Search Results for "constructivism-in-mathematics-vol-1-an-introduction-studies-in-logic-and-the-foundations-of-mathematics"

Constructivism in Mathematics

Constructivism in Mathematics

  • Author: A.S. Troelstra,D. van Dalen
  • Publisher: Elsevier
  • ISBN: 008095510X
  • Category: Mathematics
  • Page: 129
  • View: 2312
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Studies in Logic and the Foundations of Mathematics, Volume 123: Constructivism in Mathematics: An Introduction, Vol. II focuses on various studies in mathematics and logic, including metric spaces, polynomial rings, and Heyting algebras. The publication first takes a look at the topology of metric spaces, algebra, and finite-type arithmetic and theories of operators. Discussions focus on intuitionistic finite-type arithmetic, theories of operators and classes, rings and modules, linear algebra, polynomial rings, fields and local rings, complete separable metric spaces, and located sets. The text then examines proof theory of intuitionistic logic, theory of types and constructive set theory, and choice sequences. The book elaborates on semantical completeness, sheaves, sites, and higher-order logic, and applications of sheaf models. Topics include a derived rule of local continuity, axiom of countable choice, forcing over sites, sheaf models for higher-order logic, and complete Heyting algebras. The publication is a valuable reference for mathematicians and researchers interested in mathematics and logic.

Berechenbarkeit Komplexität Logik

Berechenbarkeit Komplexität Logik

Algorithmen, Sprachen und Kalküle unter besonderer Berücksichtigung ihrer Komplexität

  • Author: Egon Börger
  • Publisher: Springer-Verlag
  • ISBN: 3322832279
  • Category: Technology & Engineering
  • Page: 499
  • View: 4023
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Endlich liegt der ,,Klassiker" der Theoretischen Informatik, der Studenten und Forschern ein unentbehrliches Standardwerk ist, in neuer Auflage vor.

Logic, Mathematics, Philosophy, Vintage Enthusiasms

Logic, Mathematics, Philosophy, Vintage Enthusiasms

Essays in Honour of John L. Bell

  • Author: David DeVidi,Michael Hallett,Peter Clark
  • Publisher: Springer Science & Business Media
  • ISBN: 9789400702141
  • Category: Philosophy
  • Page: 486
  • View: 7106
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The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic (William Lawvere, Peter Aczel, Graham Priest, Giovanni Sambin); analytical philosophy (Michael Dummett, William Demopoulos), philosophy of science (Michael Redhead, Frank Arntzenius), philosophy of mathematics (Michael Hallett, John Mayberry, Daniel Isaacson) and decision theory and foundations of economics (Ken Bimore). Most articles are contributions to current philosophical debates, but contributions also include some new mathematical results, important historical surveys, and a translation by Wilfrid Hodges of a key work of arabic logic.

L'Enseignement mathématique

L'Enseignement mathématique

  • Author: N.A
  • Publisher: N.A
  • ISBN: N.A
  • Category: Mathematics
  • Page: N.A
  • View: 5905
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Vols. for 1965- include a separately paged section, Bulletin bibliographique.

Proof Theory and Automated Deduction

Proof Theory and Automated Deduction

  • Author: Jean Goubault-Larrecq,Ian Mackie
  • Publisher: Kluwer Academic Pub
  • ISBN: 9780792345930
  • Category: Computers
  • Page: 424
  • View: 2400
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Proof Theory and Automated Deduction is written for final-year undergraduate and first-year post-graduate students. It should also serve as a valuable reference for researchers in logic and computer science. It covers basic notions in logic, with a particular stress on proof theory, as opposed to, for example, model theory or set theory; and shows how they are applied in computer science, and especially the particular field of automated deduction, i.e. the automated search for proofs of mathematical propositions. We have chosen to give an in-depth analysis of the basic notions, instead of giving a mere sufficient analysis of basic and less basic notions. We often derive the same theorem by different methods, showing how different mathematical tools can be used to get at the very nature of the objects at hand, and how these tools relate to each other. Instead of presenting a linear collection of results, we have tried to show that all results and methods are tightly interwoven. We believe that understanding how to travel along this web of relations between concepts is more important than just learning the basic theorems and techniques by rote. Audience: The book is a valuable reference for researchers in logic and computer science.

Internal Logic

Internal Logic

Foundations of Mathematics from Kronecker to Hilbert

  • Author: Y. Gauthier
  • Publisher: Springer Science & Business Media
  • ISBN: 9401700834
  • Category: Mathematics
  • Page: 251
  • View: 4405
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Internal logic is the logic of content. The content is here arithmetic and the emphasis is on a constructive logic of arithmetic (arithmetical logic). Kronecker's general arithmetic of forms (polynomials) together with Fermat's infinite descent is put to use in an internal consistency proof. The view is developed in the context of a radical arithmetization of mathematics and logic and covers the many-faceted heritage of Kronecker's work, which includes not only Hilbert, but also Frege, Cantor, Dedekind, Husserl and Brouwer. The book will be of primary interest to logicians, philosophers and mathematicians interested in the foundations of mathematics and the philosophical implications of constructivist mathematics. It may also be of interest to historians, since it covers a fifty-year period, from 1880 to 1930, which has been crucial in the foundational debates and their repercussions on the contemporary scene.

The Philosophy of Mathematics Education

The Philosophy of Mathematics Education

  • Author: Paul Ernest
  • Publisher: Routledge
  • ISBN: 1135387540
  • Category: Education
  • Page: 346
  • View: 6603
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Although many agree that all teaching rests on a theory of knowledge, there has been no in-depth exploration of the implications of the philosophy of mathematics for education. This is Paul Ernest's aim. Building on the work of Lakatos and Wittgenstein it challenges the prevalent notion that mathematical knowledge is certain, absolute and neutral, and offers instead an account of mathematics as a social construction. This has profound educational implications for social issues, including gender, race and multiculturalism; for pedagogy, including investigations and problem solving; and challenges hierarchical views of mathematics, learning and ability. Beyond this, the book offers a well-grounded model of five educational ideologies, each with its own epistemology, values, aims and social group of adherents. An analysis of the impact of these groups on the National Curriculum results in a powerful critique, revealing the questionable assumptions, values and interests upon which it rests. The book finishes on an optimistic note, arguing that pedagogy, left unspecified by the National Curriculum, is the way to achieve the radical aims of educating confident problem posers and solvers who are able to critically evaluate the social uses of mathematics.

In Defense of Intuitions

In Defense of Intuitions

A New Rationalist Manifesto

  • Author: A. Chapman,A. Ellis,R. Hanna,T. Hildebrand,H. Pickford
  • Publisher: Springer
  • ISBN: 1137347953
  • Category: Philosophy
  • Page: 427
  • View: 3388
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A reply to contemporary skepticism about intuitions and a priori knowledge, and a defense of neo-rationalism from a contemporary Kantian standpoint, focusing on the theory of rational intuitions and on solving the two core problems of justifying and explaining them.

The Infinite in Mathematics

The Infinite in Mathematics

Logico-mathematical writings

  • Author: Felix Kaufmann
  • Publisher: Springer Science & Business Media
  • ISBN: 9789027708472
  • Category: Science
  • Page: 237
  • View: 3079
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The main item in the present volume was published in 1930 under the title Das Unendliche in der Mathematik und seine Ausschaltung. It was at that time the fullest systematic account from the standpoint of Husserl's phenomenology of what is known as 'finitism' (also as 'intuitionism' and 'constructivism') in mathematics. Since then, important changes have been required in philosophies of mathematics, in part because of Kurt Godel's epoch-making paper of 1931 which established the essential in completeness of arithmetic. In the light of that finding, a number of the claims made in the book (and in the accompanying articles) are demon strably mistaken. Nevertheless, as a whole it retains much of its original interest and value. It presents the issues in the foundations of mathematics that were under debate when it was written (and in some cases still are); , and it offers one alternative to the currently dominant set-theoretical definitions of the cardinal numbers and other arithmetical concepts. While still a student at the University of Vienna, Felix Kaufmann was greatly impressed by the early philosophical writings (especially by the Logische Untersuchungen) of Edmund Husser!' He was never an uncritical disciple of Husserl, and he integrated into his mature philosophy ideas from a wide assortment of intellectual sources. But he thought of himself as a phenomenologist, and made frequent use in all his major publications of many of Husserl's logical and epistemological theses.

Computable Foundations for Economics

Computable Foundations for Economics

  • Author: K. Vela Velupillai
  • Publisher: Routledge
  • ISBN: 1134253362
  • Category: Business & Economics
  • Page: 512
  • View: 5750
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Computable Foundations for Economics is a unified collection of essays, some of which are published here for the first time and all of which have been updated for this book, on an approach to economic theory from the point of view of algorithmic mathematics. By algorithmic mathematics the author means computability theory and constructive mathematics. This is in contrast to orthodox mathematical economics and game theory, which are formalised with the mathematics of real analysis, underpinned by what is called the ZFC formalism, i.e., set theory with the axiom of choice. This reliance on ordinary real analysis and the ZFC system makes economic theory in its current mathematical mode completely non-algorithmic, which means it is numerically meaningless. The book provides a systematic attempt to dissect and expose the non-algorithmic content of orthodox mathematical economics and game theory and suggests a reformalization on the basis of a strictly rigorous algorithmic mathematics. This removes the current schizophrenia in mathematical economics and game theory, where theory is entirely divorced from algorithmic applicability – for experimental and computational exercises. The chapters demonstrate the uncomputability and non-constructivity of core areas of general equilibrium theory, game theory and recursive macroeconomics. The book also provides a fresh look at the kind of behavioural economics that lies behind Herbert Simon’s work, and resurrects a role for the noble classical traditions of induction and verification, viewed and formalised, now, algorithmically. It will therefore be of particular interest to postgraduate students and researchers in algorithmic economics, game theory and classical behavioural economics.

Kurt Gödel

Kurt Gödel

Wahrheit & Beweisbarkeit

  • Author: Kurt Gödel,Eckehart Köhler,Bernd Buldt
  • Publisher: N.A
  • ISBN: 9783209038340
  • Category: Logicians
  • Page: 448
  • View: 5868
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Mathematical Reviews

Mathematical Reviews

  • Author: N.A
  • Publisher: N.A
  • ISBN: N.A
  • Category: Mathematics
  • Page: N.A
  • View: 7401
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Foundations of Constructive Mathematics

Foundations of Constructive Mathematics

Metamathematical Studies

  • Author: M.J. Beeson
  • Publisher: Springer Science & Business Media
  • ISBN: 3642689523
  • Category: Mathematics
  • Page: 466
  • View: 3330
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This book is about some recent work in a subject usually considered part of "logic" and the" foundations of mathematics", but also having close connec tions with philosophy and computer science. Namely, the creation and study of "formal systems for constructive mathematics". The general organization of the book is described in the" User's Manual" which follows this introduction, and the contents of the book are described in more detail in the introductions to Part One, Part Two, Part Three, and Part Four. This introduction has a different purpose; it is intended to provide the reader with a general view of the subject. This requires, to begin with, an elucidation of both the concepts mentioned in the phrase, "formal systems for constructive mathematics". "Con structive mathematics" refers to mathematics in which, when you prove that l a thing exists (having certain desired properties) you show how to find it. Proof by contradiction is the most common way of proving something exists without showing how to find it - one assumes that nothing exists with the desired properties, and derives a contradiction. It was only in the last two decades of the nineteenth century that mathematicians began to exploit this method of proof in ways that nobody had previously done; that was partly made possible by the creation and development of set theory by Georg Cantor and Richard Dedekind.

From Logic to Practice

From Logic to Practice

Italian Studies in the Philosophy of Mathematics

  • Author: Gabriele Lolli,Marco Panza,Giorgio Venturi
  • Publisher: Springer
  • ISBN: 3319104349
  • Category: Philosophy
  • Page: 336
  • View: 8245
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This book brings together young researchers from a variety of fields within mathematics, philosophy and logic. It discusses questions that arise in their work, as well as themes and reactions that appear to be similar in different contexts. The book shows that a fairly intensive activity in the philosophy of mathematics is underway, due on the one hand to the disillusionment with respect to traditional answers, on the other to exciting new features of present day mathematics. The book explains how the problem of applicability once again plays a central role in the development of mathematics. It examines how new languages different from the logical ones (mostly figural), are recognized as valid and experimented with and how unifying concepts (structure, category, set) are in competition for those who look at this form of unification. It further shows that traditional philosophies, such as constructivism, while still lively, are no longer only philosophies, but guidelines for research. Finally, the book demonstrates that the search for and validation of new axioms is analyzed with a blend of mathematical historical, philosophical, psychological considerations.

Die Grundlagen der Mathematik

Die Grundlagen der Mathematik

  • Author: David Hilbert
  • Publisher: Springer-Verlag
  • ISBN: 3663161021
  • Category: Mathematics
  • Page: 29
  • View: 7238
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Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

Constructive Foundations for Functional Languages

Constructive Foundations for Functional Languages

  • Author: Raymond Turner
  • Publisher: McGraw-Hill Book Company Limited
  • ISBN: N.A
  • Category: Computers
  • Page: 269
  • View: 8177
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For the Learning of Mathematics

For the Learning of Mathematics

An International Journal of Mathematics Education

  • Author: N.A
  • Publisher: N.A
  • ISBN: N.A
  • Category: Mathematics
  • Page: N.A
  • View: 9469
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The Principles of Genetic Epistemology

The Principles of Genetic Epistemology

  • Author: Jean Piaget
  • Publisher: Psychology Press
  • ISBN: 9780415168908
  • Category: Reference
  • Page: 98
  • View: 6106
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First published in 1997. Routledge is an imprint of Taylor & Francis, an informa company.

Encyclopedia of Philosophy: Kabbalah - Marxist philosophy

Encyclopedia of Philosophy: Kabbalah - Marxist philosophy

  • Author: Donald M. Borchert
  • Publisher: Thomson Gale/MacMillan Reference USA
  • ISBN: 9780028657851
  • Category: Philosophy
  • Page: 10
  • View: 8110
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Containing material from hundreds of highly distinguished contributors representing the world's top universities and institutions, the second edition has a truly global perspective. It contains more than 2,100 entries -- including more than 450 new articles. Among the many topics covered are African, Islamic, Jewish, Russian, Chinese, and Buddhist philosophies; bioethics and biomedical ethics; art and aesthetics; epistemology; metaphysics; peace and war; social and political philosophy; the Holocaust; feminist thought; and much more. Additionally, the second edition also features 1,000 biographical entries on major figures in philosophical thought throughout history.