This book has had arguably a uniquely long gestation period. The original proposal was first drafted while we attended the 10th International Congress of Logic, Methodology and Philosophy of Science in the magnificent setting of the Palazzo dei Congressi in Florence in 1995. In its initial formulation the central figure was Henri Poincaré and his philosophy of science, and more particularly what would have been the consequences for economics, as a discipline, if it had followed the path contained in Poincaré’s philosophy of science as we interpreted it at that time, based on Paschal O’Gorman’s earlier work on Poincaré’s conventionalism. A central informing thesis for us at that time was what we regarded as the high epistemic cost that economics had inflicted on itself arising from the application of a particular philosophy of mathematics from the late nineteenth century, which was further intensified in the formalistic philosophy of mathematics that dominated for most of the twentieth century in economics. In contrast to the formalism reflected in the neo-Walrasian programme and in particular in the work of Debreu for instance, our initial proposal was informed and heavily influenced by our attraction to the intuitionism of Brouwer, Heyting and later in the twentieth century in the work of Michael Dummett, which we felt provided a more adequate and epistemically satisfying philosophy for our discipline. On reflection our work in economic methodology over the last quarter of a century has been, in a variety of ways, influenced by the broad philosophical framework of Brouwerian intuitionism.
Since 1995 and the intrusions of new and varied work demands at university and national levels, along with the completion of a number of book projects and edited collections, conspired to delay sustained engagement with this book project. As a result the current book proposal was persistently relegated to the ‘for completion’ file! But we were determined to honour this project, which could only have been achieved with the extraordinary patience and forbearance of all at Routledge, for which we are deeply grateful. But in this extended intervening period our thinking also underwent change which generated a considerable amount of reconfiguration with respect to the dimensions and topics to be addressed in the central domain of interest to us, namely the relation and influence of major developments in the philosophy of mathematics and their influence on economics. It became clear to us arising from our teaching at both undergraduate and postgraduate levels, that the need for an extended reflection on the major developments within the philosophy of mathematics and their impact on economics was an emergent and pressing intellectual challenge which required inclusion as an integral component in the philosophical and methodological pedagogy of economics, particularly at the postgraduate level. While this book is not a textbook in any conventional understanding of the term, it does explore in some considerable depth a range of topics that we deem paramount in any extended intellectual understanding of the complex interaction between economics and mathematics (incorporating the processes of quantification, measurement and formalisation). We can concur with Lawson that in ‘the history of the modern mainstream, the rise to dominance of formalistic modelling practices and the manner of their “survival” in this role, constitutes a central chapter in the history of academic economics that remains largely unwritten’ (Lawson 2003: 256). It awaits the comprehensive scholarly treatment exemplified for instance in Ingrao and Israel’s (1990) outstanding work on the emergence and development of general equilibrium theory. In the remainder of this Preface we provide, albeit briefly, an outline of the rationale that motivated our initial formulation, but more particularly as we extended our consideration to embrace a more extended array of topics related to the focus of our central concern, namely the relation between the major developments in the philosophy of mathematics and their influence on economic theorising and modelling.
This book is motivated by the conviction that both philosophers of economics/economic methodologists and theoretical economists have much to gain by addressing the philosophy of mathematics. The indispensable skill set of mainstream theoretical economists includes a competent, preferably expert command of particular areas of advanced mathematics to facilitate the construction of sophisticated economic models required for the rigorous analytical exploration of complex economic systems. The challenges theoretical economists face presuppose the acquisition of increasing competence. Consequently our recourse to the philosophy of mathematics is predicated on two objectives. Firstly, it is used to critically interrogate the intricate and complex process of what is called the formalisation/mathematisation of economics. Secondly, the philosophy of mathematics opens up ranges of novel logico-mathematical techniques for the theoretical modelling of rationality on the one hand and economic systems on the other, which result in outcomes at variance with orthodox/mainstream economic theorising.
Vis-à-vis the philosophy of economics/economic methodology this work may be read as a contribution to the research agenda identified by Weintraub namely ‘to study how economics has been shaped by economists’ ideas about the nature and purpose and function and meaning of mathematics’ (Weintraub 2002: 2). In this connection we distinguish the philosophy of pure mathematics from the philosophy of applied mathematics and analyse how these evolved over the course of the twentieth century (with particular reference to the twentieth century). Thus we examine how major figures, including Walras and Debreu among others exploited divergent philosophical perspectives on applied mathematics along with their relationship to pure mathematics in their methodological defences of their mathematical economics.
This book, however, is not limited to enhancing our understanding of the intriguing, often divergent, defences of the formalisation of economics by some of its major architects. It is also engaged in the critical evaluation of these defences, thereby complementing challenging critiques by, among others, Bridel and Mornati (2009), Mirowski and Cook (1990), Ingrao and Israel (1990), Lawson (1997, 2003a) and his critical realist colleagues. Our critique is distinct in its extensive exploitation of both the philosophy of pure mathematics and the philosophy of applied mathematics. For instance, in connection with the neo-Walrasian programme, our critique is based on Brouwer’s novel philosophy of pure mathematics and his alternative mathematics which is fundamentally grounded in that philosophy. This Brouwerian mathematics is not another chapter in the vast book of advanced mathematics exploited by mainstream mathematical economists: it is a different mathematics grounded in a different logic. A central contention of this book is that this Brouwerian mathematics is crucially significant for the epistemic critique of the neo-Walrasian programme. If a Brouwerian analysis of a rigorous mathematical proof stands up to critical scrutiny the very core of the neo-Walrasian programme, i.e. the existence proof of general equilibrium is mathematically undermined.
This book is not, however, limited solely to the enrichment of our understanding of the dynamics of some of the major critical junctures in the long and intricate process of the formalisation of economics and to the epistemic critique of the often intriguing views of some of its major contributors. It is also concerned with the current state of academic economics. Vis-à-vis mainstream mathematical economics some commentators such as Dow (2002) perceive a process of increasing fragmentation. Others, for instance Lawson (1997, 2003a) maintain that current academic economics is experiencing a deep malaise. The extent of either increasing fragmentation or malaise within academic economics is a question of empirical research which is not a central focus of this book. Rather, assuming an increasing fragmentation whether slow or fast, we pose the question: what is the relevance of the philosophy of mathematics in the contemporary economic climate? We argue that the relevance of the philosophy of mathematics is both positive and negative. Firstly, developments in twentieth-century logic expose the logical limitations of the advanced mathematics exploited in academic economics even when fragmented. Simultaneously, these developments offer theoretical economists a novel range of rigorous logical tools to be used in their mathematical modelling of rationality – a range not exploited in mainstream academic economics. Secondly, we explore the thesis that Brouwerian mathematics – used in our critique of the neo-Walrasian programme – is a more appropriate mathematics than the mathematics exploited in mainstream economics. By recourse to the philosophy of mathematics we expose the descriptive inadequacy of various results of orthodox theorising. In particular we show how these results exploit non-algorithmetic theorems of advanced mathematics and argue that non-algorithmetic models of economic decision making hold only for God-like beings whose decisions are not in any way constrained by temporal considerations. Non-algorithmetic models cannot be applied to actual economic decision makers who, even with the aid of the most sophisticated computers, take real time to make decisions. On the other hand if economic modelling was limited to computable mathematics then its models of rational decision making, by virtue of the algorithmetic nature of the mathematics used, would in principle be compatible with the time constraints of actual economic decision making. In this vein our sympathies totally lie with those economists, such as Velupillai (2000), who argues for the limitation of mathematical modelling to a judicious synthesis of computable mathematics with developments in Brouwerian mathematics.
A few caveats are in order. We are not claiming that the philosophy of mathematics should colonise the philosophy of economics/economic methodology. Rather the philosophy of mathematics makes an intriguing and unique contribution to our reflections on how economics became a mathematical science and on the contemporary status of mathematical economics. Neither is there any suggestion that mathematical economics exploiting the resources of Brouwerian-computable mathematics should colonise the discipline of economics. In so far as mathematical economics influences the construction of specific economic models, these models must be empirically interrogated. While theoretical economics cannot emulate the experimental sophistication of theoretical physics, economics is an empirical endeavour and thus its models must come before the bar of experience. Whether or not its models will or will not successfully pass this indispensable constraint, is a question for sophisticated economic testing which cannot be answered by any philosophy of economics.
As with the production of any book there are many debts incurred, of which we would like to mention a small number. We would like to convey our thanks to our colleagues in the Departments of Economics and Philosophy respectively at the National University of Ireland Galway (NUIG) for their support and co-operation during the writing of this book. In particular we would like to thank Professors John McHale and Alan Ahearne and Dr Aidan Kane for their support, encouragement and assistance during their respective terms as Head of the Department of Economics at NUIG. We also recall with fond memory the friendship, generosity and intellectual insights of Professor Vela Velupillai during his sojourn as the J.E. Cairnes Professor of Economics at NUIG. Vela’s work in computable or algorithmic economics stands, in our estimation, as one of the truly pioneering contributions of the late twentieth century in the area of mathematical economics which is informed by a sophisticated philosophy of mathematics. We are also very grateful to Professor Gerhard Heinzmann and his colleagues at the Poincaré Archives (ACERHP) at the University of Nancy 2 in France for their help and hospitality to Paschal O’Gorman on the occasion of his visits there.
As always we would like to thank a number of people at Routledge, who as publishers have been exceptionally facilitating, patient and encouraging to us over the last twenty-five years. To Terry Clague we are deeply grateful for welcoming us back so warmly when contacted after a long period of absence on our part with this particular project. While Terry has moved on to other areas in Routledge, he directed us to Andy Humphries the current Publisher for Economics at Routledge and to Anna Cuthbert the current Editorial Assistant for Economics. It has been a particular pleasure to work with both of them and we would like to thank them for their advice, help and assistance in bringing this project to fruition. Finally, a very special thanks must go to Claire Noone of the Department of Economics, who with her usual extraordinary cheerfulness, expertise and efficiency managed to convert various unruly drafts of a difficult manuscript, that changed both direction and content at various junctures over its extended period of production, into a final presentable version, while overseeing the various requirements for the electronic versions for transfer to Routledge. For all of this we are deeply indebted and sincerely grateful to Claire not only for this particular project but for all our previous publications with Routledge.