Formalism – the extensive exploitation of mathematics in economic theorising – has been a distinguishing characteristic of economics since the Marginalist Revolution of the 1870s, a process that has intensified through the course of the twentieth century. The application of mathematics to economics has been particularly reflected in the domain of general equilibrium which represented the most fundamental attempt to unify the foundations of the discipline from Walras’s contribution through to its current variant of the Dynamic Stochastic General Equilibrium (DSGE) model, which is at the centre of so much of current orthodox macroeconomics. Meanwhile critics of orthodox economics have stridently criticised what they perceive as the excessive use of mathematics in general in economics, while at the frontiers of research in the discipline, new mathematical approaches have been suggested by a number of pioneering figures who are not opposed to the mathematisation of economics per se, but are opposed to the type of mathematics – classical mathematics in the main – that has been used and continues to be applied in economic theorising. From a methodological point of view, there is an urgent need for economic methodologists and economic theorists in general to direct their attention to a cognate domain of study, namely the philosophy of mathematics particularly its influence on the development of the mathematisation of economics. This is a fundamental informing rationale for this book if our understanding of the genesis, evolution and current developments at the foundational frontier of our discipline are to be methodologically engaged, analysed and understood.
Arising from the unwavering formalisation of economics throughout the twentieth century, and in our view the influence of developments in the philosophy of mathematics on this formalisation process, it is surprising that developments in the philosophy of mathematics have not played a more central role in the discourse of economic methodology. This book will address this critical relationship thereby redressing an existing gap in the methodological literature by posing the question: what can the philosophy of mathematics contribute to our methodological understanding of the trajectory of the formalisation of economics from Walras’s initial attempts to the current situation in the foundations of the discipline? One considered answer is that the philosophy of mathematics is indispensable for both the enrichment of our methodological understanding of the complex process of the mathematisation of economics and the critical evaluation of that process.
Methodologically, this work is woven about five central methodological themes. These are the implications of the philosophy of mathematics for (i) mathematical modelling in economics, (ii) the mathematical modelling of equilibrium, (iii) the mathematical modelling of rationality, (iv) the manner in which theoretical economics is conceived as an applied mathematical science, (v) the axiomatisation of economic theory. In terms of the history of economic thought, the focus is on Walras, Keynes, Debreu and Simon. In this connection we are engaged in three complementary tasks. Firstly we use the philosophy of mathematics to enrich our understanding of the divergent philosophies of economic theorising adopted by Walras and Debreu. We attempt to be as objective as possible in presenting their sophisticated and creative methodological positions. Secondly, having identified these methodological positions, we use the philosophy of mathematics to critically evaluate their respective mathematisations of economics. Finally, we use the philosophy of mathematics to suggest how creative ideas, like Keynesian conventions and Simon’s bounded rationality, may be methodologically reconstructed by recourse to the philosophy of mathematics. We assume the reader is not familiar with the philosophy of mathematics. Both within the text itself, including footnotes, and the appendix, we introduce key developments in the philosophy of mathematics which impinge on the methodological interrogation of the mathematisation of economics.
Broadly speaking we divide the philosophy of mathematics into two groups. The first group takes classical mathematics – the mathematics used by Debreu and other general equilibrium theorists – as being unproblematical and sets about defending it on logico-philosophical grounds. By the second decade of the twentieth century Hilbert and his formalist school at Göttingen were acknowledged as the principal exponents of this heterogeneous group. Hilbertian formalists defend classical mathematics by reconstructing it as an axiomatic system which is shown to be consistent, complete and decidable. The elaboration of this definitive defence of classical mathematics is known as the Hilbertian formalist programme.
In the other group we find mathematicians like Poincaré in France, Brouwer in Holland and philosophers like Dummett in England who do not take classical mathematics as it stands: some of the methods and theorems of classical mathematics are illegitimate! Thus Brouwer in the opening decades of the twentieth century developed an alternative mathematics to classical mathematics, called intuitionism. According to intuitionists the very core of classical mathematics is logically flawed and consequently classical mathematics must be replaced by intuitionistic mathematics. According to intuitionists, pure mathematics is created by the time-bound human mind and consequently is constrained by constructive, algorithmetic methods. In this way intuitionistic mathematics rejects what it sees as the infinitist extravagances of classical mathematics.
This fundamental conflict between Hilbert formalists and Brouwerian intuitionists raises a central question for the methodological analysis of the mathematisation of economics. It is frequently assumed that the mathematisation of economics is in principle not problematical: there is one and only one way to mathematise economics or indeed any other discipline, namely recourse to classical mathematics. The upshot of the foundational conflict between Hilbertian formalists and Brouwerian intuitionists is that theoretical economists have a genuine choice. Should they use intuitionistic rather than classical mathematics in their economic modelling? Is there a better fit between the subject matter of economics and intuitionistic mathematics than between the subject matter of economics and classical mathematics? In this connection we argue that intuitionists, not classical, mathematics is the more appropriate mathematics for economic theorising. Indeed we go much further. We contend that the neo-Walrasian programme of the mathematisation of economics is utterly undermined at its logico-mathematical core by intuitionistic mathematics.
For various reasons Brouwer’s attempted intuitionistic revolution in mathematics failed: the majority of mathematicians continued to adhere to the framework of classical mathematics. However, within this framework of classical mathematics, events in the foundations of mathematics in the 1930s undermined the received view of classical mathematics as an axiomatic system which is consistent, complete and decidable. These events gave birth to what we call computable mathematics, a specialised branch of classical mathematics. We argue that these events, combined with computable mathematics, have major implications for the project of mathematisation of economics in general and for the neo-Walrasian project in particular.
Prior to the astounding results of Gödel, Church and Turing – the progenitors of computable mathematics – it was widely assumed that classical mathematics is an axiomatised system which is consistent, complete and decidable. Given that assumption, there is, in principle, no problem in exploiting the full resources of classical mathematics in theoretical economics. In the 1930s, Gödel, Church and Turing, by rigorous logico-mathematical analysis, undermined that assumption: the received view of classical mathematics as an axiomatic system which is consistent, complete and decidable must be rejected. Gödel’s theorems established that there are, as it were, trade-offs between consistency and completeness in the axiomatisation of classical mathematics while Church and Turing proved that classical mathematics contains algorithmatically undecidable propositions. Contrary to the received view, classical mathematics contains both algorithmatically decidable and algorithmatically undecidable theorems.
Our methodological concern is with the implications of this post Gödel-Church-Turing view of classical mathematics for the mathematisation of economics. As we already indicated, given the received view of classical mathematics, theoretical economists can in principle exploit the full resources of classical mathematics in their theorising. Does the post-Gödel-Church-Turing view of classical mathematics undermine this thesis? That is, can economists continue to exploit the full resources of classical mathematics without any adverse economic consequences? What effect has recourse to an algorithmetical undecidable theorem on the project of the mathematisation of economics? In this connection we address three issues: (i) the implications of Gödel’s theorems for economic methodology, (ii) the implication of the use of undecidables for economic theorising; (iii) should economic theoreticians confine their mathematical modelling to computable mathematics, the algorithmetically decidable part of classical economics? Vis-à-vis (i), we argue that Gödel’s theorems undermine Debreu’s own unique ‘philosophy of economic analysis’ (Debreu 1992: 114) but that these theorems as such do not undermine Debreu’s mathematical model of a private ownership economy. In connection with our second issue, viz. the implications of recourse to undecidables in economic theorising, we argue that both Debreu’s mathematical model of an economy and his methodological defence of that model are undermined. The neo-Walrasian explanation of prices is rendered economically vacuous by recourse to undecidables. Vis à vis our third issue we address the feasibility of the programme of the mathematisation of economics when economic modelling is confined to computable mathematics, i.e. the algorithmetically decidable theorems of classical mathematics. Our thesis is that prima facie, the computable mathematical resources furnished by the Church-Turing thesis have the potential to open up an interesting research programme for theoretical economists. Whether or not this research programme will evolve into an algorithmetic revolution in economic theorising will depend on future economic research.
Vis à vis the mathematical modelling of rationality we address three issues. The first issue concerns the mathematical model of rationality used at the birth of the Marginalist Revolution, particularly that advocated by Walras. When Walras wrote and re-edited his Elements, the intellectual climate was hostile to the project of the mathematisation of economics. Walras wrote to Poincaré – the leading European mathematician at the turn of the twentieth century – for the support for his project of the mathematisation of economics. In his brief correspondence with Walras, Poincaré, while sympathetic to Walras’ project, raised a hierarchy of reservations and objections to Walras’ specific defence of his mathematical modelling of rationality. In this connection we re-examine Poincaré’s reservations and objections in light of Poincaré’s own philosophy of mathematics, particularly his suggestion that Walras’ model of rationality violates some methodological limits.
The second issue concerns the implications of computable mathematics for the orthodox modelling of rationality. In this connection we take up the theme of Simon’s bounded rationality. In particular we engage a Hahn-type defence of the orthodox model of rationality, namely Simon’s bounded rationality is no match for the orthodox theory because the potential for mathematical elaboration and sophistication is poor in Simon’s approach. We engage the thesis that computable mathematics, as is evident in Velupillai’s computable economics, is a fruitful source for a mathematical model of bounded rationality matching the mathematical sophistication of orthodoxy. Prima facie this novel research programme offers theoretical economists with new challenges and a rich reservoir of algorithmetic techniques by which these challenges can be met. Indeed we concur with Simon that Velupillai’s algorithmetic model of rationality shows how the orthodox model of rationality transcends the outer bound of what is humanly rational. Of course, as one moves from thin to thick descriptions of economic decision-making, whether or not such a computable model of rationality will be vindicated is a matter for economic research, it is not a matter for economic methodology.
The third issue we address is the re-examination of the rationality of the Keynesian notion of convention in the context of radical uncertainty. When Keynes employed the concept of convention, he introduced a complex set of ideas that continues to present challenges of both interpretation and application. In the literature to date two pivotal figures have emerged, separated in time by two hundred years, namely David Hume and David Lewis. Hume’s analysis of convention has remained a remarkable fruitful source of formative ideas on the creation, development and application of conventions in the social domain. In his seminal work Convention, first published in 1969, Lewis developed a framework broadly Humean in character but articulated within the analytic framework of game theory and centred on the notion of co-ordination. Central to the co-ordination problem is the idea that there exists a number of alternatives to choose from by which social agents can co-ordinate their actions to achieve mutual benefit. This idea informs a large portion of contemporary conventionalist theory. In this work we argue that post-Keynesian scholars should add another key figure to the line from Hume to Lewis, namely Henri Poincaré, the same mathematician to whom Walras wrote in connection with the mathematisation of economics. Within the philosophy of mathematics both pure and applied, Poincaré is known as the father of conventionalism. As Carnap remarked, ‘Poincaré was a philosopher who emphasised more than previous philosophers the great role of convention’ (Carnap 1966: 59). In this work we identify some of the principal tenets of Poincaré’s analysis of convention and we relate this analysis of convention to the post-Keynesian methodological agenda. In particular we show how Poincaré places conventions at the core of pure mathematical reasoning, thereby rejecting the received view of mathematics as being nothing but a deductive system. We argue that this Poincaré analysis of convention liberates post-Keynesians from the influential legacy of Humean sceptical philosophy on the one hand and from the intimate association of Lewis’ convention with the salient solution to co-ordination games with multiple equilibria on the other. Post-Keynesian conventional decision-making is thereby shown to be fundamentally rational.
Intimately related to the theme of which mathematics is most appropriate for economic theorising is the issue of the conceptualisation of theoretical economics as an applied mathematical science. With respect to the birth of formalism in the Marginalist Revolution we focus on Walras’ philosophy of applied mathematics. Like the majority of his contemporaries, Walras took mechanics/mathematical physics as the exemplar of an applied mathematical science. In this connection we argue that Walras developed a scientific realist philosophy of mechanics: the principles of mechanics, mathematically represented by differential equations, convey the fundamental mechanisms operating in the physical universe. Given the analogy between mechanics and his mathematical economics shown by the existence of differential equations in both, Walras maintains that theoretical economics, like mechanics, reveals the fundamental principles of the economic world. Thus, while Mirowski in his analysis of the Marginalist Revolution emphasises the energy transfer from mechanics to economics, in the case of Walras we emphasise a different transfer, viz. the transfer of his scientific realist philosophy of mechanics to theoretical economics: just as the principles of mechanics reveal the fundamental mechanisms of the physical university, the principles of mathematical economics reveal the fundamental principles of the economic world. By way of critique we examine Walras’ recourse to Poincaré in his final defence of his scientific realist reading of Walrasian economics contained in his ‘Mechanics and Economics.’ Despite Walras’ total silence on the matter, Poincaré never espoused a scientific realist reading of mechanics. Indeed we show how Poincaré’s reading of the principles of mechanics undermines Walras’ scientific realist reading.
In connection with the neo-Walrasian programme, we focus on Debreu’s Theory of Value – the paradigm of economic theorising for a generation of orthodox economists – to show how Debreu’s philosophy of applied mathematics is very specific and quite unlike that of Walras. The analogy to mechanics exploited by Walras plays no role in Debreu’s philosophy of economic theorising. Thus Debreu, unlike Walras, does not transfer a scientific realist philosophy of mechanics to theoretical economics. Rather Debreu emphasises a crucial difference between theoretical economics and mathematical physics, viz. mathematical physics, by processes of very sophisticated experimentation exemplified in research at CERN, is experimentally testable, whereas theoretical economics is not similarly testable. Debreu’s philosophy of economic theorising as an applied mathematical science has a different source to that of Walras. Its source is, we argue, the philosophy of applied mathematics developed by Hilbert and his formalist school in the 1920s. By analysing Debreu’s Theory of Value we show how he uses the Hilbertian formalist template for an applied mathematical science in his celebrated proof of the existence of general equilibrium, thereby enhancing our methodological understanding of Debreu’s economic theorising.
The theme of the philosophy of applied mathematics used in economic methodology is intimately related to the theme of axiomatisation. Various orthodox economists insist that theoretical economics is or can be reconstructed as an axiomatic system. This reference to axiomatisation is frequently seen as being methodologically unproblematical. It is assumed that there is one and only one way of axiomatising economics. This traditional approach to the axiomatisation of a scientific domain we call the Euclidean model of axiomatisation of an empirical science. In this Euclidean model one first identifies the fundamental economic axioms and, secondly, the theoretical economist uses logic cum mathematics in association with economic assumptions to derive the implications of these economic axioms. In this Euclidean model of the axiomatisation of economics the axioms are generated from the economic domain and are frequently taken to be the fundamentals of the economic domain. Our thesis is that, while Walras did use this Euclidean model, Debreu, despite his commitment to axiomatisation, did not exploit this Euclidean model in his economic theorising. Rather, in line with his explicit commitment to the Hilbertian formalist school, Debreu used a different model of axiomatisation to the Euclidean one in his neo-Walrasian research. In the 1920s Hilbertian formalists developed a novel, non-Euclidean model for the rigorous axiomatisation of any empirical science.
In this Hilbertian formalist conception of the axiomatisation of any empirical science, the axioms are not generated in the scientific domain. Thus in the case of economics, its axioms are not generated from the economic domain. In this formalist approach to axiomatisation of any empirical domain, the only axioms are those of pure mathematics! Rigorously axioms reside only in the pure mathematics side of the divide between pure and applied mathematics. The formalist canons of axiomatisation hold only in pure mathematics which is the consistent, complete and decidable servant of applied mathematics. Given this consistent, complete and decidable axiomatic system, the applied mathematician identifies an appropriate sub-domain and finds an empirico-theoretical interpretation of that sub-domain of axiomatised pure mathematics. We contend that it is this formalist Hilbertian model of axiomatisation of any empirical science, and not the traditional Euclidean model, which informs Debreu’s commitment to axiomatisation in his Theory of Value. Moreover in this Hilbertian formalist model of axiomatisation of economic theory, the tacit assumption frequently associated with the Euclidean model, viz. the axioms convey the fundamental mechanisms operating in a real economy, does not hold. The umbilical cord between axiomatised economic theory and the fundamental mechanisms of an economy is severed. Rather in Debreu’s Hilbertian formalist approach, axiomatisation is the vehicle for the rigorous presentation and expansion of economic theory. For Debreu theoretical economists, like other applied mathematical scientists, wish to attain the highest standards of rigour. This issue is not to be confused with the totally different issue of the empirical adequacy of axiomatised economic theory. Successful axiomatisation throws no light on empirical adequacy, nor is it intended to do so. Axiomatisation is the handmaiden of rigour, not empirical adequacy.
Overall, a central thesis of this work is that the philosophy of mathematics is a gale of creative destruction through the programme of the mathematisation of economics. While exposing the limitations of mathematical modelling in orthodox economics it offers alternative ways of formalising economics. Thereby it enables economic methodologists to address in novel and more precise terms the long-standing debate as to whether or not formalism in economics has gone too far. However, while emphasising the indispensable role of philosophy of mathematics in economic methodology, there is no suggestion that the philosophy of mathematics should colonise the philosophy of economics. The philosophy of economics has many mansions, one of which is constructed on the site of the philosophy of mathematics. Finally, we assume that readers are not familiar with the philosophies of mathematics exploited in the various chapters. With that assumption in mind we have attempted to balance the demands of communication with our potential readers and the demands of the accurate presentation of the complexity and sophistication of the logico-philosophical analyses in the various philosophies addressed in this work.