1. The complete analysis of economic dynamics

The obsession with the desire for a complete analysis of economic dynamics, is analogous to the classic – but forlorn – search for a ‘qualitative determination of all possible types of motions’, or to considering all possible sets and ending up with the (in)famous Russell paradox of considering all possible sets and wondering what to do about that class that is made up of those sets that are not members of themselves.

It is possible, perhaps even desirable, to separate the notion of complete analysis of any problem – not only in economics – and the idea of a complete economic dynamics, independent (at least in the latter case) of the feasibility of its complete analysis. Then Quadrio Curzio's wise injunction comes into its own (see Birkhoff above and Quadrio Curzio and Pellizzari, Reference Quadrio Curzio and Pellizzari1999, p. 240, italics added): ‘Theory always takes a specific definition of [an] economic system and chooses a select set of tools and methods for its analysis.’ That ‘set of tools and methods for its analysis’, chosen by a ‘theory taking a specific definition of [an] economic system’ is often thought of as having achieved completeness – of analysis, of formalization, etc. Thus in the admirable volume of essays in honour of Augusto Graziani (Arena and Salvadori, 2004)¹, Wynne Godley claims (ibid., p. 143) to have built a Complete Keynesian Model (albeit a simple one!). Even more remarkably, in the same volume, Jean Cartelier claims that (ibid., p. 221; italics added): ‘A complete demonstration of the existence of a Keynesian equilibrium in an Arrow-Debreu framework exists.’² Obviously Professor Cartelier, like most of the profession, has forgotten – or, more likely, is ignorant of the fact – that Glustoff (like his fellow general equilibrium theorists) ‘chose a select set of tool and methods’ to provide a so-called complete demonstration of the existence of a Keynesian equilibrium in an Arrow-Debreu framework. From the perspective of a different kind of mathematics, say computable or constructive analysis, this demonstration by Glustoff would not be considered complete³. This is precisely the point made by Quadrio Curzio – leaving aside the other injunction of taking ‘a specific definition of [an] economic system’, often unadaptable to the tools and methods of analysis being used.

To the best of my knowledge, the earliest attempt to seek a formalization of a notion of complete economic dynamics was in a classic contribution by Kuznets at the dawn of the era, which sought to integrate equilibrium theory with dynamics (Kuznets, Reference Kuznets1930). On the other hand, all attempts, in every kind of economic theory, at formulating ‘General Theories’ – be it the General Theory of Employment, Interest and Money or General Equilibrium Theory – are efforts, always falling foul of Quadrio Curzio's previously mentioned wise injunctions at building complete theories and choosing a select set of tools and methods for their analysis, from which special theories are derived.

Consider the following general form of a differential-difference equation of differential order n and difference order m (Bellman and Cooke, Reference Bellman and Cooke1963, p. 43):

Where: F and u are real functions of real variables, t; ω_i: real numbers, ∀i = 1…m, and n, m: integers.

Then it is easy to show that the endogenous macrodynamic business cycle models of Kalecki (Reference Kalecki1935), Lundberg (Reference Lundberg1937), Kaldor (Reference 185Kaldor1940), Hicks (Reference Goodwin1950) and Goodwin (Reference Goodwin1951b) can be derived from the above general form of a differential-difference equation. Now two simple questions can be posed:

1. 1. In what sense can (1) be considered a complete formulation of endogenous business cycle theories? In other words, do differential, difference and mixed differential-difference equations encapsulate, completely, all dynamic phenomena in economics?

2. 2. Even if we grant the above, is there one unique mode of analysis of such a system for which completeness can formally be defined?

In answer, first, to the second part of the first question, it is easy to construct an example of an algorithmic dynamical system that has not been shown to be reducible to the dynamics of a differential, difference or mixed differential-difference system of any generality. I have in mind here the Goodstein Process (cf. Goodstein, Reference Goodstein1944 and Paris and Tavakol, Reference Paris and Tavakol1993). As for an answer to the first part, nothing in dynamical systems theory indicates that there is any hope of characterizing the complete (sic!) dynamics of (1), when hardly any breach has been made towards even solving the second part of Hilbert's 16th Problem regarding even planar dynamical systems (Hilbert, Reference Hilbert1902).

As for the second question, first of all there are many modes of analysis to prove the existence of a closed path using the Poincarè – Bendixson theorem, as applied, for example, to Kaldor (Reference 185Kaldor1940), some of which invoke Hilbert's notion of ‘consistency implies existence’, which fall foul of Gödel's results on incompleteness (Shapiro, Reference Shapiro1997, p. 134)⁴. Moreover, it is by now well known that many special cases of (1) are subject to undecidable dynamics (Stewart, Reference Stewart1991), which implies that no constructive, algorithmic methods can lead to any kind of complete analysis, except in the world of non-applicable (i.e. non-empirical) classical mathematics.

2. If economic dynamics is algorithmic dynamics, then complete dynamics is a chimera

Change, in economics and in all the sciences, is considered synonymous with dynamics, at least since Newton and always in mathematical formalisms that invoke one or another form of dynamical systems theory – most commonly, almost trivially, in terms of differential, difference or mixed differential-difference systems of equations. These phantoms lead to the false dichotomy, in economics, mimicking that which originated in Newtonian mechanics, between statics and dynamics; between equilibrium and disequilibrium; and between stability and instability. Rarely does one think – at least not in economics – of algorithms as dynamic objects, implemented in time, imperative in their logical form and intrinsically constructive, in the strict mathematical sense of this word, especially in its Intuitionistic, Brouwerian versions. As affirmed in what he called the First Act of Intuitionism, Brouwer stated, in his Cambridge Lectures on Intuitionism uncompromisingly:

The fundamental dichotomies in algorithmic mathematics – whether it be in its computable or constructive versions – is that between the halting and non-halting behaviour of a machine implementing an algorithm, between decidable and undecidable problems that an algorithm aims to solve, against the backdrop of the further dichotomy between the solvability or the unsolvability, easily or with difficulty, of a decision problem⁶. Six decades ago, at the pre-dawn period of what eventually became the algorithmic avalanche and the dominance of the computational⁷ approach to economics, in both theoretical and empirical modes, Richard Goodwin's acute prescience led him to observe that:

In this perceptive conjecture, Goodwin suggests an equivalence between algorithmic dynamics and economic dynamics, mediated by viewing ‘the motion of an economy as a process of computing the answers to the problems posed to it’. The false dichotomies pointed out earlier are replaced by those that are intrinsic to algorithmic dynamics.

Computation, problem, solvability and dynamics are all defined algorithmically.

However, algorithmic dynamics is replete with undecidabilities and incompleteness, in the strict metamathematical senses; it is also richly endowed with uncomputabilities in recursion theoretic senses. Moreover, carrying the suggested equivalences a little further, it would be natural to ask what kind of problems can be posed to ‘such a machine’ (or the economic system) and ask whether we can classify the ‘posable’ problems as easy and hard, on a sliding scale, first separating the decidable problems from the undecidable ones. Such questions lead immediately to structuring the data, which is the bridge between theory and empirical implementation, in terms of sets of numbers that are recursive, recursively enumerable and recursively enumerable but not recursive. These, then, lead to notions of recursive separability and inseparability, from which naturally paradoxical undecidable and uncomputable dynamics emerge, encapsulable even in the formalism of dynamical systems theory (Smullyan, Reference Smullyan1993; Pour-El and Richards, Reference Pour-El and Richards1989).

The classification of the decidable problems – i.e. classification of recursive sets – into a sliding scale, ranging from easy to hard – the domain of computational complexity theory⁹ – say from polynomial time computability at the ‘easy’ end, all the way to the NP-Complete problems at the difficult end.

On the other hand, classification of undecidable problems – i.e. classification of non-recursive sets (of numbers) – requires the formalism of oracle (or relative) computations, reducibilities and a rigorous notion of complete (Davis, Sigal and Weyuker, Reference Davis, Sigal and Weyuker1994; Salomaa, Reference Salomaa1985).

The idea of classifying undecidable problems in terms of the difficulty of solving them may seem like a contradiction in terms. However, this apparent ‘contradiction’ is resolved by means of Oracle Turing Machines, or using relative computations (Rogers, Reference Rogers1967; Davis, Reference Davis1958). Thus, an undecidable problem P* is classified as being at least as difficult as another problem, P, if an algorithm, to which an Oracle is adjoined, implemented on an Oracle Turing Machine, that can solve P* can also be utilized to solve P. If the reverse process is infeasible, i.e. if the undecidable problem P, solved with an Oracle Turing Machine, cannot be used to solve P*, then the latter is, intuitively, considered ‘more undecidable’, or more difficult, than the former.

• Definition 1:

Given two sets of arbitrary – i.e. not necessarily recursively enumerable sets – nonnegative integers, S and S*, if any algorithm for solving whether an element is a member of S* implies the existence of an algorithm for solving the membership problem for S, then we denote this by:

$\begin{array}{l}S\le S^{*}\\ \text{If}S\le S^{*}\&S^{*}\le S,\text{then}{S}^{*}\equiv S\end{array}$

• Definition 2: Complete

If S* is a recursively enumerable set, and ∀S , also recursively enumerable sets, S ≤ S^*, then S* is said to be COMPLETE.

• Definition 3: m-reducible

S is m-reducible to S* iff ∃ a recursive function g s.t., ∀i, j ∈ g(i) in case g ∈ S^*

• Remark 1:

Every recursively enumerable set is m-reducible to the Halting set.

Thus the halting problem for Turing Machines is an alternative mode of tackling undecidability in the context of recursively enumerable sets and their membership problem. If the economic system's dynamics is to be interpreted as the trajectory of the path taken by a computing machine solving problems posed to it, then the data sets – in this case, economic data in the form of numbers – must be considered to be appropriately recursive. Ad hoc habits of assuming data to be generated from exogenous probability distributions, or worse, will have to be given up. In this way theory and empirical analysis are tied together within a unified algorithmic framework.

This notion of complete has, of course, nothing whatsoever to do with the loose, almost flippant, use of the word complete by economists, claiming to model this or that completely, or analyse a given model completely; not even anything to do with the notion of completeness in classical real analysis (for example the ‘completeness of a metric space’); above all, nothing at all to do with the deep notions of completeness and incompleteness in metamathematics, due to the pioneering contributions of Thoralf Skolem and Kurt Gödel in the first third of the twentieth century.

If Kaldor can imagine an Economics without Equilibrium (Kaldor, Reference Kaldor1985), I see no reason why, with a vision from an algorithmic point of view, I should not – or cannot – imagine an economics without the dichotomies that originated in the natural sciences and replace them with those that are intrinsic to a mathematics that is naturally adapted to implementing empirically the theoretical structures of economics.

3. Towards undecidable, incomplete and uncomputable dynamics

An algorithmic vision of economic dynamics – or any kind of dynamics – entails a challenge of a qualitatively different kind from those that have become almost ‘standard’ in an analytically trivial sense. I am referring here to all the hype about complex dynamics, emergence and the like. The algorithmic vision forces the serious scholar of change as dynamics to come to terms with undecidable dynamics, incomplete dynamics and uncomputable dynamics. In the world of algorithmic dynamics, richly endowed with undecidabilities, incompleteness and uncomputabilities, the notion of optimization is meaningless. Instead, the framework is that of decision problems in its metamathematical algorithmic senses.

Completeness, even in mathematics, has become a weasel word; disentangling its many splendoured variations, showing its chimerical usage by pseudo-mathematical economists, is itself a difficult task, like that faced by a mycologist trying to classify mushrooms, as Vladimir Arnold suggested:

Taking a cue from what I like to think of as Alberto Quadrio Curzio's fundamental precept – to recognize, explicitly, that ‘theory always takes a specific definition of [an] economic system and chooses a select set of tools and methods for its analysis’– I have tried to discuss what may be implied by selecting a particular set of tools and methods for the analysis of economic theory. I have tried to couple this to a vision of an economic system devoid of false dichotomies, those that have shackled economic theory for over two centuries. The price of the vision I advocate is the introduction of new dichotomies, but those that I think are natural to economic analysis in its mathematical mode.

However, anything I can contribute, to pay homage to the aristocratic scholar-gentleman who is Alberto Quadrio Curzio must, surely, be woefully inadequate from every point of view. Yet, I consider myself privileged to have been invited to do so, and – however inadequate – it is written with conviction and humility, knowing that the unattainable is also the impossible, but yet, with Tennyson's Ulysses (italics added):

4. A personal acknowledgement

My first encounter with one of Quadrio Curzio's brilliant writings was one that was bristling with implicit dynamics, although clothed in the impeccable and illuminating formalism of capital theory – and that was over three decades ago (Quadrio Curzio, Reference Quadrio Curzio and Pasinetti1980). Over the next two decades – and more – Quadrio has worked out the ‘formidable empirical content’ of the clearly set out theoretical research program (Quadrio Curzio and Pellizzari, Reference Quadrio Curzio and Pellizzari1999, Reference Quadrio Curzio and Pellizzari1996), ‘following [a] path of constructive research’, implicit in that classic, itself a continuation of a deeply original line of research, powerfully inspired in an unparalleled way by Sraffa’s classic (Sraffa, Reference Sraffa1960), that had begun at least as early as 1967 (Quadrio Curzio, Reference Quadrio Curzio1967, Reference Quadrio Curzio1975).

In a precisely definable sense of the consummation of a research program, ‘following a path of constructive research’ and endowing it with some ‘formidable empirical content’, the more mature Quadrio Curzio concluded almost with some resignation: ‘In conclusion … we recall some studies of economic dynamics just to remind us [of] the vast range and complexity of this field, and the absence of a complete analysis of economic dynamics’ (Quadrio Curzio and Pellizzari, Reference Quadrio Curzio and Pellizzari1999, p. 240; italics added). Perhaps one should read ‘absence’ as ‘impossibility’! Such a reading would be entirely consistent with the view of Siro Lombardini, summarized, I think with warm approval by Quadrio Curzio and Pellizzari (Reference Quadrio Curzio and Pellizzari1999, p. 242; italics added):

These wise caveats by Quadrio Curzio are those of a scholar with a mastery of the history of thought, a supreme economist who has grappled with the conundrums of grounding empirical content in solid economic theory, a concerned policy analyst who sees the need for understanding the evolution of institutions, and a passionate advocate of the need to foster innovation and, therefore, to grasp the role of entrepreneurship in advanced industrial economies (Quadrio Curzio and Fortis, Reference Quadrio Curzio and Fortis2005, Reference Quadrio Curzio and Fortis2008). In these attempts at forging an eclectic vision for an enlightened society, Quadrio Curzio invokes, as if they were intimate friends with whom he converses regularly, the classics and the pioneering Italian pre-Classics (for example, Cesare Beccaria and Ferdinando Galiani), our neoclassical masters, and those of the interregnum who aimed to consolidate the classical framework – John Stuart Mill and John E. Cairnes – and some who went beyond, Karl Marx, and the giants of the twentieth century, from Wicksell and Schumpeter, to Keynes and Sraffa. There were the Scottish and the French enlightenments – and also the partial Italian ones, those of an old nation yet to be forged into a new country. These partial Italian contributions, particularly the Lombard enlightenment, are reflected in Quadrio Curzio's passionate commitment to the unified ideals of European values, against a backdrop of European unity and unified policy for research and innovation within a framework of evolutionary institutions, fostering entrepreneurship (Quadrio Curzio and Fortis, Reference Quadrio Curzio and Fortis2005, Reference Quadrio Curzio and Fortis2008)¹¹

I have often found myself remembering, and reminding my own students, when we are prone to indulge in flights of theoretical fancy and atheoretical empirical fantasies, precepts elegantly and concisely formulated by Quadrio Curzio, but in the unlikely context of ‘the Gold Problem’ (Quadrio Curzio, Reference Quadrio Curzio1982, p.1): ‘History and fantasy, scientific analysis and platitudes, present-day dicta and the beliefs of centuries past overlap and mingle in such a way as to render quite complex both an objective analysis and a detached reading of contributions which seek to be objective.’ Alberto Quadrio Curzio, an aristocrat of theory, with a mastery of history and tradition – as Carlo Cipolla was, in a not unrelated field – combined majestically, as I, with my stunted scholarship and blinkered visions have understood, the visions of a Schumpeter with the theoretical purity of a Sraffa, and took that synthesis to new heights of formidable empirical and institutional scholarship. Lesser mortals like me can only wonder at, with admiration, and seek to emulate, the ideals of scholarship, institution-building and fostering of the evolution of tradition that Alberto Quadrio Curzio has made into an art, whilst also personifying its encapsulation in one, whole personality.