In Chapters Three and Four we engaged Debreu’s conception of economic theorising, or, in his own words, his ‘philosophy of economic analysis’ (Debreu 1992: 114). As already outlined, his conception of theoretical economics is rooted in Hilbert’s formalist philosophy of mathematics, both pure and applied. In Chapter Five we considered the reception of the neo-Walrasian programme within economics, taking Hahn as a representative of those fully committed to that programme and Kaldor as a representative of its critics. In this connection we highlighted the commitments to both rationality and to equilibrium in the orthodox mathematisation of economics. In Chapter Six we used specific developments in the philosophy of geometry to buttress the challenge posed to the orthodox mathematical modelling of rationality by the post-Keynesian emphasis on the role of conventions in economic decision-making. In this chapter we return to the general philosophy of mathematics by bringing two well-known critiques of the Hilbert formalist philosophy of mathematics to bear on the orthodox commitments to both rationality and equilibrium. These critiques we argue furnish methodologists with a range of insights and themes into both the analysis and the critical evaluation of these orthodox commitments. Moreover, these critiques suggest an alternative way of formalising economic theory.
The first critique of the Hilbertian formalist programme – which we call the external critique – is associated with Brouwer and culminated in a new mathematics, called constructive or intuitionistic.1 In this connection it is crucial to note that constructive/intuitionistic mathematics is not a new chapter which can be simply added on to the existing corpus of classical mathematics. Rather it is a different mathematics grounded in a different logic! Intuitionistic mathematics rejects the logic which underpins classical mathematics and replaces it with a different logic, called intuitionistic logic. The emergence of constructive/intuitionistic mathematics raises a fundamental issue for the project of the mathematisation of economics – an issue which has not received due recognition in the methodology of economics dominated by the philosophy of science. For numerous economists the project of the mathematisation of economics is unproblematical: basically economists are using mathematics as a logical tool for the rigorous analysis of economic issues. However, in light of the emergence of intuitionistic mathematics, the issue is not that simple. Theoretical economists are, prima facie, faced with a genuine choice between constructive/intuitionistic mathematics and classical mathematics in their formalisation of economics. There is more than just one way to formalise economics. This choice in turn raises a fundamental methodological question: which mathematics – intuitionistic or classical – is more appropriate to the specific discipline of economics? Alternatively what are the respective advantages and disadvantages of classical and of intuitionistic mathematics for the formalisation of economics? In this chapter we address this fundamental issue.
The second critique of the Hilbert programme, which we call the internal critique, is associated with Gödel, Church and Turing. This culminated in the construction of the first electronic, store-programmed digital computer in 1948. Fundamental to this revolutionary engineering success is what we call computable mathematics. Computable mathematics, like constructive/intuitionistic mathematics, emphasises algorithmetic mathematical procedures but, unlike constructive/intuitionistic mathematics, uses classical, rather than intuitionistic, logic. Thus one could say that computable mathematics, unlike intuitionistic mathematics, is a specialised branch which limits itself to algorithmetic procedures within the broader corpus of classical mathematics. Perhaps the most amazing result in the development of computable mathematics is the proof of the existence of undecidable propositions within classical mathematics by Church and Turing in the 1930s. Contrary to what is sometimes tacitly assumed, post Church-Gödel-Turing, classical mathematics divides into algorithmetically decidable and algorithmetically undecidable theorems. This division raises another central question for the project of the mathematisation of economics, viz. does recourse to an algorithmetically undecidable theorem in economic modelling effect the explanatory value of such a model? In this connection we present a methodological case for the thesis that Debreu’s general equilibrium explanation of prices is rendered economically vacuous by its recourse to algorithmetically undecidable theorems.
Finally, in light of the poor prognosis for the neo-Walrasian programme, we address the following question. If theoretical economists were to confine their mathematical modelling to the resources of computable economics what positive contributions could they make to the scientific understanding of the economic world? If the economic exploitation of computable mathematics could deliver a range of positive results, then perhaps it could, in Mirowski’s words, ‘serve as a new template for an entirely new approach to the formalization of economic life’ (Mirowski 2012: 180). Whether or not that is likely to happen depends on numerous contingencies, ranging from economic theoreticians’ willingness to devote time to mastering the resources of computable mathematics to the power of influential institutions, for whatever reasons, committed to the status quo.2 These contingencies aside, our concern is with, in Debreu’s terminology, the extent to which ‘the fit’ (Debreu 1986: 1262) between economic content and computable mathematics can be demonstrated to be better than the fit held to be established between the full resources of classical mathematics and economic content in orthodox theorising. Clearly this issue cannot be resolved by economic methodology: its resolution depends on the fruits or otherwise of economic theorising confined to the resources of computable mathematics. In our opinion, however, economic methodology has a significant contribution to make to this issue, especially when economic methodology is expanded to include the philosophy of mathematics. In light of the philosophy of mathematics we argue that a prima facie methodological case can be made for limiting mathematical resources to computable mathematics in economic theorising. In light of this prima facie case sufficient time should be given to the exploration of this potential alternative approach to the formalisation of economics.
Intuitionistic mathematics is associated with the Dutch mathematician, L.E.J. Brouwer.3 In the early decades of the twentieth century, Brouwer explicitly developed intuitionistic mathematics in opposition to Hilbert’s formalism. Indeed the Brouwer-Hilbert controversy in the foundations of mathematics is very well known, probably for the wrong reasons. Van Stigt sums up this infamous controversy as follows.
The Brouwer-Hilbert controversy grew increasingly bitter and turned into a personal feud. The last episode was the ‘Annalenstreit’ or, to use Einstein’s words, ‘the frog-and-mouse battle.’ It followed the unjustified and illegal dismissal of Brouwer from the editorial board of the Mathematische Annalen by Hilbert in 1928 and led to the disbanding of the old Annalen company and the emergence of a new Annalen under Hilbert’s sole command without the support of its former chief editors, Einstein and Carathéodory.
(van Stigt 1998: 3)
Early in his mathematical career Brouwer had accomplished outstanding research in the domain of topology. This research was within the parameters of classical mathematics and included his famous fixed point theorem, exploited by Debreu. Brouwer’s research was acknowledged throughout Europe, including Hilbert and his school. Thus Brouwer’s credentials as an outstanding mathematician were universally acknowledged. However, even when engaged in his PhD research in Amsterdam, Brouwer was utterly dissatisfied with contemporary approaches to the foundations of mathematics from Russell to Hilbert. According to various scholars Brouwer’s intuitionistic mathematics ‘can only be fully understood in the context of his particular philosophy of mathematics’ (van Stigt 1998: 4).
As van Stigt sums up,
Brouwer’s main concern was the nature of mathematics as pure ‘languageless’ thought-construction. He set himself the task of bringing the mathematical world around to his view, convincing them of the need for reform, and had started the programme of reconstructing mathematics on an Intuitionist basis.
(van Stigt 1998: 2)
Brouwerian scholars contextualise Brouwer’s intuitionistic mathematics in his unique romantic pessimistic and radical individualistic view of life.4 In addition to his own unique philosophy of life, scholars also identify the influences of Descartes, Kant, Poincaré and others. With his Cartesian emphasis on individual consciousness, according to Brouwer, the mind has a direct, languageless, Kantian-type intuition of time.5 This primordial intuition is pure time awareness, without any external influence. In this subjectivist philosophical context mathematics is a free creation of the mind which is developed from this single, a priori, primordial intuition of time, totally independent of external experience. In other words the time-bound individual mind creatively constructs mathematics subject to the constraint of this primordial intuition. Thus, as Heyting puts it, ‘a mathematical construction ought to be so immediate to the mind and its results so clear that it needs no foundation whatsoever’ (Heyting 1956: 6). For the Brouwerian intuitionist, the pure intuition of time is bedrock and hence does not require any further legitimation. This pure intuition of time as continuing on ad infinitum is the foundational source of the universe of discourse of pure mathematics. This mathematical universe of discourse is thus a potentially infinite sequence, i.e. a domain which is never complete. Thus for intuitionistic mathematicians mathematics, ab initio, concerns the potentially infinite: this open-ended potentially infinite domain is immediately grasped by the individual mind in its pure intuition of time. Hence mathematics is totally independent of all discourse about, or our conception of, the physical world.
Instead of engaging the various, but significant nuances and historical evolution of Brouwer’s Intuitionism, for the purposes of deepening our understanding of the methodological issues involved in the mathematisation of economics, we distinguish between strict and pragmatic intuitionism. Strict intuitionism has a negative and a positive dimension. Negatively the strict intuitionist holds that classical mathematics, including its logic, ‘is incoherent and illegitimate, that classical mathematics, while containing, in distorted form, much of value, is, nevertheless, as it stands unintelligible’ (Dummett 2000: 250). The positive thesis is that ‘the intuitionist way of construing mathematical notions and logical operations is a coherent and legitimate one, that intuitionist mathematics forms an intelligible body of theory’ (Dummett 2000: 250).
In other words strict intuitionism maintains that our inherited, classical mathematics – defended by Hilbert – is fundamentally flawed and thereby must be radically overhauled. When this is done, there is one and only one way of doing mathematics, viz. the intuitionistic way. We will be addressing pragmatic intuitionism in a later section. For the present purposes, we note one major difference between strict and pragmatic intuitionism. According to pragmatic intuitionism, classical mathematics, contrary to strict intuitionism, is not fundamentally flawed. Classical and intuitionistic mathematics peacefully co-exist: each has an authentic, but distinct, mathematical domain.
Our contention is that the neo-Walrasian programme is flawed beyond redemption within strict intuitionism. In Chapter Five we saw how Kaldor advocated the total demolition of the neo-Walrasian programme. We content that within strict intuitionism the neo-Walrasian programme is undermined at its mathematical core. In the remainder of this section we outline the grounds for this radical claim. These grounds concern (i) the universe of discourse of mathematics, (ii) logic, (iii) the concept of mathematical existence and (iv) legitimate mathematical techniques and theorems.
As we saw in Chapter Three classical mathematics insists that Cantorian set theory with its unequivocal commitment to, what philosophers call, actual, as opposed to potential, infinity is indispensable to mathematics. Consequently the universe of discourse of classical mathematics is much more extensive than the potentially infinite domain of discourse of strict intuitionistic mathematics. The actual infinite domain of discourse of classical mathematics contains a complete infinity of infinities of elements. According to the strict intuitionist such a conception of the universe of discourse of mathematics is illegitimate and illusory because it is incompatible with the fundamental basis of all of mathematics, viz. the pure, primordial, mental intuition of time. This illegitimacy is exposed by the paradoxes in the foundations of mathematics: the source of these paradoxes is the misconceived notion of a complete, actual infinite domain of discourse.6 By acknowledging that potential infinity is the only legitimate conception of mathematical infinity, the paradoxes are avoided. For the strict intuitionist, Debreu, in using classical mathematics in his economic theorising, is ipso facto operating within this illegitimate domain of discourse of actual infinity.
In conjunction with this crucial, ontological fault-line between strict intuitionists and classical mathematicians concerning the universe of discourse of mathematics another equally crucial fault-line emerges. This fault-line concerns logic.7 Prior to the development of intuitionistic mathematics it was tacitly assumed that classical logic, which originated in Aristotelian logic and which was rigorously reconstructed by Frege and Russell, was the indispensable core of all legitimate deductive inferences, including those of mathematics. This assumption was challenged by strict intuitionism. Classical logic must be rejected: it is not adequate to the potential infinite domain of discourse of mathematics. Strict intuitionists maintain that classical logic was devised for finite domains of discourse and that classical mathematicians unhesitatingly, but illegitimately, transferred it to the potential infinite domain of discourse of mathematics.
The famous Aristotelian principle of the excluded middle became a focal point of this fundamental conflict between intuitionistic and classical mathematicians. This principle states that for any object x, x is b or x is not b, where b is a well-defined property of objects. Alternatively any proposition must be either true or false. The principle clearly holds for any finite class: one can in principle identify each member of the finite class and check whether or not the property b holds. For the strict intuitionist, this principle is not transferrable to the potential infinite domain of discourse of mathematics. Take, for instance, Troelstra’s and van Dalen’s example ‘there are infinitely many twin primes (twin primes are two consecutive prime numbers which differ by two, e.g. 3 and 5 are twin primes but 7 and 11 are not)’ (Troelstra and van Dalen 1988: 6). This proposition is currently undecidable. To date mathematicians have not created a systematic method to prove this proposition and neither have they created a systematic method to prove its negation. The classical mathematician, committed to the principle of the excluded middle, maintains that, despite its current undecidability, the proposition ‘there is an infinite number of twin primes’ is in reality either true or false. The strict intuitionist responds that it is possible that mathematicians may never create a systematic method which will either prove or disprove that there are an infinite number of twin primes. Again for the classical mathematician that response is also beside the point: there either is or is not, in reality, an infinite number of twin primes. Both the classical and the strict intuitionistic mathematician agree that (a) the only way of knowing whether or not there is an infinite number of twin primes is to prove it and (b) it is logically possible that mathematics will never resolve this issue one way or the other. The classical mathematician does not find the combination of (a) and (b) perturbing: the fundamental commitment is that in reality there either is or is not an infinite number of twin primes. The strict intuitionistic mathematician is genuinely perturbed with this commitment on the part of classical mathematicians. If that classical commitment is true and (a) and (b) are true, what does that combination assume? Given (a), it assumes that a proof exists in some realm or other which transcends the realm of humanly constructable mathematical proofs! In other words the classical mathematician is assuming beings with mathematical abilities which transcend our own human deductive abilities. The moral is clear: abandon the cornerstone of traditional logic, viz. the principle of the excluded middle. A logic without the principle of the excluded middle is required. This logic, called intuitionistic logic, was rigorously formalised by Brouwer’s outstanding pupil, Heyting.8 Intuitionistic logic is the only authentic logic. Classical logic is fundamentally flawed.
This novel logic of strict intuitionism undermines the neo-Walrasian programme at its logical core. A basic thesis of the neo-Walrasian programme is that Debreu has unquestionably proved the logical possibility of general equilibrium. As we have just seen, according to strict intuitionism, classical logic must be rejected and replaced by intuitionistic logic. Contrary to Debreu’s tacit assumption, classical logic is not the criterion of logical possibility. Rather the criterion of logical possibility is furnished by intuitionistic logic. By tacitly assuming, with the formalists, that classical logic is the one true logic and failing to recognise that the one true logic is that of intuitionism, Debreu fails to prove the logical possibility of general equilibrium. In this fashion the neo-Walrasian programme crumbles at its logical core: its so-called proof of the possibility of general equilibrium is grounded in a spurious logic. As we noted in Chapter Five a number of economists and methodologists reject the neo-Walrasian programme on the grounds that, while Debreu has rigorously proved the logical possibility of general equilibrium, his proof shows that the economic conditions of this logical possibility are so unrealistic as to render it valueless to the economic analysis of real world economies.9 The objection above on the grounds of strict intuitionism is much more radical in that it challenges the tacit assumption of these methodologists, viz. Debreu has rigorously established the logical possibility of general equilibrium. According to strict intuitionism he has done no such thing. Debreu’s so-called proof is illogical. He failed to recognise that intuitionistic logic is the one and only logic available to human beings.
We now turn to the third theme where classical mathematics as understood by formalists is mistaken, namely mathematical existence. As already emphasised for the strict intuitionist, mathematical existence is mind dependent. Moreover, the intuitionist mind is fundamentally creative: based on the primordial intuition of time, it constructs its mathematical entities. Thus mathematical existence means constructability in a finite number of algorithmetic steps. Debreu, as a committed member of the Hilbertian formalist school, is utterly confused on this matter: he fails to recognise that the concept of mathematical existence is completely distinct from the concept of formal consistency. To prove that a mathematical entity exists, one must show that it is in principle algorithmetically constructable. However, by equating mathematical existence with formal consistency, Debreu’s so-called proof of the existence of general equilibrium does no such thing.10
Fourthly, the rejection of classical logic and its replacement by intuitionistic logic entails that some of the proof-techniques and various theorems cherished by classical mathematicians are spurious. Weyl, a member of the Hilbert formalist school who for a period was more than impressed by Brouwer’s intuitionistic programme, sums up as follows:11
Mathematics with Brouwer gains its highest intuitive clarity. He succeeds in developing the beginnings of analysis in a natural manner, all the time preserving the contact with Intuition much more closely than had been done before. It cannot be denied, however, that in advancing to higher and more general theories the inapplicability of the simple laws of classical logic eventually results in an almost unbearable awkwardness. And the mathematician watches with pain the larger part of his towering edifice, which he believed to be built on concrete blocks dissolve into mist before his eyes.
(Weyl 1949: 54)
In this fashion Weyl identifies two major objections to strict intuitionism. Firstly, by rejecting classical logic, the proof techniques of intuitionist mathematics become unbearably cumbersome. We saw in Chapter Three how Debreu picks up this theme by insisting on the simplicity of mathematical proofs. The strict intuitionist response is that this lack of simplicity is not too dear a price to pay for limiting mathematical proofs to constructive ones, especially since such a limitation avoids the paradoxes in the foundations of mathematics. Vis-à-vis the second objection, viz. the destruction of a large part of the edifice of classical mathematics, the strict intuitionist concurs with Heyting’s response.
As to the mutilation of mathematics of which you accuse me, it must be taken as an inevitable consequence of our standpoint. It can also be seen as the excision of noxious ornaments, beautiful in form but hollow in substance.
(Heyting 1956: 11)
Among the theorems which are noxious ornaments is Brouwer’s fixed point theorem, a mathematical cornerstone of Debreu’s equilibrium theorising.
To sum up, the strict intuitionist maintains that classical mathematics misidentifies the domain of discourse of mathematics, uses a pseudo-logic, misconstrues mathematical existence and is mistaken in its identification of genuine mathematical theorems. The neo-Walrasian programme collapses on each of these basic themes. (1) It is mistaken in its assumption that the mathematical domain of discourse is Cantor’s actual infinity. (2) It fails to recognise that classical logic is not the genuine logic of mathematics. (3) It is incorrect in its formalist commitment to the identification of mathematical existence with formal consistency. (4) The Brouwerian-type fixed point theorems of classical mathematics are not genuine mathematical theorems at all. In Heyting’s colourful language Debreu’s equilibrium theorising is a ‘noxious ornament, beautiful in form but hollow in substance’, The strict intuitionist, in the spirit of Kaldor, demolishes the neo-Walrasian programme at its logico-mathematical core.
The good news for the neo-Walrasian programme is that Brouwer’s attempted revolution failed: the vast majority of mathematicians continued their researches within the framework of classical mathematics. In this connection we briefly dwell on Bourbaki’s account of this failure.12 We focus on Bourbaki because of the manner in which he identifies the Achilles’ heel in Brouwerian intuitionism. Also historians of economic thought have argued that Bourbaki was a major influence on Debreu’s formalist conception of mathematics.13 Bourbaki concurs with Weyl that the majority of mathematicians value the simplicity of the proof procedures of classical mathematics over the more cumbersome procedures of intuitionistic mathematics and that they are more than reluctant to sacrifice the large portion of classical mathematics demanded by strict intuitionism. According to Bourbaki there is a very good reason for this reluctance, namely Brouwer’s flawed philosophical psychology.
Bourbaki, while in no way attempting to summarise ‘a doctrine as complex as intuitionism,’ maintains that ‘it consists as much of psychology as mathematics’ (Bourbaki 1994: 37). This claim is in keeping with the conventional interpretation of Brouwer’s intuitionism: his mathematical programme is grounded in, and driven by, a specific philosophical psychology.14 Indeed Bourbaki identifies ‘some of the most striking aspects’ of this philosophical psychology (Bourbaki 1994: 37). The first of these is the Brouwerian thesis that mathematics is ‘the “exact” part of our thought, based on the first intuition of the sequence of natural numbers’ (Bourbaki 1994: 37). This reading of Brouwer is not controversial and for some it may not be that striking. For others, what makes it striking is the fact that for Brouwer this is ‘only “exact” in the thought of mathematicians, and it is a chimera to hope to develop an instrument of communication between them which is not subject to all the imperfections and ambiguities of language’ (Bourbaki 1994: 37). In other words Brouwer’s philosophical psychology includes four theses.
1 Mathematical thought is a private mental activity.
2 This mental activity is totally independent of any language, either natural or formal.
3 All languages are, relative to pure mathematical thought, defective.
4 When mathematical thoughts are expressed in a language, either natural or formal, they are contaminated by the inherent defects of this linguistic process.
The consequences of this philosophical psychology are equally clear: (a) Mathematicians, no matter how gifted, can never accurately convey their mathematical thoughts through language. (b) All mathematical communication conducted in either a natural or a formal language is inherently defective. (c) All the mathematical symbols and terms developed over the centuries in the various branches of mathematics fail to perfectly convey the pure mathematical thoughts underlying these symbols and terms.
Another striking aspect of Brouwer’s philosophical psychology is its negative attitude towards logic. ‘Intuitionist mathematics attaches hardly more importance to logic than to language’ (Bourbaki 1994: 37). This is Bourbaki’s way of stating the Brouwerian conviction that formal logic is not productive in mathematics. As the Brouwerian scholar van Stigt puts it, even while approving of, his loyal student, Heyting’s explorations into intuitionistic logic, Brouwer remained true to his conviction that engagement in logic ‘is an interesting but irrelevant and sterile exercise’ (van Stigt 1998: 276).15 To mathematicians who hold that logic is absolutely indispensable to mathematical thought and that, through the combination of natural and formal languages, mathematicians successfully communicate their thoughts, Brouwer’s philosophical psychology is far from appealing. It certainly is not a sufficient reason to reject a large portion of classical mathematics. Hence, not surprisingly, Bourbaki sums up as follows.
The intuitionist school, of which the memory is no doubt destined to remain only as a historical curiosity, would at least have been of service by having forced its adversaries, that is to say definitely the immense majority of mathematicians, to make their position precise …
(Bourbaki 1994: 38)
In this connection Bourbaki acknowledges that intuitionism forced its adversaries, especially the Hilbert formalists, to make more precise their metamathematical notion of a finite procedure. His point is that, these contributions to adversaries aside, intuitionistic mathematics remains a historical curiosity. In short according to Bourbaki, Brouwer’s philosophical psychology, on which intuitionistic mathematics is based, is its Achilles’ heel.
If intuitionistic mathematics is not to remain a passing curiosity in the history of mathematics, intuitionists may be well advised to sever its links to Brouwer’s philosophical psychology.16 This it has done in two ways. One route is to retain strict intuitionism but on the basis of an alternative philosophy to that of Brouwer’s philosophical psychology. The other route is that of pragmatic intuitionism which abandons all links to philosophy and develops intuitionistic mathematics as a legitimate mathematical domain alongside the equally legitimate domain of classical mathematics.17
A different philosophical route to strict intuitionism to that of Brouwer was pioneered in a highly original way by the distinguished Oxford philosopher Sir Michael Dummett in the latter decades of the twentieth century.18 Dummett’s fundamental thesis is that
intuitionistic mathematics is pointless without the philosophical motivation underlying it … intuitionism will never succeed in its struggle against rival, and more widely accepted, forms of mathematics unless it can win the philosophical battle. If it ever loses that battle, the practice of intuitionistic mathematics itself … will become a waste of time.
(Dummett 2000: ix)
The philosophical battleground is the philosophy of language, not Brouwer’s very questionable philosophical psychology. In the philosophy of language the focus is on the meaning or the objective, intersubjective understanding of mathematical statements.
For Brouwer mathematics is a mental activity that can only be imperfectly communicated in language. I on the other hand have stressed what I take to be the perfect communicability of mathematics … as such mathematical language plays a central role in it.
(Dummett 1994a: 305)
In Dummett’s philosophy of language, there is no question of denying the reality of mental activities. Rather language is a vehicle of thought but ‘it could not serve as a vehicle of thought unless it were first an instrument of communication’ (Dummett 1994b: 262). From the point of view of a faithful follower of Brouwer, Dummett is ‘a heretic who feels no urge to recant’ (Dummett 1994a: 305).
In short for Dummett, mathematical thought, contrary to some other strict Brouwerians, is clearly articulated and communicated by a public language and, contrary to Hilbertian formalists, what is communicated by mathematical language has content, i.e. objective meaning grasped by the mathematical community. In this intersubjective linguistic context, strict intuitionism still retains a positive and a negative thesis. Positively, ‘intuitionists succeed in conferring a coherent meaning on the expressions used in intuitionistic mathematics, and, in particular, on the logical constants’ (Dummett 2000: 251). (The logical constants are the terms, ‘and,’ ‘or,’, ‘not,’ ‘if-then-’, which are constantly used in our deductive/logical inferences). According to the negative thesis ‘classical mathematicians fail to confer an intelligible meaning on the logical constants, and on mathematical expressions in general, as they use them’ (Dummett 2000: 251). Again, contrary to strictly faithful followers of Brouwer, Dummett’s way of stating strict intuitionism puts the philosophical spotlight onto logic and on how meaning is conferred on its logical terms. In short, Dummett, like Bourbaki, recognises defects in Brouwer’s philosophical psychology but, unlike Bourbaki, defends strict intuitionism on different philosophical grounds. Clearly if Dummett’s defence of strict intuitionism stands up to critical scrutiny, the neo-Walrasian programme is once again shown to be contaminated at its logico-mathematical core.
As we have just seen Dummett, unlike Brouwer, puts the philosophical spotlight on language and, in particular, its deductive inferences. Prior to Brouwer, it was accepted without question that the logic of any deductive inference is classical logic. Classical logic, which originated with Aristotle and was rigorously represented by the logical works of Frege and Russell at the turn of the twentieth century, is the one true logic governing all deductive inferences in both natural languages and the specialised language of mathematics. Classical mathematics – the mathematical tool of the neo-Walrasian programme – stands or falls with classical logic. As already discussed in Chapter Three, Kenneth Arrow correctly identified extensionality as a key characteristic of classical logic and pointed to psychological research which shows that the doctrine of extensionality is not adhered to in various choice situations. Dummett goes much further than Arrow. He puts the philosophical spotlight on the fundamental presupposition of the doctrine of extensionality and argues that this fundamental presupposition does not stand up to critical scrutiny. In other words, extensional logic, the logical cornerstone of classical mathematics, does not stand up to critical scrutiny. It must be rejected and replaced by intuitionistic logic – the one true, non-extensional logic of deductive discourse, mathematical and non-mathematical.
In the following we focus on one strand in Dummett’s many-layered philosophical argument for this radical conclusion. This strand is his use of mathematical undecidable propositions. Our reason for choosing this, rather than other strands, is that mathematical undecidables also play a central role in the negative assessment of the explanatory success of the neo-Walrasian programme by some theoretical economists.19 As we will see later, these theoretical economists demonstrate that the neo-Walrasian explanation of value (modelled on Debreu’s Theory of Value) uses mathematically undecidable propositions and argue that, given these undecidables, neo-Walrasian theorising has no explanatory value. At the moment, however, our concern is with Dummett’s recourse to mathematical undecidables to buttress his claim that classical logic and ipso facto classical mathematics (exploited in neo-Walrasian theorising) is fundamentally flawed.
This fundamental flaw lies in what Arrow identified as the doctrine of extensionality, which constitutes the logical basis of classical mathematics. As pointed out in Chapter Three the doctrine of extensionality, when applied to propositions, as distinct from terms within propositions, is based on what philosophers of language call the thesis of bivalence.20 The thesis of bivalence states that every proposition has a determinate truth-value – it is either true or false – irrespective of our ability to decide which truth-value actually holds. Prima facie, the thesis of bivalence appears plausible. Take, for instance, the proposition ‘Napoleon blinked at 11.31 a.m. on July 10th, 1810’, That proposition is historically undecidable – we assume there is no evidence one way or the other. Nonetheless, despite this absence of historical evidence, it is in fact either true or false that Napoleon blinked at that time on that date. Based on this kind and other kinds of examples, logicians tacitly assumed the thesis of bivalence in developing classical extensional logic.
According to one strand of Dummett’s philosophical argument which we simplify here, the thesis of bivalence, and thus classical logic, is undermined by mathematical undecidable propositions. We distinguish between three kinds of mathematical undecidables. Firstly, some mathematical propositions, such as the proposition there is an infinite number of twin primes, used in the previous section, is currently undecidable. Despite its current undecidability, according to the thesis of bivalence, that proposition has a determinate truth-value: in reality it is actually either true or false. Let us assume that such current undecidables do not undermine the thesis of bivalence. The second kind of undecidables we call hypothetical Gödelian undecidables. Despite the lack of consensus on its full philosophical implications, Gödel’s second incompleteness theorem proves
(i) If T is consistent then U,
Where U is a very complex sentence of T which is undecidable by T and where T is an axiomatisation of standard arithmetic.21 The mathematical sentence, U, is undecidable. However, it is hypothetical, as it is the consequent of the antecedent, ‘T is consistent,’ in (i). According to some philosophers, the hypothetical Gödelian undecidable sentence U is not sufficient to undermine bivalence. Let us also assume that these philosophers are correct. A Dummettian intuitionist, by recourse to philosophical argument, transforms the hypothetical Gödelian undecidable sentence, U, into a categorical undecidable. Unlike a current undecidable sentence which may at some future date be shown to be decidable, this future possibility of decidability is ruled out in the case of a categorical undecidable.
We focus on three layers to the transformation of the hypothetical Gödelian undecidable sentence U into the categorical undecidable U. The logical layer is the logical principle, modus ponens i.e. if p then q and p therefore q, which intuitionist logicians share with classical logicians.22 The principle modus ponens is applied to (i) above: if one can show that the antecedent ‘T is consistent’ of (i) above is true then by modus ponens its consequent, U, is shown to be categorically undecidable. The second layer is a philosophical, not a mathematico-logical, argument to establish that the antecedent of (i), viz. T is consistent, stands up to critical scrutiny and thus is philosophically ‘sound’ (Dummett 1994b: 331). A central component in this philosophical argument is our shared social arithmetical practices: the number system is indispensable to vast ranges of social interactions based on counting and numerical ordering. These social practices would be meaningless without the natural number system.23 The third layer is to argue that, since the antecedent of (i) is shown to be philosophically sound it follows that it is philosophically correct to claim that the antecedent of (i), viz. T is consistent, is true. Given this, by modus ponens the hypothetical Gödelian undecidable U is categorically undecidable.24
The thesis of bivalence in the face of categorical undecidables is, to say the least, paradoxical. On the one hand, we have the categorical undecidability of the Gödelian sentence, U. On the other hand, by the thesis of bivalence, U has in reality a determinate truth value: it is in reality either true or false. There are two ways of resolving this paradox. One is to become a strict intuitionist, i.e. to reject classical logic which is grounded in the thesis of bivalence and replace it with intuitionistic logic and intuitionistic mathematics. The other way of resolving the paradox is to retain the thesis of bivalence in the face of categorical undecidables. According to Dummettian intuitionists this second option of retaining the thesis of bivalence carries a price which they are not willing to pay, namely a resolute commitment to categorical undecidables having determinate truth-values which are forever outside the pale of linguistically bound, mathematical reasoning. In other words it requires postulating beings with logico-mathematical abilities which transcend what is possible for coherent, finitely-bound, linguistic mathematicians. As Dummett states: ‘those many people who favour classical over intuitionistic logic are therefore guilty of the presumption of reasoning as if they were God’ (Dummett 2008: 109).
To sum up, strict Dummettian intuitionists replace classical mathematics, which is logically flawed, with intuitionist mathematics. Like strict Brouwerian intuitionists, Dummettian intuitionism implies that the neo-Walrasian programme collapses on the four central themes outlined earlier. Firstly, the neo-Walrasian programme misidentifies the domain of discourse of mathematics. Secondly, it incorrectly assumes that classical logic stands up to critical scrutiny. Thirdly, it utterly misconstrues the correct analysis of mathematical existence. Finally, its Brouwerian-type fixed point theorems, central to its proof of existence of equilibrium, are not genuine theorems at all. At its logico-mathematical core, the neo-Walrasian programme, to use Dummett’s phrase, ‘is incoherent and illegitimate’ (Dummett 2000: 250).
In the introduction to this chapter we said that, prima facie, theoretical economists have a genuine choice between constructive and classical mathematics in their programme of the mathematisation of economics. Clearly that is not correct for either a Brouwerian or Dummettian strict intuitionist. For the strict intuitionist – Brouwerian or Dummettian – there is one and only one way of mathematising any theory, including economic theory, namely by recourse to intuitionistic mathematics. The choice we referred to in the introduction applies to pragmatic intuitionism. The pragmatic intuitionist ignores the philosophical grounds for intuitionistic mathematics and focuses on doing intuitionistic mathematics. According to pragmatic intuitionists, classical and intuitionistic mathematics peacefully co-exist. There is no question of, as it were, the colonial expansion of one into the other. Both have distinct, authentic domains. What we call pragmatic intuitionism is summed up by Troelstra and van Dalen in their preface to their Constructivism in Mathematics as follows. ‘The ending “-ism” has ideological overtones: “constructive mathematics is the (only) right mathematics;” we hasten, however, to declare that we do not subscribe to this ideology …’ (Troelstra and van Dalen 1988: vii). Thus theoretical economists have a genuine choice between the resources of classical mathematics and those of intuitionistic mathematics in their formalisation of economic theory. The methodological issue is the question of the relative merits of the fit between economic content and classical mathematics exploited in the neo-Walrasian programme and the fit between economics and intuitionistic mathematics. By focusing on pragmatic intuitionism we identify two themes in which the fit between economics and intuitionistic mathematics is, ceteris paribus, better than the fit exploited in the neo-Walrasian programme.
The first theme concerns the universe of discourse of intuitionistic mathematics. This domain of discourse is a very general abstract open-ended system, constructed by the human mind. In this connection it is useful to quote Poincaré – the precursor of intuitionistic mathematics. Mathematics ‘reflecting upon itself is reflecting upon the human mind which has created it, … of all its (the mind’s) creations, mathematics is the one for which it (the mind) has borrowed least from outside’ (Poincaré 1956: 36). In many of its constructions, e.g. a model of an aeroplane, the mind is obliged to exploit the specific empirical characteristics believed to be relevant to the proposed construction. However, the mind in intuitionistic mathematics leaves aside these specific empirical constraints and investigates the logico-mathematical structure of any constructable system – the logic being intuitionistic, not classical. The domain of discourse of intuitionistic mathematics is centrally concerned with what is in principle constructable, where the ‘in principle’ qualification means subject to the constraints of intuitionistic, not classical, logic. Classical mathematics does not investigate the logico-mathematical structures of an, in principle, constructable system. Its domain of discourse is different to intuitionistic mathematics – the constraints of constructability-in-principle are not relevant to its logico-mathematical investigations. Economies, however, are not natural givens. Economies are very specific, complex, constructed systems. Given that it is a constructed system, an economy, if it can be studied mathematically, the resources of intuitionistic mathematics which are tailored made for what is in principle constructable are a better fit than those of classical mathematics which are not tailored for what is in principle constructable. Orthodox theoretical economists, given their recourse to the full resources of classical mathematics, have no guarantee that their explanations are applicable to what is in principle constructable, whereas by recourse to the resources of intuitionistic mathematics economists are guaranteed that their explanations are applicable to what is in principle constructable. In particular, neo-Walrasian explanations, by recourse to Brouwerian fixed point theorems of classical mathematics, are not shown to be realisable in a logically possible constructable world, never mind our social world of finite historical time.
In the previous theme, we commenced with an in-principle constructable system. In the following theme, we start with the rigorous formalisation of intuitionistic logic by Heyting. Given this starting point, the problematic or challenge is to specify a domain of application or interpretation of this intuitionistic logic. In this second theme we focus on Kolmogorov’s ‘On the Interpretation of Intuitionistic Logic,’ first published in 1932. This short paper is divided into two sections. The first section is directed at those who ‘do not accept intuitionistic epistemological assumptions,’ while in the second section ‘intuitionistic logic is critically investigated while accepting the general intuitionistic assumptions’ (Kolmogorov 1998: 328). In line with what we named pragmatic intuitionism, Kolmogorov recognises that both classical and intuitionistic logic have distinct domains of discourse. According to Kolmogorov, classical logic (which he terms theoretical logic) ‘systematizes the proof schemata of theoretical truths’ (Kolmogorov 1998: 328). He does not elaborate on this characteristic. His primary concern is not with classical logic. Nonetheless, contrary to strict intuitionism, there is no question of claiming that classical logic is fundamentally flawed. Rather Kolmogorov’s concern is with Heyting’s rigorous formalisation of intuitionistic logic. The challenge is to find a domain of application, different to the domain of application of classical logic, for Heyting’s formalisation of intuitionistic logic or, in logical terms, to identify an interpretation of Heyting’s formal system. Kolmogorov establishes that Heyting’s formalisation of intuitionistic logic systematises a different domain to that of classical logic: it systematises ‘the schemata of solutions to problems’ (Kolmogorov 1998: 328). In other words the domain of discourse of classical logic is the domain of theoretical truths and its aim is to systematise the proof schemata of that domain. The domain of discourse of intuitionistic logic is not that of classical logic: its domain is solutions to problems, and its aim is to systematise the schemata of solutions to problems.
Kolmogorov is drawing the mathematical community’s attention away from the calculus of classical logic which up to that time dominated mathematical thinking onto what he calls the ‘new calculus of problems’ (Kolmogorov 1998: 328). This new calculus, like the calculus of classical logic, is well worth developing. ‘The proper goal of the calculus of problems consists in giving a method for the solution of problems … by means of the mechanical application of some simple computational rules’ (Kolmogorov 1998: 330). Thus the focus shifts to the computational solutions to problems. Kolmogorov states his central conclusion as follows. ‘The following remarkable fact holds: The calculus of problems is formally identical with Brouwerian intuitionistic logic, which has recently been formalised by Mr. Heyting’ (Kolmogorov 1998: 328). Thus the framework for the systematic study of constructive solutions to problems is not given by classical logic: rather the correct framework is intuitionistic logic.
Kolmogorov does not define what a problem is; neither does he define the concept of a solution to a problem. Rather he furnishes the reader with examples of problems and their solutions. In this way he hopes to avoid current philosophies of solutions to problems by focusing on what is accepted as problems and solutions in ‘concrete areas of mathematics’ (Kolmogorov 1998: 329). Clearly Kolmogorov is referring to solutions to mathematical problems. Thus the question arises: how is this related to economic theorising? Economic problems are empirical, not purely mathematical. However, by the process of mathematisation, these economic problems are represented in mathematical terms and then pure mathematics is exploited to solve these problems. Once a solution is found, the economic theoretician finds an economic interpretation for this solution. For instance, take the theoretical problem of whether or not a general equilibrium is possible in some socio-economic arrangement. This problem is represented in mathematical terms and the resources of classical mathematics are used to solve this problem. This solution to the mathematical problem is then given an economic interpretation. A Kolmogorov type objection to this neo-Walrasian approach lies in the method used to solve the mathematical problem. The objection applies to the internal mathematical step of finding a solution to the mathematical problem; it is not concerned with the final step, i.e. to find an economic interpretation to the mathematical solution. Neo-Walrasians tacitly assume that classical logic, which underpins classical mathematics, is the correct logic of solutions to problems. This assumption is not correct. The correct logic is intuitionistic, not classical. When intuitionistic logic and mathematics is used, the Debreu neo-Walrasian solution is seen for what it is: it is no solution at all. In short, in so far as economic theory has recourse to mathematics in solving its economic problems, its logic and mathematics must be intuitionistic.
To conclude this section, the external critique of Hilbertian formalism initiated by Brouwer culminated in either strict or pragmatic intuitionism. In strict intuitionism, either Brouwerian or Dummettian, the prognosis for the neo-Walrasian programme is more than bleak: theoretical economists in mathematising their discipline have no choice. The only option open is intuitionistic logic and mathematics. The situation is different for the pragmatic intuitionist. In principle theoretical economists have a genuine choice between classical and intuitionistic mathematics in their programme of mathematisation of economics. By focusing on the fact that an economy is a constructed system and that much of economic theorising is problem solving we outlined how pragmatic intuitionists could argue that the fit between economic theory and intuitionistic mathematics is better than the fit exploited in the neo-Walrasian programme.
In the previous sections we engaged the mathematisation of economics, especially the neo-Walrasian programme, from the point of view of intuitionistic mathematics introduced by Brouwer. As already pointed out, Hilbertian formalism was a central target of Brouwerian intuitionism. Since Brouwerian intuitionistic mathematics starts with a different vision of mathematics to that of Hilbert, we called the intuitionistic critique of Hilbertian formalism external. In this section we turn to what we called the internal critique of formalism. Among the key logico-mathematical figures in this section are Church, Gödel and Turing. We call this critique of formalism internal for two reasons. Firstly, unlike intuitionism, it does not have recourse to intuitionistic logic; rather it undermines the Hilbertian formalist programme on classical logical grounds. Secondly, this critique, as it were, forensically interrogates the key concepts of consistency, completeness and decidability of the Hilbert programme, culminating in surprising mathematical results. The novel logico-mathematical techniques underpinning these amazing results gave rise to our computer age. These innovative, logico-mathematical techniques also deliver novel alternative ways of mathematically modelling an economy to those exploited in the neo-Walrasian programme.
We divide the internal critique of formalism into two sections. In the following section we introduce the surprising results of Gödel’s theorems and explore their implications for the mathematisation of economics. Our conclusion is that while these theorems have adverse implications for Debreu’s own formalist philosophy of economic theorising, as such they do not adversely undermine his mathematical representation of an economy. We then return to the issue of undecidables, discussed in our reflections on the impact of intuitionist mathematics on the mathematisation of economics. According to Hilbertian formalism there are no undecidables in mathematics. On the contrary, there is a systematic method for deciding whether or not any mathematical statement is provable. The surprising result that no such systematic method or algorithm exists was established in 1936 by both Church and Turing. This proof of the existence of undecidables in classical mathematics was seismic in the foundations of mathematics. Post Church-Turing one could say that a new, non-formalist image of classical mathematics emerges. Instead of the formalist image of a consistent, complete and decidable meaningless calculus, the new image is that of a semantical system of algorithmetical decidable and algorithmetically undecidable theorems. This gives rise to the methodological issue of how the use of undecidable theorems effect economic modelling. In this connection we argue that the use of undecidable theorems in the neo-Walrasian mathematical model of a private ownership economy renders its explanation of prices economically vacuous.
A central aim of the Hilbert formalist programme was to establish the consistency, completeness and decidability of the whole edifice of pure, i.e. classical mathematics. Brouwerian considerations aside, the Hilbert school had no reason to be pessimistic about the successful completion of this challenging programme. The first suggestion that all was not well was articulated by Gödel on 7 September 1930 at a conference in Köningsberg. As Dawson remarks, at that time Gödel was virtually unknown outside of Vienna. Indeed at the conference he read a paper summing up the results of his dissertation completed the previous year in which he established ‘a result of prime importance for the advancement of Hilbert’s programme: the completeness of first order logic’ (Dawson 1988: 76). The following day, however, in the course of a discussion on the foundations of mathematics, Gödel made some remarks to the effect that number theory was incomplete! His remarks were largely ignored, with the notable exception of von Neumann. ‘(A)fter the session he drew Gödel aside and pressed him for details’ (Dawson 1988: 77). The rigorous formulation of Gödel’s incompleteness remarks was subsequently published in his famous ‘On Formally Undecidable Propositions of Principia Mathematica and Related Systems I’ in January 1931. This paper contains his two incompleteness theorems. The first incompleteness theorem ‘shows that any properly axiomatized and consistent theory of basic arithmetic must remain incomplete, whatever our efforts to complete it by throwing further axioms into the mix’ (Smith 2007: 5). Roughly interpreted, the second incompleteness theory says ‘there is no axiomatizable theory of arithmetic which can prove its own consistency’ (Epstein and Carnielli 2000: 172).
Von Neumann in March 1951, on the occasion of the award of the Einstein medal to Gödel, said ‘Kurt Gödel’s achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time’ (New York Times, 15 March: 31, quoted in Kleene 1988: 60). However, as Dawson reminds us, ‘we must also recognize the hazard in assessing the cogency of Gödel’s arguments from our own perspective’ (Dawson 1988: 75). In his brief analysis Dawson’s aim is to show that, despite Gödel’s own belief that his results were promptly accepted at that time, ‘there were doubters and critics as well as defenders and rival claimants to priority’ (Dawson 1988: 76). Without getting into such details, it is clear that Gödel’s theorems pose a serious challenge to the Hilbert programme: Hilbert’s aim of establishing by formal means the consistency and completeness of mathematics is not achievable.
Our concern here, however, is with the philosophy of economics/economic methodology, especially the mathematisation of economic theory. In this connection we make two suggestions. Firstly, Gödel’s theorems impact in an adverse way on Debreu’s commitment to Hilbertian formalism in his own philosophy of economic theorising. Clearly anyone building a comprehensive case against Debreu’s formalist philosophy of economic theorising will, among other themes, have recourse to the negative impact of Gödel’s theorems for any such formalism. Secondly, however, Gödel’s theorems do not directly impact on Debreu’s mathematical representation of the action of an economic agent as a vector in a real vector space, exploited in his Theory of Value. Gödel’s theorems, unlike strict intuitionism, do not challenge classical mathematics at its logical core. Thus the neo-Walrasian mathematical representation of the actions of an economic agent in terms of real vectors and its recourse to Brouwer’s fixed point theorem are not immediately challenged by Gödel’s theorems. One might say Gödel’s theorems impact on the image of the mathematisation of economics developed by Debreu but they do not as such negatively impact on its substance, in the sense that Debreu’s neo-Walrasian mathematical representation of an economy is not directly challenged by Gödel’s theorems.
Kleene draws our attention to another significant aspect of Gödel’s theorems, namely ‘how seminal his ideas and results have been’ (Kleene 1988: 60). One such seminal idea is the reference to undecidable propositions in Gödel’s famous paper. This is a reference to the other key aim of the Hilbert programme, namely to resolve the decision problem – the Entscheidungsproblem. The decision problem was emphasised by Hilbert as early as 1900 in his famous Paris address. In that address Hilbert maintained that ‘in mathematics there is no ignorabimus’ (Hilbert 1902: 445). As Copeland puts it,
Hilbert’s requirement that the system expressing the whole content of mathematics be decidable amounts to this: there must be a systematic method for telling, of each mathematical statement, whether or not the statement is provable in the system. If the system is to banish ignorance totally from mathematics it must be decidable.
(Copeland 2004: 47)
Church gives the following definition of the decision problem.
By the Entscheidungsproblem of a system of logic is here understood the problem to find an effective method by which, given any expression Q in the notation of the system, it can be determined whether or not Q is provable in the system.
(Church 1936b: 41)
To answer the decision problem one must have a precise definition or articulation of the interchangeable terms, systematic method/effective mechanical procedure/algorithm. Prior to Hilbert emphasising the significance and centrality of the decision problem, mathematicians did not attempt to precisely define the notion of an algorithm. Normal usage of the term sufficed. In its normal usage an effective mechanical procedure or algorithm was explained by examples and an adjoining commentary. An algorithm, based on these examples, is said to be a set of step-by-step instructions, with each step clearly specified in every detail, thereby ruling out any recourse to imagination, intuition or insight. The steps were thus executed mechanically and the number of steps taken must be finite. Today we would say that an algorithm is any procedure which can be carried out by a modern computer. However, in the 1930s there were no computers: the question ‘what precisely is an algorithm?’ had to be resolved before one could begin to construct a universal stored-programme computer.
The young British mathematician Alan Mathison Turing resolved this challenge in 1936 in his famous paper ‘On Computable Numbers with an application to the Entscheidungsproblem’. In that paper he furnished mathematicians with a precise characterisation of a systematic procedure by outlining an abstract template for a computing machine called the Universal Turing machine. Church sums up Turing’s achievement as follows. ‘The notion of an effective computation, or algorithm is one for which it would be possible to build a computing machine. This idea is developed into a rigorous definition by A.M. Turing …’ (Church 1956: 52). In a note added in 1963 to his famous 1931 paper Gödel sums up as follows.
In consequence of later advances, in particular of the fact that due to A.M. Turing’s work, a precise and unquestionably adequate definition of the general notion of formal system can now be given … it can be proved rigorously that every consistent formal system that contains a certain amount of finitary number theory there exist undecidable arithmetic propositions and that, moreover the consistency of any such system cannot be proved in the system.
(Gödel 1988: 40–41)
In his paper Turing furnished the mathematical community with a novel and accurate analysis of the received notion of algorithmetic reasoning used by mathematicians. He established that our contemporary notion of an algorithm as a process which could be carried out by a computing machine is correct. Thus his paper is the source of our contemporary notion of an algorithm. Indeed his paper was used by electronic engineers in Manchester. Their work culminated in the construction of the world’s first electronic stored-programme digital computer in 1948. Secondly, as Gödel points out, Turing’s paper eliminated any reservations one might have had about the existence of undecidables in mathematics. Thus Turing’s paper sounded the final death knell of the original Hilbert formalist programme by establishing the existence of undecidables in mathematics.
Vis-à-vis the challenge of mathematising economics, unfortunately this death knell was not heard by Debreu. As we saw in the last section, Gödel’s incompleteness theorems undermined Debreu’s own formalist philosophy of economic theorising. Turing’s results go much further: in our post-Turing mathematical world the economic substance of Debreu’s theorising is, it could be argued, also undermined. An economy is a specific, constructed finite system. Because of the finite nature of economies, a minimum condition on the proof of the existence of an economic equilibrium is that it is in principle realisable in a finite number of steps. If an equilibrium is not realisable in a finite number of steps, however large, it lies outside the domain of the finite. Debreu’s proof of the existence of equilibrium fails to meet this minimum requirement, precisely because it is grounded in algorithmetically undecidable propositions. For instance, as Velupillai has demonstrated, ‘a Walrasian equilibrium vector price is undecidable’ (Velupillai 2002: 318) and ‘the Walrasian excess demand function is undecidable, i.e. it cannot be determined algorithmetically’ (Velupillai 2006: 363).25 Prima facie any entity which is claimed to exist on grounds which include an undecidable mathematical proposition, lies outside the domain of the finite. In this connection it is crucial to note the following difference between theoretical physics and neo-Walrasian theorising. Advanced theoretical physics has at its disposal a vast and sophisticated range of experimental techniques which, at times, enable it to establish the actual existence of some of its theoretically postulated entities. In the absence of such experimental verification, theoretical physics suspends judgment on claims of actual existence. As emphasised by Debreu and Hahn, however, neo-Walrasian economics lacks recourse to that kind of experimental verification of its existence claims. Given that this kind of sophisticated experimental verification is not to be expected and that its proof of the existence of equilibrium hinges on undecidable propositions, neo-Walrasian economists have no justification for either claiming or assuming their equilibrium could exist within any finite economic domain. In short their non-algorithmetic proof of the existence of equilibrium renders their notion of equilibrium economically vacuous.
According to Debreu, his Theory of Value furnishes economists with a rigorous model which is ‘the explanation of the prices of commodities resulting from the interaction of the agents of a private ownership economy through markets’ (Debreu 1959: vii) – an explanation which was pioneered by Walras and Pareto. The thesis here is that the recourse to undecidables in this model undermines its explanatory capacity. Undoubtedly as a piece of pure classical mathematics, Debreu’s Theory of Value attains the highest standards of rigour as characterised by Hilbertian formalists. However, the combination of two crucial characteristics of classical mathematics undermines the economic interpretation of Debreu’s mathematics and thus its explanation of prices. Firstly, the universe of discourse of classical mathematics is Cantorian actual infinity which transcends the limitations of any finite domain. Indeed it transcends the potentially infinite domain of the natural numbers. Secondly, classical mathematics has recourse to non-algorithmetic methods of proof. This recourse is clearly illustrated by Poincaré. He takes Zermelo’s theorem which proves ‘that space is capable of being transformed into a well-ordered set’ (Poincaré 1963: 67). Poincaré asks the classical mathematician to actually carry out the transformation but is told that it would take too long. To this Poincaré responds, ‘then at least show us that someone with enough time and patience could execute the transformation’ (Poincaré 1963: 67). The classical mathematician responds, ‘No, we cannot because the number of operations to be performed is infinite, it is even greater than aleph zero’ (Poincaré 1963: 67). Poincaré’s concern was with pure mathematics. Our concern is with the application of this pure mathematics which contains undecidable propositions to the interaction of finitely bound agents in finitely bound markets. These economic agents process their information via language and this processing takes time. Thus this processing must be capable of being carried out in a finite number of steps. In particular, price signals must be capable of being processed in a finite number of steps. However, as the Poincaré example above indicates, the use of undecidable propositions in Debreu’s Theory of Value means that its price signals cannot be processed in a finite number of steps.
Neo-Walrasians forget that the price of the application of a piece of pure classical mathematics to the interaction of agents of a private ownership economy is eternal vigilance. Economic agents process all their information, including price signals, within the constraints of time. However, pure classical mathematicians, unlike Poincaré, have no reservations about abstracting from the limitations of time in their efforts at achieving consistency. The upshot of this is that classical mathematics divides into theorems which are algorithmetically decidable and those which are not algorithmetically decidable. In their application of classical mathematics to the interaction of economic agents, the neo-Walrasians fail to pay attention to this crucial distinction. In their model of a private ownership economy they use undecidable theorems in their explanation of prices which means that its price signals cannot be processed in any finite time horizon.
To sum up, the proof of the existence of undecidables in classical mathematics in the 1930s was seismic within the foundations of mathematics. We contend that it is also seismic for the programme of the mathematisation of economics. Prior to the amazing results of Gödel and Turing, the Hilbertian formalists were not alone in assuming that there is no ignorabimus in classical mathematics. Numerous philosophers of mathematics and practising mathematicians with no affiliation to Hilbertian formalism shared that commitment. Given that commitment to the decidability of classical mathematics, the use of the full resources of classical mathematics in an applied domain was not problematical. In that vein numerous theoretical economists saw no reason why they should not exploit the full resources of classical mathematics in their economic theorising. Indeed various, post-World War Two, pioneering theoretical economists explicitly advocated this exploitation. Gödel’s and Turing’s research, however, shattered the assumption that there is no ignorabimus in classical mathematics. Turing proved that undecidables exist within classical mathematics. This truth has fundamental implications for the mathematisation of economics. The use of undecidable propositions in economic modelling presupposes one has abstracted from any and every consideration of the linguistic-boundedness of economic information which is processed in a finite number of steps, however large. Thus any explanation of prices grounded in a mathematical model of an economy which exploits undecidables is economically vacuous: its mathematical representation of price signals is such that they cannot be processed in a finite number of steps.
Thus far we have kept three strands in the methodological critique of General Equilibrium Theory quite separate, namely (1) the critique inspired by strict intuitionism (Brouwerian or Dummettian), (2) the critique inspired by pragmatic intuitionism, and (3) the critique inspired by the Turing proof of the existence of undecidables. The critique inspired by strict intuitionism is, perhaps, the most radical. General Equilibrium Theory is fundamentally illogical. The only logic available to both the pure and applied mathematician is intuitionistic logic. Classical logic is fundamentally flawed. In contrast to strict intuitionists, pragmatic intuitionists are not just tolerant: they are pluralists. Classical mathematics, rooted in classical logic, is legitimate. Equally intuitionistic mathematics, rooted in intuitionistic logic, is legitimate. Classical mathematics pertains to a Cantorian infinite universe of discourse whereas intuitionistic logic, being the logic of problem-solving, pertains to a potentially infinite domain of discourse. Thus existence proofs using classical mathematics fails to guarantee existence in either a potentially infinite domain or a finite domain. However, recourse to intuitionistic mathematics, with its methods of proof confined to constructive ones, necessarily limits its application to what is in principle constructable and is thus more appropriate than classical mathematics to economic theorising. The third critique, inspired by Turing’s proof of the existence of undecidables in classical mathematics, is distinct from these other two critiques. Turing’s conclusion is, as already noted, based on classical, not intuitionistic, logic. Post-Turing classical mathematics divides into algorithmetically decidable and algorithmetically undecidable theorems. The third critique of General Equilibrium Theory – which is not an experimental science – maintains that its mathematical model of an economy uses algorithmetically undecidables and thus fails to have explanatory value.
In the previous section we followed Gödel in giving the laurels to Turing. It could be argued that Alanzo Church at Princeton equally deserves these laurels. In the early 1930s Church was exploring logical systems by recourse to a specific formal procedure, called the lambda calculus. Roughly speaking, a function of positive integers is lambda definable if the values of the function can be calculated by a process of repeated iteration. In the autumn of 1931, when von Neumann addressed the mathematics colloquium, Church became aware of Gödel’s theorems. According to Kleene ‘Church’s immediate reaction was that his formal system … is sufficiently different from the systems Gödel dealt with that Godël’s second theorem might not apply to it’ (Kleene 1981: 52). The significance of von Neumann’s address, however, was that Church and his students engaged Gödel’s original paper. Thus in 1934 when Gödel delivered a series of lectures on his incompleteness theorems, his audience was more than attentive and very well informed.
From the point of view of economic methodology what is significant for us is that in those lectures Gödel advanced ideas of Herbrand about the most general form of recursiveness resulting in what is known as Herbrand-Gödel general recursiveness. For our methodological purposes it suffices to say that Herbrand-Gödel general recursiveness is a generalisation of the elementary inductive-recursive definitions of addition and multiplication. Thus we have two distinct concepts, namely lambda definability and recursion. In 1936 Church published two famous papers, viz. ‘An unsolvable problem of elementary number theory’ and ‘A note on the Entscheidungsproblem’, The results of these papers became known as the Church Thesis. The Church Thesis has three dimensions. Firstly, like Turing, Church formalises the received, pre-theoretical notion of an algorithm. However, unlike Turing, he formalises it in terms of his lambda calculus. In other words the lambda calculus is also a rigorous technical replacement of the received pre-theoretical notion of an algorithm. Thus an effective calculable function is lambda definable. Secondly, Herbrand-Gödel recursiveness and lambda definability are equivalent. In other words when we analyse the received notion of an effective procedure/algorithm in terms of lambda definability or in terms of Herbrand-Gödel recursiveness we obtain the same class of functions. The lambda procedure can be translated in Herbrand-Gödel recursiveness in such a way that the two formalisations have the same outputs for the same inputs. Thirdly, like Turing, there are undecidables in mathematics. When Church’s results are combined with those of Turing we have the famous Church-Turing thesis. In simple terms the Church-Turing thesis states that the various rigorously formalised systems, such as a Turing machine, the lambda calculus and recursion theory, proposed as a rigorous explication or formulation of the inherited, informal concept of an effective mechanical procedure or algorithm, are equivalent. Thus, for instance, every Turing computable function is recursive and every recursive function is Turing computable.26
Vis-à-vis the philosophy of economic theorising, the Church-Turing thesis has nothing new to add to the negative assessment of general equilibrium outlined in the previous section. Mathematics post the Church-Turing thesis enables methodologists to highlight the role of undecidables in general equilibrium theorising. Rather the additional significance of the Church-Turing thesis lies in its expansion of the conceptual resources beyond those of a universal Turing machine which theoretical economists could exploit in addressing various challenges which are bound to arise in the course of the algorithmetic mathematisation of economics. At the pragmatic level, the Church-Turing thesis identifies a rich reservoir of algorithmetic concepts which may be exploited in specific problem situations by the theoretical economist. Despite the fact that ultimately one is operating with the same set of functions, pragmatically in a specific problem situation one may find the concepts used in a Turing machine very cumbersome, whereas the resources of recursion theory may be more easily adapted to the problem or vice versa. We call this range of algorithmetic mathematics computable mathematics.
Computable mathematics, under the rubric of the Church-Turing thesis, by its very nature avoids the undecidable theorems of classical mathematics and is thus in principle applicable to any finite open-ended constructable system. Our question is whether or not computable mathematics has the potential to enrich our understanding of real economies? Prima facie one may claim that the prospects are good and indeed are fairly well advanced, as is evident in computable general equilibrium theory pioneered by Scarf in the 1970s. In the preface to his Computation of Economic Equilibria, Scarf opens with the following:
One of the major triumphs of mathematical economics during the past quarter of a century has been the proof of the existence of a solution for the neoclassical model of economic equilibrium. This demonstration has provided one of the rare instances in which abstract mathematical techniques are indispensable in order to solve a problem of central importance to economic theory.
(Scarf 1973: ix)
This major triumph was accomplished by among others Gerard Debreu, Scarf’s ‘good friend and valued advisor’ (Scarf 1973: x). Thus according to Scarf the triumph of the proof of the existence of general equilibrium by recourse to fixed point theorems is central to his programme of the computation of economic equilibria. In this vein he points out that ‘the central problem of this monograph is the description of an efficient computational procedure for the approximation of a fixed point of a continuous mapping’ (Scarf 1973: 12). Thus Scarf’s programme is an original synthesis of General Equilibrium Theory à la Debreu and the resources of computable mathematics.
A Scarf-type computable general equilibrium theory is not what we have in mind in referring to the possibility of a computable revolution in economics. The computable revolution we wish to address is a more radical programme, where the economic theoretician is, without exception, confined to the resources of computable mathematics – a programme advocated over a number of decades by Velupillai. In the opening page of the Festschrift in honour of Professor Kumaraswamy Vela Velupillai, edited by Zambelli, it is claimed that Velupillai ‘is the founder of computable economics, a growing field of research where important results stemming from classical recursion theory and constructive mathematics are applied to economic theory. The aim and hope is to provide new tools for economic modelling’ (Zambelli 2010: I). According to Velupillai, while Scarf must be given due recognition among the pioneers or precursors of computable economics, Scarf’s approach is basically:
schizophrenic: proof of existence in one domain; implementation of an algorithm to find the provable existent entity in quite a different domain. As a result, the algorithm cannot, in general, locate the fixed point even approximately. This is so because the algorithm cannot, in any effective constructive sense, be given meaningful information about the characteristic of the entity, it is supposed to locate.
(Velupillai 2002: 316)
In Kaldor’s metaphor, once the scaffolding of algorithmetic procedures used for the purposes of approximation are removed, one is left with the non-computable classical mathematics of Debreu exemplified by Brouwer’s fixed point theorem, which cannot be coherently interpreted in the real historical time of economic systems. Instead of the dualism of non-computable existence proofs combined with computable procedures for estimating approximations, in the computable revolution proposed by Velupillai theoretical economists confine their mathematical resources to computable mathematics under the rubric of the Church-Turing thesis. As Zambelli puts it ‘any reasoning in terms of economic theory is meaningful only if one can produce an effective algorithm for the solution of a problem’ (Zambelli 2010: 34).27 In this radical programme computable economics ‘is about basing economic formalisms on recursion-theoretical fundamentals. This means we will have to view economic entities, economic actions and economic institutions as computable objects or algorithms’ (Velupillai 2000: 2). For instance just as Debreu describes the action of an economic agent by a vector in the commodity space Rι where the commodity space has the mathematical structure of a real vector space, in Velupillai’s computable economics one can describe the action of an economic agent as if it ‘were a Turing machine; in this case the commodity space would have an appropriate recursive structure …’ (Velupillai 2000: 3).
For readers not familiar with the details of recursion theory or a universal Turing machine, a useful way of approaching Velupillai’s computable economics is to start with another major precursor, namely Herbert Simon. Simon’s bounded rationality critique of the orthodox model of economic rationality is well known. In this connection Velupillai is surely correct in claiming that ‘Simon’s research programme pointed the way towards computable economics in a precise sense’ (Velupillai 2000: 25). Indeed in correspondence with Velupillai in relation to his Computable Economics Simon points out:
There are many levels of complexity in problems, and corresponding boundaries between them. Turing computability is an outer boundary and as you show, any theory that requires more power than that surely is irrelevant to any useful definition of human rationality.
(Simon 2010: 409)
Clearly, in Simon’s opinion, Velupillai succeeds in undermining the orthodox mathematical model of rationality by proving that it is located outside the outer boundary of human rationality – no mean achievement. Moreover one could argue that Velupillai’s computable economics goes further. Velupillai takes Simon’s fertile, related concepts of rationality and satisficing and re-interprets these within the sophisticated mathematical architecture of computable mathematics. In this fashion he completely undermines the orthodox assumption that Simon’s bounded rationality approach is no match for the mathematical sophistication and elaboration of the orthodox model of rationality. He does this by placing economic agents and institutions in a decision theoretic context. He then shows how to reformalise economic agents and institutions so understood in algorithmetic terms, thereby embedding the insightful concepts of bounded rationality and satisficing behaviour in the sophisticated mathematical architecture of computable mathematics. In this way his programme undermines ‘the conventional misconception that a bounded rational agent is simply orthodox’s omnipotent substantively rational agent cognitively constrained in various ad hoc ways; and that satisficing is simply a “second best” optimization outcome’ (Velupillai 2005b: 173). Rather the bounded rational agent is a Turing machine. In this model, unlike the orthodox optimising model, the theoretical economist faces up to incompleteness and undecidability.
It is not our intention to outline Velupillai’s defence of his programme of computable economics. Velupillai’s defence constitutes an original, detailed, highly sophisticated and nuanced philosophy (methodology) of economic theorising. Rather we refer to his research programme in the spirit of constructivism: it concretely demonstrates that a prima facie case actually exists for the radical programme of a comprehensive algorithmetic mathematisation of economic theorising. This research programme offers the promise of providing new and innovative theoretical foundations for a digital economy. This, in our opinion, includes the possibility of modelling an economy, conceived as vast dynamical system of interacting markets capable of self-reproduction and evolution in terms of Turing-machine, cellular automata pioneered by von-Neumann, e.g. Mirowski’s markomata.28 It also opens up the possibility of the economic exploration by computer simulation. Of course whether or not this novel research agenda of computable economics will succeed depends on whether or not it actually delivers on its initial promise. As one moves from thin to thick descriptions of economic agents and markets whether or not algorithmetic models will deliver fresh insights for both economists and economic policy decision makers is an issue to be settled by future economic research. However, the value of openness in the search for truth which has inspired the scientific mind over the ages points to the need for more detailed research in this potentially challenging research programme of algorithmetic economics.
1 As noted by Troelstra and van Dalen, there are ‘considerable differences in outlook’ between various representatives of constructivism (Troelstra and van Dalen 1988: 1). In their opening introductory chapter they give a very brief account of some similarities and differences. In this work we do not dwell on these differences. Rather, like Troelstra and van Dalen, we focus on a number of constructivist trends. The selected trends are significant for economic methodology. Of course the theoretical economist who wishes to exploit constructive mathematics will have to pay more attention to these differences among constructivists.
2 One recalls the resistance to the seventeenth century scientific revolution by various institutions, academic and otherwise.
3 The curious reader could do worse than consult van Dalen’s (2005) excellent, two volume biography of Brouwer, Mystic, Geometer and Intuitionist.
4 As for instance van Stigt points out, in his Life, Art and Mysticism Brouwer ‘rails against industrial pollution and man’s domination of nature through his intellect and against established social structures and promotes a return to “Nature” and to mystic and solitary contemplation’ (van Stigt 1998: 5).
5 Brouwer focuses on the Kantian intuition of time and ignores the Kantian intuition of space. This could be due to the influence of Poincaré on Brouwer. Poincaré’s philosophy of geometry undermined Kant’s intuitive approach to geometry.
6 More precisely, as we will see later, classical logic combined with Cantorian set theory is the source of the paradoxes.
7 In this presentation of strict intuitionism, we, à la Heyting, emphasize intuitionistic logic. As van Stigt points out, Brouwer, while approving of Heyting’s contribution to logic, ‘remained true to his conviction than engagement in logic is an interesting but irrelevant and sterile exercise’ (van Stigt 1998: 276). Brouwer, in line with J.S. Mill, holds that deductive logic merely makes explicit what is implicit in the premises and thus cannot advance mathematical knowledge. Moreover Brouwer was suspicious of all languages, natural and artificial and contemporary logic requires an artificial, ideal language. These two factors help explain Brouwer’s negative attitude to logic which may seem surprising to the contemporary reader.
8 Principles or rules of inference other than the principle of the excluded middle are rejected by intuitionists. Troelstra and van Dalen (1988: 12) give a brief list of classical logical principles rejected by intuitionists.
9 For instance, Debreu points out that in the case of perfect competition, integration theory was invented to ‘solve the problem of aggregating negligible quantities so as to obtain a nonnegligible sum … That application requires the set of agents to be large – larger than the set of integers’ (Debreu 1991: 3). To a large number of pragmatic economists this sounds like science fiction: it throws no light on a real economy.
10 In this and previous chapters we focused on the formalist’s conception of mathematical existence in terms of freedom from inconsistency. This is not the only philosophy of mathematical existence presupposed in classical mathematics. Frege, for instance, gives us a realist (Platonic) philosophy of mathematical existence, where mathematical entities subsist in a Platonic world independent of the physical world on the one hand and the mind on the other. Both the formalist and realist notions of mathematical existence are rejected by intuitionists.
11 In his 1921 article, ‘On the New Foundational Crisis in Mathematics,’ Weyl, one of Hilbert’s most outstanding pupils, claimed ‘… and Brouwer – that is the revolution’ (Weyl 1998: 99). Weyl’s conversion to intuitionism sounded the alarm bells for Hilbert. From 1925, according to Mancosu, Weyl ‘attempts to take a middle stand between Hilbert and Brouwer’ (Mancosu 1998: 80).
12 Under the influence of Hilbert’s formalism, a group of French mathematicians, writing under the pseudonym, Nicholas Bourbaki, beginning in the 1930s, attempted in their Élements de Mathématique to give an authoritative summing up of existing mathematical knowledge in the most rigorous way possible. This work was influential both inside and outside France. Vilks (1995) gives an interesting summary of the Bourbaki project’s influence on the mathematization of theoretical economics.
13 See, for instance, Weintraub and Mirowski (1994) and Weintraub (2002).
14 The Brouwerian scholar van Stigt (1998) also emphasizes this point.
15 As we already noted that J.S. Mill, for instance maintained that deductive logic does not lead to new knowledge. Thus there is more to creative mathematics than logic. If Brouwer had followed Poincaré on his intriguing analysis of the indispensable roles of both logic and intuition in mathematics, Brouwer would not have given hostage to fortune in remarking that logic is an irrelevant and sterile exercise. To distinguish Poincaré’s approach to foundational questions from his own, Brouwer places Poincaré into ‘old intuitionism’ (Brouwer 1998: 55). For an interesting account of some of the differences between Poincaré and Brouwer see Heinzmann and Nabonnard (2008). Certainly the present authors believe that the passing remark by Troelstra and van Dalen, i.e. ‘one should not, however, expect too much consistency in Poincaré’s philosophical writings’ (Troelstra and van Dalen 1988: 19), does not do justice to Poincaré’s sophisticated analysis of the indispensable role of both logic and of intuition in mathematics. If one were looking for a one sentence summary of Poincaré’s view of the indispensable role of intuition in mathematics, the following would not be too misleading: intuition is to discernment and judgment in mathematics what wisdom is to discernment and judgment in practical reasoning.
16 Some intuitionistic mathematicians would not agree with this assessment. They link Brouwerian philosophical psychology to Husserl’s phenomenological psychology.
17 We will examine the implications of pragmatic intuitionism for the mathematization of economics in the section after the next.
18 Since the late 1950s Dummett has exerted a dominant influence on British philosophy. Frege, Wittgenstein and Brouwerian intuitionism are the principal influences on Dummett’s research. We suggest McGuinness and Oliveri (1994) The Philosophy of Michael Dummett as a must for an exploration of its intriguing challenges. Dummett’s Intellectual Autobiography is contained in Auxier and Hahn (2007) The Philosophy of Michael Dummett.
19 In later sections we will meet, for instance, Velupillai’s critique of the neo-Walrasian programme.
20 One could say that the doctrine of bivalence is just another way of expressing the classical logical principle of the excluded middle. This would hold for pre-twentieth-century logicians who maintained that the principle of the excluded middle is among the fundamental principles of all thought. However, in the twentieth century axiomatisation of classical logic, the principle of the excluded middle does not appear among the axioms. Rather it is a theorem. However, this axiomatised logic is grounded on the doctrine of bivalence, i.e. every proposition has a determinate truth-value, irrespective of whether or not we know which one actually applies.
21 The curious reader should consult Wright (1994) and Dummett (1963 1994a,1994b) for more information on the range of issues at play have.
22 More precisely, intuitionistic and classical logicians agree on the outcomes of numerous applications of the modus ponens rule of inference. However, since they differ in their respective understandings of logical implication, they do not agree on the meaning of the modus ponens rule of inference.
23 For some philosophical traditions the philosophical challenge to establish that the antecedent, T is consistent, of (i) is true is not very difficult. For instance in the Kantian philosophical tradition the axioms of elementary arithmetic are synthetic a priori truths. Since truths implies consistency then the antecedent, T is consistent, is true. Neither is this challenge problematical for a strict Brouwerian intuitionist. Neither of these methods is open to strict Dummettian intuitionists. Rather in line with Frege, they argue that the number system is endemic to our thoughts about the world. The denial of the natural number system would render coherent thought impossible. This is a philosophical thesis.
24 This is a very adumbrated presentation of the strict Dummettian argument. See note 21.
25 Zambelli lists some of Velupillai’s results as follows:
• ‘there is no effective [universal] procedure to generate preference orderings’ (Velupillai 2000: 38). The generation of preference ordering is undecidable or ‘rational choice, understood as maximizing choice, is undecidable’ (Velupillai 2008c: 8)
• The Walrasian excess demand ‘X(p*) is undecidable, i.e., it cannot be determined algorithmetically’ (Velupillai 2006b: 363). Alternatively the existence of a ‘Walrasian equilibrium price vector is undecidable’ (Velupillai 2002: 318)
• ‘Uzawa equivalence theorem is neither constructive nor computable’ (Velupillai 2005: 862)
• ‘The Recursive Competitive Equilibrium of New Classical Macroeconomics, Recursive Macroeconomic Theory, is uncomputable’ (Velupillai 2008b: 10)
• ‘Scarf’s Computable General Equilibrium is neither computable nor constructive’ (Velupillai 2006b)
• ‘neither the first nor the second welfare theorems are computationally feasible in the precise sense of computability theory and constructive analysis’ (Velupillai 2007b: 27–28)
• ‘Nash equilibria of finite games are constructively indeterminate’ (Velupillai 2008b: 21), (Zambelli 2010: 37–38).
26 It should be noted that the Church-Turing thesis is neither a mathematical definition nor a mathematical theorem. This is so, because mathematics establishes relationships between and only between rigorously formulated concepts, whereas the Church-Turing thesis is primarily concerned with the inherited, but not rigorously formulated concept of an effective mechanical procedure. It is summing up a most amazing fact, viz. all of the above attempts at rigorously formalizing the informal or inherited notion of a computable function yield exactly the same class of functions (See Boylan and O’Gorman (2010) for more details).
27 Here we are emphasizing the computable aspect of Velupillai’s research programme. Velupillai himself invokes both computable and Brouwerian constructivist mathematics. According to Zambelli ‘among the purists in classical recursion theory and the purists in constructive mathematics there would be disagreement with this (Velupillai’s) somewhat unorthodox view’ (Zambelli 2010: 34). It is beyond the scope of this work to critically engage the full implications of Velupillai’s research programme, especially the manner in which he exploits both Brouwerian constructivism and computable mathematics in his theorising. In this connection one should recall that Velupillai is above all a mathematical theoretical economist whose interest is the scientific study of real economies. Since economies are finite systems and since Brouwerian constructivists insist that the principle of the excluded middle holds for finite systems, but not for potentially infinite ones, we feel that there is nothing paradoxical in the manner in which Velupillai synergizes both Brouwerian intuitionistic mathematics and computable mathematics. While at the level of pure mathematics the Church-Turing thesis does not hold in Brouwerian intuitionistic/constructivist mathematics, as Dummett points out, ‘it is indeed true that Church’s Thesis is, when expressed in a suitable form, demonstratably consistent with most intuitionistic formal systems …’ (Dummett 2000: 186). In our view one could argue that the logic used in all discourse is intuitionistic but in economics this logic is applied to a finite domain where the principle of the excluded middle is a law-like empirical truth. There is no logical paradox in that viewpoint.
28 The economic methodologist interested in this aspect of what we call computable economics should read Zambelli’s (2007) critique of Mirowski’s thesis and Rosser Jr’s (2010) attempted reconciliation. In our opinion in any new research programme it is too much to expect complete consensus on all matters on the part of those advocating the new research programme. Thus in this work we do not engage the differences between Mirowski and Velupillai. Rather we emphasize their similarities.