73 research outputs found
Uncomputability and Undecidability in Economic Theory
Economic theory, game theory and mathematical statistics have all increasingly become algorithmic sciences. Computable Economics, Algorithmic Game Theory ([28]) and Algorithmic Statistics ([13]) are frontier research subjects. All of them, each in its own way, are underpinned by (classical) recursion theory - and its applied branches, say computational complexity theory or algorithmic information theory - and, occasionally, proof theory. These research paradigms have posed new mathematical and metamathematical questions and, inadvertently, undermined the traditional mathematical foundations of economic theory. A concise, but partial, pathway into these new frontiers is the subject matter of this paper. Interpreting the core of mathematical economic theory to be defined by General Equilibrium Theory and Game Theory, a general - but concise - analysis of the computable and decidable content of the implications of these two areas are discussed. Issues at the frontiers of macroeconomics, now dominated by Recursive Macroeconomic Theory, are also tackled, albeit ultra briefly. The point of view adopted is that of classical recursion theory and varieties of constructive mathematics.General Equilibrium Theory, Game Theory, Recursive Macro-economics, (Un)computability, (Un)decidability, Constructivity
Varieties of Mathematics in Economics- A Partial View
Real analysis, founded on the Zermelo-Fraenkel axioms, buttressed by the axiom of choice, is the dominant variety of mathematics utilized in the formalization of economic theory. The accident of history that led to this dominance is not inevitable, especially in an age when the digital computer seems to be ubiquitous in research, teaching and learning. At least three other varieties of mathematics, each underpinned by its own mathematical logic, have come to be used in the formalization of mathematics in more recent years. To set theory, model theory, proof theory and recursion theory correspond, roughly speaking, real analysis, non-standard analysis, constructive analysis and computable analysis. These other varieties, we claim, are more consistent with the intrinsic nature and ontology of economic concepts. In this paper we discuss aspects of the way real analysis dominates the mathematical formalization of economic theory and the prospects for overcoming this dominance.
Defining a General Structure of Four Inferential Processes by Means of Four Pairs of Choices Concerning Two Basic Dichotomies
In previous papers I have characterized four ways of reasoning in Peirceâs philosophy, and four ways of
reasoning in Computability Theory. I have established their correspondence on the basis of the four pairs
of choices regarding two dichotomies, respectively the dichotomy between two kinds of Mathematics and
the dichotomy between two kinds of Logic. In the present paper I introduce four principles of reasoning in
theoretical Physics and I interpret also them by means of the four pairs of choices regarding the above two
dichotomies. I show that there exists a meaningful correspondence among the previous three fourfold sets
of elements. This convergence of the characteristic ways of reasoning within three very different fields of
research - Peirceâs philosophy, Computability theory and physical theories - suggests that there exists a
general-purpose structure of four ways of reasoning. This structure is recognized as applied by Mendeleev
when he built his periodic table. Moreover, it is shown that a chemist-, applies all the above ways of
reasoning at the same time. Peirceâs professional practice as a chemist applying at the same time this
variety of reasoning explains his stubborn research into the variety of the possible inferences
Continuity, Discontinuity and Dynamics in Mathematics & Economics - Reconsidering Rosser's Visions
Barkley Rosser has been a pioneer in arguing the case for the mathematics of discontinuity, broadly conceived, to be placed at the foundations of modelling economic dynamics. In this paper we reconsider this vision from the broad perspective of a variety of different kinds of mathematics and suggest a broadening of Rosserâs methodology to the study of economic dynamicsContinuity, Discontinuity, Economic Dynamics, Relaxation Oscillations
Dealing with Paradoxes of Law: Derrida, Luhmann, Wiethölter
Dt. Fassung: Der Umgang mit Rechtsparadoxien: Derrida, Luhmann, Wiethölter. In: Christian Joerges und Gunther Teubner (Hg.) Rechtsverfassungsrecht: Recht-Fertigungen zwischen Sozialtheorie und Privatrechtsdogmatik. Nomos, Baden-Baden 2003, 249-272
A NEW PHILOSOPHICAL FOUNDATION OF CONSTRUCTIVE MATHEMATICS
The current definition of Constructive mathematics as âmathematics within intuitionist logicâ ignores two fundamental issues. First, the kind of organization of the theory at issue. I show that intuitionist logic governs a problem-based organization, whose model is alternative to that of the deductive-axiomatic organization, governed by classical logic. Moreover, this dichotomy is independent of that of the kind of infinity, either potential or actual, to which respectively correspond constructive mathematical and classical mathematical tools. According to this view a mathematical theory is based on the choices regarding these two dichotomies. As an example of this kind of foundation, arithmetic is rationally re-founded on constructive mathematical tools and the model of the problem-based organization. In conclusion, constructive mathematics is not only mathematics making use of constructive tools in intuitionist logic but also organized according to around a basic problem, solved by a method discovered using intuitionist logic
A Primer on the Tools and Concepts of Computable Economics
Computability theory came into being as a result of Hilbert's attempts to meet Brouwer's challenges, from an intuitionistc and constructive standpoint, to formalism as a foundation for mathematical practice. Viewed this way, constructive mathematics should be one vision of computability theory. However, there are fundamental differences between computability theory and constructive mathematics: the Church-Turing thesis is a disciplining criterion in the former and not in the latter; and classical logic - particularly, the law of the excluded middle - is not accepted in the latter but freely invoked in the former, especially in proving universal negative propositions. In Computable Economic an eclectic approach is adopted where the main criterion is numerical content for economic entities. In this sense both the computable and the constructive traditions are freely and indiscriminately invoked and utilised in the formalization of economic entities. Some of the mathematical methods and concepts of computable economics are surveyed in a pedagogical mode. The context is that of a digital economy embedded in an information society
Some Lesson About the Law From Self-Referential Problems in Mathematics
We first describe briefly mathematician Kurt Gödel\u27s brilliant Incompleteness Theorem of 1931, and explore some of its general implications. We then attempt to draw a parallel between axiomatic systems of number theory (or of logic in general) and systems of law, and defend the analogy against anticipated objections. Finally, we reach two types of conclusions. First, failure to distinguish between language and metalanguage in mathematical self-referential problems leads to fallacies that are highly analogous to certain legal fallacies. Second, and perhaps more significantly, Gödel\u27s theorem strongly suggests that it is impossible to create a legal system that is complete in the sense that there is a derivable rule for every fact situation. It follows that criticisms of constitutional systems for failure to determine every answer are unfair: they demand more than any legal system can give. At best they are antilegal in the sense that rejection of such constitutional systems on such a ground would require the rejection of all constitutional systems
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