Complex Agent-Based Models by Mauro Gallegati

Author:Mauro Gallegati
Language: eng
Format: epub, pdf
ISBN: 9783319938585
Publisher: Springer International Publishing

3.1 From Axioms to Hypotheses

The mainstream pursues the same utopia as Hilbert’s program in mathematics, which aimed to prove that all mathematical laws could be explained by a consistent system of axioms. Gödel’s incompleteness theorem, however, demonstrates the impossibility of this program, because even the axioms are based on statements that are impossible to prove in regard to whether they are true or false with nothing but the tools derived from the axioms themselves. In the general equilibrium framework—and in its modern developments—topological approaches are used to derive the existence of a fixed point, such as the Brouwer fixed point theorem or the more general one by Kakutani, but these are pure theorems of existence, not constructive, which do not provide any indication as to how to find the fixed point.

Regarding the incompleteness theorem, (Gödel 1931) says: “In any formal system S endowed with a logical (ie, non-contradictory) set of basic axioms which are sufficiently complex as to be able to encompass arithmetic, it is possible to construct propositions that the system can’t determine: they can neither be proved nor disproved, on the basis of the axioms and rules of system deduction.” Basically, Gödel shows that we can construct a proposition P that succeeds in affirming: “P cannot be demonstrated in S,” as in the paradox of the liar: “I’m lying.” The same happens for the axiomatic system of Debreu, in which the SMD theorem proves the undecidability of the same.

To overcome the impasse of the GET and DSGE (Landini et al. 2018), it is necessary to solve the problem of axiomatic systems—not verifiable internally—and thus subject to Gödel’s theorems, and “validate” the hypotheses, as the ABM can do. The paradigm of Walras was based on an analogy with classical physics, but it was not sufficient to solve the fundamental problem in the general equilibrium framework, that is, proving the existence of the equilibrium. Mathematical economics from Walras’ Elements to the end of the 1930s of the last century was basically an analogical representation of economic facts with the classical mechanics models of physics (Mirowski 1989; Ingrao and Israel 1991). This mathematization of economics consists in changing the language from ordinary (e.g., English) to artificial (mathematics) and finding analogies between economic and physical phenomena to formulate economic theories.

The mathematical economics between the 1940s and the 1950s is the formalization of the methods for analyzing the economy. Formalization of economics consists in the mathematical specification of economic problems. Von Neumann (1937) formulated the equilibrium problem replacing the analogy with physics with a mathematical analogy (Gloria-Palermo 2010). In this mathematical economics of the second generation, the problem ceases to be economic, becoming a mathematical problem (Blaug 2003). [It is certainly not mathematics that is in crisis today—given the axioms it started with, it has done a great job—but rather the use that the mainstream paradigm makes of it.] The switch from the first to the second generation of mathematical economics is a consequence of what happened in mathematics in the 1920s–1930s: the

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