New Mathematics and Natural Computation

The emergence of non-constructivities in economics is entirely due to the unnecessary and inappropriate formalization of economics by means of 'classical' mathematics. I have made similar claims for the emergence of uncomputabilities and undecidabilities in economics in earlier writings. Here, on the other hand, I want to suggest a way of confronting uncomputabilities, and remedying non-constructivities, in economics, and turning them into a positive force for modeling, for example, endogenous growth, as suggested by Stefano Zambelli. ^107,108 In between, a case is made for economics to take seriously the kind of mathematical methodology fostered by Feynman and Dirac, in particular the way they developed the path integral and the δ-function, respectively. A sketch of a "research program" in mathematical economics, analogous to the way Gödel thought incompleteness and its perplexities should be interpreted and resolved, is also outlined, albeit briefly, in the concluding section.

Kumaraswamy Vela Velupillai⁷⁴ presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics and considers its view on the Bolzano-Weierstrass, Hahn-Banach, and intermediate value theorems, and then the implications of these arguments for such "crown jewels" of mathematical economics as the existence of general equilibrium and the second welfare theorem. He also relates these ideas to the weakening of certain assumptions to allow for more general results as shown by Rosser⁵¹ in his extension of Gödel's incompleteness theorem in his opening section. This paper considers these arguments in reverse order, moving from the matters of economics applications to the broader issue of constructivist mathematics, concluding by considering the views of Rosser on these matters, drawing both on his writings and on personal conversations with him.

The work of K. 'Vela' Velupillai has illuminated the debate on the mathematization of economics by providing a broader view of the universe of mathematics and its possible applications in economics. The theoretical and policy consequences from the peculiar mode of mathematization in economics is another important theme in Vela work. Alternative modes of mathematization are offered with a call for an "Algorithmic Economics" in the future.

The use of mathematics in physics is not restricted to non-formal use with emphasis on approximation and computability. There is also a domain of creative use of symbolic manipulation as well as of metaphoric imagination that makes applicability in the sciences so effective. The challenge to mathematical use in economics is to find ways to use it in these creative ways, including finding new discursive and writing strategies. For this to be possible, the nature of realism in economics has to be delineated since the way in which mathematics is used in economics seems to be in the "reductive" mode instead of the "applied" mode.

The problem of the computability of Walrasian competitive equilibrium is considered from the point of view of concepts of approximate equilibrium. Neither the market-clearing nor Negishi approaches to the proof of existence of Walrasian competitive equilibrium give rise to adequately robust notions of approximate equilibrium. This explains the non-computability of Walrasian competitive equilibrium. The problem lies in the economic conception of markets, in particular the inconsistent treatment of information underlying the Walrasian definition. When trade takes place at disequilibrium prices decentralized market exchange redistributes income and economic welfare, and its equilibrium is path-dependent. The set of such equilibrium outcomes, however, in contrast to the Walrasian competitive equilibrium, is constructive and computable.

Algorithmic economics helps us stipulate, formulate, and resolve economic problems in a more precise manner than mainstream mathematical economics. It does so by aligning theorizing about an economic problem with both the data generated by the real world and the computers used to manipulate that data. Theoretically coherent, numerically meaningful, and therefore policy relevant, answers to economic problems can be extrapolated more readily using algorithmic economics than present day mathematical economics. An algorithmic economics would allow mathematical economics to prove theorems relating to economic problems, such as the existence of equilibria defined on some metric space, with embedded mechanisms for getting to the equilibria of these problems. A blueprint for such an economics is given and discussed with an example.

In this paper, a case for Velupillai's Computable Economics is made. It is claimed that the methodological approach and results (theorems) of classical recursion theory and constructive mathematics should be at the foundation of theorizing in economics. The major point is that Turing machine equivalent computations set an upper bound to rational choice that cannot be circumvented with a non-rigorous and teleological "as if" assumption. Clearly, given that the optimal choices are almost always not constructible the paradigm of rational behavior as optimizing behavior is deprived of any practical, operational or theoretical meaning. It is Computable Economics that provides the proper foundation for a formally rigorous and meaningful definition of rationality.

In order to ponder the question of (space) time, we consider exotic spacetimes in the sense of differential geometry, and then present undecidability and incompleteness results about general relativity, with the nature of time and the existence of some kind of "cosmic time" as central questions. We conclude with a discussion on the possible interpretation of our results.

Non-standard analysis can be harnessed by the recursion theorist. But as a computable economist, the conundrums of the Löwenheim-Skolem theorem and the associated Skolem paradox, seem to pose insurmountable epistemological difficulties against the use of algorithmic non-standard analysis. Discontinuities can be tamed by recursive analysis. This particular kind of taming may be a way out of the formidable obstacles created by the difficulties of Diophantine Decision Problems. Methods of existence proofs, used by the "classical" mathematician — even if not invoking the axiom of choice — cannot be shown to be equivalent to the exhibition of an instance in the sense of a constructive proof. These issues were prompted by the fertile and critical contributions to this special issue.