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William A Treadwell - One of the best experts on this subject based on the ideXlab platform.

  • fuzzy set theory movement in the social sciences
    Public Administration Review, 1995
    Co-Authors: William A Treadwell
    Abstract:

    Some very familiar constructs in the realm of public administration discourse are supervisor-worker, teacherstudent, public-private, democracy-autocracy, conservatism-liberalism, patronage-merit, individualism-collectivism, and centralized-decentralized. These and other constructs and theories in public management are bantered about through iterations of paired comparisons of contrasting relationships that are framed in a language of dichotomous symbolism. Even when a construct does not appear to lend itself readily to a bi-polar structure, it will implicitly exist, as in the familiar constructs of representation, efficiency, effectiveness, equity, and property; in these examples the negative of the terms form the bipolar comparison point - no representation, no efficiency, effectiveness, no equity, no property. Our temporal process is uniquely bi-polar (Kelly 1955; Adams-Webber 1979; Rychlak 1981), where our constructs tend to be labeled by their similarity pole. When we speak of what something means, we are always referring to a relationship between similarities and differences. Kelly's personal construct theory is based on the notion of a construing process where thought is only possible because humans dichotomize experience into similarities and contrasts, as our bi-polar nature. Personal construct theory can be extended into an ontological case for fuzziness as presented by Kosko (1989) wherein: The universe consists of all subsets of the universe. The only subsets of the universe that are not fuzzy are the constructs of Classical Mathematics. All other sets - sets of particles, cells, tissues, people, ideas, galaxies - in principle contain elements to different degrees. Their membership is partial, graded, inexact, ambiguous, or uncertain. To the extent that the human construes reality as sets of bi-polar constructs, the strength of any construct would then be based on its membership gradient value at the time an event occurs. The membership gradient of any construct is a matter of perceived similarities and contrasts. At this point, it may be best to stop and back up, because we have the cart before the horse. Presenting an historical introduction to logic and fuzzy theory will better orient our frame of reference. Currently in the United States, people who are over 30 years of age were traditionally raised using a nonmetric system of weights and measures - using terms such as feet, ounces, and quarts. The younger generations have been raised using the metric system of weights and measures of meters, grams, and liters. The transition of the older generation to using the metric system has not been very successful. The old ways are still thoroughly embedded in society. There is a split between the conflicting institutionalized old way and the new. The same may hold true in the coming years between the use of Classical set theory and the rise in the use of fuzzy set theory. Classical set theory has a long documented history starting with Aristotle; fuzzy theory is a mere child of the 1960s. The dialogue between the human sciences and fuzzy set theory has been scattered, unsystematic, and slow to develop. According to Smithson (1987) "... fuzzy set Mathematics are couched in foreign and rather obtuse notation which is forbidding even to the more mathematically sophisticated behavioral scientist." Virtually all texts on the topic assume either a mathematical or computer science and engineering orientation. Smithson's book was an attempt to bridge the gap and lighten the burden of the Mathematics to illustrate the basic elements of fuzzy set theory in real-world research examples taken from cognitive psychology, social psychology, sociology, social anthropology, political science, and evaluation research. In the same spirit of Smithson, the goal of this expository article on fuzzy theory is to minimize the obtrusion of Mathematics and make the topic more palatable to a wider audience. This initial paper sets out to provide some structure for handling fuzzy concepts and illustrate their use in budgeting and decision making. …

Vladimir Trifonov - One of the best experts on this subject based on the ideXlab platform.

  • a linear solution of the four dimensionality problem
    EPL, 1995
    Co-Authors: Vladimir Trifonov
    Abstract:

    Modelling the measurement ("active observation") process makes it possible to express in strict terms the degree to which the logic of the observer determines what he "sees", and formalize the difficult concept of rational behaviour. Presented here is a rigorous formulation of several implicit assumptions of standard physics which leads to a first-order theory shown to possess a real-world model: if an observer's logic is Boolean, he is bound to perceive his spacetime as a four-dimensional pseudo-Riemannian manifold of signature 2, with an ideal big bang geometry. The connections between the type of an observer's logic and large-scale structure of the observable universe yield a testable prediction, existence of positive cosmological constant and suggest a non-standard integration-over-spacetime technique. They strongly favour non-local reality and deliver an operational explanation of the number of particle generations. The result casts some doubts (arising also from the necessity of renormalization procedures) that Classical Mathematics (i.e. the Mathematics of the topos of sets) is the "natural" Mathematics of our world, and offers a new candidate for this role, that differs from its Classical counterpart. In general, the scheme outlines a formal way to unify the logical, physical and, possibly, psychological templates of perception, which can be briefly expressed as "physics is an exponent-image of psychology".

Brian Rotman - One of the best experts on this subject based on the ideXlab platform.

  • Will the digital computer transform Classical Mathematics
    Philosophical transactions. Series A Mathematical physical and engineering sciences, 2003
    Co-Authors: Brian Rotman
    Abstract:

    Mathematics and machines have influenced each other for millennia. The advent of the digital computer introduced a powerfully new element that promises to transform the relation between them. This paper outlines the thesis that the effect of the digital computer on Mathematics, already widespread, is likely to be radical and far–reaching. To articulate this claim, an abstract model of doing Mathematics is introduced based on a triad of actors of which one, the ‘agent’, corresponds to the function performed by the computer. The model is used to frame two sorts of transformation. The first is pragmatic and involves the alterations and progressive colonization of the content and methods of enquiry of various mathematical fields brought about by digital methods. The second is conceptual and concerns a fundamental antagonism between the infinity enshrined in Classical Mathematics and physics (continuity, real numbers, asymptotic definitions) and the inherently real and material limit of processes associated with digital computation. An example which lies in the intersection of Classical Mathematics and computer science, the P = NP problem, is analysed in the light of this latter issue.

Zhaohui Luo - One of the best experts on this subject based on the ideXlab platform.

  • weyl s predicative Classical Mathematics as a logic enriched type theory
    arXiv: Logic in Computer Science, 2008
    Co-Authors: Robin Adams, Zhaohui Luo
    Abstract:

    We construct a logic-enriched type theory LTTW that corresponds closely to the predicative system of foundations presented by Hermann Weyl in Das Kontinuum. We formalise many results from that book in LTTW, including Weyl's definition of the cardinality of a set and several results from real analysis, using the proof assistant Plastic that implements the logical framework LF. This case study shows how type theory can be used to represent a non-constructive foundation for Mathematics.

  • weyl s predicative Classical Mathematics as a logic enriched type theory
    Types for Proofs and Programs, 2006
    Co-Authors: Robin Adams, Zhaohui Luo
    Abstract:

    In Das Kontinuum, Weyl showed how a large body of Classical Mathematics could be developed on a purely predicative foundation. We present a logic-enriched type theory that corresponds to Weyl's foundational system. A large part of the Mathematics in Weyl's book -- including Weyl's definition of the cardinality of a set and several results from real analysis -- has been formalised, using the proof assistant Plastic that implements a logical framework. This case study shows how type theory can be used to represent a non-constructive foundation for Mathematics.

Jean Alexandre Dieudonne - One of the best experts on this subject based on the ideXlab platform.

  • some problems of Classical Mathematics
    1992
    Co-Authors: Jean Alexandre Dieudonne
    Abstract:

    In Chapter V we shall attempt to show how, starting from Classical objects and methods, mathematicians of the nineteenth century were obliged to invent new mathematical objects and new methods in order to make further progress.

  • objects and methods in Classical Mathematics
    1992
    Co-Authors: Jean Alexandre Dieudonne
    Abstract:

    While all ancient civilisations, in order to satisfy the needs of daily life, had to develop procedures of arithmetical calculation and spatial measurement, only the Greeks, from the sixth century B.C., thought of analysing the chain of reasoning behind these procedures, and thus created an entirely new mode of thinking. In this chapter we shall try to make plain the essential features of Greek Mathematics and the unsuspected developments, so extraordinarily fruitful, which mathematicians brought to them between the Renaissance and the end of the eighteenth century.