By focusing on Walras’s and Debreu’s philosophies of economic analysis we have responded to Weintraub’s challenge to reflect on ‘how economics has been shaped by economists’ ideas about the nature and purpose and function and meaning of mathematics’ (Weintraub 2002: 2). In connection with Debreu we saw how he imposed the Hilbertian formalist philosophy of applied mathematics on to the historical developments of the neo-Walrasian programme accomplished in journals from the 1930s to the 1950s. The result of this imposition is an austere, rigorous ‘definitive statement of the theory of competitive equilibrium’ (Hahn 1961: 204). Debreu’s Theory of Value is not a definitive analysis of the actual methods used by individual economists who pioneered neo-Walrasian theorising: it has nothing to say about how these economists actually theorised. Rather this Theory of Value is a rigorous definitive axiomatic representation of the outcome of their research, namely competitive general equilibrium theory.
In this chapter we focus on the reception of Debreu’s formalisation. Some economists saw it as a monumental achievement, others were utterly hostile to it. Those who saw it as a major achievement did not form a homogeneous group.1 Among these neo-classical economists we distinguish a majority opinion from a minority opinion. A majority of neo-classical economists read Debreu’s contribution as being indispensable to their scientific investigations of actual economies. For instance, writing in the 1970s Kaldor maintains that neo-classical economists hold that Debreu’s contribution ‘is the one and only starting point’ (Kaldor 1972: 1238) for their scientific theorising. In a similar vein, and writing much later, Hahn points out that various neo-classical economists – particularly American – read it as making an indispensable contribution to their scientific descriptions of actual economies. The mention of Hahn brings us to a minority view. According to Hahn, those neo-classical economists who read general equilibrium theory as an empirical science are utterly mistaken. Whatever Debreu has accomplished and that is outstanding – it does not establish that some real economy will be or tends towards an equilibrium state.2 Indeed general equilibrium theory, contrary to numerous American economists, is not a scientific theory at all. Rather Debreu’s Theory of Value makes a major contribution to an objective, clearly articulated grammar of argumentation which enables applied economists to critically and objectively interrogate economic policy. Practically speaking ‘it is a line of first defence against madmen and witches’ (Hahn 1985: 4). Theoretically, it furnishes economists with an objective instrument in their efforts at understanding ‘the messy reality’ (Hahn 1996: 193) of actual economies.
In particular, in this chapter we compare and contrast Hahn’s own philosophy of the formalisation of economics with that of Debreu. In this connection we show how Hahn is in line with the Hilbertian formalists in maintaining that Debreu’s existence proof merely establishes the logical possibility of equilibrium and nothing more. However, in developing his own unique philosophy of the formalisation of economics as a grammar of argumentation, Hahn does not espouse Debreu’s Hilbertian formalist template for the axiomatisation of economics. Rather he has recourse to what we called the Euclidean Template. Given his grammar of argumentation, Hahn then undermines various erroneous scientific interpretations of general equilibrium theory on the part of various neo-classical economists.
It would be utterly misleading to give the impression that the general equilibrium formalisation of economics was not radically questioned by reputable economists. Because of the unambiguity of his challenge and, in particular, because of his objections to its formalisation we focus on Kaldor’s rejection of the neo-Walrasian programme. Kaldor, like Hahn, is fully aware that Debreu’s Theory of Value is ‘an elegant, exact and logically precise’ account of general equilibrium (Kaldor 1972: 1237). Unlike Hahn, Kaldor emphasises Debreu’s formalist approach to economic theorising. This formalist approach, according to Kaldor, is its Achilles’ heel. In place of Debreu’s formalist template for economic theorising Kaldor proposes an empirical template. According to Kaldor scientific theorists begin their theory construction with a summary of the known observational facts of the domain under investigation. In the case of economics, since the required summary is frequently presented in statistical fashion, economic theorists start with a stylised view of its facts. These stylised facts are statistical generalisations about observed empirical regularities. The culmination of this empirical approach is ‘a body of theorems based on assumptions that are empirically derived (from observations) and which embody hypotheses that are capable of verification both in regard to assumptions and predictions’ (Kaldor 1972: 1238). In light of this empirical view of economic theorising and Debreu’s formalist approach to economic theory it is not surprising that Kaldor concludes that ‘without a major act of demolition – without destroying the basic conceptual framework – it is impossible to make any real progress’ (Kaldor 1972: 1240).
One way of approaching the contribution of Debreu’s Theory of Value is to locate it in the broad context of the general perception of economics and in the specific context of its perception by orthodox, neo-classical economists. Vis-à-vis the broad context of the 1950s we have recourse to Friedman (1953) in his influential methodological paper, ‘The Methodology of Positive Economics’. Indeed one may feel that Friedman’s summary has a lot going for it today. According to Friedman the subject matter of economics is regarded by most ‘as vitally important’ but also that ‘it is the source of continuous and extensive controversy’ (Friedman 1953: 3–4). These controversial differences are evident in debates about economic policy, such as the minimum wage, where ‘a welter of arguments for and against’ (Friedman 1953: 5) is evident. In these policy disputes ‘self-proclaimed “experts”’ (Friedman 1953: 4) voice their conflicting opinions. In addition to Friedman’s self-proclaimed experts one could add that reputable economic advisors also contribute to these conflicting opinions. This situation in economics is in marked contrast to ‘the prestige and acceptance’ of the views of physicists, which as Friedman correctly points out derives ‘from the evidence of their works, the success of their predictions and the dramatic achievements from applying their results’ (Friedman 1953: 4).
To extricate the discipline of economics from this world of divergent opinion, Friedman argued for the clear-cut distinction between ‘normative economics’ – the domain of policy decision-making – and ‘positive economics’ which is totally independent of normative judgements. Positive economics, being a scientific discipline on par with physics, is to serve as the objective basis for consensus in policy issues. According to Wade Hands, Friedman’s piece is ‘clearly the best-known work in twentieth-century economic methodology’ (Hands 2001: 53) and according to Hausman even almost fifty years after its publication it remains ‘the only essay on methodology that a large number, perhaps a majority of economists, have ever read’ (Hausman 1992: 162). Hahn would not object to these assessments of Friedman’s influence: Friedman’s methodological piece informed the context into which Debreu’s Theory of Value was integrated. Thus numerous economists, especially American, read Debreu as making a major contribution to economics understood as a science on par with physics. In a similar vein according to Kaldor by the 1970s Debreu’s general equilibrium theory was the prevailing scientific theory of value among neo-classical economists. Neo-classical economists held ‘a deep underlying belief’ that general equilibrium theory ‘is the one and only starting point for any logically consistent explanation of the behaviour of decentralized economic systems’ (Kaldor 1972: 1238).
This methodological approach to general equilibrium theory is vehemently opposed by Hahn: general equilibrium theory is not a scientific theory in the sense in which physics is a scientific theory.3 According to Hahn the methodological claim that theoretical economics is a science ‘is not only premature and not very honest but also, perhaps worse, pretentious’ (Hahn 1993b: 163). By insisting that economics is a science, ‘the subject gets a bad name’ (Hahn 1993a: 91). One cannot imagine theoretical physicists insisting to their students that ‘everything I will teach you is in a sense false but useful’ (Hahn 1996: 191). Neither would theoretical physicists claim that Hahn’s thesis that economic theory ‘is at best a powerful aid to thought about the world, not because it provides a very satisfactory description, but because it provides clear limits to understanding’ (Hahn 1993b: 164) conveys the contribution of theoretical physics to our knowledge and technological control of the physical world. Hahn amasses a diverse body of evidence in favour of his unconventional thesis that theoretical economics is not a scientific theory. Firstly, relative to theoretical physics theoretical economics’ predictive record is very poor. For instance it is no match for quantum physics in the accuracy of its predictions. Moreover, he concurs with Debreu that economic theory cannot claim to be substantiated or confirmed by experimental testing à la physics. Like Debreu, mathematical economics is not, in this regard, akin to mathematical physics. Thirdly, unlike physics ‘economics has no laws’ (Hahn 1996: 192). Indeed Hahn loses his patience with economists who insist on the iron laws of economics. These so-called iron laws ‘are not laws which hold in all societies and at all times’ (Hahn 1993a: 91). In Hahn’s opinion this ‘babble’ (Hahn 1993a: 95) about economic laws is rooted in the erroneous belief that economics is a science. When economists claim to be scientists ‘they sooner or later babble about “the laws of economics” … The subject gets a bad name’ (Hahn 1993a: 95).
Hahn, in his forceful opposition to the conventional assumption that theoretical economics is a science, may sound like a fundamentalist hermeneutical philosopher who relishes in pointing out the impossibility of any scientific study of human interactions. This of course is not the case. Hahn is, at it were, a whistle blower who, because of his deep-seated commitment to general equilibrium theory and because of the damage caused by its presentation as a science, feels he has no option but to undermine this harmful methodology of economics. In this connection he singles out Friedman’s catastrophic methodological defence of economic theory. Friedman’s prescriptive methodology ‘may be fine for Quantum theory (predictions confirmed to a higher order of decimals) but only dangerous sloppiness and blinkered arguing can result from its use in economics’ (Hahn 1996: 185, italics ours). Rather than being a science, general equilibrium theory is an objective grammar of argumentation which is very useful in policy matters as ‘a line of first defence against madmen and witches’ (Hahn 1985: 4). In short, in order to extricate economics from the world of opinion there is no need, à la Friedman, to exaggerate the undoubted success of economic theory by misrepresenting it as a science on par with physics. Rather the success of economic theory lies in it being an objective, rigorous grammar of argumentation which enables economists to gain an objective footing in their studies of our complex, dynamic world. This objective footing, however, is not in the business of predicting, á la physics, the course of any economy. Rather economic theory protects us from self-proclaimed experts who possess all the answers to contemporary economic challenges. This is its practical usefulness. Its theoretical usefulness is that it contributes to the objective, understanding of our economic world, by furnishing economists with a rigorous grammar of argumentation.
In his Jevon’s Memorial Fund Lecture, delivered at University College London in November 1984, which he titled ‘In Praise of Economic Theory’, Hahn remarks that ‘In Praise of Theorizing in Economics’ would have been a more appropriate title. This shift in proposed title from economic theory to economic theorising marks Hahn’s shift in emphasis to the grammar of argumentation which he sees as informing the on-going, dynamic process of general equilibrium theorising. Hahn is undoubtedly correct in insisting that general equilibrium theorists ‘are not slouches’ (Hahn 1996: 187). They have forged and continue to forge innovations at the frontiers of their research. To gain an accurate appreciation of neo-Walrasian general equilibrium theory, economists and methodologists need to focus on this dynamic activity of theorising at the frontiers of research. When one examines this activity one sees that it is informed by a grammar of argumentation which gives economists an objective footing in their attempts to come to grips with ‘the messy reality’ (Hahn 1996: 193) of actual economies.
This grammar of argumentation renders theoretical economics unique. One such unique characteristic is its solid starting point. The solidity of this starting point in this grammar of argumentation is guaranteed by Debreu’s rigorous proof of the existence of equilibrium. As Hahn puts it ‘if we did not have Arrow-Debreu theory, it would be priority number one to construct it. For, while it does not describe the world, it is a solid starting point for the quest for understanding it’ (Hahn 1985: 14). This solid starting point ‘is precise, complete and unambiguous’ (Hahn 1973: 3). Arrow and Debreu ‘wrote down precisely and beyond the power of misunderstanding by a normal person, what state of an economy was to be designated as an equilibrium’ (Hahn 1985: 12). This clear, unambiguous definition of equilibrium is required for at least two reasons. Firstly, while the term equilibrium is extensively used in various applied contexts such as popular analyses of economic situations or speeches by ministers for finance, these meanings do not suffice for mathematical economics, where clearly defined terms are the norm. Thus Debreu clearly defines for all to see what he meant by the term equilibrium. As we have already seen, this unambiguous meaning of the term equilibrium is achieved by various idealisations and assumptions which are explicitly stated and not tacitly assumed.
The second reason for such a clear, unambiguous characterisation of equilibrium is that, since Adam Smith, economists have been divided on his thesis of the Invisible Hand. For many the Invisible Hand thesis is an astounding claim. How can millions of agents each solely concerned with his or her own individual interests and acting only on the sparse information of prices attain any coherent economic arrangement? Without a clear, unambiguous definition of such an equilibrium arrangement and a rigorous proof of its possibility, the debate on the Hidden Hand is condemned to remain in the subjective domain of opinion. The genius of Arrow-Debreu is that they ‘provided the first essential step in any discussion of the Invisible Hand …’ (Hahn 1985: 13). This essential first step included, as we have already indicated in the previous paragraph, a clear unambiguous definition of an equilibrium state. This essential first step culminates in Debreu’s rigorous proof that such an equilibrium state is logically possible. In other words, by recourse to very sophisticated mathematics, Debreu conclusively proves that the concept of an equilibrium state, as defined by the idealisations and assumptions of his Theory of Value, is free from contradiction.
Contrary to what some neo-classical economists assume, Hahn points out that Debreu’s monumental achievement does not prove that this logically consistent concept actually or even approximately applies to or describes an actual economy. As Hahn puts it ‘There is nothing here (Debreu’s proof) to tell us that any given economy will be in that state or that it tends to that state’ (Hahn 1985: 12). Anyone who understands the various idealisations and assumptions readily sees that Debreu’s Theory of Value is not describing an actual economy. In particular this solid starting point, contrary to the misleading image created by Friedman’s methodology, is not a body of knowledge which has been experimentally tested and vindicated by successful predictions. Rather the monumental achievement of Debreu lies in spelling out in no uncertain terms the idealisations and assumptions under which an equilibrium state is shown to be logically possible. This remarkable achievement is the solid starting point for the neo-Walrasian grammar of argumentation which will enrich our objective understanding of the messy reality of actual economies.
Hahn is well aware that some eminent economists – he explicitly names Kaldor and Kornai – are hostile to the axiomatisation of economics. Contrary to this hostility Hahn maintains that ‘any coherent general propositions are decomposable into this form …’ (Hahn 1985: 5). Taken literally, Hahn is not claiming that economic theorising is or should be carried out as an axiomatic-logico deductive system. Rather he is claiming that given any piece of economic theorising one can analyse or decompose it by recourse to the axiomatic template. The basic advantage of such a decomposition is that economists become very clear on both the axioms, and assumptions used in any piece of economic theorising. In other words axioms and assumptions pervade economic theorising and by recourse to the axiomatic template economists make their axioms and assumptions clear and explicit. Our thesis is that Hahn, in analysing a piece of economic theorising into axioms and assumptions, is using the Euclidean, as distinct from Debreu’s Hilbertian formalist, template of axiomatisation. This, we contend, is substantiated by the manner in which Hahn draws the distinction between axioms and assumptions. As we saw in Chapter Four, the distinction between axioms as fundamental truths and assumptions which lack that fundamental basis plays no role in the formalist axiomatisation of economics exploited by Debreu. For Hahn, however, this distinction between axioms and assumptions is indispensable to Hahn’s own analysis of the grammar of argumentation of economic theorising.
Few would challenge Hahn’s claim that ‘the best known and most important axiom is, of course, that of the rational agent …’ (Hahn 1985: 5). He insists that this axiom is not a Friedmanite, implausible, ‘as if’ hypothesis which neo-classical economists plucked out of thin air with no basis in the real economic world. On the contrary, the rational agent axiom is ‘a claim about this world so widely agreed as to make further argument unnecessary’ (Hahn 1985: 5). This is clearly correct within neo-classical economics: the consensus among neo-classical economists is that this axiom is, in Hahn terminology, ‘an abstract unifying principle’ of their theorising. The same consensus, however, does not prevail among so-called heterodox economists. Be that as it may, Hahn makes an additional claim, namely ‘in the first instance the axiom says that economic agents are persons’ (Hahn 1985: 6). This additional claim is crucial for Hahn. We all recognise that the proposition economic consumers and producers are human beings is self-evidently true. Hahn maintains that this fundamental, self-evident truth is an indispensable component of the rational agent axiom. If that is the case then it necessarily follows that the grammar of argumentation based on the rational agent axiom has a genuine foothold in the real economic world. By virtue of this self-evident truth, the outcome of the grammar of argumentation of neo-Walrasian theorising is not pure fiction. This self-evident truth is the umbilical cord connecting theoretical economics to the real economic world.
However, if one consults either textbooks or research papers where the rational agent axiom is used, one certainly will not find it expressed as rational agents are persons. It goes without saying that Hahn is very well aware of this. As he points out, the term rational is vigorously debated in philosophy and elsewhere. To avoid these debates, mathematical economists are very precise in their use of the term rational. In economics the rational agent axiom means the following. ‘The rational agent knows what he wants and from the among the alternatives available to him chooses what he wants’ where ‘knowing what he wants’ means that the agent ‘has a proper preference ordering over a relevant domain’ (Hahn 1985: 5). Hahn’s claim is that this precise axiom contains the self-evident truth that economic agents are persons. In this connection Hahn contrasts persons with other animals on the one hand, and with beings with no preferences at all on the other. While a person and an animal reveal their preferences in their respective behaviours, a person, unlike other animals, has opinions and beliefs about the elements in his domain of choice which he articulates in language. Indeed, a person, unlike other animals, gives reasons for his preferences. On the other hand, a being with no preferences – which of course should not be confused with a being which is indifferent between the elements of its choice set – is not a person at all. Thus the neo-classical rational agent is in the first instance a person, not an animal nor a being with no preferences at all. As we already indicated, a basic reason for Hahn’s insistence on this point is to counteract those who dismiss neo-Walrasian theorising on the grounds that it is not in touch with economic reality.
Of course Hahn readily admits that there is more to the rational agent axiom than what is conveyed by the self-evident truth that economic agents are persons. The self-evident truth that economic agents are persons is transformed into the rational agent axiom by a process of idealisation and strengthening. Indeed, as Hahn points out, the rational agent axiom is ‘idealized and strengthened by theorists beyond the point at which it commands universal consent’ (Hahn 1985: 7). In other words whereas the claim that economic agents are persons rightly commands universal consent, the rational agent axiom does not command the same universal consent. For instance in addition to the self-evident truth that economic agents are persons the rational agent axiom contains the idealisation that the agent’s preferences are transitive. This idealisation may be questioned. Indeed an integral part of Hahn’s grammar of argumentation is the rigorous mathematical interrogation of such idealisations. This interrogation is evident, for instance, in some of Sonnenshein’s research. There are, however, limits beyond which this interrogation can go. In this case the limits are set by the fundamental truth that economic agents must be capable of giving reasons for their actions. For instance, if the proposed specific kind of intransitivity, which of course must be clearly articulated in mathematical terms, made it impossible for economic agents to give coherent reasons for their actions, then that kind of intransitivity would have to be rejected.
At the level of economic axioms an indispensable dimension of Hahn’s grammar of argumentation may be summed up as follows. The theoretical economist starts with fundamental or agreed features of the economic world. These basic or self-evident truths are then sharpened by the process of idealisation. This process culminates in rigorously formulated economic axioms which serve as unifying principles in economists’ endeavours at understanding the events of the economic world. At some subsequent state economists may have sound reasons to interrogate a particular idealisation used to sharpen a fundamental or consensus truth. There are, however, limits to this process of interrogation of idealisations, namely any alternative rigorously constructed idealisation must not contradict the pre-theoretical, fundamental or self-evident truths of economics.
Hahn, moreover, is emphatic that one should distinguish between economic axioms and assumptions. Axioms are inextricably linked to fundamental truths whereas assumptions are not. However, like axioms, assumptions are not capriciously imposed. There has to be some good reason for imposing assumptions. For Hahn these reasons range from cutting down the domain of choice, to requirements of tractability and simplicity to what he calls ‘a casual empiricism’ (Hahn 1985: 6). Hahn furnishes a number of examples of his distinction between axioms and assumptions. For instance ‘agents have preferences and try to satisfy them’ is a axiom, whereas ‘universal perfect competition must count as an assumption’ (Hahn 1985: 10). His explicit reason for locating universal perfect competition as an assumption is that ‘neither introspection nor observation make it self-evident up to an acceptable margin of error’ (Hahn 1985: 10–11). Universal perfect competition is too far removed from basic truths and hence cannot be deemed to be an axiom. Take another example. Managers have preferences is an axiom, while it is assumed that these preferences are linear in expected profit.
While recourse to simplifying assumptions is indispensable to Hahn’s grammar of argumentation, he advises ‘circumspection in praising simplicity in economic theorizing’ (Hahn 1994: 250). In this connection he concludes that various neo-classical economists have not exercised due care. These economists end up proposing ‘silly models’ (Hahn 1994: 251). Hahn lays the blame for this at Friedman’s door. Under Friedman’s influence – who insists that theoretical economics is an empirical science which should be evaluated solely on its predictive merits, without any consideration being given to the realism of its assumptions – these economists engage in ‘blinkered’ theorising. For instance vis-à-vis his own research on the Solow growth model with many capital goods in which the steady state turned out to be a saddle point with the consequence that numerous equilibrium points are possible, ‘the Chicago economists’ responded by simplifying even more than what Solow had done by using ‘the silly assumption’ that ‘the economy followed an equilibrium path over infinite time’ (Hahn 1994: 251). In principle Hahn’s circumspection vis-à-vis simplicity goes quite far: ‘if at a certain stage of knowledge nothing can be said without drastic simplifying falsification, then perhaps we should keep quiet’ (Hahn 1994: 251).
In practice, however, Hahn admits that he himself has not adhered to his own reservations about simplicity in his economic theorising. Despite maintaining that ‘the perfect competition simplification has had rather disastrous effects on macro-economics’ (Hahn 1994: 252) he concedes that a good deal of his own research is based on that simplification. One is thus left with the issue of what would be contained in the outcome of neo-classical theorising if theoretical economists practised what Hahn preached about the limited use of simplicity. Be that as it may, Hahn agrees that ‘the basic assumptions of much of our theory are often of low descriptive merit’ (Hahn 1993b: 163). Given this, another indispensable dimension of Hahn’s grammar of argumentation is the project of increasing the descriptive merit of these assumptions. For instance he readily admits that increasing returns constitute a telling objection to the simplification of perfect competition. His grammar of argumentation requires the rigorous mathematical explication of increasing returns and thereby the reshaping of the original Debreu theory. Indeed he maintains that the integration of increasing returns into his grammar of argumentation means that Adam Smith’s invisible hand ‘is likely to be unsure in its operations and occasionally downright arthritic’ (Hahn 1982a: 129). The crucial point is that this is not a matter of opinion. It is the outcome of a rigorous grammar of argumentation. Overall the outcome is that the Debreu conception of an equilibrium is replaced by ‘a more feet on the ground Keynesian notion’ (Hahn 1973: 16).
Hahn is well aware that he is portrayed as ‘a dyed-in-the-wood neo-classical who considered Arrow-Debreu adequate for all economics’ (Hahn 1984: 18). Obviously Hahn rejects this portrayal. An important reason for this rejection lies in his view that neo-Walrasian theorising is an objective grammar of argumentation which starts with Arrow-Debreu but does not end there. Indeed Hahn confesses to be ‘completely agnostic’ (Hahn 1984: 18) on the final truths of economics. However, what he is not agnostic about is the grammar of argumentation: unambiguous results in economic theorising can only be achieved by this grammar of argumentation. In Hahn’s view this grammar of argumentation is grounded in ‘three commitments’ (Hahn 1984: 2), namely rationality, equilibrium and methodological individualism. As the focus of this work is on the role of mathematics in economic theorising, we have excluded Hahn’s commitment to methodological individualism. In the last two sections we introduced a brief synopsis of Hahn’s, possibly unique, conception of the formalisation of economics as a grammar of argumentation, grounded in his commitments. According to Hahn, this grammar of argumentation questions almost everything in the Debreu original formulation. However, among the items which cannot be questioned is the rigorous proof of the existence of equilibrium furnished by Debreu.4
Like Hahn, Kaldor notes that the term ‘equilibrium’ is used in all kinds of economic contexts. His disenchantment is with the specific notion of equilibrium originally formulated by Walras and rigorously developed by the mathematical economists of his own generation ‘of whom perhaps the French economist, Gerard Debreu, is now regarded as the most prominent exponent’ (Kaldor 1972: 1237). This research informs ‘the prevailing theory, as taught in regular textbooks in most of the universities of the Western World’ (Kaldor 1975: 347). This prevailing theory’s ‘declared objective is to explain how economic processes work in a decentralized market economy’ (Kaldor 1975: 347).
Kaldor draws out attention to the original aspiration of the neo-Walrasian programme, namely to establish the existence of an equilibrium set of prices that is unique, stable and satisfies the conditions of Pareto optimality. Uniqueness would establish that there is one and only one such equilibrium and stability would demonstrate that this equilibrium set of prices would maintain itself in the face of chance disturbances. Existence, uniqueness, stability and Pareto optimality constituted, in the language of Ingrao and Israel, ‘the invariant paradigmatic nucleus’ of the neo-classical interpretation of Adam Smith’s hidden hand (Ingrao and Israel 1990: 3). Debreu’s Theory of Value was seen as the first major step in the justification of this invariant nucleus. This invariant paradigmatic nucleus, however, quickly disintegrated when it was established that uniqueness and stability proofs eluded these mathematical economists. It is not unreasonable to assume that Kaldor would agree with Kirman that ‘without stability and uniqueness the intrinsic interest of economic analysis based on general equilibrium is extremely limited’ (Kirman 2006: 257). This is presumably part of the reason for Kaldor’s call for ‘a major act of demolition’ (Kaldor 1972: 1240). These significant failures, however, did not bring about the demise of the neo-Walrasian programme. Neither did it adversely impact on its influence. On the contrary, Kaldor believes that ‘the great majority of academic economists take for granted that the economy always approaches, or is near to a state of “equilibrium”’ (Kaldor 1972: 1239), with the consequence that this near equilibrium state is a major causal factor in the real economic world. Allowing for some rhetorical flourish, it is clear that Hahn would concur with this view. Moreover, like Hahn, Kaldor is very clear that this erroneous belief has not been established by pure mathematical economists. In this connection he acknowledges that Debreu describes his Theory of Value as the explanation of prices of commodities in a private sector economy. Kaldor, however, draws our attention to Debreu’s formalism and correctly concludes that, given this formalism, Debreu is not claiming that his Theory of Value explains, in either the ordinary or scientific sense of the term ‘explanation’, these prices. Rather, as Kaldor maintains, Debreu’s theory:
is not put forward as an explanation of how actual prices of commodities are determined in particular economies or in the world economy as a whole. By the term ‘explanation’ Debreu means a set of theorems that are logically deducible from precisely formulated assumptions; and the purpose of the exercise is to find the minimum ‘basic assumptions’ necessary for establishing the existence of an ‘equilibrium’ set of prices (and output/input matrixes) that is (a) unique, (b) stable, (c) satisfies the conditions of Pareto optimality.
(Kaldor 1972: 1238)
In short, for Kaldor Debreu’s conception of explanation should not be confused with the scientific notion of explanation.
In this connection Kaldor contrasts the situation in theoretical physics with that of theoretical economics. Contemporary theoretical physics is such that its basic axioms are not derivable by the scientific process of induction. However, these non-inductive, first principles of theoretical physics come before the bar of experience. As noted by Friedman, this is done by the empirical interrogation of their consequences. This contrasts with the situation in general equilibrium theory. Here the
observations which contradict the basic hypotheses of prevailing theory are generally ignored: the “theorist” and the ‘empiricist’ operate in two isolated compartments and the challenge of anomalous observations is ignored by the theorist as something that could be taken into account at the stage of ‘second approximation’ without affecting the basic hypotheses.
(Kaldor 1972: 1239)
Unlike Hahn, Kaldor does not attempt to draw a fundamental methodological distinction between economic axioms and economic assumptions. His focus is on the basic hypotheses of general equilibrium theory which he characterises as assumptions. While not attempting to fully enumerate these, he gives the reader a flavour of what he has in mind. These include perfect knowledge of all relevant prices on the part of economic agents; perfect foresight in the sense that individual agents’ future experiences will confirm, and not disappoint, their expectations on which they based their past decisions; perfect competition such that each agent can sell or buy anything in unlimited amounts without affecting market prices with the consequence that prices are the only type of information required for individual decisions. Kaldor’s key methodological point is that such like assumptions are not chosen on the basis of direct observation. On the contrary some are unverifiable while others are directly contradicted by observation. These arbitrary basic assumptions are necessitated by the demands of logical consistency.
In view of their low descriptive value, why, in Kaldor’s view, is Debreu’s Theory of Value so central to neo-classical economics? Kaldor maintains that neo-classical economists have ‘the deep underlying belief … that general equilibrium theory is the one and only starting point for any logically consistent explanation of the behaviour of decentralized economic systems’ (Kaldor 1972: 1239). In the face of the arbitrariness of its assumptions, this deep underlying belief ‘sustained’ (Kaldor 1972: 1239) general equilibrium theory. If we unpack this, we see that, in Kaldor’s opinion, neo-classical economists have, in Hahn’s phrase, two commitments, namely the commitment to decentralised economic systems and the commitment to Debreu’s Theory of Value as the one and only starting point for the explanation of the behaviour of these systems. Whether or not it is the one and only starting point, Hahn agrees that it supplies a solid starting point. Starting with the indubitable proof of the existence of equilibrium as characterised by Debreu, the neo-Walrasian programme entails the application of Hahn’s grammar of argumentation, which includes the relaxing of the unreal assumptions. According to Kaldor ‘the process of removing the “scaffolding,” as the saying goes – in other words of relaxing the unreal basic assumptions – has not yet started’ (Kaldor 1972: 1239). According to Hahn, Kaldor is just plainly wrong on this point. Hahn focuses on neo-Walrasian research which aims at a notion of equilibrium that is ‘sequential in an essential way’ which requires ‘that information processes and costs, transactions and transaction costs and also expectations and uncertainty be explicitly included in the equilibrium notion. That is what the Arrow-Debreu construction does not do’ (Hahn 1973: 16).
Kaldor, however, is not denying that such research has taken place. On the contrary he is assuming that research. Unlike Hahn, he sees neo-classical economists as introducing yet more unrealistic assumptions in that research. In this connection he draws our attention to what Hahn calls ‘silly’ models above. For instance, he points out that theoretical models which attempt ‘to construct an equilibrium path through time with all prices for all periods fully determined at the start under the assumption that everyone foresees future prices to eternity correctly’ (Kaldor 1972: 1239) is more unrealistic than the original Walrasian theory. In short, according to Kaldor, the neo-Walrasian theorist has ‘successfully (though perhaps inadvertently) demonstrated that the main implications of this theory cannot possibly hold in reality, but has not yet managed to pass his message down the line to the textbook writer and to the classroom’ (Kaldor 1972: 1240). In this sense the process of relaxing the assumption has utterly failed. The efforts at relaxing the assumptions have
ended in a ‘cul-de-sac’: it made the theory a less usable tool than it was thought to have been in its early and crude stage before the full implications of general equilibrium had been so thoroughly explored.
(Kaldor 1975: 347)
In short his ‘basic objection’ to general equilibrium theory ‘is that it starts from the wrong kind of abstraction, and therefore gives a misleading “paradigm” … of the world as it is: it gives a misleading impression of the nature and the manner of operation of economic forces’ (Kaldor 1975: 347).
This is not the place to analyse in detail the intricacies of the Hahn–Kaldor altercation.5 Our focus is on the formalisation of economics and responses to that formalisation. Within neo-classical economics we considered the response which saw the formalisation of the neo-Walrasian programme as resulting in a science, like theoretical physics, and the response of those like Hahn who utterly oppose that interpretation. We then took Kaldor as an exponent of those calling for its completion demolition. From our point of view what is interesting is the consensus among neo-classical economists that the formalisation of economics, exemplified in Debreu’s Theory of Value, is a solid starting point in their theorising, be it called scientific or grammatical, about decentralised economies. By recourse to developments in the philosophy of mathematics we will show how this consensus can be challenged. Debreu’s existence proof itself – the solid starting point of neo-Walrasian theorising – will be shown not to be as solid as originally thought. This only remaining component of the original nucleus of existence, uniqueness, stability and Pareto optimality will be logically undermined. Its Achilles’ heel will be shown to lie in the mathematics presupposed by Debreu!
1 For a detailed discussion of these see Boylan and O’Gorman (2012).
2 Thus Hahn is located in the Euclidean tradition of axiomatisation where the distinction between axioms and assumptions are crucial. See the previous chapter for details of this.
3 See Boylan and O’Gorman (2012) for a more detailed exposition and critique of Hahn’s philosophy of economics.
4 We will see in Chapter Seven how this proof may be challenged.
5 We have already addressed that challenge in Boylan and O’Gorman (2012).