenomenon, namely Yoshimura (1997) and Goles et al. (2001). Here I will focus on the latter, which is the only one making a systematic use of mathematics.

Goles et al. (2001) describes a hypothetical setting in which both the cicadas and their predators are present, with life cycles of fixed lengths Y, X respectively. Mutant cicadas and mutant predators having different life-cycles may emerge and supersede the existing ones just in case they are fitter to survival in a sense to be specified. On the basis of a suitable fitness measure it follows that, if the predators’ and cicadas’ life-cycles lie within prescribed bounds and the cicada’s life-cycle is a prime number when measured in years, there are no predators or cicadas fitter than the existing ones. The prime-numbered length of Magicicada’s life-cycle is thus explained by a correlation between primality and fitness: Baker takes this to be a genuine mathematical explanation.

I will now illustrate in some detail how mathematics is employed to deliver it (I follow Goles et al. (2001, pp. 34–35)). Suppose the predator’s life-cycle lasts X years and the cicada’s lasts Y years. Assuming predator and cicada co-emerge at an initial time t0, they will also co-emerge XY years later. Attention may be restricted to just one period of XY years. Now for every year between t0 and XY three possibilities arise: (i) either predator and cicada occur simultaneously or (ii) predator occurs without cicada or (iii) cicada occurs without predator.

The number of times predator and cicada occur simultaneously over a period of XY years is4 XY/lcm(X, Y), but since XY = lcm(X, Y)gcd(X, Y), this number is simply the greatest common divisor of X and Y, i.e. gcd(X, Y). Then the number of times predator occurs without cicada is Y–gcd(X, Y) and finally the number of times cicada occurs without predator is X–gcd(X, Y). We thus have the following three cases:
  1. (i)

    For gcd(X, Y) times the predator ‘wins’ in terms of fitness (because it emerges exactly when it can feed upon cicada), while the cicada ‘loses’ (because it emerges when the predator is present);

     
  2. (ii)

    For Y–gcd(X, Y) times the predator loses (because the cicada has not emerged yet) while the cicada has neither losses nor gains (it simply did not emerge);

     
  3. (iii)

    For X–gcd(X, Y) times the cicada wins (because the predator is not present when it emerges) while the predator has neither losses nor gains.

     
These observations lead to fitness measures for both cicada and predator, which are computed using the following stipulations:
  1. i.

    For gcd(X, Y) times assign 1 to predator and – 1 to cicada;

     
  2. ii.

    For Y–gcd(X, Y) times assign – 1 to predator and 0 to cicada;

     
  3. iii.

    For X–gcd(X, Y) times assign 0 to predator and 1 to cicada.

     
The fitness measure of the cicada is obtained by summing together its fitness gains and losses over a period of XY years and dividing the result by the number of generations emerging during that period (i.e. X). Thus, the cicada’s fitness measure is:
$$ {\text{F}}_{\text{Y}} = 1 - \left( {2gcd\left( {\text{X}}, \; {\text {Y}} \right)} \right){\text{X}}^{ - 1} $$
The predator’s fitness measure is obtained in a similar way and it is:
$$ {\text{F}}_{\text{X}} = \left( {2gcd\left( {\text{X,}} \, {\text {Y}} \right)} \right){\text{Y}}^{ - 1} -1 $$
Goles et al. (2001) impose bounds on X and Y: in particular, they require (a) 2 ≤ X ≤ L/2 and (b) 2 + L/2 ≤ Y ≤ L (with L a constant to be fixed on the basis of ecological considerations)5. Using these bounds, it can be seen that if Y, the Magicicada’s life-cycle, is a prime number YP, then the predator’s life-cycle XP is strictly smaller than YP, so gcd(XP, YP) = 1. Now note that the predator’s fitness is an increasing function of gcd(X, Y). But no change in XP would increase gcd(XP, YP) (since the only divisor of YP which is smaller than YP is 1) and so the predator’s fitness measure cannot increase.6 This means that XP is selected as the predator’s life-cycle. Now, having fixed XP, no YR ≠ YP improves the fitness of YP. The reason is that the cicada’s fitness measure is a decreasing function of gcd(X, Y) and this is already 1, i.e. the smallest value it can take. Thus no variation of YP can increase the cicada’s fitness measure. It follows that YP is selected as the cicada’s life-cycle. In other words, if predator and cicada compete to improve their fitness, their competition stabilizes on the couple (XP, YP) (cf. Baker (2005, p. 232)), where YP is a prime number. If the inequalities in (b) are strict and L=18, the only possibilities for YP are 13 and 17, the life-cycle lengths of the southern and northern variety of Magicicada respectively.

In sum, the fact that Magicicada has a life-cycle of a prime number of years may be explained by showing that, on a bounded prey-predator model, the competition to improve fitness stabilizes on a couple of the form (XP, YP), with YP prime. Note that we can reach this result just by assuming that (1) the fitness of cicada is a decreasing function of gcd(X, Y) and (2) the fitness of predator is an increasing function of gcd(X, Y) (this fact will be exploited in Sect. 4).

3 Baker’s Claim

The reason why the Magicicada’s life-cycle has a particular length can be explained on the basis of some elementary number-theoretical facts and a number-theoretical property possessed by that length, in presence of suitable ecological hypotheses. I agree with Baker that it is essential, in order to properly articulate this explanation, to resort to formal notions like primality. I however disagree that this involves ontological commitment to mathematical entities. I’m going to show this in Sect. 4, while I’ll explain why I think mathematics is nonetheless essential to the explanation in Sect. 5.

For the moment, let me clarify what Baker takes the Magicicada example to show. Baker stresses the fact that number-theoretical properties, in particular ‘being prime’, enter the explanation of a biological phenomenon (in fact, the very formulation of the explanandum). But, according to Baker, number-theoretical properties are intrinsic properties of certain mathematical objects, i.e. the natural numbers.7 Since these properties are needed to deliver the required explanation and, at the same time, are properties of numbers, if explanatory power is ontologically committing, then ontological commitment to numbers follows. As a result, the mathematical explanation that Magicicada’s life-cycles have prime-numbered lengths provides support for mathematical realism, at least with respect to the natural numbers.

In a recent paper (Bangu (2008)) Sorin Bangu has argued that Baker’s example is problematic because it begs the question in favour of the mathematical realist by presupposing that an explanandum formulated in mathematical terms (i.e. the prime-numbered length of Magicicada’s life-cycle) is true.8 Although Bangu makes a subtle objection, the truth of the explanandum can be conceded to Baker9 while this does not block the possibility of raising another stronger objection, namely that no postulation of mathematical objects is involved in any part of the explanation of the biological phenomenon described in Baker (2005). This possibility is excluded by Baker, who takes it to be uncontroversial that mathematical objects play a part in the explanation10 he illustrates. I don’t think this is correct because I don’t think the property of being prime, or the other mathematical relations used in the mathematical treatment of Baker’s example, should be interpreted as properties and relations of certain distinctively mathematical entities.

They rather are properties and relations of time intervals corresponding to life-cycles (which, as will be seen, can be studied non-numerically). To see this, observe that in the model of Goles et al. crucial attention is paid to how often predators’ and cicadas’ life-cycles occur simultaneously. If one depicts the respective life-cycles as segments on a line, the problem is that of considering how often sequences of consecutive life-cycles of predator and cicada respectively determine time-intervals of equal length. When this happens, one can say that a divisibility relation between life-cycles and sequences of life-cycles obtains. But then a form of divisibility can be recovered in non-numerical terms and so a non-numerical version of primality can be recovered too. It suffices to say that a time interval T composed of basic units is prime if, except for the basic unit, none of the shorter time intervals or its multiples is congruent to T. Proceeding along these lines, simple number-theoretical facts depending on divisibility properties, e.g. the theorem that gcd(X, Y) = 1 if X < Y and Y is prime, can be expressed without mentioning numbers or numerical concepts and Baker’s biological explanation can be given without invoking numerical facts.

It follows that it is not the postulation of mathematical objects like numbers that drives the explanation Baker is interested in, but rather the fact that certain formal properties like ‘being prime’ can be used to describe empirical relations between life-cycles measured in years. This conclusion can be made formally more precise, as I will now show.

4 Empirical Relations and Numerical Models

The model developed in Goles et al. (2001) to explain the length of Magicicada’s life-cycles is designed to describe the pattern of co-emergence of cicadas and their predators during a finite number of years.

As suggested in the previous section, patterns of co-emerges can be suitably studied on a finite one-dimensional geometry of time-intervals. It suffices to have a segment S partitioned into XY congruent intervals: each interval represents a 1-year period and consecutive intervals, whose points are endowed with an ordering, represent temporal periods of several years, which may be used to describe life-cycles. In order to carry out the numerical reasoning in Goles et al. (2001) it suffices to assign the first XY+1 positive integers to the points of S.

In this context, the length of each interval in the partition is 1 and the length of any finite sequence of consecutive intervals is given by nm, where n is the number assigned to its last point (relative to a fixed ordering of points) and m the number assigned to its first point. Once the numerical assignment is in place, it is possible to introduce the numerical notion of greatest common divisor and thus describe divisibility relations between the lengths of sequences of intervals over S.

Although this can be done numerically, it is not necessary to appeal to numbers. Divisibility relations between life-cycles, i.e. time-intervals, can be represented using the geometry of S alone. One may simply say that a sequence X of consecutive intervals divides a sequence Y if successive congruent copies of X determine an interval congruent to Y (the successive juxtaposition of congruent copies of intervals can be described without mentioning numbers to define iterations, as will be seen in a moment). For example, in Fig. 1ac divides ae.
Fig. 1

ac divides ae

In this setting the greatest common divisor of X and Y is the longest sequence of intervals which divides, in the above geometrical sense, both X and Y. Finally, the fact that the greatest common divisor increases or decreases can be expressed by considering the ordered endpoints of sequences of intervals.11

These remarks suggest two things: first, that the basic notions and results in Goles et al. (2001) can be given in a non-numerical fashion; second, that the numerical formulation of these notions and results can be recovered from non-numerical conditions. In order to establish these points it suffices to provide a set of axioms that describes a finite sequence of equal intervals and whose models can be represented on a numerical structure of the following form:
$$ {\text{\bf N}} = \langle {\text{N}}, \le_{N} \rangle $$
where N is a finite set of positive integers and ≤N a quaternary relation on positive integers which holds of numbers l, m, n, o (lm, no) when:
$$ m - l \le o - n $$
(1)

The numerical structure is related to the geometry of equal intervals as follows: (1) says that the segment of endpoints m, l is not longer than the segment of endpoints o, n. If the converse of (1) holds we have congruent segments. In addition, the ordering of points may be defined by letting m = l in (1) and consecutive points are the ones whose numerical assignments determine the least positive difference. With positive integers and some additive structure on them we can talk about numerical multiples and greatest common divisors of two numbers, thus reducing the reasoning in Goles et al. (2001) to the numerical representation of certain facts about a geometry of time-intervals.

Thus, if axioms exist that can be interpreted on relations between time-intervals and whose models are representable on structures like N, it is possible to conclude that the numerical reasoning leading to the derivation of the length of Magicicada’s life-cycles depends on the preliminary availability of suitable empirical relations: it is because these empirical relations are in place that it is meaningful to use numbers in order to reason about life-cycles.12

The axioms needed to reach this conclusion have been firstly presented in Suppes (1972), which provides a characterization of what are known in measurement theory as finite, equal-interval difference structures.

Suppes’ axioms have models that can be set-theoretically described as relational structures like \( {\mathbf{A }} = \langle {\text{A}},{\text{D}}\rangle \), whose domain A is finite and nonempty and on which the quaternary relation D is defined. It is assumed that, for any a, b, c, d in A the following conditions hold:
  1. 1.

    The relation D is a weak ordering of A×A (transitive and connected);

     
  2. 2.

    abDcd implies acDbd;

     
  3. 3.

    abDcd implies dcDba;

     

Definition 1

a <Db iff aaDab and (abDaa);

Definition 2

aJb iff for any c, a <Dc implies c = b or b <Dc.
  1. 4.

    aJb and cJd implies ab ≈ cd (where the last relation is defined by abDcd and cdDab)13

     

Axiom 4 is the crucial assumption in this context: it states that the relation J, which holds between consecutive points in A, determines equal, i.e. congruent, intervals.14

When Magicicada’s life-cycles are modelled by means of Suppes’ axioms (plus an additional assumption introduced below), it can be seen that the numerical properties exploited in Goles et al. (2001) are in fact the counterpart of empirical properties of time-intervals. Furthermore, it is possible in this framework to reproduce the reasoning underlying the prey-predator model of Goles et al. (2001) in a purely non-numerical fashion.

To see this, suppose life-cycles satisfy the above axioms: then they can be described as a structure \( {\mathbf{A }} = \langle {\text{A}},{\text{D}}\rangle \), with A finite. Thus A has a first and a last element relative to J: call them a and z.

If the first and last element do not coincide, there is b such that aJb: then ab determines a basic ‘unit’ interval. In this case, it is possible to define ‘multiples’ of this interval by letting the points of A operate on the points of A through the following definitions, which introduce something like a ‘multiplication’ between points15:
$$ a\left( b \right) = b $$
$$ b\left( b \right) = {c \; \text {such}} \; {\text {that}\; ab \; \approx \; bc} $$
$$ c(b) \; = \; d \, {\text {such}} \, {\text {that}} \, ab \; \approx \; bc \; \approx \; cd $$
and so on. Geometrically, x(y), if it is defined, is a point which determines a segment congruent to x copies of ay and with a as an endpoint. For at least some of these definitions to make sense, it is necessary to assume the existence of at least some points obtained by applying points to points. This can be done by introducing the following:
  1. 5.

    There are u, v such that a <DuDv <Dz and, for all y(yDv → there is w such that w = u(y)).

     

Among the points between a and v(v) it is possible to single out prime points. A point x is prime if, for all a <DuDv and all y <Dx, u(x) ≠ x. Axiom 5 implies that there exist at least three points and this makes it possible to prove the existence of at least one trivial prime point. In fact, let a be the first element of A and aJb. By 5, b(b) is defined. By definition, this is a prime point (if enough point multiplications are defined in a model, then further prime points can be found systematically when the model is given, using an algorithm like Eratosthenes’ sieve).

It follows from the definition of prime point that, if y is prime, it is not covered by any segment ax with x <Dy. It also follows from Suppes’ axioms that a and y are separated by a finite number of equal intervals, thus ay is a ‘multiple’ of the ‘unit’ interval. The gcd of two intervals ax, ay is the longest interval such that multiples of it reach both x and y, but we may consider as gcd only the greater endpoint of the relevant interval, so we have that, if y is prime and x smaller than y, then gcd(x, y) = b. This is the first half of the reasoning involved in Goles et al. (2001) illustrated in Sect. 2. We know that no point y smaller than a prime point x can increase their gcd beyond b. We also know that, if y is fixed, gcd(x, y) is not going to decrease below b.

But, in terms of time-intervals, gcd(x, y) is an index of how frequently cicada and predator co-occur. Under the assumption that, in a competition between cicada and predator, the former will try to lower the index and the latter to raise it, it is possible to conclude from the above facts that the competition will stabilize on a prime point y.16 In essence, the mathematical reasoning of Sect. 2 can be reconstructed within a non-numerical theory. This result contradicts the claim that the explanation of Magicicada’s prime-numbered life-cycles supports mathematical realism, since such explanation can be given in non-numerical terms.

5 Mathematical Concepts

The previous discussion has shown that what is done in terms of number-theoretical properties in Goles et al. (2001) can be done in terms of time intervals and empirical relations between them. An explanation of the length of Magicicada’s life-cycles can be provided without assuming the existence of any mathematical objects. As a result, the Magicicada example cannot be used to support mathematical realism, against what Baker concludes in Baker (2005) (see also Baker (2009)).

If this is the way things stand, it may be asked whether mathematics really plays a significant role in the given example, or it is just ‘functioning as a descriptive or calculational framework for the overall explanation’ (Baker (2005, p. 234)). I suspect Baker would opt for the second alternative.

On the contrary, I believe mathematics plays an important role in the Magicicada case, but I don’t think this role is correctly characterized in terms of mathematical entities. Rather, I think it should be characterized in terms of the concepts that are introduced to study life-cycles in a way that constrains the selection of a certain length.

Although the reconstruction I proposed of the results of Goles et al. (2001) in a non-numerical fashion does eliminate references to numbers, it does not eliminate the mathematical form of the reasoning leading to the derivation of a prime life-cycle length. This reasoning and the formal notions employed in it are necessary to direct the analysis of the given biological problem. Let me clarify this point by illustrating the conceptual structure of the model proposed in Goles et al. (2001).

The initial hypothesis is that cicadas’ life-cycles are influenced exclusively by prey-predator dynamics. Then the problem is to find a way of describing prey-predator dynamics from which one can deduce a prime-numbered length.

This task amounts to the isolation of a type of space of life-cycle lengths so constrained that only some particular position within it (a prime point, in the model from 4) can be occupied.

Given that only prey-predator dynamics and life-cycles matter, it is necessary to look at the simultaneity relations between synchronized life-cycles of cicadas and predators. Taking the space to be one-dimensional (because time-intervals are being considered), it is seen that the life-cycles may superpose any number of times, depending on their lengths. The issue becomes then to identify patterns of superpositions that have certain special properties, by means of which they can be singled out. The relevant patterns are the ones determining the least number of superpositions over a fixed finite period, since these are most advantageous to cicadas. Then, under suitable restrictions on the length of life-cycles (conditions (a), (b) from Sect. 2) it is possible to deduce that patterns generated by a prime point will be selected. The reasoning is general17 and could be articulated without mentioning any specifically biological assumption, i.e. by just giving constraints that determine whether a point in the space is or is not selected. Because of this and the central role played by a system of formal concepts (primality, divisibility, gcd, etc.) that can be characterized axiomatically, it seems to me correct to qualify the non-numerical reasoning underlying Goles et al. (2001) as mathematical in character. If this conclusion is accepted, then the elimination of the numerical treatment of the Magicicada case is not to be identified with the elimination of mathematical reasoning from it.

In other words, if one accepts that the mathematical character of the explanation resides in the form of the reasoning being used and in the concepts invoked, rather than in the reference to certain particular entities, then one can meaningfully talk about an explanation making an essential use of mathematics without forcing any ontological commitment to mathematical entities.

6 Baker’s View Reconsidered

The previous observations have two important consequences: a more specific one, relevant to Baker’s view of mathematical explanation, and a more general one, which is relevant to nominalism in the style of Field (1980) or Melia (2000, 2002). In this section I will focus on Baker, while leaving the issue of nominalism for the next one.

As far as Baker’s position is concerned, my discussion of the Magicicada example reveals two problems with it. The first problem is that it takes as uncontroversial a postulation of mathematical objects that may be dispensed with altogether, while the second problem arises from his insistence that genuine mathematical explanations require positing the existence of mathematical objects. In my opinion this causes Baker to overlook the centrality of mathematical concepts in constructing explanations, which is independent of ontological commitment to particular abstract entities. This criticism stands even in the face of the arguments provided by Baker to strengthen his position on mathematical explanation against various possible objections to it and presented in Baker (2009).

I cannot give here a detailed account of this paper but I wish to briefly explain how some important arguments in it can be dismissed in the light of my treatment of the Magicicada example. A general strategy to deny that ontological commitment to mathematical entities is involved in this context consists in pointing out that there is some degree of arbitrariness in the introduction of numerical objects, numerical properties (primality) or a numerical theory (arithmetic) respectively (see Baker (2009, p. 615–619)). Baker argues against this line of argument that: (i) the introduction of numerical objects is forced by the fact that life-cycles are measured by the unit of 1 year; (ii) the introduction of the numerical property of primality is forced by the need to study the co-emergence of cicadas and their predators; (iii) the introduction of arithmetic is needed in order to exploit results about prime numbers.

On my treatment of the Magicicada case (i) is not a problematic objection because I describe life-cycles simply in terms of equal intervals of time; (ii) is not problematic because primality can be defined within the modification of Suppes’s theory provided in Sect. 4; (iii) is not problematic because, with the non-numerical definition of primality in place, one can deduce from it the desired explanation without invoking a numerical theory. This shows that the notions of unit, primality and divisibility property are not arbitrary but can nonetheless be articulated in non-numerical terms.

It should also be stressed that these notions, insofar as they are relevant to explaining the lengths of the Magicicada life-cycles in the model of Goles et al. (2001), are less general than purely number-theoretical notions.

The prey-predator model in Goles et al. (2001) yields the desired result only when certain bounds are imposed on the length of life-cycles and the synchronization of prey and predator is assumed. Without the first assumption the couples of life-cycles of the form (jY (for predator), Y (for cicada)), with Y prime and j a positive integer18 (cf. Goles et al. (2001, p. 35)), would be unstable. This would block the deduction that the life-cycle for Magicicada must have a prime-numbered length, because some prime-numbered lengths might not be selected by the prey-predator model.

In addition, without the synchronization assumption the co-emergence rate of predator and cicada could be zero even if they both had life-cycles of an even number of years. These remarks make apparent that the mathematical explanation of Magicicada does not rely on general number-theoretical results, but on their adaptation to the empirical problem at hand. Their applicability to this problem depends therefore on the fact that they can be adjusted to the relevant empirical setting and this is partly shown by the representation theorem for the structures satisfying the axioms of Suppes (1972). The theorem links these structures to numerical models that are much more restricted and simpler than a model for elementary arithmetic. It is in virtue of this restriction that numbers can be meaningfully applied to deal with the small finite structures underlying the results in Goles et al. (2001). In other words, it is the representation theorem provable from Suppes’ axioms that warrants the applicability of numerical reasoning in the Magicicada case. This is the main reason why I think that the non-numerical explanation sketched in Sect. 4 does not suffer of any particular shortcomings with respect to Baker’s numerical alternative. While the structure of the explanation is the same in both cases, the numerical explanation works precisely because it rests upon a representation of the relevant empirical setting.

7 Nominalization

The strategy I adopted to deal with the Magicicada case is in essence the strategy presented in Field (1980) to nominalize scientific theories. The idea is to begin with a given numerically presented theory and then identify a set of non-numerical axioms whose models can have suitable numerical representations. Thanks to the representations it is possible to show how the ordinary numerical treatment of the initially given theory can be recovered from a non-numerical system of axioms. What is obtained here is the isolation of the empirical conditions that underlie the introduction of certain numerical concepts and methods. It is possible to read this result as an account of the applicability of numerical theories to empirical ones, which is one of the objectives Field seeks to achieve (see Field 1980, p. 6). However, his main goal is to show that scientific theories can dispense with mathematics. As remarked in Sect. 1, this conclusion is not supported by the use of representation theorems alone, but also by an argument to the effect that mathematical theories are conservative over empirical theories (i.e. one cannot obtain new empirical consequences from an empirical theory by adjoining to it a mathematical theory).

My use of (part of) Field’s strategy in the present context is not strictly meant to provide an extension of Field’s results. I do extend them insofar as I can give an analysis of the applicability of numbers to an empirical problem from biology in terms of non-numerical conditions. I also extend Field’s results by establishing that this analysis, together with the non-numerical explanation of the life-cycle’s length of Magicicada found in Sect. 4, shows that no ontological commitment to numbers is involved in the explanation.

However, I depart from Field’s view by emphasizing the significance of mathematical concepts, as opposed to objects, in the construction of the prey-predator model of Goles et al. (2001), as clarified in Sect. 5.

My point here is that we do not need to think in every case of the applicability of mathematics as a practice that involves a postulation of abstract objects, thus raising the issue of ontological commitment. It seems to me that, simple as it is, the Magicicada case shows how mathematics can play a relevant explanatory role in a non-ontological way, by providing the means to conceptualize the structure of an empirical phenomenon in such a way that its explanation can be presented as a derivation from certain formal conditions (in this case the reasoning described at the end of Sect. 4, which is a non-numerical counterpart of the reasoning in Goles et al. (2001)).

Thus I am not really disagreeing with Field’s denial that the postulation mathematical entities should be required by scientific theories. I am rather disagreeing with the preliminary presupposition that we should regard any application of mathematics as essentially an ontological matter.

For the same reason I do not disagree with the recent, penetrating defence of nominalism offered in Melia (2000, 2002) and stressing the fact that ‘though the postulation of mathematical objects may increase the attractiveness or utility of our scientific theories, the way in which they do so is unlike the way in which the postulation of theoretical physical entities increases the utility of our scientific theories’ (Melia (2002, p. 75)).

I am quite sympathetic with Melia’s view insofar as it is critical of conceptions of applied mathematics that revolve around the idea of ontological commitment to abstract entities. On the other hand it does not seem to me necessary to think that the applicability of mathematics should be understood centrally in terms of ontological commitment. To accept this perspective means to overlook the cases in which mathematics works as a guide to orient and shape our thinking and intuitions. The Magicicada example provides an example of this function of mathematics and suggests that a sufficiently rich analysis of applicability should take into account the conceptual role of mathematical ideas and avoid the one-sidedness coming from an exclusive focus on ontological issues.

Footnotes

  1. 1.

    For example, Malament (1985) expresses skepticism about the possibility of extending Field’s results to quantum mechanics, while Balaguer (1996) and Balaguer (1998) sketch a way of doing it.

  2. 2.

    This possibility for the realist to pursue this strategy is already outlined in Field (1989, p. 14–20).

  3. 3.

    That explanatory power warrants ontological commitment is assumed by Baker for the sake of argument. I intend to show in the next sections that this is not the case.

  4. 4.

    In what follows, lcm and gcd are, respectively, the least common multiple and greatest common divisor of X, Y.

  5. 5.

    A justification for the choice of bounds can be found in Goles et al. (2001, p. 35).

  6. 6.

    Goles et al. assume that a mutant predator will supersede an existing predator only if it is fitter.

  7. 7.

    According to Baker ‘[…] number theory deals with intrinsic mathematical properties of the natural number’ (Baker 2005, p. 236), so the use of number-theoretical facts within an explanation involves references to the natural numbers as abstract mathematical entities.

  8. 8.

    This amounts to saying that the explanandum is satisfied by mathematical objects. For details, see Bangu (2008, p. 18).

  9. 9.

    However it is not conceded that the mathematical references in the explanandum should be interpreted realistically. The explanandum refers to a property of the length of Magicicada’s life-cycles measured in years, not to a purely number-theoretical property. It will be seen below that this property can in fact be formulated non-numerically and represented numerically by primeness.

  10. 10.

    ‘I do not see how one can coherently deny that mathematical objects play a part in the explanation’ (Baker 2005, p. 234).

  11. 11.

    Note that one could dismiss a geometrical language and describe congruence, divisibility and juxtaposition as empirical relations between time-intervals.

  12. 12.

    A ‘meaningful’ use of numbers is one that can be explicitly related to some appropriate empirical content. Meaningfulness is required if numerical results are to be applied to investigate the structure of empirical phenomena. I am not assuming that meaningfulness is only established by proving a representation theorem, although this is what happens in my discussion of the Magicicada case.

  13. 13.

    The relation D can be seen as an ordering of intervals whose endpoints are in A.

  14. 14.

    For a proof, see Suppes (1972, pp. 56–58).

  15. 15.

    This operation is not supposed to correspond to arithmetical multiplication in a numerical structure like N, isomorphic to the given non-numerical structure.

  16. 16.

    The bounds on life-cycles and their lengths are then expressed using the endpoints of the relevant intervals.

  17. 17.

    This is confirmed by the fact that Goles et al. (2001) go on to generalize their prey-predator model so that ‘prime numbers are selected by competition between neighboring residents in a spatially extended system’ (Goles et al. 2001, p. 35).

  18. 18.

    Otherwise gcd(jY, Y) would decrease if Y changed to Y1, in which case the prey, whose life-cycle is Y1, would be fitter than the prey with life-cycle Y. But if Y is prime, Y1 is an even number, so primality is no longer related to fitness maximization.

Notes

Acknowledgments

I wish to thank Alan Baker, Sorin Bangu, Mary Leng and David Liggins for helpful comments on previous drafts of this paper. Its clarity and structure have also benefited from valuable suggestions and remarks from two anonymous referees.

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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.University of East Anglia, School of PhilosophyNorwich, NorfolkUK

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