1,328 research outputs found

    A Primer on the Tools and Concepts of Computable Economics

    Get PDF
    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

    Decision problems for 3-manifolds and their fundamental groups

    Full text link
    We survey the status of some decision problems for 3-manifolds and their fundamental groups. This includes the classical decision problems for finitely presented groups (Word Problem, Conjugacy Problem, Isomorphism Problem), and also the Homeomorphism Problem for 3-manifolds and the Membership Problem for 3-manifold groups.Comment: 31 pages, final versio

    Understanding the Role and Importance of Design Problems in Creativity Research

    Get PDF
    The overall objective of this research is to address the need for using similar conceptual design problems in experiments in engineering design creativity. This is accomplished by addressing three sub-objectives i) to identify the pattern of design problem usage, ii) to enable comparison between two conceptual design problems based on their natural language representations and iii) to analyze the impact of design problems on effectiveness of example interventions used in user studies in engineering design creativity. Design problems are an essential component of experiments in creativity research.The requirements of experiment’s design sometimes limit problem sharing between researchers or studies conducted by them. For understanding and identifying the design problem usage pattern, two network representations of design problems, connected to each other by authors and papers using them has been used. Both networks indicate that several problems have been used for creativity experiments and suggest the need for using same or ‘similar’ design problems to reduce between-study differences in design problem usage.This addresses the first objective of identifying pattern of design problem usage in creativity research. Problem similarity is assessed using two methods. The first method is based on identification of five structural elements of a design problem namely goals of a problem, functional requirements, non – functional requirements, reference to an existing product and information about end user. The protocol for identifying these elements in problem statement and then comparing design problems is illustrated through two examples. The second method for similarity assessment is based on Latent Semantic Analysis (LSA) of problem statements. LSA provides an objective method to compare semantic similarity of problem statements. Both methods help address the research objective of comparing problems based on their representation but fail to evaluate problem solvability. For understanding whether design problems influence the effectiveness of examples used as interventions, a meta-regression model between effect size and problem size has been used. Regression models suggest that problem size might have a linear relationship with effectiveness of examples for quantity of ideas produced by treatment group participants but enough evidence did not exist to suggest similar relationship for metrics quality and novelty. This addresses the sub-objective of design problems affecting the effectiveness of methods tested in experiments and overall objective of the need for using similar problems in creativity research

    From Solvability to Formal Decidability: Revisiting Hilbert’s “Non-Ignorabimus”

    Get PDF
    The topic of this article is Hilbert’s axiom of solvability, that is, his conviction of the solvability of every mathematical problem by means of a finite number of operations. The question of solvability is commonly identified with the decision problem. Given this identification, there is not the slightest doubt that Hilbert’s conviction was falsified by Gödel’s proof and by the negative results for the decision problem. On the other hand, Gödel’s theorems do offer a solution, albeit a negative one, in the form of an impossibility proof. In this sense, Hilbert’s optimism may still be justified. Here I argue that Gödel’s theorems opened the door to proof theory and to the remarkably successful development of generalized as well as relativized realizations of Hilbert’s program. Thus, the fall of absolute certainty came hand in hand with the rise of partially secure and reliable foundations of mathematical knowledge. Not all was lost and much was gained

    Computation in Economics

    Get PDF
    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
    • 

    corecore