1. Mappings and Applications

In this article, I wish to call into question the generality of a certain picture of applied mathematics that is very influential in the recent philosophical literature. This picture elaborates on the natural, intuitive idea that mathematics is applied whenever a link is established between a nonmathematical domain and a mathematical domain. The idea can be given a first articulation as follows: “some mathematical structure is either designed to, or otherwise found to, accurately capture the important structural relations of an empirical set up, and we can thus read off important facts about the empirical set up from the mathematics” (Bueno and Colyvan Reference Bueno and Colyvan2011, 346). As soon as it is possible to present the two domains to be linked in an application as structures, the applicability of mathematics amounts to the possibility of establishing a correspondence between certain properties of an empirical structure and the properties of a suitable mathematical structure. The relevant structural correspondence is usually determined by a structure-preserving mapping: as a result, to apply mathematics is to apply mathematical structures via suitable mappings.

A very simple illustration of this account of applications is provided by the measurement of subjective preference over a countable (i.e., finite or denumerable) set S of alternatives. Suppose that an agent a can rank the elements of S on a complete preorder,Footnote ¹ denoted by R, thus determining an empirical structure , which may be taken to describe the choice behavior of a over S. In particular, means that a does not strictly prefer y to x, while, if both and hold, a is indifferent between x and y, and if and not holds, then a strictly prefers x to y.

Because R is a complete preorder, it is possible to assign a positive rational number to each element of S in such a way that the following biconditional holds:

where u(x) is the positive number assigned to x, called the utility of x. If is the set of the ordered, positive rationals, what the above biconditional says is that the function preserves the preferences of a and represents them by means of a numerical ranking. Technically, u is a strong homomorphism between an empirical setup (a set of alternatives ordered by subjective preference) and a numerical structure.Footnote ² The mapping u yields a numerical description of a nonnumerical ranking of alternatives: one may thus straightforwardly use it to extract empirical information from a nonempirical structure.

The availability of a mapping like u is not in itself terribly useful, though, since a utility representation simply produces a numerical copy of an empirical ordering that is independently given as soon as a’s preference over S is revealed. Nonetheless, the existence of a numerical representation of individual preference provides the foundation for subtler ways of applying mathematics to its study.

For example, if a richer structure is assumed on S and stronger formal properties are imposed on R, it is possible to prove the existence of a continuous, real-valued utility representation u.Footnote ³ Continuity plays an important role in the presence of certain restrictions on the domain of u: if, for example, u is defined on a compact subset of the empirical domain S, then it can be deduced that a has a most preferred alternative over this subset since in this case the function u attains a maximum there. If, in addition, it is assumed or can be established that u is convex, the uniqueness of the maximum follows. These results already go beyond the mere possibility of describing an individual configuration of preferences: they show that, if a prescribed utility model exists, then utility maximization is a well-posed problem relative to the preference that it represents.

This example illustrates the fact that an account of applications based on mappings cannot exclusively focus on the correspondence between an empirical and a mathematical structure: it must at least take into account the various formal properties involved in the characterization of the empirical setup and pay attention to the properties of the mapping that links it to a mathematical structure. One may proceed further and include cases in which more than a single mapping is involved in the application process: for example, one may have a mapping that embeds the empirical structure into a mathematical model and a distinct mapping that induces additional structure from the mathematical model into the empirical structure. The need for these requirements as parts of a general mapping-based account of applications has been emphasized in Bueno and Colyvan (Reference Bueno and Colyvan2011), which provides what the authors take to be an expanded and supplemented version of previous mapping-based accounts of applications, notably the recent one presented in Pincock (Reference Pincock2004).Footnote ⁴

The foregoing discussion highlights the fact that, when articulated in sufficient detail, mapping-based accounts of applications do indeed provide a rich framework within which the scientific use of mathematics can be understood. These accounts have already proved to be of significant philosophical interest in a variety of contexts, from classical field dynamics (discussed in Field Reference Field1980) and quantum mechanics (discussed in Balaguer Reference Balaguer1996) to psychological measurement (Krantz et al. Reference Krantz, Luce, Suppes and Tversky1971) and economic theory (e.g., the examples in this section). This seems to suggest that the philosophical understanding of the process whereby mathematics is applied is well understood and that it revolves around the dominant idea of structure preservation. The main aim of this article is to challenge this conclusion by drawing attention to the existence of applications that radically differ from the ones whose salient features must be spelled out in terms of mappings between structures.

2. Applying Mathematical Concepts

The main reason why a mapping-based account of applications, however detailed and sophisticated, cannot be comprehensive already emerges if one carefully examines the economic example from the previous section. In that example, it was apparent that the mere availability of a utility representation for subjective preference did no more than making it possible to present empirical information in a numerical format, while it did not, as such, qualify any fruitful or productive feature of the application process.

Fruitful features could be discerned when the empirical structure was assumed to be sufficiently rich or when it became meaningful to use certain resources of the mathematical structure in order to gain additional information about the empirical structure. This happened only after S was assumed to be a metric space and the existence of a maximum of u was used to deduce that an optimization problem could be solved in S. In this case, although the correspondence between an empirical structure and a mathematical structure plays an important role, its applicative relevance is subordinated to the isolation of relevant formal properties of the empirical structure and their use as constraints on the corresponding mathematical model (e.g., the restriction of u to a compact subset of S implies the existence of a maximum in the numerical image of that subset). The general fact illustrated by the utility example is that the isolation and selection of salient formal properties characteristic of an empirical structure is a nontrivial conceptual move on which the application of mathematics may fundamentally, and sometimes entirely, rely but that cannot be described in terms of mappings because it is required and accomplished before the representation into a mathematical model takes place.

In other words, structure preservation and the choice of a mathematical model can sometimes be effective only after a specific empirical structure has been selected. Incidentally, this implies that the introduction of mathematical structures may be naturally productive in those contexts in which it is assumed that the corresponding empirical structures already satisfy sufficiently strong formal properties. In other words, the successful use of a mathematical structure may sometimes be no more than a consequence of the way in which an abstract characterization of the corresponding empirical setup has been constructed.Footnote ⁵

Independently of whether this is the case, the isolation and deployment of formal constraints remains a substantive step in the application process that does not rely on the use of mappings, even though it may occur in types of application in which mappings are involved. Mapping-based accounts of applications must therefore include mapping-independent moves as part of the application process, if they are to be realistic. The stronger conclusion, which I will try to establish in the following sections, is that there exist empirical problems in which the selection of salient formal properties of an empirical setup is all that is needed to apply mathematics: mappings into a distinct mathematical structure are not required; that is, mapping-based accounts are largely irrelevant. This is particularly clear when the empirical problem at hand provides only a heavily underdetermined picture of the associated empirical structure, for in this case even the presentation of an empirical structure is less relevant to the application process than the deductive analysis of certain significant consequences of its assumed formal properties. In the next two sections, I will put some flesh on these very general remarks by illustrating them in the context of an application of mathematics to a design problem in voting theory.

3. Preference Aggregation

The problem that I want to discuss concerns the characterization of a class of voting rules. A voting rule (think of, e.g., simple majority) may be regarded as a procedure that computes an aggregate ranking of competing candidates on the basis of the individual rankings or individual preferences expressed by the voters.Footnote ⁶ A voting rule applies to any combination of these preferences and generates from them an aggregate ranking, that is, a complete preorder of the candidates. This way of describing a voting rule is very general and applies to many possibilities, including the rules that identify, under any circumstances, the aggregate ranking of any two candidates with the individual ranking of a fixed voter. Any such rule is called a dictatorship.

Despite their obvious negative features, dictatorships have two properties, sometimes called Binary Independence and Unanimity, which may be considered plausible requirements of any voting rule. Binary Independence says that the aggregate ranking of any pair (x, y) of candidates is uniquely determined by the list of individual rankings of that pair. Unanimity says that, whenever every agent agrees on the same strict ranking of a pair (x, y), this coincides with the aggregate ranking of the same pair. Unanimity specifies how the complete agreement of the voters is to determine the aggregate preference over a pair, while Binary Independence expresses the condition that two distinct candidates should be judged on their relative merits alone. The design problem that now arises is whether it is possible to retain Binary Independence and Unanimity without choosing a voting rule to be a dictatorship. The surprising answer is that, as soon as at least two voters and three candidates are involved, this is not possible. In short:

Arrow’s theorem, together with its proof, is the (negative) mathematical answer to the question whether a nondictatorial design is compatible with the joint assumption of Binary Independence and Unanimity. In order to reach this answer, it is necessary to find a way of studying voting rules of which only certain properties are known and a way to establish a correlation between these properties and the concentration of voting power. Both problems are solved by the use of mathematical concepts and arguments.

4. Articulating the Problem

On a mapping-based account of applications, the starting point of an application of mathematics consists in the presentation of an empirical setup endowed with a certain structure. In the context of preference aggregation, the immediate object of interest is not an empirical setup (i.e., a given object open to empirical investigation) but a type of procedure, that is, an election rule whose design is only partially specified. One may try to turn such a procedure into an empirical structure by identifying it with the complete list of its outcomes for each possible profile of individual preferences. This is, in effect, not enough because an election rule will be fully determined only once one knows the totality of its outcomes for the totality of profiles determined by any finite number of voters and candidates. This generates a space of election profiles and outcomes of which too much is unknown for it to be helpful in finding out how Binary Independence and Unanimity constrain a voting procedure. A representation of this space by a suitable mapping or family of mappings would not lead to any progress in the solution of the design problem; it would merely lead to its translation into an equivalent problem.

The difficulty here is that it is unfeasible to handle an election rule by brute force, identifying it with the totality of its outcomes under all possible circumstances. However, because two conditions are assumed to hold for the type of election rule being considered, it is natural to think of constructing arguments that identify the properties of election rules and only rely on the two conditions that they are known to satisfy. The next step is to decide what properties must be sought. Because dictatorships satisfy Binary Independence and Unanimity and the question is whether there exist alternative procedures that do, the relevant properties must concern the concentration of power in the hands of groups of voters. In fact, Unanimity is already a property of this type because it asserts that the totality of voters can impose any admissible election outcome over a list of candidates.Footnote ⁷ The question is whether the same prerogative belongs to smaller groups of voters and, in particular, whether it must always belong to a single voter.

The relevance of this question lies in the way it changes the approach to the voting problem at hand: instead of working on the combinatorial structure of preference profiles, one can focus on the way in which the voting rule divides up any family of voters into groups that detain the power to impose election outcomes. Note that this train of thought changes the position of the problem: instead of beginning with an election rule, which may be identified with a space of empirical structures, one uses information coming from the properties of the election rule in order to turn it into information concerning the concentration of power among the voters. One may semantically regard this approach at most as a transition from a given empirical structure to an induced empirical structure, not as the mapping of an empirical structure into a mathematical model. In fact, semantics does not play a more prominent role in the solution of the voting problem than deduction based on the elements of the problem itself, as I will now clarify.

5. Tackling the Problem

The previous section has shown that a mapping-based approach will not help elucidate the mathematical treatment of the voting problem that is negatively solved by Arrow’s theorem. The reason is simply that the type of design to be studied mathematically is not given as an object that already exhibits salient features whose representation over a mathematical model could then be exploited to solve the voting problem. The real issue is to find a way of studying the empirical problem itself. In the light of the foregoing discussion, the way to do it is deductively to determine certain properties of voting coalitions that are induced on the family of voters by the conditions of Binary Independence and Unanimity.Footnote ⁸ Because the voting problem is underdetermined, in the sense that no voting rule is explicitly described and no number of voters and candidates fixed (only two properties of an arbitrary procedure are specified, and only a lower bound is given for the finite number of voters and candidates involved), its treatment must be essentially deductive; that is, it cannot rely on the description of, say, an election or a family of elections but must consist in the construction of arguments that link what is known about any procedure of the prescribed type to what must happen to the associated coalitions of voters.

The required link is not obtained by means of a mapping but through the introduction of the empirical concept of a decisive coalition of voters:

This definition pins down a specific aim to be pursued in the study of a type of election rule: the aim is to describe how the type of rule constrains (strong) decisiveness.Footnote ¹⁰ If decisive coalitions are forced to satisfy certain properties, then these properties can be used to obtain general results about the concentration of voting power.

It can be quickly seen that the definition of decisiveness entails a number of immediate consequences of interest. First, it implies that Unanimity is equivalent to the existence of a decisive coalition, due to the fact that any enlargement of a decisive coalition is decisive. Second, it implies that a coalition without members cannot be decisive (i.e., no aggregate, pairwise rankings can be imposed vacuously): the reason is that, because this coalition has no members, it is consistent to assume that its members should both rank y over x and x over y, thus producing the cycle in the election outcome. Because of this, the smallest possible coalition must include at least one agent. If it coincides with one, this agent is a dictator. That this outcome is unavoidable follows from three less immediate properties of decisive coalitions,Footnote ¹¹ namely,

1. 1. If a coalition is decisive over a pair of alternatives, then it is strongly decisive over every pair.

2. 2. The common members of two decisive coalitions form a decisive coalition.

3. 3. If the totality of agents splits into two coalitions without common members, one of them is decisive.

Given these properties, it suffices to consider the coalition formed by the agents that appear in every decisive coalition. Because there are only finitely many of them, these agents form the smallest decisive coalition, by property 2. Such a coalition must have at least one member δ but cannot have more than one member. For, if it had another member ε, property 3 would be contradicted since neither the coalition with δ only nor the coalition with every other voter could be decisive. As a result, the smallest decisive coalition has no members other than δ; that is, δ is a dictator.

6. Deductive Analysis

The last two sections have described one direct way in which it can be shown that the problem of finding an election rule that satisfies Binary Independence and Unanimity but is not a dictatorship cannot be solved. This result can be obtained by introducing the empirical definition of decisive coalition and carrying out the deductive analysis of its consequences, given the available information about the type of election rule under investigation. The crucial point to be noted is that the proof of Arrow’s theorem is made possible not by the identification of an empirical structure and its embedding into a mathematical model but by the deployment of conceptual resources (the definition of decisiveness and the properties of decisive coalitions) that establish a characterization of all binary independent and unanimous election rules. What matters here is precisely the isolation of a concept that offers a way of studying these rules and proves sufficient to determine their fundamental feature. The arguments from the previous section (as well as the appendix in which properties 1–3 of decisive coalitions are established) can be articulated in a direct way, that is, without abandoning their empirical motivation and switching to a purely mathematical interpretation of the inferences being made. This suggests that, in the current example, the search for a fruitful way of thinking about election rules and of constructing arguments that spell out their properties is more important than the choice of a particular mathematical representation of the problem at hand. It would be incorrect, however, to conclude for this reason that my discussion of Arrow’s theorem shows that this result can be established in a nonmathematical way: what it does show is that mathematics can enter applications as a principle of argument construction and deductive analysis.

In fact, it would not be difficult to translate in set-theoretical language the results of the previous section: they can be concisely summarized by the statement that any binary independent and unanimous election rule induces an ultrafilter of decisive coalitions on the family of voters.Footnote ¹² Because this family is finite and it can be proved that an arbitrary ultrafilter over a finite set is generated by a singleton, one can run the very same argument directly on decisive coalitions to show that they include a dictator, that is, that one of them coincides with one agent. In other words, Arrow’s theorem is arrived at by working on the available data of a voting problem to construct an ultrafilter-style argument.

7. Arguments, Not Mappings

Having provided a detailed account of the way in which a negative answer is obtained for the problem of aggregating preferences by a nondictatorial rule when Unanimity and Binary Independence are assumed, I wish to further clarify why this application of mathematics does not invoke the mappings that are central to the picture of applications discussed in section 1. That picture requires mappings at two stages of the application process: in the first stage, a transition from an empirical structure to a mathematical structure is effected; in the second stage, a particular statement, whose interpretation is given relative to the mathematical structure, is reinterpreted in empirical terms. Mappings allow this back-and-forth movement by linking the structures involved in one or the other direction. The voting design problem answered by Arrow’s theorem is presented as soon as the empirical conditions (concerning preferences, candidates, and the properties of aggregation rules) that make up its formulation are stated. In order to deal with this problem, there is no particular need to specify an empirical structure associated with it: what is important is to find a way of deducing relevant information from its formulation. This is not achieved by introducing a mathematical structure into which some alleged empirical structure is to be embedded but by introducing the empirical definition of a decisive coalition: this definition fixes a specific approach to the problem; that is, it turns it into a systematic investigation of the properties of decisive coalitions. These latter properties are to be interpreted empirically since they are meant to be meaningful for groups of voters (there would otherwise be no reason to spell them out). Again, no mappings between structures are involved since what is being carried out is the deduction of formal properties on the basis of an empirical definition and additional constraints given by the formulation of an empirical problem. Finally, the formal properties so deduced lead to a conclusion that, in turn, has a direct empirical interpretation. The argument leading to this conclusion is a formal deductive argument concerning the properties of groups of voters: its conclusion does therefore refer to groups of voters and does not need to be endowed with a preliminary mathematical interpretation, later to be translated into empirical terms. Since the aim is to shed light on voting procedures and coalitions, the only relevant interpretation of the steps in the proof of Arrow’s theorem is the empirical one, which does not have to be reached through the mediation of a mapping linking a mathematical structure to an empirical setup.

This is not to deny that one may regard the proof of Arrow’s theorem as an application of a theorem about ultrafilters to voting theory but to stress the fact that the theorem is applied in the precise sense that the argument used in its proof can be run with respect to decisive coalitions because these very coalitions satisfy formal properties sufficient to run it. It is crucial to remark that the argument leading to Arrow’s theorem is not an attempt to link a mathematical structure (i.e., an ultrafilter) to an empirical structure but rather the proof that the statement of an empirical problem entails the defining properties of an ultrafilter, which can in turn be exploited to address the problem itself. It is the deduction of these properties for coalitions, not the set-theoretical object ultrafilter, which makes this possible.Footnote ¹³

A reference to ultrafilters is useful in this context only because it pins down a formal argument schema that, together with its variations, can be adopted to systematically study preference aggregation, without this having to require a back-and-forth semantic move between ultrafilters as set-theoretical constructs and families of decisive coalitions. The applicative significance of this possibility goes beyond Arrow’s theorem because it gives rise to further ways of applying ultrafilters or their generalizations to the study of election rules. For example, approximations of Arrovian procedures can be constructed by defining them in terms of large families of decisive coalitions that do not coincide with ultrafilters (Brown Reference Brown1974), distinct dictatorships’ results can be unified by showing that they all depend on the same ultrafilter argument (e.g., Batteau, Blin, and Monjardet Reference Batteau, Blin and Monjardet1981), or election rules weaker than dictatorships can be directly recovered by weakening the ultrafilter properties of a family of decisive coalitions (e.g., Weymark Reference Weymark1984). None of these strategies relies on mappings: they are all based on a systematic, deductively established correlation between certain formal properties of decisive coalitions and corresponding properties of voting rules.Footnote ¹⁴

8. Empirical Problems and Formal Properties

When it is possible to prove the existence of a structure-preserving mapping that embeds an empirical setup into a mathematical model, this is because the formal properties of the empirical setup suffice to establish the embeddability relation.In this respect it is natural to think, as in section 2, that the applicability of mathematics depends essentially on the identification of the formal properties of an empirical setup, in view of which the latter setup becomes mathematically tractable because it can be linked to a particular mathematical structure.Footnote ¹⁵

Applications that are based on structure-preserving mappings thus rely on a fundamental step that is preliminary to the introduction of mappings and consists in the specification of formal properties possessed by an empirical structure. The previous sections have shown that this specification has in fact an independent value in empirical investigation, even though in several cases its importance is closely connected to the introduction of mappings.

In other words, it may happen that the specification of formal properties suffices for the resolution of an empirical problem, once it is carried out. The examples from voting theory on which I have focused clarify how nontrivial and fruitful it may be to identify the desired formal properties: their isolation is typically closely related to the introduction of a suitable empirical definition, which throws a new light on the data of the problem at hand and ultimately leads to its solution and generalization. Mappings do not appear in this context simply because the tractability of the problem of interest depends entirely on an advantageous choice of concepts, from which arguments can be constructed that yield a deduction of the solution to the problem itself, based solely on its formulation. Mathematics plays a role here not so much as a source of structures but as a source of concepts and forms of reasoning, which can be applied as soon as the target empirical problem is framed in a way that allows their deployment.

In such contexts, the central and most difficult step in the application process consists in reaching a presentation of the relevant empirical problem in terms of concepts that make it possible to analyze it by means of suitable arguments. In other words, the issue is to find a fruitful way of thinking about an empirical problem that, as it is given, does not constrain any particular approach to its investigation and resolution: this means that it is crucial to find a way of working with the empirical elements of the problem directly, in order to deduce a solution from them. When this is done, mappings between structures are not invoked or required: as a consequence, not every application of mathematics is an application of mathematical structures via mappings.

In particular, since mappings result from the isolation of the formal properties of an empirical structure, they are to be regarded as one, and only one, of the possible effects of this move, which in general may let the application process unfold in different ways. In other words, when mappings joining structures are assumed as the cornerstone of a general characterization of applied mathematics, it becomes more difficult to discern a stage in the application process at which divergences, only some of which ultimately issue in the introduction of mappings, occur. It follows that one cannot be guided by the thought of the universal validity of mapping-based accounts of applications without at the same time being prevented from making progress in the understanding of how mathematics is actually applied.