Constructivism in Mathematics

Author: A.S. Troelstra,D. van Dalen

Publisher: Elsevier

ISBN: 008095510X

Category: Mathematics

Page: 129

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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.

Constructivism in Mathematics

Author: A.S. Troelstra,D. van Dalen

Publisher: Elsevier

ISBN: 0080570887

Category: Mathematics

Page: 355

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These two volumes cover the principal approaches to constructivism in mathematics. They present a thorough, up-to-date introduction to the metamathematics of constructive mathematics, paying special attention to Intuitionism, Markov's constructivism and Martin-Lof's type theory with its operational semantics. A detailed exposition of the basic features of constructive mathematics, with illustrations from analysis, algebra and topology, is provided, with due attention to the metamathematical aspects. Volume 1 is a self-contained introduction to the practice and foundations of constructivism, and does not require specialized knowledge beyond basic mathematical logic. Volume 2 contains mainly advanced topics of a proof-theoretical and semantical nature.

Set Theory and its Philosophy

A Critical Introduction

Author: Michael Potter

Publisher: Clarendon Press

ISBN: 0191556432

Category: Philosophy

Page: 360

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Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science.

Philosophies of Mathematics

Author: Alexander L. George,Daniel Velleman

Publisher: Wiley-Blackwell

ISBN: 9780631195443

Category: Science

Page: 244

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This book provides an accessible, critical introduction to the three main approaches that dominated work in the philosophy of mathematics during the twentieth century: logicism, intuitionism and formalism.

Foundations of Constructive Analysis

Author: Errett Bishop

Publisher: Ishi Press

ISBN: 9784871877145

Category: Mathematics

Page: 404

View: 7216

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This book, Foundations of Constructive Analysis, founded the field of constructive analysis because it proved most of the important theorems in real analysis by constructive methods. The author, Errett Albert Bishop, born July 10, 1928, was an American mathematician known for his work on analysis. In the later part of his life Bishop was seen as the leading mathematician in the area of Constructive mathematics. From 1965 until his death, he was professor at the University of California at San Diego.

Varieties of Constructive Mathematics

Author: Douglas Bridges,Fred Richman

Publisher: Cambridge University Press

ISBN: 9780521318020

Category: Mathematics

Page: 149

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A survey of constructive approaches to pure mathematics emphasizing the viewpoint of Errett Bishop's school. Considers intuitionism, Russian constructivism, and recursive analysis, with comparisons among the various approaches included where appropriate.

The Foundations of Mathematics

Author: Kenneth Kunen

Publisher: N.A

ISBN: 9781904987147

Category: Mathematics

Page: 251

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Mathematical logic grew out of philosophical questions regarding the foundations of mathematics, but logic has now outgrown its philosophical roots, and has become an integral part of mathematics in general. This book is designed for students who plan to specialize in logic, as well as for those who are interested in the applications of logic to other areas of mathematics. Used as a text, it could form the basis of a beginning graduate-level course. There are three main chapters: Set Theory, Model Theory, and Recursion Theory. The Set Theory chapter describes the set-theoretic foundations of all of mathematics, based on the ZFC axioms. It also covers technical results about the Axiom of Choice, well-orderings, and the theory of uncountable cardinals. The Model Theory chapter discusses predicate logic and formal proofs, and covers the Completeness, Compactness, and Lowenheim-Skolem Theorems, elementary submodels, model completeness, and applications to algebra. This chapter also continues the foundational issues begun in the set theory chapter. Mathematics can now be viewed as formal proofs from ZFC. Also, model theory leads to models of set theory. This includes a discussion of absoluteness, and an analysis of models such as H( ) and R( ). The Recursion Theory chapter develops some basic facts about computable functions, and uses them to prove a number of results of foundational importance; in particular, Church's theorem on the undecidability of logical consequence, the incompleteness theorems of Godel, and Tarski's theorem on the non-definability of truth.

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

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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.

Proof Theory and Automated Deduction

Author: Jean Goubault-Larrecq,Ian Mackie

Publisher: Kluwer Academic Pub

ISBN: 9780792345930

Category: Computers

Page: 424

View: 7841

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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

Foundations of Mathematics from Kronecker to Hilbert

Author: Y. Gauthier

Publisher: Springer Science & Business Media

ISBN: 9401700834

Category: Mathematics

Page: 251

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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.

Computability

Computable Functions, Logic, and the Foundations of Mathematics

Author: Richard L. Epstein,Walter Alexandre Carnielli

Publisher: N.A

ISBN: 9780981550725

Category: Mathematics

Page: 370

View: 3631

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This classic presentation of the theory of computable functions includes discussions and readings about the crisis in the foundations of mathematics in the early 20th century, while presenting the basic ideas of whole number, function, proof, and real number.

The Philosophy of Mathematics Education

Author: Paul Ernest

Publisher: Routledge

ISBN: 1135387540

Category: Education

Page: 346

View: 5978

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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

A New Rationalist Manifesto

Author: A. Chapman,A. Ellis,R. Hanna,T. Hildebrand,H. Pickford

Publisher: Springer

ISBN: 1137347953

Category: Philosophy

Page: 427

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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.

Foundations of Constructive Mathematics

Metamathematical Studies

Author: M.J. Beeson

Publisher: Springer Science & Business Media

ISBN: 3642689523

Category: Mathematics

Page: 466

View: 1024

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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.

Handbook of Educational Psychology

Author: David C. Berliner,Robert C. Calfee

Publisher: Routledge

ISBN: 1136500316

Category: Education

Page: 1082

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Sponsored by Division 15 of APA, the second edition of this groundbreaking book has been expanded to 41 chapters that provide unparalleled coverage of this far-ranging field. Internationally recognized scholars contribute up-to-date reviews and critical syntheses of the following areas: foundations and the future of educational psychology, learners’ development, individual differences, cognition, motivation, content area teaching, socio-cultural perspectives on teaching and learning, teachers and teaching, instructional design, teacher assessment, and modern perspectives on research methodologies, data, and data analysis. New chapters cover topics such as adult development, self-regulation, changes in knowledge and beliefs, and writing. Expanded treatment has been given to cognition, motivation, and new methodologies for gathering and analyzing data. The Handbook of Educational Psychology, Second Edition provides an indispensable reference volume for scholars, teacher educators, in-service practitioners, policy makers and the academic libraries serving these audiences. It is also appropriate for graduate level courses devoted to the study of educational psychology.

Philosophy of Logic

Author: N.A

Publisher: Elsevier

ISBN: 9780080466637

Category: Mathematics

Page: 1218

View: 8939

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The papers presented in this volume examine topics of central interest in contemporary philosophy of logic. They include reflections on the nature of logic and its relevance for philosophy today, and explore in depth developments in informal logic and the relation of informal to symbolic logic, mathematical metatheory and the limiting metatheorems, modal logic, many-valued logic, relevance and paraconsistent logic, free logics, extensional v. intensional logics, the logic of fiction, epistemic logic, formal logical and semantic paradoxes, the concept of truth, the formal theory of entailment, objectual and substitutional interpretation of the quantifiers, infinity and domain constraints, the Löwenheim-Skolem theorem and Skolem paradox, vagueness, modal realism v. actualism, counterfactuals and the logic of causation, applications of logic and mathematics to the physical sciences, logically possible worlds and counterpart semantics, and the legacy of Hilbert’s program and logicism. The handbook is meant to be both a compendium of new work in symbolic logic and an authoritative resource for students and researchers, a book to be consulted for specific information about recent developments in logic and to be read with pleasure for its technical acumen and philosophical insights. - Written by leading logicians and philosophers - Comprehensive authoritative coverage of all major areas of contemporary research in symbolic logic - Clear, in-depth expositions of technical detail - Progressive organization from general considerations to informal to symbolic logic to nonclassical logics - Presents current work in symbolic logic within a unified framework - Accessible to students, engaging for experts and professionals - Insightful philosophical discussions of all aspects of logic - Useful bibliographies in every chapter

Kant, Science, and Human Nature

Author: Robert Hanna

Publisher: Clarendon Press

ISBN: 0191536539

Category: Philosophy

Page: 504

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Robert Hanna argues for the importance of Kant's theories of the epistemological, metaphysical, and practical foundations of the 'exact sciences'—- relegated to the dustbin of the history of philosophy for most of the 20th century. Hanna's earlier book Kant and the Foundations of Analytic Philosophy (OUP 2001), explores basic conceptual and historical connections between Immanuel Kant's 18th-century Critical Philosophy and the tradition of mainstream analytic philosophy from Frege to Quine. The central topics of the analytic tradition in its early and middle periods were meaning and necessity. But the central theme of mainstream analytic philosophy after 1950 is scientific naturalism, which holds—-to use Wilfrid Sellars's apt phrase—-that 'science is the measure of all things'. This type of naturalism is explicitly reductive. Kant, Science, and Human Nature has two aims, one negative and one positive. Its negative aim is to develop a Kantian critique of scientific naturalism. But its positive and more fundamental aim is to work out the elements of a humane, realistic, and nonreductive Kantian account of the foundations of the exact sciences. According to this account, the essential properties of the natural world are directly knowable through human sense perception (empirical realism), and practical reason is both explanatorily and ontologically prior to theoretical reason (the primacy of the practical).

The Infinite in Mathematics

Logico-mathematical writings

Author: Felix Kaufmann

Publisher: Springer Science & Business Media

ISBN: 9789027708472

Category: Science

Page: 237

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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.