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Mathematical discourse andcross‐disciplinary communities:The case of political economyRobert Pahre aa School of Public Policy , University of Michigan , AnnArbor, MI, 48109, USAPublished online: 19 Jun 2008.

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SOCIAL EPISTEMOLOGY, 1996, VOL. 10, NO. 1, 5 5 - 7 3

Mathematical discourse and cross-disciplinary communities: the case ofpolitical economy

ROBERT PAHRE

Mathematicians are only dealing with the structure of reasoning, and they do not really care what theyare talking about. They do not even need to know what they are talking about' (Feynman, 1965, p. 55).

Abstract

This paper explores the role of symbolic languages within and between positivistdisciplines. Symbolic languages, of which mathematics is the most importantexample, consist of tautologically true statements, such as 2 + 2 = 4. These mustbe operationalized before being useful for positivist research agendas (i.e. twoapples and two oranges make four fruit). Disciplines may borrow either thesymbolic languages of another discipline or the symbolic language and theaccompanying operationalizations. The choice has important theoretical effects,and affects the kind of interdisciplinary community created. The game theorycommunity is an example of a community based on the interdisciplinary exchangeof symbolic language only, while the ' political economy' community exchangesboth symbolic languages and operationalizations.

Mathematics plays a vital role in all the natural and social sciences. Even high schoolstudents learn that many central propositions of these sciences are best statedmathematically. In addition, mathematics may play a special role in forming scientificconsensus, for the unambiguity of these 'symbolic generalizations' means that they canbe shown to scientists who might disagree with a particular claim when presented innatural language (Fuller, 1988, p. 219). The use of mathematics may even serve as aspur to innovation. James Buchanan (1966, p. 173) claims that mathematics''contribution to the productivity of economists, at the margin, may be questioned, butthe integral of the product function must be large indeed'. It is worth noting that

Author: Robert Pahre, School of Public Policy, University of Michigan, Ann Arbor, MI 48109, USA. Anearlier version of this paper was presented at the Group Research on the Institutionalization and Pro-fessionalization of Knowledge Production (GRIP) Conference at the University of Minnesota, Minneapolis,April 1994. The paper has benefited from comments by Robert Axelrod, Bear Braumoeller, Lars-ErikCederman, Don Herzog, Ted Hopf, Simon Jackman, Ann Lin, Raymond Mclnnis, and especially EllenMesser-Davidow. I would also like to thank GRIP Conference participants for their reaction.

0269-1728/96 $12-00 © 1996 Taylor & Francis Ltd

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Buchanan's causal use of the language of calculus highlights how deeply mathematicspenetrates the way in which economists think about the world.

Mathematics is also important for the philosophy of science. Heavily mathematicalfields such as physics play a special role in the philosophy of science, and serve as a pointof reference for examining the validity of scientific laws or scientific advance (Lakatos,1978). It does not surprise us to see an exposition of physical laws revel in the beauty ofmathematics (Feynman, 1965), nor is it a surprise to see a philosopher address themathematics of physics as part of her critique of these physical laws (Cartwright, 1983).

Like the sciences themselves, cross-disciplinary communication in the sciences oftentakes a mathematical form. To take one obvious example, Newton's calculus spreadfrom physics to other disciplines and is now essential for all the sciences. Many arguethat scientific borrowing moves down a disciplinary hierarchy, from the 'harder'disciplines to the 'softer' ones, where hardness is defined in large part by reliance onmathematics (Sherif and Sherif, 1969a; Klein, 1990, pp. 85-86).

Despite its importance, we rarely think much about the role of this discourse. Thestudy of mathematical discourse is lacking even in the growing field of boundary studieswithin the sociology of knowledge, which explores the permeation of disciplinaryboundaries and the creation of interdisciplinary fields (Campbell, 1969; Dogan andPahre, 1990;Fuller, 1993, ch. 6; Klein, 1993; Sherif and Sherif, 1969b). To fill this gap,I will consider here the role of mathematics in scientific exchange.

I examine two questions stemming from this exchange. First, what is the logicalstatus, or truth-value, of a given symbolic borrowing? In other words, if we know asymbolic or mathematical statement to be ' t rue ' in one discipline, should we alwaysexpect it to be true in another? Discussing the truth-status of a borrowing requires thatwe have some notion of the epistemology within which scientists make truth claims.That epistemology will, then, shape bo th truth claims and discourse. For simplicity, theanalysis here will be limited to disciplines with a broadly positivist epistemology.

Second, what kinds of interdisciplinary communities result from the exchange ofsymbolic language? If borrowings are unidirectional, from mathematics to physics tochemistry to biology to economics to the other social sciences, those at the start of thefeeding chain will probably not care how others consume their mathematics. On theother hand, bilateral exchange might help to create scientific communities, inter-disciplinary syntheses, or even new disciplines.

These two issues, truth-status and community, are interrelated. When we look atscientific communities, we often observe miscommunication across disciplinary bound-aries (Becher, 1994). Miscommunication stems in part from the fact that the truth-status of the borrowing may or may not depend on the truth-status of the language in thelender discipline. If the truth-status of a borrowed statement does not depend on itstruth-status in the lender field, then the lender field may be surprised to see othersborrowing obsolete theory or 'misusing' good theory. This makes misunderstandingespecially likely and interdisciplinary communication particularly difficult.

As this overview may suggest, this paper belongs to a larger agenda that tries todevelop a sociology of knowledge that tries to understand the intellectual content ofscience as we study the social structure of science (cf. Schmausrf a/., 1992;Pahre, 1995).Despite the value of the effort, sociological attention to the intellectual content of sciencehas fallen by the wayside in recent years, criticized as an 'intellectualist' history ofscience that ignores the social constraints on science. I will aim here for a criticalevaluation of the content of science while studying the social organization of scientificcommunities; because of space limitations, I will set social constraints aside.

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My approach is speculative, with a few illustrations of the argument. Almost allillustrations come from political economy broadly defined (that is, to include the use ofeconomic methods such as game theory to study politics). Political economy makes avery good case, for it is a mathematical hybrid that makes extensive use of severaldifferent kinds of mathematical exchange. This also happens to be my primary field ofresearch, so this perspective makes the paper resemble an anthropologist's method ofparticipant-observation.

1. Symbolic (non-natural) languages

Most scientific disciplines make use of some symbolic language. By a symbolic language,I mean any symbolic system of representation other than natural languages. This meansmathematics above all but includes any other symbolic system, such as computerlanguages, Boolean logic, transformational grammar, or a system such as Claude Levi-Strauss's Les structures ele'mentaires de la parente.

While discussing symbolic languages, I will use both natural language and a symboliclanguage.1 I find a symbolic language useful in thinking about these things, and hopethat others do so as well. The symbolic language I use is secondary to the argument here,but it is intended to be a first step in trying to develop a language for exploring questionsof interdisciplinary discourse.

To develop this symbolic language, I conceive of all the 'stuff' of science aspropositions. Clearly, an assumption is a proposition, as is an hypothesis or a conjecture.A proof consists of the proposition proved and the assumptions on which that proof rests.Thus, a mathematical proof is simply a set of logically interconnected propositions.Evidence, too, is a set of propositions. Conceived of in this way, a theory such asMarxism is a collection of propositions serving as assumptions ('a class is defined by itslocation in the relations of production', 'labor is the source of all value'), propositionsderived from these that serve as hypotheses (' revolution is most likely during economiccrisis'), and a set of discovered evidentiary propositions that are to be compared withthe hypotheses ('the German Social Democrats did not turn revolutionary during theGreat Depression'). Clearly the set of all such propositions in any specialized field, muchless a discipline, is extraordinarily large.

Symbolic statements are a subset of all propositions in a discipline. By its nature, anysymbolic statement is tautologously true. For instance, the symbolic statement that' 2 + 2 = 4 ' is a mathematical truth derived from numbers theory, and is true if thelogical system in which it is found is internally consistent.2 This statement would bemeaningful in a mathematical sense even if we never applied it to real-world problemssuch as adding two apples and two oranges to obtain four fruit.3

It may seem strange for empirical sciences to make widespread use of tautologicallytrue statements. Using a symbolic language for empirical problems requires first that weoperationalize the propositions of that language. If we falsify a hypothesis derived fromoperationalizations of this symbolic statement, we do not falsify the symbolic statement.Instead, our attention is directed to the operationalizations, which might be faulty,incomplete, or wrong. For instance, if I have one coin collection and inherit anothercoin collection, I still have only one coin collection. I do not conclude that the symbolicstatement '1 + 1 = 2' is false, but conclude that I must modify my operationalizationsto take collective nouns into account.

To take an example from social science, in the 1950s several scholars tried to

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operationalize the mathematics of game theory to develop a measure of'power'. Thegeneral approach was to imagine all the different possible coalitions that a group ofplayers might form and then see whether a given player was on the winning side (theBanzhaf measure) or was critical to the winning side (the Shapley-Shubik measure).The frequency with which that player ' won' in either sense measures power. Empiricalresearch later showed that none of these measures worked very well (Riker 1992). Thisresearch does not show game theory to be 'false', but shows merely that certainoperationalizations of the mathematical theory are false when applied to the field ofpolitics. For instance, imagining all possible coalitions is probably a poor way to thinkabout real political coalitions, the possibilities of which are much more constrained.

Since any symbolic language is protected from falsification by the fallibility of itsoperationalizations, a theory embodying these languages is unfalsifiable for the samereasons that the 'hard core' in a Lakatosian (1970) research program is unfalsifiable.Yet this is not a satisfactory objection to such languages. We can only criticize a givenscholar for non-falsifiability if that person operationalizes a symbolic language in a non-falsifiable way. The theory of voting was long guilty of such non-falsifiability (Green andShapiro, 1994; Lohmann, 1995). After showing that voting was irrational because ithad some cost to the individual and had no real benefit (no one person influences anelection), theorists hypothesized that voters received some utility from 'duty' — that is,people voted because it made them feel good to act according to internalized socialnorms. This was a non-falsifiable operationalization of the underlying mathematicaltheory, since theorists could ' explain' any apparently irrational voter by claiming thatthe 'duty' which that voter felt must have been sufficiently large to induce the personto vote. This, of course, is a non-falsifiable kind of operationalization — but it does not'falsify' the underlying symbolic language of game theory.4

Since they are unfalsifiable, we might think that such statements are trivial andremoved from reality.5 This is mistaken. Substantively empty symbolic languagesare immensely important in individual disciplines and in cross-disciplinary discourse.To take the most striking example, symbolic statements from statistics — the descriptionof a normal distribution, sampling rules, Bayesian inferences, and hundreds more — aresubstantively empty: it matters not whether one is counting gold mine production,deaths in war, or quasar emissions. Many such statements, such as the definition of amean or the derivation of a regression estimate, are tautologously true. None the less,these symbolic statements are enormously useful and have spread from discipline todiscipline.

If we grant my distinction, then any use of symbolic language in the sciences consistsof at least two parts: a set of symbolically true propositions and a set of statementsoperationalizing these symbolic propositions.6 These operational statements tell us thatin Newton's symbolically true statement 'F = ma', F represents 'force', m represents'mass', and a represents 'acceleration'. The choice of operationalizing statements islogically unrelated to the substance of the symbolic statements. We might choose F torepresent, say, the political importance of a demonstration on the Mall in Washington,m to represent the number of people at the demonstration, and a to represent the levelof media attention the demonstration receives.

Symbolically, any theory T using symbolic language will consist of symbolicpropositions Ps and operational propositions Po; T is a set whose members are, at aminimum, defined thus: T = {Ps, Po}. The various propositions of a theory are also usedto generate hypotheses, and I shall label the set of hypotheses H. For any well-definedtheory there is a relationship, or mapping, from the sets of propositions into the set of

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hypotheses.7 Symbolically, we can represent this as T = [Ps xPo->H]. If its statementsare internally consistent, Ps will be symbolically true and non-falsifiable, as describedabove.8 Elements of Ps are rejected only if no Po makes them useful.

2. Data, discourse and discipline

The set of propositions making up the data of a discipline both constitute and define thatdiscipline (Pahre, 1995).9 The data are not logically dependent upon the relationshipbetween propositions and hypotheses, so let us conceive them as a separate set within thetheory. Thus, I understand the role of any set of data D in a theory as follows:T={PsxPo^H;D}.

In this framework, interdisciplinary discourse means copying a set of propositionsfrom one discipline to another.10 For instance, a discussion between anthropologists andeconomists about the nature of non-market economies consists of some propositionsbeing copied from anthropology into economics ('reciprocity in social relationsinfluences economic exchange') and some propositions being copied from economicsinto anthropology ('scarcity raises the exchange-value of goods whatever the socialcontext'). Discursive features that are important to meaning, such as the sequencing ofstatements, are merely additional propositions in this exchange.

Interdisciplinary discourse is constrained by some notion of germaneness in theborrowing discipline. Sociology will not borrow the propositions of particle physicssimply because they are true, for they are (presumably) irrelevant to sociologicalconcerns. To be germane, a borrowing must explain some subset of the new discipline'sexisting set of data.11 Consequently, the truth-status of a proposition such as anhypothesis depends on whether it contradicts other propositions such as evidencegermane to the discipline. No theory seeks to explain everything, and we shouldevaluate its (partial) truth-status compared with what it seeks to explain.

Sometimes, these hypotheses are consistent with the data, while in other cases, theyare inconsistent. Hypothesis testing consists of setting a set of evidence next to a set ofhypotheses and looking for logical contradictions. If the hypotheses perfectly describethe data, and if no hypothesis describes data that are not in the set D, thenH(\D = H\)D = H=D. This will not normally be true in the real world of scientifictheories, for there is inevitably disjuncture between a field's data and the hypotheses thefield uses to explain the data. This disjuncture helps to define the field's research agenda.

When we think of symbolic languages as set forth above, there are two main forms ofinterdisciplinary discourse (borrowing). The first kind borrows results; in the abovelanguage, it borrows H. Logically this H will depend on the prior mapping{PsxPo->H}. This kind of borrowing leads to research communities practisingcumulative knowledge.

The second kind of borrowing takes the symbolic language Ps without theoperationalizations Po. Then, the first thing a borrower will do is to develop a new setof operationalizing propositions Po', which then produces a new mapping{PsxPo'->H'}. This is the language of metaphor. Metaphorical communities mayemerge, but successful innovation in a metaphorical community is a less predictablematter, so such communities do not practise the accumulation of knowledge in thenormal sense. Since metaphors may or may not work, using them is a risky strategy forinnovators — but the possible pay-off is high because the metaphor is truly novel withinthe borrower community.

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In both cases, I assume that there are no problems of interpretation: that is, bothborrower and lender can agree on what a given set of propositions is and what it means{contra Quine). This is a simplification. Even in highly mathematical fields, thestandards of proof, the interpretation of proof, and the rhetoric of mathematicallanguages can vary widely, and such standards are social constructions (see Livingston,1993; Lakatos, 1970; McCloskey, 1985, chs 8-9). In the social sciences, the standardsof mathematical proof are very different in economics than in sociology or politicalscience. For that matter, the standards are different in the highly technical journalEconometrica, where — ironically enough — authors can often rely on informal shortcutssuch as 'clearly', 'proof follows easily', or even 'proof omitted'. The standard ofacceptability for such shortcuts is a social construction.

Different disciplines will also see the importance of a mathematical statement throughdifferent lenses, even when they agree what that statement means. For instance, theCoase Theorem is an important part of neoclassical economics, deemed worthy of aNobel prize. It says that economically efficient outcomes will occur whenever peoplehave well-defined property rights, freedom of contract, complete information, andcostless transactions. Economists normally interpret this as saying that markets areefficient, and that government intervention is unnecessary. Political scientists normallyinterpret the theorem as saying that government intervention is necessary, not only toestablish property rights but also to remedy market imperfections in a world ofincomplete information and costly transactions.12 Such differences of interpretationmay affect the interdisciplinary exchange of mathematics.

3. Borrowing results

Let me begin with the first of my two types, the exchange of results along with theimplicit borrowing of the prior mapping {Ps X Po^-H}. Using our symbolic language,let us characterize the lender theory of {Ps xPo-+ HL}. Borrowing this theory transformsa borrower field from {HB; Z)B} to {HB; {iDs x Po -> Hh}; DB'}. If the borrower theory didnot explain something new, there would be no reason to borrow it, so DB c Z)B'. Here,we have added the lender theory to the borrower field, where it helps to explain moredata than the borrowers could previously explain. This creates a set of newly-explaineddata, which we may define as the set of data now part of the field that was not in the fieldbefore (that is, DBN = {DeDB':D<£DB}).

This set of newly-explained data may play a dual role; because the symboliclanguage, operationalizations, and hypotheses are identical in lender and borrowerdisciplines, any data that these hypotheses explain now fit as easily in the lenderdiscipline as in the borrower. Thus, instead of exporting theory to the borrower, we mayas easily export data to the lender, transforming the lender theory from{Ps x PO^HL;DL} to {Ps x P0->#L;Z)L '}, where Z>L' = {DL U DBN}.13 In other words,whatever new data the borrowing field can explain with the aid of the new theory canbe re-exported to the lender field and count as explained data there.14

When this occurs, observers may describe an ' imperialist' discipline that incorporatesthe subject matter of neighbouring fields. Among the social sciences, economics is mostoften accused of imperialism. One widely exported theory is the mathematical theory ofpublic goods, developed by Paul Samuelson and others in the early 1950s. In the early1960s, Mancur Olson (1965) saw that the results would also apply to the formation ofinterest groups, a traditional topic of political science. His application of the theory to

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these collective action problems transformed the subfield, and called into seriousquestion previously dominant theories such as David Truman's more sociologicalanalysis.

Throughout the 1960s and 1970s, increasing numbers of economists conquered largerand larger pieces of political science, often using collective action theory as a vanguard.One narrower area into which they have moved is 'endogenous tariff theory' — atheory that explains policies such as tariffs instead of treating them as exogenous toeconomic theory. The theory of collective action had some success explaining tariffs,and led researchers to examine questions such as the number of members in a potentialinterest group and the concentration of resources among these members. Once theybegan to think about the topic in depth, economists confronted political scientists' moreinductive theories about the data. They imported many such theories into economics,where they became stylized acts that served as the cornerstones of deductive economicmodels. As a result, important economic models of tariff formation rest on key variablesfrom political science, such as elections, parties, interest groups, or administrativeagencies. This exchange also sparked the interest of many political scientists, whoborrowed theoretical findings from international economics and public choice theory.The pattern by which the mathematical field exports theory and imports data haspersisted for some time, as in other areas of political economy.15 Ironically enough, thispattern has tended not to include the export of mathematical political science back todeductive economics, although the deductive 'new institutionalism' is a very promisingavenue of research for economists interested in these matters (Nelson, 1988).

Now let us consider the truth status of such exchanges. The initial borrowing causesno particular problem. After all, the symbolic language, operationalizations, and theresults are already tightly linked in the lender theory. The borrower retains these linkswhen it borrows results.

Further developments in both disciplines may change matters. If a discipline falsifiesa borrowed theory, these data create serious problems for the lender discipline. If theborrowers use the same operationalizations as the lender field, there is no logicalfoundation for a lender discipline to ignore anomalies discovered outside the discipline.16

For instance, data showing that the concentration of economic resources does not affectpolitical outcomes should count as anomalies not only for endogenous tariff theory butalso for the theory of public goods on which it rests.17 For these reasons, both lender andborrower are better off if they form an intellectual community and continue to exchangefindings (Sherif and Sherif, 1969a).

Such a community would help both disciplines to make sure that evidence andhypothesis mesh where appropriate. Todd Sandier (1992, pp. 163-164) rightly criticizesseveral political scientists who test economic models of collective action for failing tospecify which model of collective action they are testing.18 Since there are several suchmodels, with different assumptions and predictions, the point is critical.

Another possibility is that such evidence might lead us to conclude that we shouldlocalize a given theory's domain, applying it to one topic and not the other. Of course,this approach is not without its problems. If we decide to exclude only those domainswhere the theory doesn't work, then we have used impermissible circular reasoning todefine the domain where the theory will work. This also has implications for the creationof scientific communities. If borrowers are using only part of a theory, and if otherborrowers may be using different parts, intellectual divisions can develop amongborrower fields despite the original close links that stem from borrowing logicallycoherent results.

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Divisions can reach the point where those who borrow results will not conductresearch useful for the lender discipline. This happens when the borrower adds to (orchanges) the operationalizing propositions of the lender theory. As an example, let ustake the application of key theorems of international economics to political behavior.The Stolper-Samuelson theorem examines a world of two countries and two factors ofproduction (labor and capital, for example). It says that the relatively abundant factorin each country will gain from international trade, while relatively scarce factors will beharmed by international trade but benefit from protection. In other words, the theoremsays that the North American Free Trade Agreement (NAFTA) will benefit capital andhurt labor in the capital-abundant USA and benefit labor and hurt capital in labor-abundant Mexico.19

Ronald Rogowski (1989) borrowed the Stolper-Samuelson theorem, adding thenovel proposition that Stolper-Samuelson winners will form political coalitions witheach other, while losers will form coalitions with other losers. He also argues thatpolitical coalitions may follow these trade-based coalitional lines even when non-tradeissues are involved. Symbolically, Rogowski's theory Tn adds Rogowski's newpropositions about coalition formation Pn to the Stolper-Samuelson theory 7^ss. Wemay represent this as TK = {Tsa x PR->HR} where Tss = {Ps x Po^-Hss}. Rogowski'sproblems come when he takes the two-factor Stolper-Samuelson model and oper-ationalizes it as if the result remains unchanged with three factors. There is no reasonwithin Stolper-Samuelson to believe that this works, and there is reason to believe thatmatters are more complicated than this (Learner, 1980). By changing Po — or byadding a logical contradiction between Po and P s — the hypotheses //gs fromStolper-Samuelson no longer follow from the mapping {Ps x PO^HSS}. Rogowski canno longer claim to be applying Stolper-Samuelson. He may be right about the role ofcoalitions in politics, but this claim must rest on another theoretical foundation.

Since Rogowski is no longer applying Stolper-Samuelson, if he finds data that falsifyhis theory, they are of no logical consequence for models of international trade.Similarly, any empirical successes do not logically count as successes of Stolper-Samuelson. His work may be interesting to international economists as a study ofpolitical economy but it has no implication for the lender theory. Miscommunicationmay result from such exchanges, since each discipline may easily misunderstand thelogical standing of its theory in the subject matter of the other.

The analysis here has another implication for borrowers that helps us to understandinterdisciplinary miscommunication. One alleged pitfall for borrowers is the risk ofborrowing an obsolete or even falsified theory. A borrower might also find that a theorybecomes obsolete under one's feet. Steve Fuller (1988; p. 194) describes such a case inthe history of economics:

classical political economy's model reasoner was originally drawn from the rational egoist psychologythen current in the eighteenth century; yet once psychology surrendered the model (in the earlytwentieth century) economics also did not immediately follow suit. Whereas the model was completelyundermined in psychology once the significance of unconscious irrational factors on behavior wasappreciated economists have so far attacked the model largely on practical grounds... economics shouldhave given up the rational egoist model immediately after psychology did or suffer the consequences.

Immanuel Wallerstein (1991, pp. 93-94), echoing Gunnar Myrdal, makes a similarcomplaint about the same borrowing.

Despite such claims, borrowing obsolete theory presents less of a problem than iscommonly supposed. To see this, let us take the most extreme example, when somesuperior theory completely subsumes the inferior theory. Let us suppose that all of the non-

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falsified hypotheses of the Stolper-Samuelson model also appear in a more complexmodel of international trade that we can call 'Ricardo-Viner',20 i.e. HSS<=HRV.Moreover, suppose that Ricardo-Viner is a 'better' theory, in that the theoreticalcomplexities it includes explain more economic data than Stolper-Samuelson. Thus,Rogowski faced the choice of borrowing either Tss = {Ps xP0-+ Hss} orTRV = {Ps' x Po' -*• Hnv}, where Hss cz HRV. Even in this extreme case, it is legitimate toborrow the 'wrong' theory (//ss) and not the superior HRY. This is true if the scholaradds auxiliary assumptions (PA) to the borrowed theory, and if the set of auxiliaryassumptions is different depending on which theory is borrowed. In particular, this willbe true if the additional auxiliary assumptions necessary to operationalize the superiorlender theory are inappropriate for the borrower discipline, even if the auxiliaryassumptions necessary to operationalize the inferior lender theory work were in theborrower discipline.

In symbolic terms, a borrower can use the theory {Tss xPA-+ HB} instead of the'better' theory {TKV x PA'->HB'} if PA 4= PA. This is possible because it might be truethat HB is ' more true' than HB, although Tss is ' less true' than TRV. In other words,the theory might be better within the borrowing field without the improvements knownto the lender field.21

The example is less far-fetched than it might sound. When used to explain politicalbehavior (Magee, 1980), Stolper-Samuelson predicts political cleavages based onclass — that is, labor versus capital. On the other hand, Ricardo-Viner predictscleavages based on industry — labor and capital in the steel industry against laborand capital in the auto industry, for instance. Even if Ricardo-Viner better explainseconomic behavior such as prices and output, the political system might encourageorganization along the 'wrong' cleavage. For example, a country such as Sweden thathas an electoral system based on proportional representation might reward partyorganizations that represent classes such as labor, capital, farmers, and the petitbourgeoisie. In such countries, Stolper-Samuelson may well explain political cleavages.In a different political environment, such as the USA, the political system might rewardindustry-based lobbies that can cross party lines. Here, Ricardo-Viner may explaincleavages. Adding an auxiliary proposition that different political institutions createdifferent incentives for political organization greatly affects the validity of the lendertheory. As a result, the economic theory that does a poor job predicting price changesmay do a better job explaining politics.22

What does all this mean for interdisciplinary discourse? A scholar at the researchfrontier in economics might be right that one cannot learn anything from someone whoborrows obsolete theories from one's field. Even so, it may be legitimate for people toborrow obsolete theory into other disciplines. On the other hand, borrowers likeRogowski might use non-obsolete theories, but in a way that breaks the logicalconnection to the lender. Here, the lender field might think that the borrower's work isrelevant, missing the logical break within the borrowing. Clearly, possibilities forinterdisciplinary confusion abound.

In contrast to this pessimistic picture, discourse might also contribute to synthesis andscientific advance. In such cases, a transdisciplinary community might eschew myhypothetical choice between {Hss x PA -»• HB} and {i/BV x PA -> HB'}, preferring todevelop a transdisciplinary theory better than either. Symbolically, the goal of thisendeavor would be {HRV x PA" -> HB"}, whereby some standard, the truth-value (V) ofthese theories is such that V(HB") >- V(HB), V(HB). Where such a transdisciplinarycommunity exists, discussion will centre on the exchange of propositions to develop

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better hypotheses for a given subject matter. Inevitably, the set of relevant data willchange and will not be identical to the set of propositions deemed germane in eitherparent discipline. This is the creation of a new, hybrid community based on a commonsubstantive interest.

While this sounds idyllic, it must be remembered that scientific research is subject todiminishing marginal returns.23 Since it uses all the theoretical propositions from thetwo disciplines to explain data in the union of two intersecting sets, it is less efficient thaneither of the preceding theories. We should like to make the new theory more powerful,and more parsimonious, by eliminating theoretical propositions. Unfortunately,borrowing results gives us no guidance in this matter: for if we have borrowed properly,there are no logical contradictions that might force us to eliminate contradictoryhypotheses. As I will discuss below, this is quite different from borrowing symboliclanguage alone, where logical contradictions are to be expected and can spur scientificadvance.

Further advance can probably only come through 'mitosis', by which the hybridcommunity divides into two new subfields, which I shall arbitrarily label 1 and 2, eachwith its own symbolic propositions (Psl and PS2) and operationalizations (Po l and Po2).Symbolically, I think that the logical structure of this division has to look somethinglike this: PslcPs, PS2<=PS, PS1 + PS2, PO1<=PO, P02^P0, POi * o2> a n d « isprobably also true that Psl (\ PS2 +- 0, Poi H PO2 + 0. This creates two newfields, 7\ = {PS1 x P01 -+ Hx; D J and T2 = {PS2 x PO2 -> H2; D J . In this case we wouldrecognize two fields with some historical or genetic relationships, such as astronomy andphysics, organic and inorganic chemistry, or history and political science. Byspecializing, each subfield overcomes the problem of diminishing marginal returns, atleast for some time.

Ironically, mitosis was the historical fate of our running example of a cross-disciplinary community political economy. While Smith, Ricardo, Malthus, Marx, andother classical economists labelled their field 'political economy', economics andpolitical science has been cloven from politics since Alfred Marshall. As each establishedan independent disciplinary identity, each consisted of a mostly distinct set ofpropositions. In such a case, the process of borrowing results that I have described herecan begin anew (cf. Dogan and Pahre, 1990, ch. 23).

4. Borrowing symbolic language

An important alternative to borrowing results is the borrowing of a symbolic languagewithout borrowing operationalizations or hypotheses. Take the lender theory!TL = {P& xP0->Hf). Odd as it may seem, a borrower might find some insight in thesymbolic propositions, without any substantive connection to the existing oper-ationalizations. To take an absurd example, one of our mutual ancestors decided oneday to apply the symbolic language of hand- or foot-counting (1 + 1 = 2) to objects suchas relatives, berries, or gazelles. This arithmetic pioneer learned that the rules ofaddition do not vary according to the nature of the object counted.

Similar things happen with each new mathematical technique. Symbolically, then,the borrower might propose TB = {Ps x Po' -> HB}, changing the operationalization ofa symbolically true statement.

One interesting example links classical physics and contemporary economics.Newton's inverse-square law says that the force between two bodies is proportional to

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the product of their masses divided by the square of the distance between them;formally, F= Gmm'/r2. Working independently, Tinbergen (1962) and Poyhonen(1963) saw this symbolic statement as a useful way to estimate trade volumes betweencountries. Their initial insight was to relabel the algebra and create a ' gravity model'of international trade in which Newton's F is reinterpreted as bilateral trade volume, mand m' become the GNP of any two countries, and r is the distance between thesecountries. The results were remarkably good: this formula describes the trade databetter than any other we know, especially after making some minor additions such asadding national population as a second measure of'mass' (Linnemann, 1966).

These'gravity models' work despite having no connection at all to economic theoriesof international trade (Deardorff, 1984). Their theoretical contribution lies inuncovering an empirical regularity that many economists have tried to ground in sometheoretical explanation (i.e. Learner and Stern, 1970, ch. 6; Linnemann, 1966;Anderson, 1979). Thus, it now belongs to a body of theoretical claims in economicsdespite its origins from outside the discipline's neoclassical paradigm.

Newton's inverse-squares formula has also been extended to other domains in whichdata are organized spatially.24 For instance, the Australian Election Commission hasused inverse-squares to interpolate early election returns for districts with missing data,using nearby districts for which data exist (Maley and Medew, 1991). In fact, severalpersons with whom I have discussed this paper believe that inverse squares capturessomething profound about spatial relationships in our world.25

This is clearly a very different kind of borrowing to that described in the previoussection. First, it is a spur to creativity instead of the foundation for cumulative advance.Its value lies largely in its role as a source of inspiration. Second, we all know thatNewton's inverse-squares law is obsolete with physics, but this obsolescence is irrelevantfor economics and political science. In fact, it is never unacceptable to borrow anyinternally consistent symbolic language, no matter how obsolete that language may bein the lender discipline. It is not at all necessary for economists to keep up with state-of-the-art physics, and physicists will learn nothing from the economists. I dare say thatphysicists will find the entire exercise rather odd.26

Economists seem especially prone to borrowing mathematics from the naturalsciences. For instance, the chemical equation for an ideal gas is the basis for themonetarist identity that MV = PT.27 Economists are quite open about their sources ofinspiration. One has privately described the motivation of a certain paper as a(successful) attempt to find some economic application of mathematical catastrophetheory. While formulated without empirical referent, the resulting paper has usefulapplications in exploiting the catastrophe metaphor to find examples of discontinuousand 'catastrophic' economic change in industries.

Important as the process is, the precise origin of the economists' intuition remainsmysterious. Why, indeed, does it seem plausible that distance works in an inverse-squares way in international trade as it does in physics? Why is the analogy betweenmonetary velocity and the pressure of a gas less satisfying, even if we recognize thecreative connection?

If we knew the answers to such questions in advance, research would not be necessary.Borrowing a symbolic language is inherently a chancy affair, since one severs theconnections to reality inherent in a language's operationalizing propositions. Thespread of game theory from economics to political science illustrates some successesand failures of borrowing symbolic languages.

Game theory was originally a mathematical theory. The mathematician John von

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Neumann and the economist Oskar Morgenstern introduced it to the social sciencesin their seminal Theory of Games and Economic Behavior (1944). Despite substantialopposition from traditionalists in each field, the theory spread through economics(Shubik, 1992), political science (Riker, 1992), operations research (Rider, 1992) andother disciplines. Early applications of game theory to political science relied heavily onthe logic of two-person zero-sum games. As it turns out, that is probably the gametheoretic field least useful for political science — although there was no way to know thisat the time. Theorists localized the use of two-person zero-sum games to the narrowrange of political activity to which they apply, mostly areas of military science andstrategy. In contrast, non-zero-sum games proved to be a very fruitful area of research,so much so that Jon Elster has even described political science as the study of ways toescape the Prisoners' Dilemma.

The spread of such symbolic languages can act as a spur to other intellectualdevelopments. One interesting possibility is the unexpected contact between theoriginal borrowers' use of the metaphor and other scholars in the borrower discipline.For instance, the first political scientists to apply game theoretic models of cheap-talk'(Farrell and Gibbons, 1989) from economics to political science were formal modellersof legislative behaviour (Austen-Smith, 1990). Once known in political science, somenormative political theorists discovered the topic. They have since set these models nextto political philosophy such as Jiirgen Habermas's critical theory (i.e. Dryzek, 1992;Johnson, 1993). It is not hard to imagine an interesting research frontier playingdeconstructionists' interest in discourse against game theorists' models of talk.28

A new term is needed for a community arising from symbolic exchange; 'hybrid' or' multidisciplinary' are inadequate to the task. Let me propose the term ' metaphoricalcommunity'. Cross-disciplinary discourse takes place in two different forms in suchcommunities. The first form is found in the initial phase of community-building. Here,a discipline borrows the symbolic language alone. This borrowing has no implication forthe lender discipline, so we should expect little or no cross-disciplinary exchange.

In the second phase, there are multiple borrower communities. It is always possiblethat these borrower communities will ignore each other, in which case the story endsthere. Although these communities share a symbolic language without sharingsubstantive interests, some will remain in contact and repeatedly provide each otherwith possible inspiration. This is the pattern in chaos theory, as the Lorenz equations,Mandelbrot sets, fractals, and other symbolic innovations spread through mathematics,physics, meteorology, biology, economics, political science, and other disciplines(Gleick, 1987). It has persisted in the chaos field, institutionalized in conferences andsome institutes. Consequently, new 'chaotic' mathematics (or computer simulations)can spread from subfield to subfield, linked by some prior borrowing of chaos symboliclanguage.

This second phase may also see the gradual disappearance of the borrowing, asscholars learn that it is not as useful as it first appeared to be. Indeed, symboliclanguages are at the core of any number of mathematical 'fads'. Well-known examplesof such fads in the social sciences over the last 50 years include factor analysis, gametheory, non-linear dynamics, operations research, catastrophe theory, and chaostheory. Rather than decrying these fads, I think we should see them as attempts toborrow a symbolic language as a metaphor under a variety of operationalizations. Somewill succeed, many will not. These fads often create communities of scholars convincedthat the new tool can explain everything. The potential pay-off is high because theborrowed metaphor is truly novel within the created community. In contrast, those that

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borrow results build more narrowly on existing work, so that their contributions willappear less novel. Of course, there is no particular reason why a given metaphor shouldfit the borrower discipline, so we would expect most fads to die quickly — but a few willbe of lasting value.

To return to our symbolic language, I think that the structure of a metaphoricalcommunity may be sketched as follows. Suppose that two disciplines, named 1 and 2,shared some (borrowed) symbolic language. Let us label these common propositions Pc.Each discipline then conducts research, adding symbolic statements to its repertoire, orsynthesizing older symbolic statements with the new borrowing. The resulting set ofsymbolic propositions in the two disciplines we may label Pcl and PC2, where Pc a Pcl,Pc c Pci, Pcl 4= Pci. Where Pcl fl PC2 = P& the two borrowers' developments of thecommon symbolic language are mutually exclusive. This is unlikely, of course. It willoften be true that one discipline's symbolic development of Pc might provokemetaphorical insight in the second discipline. This is especially useful because anystatements that are consistent with Pc in discipline 1 will obviously be consistent with Pc

in discipline 2, making it easier to assimilate innovations. (They might still beinconsistent with any P s #= Pc in discipline 2, of course.) In short, some 'new symbols'in discipline 1 are available as a metaphor to discipline 2; symbolically, these newsymbols are PN1, where PN1 c: Pcl, Pm |~| Pc = 0.

These metaphorical communities can be destructive of traditional ways of organizingand communicating knowledge. Their symbolic languages can also be very destructiveof the borrower discipline. Consider the logical status of the symbolic metaphor orborrowing (.PSM) added to the old symbolic language of the borrower discipline (PSB),which coexists with the borrower's original operationalizations of its own symboliclanguages (P0B) and the propositions needed to operationalize the new symbolicmetaphor (POM)- I n other words, we see the old borrower theory,TB = {^SB x ^OB -* HB; DB}, become TB* = {{PSB, PSM} x {P0B, POM} -> HB*; DB}.

Logical contradiction comes easily to this new theory. The symbolic language of themetaphor (PSM) may not be logically consistent with the original symbolic language ofthe borrower field (PSB)> and this will be evident when stated in symbolic or logicalterms. Also, some old symbolic propositions (PSB) may meet odd operationalizations ofthe new symbolic language (-POM)- Simultaneously, the metaphor also adds explanatorypower to the borrower field that is too valuable to reject out of hand. Some propositions— whether they come from PSB or PSM or both is irrelevant — will have to go. Theresult is a process of innovation like that described by Feynman (1965, p. 166):

The question is, what to throw away and what to keep. If you throw it all away that is going a littlefar, and then you have not much to work with... To guess what to keep and what to throw away takesconsiderable skill. Actually it is probably merely a matter of luck, but it looks as if it takes considerableskill.

We may call this process 'creative destruction'.29

The new theory will include a large portion of these symbolic and operationalizingpropositions, producing sets of hypotheses and data that are also subsets of the union ofthe 'old' sets of hypotheses and data. By including only some original propositions, thenew theory gains parsimony; if the borrowing is done well, the new theory should alsoeliminate anomalies and perhaps add explanatory power. Formally, we get a newtheory TN = {{PSL*,PSB*}x{P0B*,P0M*}^HN;DN} where PSL$PSL, PSB$PSB,P0B* §: P0B , -POM* ^ ^OM (that is, each asterisked set is a subset of the original set, andeach has eliminated some elements). This gives us a stripped-down novel theory. The

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new theory may explain a larger evidentiary domain than the union of it parents (ifZ)N r> {DB U -OM}) -3° I{ should also be more parsimonious, because we should expectthat //N ^ {HB U HM) • Again, if we believe that the law of diminishing marginal returnsapplies to science, this describes a more powerful theory.

One example is the study of international co-operation in the field of political science.Traditional international relations theory, an approach known as (neo-)Realism, findsinternational co-operation surprising. Theorists developed a variety of ad hoc explana-tions of international co-operation, such as shared value communities, geographicproximity, domestic political norms, the size of nations, altruism, and the like (Deutsch,1957; Holsti etal., 1973). In the late 1970s, the field borrowed parts of game theory fromeconomics and other fields. In particular, international relations theory becameenamored of the game of Prisoners' Dilemma, which soon proved to be a usefulmetaphor for a variety of problems in international relations (Snidal, 1986). Prisoners'Dilemma suggested that co-operation is perfectly normal under some circumstances,31

while also explaining the failure of co-operation in other circumstances.This metaphor proved to be far more parsimonious than the earlier, more ad hoc,

explanations of international co-operation and conflict. Although Prisoners' Dilemmadid not preclude any of the earlier explanations — for it is logically consistent with anyof them — its power made the field question how many earlier explanations were reallynecessary or useful. Some theories, like shared value communities, have faded from theliterature. They have not faded because they are wrong, but because the borrowedmetaphor proved to be more parsimonious. Others among the older explanations, suchas the importance of international norms, remain active at the research frontier, in nosmall part because norms are comprehensible within a Prisoners' Dilemma framework.

As a result, the metaphor proved destructive of many bodies of theory, but ourunderstanding of international co-operation today is better than our understanding of30 years ago. A purely cumulative model of scientific advance would not be surprisedthat our understanding has improved over the course of 30 years, but might be surprisedthat this improvement was accompanied by the destruction of older theories that are notincorrect or inaccurate. Borrowing symbolic languages lets us do this in a way that isimpossible when we borrow results.

At the same time, some things are lost. There are things that Prisoners' Dilemmacannot explain, such as the apparent convergence of domestic political norms stemmingfrom international co-operation. Such concerns fade from the literature, although theymight well be rediscovered later. Condorcet's mathematical analysis of the inde-terminacy of social decision rules was a response to the indeterminacy of the FrenchRevolution, but his analysis was not rediscovered until the growth of public choicetheory in political science and economics from the 1950s.

This pattern of exchange, merger, creative destruction, and lost knowledge changesthe geography of the social sciences (White, 1956; Campbell, 1969; Dogan and Pahre,1990; Easton, 1953, pp. 100-106). Such processes mean that it should no longer surpriseus that it i s ' more interesting as well as more productive, for the economist who workswith non-market decisions to communicate with the positive political scientist, the gametheorist, or the organizational theory psychologist than it is for him to communicatewith the growth-model macro-economist, with whom he scarcely finds any commonground' (Buchanan, 1966, p. 181). Nor is it surprising to learn that political scientistssit at 'separate tables' (Almond, 1990) with different subfields, methodologies, schoolsand sects unable to communicate with one another.

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5. Conclusions: perspiration and inspiration

In this paper it has been argued that there are two different ways in which disciplinesborrow symbolic languages from one another. Loosely speaking, we may think of thesepaths as characterized by either perspiration or inspiration. Borrowing results requiresperspiration: to avoid logical inconsistencies a scholar must have an adequate passiveknowledge of the lender discipline. The scholar also needs to know whether the theoryis obsolete or state-of-the-art, taking appropriate care when borrowing obsolete theories.Finally, the scholar may wish to be able to present a data analysis to the lenderdiscipline.

Borrowing a symbolic language alone requires mere inspiration: the borrower needknow nothing of the lender discipline at all. A scholar seeking inspiration may wish tohave a cursory familiarity with many fields, reading widely in search of promisingmetaphors.

Most approaches to cross-disciplinary graduate training emphasize perspiration.Many disciplinary programs require that students study some cognate discipline,apparently in the belief that a year's study will provide an acceptable passiveknowledge of the other field. Similarly, 'bridging' programs within universities or fromMacArthur and other foundations provide faculty with one or two years' study inanother discipline — again, the idea is to develop a passive knowledge of the field thatpermits the intelligent borrowing of results.

On the other hand, we do not think about how to train graduate students or ourselvesto seek outside inspiration. A few models exist: the Santa Fe Institute, for instance, hasa summer school that trains people from a variety of backgrounds in non-lineardynamics, complexity theory, and 'chaos'. The Institute also supports extended visitsand sabbaticals for scholars from many disciplines.

This institutional model seems a good way to support an existing metaphoricalcommunity. Even so, it does little to inspire the creation of new metaphoricalcommunities. The 'creative destruction' of metaphorical borrowing is, like the moretraditional cumulative knowledge of borrowing results, an important part of scientificadvance.

Notes

1. It is hard not to be recursive in discussing a topic such as scientific discourse. This is perhaps especiallytrue in the social sciences, where we study knowing subjects who already have meaningful interpretationsof one another's behavior (Harbers and de Vries, 1993; Lynch, 1993). For that matter, we mayactually desire recursiveness and some of its attendant irony as part of the analysis (cf. Hofstadter, 1979;Woolgar, 1983). While I use a symbolic language to discuss the use of symbolic languages ininterdisciplinary communication, my symbolic language has itself been borrowed from other disciplines.In particular, I was struck by the resemblance between the propositional notation in Lakatos (1970) andthe mathematical language of Takayama (1985).

2. Alas, no system of such statements can be both complete and consistent, by Godel's theorem. I willcheerfully ignore all such problems here. Some of these epistemological issues have important implicationsfor understanding the sociological organization of science, see Pahre (1995) for a review.

3. I simplify. For an argument that even mathematics is an empirical science, see Lakatos (1978); for acritique of mathematical language in physics-absent causal law, see Cartwright (1983); for a study of therhetoric of mathematics in economics, see McCloskey (1985,1990). See also Hofstadter (1979, ch. 2) andFriedman (1953).

4. For a slightly different version of the argument, see Snidal's (1986) discussion of Snyder and Diesing'sConflict Among Nations (1977). Snyder and Diesing use information about outcomes to operationalize asimple version of game theory, and then use the theory to predict the outcomes. Snidal argues, correctly,that this is a descriptive use of game theory, not an explanatory one.

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5. The question of'reality' is especially relevant for those mathematical theories that make use of theinvention of non-existent qualities. Again, our evaluation of such languages should be more nuanced.Non-existent objects can play an important role in science. For instance, the concept of the 'potentialenergy' of a rock on the edge of a cliff is used to balance the energy equations describing that rock's fall,but the ' potential energy' itself is a purely metaphysical idea. (For an opposing view, see Cartwright,1983.) The social sciences have developed a number of similarly valuable concepts describing non-existent qualities, most importantly the concept of a 'utility function' in economics.

6. There may be operationalizing statements that are difficult to distinguish from theorems, and vice versa.For a good example of the difficulties, consider the Duality Principle in Projective Geometry (my exampleis from Lakatos, 1978, p. 68). The Duality Principle states that any valid statement about points and lineson a projective plane implies a second valid statement in which the words 'point' and 'line' areinterchanged. For instance, the statement that 'any two distinct lines in the same plane determine aunique point' implies that 'Any two distinct points in the same plane determine a unique line'. In somemetalanguage in which we proved the underlying statements without use of the words 'point' and 'line',then we could operationalize these statements by filling them with either of the two orderings of the set{point, line}.

7. I choose to think of this as a mapping in this way in part because there is not a unique 1:1 relationshipbetween symbolic and operationalizing propositions: any symbolic statement may be operationalized bymore than one proposition, and any operationalizing proposition may operationalize more than onesymbolic statement.

8. In the language of Lakatos (1970) sophisticated falsificationism, Ps is the negative heuristic, and Po is thepositive heuristic. Lakatos would include other, non-symbolic, propositions in the negative heuristic, butthis does not change the status of the symbolic propositions that interest me here.

9. For other discussions of discipline, see Becher (1990, 1994), Fuller (1988), Pinch (1990).10. Needless to say, this is not the normal postmodernist meaning of the word 'discourse'. Even so, there is

more overlap between our two usages than would appear at first glance, since both entail properties thatI would label logical consistency and topical germaneness.

11. I will not discuss here how disciplines construct their substantive focus (Pahre, 1995) nor their boundaries(Gieryn, 1983). Note that the analysis assumes boundaries and fields defined prior to discourse, so it isincomplete for any history of the social sciences. Histories of predisciplinary social sciences generally tracethe social and political needs that produce a given discipline and its boundaries (Wagner et al., 1991;Wallerstein, 1991).

12. Another Nobel laureate, Douglass C. North, sides with the political scientists on this issue and has madethe point central to his neoclassical theory of the state. See especially North (1981).

13. Note that the borrower need not have the active knowledge of the lender discipline that would enable herto develop 'interesting' theoretical advances of the form {Ps' x Po' H'}. Even so, borrower data mayend up contributing to lender advance.

14. In fact, it is usually easier for the mathematical lender field to assimilate new evidence than it is for thenon-mathematical borrower field to assimilate the mathematics. This is probably just a matter oftraining, since all scholars have some experience of handling evidence but not all have mathematicaltraining. For evidence on the (non)use of mathematical scholars in public choice, see Downing andStafford (1981).

15. The pattern dates to at least the late 1950s. Indeed, Buchanan (1966, pp. 171-173) argues that politicalscience's most important contribution to economics is its data, while economics' greatest contribution topolitical science is its theory. He claims that 'it is becoming increasingly evident that the importanttheoretical advances in the explanation of political phenomena have been made primarily by those whoapproach the subject matter as economists' (Buchanan, 1966, p. 177). Buchanan won a Nobel Prize forbeing a leader in exactly this area.

16. Snidal (1986) makes a similar argument, describing this difference in terms of a distinction between gametheory as analogy and game theory as model. His distinction hinges on whether empirical extensions ofthe theory are logically entailed by the propositions of the theory, in which case both prediction andfalsification are possible, or whether extensions are merely inductive conjectures. Proving such conjectureswrong forces the borrower to change the conjectures, but has no necessary consequences for theunderlying theory.

17. Ironically, theoretical developments in public goods theory have undercut (with some complications) the'concentrated resources' argument to which I refer here (Bergstrom et al., 1986; Comes and Sandier,1994). Thus, the evidence from political science is not anomalous for public goods theory after all. Oddlyenough, endogenous tariff theorists have not updated their theory, so the evidence would be anomalousfor them.

18. Psychologists testing public goods theory are even less precise, often finding 'anomalies' that actuallyconfirm a properly-specified theory (Isaac et al., 1985). Compare Lohmann's (1995) critique of Greenand Shapiro's (1994) review of the evidence allegedly anomalous for mathematical rational choicetheory.

19. Lest this example makes international trade theory appear to be hopelessly simple, let me note here thatthere are other kinds of economic models that provide more nuanced predictions of the effects of NAFTA.

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20. Incidentally, this is not true of the real Ricardo-Viner theory.21. By itself, this argument should give the social sciences pause before borrowing postpositivist epistemologies

from philosophy, even though strict logical positivism is now obsolete philosophy.22. One must be very careful about taking this argument normatively. Certainly I do not recommend

borrowing inferior theories because they are inferior! Yet if one is careful about the operationalizingstatements and about exactly why borrowing the superior theory would be inappropriate or impossible,there is some small room for intentionally borrowing inferior theories.

23. With a few additional assumptions, the law of diminishing marginal returns follows from Occam's Razor.Observation of scientific practice confirms the existence of diminishing returns. If scientific fields were notsubject to diminishing returns, we would observe only one unified scientific discipline.

24. Of course, the world itself does not organize spatially the data it gives us. Someone has collected the datafor some purpose and has chosen to organize it spatially. This person's choices are a good example of howscientists construct fields, but I will set aside such issues here.

25. I am sorry to report that none of these people have been able to explain to me what this profoundrelationship is.

26. Issues of scientific hierarchy and prestige intrude into the physics-economics example, but are notnecessary parts of the oddity. Several fields of similar status might exchange symbolic languages in alldirections, with each finding the other fields' applications odd. Something of this sort characterizes partsof the multidisciplinary community applying complexity theory (chaos theory) today.

27. McCloskey (1985, p. 73) rightly points out that this equation is both irrefutable and useful in bothchemistry and economics —just the kind of symbolic tautology that I describe.

28. Indeed, this essay's juxtaposition of symbolic language and scientific discourse is a different tack on thesame research agenda, transposed from political science to the sociology of science.

29. The origin of this term is, of course, Schumpeter's analysis of technical innovation in capitalism, by whicheach innovation destroys prior investments by making their machinery obsolete. This seems to me to bea pretty good metaphor for a lot of scientific innovation.

30. Alas, it may only be true that DN {DB U DM}. This was true of Newtonian mechanics, which unifiedmotions on earth and in heavens within a parsimonious mathematical theory, yet could not predict thelocation of planets as accurately as the Ptolemaic system until Laplace's Celestial Mechanics of 1799 (Fuller,1993, p. 135).

31. In particular, when nations interact indefinitely, when they can easily monitor one another's behavior,and when they are willing to punish each other for any non-co-operative moves. See especially Axelrod(1984).

References

ALMOND, G. A. (1990), A Discipline Divided: Schools and Sects in Political Science, Sage Publications, NewburyPark, CA.

ANDERSON, J . E. (1979), 'A theoretical foundation for the gravity equation', American Economic Review, 69,pp. 106-116.

AUSTEN-SMITH, D. (1990), 'Information transmission in debate', American Journal of Political Science, 34,pp. 124-152.

AXELROD, R. (1984). The Evolution of Cooperation, Basic Books, New York.BECHER, T. (1990), 'The counter-culture of specialisation', European Journal of Education, 25, pp. 333-

346.BECHER, T. (1994), 'Interdisciplinarity and community', in R. Barnett (ed.), Academic Community: Discourse

or Discord, Jessica Kingsley, London, pp. 55-71.BERGSTROM, T., BLUME, L. and VARIAN, H. (1986),' On the private provision of public goods', Journal of Public

Economics, 29, pp. 25-49.BUCHANAN, J . M. (1966), 'Economics and its scientific neighbours', in S. R. Krupp (ed.), The Structure of

Economic Science, Prentice-Hall, Englewood Cliffs, NJ, pp. 166-183.CAMPBELL, D. T. (1969), 'Ethnocentrism of disciplines and the fish-scale model of omniscience', in M. Sherif

and C. W. Sherif (eds), Interdisciplinary Relationships in the Social Sciences, Aldine, Chicago, pp. 328-348.CARTWRIGHT, N. (1983), How the Laws of Physics Lie, Clarendon Press, Oxford.CAVES, R. E. (1976), 'Economic models of political choice: Canada's tariff structure', Canadian Journal of

Economics, 9, pp. 278-300.CORNES, R. and SANDLER, T. (1994), 'The comparative static properties of the impure public goods model',

Journal of Public Economics, 54, pp. 403-421.DEARDORFF, A. V. (1984), 'Testing trade theories and predicting trade flows', in R. W.Jones and P. B.

Kenen (eds), Handbook of International Economics, vol. 1, Elsevier, Amsterdam, pp. 467-517.DEUTSCH, K. W. (1957), Political Community and the North Atlantic Area: International Organization in the Light of

Historical Experience, Princeton University Press, Princeton.

Dow

nloa

ded

by [

Uni

vers

ity o

f T

asm

ania

] at

17:

35 2

9 N

ovem

ber

2014

72 DISCOURSE SYNTHESIS

DOGAN, M. and PAHRE, R. (1990), Creative Marginality: Innovation at the Intersections of Social Sciences, WestviewPress, Boulder.

DOWNING, P. B. and STAFFORD, E. A. (1981). 'Citations as an indicator of classic works and majorcontributions in social choice', Public Choice, 37, pp. 219-230.

DRYZEK, J . S. (1992), 'How far is it from Virginia and Rochester to Frankfurt? Public choice as criticaltheory', British Journal of Political Science, 22, pp. 397-417.

EASTON, D. (1953), The Political System: An Inquiry into the State of Political Science, Alfred A. Knopf, New York.FARRELL, J. and GIBBONS, R. (1989), 'Cheap talk can matter in bargaining', Journal of Economic Theory, 48,

pp. 221-237.FEYNMAN, R. (1965), The Character of Physical Law, MIT Press, Cambridge.FRIEDMAN, M. (1953), 'The methodology of positive economies', reprinted in F. Hahn and M. Hollis (eds),

Philosophy and Economic Theory, 1979 Oxford University Press, Oxford.FULLER, S. (1988), Social Epistemology, Indiana University Press, Bloomington.FULLER, S. (1993), 'Disciplinary boundaries and the rhetoric of the social sciences', in E. Messer-Davidow,

D. R. Shumway and D. J. Sylvan (eds), Knowledges: Historical and Critical Studies in Disciplinarity, Universityof Virginia Press, Charlottesville, pp. 125-14:9.

GIERYN, T. F. (1983),' Boundary work and the demarcation of science from non-science: strains and interestsin professional ideologies of scientists', American Sociological Review, 48, pp. 781-795.

GLEIOK, J . (1987), Chaos: Making a New Science, Penguin Books, New York.GREEN, D. P. and SHAPIRO, I. (1994), Pathologies of Rational Choice Theory: A Critique of Applications in Political

Science, Yale University Press, New Haven.HARBERS, H. and DE VRIES, G. (1993), 'The empirical consequences of the "Double Hermeneutic"', Social

Epistemology, 7, 183-192.HOFSTADTER, D. R. (1979), Godel, Escher, Bach: An Eternal Golden Braid, Vintage Books, New York.HOLSTI, O. R., HOPMAN, P. T. and SULLIVAN, J . D. (1973), Unity and Disintegration in International Alliances:

Comparative Studies, Wiley, New York.ISAAC, R. M., MCCUE, K. F. and PLOTT, C. R. (1985), 'Public goods provision in an experimental

environment', Journal of Public Economics, 26, pp. 51-74.JOHNSON, J . (1993), 'Is talk really cheap? Prompting conversation between critical theory and rational

choice', American Political Science Review, 87, pip. 74-86.KLEIN, J . T. (1990), Interdisciplinary: History, Theory, and Practice, Wayne State University Press, Detroit.KLEIN, J . T. (1993), 'Blurring, cracking, and crossing: permeation and the fracturing of discipline', in E.

Messer-Davidow, D. R. Shumway and D.J. Sylvan (eds), Knowledges: Historical and Critical Studies inDisciplinarity, University of Virginia Press, Charlottesville, pp. 185-211.

LAKATOS, I. (1970), 'Falsification and the methodology of scientific research programmes', in I. Lakatos andA. Musgrave (eds), Criticism and the Growth of Knowledge, Cambridge University Press, Cambridge, pp.91-196.

LAKATOS, I. (1978), Mathematics, Science and Epistemology: Philosophical Papers, vol. 2, edited b y j . Worrall andG. Currie, Cambridge University Press, Cambridge.

LEAMER, E. E. (1980), 'The Leontief Paradox, reconsidered', Journal of Political Economy, 88, pp. 495-503.LEAMER, E. E. and STERN, R. M. (1970), Quantitative International Economics, Aldine Publishing Company,

Chicago.LINNEMANN, H. (1966), An Econometric Study of International Trade Flows, North Holland, Amsterdam.LIVINGSTON, E. (1993), 'The disciplinarity of knowledge at the math—physics interface', in E. Messer-

Davidow, D. R. Shumway and D.J. Sylvan (eds), Knowledges: Historical and Critical Studies in Disciplinarity,University of Virginia Press, Charlottesville., pp. 368-393.

LOHMANN, S. (1995), 'The poverty of Green and Shapiro', Critical Review, 9 (1-2), Winter-Spring.LYNCH, W. (1993), 'What does the double hermeneutic explain/justify?', Social Epistemology, 7, pp. 193ff.

(April-June).MAGEE, S. P. (1980), 'Three simple tests of the Stolper-Samuelson Theorem', in P. Oppenheimer (ed.),

Issues in International Economics, Proceedings of the Oxford International Symposium in Honor of Harry G. Johnson,Oriel Press, London, pp. 138-153.

MALEY, M. and MEDEW, R. (1991), 'Some approaches to election night forecasting in Australia', AustralianJournal of Political Science, 26, pp. 51-62.

MAYER, W. (1984). 'Endogenous tariff formation', American Economic Review, 74, pp. 970-985.MCCLOSKEY, D. N. (1985), The Rhetoric of Economics, University of Wisconsin Press, Madison.MCCLOSKEY, D. N. (1990), If You're So Smart: The Narrative of Economic Expertise, The University of Chicago

Press, Chicago.MESSER-DAVIDOW, D. R., SHUMWAY, D. R. and SYLVAN, D.J . (eds) (1993), Knowledges: Historical and Critical

Studies in Disciplinarity, University of Virginia Press, Charlottesville.NELSON, D. (1988), 'Endogenous Tariff Theory: a critical survey', American Journal of Political Science, 32,

pp. 796-837.NORTH, D. C. (1981), Structure and Change in Economic History, W. W. Norton, New York.OLSON, M. (1965), The Logic of Collective Action, Harvard University Press, Cambridge.

Dow

nloa

ded

by [

Uni

vers

ity o

f T

asm

ania

] at

17:

35 2

9 N

ovem

ber

2014

MATHEMATICAL DISCOURSE AND CROSS-DISCIPLINARY COMMUNITIES 73

PAHRE, R. (1995), 'Positivist discourse and social scientific communities: towards an epistemologicalsociology of science', Social Epistemology, 9, pp. 233-256.

PINCH, T. (1990), 'The culture of scientists and disciplinary rhetoric', European Journal of Education, 25,pp. 295-305.

PINCUS, J . J . (1975), 'Pressure groups and the pattern of tariffs', Journal of Political Economy, 83, pp. 757-778.PÖYHÖNEN, P. (1963), 'A tentative model for the volume of trade between countries', Weltwirtschaftliches

Archiv, 90, pp. 93-99.RIDER, R. E. (1992), 'Operations research and game theory: early connections', History of Political Economy,

24 (suppl.), pp. 225-239.RIKER, W. H. (1992),' The entry of game theory into political science', History of Political Economy, 24 (suppl.),

pp. 151-163.ROGOWSKI, R. (1989), Commerce and Coalitions: How Trade Affects Domestic Political Alignments, Princeton

University Press, Princeton.SANDLER, T. (1992), Collective Action, University of Michigan Press, Ann Arbor.SCHMAUS, W., SEGERSTRALE, U. and JESSEPH, D. (1992), 'Symposium on the 'Hard Program' in the sociology

of scientific knowledge: a manifesto', Social Epistemology, 6, pp. 243-265.SHERIF, M. and SHERIF, C. W. (1969a), 'Interdisciplinary coordination as a validity check: retrospect and

prospects', in M. Sherif and C. W. Sherif (eds), Interdisciplinary Relationships in the Social Sciences, Aldine,Chicago, pp. 3-20.

SHERIF, M. and SHERIF, C. W. (eds) (1969b), Interdisciplinary Relationships in the Social Sciences, Aldine, Chicago.SHUBIK, M. (1992), 'Game theory at Princeton, 1949-1955: a personal reminiscence', History of Political

Economy, 24 (supp.), pp. 151-163.SNIDAL, D. (1986), 'The Game Theory of international polities', in K. A. Oye (ed.), Cooperation under Anarchy,

Princeton University Press, Princeton, pp. 25-57.SNYDER, G. and DIESING, P. (1977), Conflict Among Nations, Princeton University Press, Princeton.TAKAYAMA, A. (1985), Mathematical Economics, Cambridge University Press, Cambridge.TINBERGEN, J. (1962), Shaping the World Economy: Suggestions for an International Economic Policy, Twentieth

Century Fund, New York.VON NEUMANN and MORGENSTERN, O. (1994), Theory of Games and Economic Behavior, Princeton University

Press, Princeton.WAGNER, P., WITTROCK, B. and WHITLEY, R. (eds) (1991), Discourses on Society: The Shaping of the Social Science

Disciplines, Kluwer, Dordrecht.WALLERSTEIN, I. (1991), Unthinking Social Science: The Limits of Nineteenth-Century Paradigms, Polity Press and

Basil Blackwell, Cambridge.WHITE, L. D. (ed.) (1956), The State of the Social Sciences, University of Chicago Press, Chicago.WOOLGAR, S. (1983), 'Irony in the social study of science', in K. D. Knorr-Cetina and M. Mulkay (eds.),

Science Observed: Perspectives on the Social Study of Science, Sage Publications, London, pp. 239-266.

Dow

nloa

ded

by [

Uni

vers

ity o

f T

asm

ania

] at

17:

35 2

9 N

ovem

ber

2014