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

## Constructivism in Mathematics

**Author**: A.S. Troelstra,D. van Dalen**Publisher:**Elsevier**ISBN:**008095510X**Category:**Mathematics**Page:**129**View:**8191

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**View:**6824

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**View:**9615

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.

## The Foundations of Mathematics

**Author**: Kenneth Kunen**Publisher:**N.A**ISBN:**9781904987147**Category:**Mathematics**Page:**251**View:**6983

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.

## Foundations of Constructive Analysis

**Author**: Errett Bishop**Publisher:**Ishi Press**ISBN:**9784871877145**Category:**Mathematics**Page:**404**View:**9529

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.

## Philosophies of Mathematics

**Author**: Alexander L. George,Daniel Velleman**Publisher:**Wiley-Blackwell**ISBN:**9780631195443**Category:**Science**Page:**244**View:**7061

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.

## Practical Foundations of Mathematics

**Author**: Paul Taylor**Publisher:**Cambridge University Press**ISBN:**9780521631075**Category:**Mathematics**Page:**572**View:**8775

Practical Foundations collects the methods of construction of the objects of twentieth-century mathematics. Although it is mainly concerned with a framework essentially equivalent to intuitionistic Zermelo-Fraenkel logic, the book looks forward to more subtle bases in categorical type theory and the machine representation of mathematics. Each idea is illustrated by wide-ranging examples, and followed critically along its natural path, transcending disciplinary boundaries between universal algebra, type theory, category theory, set theory, sheaf theory, topology and programming. Students and teachers of computing, mathematics and philosophy will find this book both readable and of lasting value as a reference work.

## Varieties of Constructive Mathematics

**Author**: Douglas Bridges,Fred Richman**Publisher:**Cambridge University Press**ISBN:**9780521318020**Category:**Mathematics**Page:**149**View:**1338

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.

## 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:**9334

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

**Author**: N.A**Publisher:**N.A**ISBN:**N.A**Category:**Mathematics**Page:**N.A**View:**2675

Vols. for 1965- include a separately paged section, Bulletin bibliographique.

## Proof Theory and Automated Deduction

**Author**: Jean Goubault-Larrecq,Ian Mackie**Publisher:**Kluwer Academic Pub**ISBN:**9780792345930**Category:**Computers**Page:**424**View:**7773

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**View:**1149

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:**4536

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:**7377

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**View:**7172

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.

## Handbook of Educational Psychology

**Author**: David C. Berliner,Robert C. Calfee**Publisher:**Routledge**ISBN:**1136500316**Category:**Education**Page:**1082**View:**5393

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.

## Kant, Science, and Human Nature

**Author**: Robert Hanna**Publisher:**Clarendon Press**ISBN:**0191536539**Category:**Philosophy**Page:**504**View:**3172

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).

## Teaching Science for Understanding

*A Human Constructivist View*

**Author**: Joel J. Mintzes,James H. Wandersee,Joseph D. Novak**Publisher:**Academic Press**ISBN:**9780080879246**Category:**Psychology**Page:**384**View:**9655

Teaching Science for Understanding

## Computable Foundations for Economics

**Author**: K. Vela Velupillai**Publisher:**Routledge**ISBN:**1134253362**Category:**Business & Economics**Page:**512**View:**4083

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.