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
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 what we have decided to call a Classical Situation’ which gave rise to economics acquiring ‘the status of a recognized field of tooled knowledge’ (Schumpeter 1954: 143). But as he observed ‘the sifting and co-ordinating works of that period’ not only deepened and broadened the ‘rivulet that flowed from the schoolmen and the philosophers of natural law’ from an earlier period, they ‘also absorbed the waters of another and more boisterous stream that sprung from the forum where men of affairs, pamphleteers, and, later on teachers debated the policies of their day’ (ibid.: 143). This ‘more boisterous stream’ of material was produced by what Schumpeter called Consultant Administrators, whose primary preoccupation was the production of factual investigations to serve the purposes of the emergent nation states. They were connected by a common informing principle, notwithstanding their different backgrounds, namely ‘the spirit of numerical analysis’ (ibid.: 209).4
This commitment to ‘the spirit of numerical analysis’ was greatly enabled in the course of the seventeenth and eighteenth centuries by the development of specialised courses, particularly at the German universities, which focused on the presentation of purely descriptive data pertinent to the needs of public administration and policy.5 But notwithstanding these developments in Europe, particularly in Germany, the decisive momentum to pursue the art of statecraft through the schematic collection and presentation of data was provided by a pioneering group in England, the pivotal figure of which was unquestionably William Petty (1623–1687). His work and that of his like-minded colleagues represented the crucial impulse for the quantification of economic, social and demographic phenomena. This quest for quantification, under the rubric of ‘political arithmetic’, represented a critical phase in the search for the schematisation and later progressive formalisation of the newly emerging discipline of political economy. We would argue that Petty deserves a distinguished place in the intellectual origins of the quest for the systematic quantification of socio-economic phenomena which provided the platform that facilitated, at least in part, the later mathematisation of our discipline. Consequently we outline in some detail Petty’s career and his seminal contribution in the form of his ‘political arithmetic’ as he chose to designate it.
Petty’s career was both mercurial and meteoric and displayed a promiscuous disregard for disciplinary boundaries, a salient characteristic of the seventeenth century. A prodigious talent, whose occupations and activities included, amongst others, that of ‘anatomist, physician, professor of music, inventor, statistician, Member of Parliament, demographer, cartographer, founding member of the Royal Society, industrialist and author’ (Murphy 2009: 22). He was born into a humble family in Romsey, Hampshire, who were involved in the clothing trade, a pursuit in which Petty himself saw little future. Abandoning this environment he went to sea as a cabin-boy at the age of thirteen. Sustaining an injury at sea he was put ashore in Normandy, an event that would have major consequences for his future career. He applied for admission to the Jesuit-run University of Caen, where he received an excellent education in Latin, Greek, French, Mathematics and Astronomy. He would later attest to his early academic achievements: ‘At the full age of fifteen years I had obtained the Latin, Greek and French tongues, the whole body of common arithmetick, the practical geometry and astronomy, conducing to navigation dialling, with the knowledge of several mathematical trades’ (Petty 1769: iv).
Clearly Petty, whom Karl Marx claimed as ‘the founder of political economy’, perceived himself even at this early age as a precocious linguist, navigator and an accomplished mathematician. For our purposes the ‘founder’ of the emerging discipline of political economy was indeed imbued with the ‘spirit’ of mathematics, which would find expression for Petty’s later work in the form of quantification rather than formal mathematics as we currently understand it.
Nor was Petty’s education in mathematics at Caen the end of his exposure to mathematics and mathematicians. When he left Caen in 1640 he ‘seems to have spent some three years in the Royal Navy’ (Hutchison 1988: 27). By 1643 he returned to the Continent and pursued the study of medicine in Holland where he became friendly with John Pell, Professor of Mathematics at Amsterdam (Malcolm and Stedall 2005). Petty moved to Paris to continue his study of anatomy, and through the good offices of Pell he was introduced to Thomas Hobbes. For over a year in 1645–1646, Petty became Hobbes’s secretary and research assistant, whose intellectual influence on the young Petty was to prove very significant. Through Hobbes, Petty became acquainted with some of the leading scientists, mathematicians and philosophers in Europe. He participated in the Paris-based circle of Father Marin Marsenne, which included, among others, Fermat, Gassendi, Pascal and Descartes. From this experience Petty combined not only the Baconian inspired methodology of Hobbes, but presumably also the insights gleaned from exposure to this distinguished circle as to the potential of mathematical analysis.6
Following his sojourn in Paris, Petty returned to England in 1646 to continue his medical studies at Oxford. Pioneering the study of anatomy, Petty’s medical academic career progressed rapidly, being appointed to the Professorship of Anatomy in Oxford in 1650 and later to the Vice-Principalship of Brasenose College in Oxford. In 1651 he was appointed Professor of Music at Gresham College in London, though there appears some ambiguity as to the precise remit of this particular professorship.7 Gresham College was a new institution dedicated to the mechanical and experimental arts, and it has been suggested that Petty ‘may have taught music in its mathematical rather than aesthetic aspect’ (Letwin 1963: 188 fn. 2). While at Oxford Petty forged a close friendship with Samuel Hartlib, who would exert considerable influence on Petty’s Baconian methodology, and through Hartlib he became acquainted with Robert Boyle. The extended circle of Petty’s friends and acquaintances formed a very influential circle, or ‘Invisible College’ as it became known, many of whom would later contribute to the establishment of the Royal Society of London for the Improving of Natural Knowledge in 1662.8 The Royal Society was dedicated to the Baconian programme of empirical observation and experiment as the basis of scientific knowledge initially in the natural world and later by extension in the social domain.
Having scaled the heights of the academic world by 1651, at the age of 28, a long and potentially outstanding academic career seemed assured for Petty. However in the same year he secured a leave of absence from Oxford which was to last for two years. In the event Petty was never to return to academic life and his career took a very different direction which would have important consequences for the development of his method of political arithmetic and his claim as the founder of that particular approach to the broader discipline of political economy. The circumstances surrounding his change of career direction were embedded in a mixture of domestic politics, colonial policy in Ireland and a deep personal ambition to amass a personal fortune if the opportunity presented itself.
The opportunity did arise following the end of the English Civil War in the midst of Cromwell’s brutal conquest and decimation of Ireland. Jonathan Goddard, one of the members of the ‘Oxford Club’ and a physician by training, was appointed as chief physician of Cromwell’s army in Ireland. After two years in Ireland Goddard returned to England as Warden of Merton College, Oxford in 1851. Petty replaced Goddard in 1652 as chief physician and would spend the next seven years in Ireland. In fact Ireland would preoccupy Petty for the remainder of his life, one way or another. Ireland would also be the site of one of Petty’s major achievements in the application of his method of political arithmetic, as it would also be the source of his acquiring the personal fortune he desired. The circumstances which facilitated this were hardly auspicious. By the end of Cromwell’s campaign of ‘appeasement’ of Ireland, great tracts of land were ‘lying unoccupied, or depopulated by the butcheries of Cromwell’s army, and was thus “up for grabs,”’ which as Hutchison sardonically notes ‘the slang is not inappropriate’ (Hutchison 1988: 28). Cromwell’s campaign in Ireland was financially constrained and as a consequence his principal means of payment at the end of the campaign to his financial creditors, mainly the soldiers and officers of his army and sundry supporters, was by allocating to them various portions of the conquered and confiscated Irish land. But in order to implement these allocations an extensive survey was deemed necessary. This job was conducted by Dr Benjamin Worsley, the surveyor-general. However, Petty seizing the opportunity launched an insidious campaign against Worsley’s tardiness in completing the survey, and offered his services to complete the survey in a mere thirteen months. Petty replaced Worsley and with stunning efficiency, in what were very difficult circumstances which included a difficult physical terrain, frequent attacks by an alienated and hostile native population, and Worsley’s continued interference, the survey was nevertheless completed within Petty’s agreed time-frame by March 1656. It became known as the Down Survey and stands as one of the most outstanding exercises in quantitative surveying of its time, setting a benchmark for the methodology, precision and logistical implementation for later exercises of its type. As one of Petty’s biographers’ described it:
The organisation of the Down Survey was on any account a remarkable feat of foresight, administrative ability and penetrating common sense … Only a man of boundless self-confidence would have undertaken such a project, only a man of infinite resource could have designed the bold plan of campaign and improvised the great organisation which it required, and only an administrator of genius could have ensured the practical execution within the stipulated period.
(Strauss 1954: 65)9
Petty invested his earnings from the Down Survey project in the purchase of debentures or land claims from soldiers who mismanaged their newly acquired land allocations. Other Irish lands he acquired as payment-in-kind and still other land he purchased outright. By 1660 he owned estates amounting to 100,000 acres, which at the time made him one of the largest landowners in Ireland. A great deal of his property was, however, seriously encumbered. In addition he was continuously accused of having acquired much of his Irish land by fraudulent means. As a result, Petty, having returned to London in the late 1650s, spent a great deal of the remainder of his life ‘chained to his possessions and his lawsuits, unwilling to buy ease at the cost of surrendering an inch of land’ (Letwin 1963: 120).10
Notwithstanding the problems Petty endured as a result of his investment ventures in Ireland, fraudulent or otherwise, it did not deter him from pursuing an extremely active life on his return to London in the late 1650s and producing an extensive and influential volume of writings.11 Following Hull (1900), Petty’s main economic writings can be arranged chronologically into three groups. Firstly, there were the two major works of the early 1660s, A Treatise of Taxes and Contributions (1662) and the much shorter work Verbum Sapienti, written in 1665 and published posthumously in 1691. The Treatise, ostensibly a work on public finance, engaged fundamental questions of economic theory, including value, price and money. The Treatise remains perhaps his most important work and had Petty written little else, he would arguably be assured of a secure place in the history of economic analysis. Secondly, there were the two major works of the 1670s, which for our purposes highlight Petty’s innovative quantitative approach to socio-economic phenomena. Those were The Political Anatomy of Ireland, written in 1671–1672 and published in 1691, and his Political Arithmetic, written in 1672–1676 and published in 1690. The latter was primarily concerned with a comparative assessment of the economic strengths and weaknesses of England relative to that of her principal rivals at the time, France and Holland. Thirdly, there were a number of works on population estimation, which were to complement and extend the work of his friend John Graunt, whose path-breaking work Natural and Political Observations mentioned in a following Index, and made upon the Bills of Mortality was published in 1662. Other works in this third group included Another Essay in Political Arithmetic (1683) and his pamphlet Sir William Petty’s Quantulumcunque concerning Money, 1682 (1695) and An Essay Concerning the Multiplication of Mankind (1686). Authoritative accounts of Petty’s writings, particularly his economic writing have been provided in the work of Hutchison (1988), Roncaglia (1985, 2005), Aspromourgos (1986, 1988, 1996, 1998, 1999, 2000, 2001a, 2001b, 2005), Amati and Aspromourgos (1985) and Ullmer (2004). Petty’s economic writings, while covering a wide range of issues, can be grouped around a number of major topics. These include national income accounting; the theory of the velocity of money, the theory of the rate of interest; and the theory of value and distribution. We do not propose to address these or Petty’s other economic writing as these are covered extensively in the writings cited above.
Our interest is with Petty’s methodological commitments and the extent to which these are reflected in his practice in the applied domain. More particularly we are interested in establishing to what extent Petty may be credited with being a major figure in the genesis of the quantitative and mathematical approach in economics at a crucial period in the emergence of the European scientific revolution. The debate on Petty’s method has focused on the respective influences of Francis Bacon on the one hand and on the other that of Thomas Hobbes. Advocates of the influence of Bacon on Petty’s approach to political economy will identify Bacon’s method as being characterised by the over-arching framework of induction, for which textual evidence, albeit limited, can be found in Petty. Certainly Bacon himself appeared quite clear about his inductive approach to knowledge. In a letter to King James I of England in 1620 with reference to his Novum Orgarum, Bacon writes that ‘The work … is no more than a new logic, teaching to judge and invent by induction, (as finding syllogism incompetent for sciences of nature)’ (Bacon 1963: 119–120). Petty’s scientific method, according to Ullmer, at least in Petty’s own estimation, ‘was the unvarnished scientific approach developed and promoted by Sir Francis Bacon in the late 16th and 17th centuries’ (Ullmer 2011: 2). In contrast the influence of Hobbes on Petty, which represented an approach based on deduction from a priori premises, was identified by Quentin Skinner in his discussion of Hobbes’s influence on an array of scholars, including Petty, in mid-seventeenth-century France and England (Skinner 1966). In the economic domain Aspromourgos is a leading advocate of the dominant influence of Hobbes rather than Bacon on Petty’s methodological approach to economics. Mary Poovey’s depiction of Petty’s methodological approach is characterised as a ‘complex theoretical amalgam’, which is arrived at by ‘mixing Baconian induction with Hobbesian deduction’ (Poovey 1998: xviii). Ullmer proffers the view that ‘there has been confusion and disagreement among historians of economic thought on how to characterize Petty’s scientific method’ (Ullmer 2011: 2).12
But with respect to Petty’s claim as a progenitor of the quantitative and mathematical approach to economics, three components of Petty’s life and work can be pointed to in order to indicate a substantive and deep commitment to the use and application of mathematics and quantification in economics. There was firstly, already noted in this section, Petty’s exposure to mathematics at the Jesuit University at Caen in the 1630s. Secondly, there was his acquaintance with the Mersenne Group in Paris in the 1640s and his interaction with some of the most distinguished mathematicians, scientists and philosophers of the day. Thirdly, there was the influence of Hobbes for whom Petty worked as secretary during these years. Ullmer’s assessment is surely correct when he indicates ‘that the deductive aspect of Petty’s scientific model owes much to his experience on the continent’ (Ullmer 2011: 11).
This unique exposure during the 1630s and 1640s was not lost on Petty. He developed a clear and deep appreciation of the potential power of mathematics, not only in the natural sciences but also in the socio-economic domain, as providing the presiding framework for reliable systematic analysis. Aspromourgos, quoting from a contemporary of Petty’s, suggests that Petty’s methodological approach to economics could be summed up as follows: ‘that Mathematical Reasoning is not only applicable to Lines and Numbers, but affords the best means of judging in all the concerns of humane Life’ (originally quoted in Hull 1899: vol. II, 513n). But Aspromourgos correctly points out that Petty ‘was sophisticated enough to recognise that even the most precise methods of inquiry would be as vacuous in execution as the concepts to which they were applied’ (Aspromourgos 1996: 57). This assessment perceptively highlights two of Petty’s central methodological tenets which underlay his method of approach to economics, namely the power and precision of mathematics on the one hand and on the other the fragility of the empirical objects to which mathematics would be applied. Petty was quite clear as to the potency and rigour of mathematics, by which he primarily meant ‘arithmetick and geometry’, as being ‘the best grounded parts of speculative knowledge’ in addition to being ‘sure guides and helps to reason, and especial remedies for a volatile and unsteady mind’ (Petty 1648: 144–145).
But if mathematics is the framing model that provides the most reliable methods for reasoned inquiry, the second presiding principle that pervades Petty’s writings is the status of the empirical objects and their quantification without which the use, or misuse, of mathematics will only be to ‘unprofitably apply without resolving needless questions, and making of new difficulties’ (Petty 1648: 13–14). At the level of his methodological commitments, Petty maintained a clear distinction between the benefits of mathematics as the basis of reasoned argument but at the same time the necessity of empirical data was paramount. In the depiction of the scientific method he wishes to apply in the socio-economic domain, under the rubric of his ‘political arithmetic’, he is quite clear as to the requirements and necessity for its empirical foundations. In the following oft-quoted passage he stated:
The Method I take to do this, is not very usual; for instead of using only comparative and superlative Words, and in intellectual Arguments, I have taken the course … to express myself in Terms of Number, Weight or Measure; to use only Arguments of Sense, and to consider only such Causes, as have visible Foundations in Nature; leaving those that depend upon the mutable Minds, Opinions, Appetites, and Passions of particular Men, to the Considerations of others.
(Petty 1690, repr. 1899: 244; italics in original)
Notwithstanding the seductiveness of the relative clarity of the above quotation in depicting Petty’s ‘not very usual’ method of approach, it does convey the full richness or complexity of his methodological commitments. It is not possible to elaborate at length within the confines of a short chapter the full extent of his approach to political economy, including his innovative contribution of political arithmetic, nevertheless a brief sketch of his novel approach must suffice to convey his status as a precursor of the quantitative approach to socio-economic phenomena. Casting Petty as either an uncritical advocate of Baconian induction on the one hand or a Hobbesian deductivist on the other is not, we believe, the most fruitful path to pursue. Rather we would argue that based on a careful examination of Petty’s writings on his methodological approach, albeit scattered across a wide array of his writings and over an extended period of time, a more useful way of approaching Petty is to posit his methodological approach as a quest for an objectivist basis for the quantification and analysis of socio-economic phenomena, thereby undermining the unconstrained intrusion of the subjectivist musings of ‘the mutable Minds, Opinions, Appetites, and Passions of particular Men.’ There is a consistency of continuity in his thinking from his Treatise of Taxes and Contributions (1662), and the three discernible models of production contained in this work which he argued should be analysed in physical terms rather than in nominal monetary terms given his mistrust of the vagaries of changing nominal values of monetary measurements as a secure basis for socio-economic measurement in the first instance. In the Treatise of Taxes he provided ground-breaking analysis of a theory of surplus, the social division of labour and his insightful analysis of the division of labour among other topics. This analysis found favour with Marx, who would confer on Petty the accolade of ‘the founder of political economy’ and with considerable justification in contrast to the ‘vulgar’ political economists as Marx would designate his successors.
It is clear that Petty saw mathematics as the most compelling, robust and reliable method of rational inquiry, arising as noted earlier from his own solid education in mathematics in Caen conjoined with his interaction with some of the leading mathematicians in Europe during his sojourn in Paris as Hobbes’s assistant. But Petty is clear that the application of ‘those most excellent sciences’ of mathematics must be related to ‘empirical objects’, for without such ‘empirical objects’ the use or misuse of mathematics will only be to ‘unprofitably apply without resolving needless questions, and making of new difficulties’ (Petty 1648: 13–14). Petty is here advocating the constructive use of mathematics which will deliver concepts with clarified precision which when combined with reliable knowledge of the empirical world would lead to the emergence and development of new empirical sciences or what Petty termed ‘mixt mathematical arts.’ As far back as 1648 Petty had written that mathematics, by which he meant arithmetic and geometry, was ‘the best grounded parts of speculative knowledge and of … vast use in all practical art’ (Petty 1648: 146). Later he went on to elaborate that with respect to the method to be applied in analysing well-defined concepts, he favoured what he termed the ‘algorithme of Algebra’, which he explained was ‘a kind of Logick’ with the aid of which ‘not only numbers but the several species of things’ could be manipulated and analysed. This ‘kind of Logick’ which facilitated the application of algebra was made possible by designating the array of different ‘species’ by letters or similar type characters or more generically by an agreed set of symbols. We concur with Aspromourgos’s broad conclusion that this ‘is the particular mathematical method which in Petty’s opinion is paradigmatic for politico-economic analysis: a calculus of symbolic forms which finds concrete expression in arithmetic’ (Aspromourgos 1996: 59). Petty’s aim appears to be informed and framed within the framework of the ‘Algorithme of Algebra’, which was to be applied not only to purely mathematical matters, but in his economic work the principal domain of application was to be the realm of policy. This was to be achieved by the translation of as many terms as possible in this domain into number, weight and measure, which provided the more secure empirical grounding of these terms and then to be analysed mathematically. This was the kernel of what Petty termed his ‘political arithmetic’.
Was there then an underlying set of informing ideas that provided a coherent methodological basis for Petty’s political arithmetic? We would argue that there was and it included an approach that would justify the identification of Petty as a significant precursor, perhaps the earliest, in the genealogical line in the quest for the mathematisation of socio-economic phenomena. The principal components of his methodological approach could be summarised as follows. Firstly, his overall approach to the systematic inquiry of economy and society was implicitly and explicitly based on mathematics, which as we have indicated earlier is hardly surprising given his early education and later his acquaintance with leading mathematicians. Secondly, as he makes clear in the Preface to his Political Arithmetic (1690), there is the emergence of a clear commitment to an objectivist approach which sought to undermine the damaging effect of exaggeration, hyperbole and the influence of unconstrained subjectivism in the domain of the socio-economic. This is patently clear from his desire to deploy his newly articulated method, a method ‘not yet very usual’, which through the primacy of ‘Number, Weight, or Measure’ would undermine and replace ‘comparative and superlative words’, through the discipline and constraints of treating matters in terms of ‘Number, Weight, or Measure.’ Thirdly, this primacy of ‘Number, Weight, or Measure’ as advocated by Petty reflected his position as an empiricist which is a more accurate and correct designation of Petty rather than his depiction as an unreconstructed Baconian inductivist. Petty was acutely aware of the need to ground, where possible, the concepts to be encountered in the domain of socio-economic policy in a solid empirical base. But these empirically established concepts and ideas were to be negotiated and analysed through the medium of already constructed theoretical concepts whose function was to provide a coherent explanatory framework where possible.
Petty’s political arithmetic was an innovative, challenging and subtle methodological framework, which was motivated by pragmatic considerations with a view to enhancing the ‘art of statecraft.’ But as a methodological pioneer Petty constructed a conceptual framework which in its economic content sought firstly to furnish as precise as possible empirical estimates which in principle, if not always in practice, reflected the theoretical concepts used to provide a coherent explanatory framework informed by the methodological principles identified above, and to be used in the interest of policy-making and implementation. Petty was also alert to the value of comparative studies of the historical economic circumstances and performance in actual economic case studies, as reflected in his detailed analysis of Holland and Ireland in his Political Anatomy of Ireland (1691). In the absence of the framework contained in his political arithmetic he saw governments pursuing policy which in his view was metaphorically equivalent to throwing dice, a metaphor he invoked on numerous occasions in his writings. For him this was surely not the basis for conducting policy exercises on experimentation in the socio-economic domain. In his own words Petty provided a succinct and insightful summary of his methodological objectives: ‘God send me the use of things, and notions, whose foundation are sense and the superstructures mathematical reasoning; for want of which props so many Governments does reel and stagger, and crush the honest subjects that live under them’ (Lansdowne 1927: vol. 1, iii).
In this brief statement we see the three principal pillars of Petty’s political arithmetic as a guiding methodological framework for the emerging discipline of political economy: the imperative of the empirical foundations; the application of ‘mathematical reasoning’; and all with a view to informing and guiding public policy. Had Petty’s political arithmetic been nurtured and developed in the ensuing decades political economy may have been spared some wrong turnings at critical junctures in its troublesome methodological trajectory. Be that as it may, it can be argued that Petty deserves to be acknowledged as a pioneering and significant figure in any extended contextualisation of the complex process of the mathematisation of economics. Following his death, his political arithmetic was not to enjoy the attention or development that might have been expected or deserved.13 In any event by the beginning of the eighteenth century the centre of activity was moving to France and the emergence of major developments in that country. France was to play a pivotal role in the process of the mathematisation of economics in one way or another during the course of the eighteenth century within the context of the Enlightenment, a role that extended into the nineteenth century culminating in the contribution of Leon Walras, whose work represents for many the agreed starting point in the conventional narrative of the mathematisation of economics. In the next section, however, we will provide an account of the French intellectual and technical contribution to the project to ‘mathematise’ economics, which was initiated long before Walras.
Hutchison (1988), in his seminal study of the period 1662–1776, distinguishes between two distinct phases which witnessed the production of important publications in the domain of ‘trade’ and ‘commerce’ which eventually transmogrified into the term ‘political economy.’ The first phase, for Hutchison, was that of the 1690s and was dominated by English contributions, which as he argues was ‘given such a decisive initial impetus by Petty’ and which ‘culminated in a remarkable concentration of important writings.’ However, by the turn of the century and for the first forty years of the eighteenth-century the ‘English advance rather suddenly and markedly slowed down’ and ‘Paris, to a significant extent, replaced London as the place of publication of major works’ (Hutchison 1988: 185). However, from the late 1740s a remarkable number of pivotal works were produced which had a distinctly international character, mainly European, which were marked by the ‘convergence upon … certain fundamental ideas and theories and … by the emergence of a new kind of systematization’ (ibid.: 185). Here Hutchison makes an interesting distinction between two phases in the eighteenth-century: the first he terms the ‘international mid-century efflorescence’ (1747–1755), which was followed by the second phase, ‘the French pre-eminence’ (1756–1770).
The international dimension of the first phase is justified with even a cursory perusal of the works produced in this phase. In 1747 Francis Hutcheson, who was a teacher of Adam Smith, published his Introduction to Moral Philosophy. The following year Montesquieu’s pivotal book, L’Esprit des Lois was published, and while not a contribution to political economy per se, it provided an influential philosophical framework with which to approach political and social analysis and represented a major intellectual contribution from the culture of the Enlightenment.14 Between 1749 and 1752 major contributions came from different parts of Europe, which included from Italy Galiani Della Moneta (1751), from Scotland Hume’s Political Discourses (1752) and from England Josiah Tucker’s early economic writings. But it was in 1755 that saw additional major contributions that shaped the intellectual, conceptual configuration and content of the emerging discipline of political economy. These included Tucker’s most important work, The Elements of Commerce, along with Johann von Justi’s Staatwissenschaft, considered one of the most engaging contributions from the German cameralist tradition.15 But what has come to be regarded as one of the most important and innovative works in the history of economic thought made its appearance in French in 1755, written by the enigmatic but extraordinary accomplished Irishman Richard Cantillon, whose Essai sur la Nature du Commerce en Géneral, holds a distinguished place in the pantheon of canonical works in the history of economics.16 But these developments, as Hutchison acknowledges, owe a great deal to the writers of the later seventeenth-century with Petty as a central figure. As he noted:
it should be emphasized and re-emphasized, that neither the mid-century efflorescence, nor the great phase of French pre-eminence which followed directly after, could have happened without the intellectual platform provided by the work of Petty and writers, mostly English, of the latter part of the seventeenth-century.
(Hutchison 1988: 187).
Developments in eighteenth-century France and in particular from the mid-century period of French pre-eminence in Hutchison’s terminology were increasingly embedded, informed and shaped by the Enlightenment. The momentum to pursue a radical intellectual programme of the systemic analysis of a reconfigured society and economy was central to this endeavour. Integral to this programme was the commitment to the application of mathematics to the social and economic domains. Ingrao and Israel (1990) in their outstanding and authoritative study of general equilibrium theory succinctly capture the situation at this time when they noted that the: ‘Historiography of philosophical thought has long identified the “mathematization” of the social sciences as one of the major themes of contemporary culture generated and molded in the rich melting pot of the Enlightenment’ (Ingrao and Israel 1990: 34).
The use of the term ‘melting-pot’ to describe the Enlightenment is indeed resonant of the vibrancy, intellectual ferment and the contention of ideas that constituted the Enlightenment, an altogether too vast a spectrum to be engaged here.17 However for our purposes in this chapter in attempting to trace a ‘lineage’ in the complex trajectory of the ‘mathematization’ of political economy, a number of key strands of developments within the ‘melting pot’ are central to our endeavour.
Critical to the contextualisation of developments in France at this time is the role and influence of Newton’s contribution, or more specifically the dissemination of Newton’s ideas within the scientific and intellectual communities of eighteenth-century France and Europe in general.18 As is well documented the conflict between Newtonianism and the Cartesian heritage in France included the use of Newtonianism as a weapon to gain intellectual primacy among the elite academic, philosophical and political communities. Voltaire in his Elements de la Philosophie de Newton (1838) deployed Newtonian physics as the intellectual counterpoint against the metaphysical components of Cartesianism (Rattansi 1982). The ‘Newton Wars’ as they came to be called were launched early on in the Enlightenment and they remain a vibrant area of current research (Israel 2006; Shank 2008). Of crucial interest here was the divergent reception of Newtonianism in France and England. For Ingrao and Israel, Newtonianism ‘was to assume totally original features in France.’ But interestingly they argue this arose in no small part from the interaction of Newtonianism with Cartesian deductivism which they argue led to a ‘moderate rehabilitation’ of ‘metaphysics’, a process which tempered the ‘empiricism of Newtonian natural science’, particularly in the work of D’Alembert. The outcome of this process of interaction between the contending approaches in France resulted in their adoption of a physico-mathematical approach which prioritised the development of the mathematical mode of presenting Newton’s body of work, which imparted a powerful impetus within the French intellectual and scientific communities to pursue this model of approach to most, if not all, realms of knowledge. Ingrao and Israel capture very perceptively the far-reaching consequences of the success of Newton’s physico-mathematical approach as developed in France and the benchmark status conferred on it with the philosophy of the Enlightenment when they write:
It is therefore hardly surprising that for a scientist like D’Alembert – but also for so many other thinkers of the Enlightenment or cultured men inspired by its philosophical ideas – the scientific intellectual became the model intellectual and the scientific community the model for scholarly communities. In the reformist view of the values and decrepit institutions of absolutism, Newton’s scientific philosophy and the model of the scientific intellectual established in France became points of reference for an ideal renewal of the whole of society. In its new Newtonian garb, science puts itself forward as the center of society and driving force of reform, promising new horizons in all fields of knowledge to which the new methods of scientific thought could be applied. This scientistic (in the full and broad sense) vision was thus projected beyond the confines of traditional science and under the urgent promptings of institutional, economic and social problems – first under the Ancien Regime and then during the Revolution – the question of the scientific government of society and the economy achieved full status also in theoretical terms.
(Ingrao and Israel 1990: 35–36, italics in original).
Meanwhile in England the version of Newtonianism that was pursued was dominated by small-scale empirical studies, which ‘led them into repetitive and fragmentary studies’ (ibid.: 35).19 These developments were reflected in the status of the prestigious institutions in both countries, with the Royal Society going into decline at this time, while the French Académie des Sciences went on to become the leading learned society in Europe.
Newtonianism clearly represented a presiding and major source of intellectual influence in France during the course of the eighteenth century. Mainly due to the influence of Voltaire, arising in particular from his Eléments de la Philosophie de Newton (1738), Newtonian philosophy of science dominated the larger cultural and social milieu of French intellectual life, becoming in effect the recognised method for scientific reasoning, particularly in its physico-mathematical guise. This was cogently articulated by Ernst Cassirer in his classic study of the Enlightenment:
The philosophy of the eighteenth century takes up this particular case, the methodological pattern of Newton’s physics, though it immediately begins to generalize. It is not content to look upon analysis as the great intellectual tool of mathematico-physical knowledge; eighteenth century thought sees analysis rather as the necessary and indispensable instrument of all thinking in general. This view triumphs in the middle of the century. However much individual thinkers and schools differ in their results, they agree on this epistemological premise.
(Cassirer 1951: 12).
The ‘universal twist’ imparted by Newtonian science arose from its success in articulating and representing the universe as perceived by Newton through the medium of mathematical analysis. The impetus thus imparted by Newton’s approach and methods of analysis stimulated hopes that similar types of results could be achieved across all areas of knowledge including the human and social branches (Cohen 1994). But the difficulties of an uncritical generalised application of ‘rational mechanics’ to all branches of knowledge were identified from early on. As was the emergence of an ‘unresolved tension’ in Enlightenment thinking between the promise of Newtonian science to provide scientific criteria of verification as alternatives to metaphysical or ‘revealed truth’ thinking. Similarly the difficulties encountered in its application to fields of knowledge outside those of natural phenomena were evident to many writers, including some who were major advocates of Newtonianism (Ingrao and Israel 1990).
But the influence of Newtonianism in France was greatly buttressed and enhanced by writers of immense influence within the broader culture of the Enlightenment. Montesquieu was certainly influenced by the overall approach of Newtonian physics, an influence that can be detected in his influential Esprit de Lois of 1748, whose central focus was on the analysis of the relative merits of alternative types of institutions of governance such as monarchy, republicanism, and despotism. But he also grappled with the more fundamental processes of social harmony and the achievement of equilibrium among the contending forces and clash of interest with the body politic and the broader society. These concepts and issues paralleled the analysis in ‘rational mechanics’, and would feature as pivotal in the development and content of mathematical economics later in the nineteenth and twentieth centuries. Montesquieu is often credited with being the father of sociology as it later came to be termed, but more fundamentally the Esprit des Lois could be viewed as an early attempt to identify the processes and mechanisms that operated within society and how they contributed to the achievement of an ‘equilibrium’, albeit imperfect, which would provide the conditions for enhanced political freedom in a progressive society. Montesquieu can also be attributed with the early search for the existence of laws in the social domain, reflecting again the influence of Newton’s work in physics. But Montesquieu’s writing on this topic is a complex story which involved the inherited influence of the natural law theorists of the seventeenth century, such as Pufendorf (1632–1694), and a more empiricist approach to social laws, which would later dominate Comte’s positivism in the nineteenth century. While neither the earlier natural law theorists nor later Montesquieu and his followers forged any attempt to represent social laws in mathematical terms, they imparted to the general culture of the Enlightenment the desirability of pursuing the quest for establishing social laws which held out the prospect, if not the promise, of articulating laws parallel or analogous to those provided by Newton in the physical domain.
While Montesquieu’s Esprit des Lois, was not in any way a contribution to political economy, it set critical parameters for a new philosophical approach to social theorising and the search for ‘laws’ in the social domain. However, the explicit articulation of a more scientific approach to the analysis of socio-economic phenomena emerged with the Physiocratic School of Political Economy in France. Its leader was Francois Quesnay (1694–1774) who along with Turgot (1727–1781) represent the pivotal figures in the development of political economy at this time. We do not propose to address their pioneering contributions in the many areas of political economy to which they contributed, since these are comprehensively treated in the extensive literature on the history of economic thought.20 It must be noted however that Quesnay was imbued with the Enlightenment philosophical commitment to the progress of humanity in all respects, central to which was the provision of the goods and services necessary for material well-being, and this was to be achieved by enlightened reason based on a science of natural order informed by the contribution of natural law. Quesnay came to view economic science, with its central focus on the production and distribution of goods and services, as ‘a great science, and the very science upon which the government of society rests’ (Ingrao and Israel 1990: 44).
Quesnay’s methodological approach to this ‘great science’ was not going to be based on the outcome of history as a primary source of insight: ‘We seek no lessons from the history of nations or of mankind’s blundering, which represent for us only an abyss of disorder’ (ibid.: 43). Nor was he attracted to the use of algebra as a mode of representing his economic ideas, and like many of his contemporaries his view of mathematics primarily embraced arithmetic and geometry as the central components of that discipline. He did however assign great importance to extensive and rigorous calculation, which emanated in the construction of his celebrated ‘Tableau Économique’, first published in 1760. This was an elaborate arithmetical scheme representing production and distribution within an economy using hypothetical data. It was also to facilitate the investigation of taxation measures on the part of the central authority with a view to evaluating the impact on the national well-being of the nation. Similarly the impact of external events on the economy could be investigated and evaluated. While the Tableau is viewed as the seminal contribution of Quesnay and the Physiocratic movement, Quesnay’s elaborate calculations were related to a systematic conceptual structure, which represented an innovative model of economic growth in which Quesnay demonstrated how surplus emerged in the production system and how the allocation of this surplus could contribute to future economic growth. The unique feature of the Physiocratic model of economic growth was that it referred exclusively to land as the exclusive source of surplus. Quesnay in fact argued that surplus could not be produced in the manufacturing sector. Within his conceptual framework, albeit limited to the agricultural sector, Quesnay also provided original insights into the role of the entrepreneur and the crucial role of the provision of ‘advances’ to producers as part of his explanation of the economic growth process.
Quesnay’s contribution to the formalisation of economics, narrowly interpreted, may be deemed to be modest but his contribution to the quantitative schematisation is altogether more significant; Ingrao and Israel point out that:
On more than one occasion Walras, though little given to generosity in acknowledging his ‘forerunners,’ evinced a predilection for the Economistes, who he indicated as the school that had laid the most correct foundations for the subsequent mathematical theory of general equilibrium
(Ingrao and Israel 1990: 45)
The Tableau was a major source of inspiration in the twentieth century to Leontief’s input-output analysis of the United States economy in 1941 (Leontief 1941: 9), which generated the extensive input-output analysis that featured as a major strand of quantitative economic research in the post-World War II period. The Physiocratic contribution, and in particular that of Quesnay, laid the foundations for the intensification of the schematic quantification and increasingly formal analysis of the economy and society that would follow in the latter part of the eighteenth and into the nineteenth centuries.21
Closely associated with Quesnay was his contemporary Turgot (1717–1781), who contributed many original insights to the emerging discipline of economics. Though never the creator of a large-scale framework of analysis compared to Quesnay, he shared the commitment to the systematic measurement of socio-economic phenomena as the necessary basis for the pursuit of informed and rational policy decisions. Turgot’s major work, the Réflexions sur la formation et la distribution des richesses (1769–1770), which was translated in 1793 as Reflections on the Production and Distribution of Wealth, developed his theory of capital and its role in the economic growth process. As a result of Turgot’s work, capital was now perceived as a crucial input along with land in the structure of production and economic growth. He also developed a model of interest rate determination based on the supply and demand of loanable funds, and traced the role of low interest rates in facilitating economic growth. Turgot did not contribute any formalisation of his many original contributions, but perhaps his more indirect influence on the process of formalisation may have come from his close friendship with Condorcet who was to contribute an explicit momentum to the Enlightenment project of the analytic schematisation of the socio-economic domain.22
In the process of the ‘mathematisation’ of economics within the broader culture of social reform and progress of the Enlightenment, Condorcet (1743–1794) occupies an interesting and significant position (Baker 1975; Rothschild 2001). A mathematician of outstanding ability and a committed adherent of the Enlightenment values, he was well placed to engage the potential application of the physico-mathematical approach to provide insight into the framing of a more rational order in the socio-economic domain. His originality and innovative contribution at a technical level was the central position he gave to the application of the probability calculus to socio-economic analysis, rather than the choice of the infinitesimal calculus of Newtonian analysis. If the probability calculus was Condorcet’s technical technique of choice, this was informed and embedded within his larger conceptual framework which attempted to reconcile a science of society which would combine a rigorous objective status, similar to that of the physico-mathematical sciences. As in the physical domain, scientific laws would be articulated and would elaborate the empirical regularities, systematically observed. However, at the same time, it was to be recognised that the ‘material’ to be analysed was the outcome of the subjective choice of free social agents based on their autonomous capacity to choose. In this latter ‘subjectivist’ component of his conceptual framework Condorcet was greatly influenced by Turgot. Condorcet’s choice of the interesting term ‘social mathematics’ was perceptive in conveying what was central to his conception of the new sciences of society, the conjoining of the rigour of mathematics to the complex domains of the socio-economic world.
From 1780 he devoted his efforts primarily to the articulation of the social sciences, which he argued needed to pursue a path of development that would lead to the achievement of a new intellectual standing. But this would have to combine both a well-established empirical basis conjoined with the rigour of the physico-mathematical sciences, a theme that was resonant of Petty’s earlier efforts and which was now a presiding and integral part of the Enlightenment culture. In a telling speech delivered in his Discours de Récéption delivered at the Académie des Sciences in 1782 and quoted in Ingrao and Israel (1990), Condorcet’s provides a potent statement of his position and ambitions for the future of social sciences and the approach to be pursued:
These sciences, whose object is man himself and whose direct end is man’s happiness, are now almost established and their developments will be as certain as that of the physical sciences. The cherished idea that our grandchildren will surpass us in wisdom is no longer an illusion. Whoever reflects upon the nature of the moral sciences cannot, in fact, but see that, supported by factual observation like the physical sciences, they must follow the same method, acquire an equally precise and exact language, and attain the same degree of certainty.
(Ingrao and Israel 1990: 50)
During the course of the 1780s and into the 1790s, before his premature death in unfortunate circumstances in 1794, Condorcet produced a number of important works which elaborated his project of ‘social mathematics’, which was inclusive of all the problems of the socio-economic realm that needed to be addressed, including the political and moral sciences along with political economy which he viewed as a particular discipline within the larger domain of the social sciences. Condorcet’s project maintained the stream of thought that distinguished between the normative and descriptive dimension of Enlightenment thinking. But for a variety of complex reasons Condorcet’s attempts to mathematise the ‘social sciences’ in line with his innovative conceptual and mathematically probabilistic approach did not emerge victorious, due in part to the tensions and weaknesses within his conceptual framework but also in part to the major political and cultural changes that emerged with the decline of the hegemony of the revolutionary period and the arrival of the Napoleonic era.
In the period following this political and cultural shift the development of ‘social science’ took two different paths of development. The lineage based on the foundations of ‘political arithmetic’, or later ‘economic arithmetic’, along with Condorcet’s social mathematics continued to enjoy support and further development at least for a comparatively short period, particularly in the class of Moral and Political Sciences of the Institut de France, which was established following the suppression of the Académie des Sciences, where it still found a hospitable environment. However, by the end of the eighteenth century critical voices were raised against the suitability of the Newton-inspired mechanical model and its corollary of the physico-mathematical methods of analysis to provide an adequate explanatory framework for a ‘science of man.’ These criticisms began to emerge also in certain areas of the natural science community such as the biological study of natural-life processes. But the more serious, some might even term it the fatal, blow to the programmatic framework of the mathematisation of the socio-economic domain informed by the Enlightenment values and in particular Condorcet’s social mathematics came with the increasing demise of leading mathematicians of the day such as the towering figures of Lagrange and Laplace who had shown genuine interest and support for Condorcet’s social mathematics. However, on Lagrange’s death Laplace assumed the leading role within the scientific community and in particular with the very prestigious Class of Geometry in the Institut de France. Whatever the basis for his earlier support for Cordorcet’s social mathematics, his opposition and intensifying hostility became quickly apparent which found expression in his ‘purging’ of the remnants of the project and people associated with it.
The intellectual consequences of these developments when conjoined with the instalment of the new Napoleonic political order was to reorientate the trajectory of the development of the social sciences. The idealism, intellectual innovativeness and the inclusiveness of the social sciences were undermined. The focus of Newtonian philosophy was to be retrieved as it were as the exclusive focus on natural phenomena and based on the methods of the physico-mathematical sciences. If the social sciences were to pursue their development, then it was to be under the shadow of the correct application of the canonical methods of the physico-mathematical sciences and methods, as reflected for instance in the infinitesimal calculus predicated on the deterministic framework of analysis. The courageous attempts by Condorcet to provide a ‘probabilistic’ foundation of analysis in response to the reality of the socio-economic world which he perceived was effectively to be discarded. In the domain of analysis the new presiding ethos was conformity to the dominant mode of a deterministic methodology dictated by the Newtonian-inspired analytical framework and its arsenal of approved methods and techniques. In the specific domain of political economy, the prestige accorded to Adam Smith’s Wealth of Nations, with its mixture of historical examples, components of inductive reasoning and the use of pragmatic case-studies and well-marshalled empirical supporting data was perceived as the paradigmatic way forward. This was in contrast to the search for schematic search for the rational ordering of society based on the articulation of ‘social laws’, both explanatorily descriptive laws along with normative rules to guide a rational and progressive society, the framework for which had been provided in large part in the work of Montesquieu, the Physiocratic contribution and that of Condorcet.
Apart from the later strident hostility of Laplace to the project of the ‘mathematisation’ of areas outside the natural domain, a major source of opposition within political economy in France came from the emergent French Liberal School of political economy and in particular from J.B. Say (1767–1832), its effective intellectual leader.23 This school of thought was the dominant influence on French economic thought for most of the nineteenth century, and adherents to its central doctrines could be found in the United States and Italy and among prominent British and German economists. For our purposes of attempting to outline the continuity of linkages, or lack thereof, that sustained the intellectual mind-set and the pursuit of the mathematisation/formalisation of economics, the emergence of the French Liberal (or Classical) School represented a major intellectual resistance to this project. Say was personally opposed to the importation of mathematics into the shifting view of ‘social science’, including political economy, and spear-headed the opposition to the project. The contributing factors to account for his opposition and ensuing influence are varied and complex. He was arguably picking up on the shift in the conception of the proper approach to the study of the social realm and the need to liberate this domain from the dominance of the physico-mathematical approach to all realms of knowledge. In this the opposition of Laplace, referred to above, may have been an important contributing factor. By the end of the eighteenth century the pathways of the future development of the social sciences were diverging in France. Say and his colleagues, collectively known as the ‘Idéologues’, including Destutt de Tracy, Garnier and later such notables as Bastiat, Rosso, Courcelle-Seneuil and many others who aligned themselves with the French Liberal School, criticised the Physiocratic legacy along with that part of the Enlightenment inheritance that pursued the schematisation of political economy informed by the aims and methods of Newtonianism and the use of physico-mathematical sciences. Their critique of the Physiocrats and associated endeavours included the overly abstract conceptual schemes of analysis, the inadequacy of extensive observational studies and their propensity for excessive speculative analytical thinking. Ingrao and Israel (1990), quoting the work of Moravia (1974) who provides an intriguing and insightful account of Say’s conception of what should inform the construction of political economy:
Neither mathematical calculation nor the abstract disciplines in general are able to provide a really valid explanatory heuristic model. Rational mechanics, algebra, and logic achieve certain results only insofar as they are based on ideal data divorced from living reality. Now, political economy proceeds, and can only proceed, on the terrain of real facts. It must take into account all the concrete data, even when imponderable or variable, to which the exact sciences (statistics and political arithmetic among them) pay no heed. The sciences Say regards as analogous to political economy are of another type. As the science of the living corps social, he regards it as qualitatively no different from physiology, the science of the individual body.
(Moravia 1974: 782–783, quoted in Ingrao and Israel 1990: 59)
Say’s opposition to the mathematisation of economics was in the opinion of Ingrao and Israel pushed, in their estimation, to an extreme position of rejection by Say ‘to the point of the idiosyncratic rejection of mathematics tout courts’ (Ingrao and Israel 1990: 60). But ‘idiosyncratic’ may not quite convey what was in effect a committed and fundamental philosophical and methodological position on the part of Say and the French Liberal School.
Say’s own pronouncements convey on the one hand what appears as a more modest aim on his part with respect to the advancement of political economy, namely the promotion and advocacy of Adam Smith’s Wealth of Nations. Smith’s work was viewed as providing a more convivial intellectual approach to the grounding of political economy in a philosophy of inter-personal relations which generated outcomes that reflected pathways to social harmony and the achievement of something approximating to overall equilibrium. Within the Wealth of Nations, Smith extolled the acceptability and virtues of laissez-faire, free trade, competition and specialisation in production, as the economic and social means of achieving the larger goals of social harmony, individual liberty and political stability, characteristics of an ordered and civilised society for the new ‘commercial society’ which was emerging with the Industrial Revolution.
The Wealth of Nations synthesised brilliantly a great deal that had gone before, not only in Britain but more significantly in France with which he was acquainted. It also contained many original insights and attempts to articulate particular topics. But notwithstanding these achievements, it was hardly a paragon of organisation and clarity and was in many respects an inconsistent and understandably an incomplete work. This was how it was viewed by contemporaries following its publication and these included Say among others. The need to address its weaknesses and to provide modifications, extensions and corrections to Smith’s magnus opus became a major programme of activity among leading figures in political economy, and in the struggle for apostolic succession as to who would succeed Smith by virtue of their contributions to a more clarified articulation of his theories redounded arguably to three principal contenders, namely Say in France, and Malthus and Ricardo in Britain. All three were informed by different perspectives on the future of political economy. We do not propose to pursue here the specific differences of approach of the three figures with respect to the content of their economic theories.24 Of interest to us is the differences that quickly emerged at the methodological level, particularly between Say and Ricardo. Say opposed Ricardo’s adoption of a deductivist methodology, which became the latter’s presiding approach to economic theorising, an approach which exerted very considerable influence during the course of the first third of the nineteenth century. While Ricardo did not himself use mathematics in his economic writing, his committed and consistent use of a powerful deductivist structure of argumentation facilitated the translation of his work into a mathematised mode of presentation, which was undertaken later within mathematical economics (Pasinetti 1974).
In Say’s own words he indicts not only Ricardo and the Ricardians but the legacy of Quesnay and the Physiocrats:
Without referring to algebraic formulas that would obviously not apply to the political world, a couple of writers from the eighteenth century and from Quesnay’s dogmatic school on the one hand, and some English economists from David Ricardo’s school on the other hand, wanted to introduce a kind of argumentation which I believe, as a general argument to be inapplicable to political economy as to all sciences that acknowledge only experience as a foundation.
Continuing his specific critique of Ricardo, he accuses him of setting truth
in a hypothesis that cannot be attacked because, based on observations that cannot be questioned, he imposes his reasoning until he draws the last consequences from it, but he does not compare its results with experience. Reasoning never waivers, but an often unnoticed and always unpredictable vital force diverts the facts from our calculation. Ricardo’s followers … considered real cases as exceptions and did not take them into account. Freed from the control of experience, they rushed into metaphysics deprived of applications; they have transformed political economy into a verbal and argumentative science. Trying to broaden it they have led only to its downfall.
(Say 1971: 15)
Ricardo would later be accused of perpetrating the ‘Ricardian vice’ through the deployment of assumptions that lacked empirical adequacy and the derivation of conclusions that purported to accurately depict the future empirical course of events. In the above depiction of Ricardo and his followers, Say is accusing them of a Ricardian reductionism decoupled from experience that misdirected them into the realm of metaphysics. Say and the emergent French Liberal School along with its doctrines, theories and overall vision of political economy prevailed in France, and as the nineteenth century progressed its dominance became firmly established. Its influence was pervasive, particularly within other higher educational domains, including within scientific and learned societies, and in the direction followed by the leading economics journals of the day (Arena 2000). The intellectual, academic and educational environment became less and less hospitable to the project of the mathematisation of the social sciences, and in particular the economic domain. Notwithstanding the complex factors that led to the tectonic intellectual shift in the nineteenth century, particularly in France, by the 1870s what was to emerge was in the form of Walras’s contribution, which is generally accepted within the conventional interpretative narrative as the pivotal development that launched the resuscitation of the mathematisation of economics that would eventually lead to its intellectual dominance within the economic academy in the course of the twentieth century. In the next section we will provide a brief overview of what is now a well-documented narrative over the course of the twentieth century even as the explanation for the dominance of mathematics remains an understudied topic as both a historical process and even more significantly the study of the methodological implications of the importation of mathematics into economics particularly given the changing character of mathematics itself.
Notwithstanding the formidable cultural, sociological and technical forces that had emerged by the late eighteenth century in France against the endeavours to ‘mathematise’ the social domain, which had congealed into a very influential position during the course of the nineteenth century, the commitment to continue the efforts to mathematise the social sciences continued. The efforts to sustain this were now represented by individual contributions and lacked the approval or support of the official establishment or academic community. However, the source and strength of the opposition to this commitment grew stronger and more influential. The almost total withdrawal of intellectual and academic support of the scientific community for the project of furthering the mathematisation of the social sciences, conjoined with Laplace’s insistence on the prescriptive correctness of the paradigm of mechanics as the correct conceptual approach, with its corollary of a deterministic methodology based on the use of infinitesimal calculus, provided the presiding mathematical framework for scientific analysis. More specifically within political economy the emergence of the French Liberal School and the influence of J.B. Say during the course of the nineteenth century and the explicit philosophical opposition to the methodological pursuit of the formalisation of political economy made for an extremely hostile and inhospitable environment within which to pursue the mathematisation of political economy or the social sciences in general.
Against this background the accepted interpretive narrative of the period identifies a small number of individuals in France who at this time made original contributions that have earned them a place of some distinction in maintaining the link with the efforts to further the mathematisation of the social sciences. This was something of a solidary track for these writers who are generally designated as being ‘precursors’ of Walras, whose seminal work emerged in the 1870s and represented a watershed in the evolution of mathematical economics. These ‘precursors’ need not detain us beyond some very brief comments to acknowledge their place in sustaining the lineage of mathematical economics in difficult circumstances. A number of them belonged to what is referred to as ‘engineer economists’, which were not uncommon in nineteenth-century France, arising from a certain pre-eminence conferred on engineering within the French academic tradition from the eighteenth century and into the nineteenth century. Their mathematical training as engineers along with their technical contributions to the development of the physical dimensions of the French economy and its implication for investment, public finance and in particular taxation, all provided the conditions that led to their involvement in the intersection of engineering and economics.
Achylle-Nicolas Isnard (1749–1803) was such an engineer-economist whose main contribution was contained in his Traité des Richesses of 1781.25 In this he dissented from the central Physiocratic doctrine, which maintained the primacy of the agricultural sector as being the only source of economic activity that could produce a net surplus for the economy. Isnard, in contrast, argued that industrial activity was also capable of producing a surplus, which had significant implications for distribution within the economy in that this surplus accrued not only to landowners, as argued by the Physiocrats, but to all the owners of the productive resources that had contributed to the production of the surplus, with their remuneration being determined by the scarcity of their resources. Walras would later acknowledge Isnard’s contribution to his own thinking in the formulation of general equilibrium theory.
Jules Dupuit (1804–1866), like Isnard, was also a good example of the engineer-economist tradition in nineteenth-century France (Ekelund and Hébert 2000). While civil engineering was his central occupation, rising to become the chief municipal engineer in Paris in 1850, his interest in political economy arose from his interest in industrial development and issues arising from this wide-ranging topic. Dupuit’s central claim to fame is based on his contribution to the concept of utility, more particularly its measurement which Dupuit regarded as pivotal for political economy (Stigler 1965). Dupuit differed from Say and the English classical economists on this topic and provided a number of original insights which essentially combined the idea that utility was to be thought of as being fundamentally linked to the quality of a good or service that satisfied some desire while at the same time acknowledging as an integral part of his analysis that utility varied from individual to individual along with the quantities of the good or service consumed by the economic agent. The implications of these insights and their formalisation by Dupuit would provide the foundations of the neoclassical paradigm of economics that would emerge in the last third of the nineteenth century, including that of Walras’s general equilibrium theory. Dupuit represents in his work, mathematically articulated, ‘an important link between the ideas of the first generation of expounders of economic arithmetic and social mathematics and the generation of those scholars who were to contribute most directly to the founding of general economic equilibrium theory’ (Ingrao and Israel 1990: 77). His methodological approach was informed by the Newtonian framework and his desire to provide a formal foundation for political economy based on the physico-mathematical sciences, an ambition that was shared by Walras. Dupuit’s work greatly facilitated the ‘mathematisation’ of the new mainstream paradigm that was to emerge later in the nineteenth century, including that of Walras.
Two other names that have earned their place in the designated ‘gallery’ of precursors to Walras in nineteenth-century France include Nicholas-Francois Canard (1750–1833) and Augustin Cournot (1801–1877). Canard was trained in science and was recognised as a mathematician of some standing. His interest in socio-economic issues arose from his interest in particular issues which involved social mathematics, i.e. his attempt to provide a mathematical solution to the problem of the outcomes of the legal judgments of the court system in France and a number of issues of political economy (Baumol and Goldfeld 1968). Unlike the Physiocrats, Canard subscribed to the doctrine that labour was the source of all wealth. Canard distinguished between three categories of income: landed income; industrial income; and income on movable property. From the dynamics of the circulation of income and money and heavily influenced by the laws of liquid dynamics and the theory of the circulation of blood, Canard contributed original and influential insights on the achievement of equilibrium. He also addressed the topic of price determination and is credited with the first application of marginal analysis, all of which were informed by his ambition and optimism in applying the methods of physico-mathematical methods to political economy, sustaining the commitment to the ‘mathematization of political economy’ (Larson 1989). Canard experienced in his lifetime sustained opposition from a variety of sources, not least from Cournot, arguably the most celebrated of Walras’s precursors.
Augustin Cournot (1801–1877) would, within the historiography of political economy, be regarded as the premier mathematician-economist within the French tradition who made major analytical contributions to the emerging framework that would later feature prominently in the neoclassical paradigm.26 In the domain of mathematical economics, he was the first to formulate the method of applying functional analysis to problems in the socio-economic realm. The use of functional analysis was attractive to Cournot by virtue of the fact that it required, in his analysis, the specification of no more than the most generalised properties of the functional forms which were to describe the underlying law reflected in the articulated functional equations. He also recognised that functional analysis facilitated the relationship between magnitudes that were incapable, for a variety of reasons, of being expressed numerically or the relationship between functions where the underlying law was not capable of being expressed mathematically. While committed to rational mechanics as the paradigmatic framework for use in the economic domain, Cournot was clear in his distinction between the capacity of the generalised mathematically articulated framework such as rational mechanics to provide mechanics with general theorems. However, when it came to their applications they required considerable insight and experience to generate plausible numerical estimates. By analogy, Cournot held that in the application of mathematics to economic problems the same requirements would hold and in the economic domain to an even greater degree. Consequently, he was cognisant of the need for successive applications and approximations arising from the variation in numerical results (Touffut 2007). In the exercise of the application and the pursuit of numerical approximations, Cournot saw a very useful role for the use of statistics, though in general he was not particularly partial to the use of statistics or the application of probabilities in the socio-economic domain.
Cournot differed from the Physiocratic writers and others imbued with the reformist ethos of the Enlightenment in that he eschewed ‘normative ambition’ from his work, and that, according to Ingrao and Israel, for two explicit reasons. One was ‘the limits he himself saw to the application of formal methods to the study of economics.’ For Cournot and those who shared his view, only ‘certain types of problems are susceptible of formalisation, one of these being the theory of wealth.’ The second reason was the problem of deriving normative prescriptions from abstracted theoretical analysis or in his own words the ‘immense step in passing from theory to governmental applications’ (Ingrao and Israel 1990: 81). Both reservations on Cournot’s part provide, in the light of hindsight, salutary reservations that later contributors to the ‘mathematisation’ of economics should perhaps have been more attentive. Apart from his engaging methodological nuances and indeed his reservations about even the possibility of pursuing a general equilibrium approach in economics, his contributions to economic theorising as later incorporated into the neoclassical paradigm included his contributions to the theory of demand, relating the quantity demanded of a good or service to its monetary price in a single market, and his celebrated theory of market forms, all of which were incorporated into the canon of economic analysis and are well known to students of economics. In addition he articulated such seminal concepts as the elasticity of demand, marginal revenue, marginal cost, monopoly, duopoly, perfect competition and many others. Notwithstanding his theoretical creativity he remained an isolated figure, whose economic work during his own lifetime encountered the various hostile forces, identified earlier, to the project of the ‘mathematisation’ of economics. It was only later, with the extensive incorporation of mathematics into the economic academy in the twentieth century and to the fact that Cournot was arguably the writer that exerted the most profound influence on Walras, that his contribution to the mathematisation of economics was duly acknowledged.
The ‘precursors’ as represented by Isnard, Dupuit, Canard and Cournot within the accepted historiographic narrative were critical to maintaining the commitment to the values of Newtonian rational mechanics and the larger cultural ethos of the Enlightenment during the course of the nineteenth century even in the face of the formidable opposition which emerged against the project of mathematising the socio-economic realm. The French contribution, which we have concentrated on here for justified reasons of its centrality and substantive contributions, could be said to have culminated in the work of Léon Walras (1834–1910) and his formulation of a mathematical theory of general equilibrium and its subsequent influence in the twentieth century. In this Walras was ‘standing on the shoulders of giants’, from his absorption and commitment to Newtonian rational mechanics as the normative paradigm for the analysis of the socio-economic system to the various insights provided by his ‘precursors’, in particular Cournot. Mention must also be made of Walras’s successor at the University of Lausanne, Vilfredo Pareto (1848–1923), a Frenchman by birth but raised in Italy. Pareto, like Walras, was also imbued with the commitment to the mechanical framework as the correct framework to further the analysis of the socio-economic domain and in particular to ground it in a theory of equilibrium using methods and techniques of the physico-mathematical sciences. For Pareto, his aim was to anchor economics in the analytical framework of rational mechanics and to buttress it with empirical data consistent with the framework (Tarascio 1968; Crillo 1978; Cunningham Wood and McLure 1999; McLure 2001).
But Walras is unquestionably the central figure who emerges from this complex trajectory in the evolution of economics and in particular the development of the mathematisation of the discipline. Not that the reception of his work in his own life-time would reflect his pivotal position as now enshrined in the canonical narrative. Recognition of his contribution and the conferring of status, which Walras so desired in his own lifetime, would only come later. Since then and particularly during the course of the twentieth century, a voluminous corpus of work on Walras has emerged largely related to his contribution to general equilibrium theory.27 But something of the flavour of the indifference and active hostility that Walras encountered is vividly conveyed when he first presented his efforts at providing a mathematical interpretation of the economic system at the Institut de France’s Académie des Sciences Morales et Politiques. Pierre Émile Levasseur (1828–1911), French economist and historian, displayed the more vitriolic end of the spectrum of critique and dismissal of attempts to apply mathematics to socio-economic phenomena, which he viewed as methodologically futile. He argued against Walras of the danger:
that lies in the desire to bring together, as a unit, at any cost, things that are complex by their nature, as in wishing to apply to political economy a method that is excellent for the physical sciences but could not be applied indiscriminately to an order of phenomena whose courses are so variable and complex and that above all involve one eminently variable cause that can absolutely not be reduced to algebraic formulae: human freedom.
(Levasseur in Walras 1874: 119)
The reaction to Walras’s endeavours to formulate a mathematical economics based on the equations of exchange within an interdependent system met with similar criticism, even if less dismissive in tone than that of Levasseur, from many of his contemporaries. This forced Walras to secure the support of leading mathematicians and scientists of his day. The most celebrated interchange, albeit brief in content, was unquestionably with the leading French mathematician of his time, Henri Poincaré. Given that in this book we commence our analysis of the relationship between developments in the philosophy of mathematics and economic theorising with Walras, and more specifically with the Walras-Poincaré correspondence, which is the subject of the next chapter where we provide a philosophical analysis of the implications of this correspondence, we will not pursue this topic here.
Walras’s campaign to win over the mathematicians, philosophers and scientists of his day following the publication of his Élements d’economie politique pure in 1874–1877 was less than successful and even into the early part of the twentieth century the prospects for the project of mathematising the social sciences, with economics being the front-runner, was widely believed to be unworthy of support, if not impossible to achieve. Notwithstanding these setbacks, the mathematisation project was in the course of the 1920s and 1930s to receive major stimulus from contributions which now came from a number of other countries including Britain, Germany, Sweden and in the post-World War II period were to be centred in the United States, to where many of the leading European contributors emigrated during the course of the 1930s under the threat of Nazism. During the nineteenth century individual contributors from a number of countries other than France contributed to the use of mathematics in economics. These included from Germany, Johann Heinrich von Thünen (1783–1850) and Hermann Heinrich Gossen (1810–1858); from Britain, William Stanley Jevon (1835–1882), Francis Ysidro Edgeworth (1845–1926), an Irishman, and Alfred Marshall (1842–1924); from Sweden Knut Wicksell (1851–1926) and from the United States Irving Fisher (1867–1947).28
In a rather idiosyncratic account of the march of the ‘mathematisation’ project in economics, Gerard Debreu, himself a major contributor to twentieth-century mathematical economics and whose philosophy of mathematics will be examined in a later chapter of our book, provides what we may call the ‘scientific accidents’ theory of the growth of mathematical economics (Debreu 2008). He argues, without elaboration, that the ‘steady course on which mathematical economics has held for the past four decades sharply contrasts with its progress during the preceding century, which was marked by several major scientific accidents’ (ibid.: 454). He cites three such ‘scientific accidents.’ The first was the publication of Cournot’s Recherches sur les principes mathématiques de la théorie des richesses in 1838, which by both ‘its mathematical form and economic content … stands in splendid isolation in time.’ The second ‘scientific accident’ he considers to be the risk that the University of Lausanne took in appointing Walras to a professorial position in 1870, who as Debreu notes ‘had held no previous academic appointment, he had published a novel and a short story … and had not contributed to economic theory before 1870, and he was exactly 36’ (ibid.: 454). Hardly a ringing endorsement as far as Debreu was concerned, particularly given what was to follow. Similarly he includes the appointment of Pareto as Walras’s successor as part of the risk that Lausanne continued to take, but one that paid off when Pareto published his Cours d’economie politique (1896–97), followed by his Manuel d’economie politique (1909) and his article ‘Economie mathématique’ (1911). The third ‘accident’ for Debreu refers to ‘the contemporary period of development of mathematical economics’ which was ‘profoundly influenced’ by John von Neumann and in particular his article of 1928 on games and his 1937 paper on economic growth.
The seminal papers by Wald (1935, 1936a, 1936b) provided further stimulus to the intensification of the mathematisation of economics, as did the introduction of algebraic topology into economic theory when von Neumann generalised Brouwer’s fixed point theorem as part of a proof of the existence of an optimal growth path in his growth model. The mainstay of mathematical economics was differential calculus and linear algebra, which had been incorporated into the classic works of John Hicks’s Value and Capital (1939), Maurice Allais’ A la recherche d’une discipline économique (1943) and Paul Samuelson’s Foundation of Economic Analysis (1947). But with the publication of von Neumann and Oskar Morgenstern’s Theory of Games and Economic Behaviour (1944), new innovations in mathematical forms were introduced into mathematical economics in the form of convex analysis, which was to become central, initially in activity analysis and linear programming and later in mainstream economic theorising. By 1954 proofs of the existence of general equilibrium based on fixed point theorems, albeit under very stringent assumptions, had been delivered. The trajectory and development of the neo-Walrasian programme, as it came to be termed with respect to the development of general equilibrium theory, is comprehensively analysed in Weintraub (1983, 1985).29 Rehearsing the details of the changing mathematical techniques and methods that came to dominate mainstream economics during the course of the twentieth century is beyond the scope of this section and will not be pursued further here. The remainder of our book will address what we consider to be the major shifts within the philosophy of mathematics in the twentieth century, with the primary focus being on a philosophical analysis of these shifts and their implications for how economic theorising was influenced by these shifts.
The rise of mathematical economics and its influence on the perception, curricular content and internal status of individuals within the discipline has been both dramatic and profound during the course of the twentieth century. Even as this dominance is now being challenged from a variety of sources ranging from those who are opposed to the overuse or misuse of mathematics in economics to a more fundamental dimension by those who argue that the problem is the use of the wrong type of mathematics which has dominated in economics with disastrous results. This latter critique has two dimensions: one methodological as reflected in the extensive corpus of work produced by Tony Lawson of Cambridge University (Lawson 1997, 2003a, 2003b, 2015);30 in this context mention must also be made of the work of Philip Mirowski, a strident critic of mainstream neoclassical economics, who in recent years had advocated an alternative to the mainstream paradigm in the form of a theory of markets viewed as automata, a view inspired by the work of John von Neumann (Mirowski 2002, 2007, 2012). A second line of critique is the mathematically based work of Vela Velupillai, which represents a rigorously articulated and challenging perspective to the mainstream framework of mathematical economics (Velupillai 2000, 2005b, 2010).31 This latter contribution must be viewed within developments in twentieth-century philosophy of mathematics, including Brouwerian intuitionism and the work of Turing, Church and Bishop in the field of computation. In later chapters of our book we will examine aspects of these developments in the philosophy of mathematics in the course of the twentieth century.
With the spectacular rise to dominance of the mathematical formalisation of economics particularly during the twentieth century, this dominance is now coming under increasing critical scrutiny within the discipline.
Notwithstanding the variegated sources of critique and contra-voices being raised against the rapid rise to dominance of mathematical economics, more pertinently a particular type of mathematical economics, a pertinent question has emerged which is of considerable interest in the historiography of twentieth-century economics. This question is encapsulated in the title of Weintraub’s book, How Economics became a Mathematical Science (2002). If this issue is to be adequately addressed, considerable work will need to be undertaken in at least three dimensions: historiographical examination of the evolution and lineage of quantification, formalisation and the importation of mathematics into economics; a rigorous analysis of the philosophical/methodological principles that underlie the mathematical frameworks advocated and deployed in socio-economic analysis along with a critical examination of their contribution to our understanding of the economic systems and their working; and finally the sociological/cultural background of larger external forces that have and will continue to exert very considerable influence on the shaping of socio-economic analysis as they have done in the past.
In this chapter we have attempted to provide no more than a skeletal outline of the historical evolution, from Petty’s ‘political arithmetic’ in the seventeenth century to developments in the twentieth century. The modest aim of the chapter was to convey the central idea that concern with quantification/formalisation is not of recent origin and to portrait it otherwise would be to display an extremely parochial view in time and place. A second subsidiary aim was to provide a context against which to view the developments in the philosophy of mathematics during the course of the twentieth century and its implications for economics from a philosophical perspective. The remainder of the book will address these developments in the philosophy of mathematics during the twentieth century, so that serious students of economics will be motivated to explore further and reflect at some length on what kind of discipline economics has become under the influence of mathematics in order to establish both the benefits and the costs of that influence.
1 Hodgson has compiled and edited an excellent collection of papers dealing with the relationship of Mathematics and Economics organised around a number of thematic areas covering the post-1945 period.
2 In his 1952 paper, Samuelson at the outset stated that he came ‘not to praise mathematics, but rather to slightly debunk its use in economics. I do so out of tenderness for the subject, since I firmly believe in the virtues of understatement and lack of pretension.’ At the end of the paper Samuelson noted that ‘one of my older friends’ who complained to Samuelson that, ‘These days one can hardly tell a mathematical economist from an ordinary economist’, which reflected the perception of an older generation that by 1952 the take-over by mathematics was near to completion, if not complete!
3 Tracing this relationship and providing an adequate intellectual context would warrant a major study in its own right. All that can be provided in this chapter is a skeletal outline of some of the principal developments in this complex relationship.
4 Schumpeter in fact refers to them as ‘Econometricians’, which may strike present-day readers as somewhat odd. He further insists that ‘their works illustrate to perfection what Econometrics is and what Econometricians are trying to do’ (Schumpeter 1954: 209). He refers the reader in a footnote to the fact that ‘The word Econometrics is, I think, Professor Frisch’s’, and Schumpeter directs the reader to the first number of Econometrica, published in January 1933, as the new journal of the Econometric Society which had been founded two years earlier. In a short Editorial to the first issue of Econometrica, Frisch provides a succinct and perceptive overview of his understanding of Econometrics. Schumpeter’s insistence on the use of the word ‘Econometricians’ to describe the work of eighteenth-century contributors in the light of Frisch’s Editorial appears somewhat off to say the least.
5 Hermann Conring (1608–1681) in the seventeenth century was a leading pioneer in the provision of such courses (Lindenfeld 1997). Schumpeter in fact claims that Contring ‘is usually credited with having been the first to give lectures of this kind’ (Schumpeter 1954: 209). Later in the eighteenth century Gottfried Achenwall (1719–1772), an economist and statistician, and professor of philosophy at Gottingen (1750–1772), gave similar courses. Ackenwall was claimed by German academics as the ‘Father of Statistics’, a claim disputed by British writers who would defend William Petty’s right to that title. If Achenwall was not the originator of the science of statistics, he was certainly one of the first; if not the actual first, to articulate and define its purpose.
6 Hobbes had served as secretary to Bacon, during the middle of the 1620s, and translated a number of the latter’s essays into Latin (Martinich 2005).
7 Petty’s appointment in Gresham College was largely due to his good friend John Graunt (1620–1674), who for many is regarded as the ‘founder of statistics’. Whatever of this overarching claim he was certainly the pivotal figure in England or elsewhere in the development of population statistics or demography. In a tercentenary tribute, it was stated of Graunt that he ‘was the first person to whom it occurred that numerical information on human populations could be of more than ephemeral interest’ (Sutherland 1963: 554).
8 The ‘Invisible College’ was more specifically the ‘Oxford Club’ of experimentalists imbued with the Baconian approach to scientific knowledge which formed around John Wilkin (1614–1672), who was Warden of Wadham College in Oxford from 1648 to 1659. He attracted an extremely talented group of scientists, mathematicians, and philosophers and distinguished physicians among others to the so called ‘Oxford Club’. These included, among others, Jonathan Goddard, John Wallis, Seth Ward, Ralph Bathurst, Robert Boyle, William Petty, Thomas Willis, Christopher Wren and Matthew Wren.
9 In addition to Strauss (1954), earlier profiles and biographies of Petty include John Aubrey, Brief Lives, edited and annotated in Bennett (2015) and Fitzmaurice (1895). It should be noted that Aubrey’s Brief Lives has an interesting publication career in its own right since they were first compiled in the seventeenth century. The 2015 edition cited here represents the most complete and scholarly compendium in two volumes and is the first annotated critical edition of Brief Lives by the leading authority on Aubrey’s work, Dr Kate Bennett of Magdalen College, Oxford, and are published by Oxford University Press.
10 For an extended historical treatment of the Cromwellian period and the following Restoration period see Barnard (2000), Dennehy (2008), Jordan (2007). For Petty’s more elaborate and outlandish proposals for the colonisation of Ireland or what became known as the ‘transmutation of the Irish’, see McCormick (2008), and Fanning (2015). For a comprehensive and detailed scholarly account of Petty’s career, particularly in Ireland, and the intellectual origins, commitment and use of political arithmetic by Petty, see McCormick’s excellent study (McCormick 2009). See also Fox (2009) for an interesting account of Petty’s involvement in Ireland.
11 While the legal defence of his Irish estates did indeed preoccupy an inordinate amount of time and effort on Petty’s part, on his return to London in the late 1650s he delved back into a very active life mainly focused on politics, science and writing. He served briefly as a Member of Parliament, and having re-established his contacts with his scientific acquaintances he was centrally involved in the establishment of the Royal Society of London for the Improving of Natural Knowledge in 1662, and remained an active member for the remainder of his life. Four years before his death, Petty was one of the founding members of and first President of the Dublin Philosophical Society in 1683. This was the period, following his return to London and over the next three decades, that Petty would produce most of his major works, even though several of them were not published until after his death. For a bibliography of Petty’s writings see Keynes (1972).
12 Pursuing the issue as to whether Petty’s methodological approach was predominantly informed by either an inductive or deductive approach has not proved particularly fruitful or insightful. A careful reading of Petty’s work would clearly indicate, in our estimation, that methodologically Petty was much more indebted to Hobbes than to Bacon.
13 For a brief account of Petty’s ‘political arithmetic’ following his death see Aspromourgos (1996), but for a more extended examination see Hoppit (1996).
14 For some recent studies of the broader intellectual culture of the Enlightenment, see Dupré (2004), Pagden (2013), and Ferrone (2015).
15 On the Cameralist tradition, see the older study by Small (2001 [1909]) and Wakefield (2009).
16 On Cantillon’s life and seminal contribution to economics, see Murphy (1986) and Brewer (1992).
17 For a more contested view of the Enlightenment, see the challenging work of Jonathan Israel in particular Israel (2001, 2006, 2010, 2011).
18 For interesting overviews on this topic, see Guerlac (1981), Dobbs and Jacob (1995).
19 There is an extensive literature on this period, or what is sometimes referred to as ‘Newtonian Britain’. See Stewart (1992) for an informative overview of this topic and period in England.
20 See Schumpeter (1954), Pribam (1983), Hutchison (1988), Screpanti and Zamagni (1995), Robbins (1998), Roncaglia (2005) and Agnar (2011).
21 On Quesnay, see the classic work by Meek (1962), along with Kuczynski and Meek (1972), Eltis (1975a, 1975b), and Vaggi (1987).
22 Turgot’s work is comprehensively covered in Groenewegen (1977, 2002), Meek (1973), Palmer (1976), and Brewer (1987). For a broader and insightful intellectual perspective on Physiocracy, see Hochstrasser (2006).
23 Over the last twenty years there has been a renewed interest in the work of Jean-Baptiste Say within French political economy more generally. See Palmer (1977), Faccarello (1998), Steiner (1998), Forget (1999), Whatmore (2000), Arena (2000), Kates (2003), Hollander (2005).
24 For an analysis of these differences see the references in Note 14 above. For an intellectual overview of political economy in the nineteenth century see Rothschild (2011).
25 For an analysis of Isnard’s work and his contribution to Walras’s general equilibrium theory, see Jaffé (1960), Theocharis (1983) and van der Berg (2005).
26 Cournot’s work is examined in all the major works on the history of economic thought, for instance, those in Note 20 above.
27 Pre-eminent among the many writers who have contributed to Walrasian scholarship has been the work of William Jaffé and more recently Donald A. Walker. See among other works of these writers the following: Jaffé (1965, 1983), Walker (1996, 2006). See also Van Daal and Jolink (1993), Jolink (1996), Arrow and Hahn (1971), Morishima (1977), Mas-Collel (1985). For an interesting account of the history of the proof of the central theorem of general equilibrium, see Düppe and Weintraub (2014).
28 See Ekelund and Hébert (2014) for an overview of their contributions and the excellent individual entries for these economists in Durlauf and Blume (2008).
29 There is an extensive literature on general equilibrium theory. In addition to the outstanding study by Ingrao and Israel (1990), see Kirman (1998), Petri and Hahn (2003) and Bridel (2011). For a useful compendium of entries pertaining to general equilibrium see Eatwell, Milgate and Newman (1989).
30 Over the last thirty years Lawson has produced an extensive and impressive volume on work in which he has emerged as the leading advocate of critical realism and the centrality of social ontology as the most appropriate philosophy to underlie economics and the social sciences more generally. See more recently Lawson (2015) and for evaluations of his work see Boylan and O’Gorman (1995), Fleetwood (1999) and Fullbrook (2009).
31 Velupillai has produced an astonishing and challenging volume of work over the course of his career in his relentless search for an ‘alternative’ mathematics for the formalisation of economics based on a deep and reflective reading of the philosophy of mathematics. For tributes to and assessment of his contributions see Zambelli (2010) and the Special Issue of the journal New Mathematics and Natural Computation, 8 (1), 2012: 1–152.