In the previous chapters we considered a variety of chords linking issues in economic methodology to issues in the philosophy of mathematics. In this chapter we attempt to weave these chords together. In this attempt we pose the general question: what is the contribution of the philosophy of mathematics to the philosophy of economics/economic methodology? What value is added to the philosophy of economics by making the philosophy of mathematics an indispensable branch? We are shifting the focus from the very specific issues discussed in the previous chapters to a more macroscopic viewpoint.
The philosophy of mathematics makes an invaluable contribution to the critical reading of pioneering works in the formalisation of economics, such as Walras’s Elements or Debreu’s Theory of Value. We are fully aware that this practice is obsolete in undergraduate courses and indeed that some eminent economists see no economic value in it. Rather this practice is more central to both the history of economic thought and to the philosophy of economics. The philosophy of mathematics enriches any objective hermeneutical analysis of such texts, in the contemporary sense of hermeneutics. An integral part of such hermeneutical studies is the contextualisation of such canonical texts in their own socio-historical contexts which, given the texts in question, includes the philosophy of mathematics. In this connection we explored the philosophico-mathematical context of the end of the nineteenth and the beginning of the twentieth centuries by using Poincaré’s philosophy of applied mathematics to unearth some of the distinguishing characteristics of Walras’s own scientific realist defence of theoretical economics as an applied mathematical science on par with the applied mathematical science of mechanics. As already noted, the choice of Poincaré’s philosophy of mathematics was motivated by two facts. Firstly Poincaré was then recognised as a towering, and for some, the towering, figure in mathematics across Europe. Secondly Walras wrote to Poincaré for support for the project of the formalisation of economics in what Walras perceived as a hostile environment. By the 1950s when Debreu came to write his Theory of Value this hostility had largely disappeared. Moreover, under the shadow of Hilbert’s formalist school, the conceptions of pure and of applied mathematics were dramatically altered from the Walras-Poincaré period. Debreu locates his Theory of Value in this new Hilbertian mathematico-philosophical climate. The uniqueness and sophistication of Debreu’s mathematico-economic commitments are explored by recourse to the Hilbertian formalist school’s understanding of the role of mathematics in the applied mathematical sciences.
In addition to enhancing our understanding of the unique philosophies of economic theory of Walras and Debreu, the philosophy of mathematics throws new light on some other developments in twentieth-century economics. In particular we focused on post-Keynesian economics and its emphasis on the role of conventions in economic decision-making in the face of radical uncertainty. In line with Keynes’s advice to engage other domains where recourse to conventions occurs, various scholars have focused on either David Hume’s or David Lewis’s intriguing reflections on conventions. This debate on the rationality of conventions in post-Keynesian economics is enhanced and extended by reflecting on another major contributor to the philosophy of conventions, namely Poincaré. By the turn of the twentieth century Poincaré, in his philosophy of mathematics, had pioneered his famous thesis of geometrical conventionalism, a thesis which incidentally was critically engaged by Bertrand Russell. By focusing on mathematical research at the frontiers of pure geometry, Poincaré showed how the received axiomatic image of mathematical rationality is fundamentally incomplete. Mathematical explorations at the frontiers of geometrical research show that in order to develop any metrical geometry, i.e. a geometry which engages distance, the pure mathematician must have recourse to convention. Thus any complete portrait of mathematical rationality must include recourse to convention. By bringing this amazing result to bear on Davidson’s emphasis on the non-ergodic dimension of the economic world, we show how economic decision-making in the face of radical uncertainty is rational in a way which does not reduce its rationality to the orthodox characterisation of rationality.
More generally, recourse to the philosophy of mathematics alerts us to the disputed nature of various logico-mathematical concepts used in economic theorising. For instance, logico-mathematical geniuses of the calibre of Brouwer, Frege and Hilbert fail to agree on how the logical concept of existence is to be understood and used. Thus where Debreu claims to prove the existence of equilibrium we saw that he understood existence as defined by Hilbert, namely freedom from contradiction. This notion of existence was challenged by both Frege and Brouwer: there is more to the logical concept of existence than freedom from contradiction. Pragmatic minded economists, some perhaps under the influence of Hahn who advised young economists to ignore methodology, might want to claim that the disputed nature of much of the philosophy of mathematics is all the more reason for ignoring it. One would, as it were, be immersed in dispute after dispute, like the above. Certainly philosophers of mathematics wash their dirty linen in public, a practice which is perhaps less common in economics. That of course is not to say that total consensus reigns among orthodox economists: as we saw in Chapter Five, Hahn, for instance, castigated some of his American colleagues. Disputes like the above in the philosophy of mathematics, however, reflect fundamental differences between the very best logico-mathematical minds. The fact that brilliant logico-mathematical minds, such as Brouwer, Frege, Hilbert and Russell, to mention but a few, fail to agree on fundamentals cannot be easily dismissed. In particular pretending to have mathematical certainty in one’s economic theorising, where such a certainty is not available, rationally leaves a lot to be desired. The integration of the philosophy of mathematics into economic methodology plays a key role here in that it makes explicit what is implicit in the mathematical choices of economic theoreticians and thereby enriches our appreciation of those choices.
In addition to making a unique and indispensable contribution to the methodological analysis of the complex process of the formalisation of economics, particular developments in the philosophy of mathematics offer philosophers of economics a range of critical tools for their task of the objective evaluation of the orthodox formalisation of economics. In this connection we focused on three major developments in the philosophy of mathematics, viz. the emergence of the twentieth-century modal logic, Brouwerian intuitionism and the Church-Turing thesis. The development of modal logic buttresses Arrow’s reservation vis-à-vis the adequacy of classical mathematics in the orthodox modelling of economic decision-making. Arrow correctly draws our attention to the fact that classical mathematics is grounded in classical logic which is extensional. Research in empirical psychology, however, shows that decision-making is intentional. Arrow’s reservation raises the critical methodological issue of the empirical adequacy of the representation of economic decision-making in its recourse to extensional classical mathematics. We considered two responses in the philosophy of mathematics available to economic methodologists. The first response is to argue that extensional logic is the one and only true logic. Hence any decision-making which does not conform to extensional logic is illogical and therefore irrational. We draw the reader’s attention to the lack of consensus on this philosophical thesis of the reduction of the whole of logic to classical extensional logic. Some who disagree with this reductionist thesis have developed modal logic to rigorously analyse the domain of the intentional. For these there is no objection to the rigorous analysis/formalisation of economic decision-making in terms of non-extensional modal logic. In short the results of empirical psychology combined with the development of modal logic suggest an alternative path to the formalisation of economic decision-making. Classical mathematics may be used in some economic modelling but not in the rigorous formalisation of economic decision- making: here the economic theoretician would have recourse to the rigorous resources of modal logic.
The critique of the orthodox formalisation of economics sourced in the development of intuitionism is much more radical. As we saw, strict intuitionism, either in its original Brouwerian interpretation or in its philosophical reformulation by Dummett, challenges the logico-mathematical core of classical mathematics. Much of classical mathematics has to be abandoned and replaced by intuitionistic mathematics. Consequently the orthodox formalisation of economics is based on a mathematics which does not stand up to critical scrutiny and thus must be abandoned. Of course, given the project of the formalisation of economics, recourse to classical mathematics was, prior to intuitionism, obviously unchallenged: classical mathematics was the only rigorous mathematics available. For the strict intuitionist, however, intuitionistic-constructive mathematics, rather than classical mathematics, is the one and only rigorous mathematics to be used in any mathematical modelling. Thus for the strict intuitionist there is no objection in principle to the formalisation of economics: rather the fundamental error of the orthodox formalisation lies in its exploitation of the pseudo-theorems of classical mathematics. Pragmatically we recognise that the chances of a strict intuitionistic revolution in the formalisation of economics taking place are not good. In addition to the subtleties of a defensible strict intuitionistic philosophy of mathematics, the lack of familiarity of intuitionistic logic on which intuitionistic mathematics is based constitutes a major practical obstacle. Our brief, however, was not to address pragmatic obstacles. Rather our brief was to explore the methodological impact of developments in the philosophy of mathematics on the formalisation of economics. If strict intuitionism stands up to critical scrutiny, the orthodox formalisation of economics is utterly undermined at its mathematical core.
Strict intuitionism was not the only response to the original Brouwerian failed revolution in mathematics. As we saw others opted for, what we called, pragmatic intuitionism. In pragmatic intuitionism classical and intuitionistic mathematics peacefully co-exist. Both are distinct, legitimate mathematical endeavours: classical mathematics is concerned with an actual infinite domain which uses classical logic and both constructive and non-constructive methods of proof, whereas intuitionistic mathematics is concerned with a potential infinite domain which uses intuitionistic logic and only constructive methods of proof. Given pragmatic intuitionism, another fundamental methodological question arises, viz. which mathematics, classical or intuitionistic, is best suited to theoretical economics? In particular what precisely is the methodological case for limiting theoretical economists to restrictive intuitionistic mathematics in their theorising when theoretical physicists are not similarly constrained? After all both theoretical economists and theoretical physicists study highly complex, dynamical systems: hence both should be open to investigation by the same set of mathematical tools, i.e. the tools of classical mathematics used in theoretical physics.
As acknowledged by Debreu, Hahn and others the analogy to theoretical physics is not the most apt. Firstly, relative to the predictive success of theoretical physics, theoretical economics fares rather badly. Secondly, unlike theoretical physics, theoretical economics lacks the sophisticated experimental resources which are a distinguishing characteristic of advanced physics. For instance, when mathematical physics postulates the existence of some theoretical entity, experimental physics extend the boundaries of what is experimentally detectable with a view to, as it were, tracking down the postulated theoretical entity. In the absence of experimental success in tracking down the postulated entity, empirical minded physicists suspend judgement on the existence claim. Given that the situation in theoretical economics is not like that in theoretical physics, what precisely is the situation in theoretical economics? According to formalists like Debreu the situation is as follows. Given the absence of these sophisticated experimental techniques, the theoretical economist uses various theorems of advanced classical mathematics and, by recourse to a range of assumptions, gives these theorems economic interpretations. In this manner, for instance, the possibility of general equilibrium is rigorously established. Much of the methodological opposition to this form of theorising focuses on the economic unrealisticness of the assumptions used. The philosophy of mathematics, as one would expect, does not address these assumptions per se. Rather pragmatic intuitionism opens up a different front in the attack on this orthodox formalisation of economics. It undermines the orthodox use of sophisticated, non-constructive theorems of advanced mathematics (such as Brouwer’s fixed point theorem) in its efforts at rigorously proving economic possibilities (such as general equilibrium). Like strict intuitionism, pragmatic intuitionism demonstrates that any such possibility cannot, by virtue of the non-constructive theorems used, be established in any finite system and thus cannot be given an economic interpretation.
For pragmatic intuitionists there is no objection in principle to mathematical modelling using the full resources of classical mathematics. Let us thus consider an economic model which proves the existence of some possible economic arrangement, based solely on a piece of non-constructive, classical mathematics. A core step in this economic model is the pure mathematical proof of the existence of an element in the piece of non-constructive classical mathematics exploited in the economic model. The first point to note is that, by virtue of the classical mathematics used, this mathematical element is an element of an actual infinite class of an infinity of classes. This is so because the mathematical domain of classical mathematics is an infinity of infinite classes. Secondly, given the mathematical proof is non-constructive, there is no mathematical way of establishing that such a mathematical entity is in principle accessible to any finite combination of finite systematic processes. Thus there is no mathematical way of demonstrating that the element belongs to some finite subset of the infinity of infinite sets in which it is proven to exist. Given that any economy is necessarily finite and also given the absence of possible experimental vindication of existence claims in theoretical economics, is there any economic point to this kind of mathematical modelling in theoretical economics? Thirdly, the non-constructive mathematics used in this kind of economic modelling necessarily abstracts from any reference to any time frame: any possibility established by a piece of non-constructive classical mathematics has no measurable time frame built into it. As already noted by virtue of the meaning of non-constructive, the established possibility is in principle inaccessible to any finite combination of systematic rule governed procedures. But the implementation of any systematic rule-governed procedure takes time, however short. Thus by virtue of being non-constructive, the logical possibility established has no time constraints imposed on it.
Contrast this with an economic model which proves the existence of some economic possibility using only the resources of intuitionistic mathematics. Since its domain of discourse is potential infinity and since its methods of proof of existence are necessarily constructive, any mathematical entity proven to exist is in principle accessible in a finite number of systematic steps. Thus the mathematical entity is conclusively established as being in a finite subset of the potential infinite domain of intuitionistic mathematics. Moreover possibilities established by intuitionistic mathematics have, by the very definition of constructable, a time tag built into them: each finite step takes time to implement. In view of its origins in the intuition of time, it is not surprising that some sequential time frame or other is built into the established possibilities of intuitionistic mathematics.
The difference between economic modelling based solely on the resources of intuitionistic mathematics as opposed to the full resources of classical mathematics is clearly illustrated in the mathematical modelling of economic decision-making. Any theorem of classical mathematics which is based on a non-constructive proof cannot model the inherent temporality of economic decision-making. Any piece of classical mathematics which has any non-constructive dimension to its proof used to model economic decision-making is in principle inaccessible to any finite combination of rule-governed processes. This necessarily follows from the non-constructive element in the piece of mathematics used to model economic decision-making. By contrast, any outcome of a piece of intuitionistic mathematics which may be a candidate for the modelling of economic decision-making is in principle accessible to some finite combination of rigorous rule-governed steps. Since the implementation of a finite rule-governed step takes time, a time tag is necessarily built into a model based on intuitionistic mathematics whereas this time tag is explicitly excluded by any classical mathematical model which has a non-constructive dimension to its mathematical proof. In short, the resources of intuitionistic mathematics can capture the temporal nature of economic decision making, whereas those of non-constructive classical mathematics cannot capture this temporal dimension.
To sum up, intuitionistic philosophy of mathematics plays a negative and a positive role in the methodological debate focused on the formalisation of economics. Negatively it exposes the illegitimacy of the orthodox attempt to give some transfinite and transtemporal possibilities of classical mathematics, finite and temporal economic interpretations. It thereby exposes the economic vacuousness at the mathematical core of the neo-Walrasian programme – the paradigm of the formalisation of economics for some orthodox economists. Positively it offers theoretical economists new mathematical resources, viz. the resources of intuitionist mathematics, in their efforts at mathematically modelling economic decision-making and economies. This intuitionistic mathematics necessarily eschews the transfinite and transtemporal possibilities at the core of classical mathematics.
The third major event in the philosophy of mathematics exploited in previous chapters is the Church-Turing thesis which, as we saw in Chapter Seven, arose in the wake of Gödel’s incompleteness theorems. Simply put Gödel showed that a strict proof of the consistency of advanced axiomatised mathematics, in Tarski’s phrase, ‘meets with great difficulties of a fundamental nature’ (Tarski 1965: 137). Moreover the situation vis-à-vis completeness is worse: it is possible to set up mathematical problems that can neither be positively nor negatively decided within pure axiomatised mathematics. As Tarski pointed out these results are not premised on the assumption of some current imperfection or other in our axiomatic system which future mathematicians could eliminate. Rather ‘never will it be possible to build up a consitent and complete deductive theory containing as its theorems all true sentences of arithmetic or advanced geometry’ (Tarski 1965: 137, italics ours). Without getting involved in the intriguing historical details, we can sum up by saying that ‘these exceedingly important achievements’ (Tarski 1965: 138) were accomplished over a short period in the 1930s by Gödel, Church and Turing.
As we pointed out in Chapter Seven, these achievements in the foundations of mathematics gave birth to our computer/digital age. From the point of view of the formalisation of economics, these achievements put the spotlight on to what we called computable mathematics. This is a specialised branch of classical mathematics which is limited to algorithmetic methods. Computable mathematics is not to be confused with intuitionistic mathematics: computable mathematics is based on classical logic whereas intuitionistic mathematics is based on intuitionistic logic. Both, however, focus on finite, systematic, rule-governed procedures. The Church-Turing thesis, as it were, collects ranges of specialised computable mathematics under one umbrella, thereby opening up an extensive reservoir of algorithmetic mathematics to the economic theoretician. In particular, given the Church-Turing thesis, the following methodological question vis-à-vis the formalisation of economics arises: given the uniqueness of economic theorising should economic theoreticians confine their mathematical modelling to the resources of computable mathematics? This methodological question is far from simple: there are at least three layers to it. Firstly, what are the advantages of computable mathematical modelling for economists over modelling using the full resources of classical mathematics? Secondly, what are the advantages of computable mathematical modelling over modelling confined to intuitionistic mathematics? Thirdly could a case be made for a judicious combination of computable and intuitionistic mathematics in economic modelling? Vis-à-vis the first layer, i.e. the advantages of computable mathematical modelling over modelling using the full resources of classical mathematics, the answer is that these advantages are the same as those outlined above for economic modelling based on intuitionistic mathematics as understood in pragmatic intuitionism. This overlap of common advantages is not surprising: both pragmatic intuitionistic mathematicians and computable mathematicians share a commitment to finite, rule-governed, systematic, mechanical procedures. The difference between them lies in their logical commitments. Thus the methodological issue in layer two, viz. the advantages of intuitionistic over computable modelling, will centrally include an analysis of which logic, classical extensional logic or intuitionistic non-extensional logic, is most appropriate to actual economic reasoning. To fully address this issue one would require a volume on the philosophy of logic. However, as we have already seen, Arrow points to empirical evidence which suggests that natural language based decision-making is not extensional. In addition we suggested that the resources of modal logic could prove fruitful. However, given that computable mathematics is based on classical extensional logic and that economic decision-making is conducted in natural languages which are not extensional, prima facie either intuitionistic or modal logic or a combination of both is better suited to the analysis of economic decision-making rather than classical extensional logic. The third layer, viz. whether a judicious combination of computable and of intuitionistic mathematics could be used in economic theorising is an intriguing one, which we have not addressed. Given Solow’s dictum ‘problems must dictate methods, not vice versa’ (Solow 1954: 374) the judicious choice from either may prove fruitful. For instance, adherence to the canons of logic is an indispensable ingredient of all rationality, including economic rationality. Thus theoretical economists would exploit intuitionistic mathematics in their modelling of rational economic decision-making. On the other hand, since all economies are finite and, as we saw in Chapter Three, classical logic was devised for finite systems, theoretical economists could use computable mathematics in their modelling of economies.
Clearly these speculations are controversial and require further scrutiny. However what is not controversial is that the philosophy of mathematics brings the age of innocence in the orthodox formalisation of economics to a close: it challenges the orthodox assumption that the recourse to the full range of classical mathematics in economic modelling is unproblematical. In particular the philosophy of mathematics exposes how the exploitation of the full resources of classical mathematics in the neo-Walrasian programme – the paradigm of formalisation in orthodox economics – is fundamentally flawed. In this vein the philosophy of mathematics plays a key role in identifying an insurmountable paradox in the use of the full resources of classical mathematics in modelling real economies. This paradox is rooted in two fundamental truths, one mathematical and the other economic. The mathematical truth is that the domain of classical mathematics is transfinite: it is an infinity of infinite sets which abstracts from any possible time constraint. Moreover, given that various mathematical theorems exploited in these economic models are based on non-constructive proofs, there is no mathematical constructable route within these transfinite models to any finite sub domain. The economic truth is that, whatever an economy is, it is a finite temporal system. Numerous orthodox economic models, however, by virtue of the classical mathematics they exploit are rooted in this transfinite mathematical domain. Furthermore, since orthodox economic theorising lacks the sophisticated experimental techniques characteristic of theoretical physics, there is also no empirical route from its transfinite models to any finite region. In short there is no route, neither mathematical nor empirical, connecting various elegant and sophisticated models of advanced orthodox theorising based on non-constructive proofs to any finite, temporal region. Ironically this inability to offer any insight into any finite temporal system is rooted in the mathematics celebrated by these orthodox economic theoreticians.
Despite the fact that events in the philosophy of mathematics put the final nails in the coffin of orthodox economic modelling as exemplified in the neo-Walrasian programme, the programme of formalisation of economics per se remains healthy. Its health lies in the wealth of constructive proofs in both intuitionistic and computable mathematics: the constructive nature of their proofs guarantee that their theorems are in principle applicable to finite, temporal systems. Thus intuitionistic and computable mathematics open up rich reservoirs of novel mathematical results for economic modelling. Whether or not such an alternative programme or programmes for the formalisation of economics will succeed is, as we have repeated time and time again, a matter for detailed economic research in an economic climate which aspires to empirical adequacy. In short, the philosophy of mathematics makes an indispensable contribution to both the analysis of the manner in which theoretical economics was formalised and to the critical evaluation of that process, while simultaneously opening up alternative ways of mathematically modelling actual economies in a more realistic way.