Uses of mathematics

Answer 1

well... Mathematics plays a very important role in economics. This role has been significant for almost a century, and has been increasing in importance particularly in recent years. A comparison of academic journals now with, say, fifty years ago reveals a tremendous increase in mathematical expression; Backhouse (1998) reports an increase in the incidence of algebra in articles in the two leading economics journals from 10% in 1930 to 75% in 1980 (see further Grubel and Boland, 1986, Mirowski, 1991, and McCloskey, 1994: Ch.9). The same is true also of textbooks at all levels.Mathematics is thus increasingly important in terms of the expression and communication of ideas in economics. This in itself is a matter of interest, particularly with respect to the public understanding of economics. Further, to the extent that public understanding of mathematics is limited, so too will be the public understanding of economics. This applies at a variety of levels, from school pupils making subject choices to policy makers' understanding of policy advice.Grubel and Boland (1986) report a concern, identified through a questionnaire survey of leading economists, with the increasing use of mathematics in published research and graduate teaching. However the survey results also implied that emphasis on pure mathematics was a rational response to incentives within the profession. Concern with the extent of mathematical expression may derive from the view that it crowds out other modes of expression, ie the issue is over how ideas are communicated. But there is a more fundamental issue concerning the role of mathematics in economics, namely its potential effect on content. One interpretation is that economics has been undergoing technical change, employing more mathematics and more sophisticated statistical techniques, which has improved the productivity of the discipline; the change in content is thus one of undoubted improvement. But concerns have been raised that mathematisation has proceeded at the cost of attention to matters which cannot be expressed mathematically, ie the alternative modes of communication can actually allow analysis in areas closed to mathematics. The issue is thus the fundamental one of what we understand by the discipline of economics and what it can achieve. This issue too feeds back into the issue of the public understanding of economics as a discipline.Mathematics is increasingly significant for economics in a third sense, namely its role in the economy itself. Increasingly activity in financial markets (particularly in derivatives trading) is governed by mathematical models. The importance of economists' role in developing these models is attested to by the award of the Nobel prize for economics to Black and Scholes for their ill-fated finance model. In what follows, we do not focus on this role of economics, other than to refer to it in illustration.We will focus here, rather, in the third section on the effect of mathematisation on the content of economics. This is preceded by a brief account of the history of the role of mathematics in economics. Some implications for the public understanding of economics are considered in the fourth section. The paper is completed by considering two case studies of the use of mathematics in economics, both of which focus on the economic effects of education. The first case study considers growth theory, which analyses the effect of education on rates of economic growth, ie at the macro level. The second focuses on the micro level, considering the effect of education on individual earnings. These case studies will be used to illustrate some of the points raised in the two sections on the effect of mathematics on the content and public understanding of economics, respectively.2 HISTORY OF MATHEMATICS IN ECONOMICSMathematics first took on a significant role in economics in the last century in the build-up to what is commonly referred to as the Marginalist Revolution. This was a period in which Classical concerns with production, growth, and the distribution of the fruits of growth among social classes, were being replaced by concern with market exchange. The focus thus shifted from the level of the economy and social classes to the level of the individual. Leon Walras, in particular, set out to establish the conditions for a successful co-ordination of market exchange, and he did so mathematically. Along with Augustin Cournot, he is responsible for the introduction of the systematic application of mathematics to economics.At the same time, there was a concern that economics should be seen as a discipline on a par with the physical sciences (see Mirowski, 1989; Drakopoulos, 1991). Walras's father, like many other economists of the time, saw mathematics as the vehicle for achieving this goal. Further, just as the physical sciences were being built up in axiomatic fashion on the basis of units of energy, etc, economics was being built up axiomatically on the basis of units of utility. The motivation of individuals in the economy engaging in market exchange is understood as the maximisation of utility, a human motivation which clearly lends itself to mathematical treatment. Walras (1965: 145) went further: 'It is only with the aid of mathematics that we can understand what is meant by the condition of maximum utility'. And indeed, the term 'Marginalist Revolution' refers to the mathematical result of the marginal conditions for market equilibrium, as derived by calculus.What was seen as standing in the way of the elevation of economics to the status of the physical sciences was measurement problems. How were 'utils' to be measured? The main force behind the introduction of mathematics to British economics, W. Stanley Jevons (1871: 25) expressed this concern as follows:'But I do not hesitate to say . . . that Political Economy might be gradually erected into an exact science, if only commercial statistics were far more complete and accurate than they are at present, so that the formulae could be endowed with exact meaning by the aid of numerical data'.Whether 'utils' could or could not even in principle be measured remained a matter for debate. But given the difficulties in practice, the mathematical development of economics proceeded at a pure rather than applied level, in contrast to the physical sciences.Such a development inevitably evoked a reaction. The Historical School in Germany (which provided the origins of the subsequent massive data-collection exercise in the US) argued that theory should emerge by induction from data, rather than being developed purely deductively. Their focus was thus on data collection rather than theory, mathematical or otherwise. The Austrian school, led by Carl Menger, took a deductivist approach, but deliberately shunned mathematics. The focus of their work was on the dynamics of the economic process, particularly the activities of entrepreneurs, rather than on market equilibrium. Their argument against the capacity of mathematics to assist such analysis was that humans are creative, purposeful beings, whose behaviour cannot adequately be represented deterministically. Further, their subjectivist approach led them to view published data as an inadequate reflection of the perceptions (rather than actuality) which prompted action. Here we have an important argument about the relationship between mathematics and the content of economics, to which we will return in the next section.In the UK, the leading figure in the subsequent development of economics was Alfred Marshall. He was influential particularly in his efforts to promote economics as a discipline and to project it as a unified social science, in spite of the debates then raging between the mathematical pure theorists, the empiricists and the non-mathematical pure theorists. It is significant that his Principles (Marshall, 1890) restricted mathematical reasoning to footnotes, so that the argument in the text was purely verbal. This is regarded as indicative of his views on the limitations of mathematics in economics; although he was more willing than the Austrians to engage in mathematical theorising, the focus of his research, like theirs, was on economic process rather than equilibrium. Further, his use of deductive reasoning (including his use of mathematics) was explicitly restricted to short chains of reasoning, ie partial analysis. Unlike Walras, he did not aspire to construct a complete mathematical system (Marshall, 1890/1920: 773).John Maynard Keynes was, like Marshall, trained initially as a mathematician, and also had reservations about its scope in economics. While he used mathematics to a limited extent, he argued that its capacity to capture the content of economics was limited, and thus so also was its application in empirical work. We will consider his arguments more fully in the next section. But his major impact on economics was to start off a new research agenda which gave particular impetus to the role of mathematics and its application in empirical work. Keynes provided the foundations for modern macroeconomics, which focuses on the economy in aggregate, rather than individuals. He also set out a policy agenda for government which required that the theory be tested and applied empirically. On this basis, increasingly elaborate mathematical models of the economy were constructed, aided by advances in computer technology and by the collection of data series, deriving from the ideas of the Historical School.Methodological issues arose over the meaning of these aggregate models. In particular, the controversial suggestion was made by Friedman (1953) that predictive success should be the sole criterion for theory choice; theorists should not seek to explain, ie theories should not be regarded as representing causal processes. The form, as well as the extent, of mathematical representation, by implication, was secondary to empirical predictive success (although it was assumed that prediction would be based on mathematical models of some form).But, by this stage, mathematics had in many ways taken over from physics as the model for the discipline of economics (McCloskey, 1994: Ch.1). This was evident in the force behind the further development of economics in the form of application of the principles of formal axiomatic systems. Macroeconomics had emerged as a mathematical system quite separate from microeconomics. Not only did they address different questions (failure of markets to co-ordinate and co-ordination success, respectively), but macroeconomics seemed to flout the axioms of individual behaviour on which microeconomics was founded. As a result, developments in macroeconomics over the last three decades can be understood as attempts to build up an over-arching general equilibrium system based on common axioms of individual behaviour.Modern economics thus relies heavily on mathematics. But measurement problems, and more fundamental methodological problems, have created a bifurcation between pure theory and applied theory. While the former constructs sophisticated mathematical analysis of individual behaviour based on utility maximisation principles, with an emphasis on existence proofs, the latter focuses more on the reduced forms for which there are corresponding data. Given the different aims of the two activities, the mathematics employed in pure theory will thus tend to differ from that employed for the purposes of statistical testing. Backhouse (1998) refers to the increasing propensity to separate the 'pure theory' part of an investigation from the 'empirical' part even within individual articles.The debate continues as to the extent to which this bifurcation between theory and application is problematic and whether it is surmountable (see Mayer, 1993; Backhouse, 1998). But it should be noted that this bifurcation is representative of what is called mainstream, or neo-classical, economics in the Walrasian tradition. Theory which has developed in the tradition of Menger, Marshall and Keynes makes much more limited use of mathematics on methodological grounds. Clearly this poses problems of communication within economics, in that the mainstream embraces mathematics as the preferred mode of expression, and indeed models itself on mathematics.With this history in mind, we turn now to address more directly the methodological issues raised by the mathematisation of economics.3 SIGNIFICANCE OF MATHEMATICS FOR ECONOMICS3.1 Mathematics and FormalismThe role of mathematics in economics can usefully be discussed in relation to the role of formalism in economics. While the two terms are often used interchangeably (e.g. Krugman, 1998), an argument need not be mathematical to be formal (see Chick, 1998). Further, it has been argued recently (by Backhouse, 1998) that, unlike mathematics, formalism entails the tighter condition of fixity of meaning. Weintraub (1998) has demonstrated the changing meaning of terms in mathematics. In particular, formalism also includes the notion of rigour; but scientific rigour may itself be subject to different meanings. Thus, while, at the turn of the century, scientific rigour referred to testing against empirical evidence, it is now associated more with mathematical axiomatisation.We have noted above the greater extent of mathematical formalism in pure than in applied economics. But formalism also has consequences for empirical testing; it requires the notion of fixity of meaning applied to data too. This allows for reference to 'the facts' as objectively measured phenomena with fixed meaning independent of theory. Indeed Mirowski (1991) argues that the very act of measurement imposes a mathematical structure. For example, the conventional market diagram presumes homogeneity of commodity space which is not in fact fixed in nature; Mirowski argues that the degree of homogeneity will vary depending on the changing social perception of market activities. Insofar as economics embraces formalism, therefore, it embraces a particular general approach to mathematics which derives from logical positivism and has implications at both the pure and applied levels. Within this general approach, there are then different uses made of mathematics depending on whether the research is pure or applied.The benefits of formalism for economics are seen (Backhouse, 1998) as:enabling a cumulative growth of knowledge since formal arguments may be readily understood by subsequent generations (note the necessity for fixity of meaning)providing an engine for discovery; a strong advocate of mathematisation of economics, Gérard Debreu puts it as follows (as quoted by Backhouse, 1998: 1852):[Mathematics] ceaselessly asks for weaker assumptions, for stronger conclusions, for greater generality. In taking a mathematical form, economic theory is driven to submit to those demands. . . . Mathematics also dictates the imperative of simplicity'.But, as Backhouse (1998) points out, the process of mathematisation itself may change the meaning of economic terms. He refers to the change from Adam Smith's (1759; 1776) notion of self interest brought about by its incorporation within a formal general equilibrium system; the social content of Smith's self-interest was lost in the atomistic axioms of general equilibrium theory. The application of formalism to the argument was represented as scientific progress by Arrow and Hahn (1971). Similarly, Keynes's theory of expectations under uncertainty changed meaning when formalised in the Rational Expectations Hypothesis. Formalisation inevitably eliminated Keynes's emphasis on unquantifiable risk. Yet Lucas (1980) represented this too as technological advance. The benefit of formalism in terms of promoting the growth of knowledge relies on meaning remaining unchanged.The first benefit from formalism noted above is a paraphrase of an expression used by another advocate of formalism in economics, Frank Hahn. In response to Joan Robinson who saw mathematics as having only a limited role in economics, Hahn (1989) would dismiss her arguments as things which 'cannot be said'; they lay outside the purview of the formal structure, ie they could not be demonstrated to be true within that structure. Thus an important issue of whether the change of meaning in the formalisation of Smith's self interest or Keynes's expectations under uncertainty eliminates something important which nevertheless 'cannot be said'. This brings us back to the arguments referred to in the previous section about the limitations to the scope for the mathematisaiton of economics.The critics of mathematisation based their critiques on their understanding of the subject matter of economics. This represents a significant methodological departure from the more general trend we had identified of mathematisation itself being a guiding principle in the drive to establish economics as a science. Whether the subject matter of the physical sciences is amenable to the kind of logical positivism which has dominated economics is a question in itself. But the question is clearly a significant one for a social science where the objects of study are creative, purposeful, social beings who act within an evolving institutional environment. How far can human behaviour be represented as conforming (albeit stochastically) to deterministic principles? While the issue is most stark when formalist economic reasoning is applied to highly personal matters, such as the family (see Becker, 1991), it has general application to all human activity which has economic content.Drawing inspiration from Keynes, Chick (1998) argues that real economies are open, organic systems which cannot be fully understood by means of closed, formal theoretical systems, and the mathematics of general equilibrium systems requires closure, and it requires that interactions between the units of analysis to be deterministic, ie it requires atomism. While formalism in the sense of rigour, precision and clarity of reasoning is a necessary feature of science, formalism in the sense of mathematical expression is not. One of the perceived virtues of mathematics as noted above is its precision; yet Keynes, following Marshall, pointed out the virtues also of vagueness:Much economic theorising to-day suffers, I think, because it attempts to apply highly precise and mathematical methods to material which is itself much too vague to support such treatment (Keynes, quoted in Chick 1998: 1864)The Black-Scholes episode provides a pointed illustration of this argument. Mathematical models of risk in financial markets require that all risk is quantifiable. But the financial crisis which caused the Black-Scholes system to collapse was not amenable to frequency distribution analysis - but nor was it random - so it lay outside the model.The general issues to be addressed then are:how to theorise about those aspects of the economic system which cannot be so approximated.For formalist pure theorists, the first is not an issue, and thus neither is the second. The driving force is mathematisation. There may even be a frank admission of lack of correspondence with the real world (see, for example, Hahn, 1973). But for formalist applied theorists (and in the extreme the application to market activity, as in the Black-Scholes case), the issues are real. Attempts are continually made to increase the realism of the models, within what is possible formally. In practice, too, the formal analysis of 'official discourse' is supplemented by informal methods in 'unofficial discourse' which remains unacknowledged because it goes against the espoused principles of formalism (see McCloskey, 1983). The limits to formalism are made much more explicit by non-formalist economists, and the application of non-mathematical-formalist methods is justified in terms of the nature of the subject matter.Critics of mathematical formalism open up the possibility of different types of mathematics. Since the argument really starts with logic, we turn now to consider the issue in terms of different approaches to logic.3.2 Mathematics and logicKeynes arrived at his views on mathematics in economics via logic:Constitutive of Keynes's philosophy is a crucial principle that flows unabated through all his writings. It is the proposition that qualitative logical analysis (i) precedes quantitative or mathematical analysis, and (ii) determines the scope of its application. Translated into a slogan, it becomes 'first logic, then mathematics if appropriate'. And logic in Keynes's philosophy ultimately rests on the use of direct judgment or intuition (O'Donnell, 1990: 35, emphasis in original).Keynes's first work (Keynes, 1973) addressed the problem of induction, in reaction against Russell and Whitehead's attempts to construct mathematical logic on rationalist grounds. He was concerned with how we establish reasonable grounds for belief in the absence of the conditions for certainty. Certainty for Keynes was the special case, only possible within a closed, atomic structure (in mathematics or in reality), ie those to which classical logic apply. Use of mathematics (based on classical logic) therefore requires justification in terms of the degree to which the case approximates to a closed, atomic structure. Keynes supported the use of mathematics as an aid to thought but argued that the onus was on the economist to demonstrate that its application was appropriate to the subject matter (see Dow, 1995).Axiomatisation is a type of formalism which relies particularly on classical logic, and which characterises the formalist approach to pure economic theory. Correspondence between theory and reality occurs only at the level of the axioms and at the level of the propositions which emerge from the application of deductive logic. There has been much discussion of the realism or otherwise of the axioms (see for example Hausman, 1992); the issue of testing we will address in the next subsection.But there is in addition the question of the logical structure which lies between axioms and testing. Keynes argued that, in the face of uncertainty, we employ what he called ordinary logic (or human logic). This logic, unlike classical logic, is non-demonstrable. It involves building up evidence and constructing indirect knowledge (for a scientist, we would call this knowledge theoretical) as far as possible. But, since this would in general be insufficient as a basis for action, we supplement this knowledge with the aid of convention and also intuition, or imagination.What this implies for the methodology of economics is reliance on a variety of methods, some of which will be non-mathematical. Only if all methods are mathematical in the classical-logic sense would the methods be commensurate, ie they could be put together to form a single mathematical system. The methods thus are not in general commensurate and judgment must be employed in order to form a basis for action. Keynes is quite explicit about what he sees as the danger of mathematics in economics:It is a great fault of symbolic pseudo-mathematical methods of formalising a system of economic analysis . . . that they expressly assume strict independence between the factors involved . . . ; whereas, in ordinary discourse . . . we can keep "at the back of our heads" the necessary reserves and qualifications . . . in a way in which we cannot keep complicated partial differentials "at the back" of several pages of algebra which assume that they all vanish (Keynes, as quoted by Chick, 1998: 1864).There remain possibilities for exploring the use of mathematics not based on classical logic. Thus, for example, fuzzy mathematics would appear to address some of the concerns of open-system theorising. The mathematics of chaos theory attracted attention for some time because it allowed a formal analysis of disequilibrium behaviour. But it has proved to be unsatisfactory in offering only the chaotic dual of stability. More promising would be the mathematics for analysing the self-organising systems of chemistry (see Prigogine and Stengers, 1984) which better captures the capacity of social systems to adapt to episodes of particular instability (see Chick, 1995). There may be other developments in mathematics outside the mould of classical logic which could assist the social sciences. But, since the requirement is for a mathematics which can handle open systems, it is inevitable that it will not provide the complete answer to economic methodology; if our subject matter evolves and is creative and purposeful, there is inevitable uncertainty on the part of the economist as much as economic agents. Thus, while Anderson et al. (1988) put forward mathematical techniques for dealing with complex systems, Chick (1998: 1863) points out that:In a complex system, results obtained through a narrow focus do not have general validity . . . . [T]he more complex the system, the greater our ignorance of all the interactions taking place. Neither the actions of agents within the system nor the study of the system from outside can be fully informed. Perfect knowledge is not available.The focus of this discussion has been on theory and the logic employed. But much of the discussion of mathematics in economics focuses on the testing of economic propositions, ie on the design and use of econometrics. We consider the particular issues raised by econometrics in the next subsection.3.3 Mathematics and econometricsEconometrics is the name given to the set of statistical techniques employed to test economic theories, or, increasingly, as a means of presenting 'the facts'.We have noted the difficulties faced by pure theorists in identifying empirical counterparts to theoretical concepts, such as utility. A variety of developments offered solutions to this problem, by avoiding it. Thus, while it was impossible to test directly the detailed mathematical reasoning behind the negative relation between price and quantity demanded, evidence of such a relation was seen as adequate justification for the underlying reasoning, however unquantifiable (Samuelson's Revealed Preference theory). Similarly, while the testing of macro-economic relationships required the existence and uniqueness of equilibrium, something which could not be tested directly, the observed relative stability of economies was taken as justification (Samuelson's Correspondence Principle).Some econometric models have been extremely elaborate, involving systems of hundreds of equations representing relationships in different sector of the economy. They are still restricted to those variables which are identifiable and measurable. These large models are now out of fashion, given their poor record in prediction, and the norm is more narrowly-defined models. The testing takes the form of estimating relationships which are more or less reduced-form. The assumption is that the posited structure of relationships is stable over the estimation period and that the data are drawn from a probability distribution.Keynes had argued that the econometrician must justify application of econometric techniques with reference to the subject matter. He argued that, since economic structure in general evolves, econometric techniques cannot in general be applied. His logic was based on the argument that the general case for social systems in particular is not one for which probabilities can be quantified, so that a more general concept of probability is required.In practice, econometrics is not entirely formal, although there have been attempts to formalise the process (see Backhouse, 1998). In selecting data, and in the formulation of relationships to be tested, economists employing econometrics bring to bear a whole variety of additional considerations. Nevertheless tremendous store is put by the formal outcome of the econometric process, and the informal input suppressed (see McCloskey, 1986: 94).The purpose of much of applied economics is to provide a basis for policy advice. To the extent that mathematical formulation inevitably rules out considerations which cannot be expressed mathematically, the policy application of mathematical models poses important questions. All theory must abstract, but is the type of abstraction required by mathematical expression particularly significant? This is a large, controversial question on which much of the previous discussion bears. Much depends on whether the outcome of the mathematical model and its empirical estimation are regarded as definitive or only partial, to be considered along with other types of knowledge arrived at using different methods.But we turn now to consider this issue in a rather different light, namely how mathematisation impacts on the public understanding of economics, considering first policy makers, then students and then the general public.4 MATHEMATICS AND THE PUBLIC UNDERSTANDING OF ECONOMICS4.1 Policy MakersThe mathematical basis for much of economic policy advice was most evident in the heyday of the large econometric macro models. The UK government was advised on economic policy by the 'Seven Wise Men', most of whom were associated with one or another macro-econometric model. The predictive power of each model was a matter for public discussion. Monetary policy is now the responsibility of the independent Bank of England's Monetary Policy Committee, the minutes of whose monthly deliberations are published. The Bank staff input on the basis of mathematical models, as discussed in the Bank of England Quarterly Bulletin, is clearly significant. Now the Bank has published a volume which explains the nature and use made of mathematical models (Bank of England, 1999).Perhaps the most significant element of the policy-maker's understanding of economics, as it is affected by the extensive use of mathematics, is the understanding this conveys about the nature of economics and its capacity for generating predictions. For all the caveats (ceteris paribus effectively means the economic structure remaining as it was during the estimation period, and no exogenous shocks occurring), an impression is given by mathematical models that they are scientific and constitute the economists' best basis for prediction. The use of models in the policy-making context thus serves a rhetorical purpose in accord with the aim of putting economics on a par with the physical sciences (McCloskey, 1986, 1994).The large multi-equation models of the 1980s did not predict well; even though they were not complex in the formal sense of allowing a significant degree of interaction between agents, they were complex in terms of scale. Whitley (1997) explains the rationale behind a greater emphasis in the Bank of England on a range of partial models. Bank policy is now based on an inflation forecast which incorporates predictions on the basis of a range of models. The forecast now takes the form of a fan-chart which effectively ranges the forecasts each within the narrow fan of its own stochastic range, the outcome being a large fan; the width of the fan reflects the level of 'uncertainty' attached to the forecast range; note that this uncertainty is quantified.An agency like the Bank collects a wide range of intelligence, much of which must remain within the category of 'vague': the sense of the markets, the propensity to innovate, the mood of public sector unions, etc. Yet these matters are of central importance to any inflation forecast. The latest Bank document explains that survey data are fed into the decision-making of the Monetary Policy Committee, alongside formal projections, as a check for the consistency of those projections. Thus, while the 'official rhetoric' form of the inflation forecast suggests quantifiability, the 'unofficial rhetoric' of actual policy-making incorporates unquantifiable elements of judgment, as the subject-matter dictates.4.2 The General PublicIt is no wonder that the public have a conflicting impression of economics as, on the one hand, scientific and, on the other hand, indecisive. There is a range of hackneyed jokes to this effect. Because the official rhetoric implies a degree of precision which is unattainable in practice, economics disappoints. Economists feel themselves misunderstood. The caveats are there; the economy is too complex a system to reasonably expect accurate forecasts; there are bound to be differences of opinion. Yet the public expects economists to agree on scientific results in the same way as physicists. I would suggest that it is no accident that this increase in public misunderstanding of economics has coincided with the increased mathematisation of the discipline.It is with reference to the public understanding of economics that Krugman (1998) in fact makes his case for mathematical formalism. He argues that formalising arguments, eg within an accounting framework, yields useful results which do not seem to be intuitively clear to the media. Indeed he argues that economics can only progress with the aid of mathematics. But, as a separate issue, he argues that economists should put more emphasis on translating the result of mathematical theories into lay terms in order to communicate more effectively with the public.4.3 StudentsA cohort of the public of particular interest to academic economists is the body of economics students and potential students of economics. There has been some extensive study in the US focusing on the teaching of economics which has bearing on the role of mathematics. The various studies, covering both undergraduate and postgraduate training, demonstrate a consciousness of the increased mathematisation of the subject (see Colander and Klamer, 1987, Krueger et al., 1991, Hansen et al., 1991 and Kasper et al., 1991). The studies note that the mathematics training of incoming students has not kept up with the requirements of economics. They also note the dissaffection brought about among students by the preoccupation with formal technique at the expense of application to real-world issues, as well as the concern of staff that the students are being given too narrow a training for future employment.In the UK the matter of attitudes among economics students has only been addressed relatively recently. The latest issue of the Newsletter of the Royal Economic Society contains an account of a survey of A-level students which indicates that students with mathematical ability are more inclined to choose to take economics at university in preference to Arts subjects, but not Science subjects. But otherwise there has been little study of this area. In the UK, unlike the US, course content and methodology are monitored, but not transparently. Thus, for example, the ESRC monitors postgraduate provision, and has the ability to influence programmes through the allocation of its student awards. At the undergraduate level, the new quality assurance system will effectively be influencing how economics is taught. But so far there has been only limited public debate.We turn now to consider two related areas of economics for illustration of the arguments developed so far.5 CASE STUDIES OF THE USE OF MATHEMATICS IN ECONOMICS5.1 Education and GrowthTraditional, neo-classical growth theory builds on micro-foundations drawn from axioms of optimising individual and firm behaviour to construct a reduced form mathematical relationship between the inputs of labour and capital and income, at the macro, or economy, level. Given population growth (and thus growth of the labour supply) and the state of technology, increased growth in income can then only come from an increase in the capital stock. This theory has been subjected to empirical testing by estimating the coefficients of the posited relationship and testing for goodness-of-fit with respect to actual growth rates.These studies have suggested that the state of technology is empirically important (ie labour and capital did not fully explain growth rates), putting a focus on technological change, something which had been treated as exogenous to the closed, formal system representing the economy. In other words, the requirements for mathematical tractability had required that something which is difficult to represent deterministically, and indeed to measure, was excluded from the analysis.More recently, attention has shifted from trying to endogenise technical change in general (ie independent of capital and labour inputs) to endogenising the other contributors to productivity, notably labour productivity. This is the post-neoclassical endogenous growth theory to which Gordon Brown has expressed allegiance (see Aghion and Howitt, 1998, for an account of the literature). Its policy significance is that, while technological change in the long-run is available to all economies (so that all economies' growth rates would be expected to converge as technological change is globalised), labour productivity is something which is amenable to policy manipulation, allowing different growth rates across economies. Thus the empirical assessment of the relative merits of the two approaches rests on the empirical judgment as to whether international growth rates are converging or not. The fact that there is not a consensus on this judgment illustrates the intrinsic difficulties of empirical testing in economics.According to the endogenous growth approach, labour productivity may increase because of learning-by-doing (ie as a by-product of employment), or it can increase through education outside employment. A series of mathematical models has been developed, which can be grouped around the idea that education provides a one-off increase in labour productivity, raising the rate of economic growth, or the idea that it also increases the capacity to absorb technological change into the production process. The aim is to determine the optimal level of education expenditure in terms of which would yield the highest rates of economic growth.The models inevitably require a series of assumptions to be made. Thus, for example, the Lucas (1988) model portrays education as an investment decision by the individual on a par with capital investment; time spent in education means time not in employment (just as capital expenditure precludes consumption expenditure). Education yields the same increase in productivity across the board, and at all levels of education. The decision is based on a rate of time preference and a co-efficient of risk aversion, but, since these are unidentifiable in aggregate, empirical application simply focuses on the coefficient of the labour variable in the reduced form equation. Other models have attempted to increase the degree of realism relative to the Lucas model, allowing for example for example for decreasing returns to education, interplay with the coefficient of technological progress and inequality between education levels of workers. Inevitably this has increased the complexity of the mathematical model. But measurement difficulties mean that these finer points cannot be assessed empirically. Aghion and Howitt (1998: 435) point out that:formal theory is ahead of conceptual clarity. . . . [T]he real question is one of meaning, not measurement. Only when theory produces clear conceptual categories will it be possible to measure them accurately.The presumptions then are formalist. Once the meaning of terms is agreed, it is fixed; theory can then be tested against the facts which can be measured as long as the definition is clear. There is no room for analysis outside the formal mathematical model.Even if meaning were clear, however, measurement issues would not be insubstantial. There is a more general issue of the capacity of econometric techniques to discriminate between theories. The endogenous growth theories are put forward as an alternative to neo-classical theories on the basis of the pure theory model which precedes the econometrics. But since the econometrics consists basically of correlation analysis applied to a reduced form of the theory which involves a similar range of variables to neo-classical theory it is not at all clear what can be distinguished. The doyen of the neo-classical approach, Robert Solow (1994) argues that his treatment of technical change as exogenous does not mean that it cannot be analysed (his model is partial rather than general) and that such analysis must take account of the unquantifiable uncertainty associated with the innovation process. He sees the extraction of workable hypotheses from case studies as a more promising avenue than the endogenous-growth theory foundation on the intertemporally-optimising representative agent. The endogenous growth theories are constructed in aggregate terms, referring to the aggregate 'representative' individual (as having a particular degree of risk-aversion, for example). But they draw on micro-foundations based on the axioms of rational (optimising) individual behaviour. We turn now to consider a literature which focuses on this behaviour (without being concerned with its implications for economic growth).5.2 Education and earningsThe micro-economic basis of endogenous growth theory refers to the individual decision about the degree of education to undertake (see Willis, 1986, for an account of the literature, and Mincer, 1974 for the classic source). This choice is based on an assessment of earnings foregone during education relative to the increase in earnings which would result from education, ie a form of present-value calculation. The benchmark is long-run competitive equilibrium, where supply and demand for workers at each schooling level are equated and no worker wishes to alter her schooling level. For each worker in equilibrium, the present value of education represents a return equal to the alternative return on foregone earnings, the interest rate.While the theory is developed mathematically in the standard terms of individual optimisation, the empirical literature is explicitly couched in different terms, but carrying forward many of the assumptions of the theoretical literature. Thus, for example, in building up his exposition of the literature, Willis (1986: 529) posits an earnings function, whereby earnings are shown as a function of years of education and years of employment over a lifetime. Rather than deriving from theory, the functional form is arrived at as the best statistical fit. The residual term has mean zero, so that, on average, earnings are fully explained by the education and employment periods. As well as all the assumptions underpinning the use of these two variables, it is assumed that the data sample are taken from a population in long-run equilibrium. Willis explains the elaboration of theory as efforts are made successively to relax these assumptions, and the interplay between theoretical formulation and statistical estimation. This interplay is primarily one of confirmation, since the statistical limitations on dealing with micro-level diversity are significant. The conclusion is that empirical work supports the human capital approach to education choices (the same approach which underpins endogenous growth theory).But two significant provisos need to be specified about what this tells us about human capital theory. One is the specific point about rational choice theory which is that it does not readily adapt to disequilibrium expression; if agents are optimising in rational fashion, then how can failure to do so be explained? The information is assumed to be available to make rational choices and any mistakes are random. That few would claim this to be the case in practise raises significant questions about the interpretation of the data. Second, the empirical analysis is essentially based on correlation, and thus tells us nothing about causation. The results of the empirical studies are in fact very illuminating of the characteristics of different cohorts of workers. But the connection with the pure mathematics of human capital theory does not follow with any necessity at all.6 CONCLUSIONWe have discussed how the use of mathematics has increased significantly in economics, and the issues this has raised. There are issues at the level of communication of ideas, among economists, and between economists and policy-makers, the general. public and students. Communication is of great importance. But communication is based on a shared view of the nature and scope of the discipline. There is therefore a more fundamental issue about whether and in what way mathematisation has changed the nature and scope of economics.Mathematical tools have allowed many advances in economic theory. But at the same time, the difficulty in combining pure theory with applied economics has allowed the two strands to proceed according to different agendas. Even so, there are elements in common (presumption of equilibrium, fixity of meaning of terms and of the objects of measurement, etc) which provide the basis for mathematical treatment, but which nevertheless are controversial. Much of this issue boils down to the question of how far a study of complex social systems is amenable to the (mathematical) methods of analysis adopted by the physical sciences.These issues are of continuing importance for the direction taken by the discipline both in academia, but also in the policy arena and more generally in terms of the public understanding of economics.thank you... :-)