Abstract
One of the many remarkable qualities of Stefano Zambelli is the breadth of his research interests. From a brief examination of Zambelli’s scientific contributions, projects development activities, and research breakthroughs, it is readily noticeable that he has not worked in a single narrow area. Instead, his research efforts have resulted in an unusually broad set of theoretical and empirical findings across several strands of the economics literature. Among these strands, Zambelli’s (1994) manifesto on the logic of computability signalled a sea change in his approach to research helping to build the program known as computable economics. In this chapter, written to honour a fraternal friend and a praeternatural scientist, the role of computability is briefly discussed, so that to provide some expository remarks concerning the main interpretations of the foundations of mathematics for a post-Neoclassical age in economic theory.
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Notes
- 1.
It is worth noting the valuable contribution written by K. Vela Velupillai (2005) entitled The Unreasonable Ineffectiveness of Mathematics in Economics, whose discussion so much recalls the famous title of a seminal paper published in 1960 by the theoretical physicist Eugene Wigner (1960) titled The Unreasonable Effectiveness of Mathematics in the Natural Sciences and is framed therein.
- 2.
During the twenties of the last century, Hilbert attempted a single rigorous formalisation of all of mathematics, named Hilbert’s program, on an axiomatic basis. He was particularly concerned with the following three questions: (i) is mathematics complete in the sense that its every statement can be proved or disproved?; (ii) is mathematics consistent meaning that no statement can be proved both true and false?; and (iii) is mathematics decidable in the sense that there exists a formal method to determine the truth or falsity of any mathematical statement? Hilbert believed that the answer to the previous questions was affirmative. On the other hand, thanks to Gödel’s (1986 [1931]) incompleteness theorem and the undecidability of first-order logic demonstrated by Church and Turing, we know nowadys that Hilbert’s aspiration will never be fully realised (i.e., Church’s and Turing’s theses or, in other words, the so-called Church–Turing thesis, also known as computability thesis). This makes it an endless task of finding ‘possible’ partial answers to Hilbert’s questions (for insights, see Gandy 1988).
- 3.
The fulcrum of Gödel’s incompleteness theorems that brought down Hilbert’s program on which formalism in mathematics was grounded, lies in the self-referentiality of the problem, most notably the arithmetic that aims to find its foundation in itself, even justifying itself in arithmetic form. Broadly speaking, one can encounter situations of self-referentiality whenever the symbol of an object is confused with the object itself, that is, blurring the difference between the ‘use’ of a word and the ‘mention’ of the word. In cases like this, logic inevitably brings out elements of undecidability.
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Acknowledgements
The authors are indebted to their colleague Ragupathy Venkatachalam for help in interpretations and analysis. Furthermore, the authors are both very grateful to Shu-Heng Chen for many really enjoyable and enlightening conversations, as well as intellectually and ethically challenging thoughts and ideas that they have benefited from. Out of any rhetoric and stereotypes, finally, the authors are deeply obliged to Stefano Zambelli whose very special encouragements and magnanimity in the last two decades have allowed them—above all—to deepen the knowledge of a brotherly friend and a praeternatural scientist. Needless to say none of them are responsible for any remaining infelicities.
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Bucciarelli, E., Mattoscio, N. (2021). Recasting Stefano Zambelli: Notes on the Foundations of Mathematics for a Post-Neoclassical Age in Economics. In: Velupillai, K. (eds) Keynesian, Sraffian, Computable and Dynamic Economics . Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-58131-2_3
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