Philosophy of Mathematics and Economics

Image, Context and Perspective

Thomas A. Boylan, Paschal F. O'Gorman

• Publisher: Routledge
• Year: 2018
• Print ISBN: 9780367592431
• eBook ISBN: 9781351124546
• Edition: 1
• Topic: Business
• Subtopic: Business General
• Index: Business

With the failure of economics to predict the recent economic crisis, the image of economics as a rigorous mathematical science has been subjected to increasing interrogation. One explanation for this failure is that the subject took a wrong turn in its historical trajectory, becoming too mathematical. Using the philosophy of mathematics, this unique book re-examines this trajectory.

Philosophy of Mathematics and Economics re-analyses the divergent rationales for mathematical economics by some of its principal architects. Yet, it is not limited to simply enhancing our understanding of how economics became an applied mathematical science. The authors also critically evaluate developments in the philosophy of mathematics to expose the inadequacy of aspects of mainstream mathematical economics, as well as exploiting the same philosophy to suggest alternative ways of rigorously formulating economic theory for our digital age. This book represents an innovative attempt to more fully understand the complexity of the interaction between developments in the philosophy of mathematics and the process of formalisation in economics.

Assuming no expert knowledge in the philosophy of mathematics, this work is relevant to historians of economic thought and professional philosophers of economics. In addition, it will be of great interest to those who wish to deepen their appreciation of the economic contours of contemporary society. It is also hoped that mathematical economists will find this work informative and engaging.

1 Economics and mathematics

Image, context and development

Introduction

In a recently published anthology, entitled Mathematics and Modern Economics, the editor, Geoffrey Hodgson, provides what would arguably be a widely accepted overview of many, but not all, practising economists of the relationship between mathematics and economics:¹

Today it is widely believed that economics is a mathematical science and the extensive use of mathematics is vital to make economics ‘scientific.’ Even if this is questioned, it must be conceded that anyone trying to grapple with economic concepts, their development and their applications must have at least some rudimentary knowledge of mathematics and statistics. It is also undeniable that some mathematical formalizations have played a key role in the development of economic ideas, from the link between marginal utility and calculus to the analysis of strategic interaction in game theory. Few would deny that such formalizations have enhanced our understanding. Economics stands way above the other social sciences in its degree of utilisation of mathematics, and consequently in terms of claims of its purported rigour.

(Hodgson 2012: xiii)

Following this account, Hodgson immediately notes that for ‘many practising economists, this will be the end of the story’. For those who subscribe to the above account, economics ‘is essentially and unavoidably mathematical’ which confers on it a number of salient characteristics, including the following: ‘Mathematics means precision. Mathematics means rigour. Mathematics means science’ (ibid.: xiii). Faced with the prospect of the conceptual enhancements to be gleaned from the importation of mathematics of this trinity of precision, rigour and science into the domain of social and economic theorising, those opposed to this expanding colonisation of economics by mathematics are deemed to be either a species of ‘pre-scientific Neanderthals’, who clearly lack an adequate understanding of the role of mathematics in the development of human knowledge in general and of science in particular, or serious doubts ‘may also be cast on their mathematical capability’. As Hodgson summarises his condensed account of the presiding account for many practising economists, ‘Doing economic theory is doing mathematics. Hence calls for less mathematics are misguided calls for “less theory”’ (ibid.: xiii).

But Hodgson, a leading scholar in institutional and evolutionary economics and a prolific writer on economic methodology including the issue of formalism in economics, is quick to point out the inadequacies of the above account of the relation of mathematics and economics, as captured in his cryptic comment, ‘if only things were so simple.’ The ‘many practising economists’ that he refers to as subscribing to the above account would find Hodgson’s Introduction to this valuable and extensive Anthology very informative, not to mention the substantive contents of many of the excellent readings he has selected for inclusion. These readings provide an excellent starting point for the variegated agenda that has emerged around the relation of mathematics and economics. The readings and the issues contained in this Anthology are confined to the post-1945 period and extend to the financial crisis of 2008. This was of course a pivotal period in the consolidation of the formalisation of economics through the use of mathematics. Taking a longer view, the post-1945 period appears parochial in time given the extended historical relationship between mathematics broadly interpreted and economics. A less restricted time-horizon would greatly extend the period of coverage if the complex interaction of the mathematical mode of reasoning within the socio-economic domain is to be adequately portrayed. If we relax our conception of mathematics as we currently deploy it in the twentieth-first century, complete with it all its associated resonances of ‘precision’, ‘rigour’ and ‘science’, then a more appropriate historical starting point would be the seventeenth century. This extended historical horizon would provide the framework for a more satisfactory and adequate understanding of the complex network of issues involved in the intellectual, institutional and philosophical interactions at play in the relationship between mathematics and economics.

Even a cursory examination of this complex relationship between mathematics and economics over this extended period makes clear that the relationship between the two domains has followed an erratic and contentious path of mutual reinforcement and even constructive development. But to claim this interpretation over a four-hundred-year period is not to downplay, much less to deny, that the relationship has neither been smooth nor straightforward. On the contrary, the relationship has been punctuated by periods of opposition and hostility to the increasing encroachment of mathematics into economics. Nevertheless, it would perhaps appear strange to a disinterested observer that the issue of the use of mathematics in economics has remained a source of contention and to many an issue of fundamental contention in the twenty-first century. The question can be posed as to what constitutes the possible sources of this tension, or methodological fault-line, between the proponents of the use of mathematics in economics and their critics. While there is a large array of possible sources, we identify three domains that would, in our view, be germane to the pursuit of insight into this complex dynamic between mathematics and economics. There is firstly, the general philosophical question which has, over the last four hundred years, been a presiding presence at the centre of European social thought, namely whether there exists discoverable social laws of development, akin to those in the physical sphere. If there are such discoverable social laws, what role would or could mathematical thinking contribute to their elucidation? Depending on one’s disposition to this question could greatly influence how one might treat the employment of mathematics in socio-economic inquiry. In his book, The Nature of Social Laws (1984), Robert Brown poses an interesting set of ancillary questions relative to this question. The question Brown poses is ‘why the efforts by so many people during the last four-hundred years, to discover laws of society have not been better rewarded?’ (ibid.: 6). He poses the following intriguing set of issues:

Is it because their character has been misconceived? Or is it simply that they have been sought in the wrong area of social life? Do they exist unrecognized, or is the long search for social laws the unhappy outcome of a gross misunderstanding? Are these laws of society with which we are all familiar and which are not difficult to state? Or are there reasons of logic, or fact, or both, which ensure that social laws do not – perhaps cannot – exist?

(Brown 1984: 6).

These questions or variations of them, no doubt, will continue to present a very challenging array of issues to both social theorists and philosophers of society. Secondly, there are issues with respect to what we may call the specificity of economics as a social science that may militate against a dominant role for the application of the mathematical mode of analysis to the economic domain, or at least to key parts of that domain. In other words can economics lay claim to some form of exceptionalism, arising from either a range of ontological considerations or epistemic issues, or are there generic conditions that apply to all the social sciences? In recent years some attention has been paid to this issue of the unique or distinctive features of economics as a discipline, but this hasn’t been explicitly related to the issue of the application of mathematics and its implications (Hausman 1992). Thirdly, we identify what is the main concern of our book, namely the influence on and the implications for economic theorising arising from the major developments in the philosophy of mathematics from the late nineteenth century. These developments in the philosophy of mathematics arose from the ‘foundations of mathematics’ debate which occurred at this time and continued into the twentieth century. Our motivation in engaging this topic is predicated on the assumption that if insight into the current state of economics, particularly the relationship between mathematics and economics, is to be achieved, then an extended and more adequate appreciation and understanding of the outcomes of developments in the philosophy of mathematics, along with their implications for economics, must become an integral part of the self-referential methodological understanding of economics as a discipline.

Against the background of these fundamental methodological questions, the relationship of changes in the philosophy of mathematics on economic methodology is the central focus of this book. There is little disputing of the trend in the intensification of the use of mathematics in economics particularly during the course of the twentieth century. This is particularly reflected in two areas where considerable empirical examination has been undertaken. One concerns the increasing volume of articles in the leading journals in the discipline which deploy mathematics as their principal mode of analysis while the second area concerns the reconfiguration of the curricula, at both undergraduate and postgraduate levels, to accommodate the increasing demands for courses with a mathematical orientation. In a number of studies that sought to quantify the extent of mathematics in economics, including Stigler (1965), Anderson, Goff and Tollison (1986), Grubel and Boland (1986) and Debreu (1986), the trend in the dramatic extension of mathematics was clearly evident. Mirowski (1991) reports on an extensive survey and review of the journal literature for the period 1887–1955. This exercise included four major journals, described as ‘representative general journals of the fledging economics profession in the three countries of France, Great Britain and the United States’. The four journals included were the Revue D’Economie Politique, the Economic Journal, the Quarterly Journal of Economics and the Journal of Political Economy. The survey was not based on a sample but included an examination of every volume within the period in question. Mirowski finds that over the period 1887 to 1924 ‘most economics journals look very much alike when it comes to mathematical discourse’ (ibid.: 150), with the journals devoting in general no more than 5 per cent of their pages to mathematical discourse. However, between 1925–1935 a very noticeable change occurred with regard to the intensification of mathematical discourse within the discipline. In fact Mirowski characterises this decade 1925–1935 as the second major ‘rupture’ which marked a critical inflection point in the rise of mathematics in economics, which took place as he notes ‘in the decade of the Depression’ (ibid.: 151). This trend was to be hugely consolidated and extended in the post-War period (Mirowski 2002).

When Stigler and his collaborators revisited this topic in the 1990s and analysed the application of mathematical techniques in a number of major economic journals, they found a decrease from 95 per cent in 1892 to 5.3 per cent in 1990 in articles that did not deploy either geometrical representation or mathematical notation in their analytical expositions (Stigler et al. 1995). This pronounced trend in the expanding use of mathematics in economics during the course of the twentieth century and in particular in the post-World War II period is characterised as the ‘formalist revolution’, which according to numerous studies was essentially consolidated by the late 1950s (Backhouse 1998; Blaug 1999; and Weintraub 2002). The publication of the lengthy report of Bowen (1953) on graduate education in economics for the American Economic Association, which advocated substantial extension of mathematical training, lent very considerable support to the reconfiguration of the graduate curriculum, as did the views of individual influential economists, such as Samuelson, who in 1952 provided a pragmatic but nuanced assessment of the desired relationship between economic theory and mathematics (Samuelson 1952).²

A related area where the influence of the increasing incorporation of mathematics into economics is clearly evident is in the domain of curriculum structure and design. This for some contemporary commentators has led to an unbalancing of the economics curriculum in favour of mathematics at the expense of other areas of economic studies. This is particularly pronounced in the areas of the more historical, discursive and qualitative areas of discourse within the discipline. The dramatic contraction, if not the complete demise, of economic history along with courses in the history of economic thought clearly illustrates this development in recent years. In passing, mention could be made in this context of the increasing retreat from the provision of cognate courses in politics, sociology and geography (urban and spatial analysis) that previously enhanced the intellectual contextualisation for the study of economics. While the issues of curriculum structure, design and development are not central to the aims of this book, they nevertheless represent an interesting locus of the tensions underlying the contents and design of what constitutes an adequate or, even more pejoratively, a proper curriculum for students of economics. The concern is that the teaching of economics has become disproportionately dominated by inclusion of an expanding volume of mathematics, mathematical statistics and econometrics leading to the exclusion of valuable material when viewed from a broader intellectual perspective. Given the finite number of hours available for the development and delivery of economic courses within the prescribed curriculum structure, the relation between the mathematical and the ‘non-mathematical’ contents has the property of a zero-sum configuration. The inclusion of more historical and qualitative material in the curriculum is not in principle incompatible with the position that mathematics has a pivotal role and that its influence will become more pronounced in the future. The issue is one of seeking balance and some may argue that the central concern is maintaining the intellectual integrity of the discipline.

Closely related to this issue of overall curriculum development and the search for a better balance between different dimensions of the discipline, in particular as between the desirability of ‘more’ or ‘less’ mathematics, is the very recent emerging debate concerning the crucial question as to what kind of mathematics should be taught (Velupillai 2000, 2005a, 2005b; Potts 2000; Colander et al. 2008). This agenda raises altogether more fundamental issues, both philosophical and methodological, and will be examined in later chapters of this book. A great deal of the future course of the role of mathematics in economics, or more precisely what kind of mathematics will or should play a pivotal role in the future, will hang on the outcome of developments currently underway in developing a different kind of mathematics that is both philosophically and methodologically better suited to the domain of economics.

To return to the longer view. In this chapter we will provide a short account of what is both historically and methodologically a very complex, challenging and extensive set of issues. The aim is to do no more than provide a context, which emphasises the historical dimension of the connection between the quest for initially the quantification of socio-economic phenomena and the ensuing search for ‘social laws’ which sought to be articulated in a mathematical mode in an attempt to capture the essential features of the pivotal socio-economic relationships and underlying mechanisms. The search for such relationships goes back, we would argue, to the seventeenth century and has gone through a number of critical phases, each with its own emphasis as to what they perceived to be the desired aim of their central endeavours.

The structure of this chapter will reflect our attempt to provide an overview of these phases of the interactive development of the relations between the construction of political economy as a discipline and its quest for a format of presentation which would establish its greater coherence, intellectual rigour, scientific status and relevance to socio-economic policy. The phases involved could be gathered broadly around the following periodisation: from the middle of the seventeenth century the articulation of ‘political arithmetic’ by William Petty and his followers launched the process of quantification and measurement of socio-economic phenomena informed by a combination of pragmatic policy concerns on the one hand, and the influence of Newtonian empirical data-gathering on the other. During the course of the eighteenth century and under the influence of the French Enlightenment, the emphasis shifts to the search for ‘laws’ in the socio-economic domain and the desire to formulate these laws in mathematical terms if possible. But it is during what we will call the ‘long nineteenth century’ that the momentum for the mathematisation of economics gathered pace and continued relentlessly into the twentieth century under the influence of the Walrasian and later the Neo-Walrasian programmes centred on general equilibrium theorising. The final phase represents developments in the post-World War II period, which arising from crucial developments in the philosophy of mathematics from the 1930s has thrown up fundamental problems for the methodology of economics in the domain of mathematical economics. In attempting to address this extensive historical span of time and the complex array of themes and issues associated with each specific period within the context of a chapter, we must of necessity be highly selective with respect to both topics and the role of individual contributors. This reflects our limited aim, in the context of a single chapter, of providing no more than an overarching overview of both the longevity and complexity of the critical relationship of economics and mathematics in the course of their extended historical interaction.³

Political arithmetic: the emergence of quantification of socio-economic phenomena

In his insightful study of the history of economics, Stark (1944) posed three interesting questions with respect to the problems of the historical development of economics, which in principle could be applied arguably to every discipline. Stark’s delineation of the issues was as follows:

The historical interpretation and explanation of theories put forward in the past is the first and foremost task which the historian of political economy has to fulfil. But besides the great problem, which might be called his material problem, he is confronted with several others more or less formal in character. Three of them are of outstanding importance. They are indicated by the following questions: When did political economy arise? What were the phases in its evolution? How can it be defined and divided from other fields of thought? … The first problem – the problem of origin – naturally and necessarily arises with regards to any science, but it is especially intricate in political economy.

(Stark 1944: 59).

If we replace ‘political economy’ with the ‘mathematicisation of economics’ in the above quotation, the issues identified by Stark apply with equal force and relevance. While there is widespread agreement that political economy was ‘a creation of the European Enlightenment – more specifically, at first, of the French and Scottish Enlightenments’ (Tribe 2003: 154), the genesis of the quantification of economic phenomena preceded the Enlightenment and was a product of the seventeenth century.

Schumpeter (1954) was quite clear that in the course of the eighteenth century ‘economics settled down into ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• Introduction
• 1. Economics and mathematics: Image, context and development
• 2. Walras’ programme: The Walras–Poincaré correspondence reassessed
• 3. The formalisation of economics and Debreu’s philosophy of mathematics
• 4. The axiomatic method in the foundations of mathematics: Implications for economics
• 5. Hahn and Kaldor on the neo-Walrasian formalisation of economics
• 6. Rationality and conventions in economics and in mathematics
• 7. The emergence of constructive and computable mathematics: New directions for the formalisation of economics?
• 8. Economics, mathematics and science: Philosophical reflections
• Appendix
• Bibliography
• Index