2,121 research outputs found
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.
Separability, Contextuality, and the Quantum Frame Problem
We study the relationship between assumptions of state separability and both
preparation and measurement contextuality, and the relationship of both of
these to the frame problem, the problem of predicting what does not change in
consequence of an action. We state a quantum analog of the latter and prove its
undecidability. We show how contextuality is generically induced in state
preparation and measurement by basis choice, thermodynamic exchange, and the
imposition of a priori causal models, and how fine-tuning assumptions appear
ubiquitously in settings characterized as non-contextual.Comment: 21 p
Computation in Economics
This is an attempt at a succinct survey, from methodological and epistemological perspectives, of the burgeoning, apparently unstructured, field of what is often – misleadingly – referred to as computational economics. We identify and characterise four frontier research fields, encompassing both micro and macro aspects of economic theory, where machine computation play crucial roles in formal modelling exercises: algorithmic behavioural economics, computable general equilibrium theory, agent based computational economics and computable economics. In some senses these four research frontiers raise, without resolving, many interesting methodological and epistemological issues in economic theorising in (alternative) mathematical modesClassical Behavioural Economics, Computable General Equilibrium theory, Agent Based Economics, Computable Economics, Computability, Constructivity, Numerical Analysis
Decidable Reasoning in Terminological Knowledge Representation Systems
Terminological knowledge representation systems (TKRSs) are tools for
designing and using knowledge bases that make use of terminological languages
(or concept languages). We analyze from a theoretical point of view a TKRS
whose capabilities go beyond the ones of presently available TKRSs. The new
features studied, often required in practical applications, can be summarized
in three main points. First, we consider a highly expressive terminological
language, called ALCNR, including general complements of concepts, number
restrictions and role conjunction. Second, we allow to express inclusion
statements between general concepts, and terminological cycles as a particular
case. Third, we prove the decidability of a number of desirable TKRS-deduction
services (like satisfiability, subsumption and instance checking) through a
sound, complete and terminating calculus for reasoning in ALCNR-knowledge
bases. Our calculus extends the general technique of constraint systems. As a
byproduct of the proof, we get also the result that inclusion statements in
ALCNR can be simulated by terminological cycles, if descriptive semantics is
adopted.Comment: See http://www.jair.org/ for any accompanying file
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