Foundations of Constructive Analysis

Author: Errett Bishop
Publisher: Ishi Press
ISBN: 9784871877145
Category: Mathematics
Page: 404
View: 1843
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.

Foundations of Constructive Mathematics

Metamathematical Studies
Author: M.J. Beeson
Publisher: Springer Science & Business Media
ISBN: 3642689523
Category: Mathematics
Page: 466
View: 3699
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.

Foundations of Discrete Mathematics

Author: K. D. Joshi
Publisher: New Age International
ISBN: 9788122401202
Category: Combinatorial analysis
Page: 748
View: 6943
This Book Is Meant To Be More Than Just A Text In Discrete Mathematics. It Is A Forerunner Of Another Book Applied Discrete Structures By The Same Author. The Ultimate Goal Of The Two Books Are To Make A Strong Case For The Inclusion Of Discrete Mathematics In The Undergraduate Curricula Of Mathematics By Creating A Sequence Of Courses In Discrete Mathematics Parallel To The Traditional Sequence Of Calculus-Based Courses.The Present Book Covers The Foundations Of Discrete Mathematics In Seven Chapters. It Lays A Heavy Emphasis On Motivation And Attempts Clarity Without Sacrificing Rigour. A List Of Typical Problems Is Given In The First Chapter. These Problems Are Used Throughout The Book To Motivate Various Concepts. A Review Of Logic Is Included To Gear The Reader Into A Proper Frame Of Mind. The Basic Counting Techniques Are Covered In Chapters 2 And 7. Those In Chapter 2 Are Elementary. But They Are Intentionally Covered In A Formal Manner So As To Acquaint The Reader With The Traditional Definition-Theorem-Proof Pattern Of Mathematics. Chapters 3 Introduces Abstraction And Shows How The Focal Point Of Todays Mathematics Is Not Numbers But Sets Carrying Suitable Structures. Chapter 4 Deals With Boolean Algebras And Their Applications. Chapters 5 And 6 Deal With More Traditional Topics In Algebra, Viz., Groups, Rings, Fields, Vector Spaces And Matrices.The Presentation Is Elementary And Presupposes No Mathematical Maturity On The Part Of The Reader. Instead, Comments Are Inserted Liberally To Increase His Maturity. Each Chapter Has Four Sections. Each Section Is Followed By Exercises (Of Various Degrees Of Difficulty) And By Notes And Guide To Literature. Answers To The Exercises Are Provided At The End Of The Book.

Techniques of Constructive Analysis

Author: Douglas S. Bridges,Luminita Simona Vita
Publisher: Springer Science & Business Media
ISBN: 0387381473
Category: Mathematics
Page: 215
View: 4178
This book is an introduction to constructive mathematics with an emphasis on techniques and results obtained in the last twenty years. The text covers fundamental theory of the real line and metric spaces, focusing on locatedness in normed spaces and with associated results about operators and their adjoints on a Hilbert space. The first appendix gathers together some basic notions about sets and orders, the second gives the axioms for intuitionistic logic. No background in intuitionistic logic or constructive analysis is needed in order to read the book, but some familiarity with the classical theories of metric, normed and Hilbert spaces is necessary.

Handbook of Analysis and Its Foundations

Author: Eric Schechter
Publisher: Academic Press
ISBN: 9780080532998
Category: Mathematics
Page: 883
View: 1634
Handbook of Analysis and Its Foundations is a self-contained and unified handbook on mathematical analysis and its foundations. Intended as a self-study guide for advanced undergraduates and beginning graduatestudents in mathematics and a reference for more advanced mathematicians, this highly readable book provides broader coverage than competing texts in the area. Handbook of Analysis and Its Foundations provides an introduction to a wide range of topics, including: algebra; topology; normed spaces; integration theory; topological vector spaces; and differential equations. The author effectively demonstrates the relationships between these topics and includes a few chapters on set theory and logic to explain the lack of examples for classical pathological objects whose existence proofs are not constructive. More complete than any other book on the subject, students will find this to be an invaluable handbook. Covers some hard-to-find results including: Bessagas and Meyers converses of the Contraction Fixed Point Theorem Redefinition of subnets by Aarnes and Andenaes Ghermans characterization of topological convergences Neumanns nonlinear Closed Graph Theorem van Maarens geometry-free version of Sperners Lemma Includes a few advanced topics in functional analysis Features all areas of the foundations of analysis except geometry Combines material usually found in many different sources, making this unified treatment more convenient for the user Has its own webpage:

Enzyklopädie Philosophie und Wissenschaftstheorie

Bd. 4: Ins–Loc
Author: Jürgen Mittelstraß
Publisher: Springer-Verlag
ISBN: 3476001369
Category: Philosophy
Page: 595
View: 1569
Das ganze Wissen der Philosophie und Wissenschaftstheorie. Lückenlos belegt das größte allgemeine Lexikon zur Philosophie in deutscher Sprache den heutigen Kenntnisstand. Erweitert auf acht Bände dokumentiert die 2. Auflage insbesondere die jüngsten Entwicklungen in Logik, Erkenntnis- und Wissenschaftstheorie sowie Sprachphilosophie. Jetzt liegt der vierte Band in Neuauflage vor mit über 100 zusätzlichen Einträgen, u. a. zu Intelligenz, Interdisziplinarität, Isotropie, Kognitionswissenschaft, Komplexitätstheorie, Konvention, Lebenswissenschaften und einer Vielzahl neuer Personenartikel.

Vom Kontinuum zum Integral

Eine Einführung in die intuitionistische Mathematik
Author: Rudolf Taschner
Publisher: Springer-Verlag
ISBN: 365823380X
Category: Mathematics
Page: 216
View: 921
Konstruktive Analysis wird in diesem Buch mit anschaulichen Graphiken und bestechenden Beispielen so vorgestellt, dass sie bereits mit elementaren Schulkenntnissen als Voraussetzung verstanden wird. Sie stellt eine höchst attraktive Alternative zur konventionellen, auf den willkürlich gesetzten Axiomen der Mengentheorie fußenden formalen Mathematik dar. Und sie führt zu spektakulären Einsichten über Stetigkeit und gleichmäßige Stetigkeit, über gleichmäßige Konvergenz und über die Vertauschung von Limes und Integral, die der konventionellen Mathematik gänzlich verwehrt sind.

Principia Mathematica.

Author: Alfred North Whitehead,Bertrand Russell
Publisher: N.A
Category: Logic, Symbolic and mathematical
Page: 167
View: 1802

The Foundations of Mathematics

Author: Ian Stewart,David Tall
Publisher: OUP Oxford
ISBN: 0191016489
Category: Mathematics
Page: 432
View: 3696
The transition from school mathematics to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory. The authors have many years' experience of the potential difficulties involved, through teaching first-year undergraduates and researching the ways in which students and mathematicians think. The book explains the motivation behind abstract foundational material based on students' experiences of school mathematics, and explicitly suggests ways students can make sense of formal ideas. This second edition takes a significant step forward by not only making the transition from intuitive to formal methods, but also by reversing the process- using structure theorems to prove that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is exemplified by a new chapter on the theory of groups. While the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields. This links intuitive ideas in calculus to the formal epsilon-delta methods of analysis. The approach here is not the conventional one of 'nonstandard analysis', but a simpler, graphically based treatment which makes the notion of an infinitesimal natural and straightforward. This allows a further vision of the wider world of mathematical thinking in which formal definitions and proof lead to amazing new ways of defining, proving, visualising and symbolising mathematics beyond previous expectations.

Foundations of classical analysis

Author: Georg Kreisel,Stanford University
Publisher: N.A
Category: Mathematics
Page: N.A
View: 9713

Varieties of Constructive Mathematics

Author: Douglas Bridges,Fred Richman
Publisher: Cambridge University Press
ISBN: 9780521318020
Category: Mathematics
Page: 149
View: 3223
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.

Apartness and Uniformity

A Constructive Development
Author: Douglas S. Bridges,Luminiţa Simona Vîţă
Publisher: Springer Science & Business Media
ISBN: 3642224156
Category: Computers
Page: 198
View: 2669
The theory presented in this book is developed constructively, is based on a few axioms encapsulating the notion of objects (points and sets) being apart, and encompasses both point-set topology and the theory of uniform spaces. While the classical-logic-based theory of proximity spaces provides some guidance for the theory of apartness, the notion of nearness/proximity does not embody enough algorithmic information for a deep constructive development. The use of constructive (intuitionistic) logic in this book requires much more technical ingenuity than one finds in classical proximity theory -- algorithmic information does not come cheaply -- but it often reveals distinctions that are rendered invisible by classical logic. In the first chapter the authors outline informal constructive logic and set theory, and, briefly, the basic notions and notations for metric and topological spaces. In the second they introduce axioms for a point-set apartness and then explore some of the consequences of those axioms. In particular, they examine a natural topology associated with an apartness space, and relations between various types of continuity of mappings. In the third chapter the authors extend the notion of point-set (pre-)apartness axiomatically to one of (pre-)apartness between subsets of an inhabited set. They then provide axioms for a quasiuniform space, perhaps the most important type of set-set apartness space. Quasiuniform spaces play a major role in the remainder of the chapter, which covers such topics as the connection between uniform and strong continuity (arguably the most technically difficult part of the book), apartness and convergence in function spaces, types of completeness, and neat compactness. Each chapter has a Notes section, in which are found comments on the definitions, results, and proofs, as well as occasional pointers to future work. The book ends with a Postlude that refers to other constructive approaches to topology, with emphasis on the relation between apartness spaces and formal topology. Largely an exposition of the authors' own research, this is the first book dealing with the apartness approach to constructive topology, and is a valuable addition to the literature on constructive mathematics and on topology in computer science. It is aimed at graduate students and advanced researchers in theoretical computer science, mathematics, and logic who are interested in constructive/algorithmic aspects of topology.

The Number Systems

Foundations of Algebra and Analysis
Author: Solomon Feferman
Publisher: American Mathematical Soc.
ISBN: 0821829157
Category: Mathematics
Page: 418
View: 8919
The subject of this book is the successive construction and development of the basic number systems of mathematics: positive integers, integers, rational numbers, real numbers, and complex numbers. This second edition expands upon the list of suggestions for further reading in Appendix III. From the Preface: ``The present book basically takes for granted the non-constructive set-theoretical foundation of mathematics, which is tacitly if not explicitly accepted by most working mathematicians but which I have since come to reject. Still, whatever one's foundational views, students must be trained in this approach in order to understand modern mathematics. Moreover, most of the material of the present book can be modified so as to be acceptable under alternative constructive and semi-constructive viewpoints, as has been demonstrated in more advanced texts and research articles.''

The Foundations of Mathematics in the Theory of Sets

Author: John P. Mayberry
Publisher: Cambridge University Press
ISBN: 9780521770347
Category: Mathematics
Page: 424
View: 4635
This 2001 book will appeal to mathematicians and philosophers interested in the foundations of mathematics.

Constructive Adpositional Grammars

Foundations of Constructive Linguistics
Author: Federico Gobbo,Marco Benini
Publisher: Cambridge Scholars Publishing
ISBN: 144383128X
Category: Language Arts & Disciplines
Page: 280
View: 7807
This book presents a new paradigm of natural language grammar analysis, based on adposition as the key concept, considered a general connection between two morphemes – or group of morphemes. The adpositional paradigm considers the morpheme as the basic unit to represent morphosyntax, taken as a whole, in terms of constructions, while semantics and pragmatics are treated accordingly. All linguistic observations within the book can be described through the methods and tools of Constructive Mathematics, so that the modelling becomes formally feasible. A full description in category-theoretic terms of the formal model is provided in the Appendix. A lot of examples taken from natural languages belonging to different typological areas are offered throughout the volume, in order to explain and validate the modeling – with special attention given to ergativity. Finally, a first real-world application of the paradigm is given, i.e., conversational analysis of the transcript of therapeutic settings in terms of constructive speech acts. The main goal of this book is to broaden the scope of Linguistics by including Constructive Mathematics in order to deal with known topics such as grammaticalization, children’s speech, language comparison, dependency and valency from a different perspective. It primarily concerns advanced students and researchers in the field of Theoretical and Mathematical Linguistics but the audience can also include scholars interested in applications of Topos Theory in Linguistics.

Über die konstruktive Behandlung mathematischer Probleme. Von Matrizen zu Jordan-Tripelsystemen

282. Sitzung am 5. November 1980 in Düsseldorf
Author: Hans Zassenhaus
Publisher: Springer-Verlag
ISBN: 3322881938
Category: Mathematics
Page: 74
View: 885

Provability, Computability and Reflection

Author: Lev D. Beklemishev
Publisher: Elsevier
ISBN: 9780080957463
Category: Mathematics
Page: 160
View: 435
Provability, Computability and Reflection

Internal Logic

Foundations of Mathematics from Kronecker to Hilbert
Author: Y. Gauthier
Publisher: Springer Science & Business Media
ISBN: 9401700834
Category: Mathematics
Page: 251
View: 1865
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.

One Hundred Years of Intuitionism (1907-2007)

The Cerisy Conference
Author: Mark van Atten,Pascal Boldini,Michel Bourdeau,Gerhard Heinzmann
Publisher: Springer Science & Business Media
ISBN: 3764386533
Category: Science
Page: 422
View: 1851
Intuitionism is one of the main foundations for mathematics proposed in the twentieth century and its views on logic have also notably become important with the development of theoretical computer science. This book reviews and completes the historical account of intuitionism. It also presents recent philosophical work on intuitionism and gives examples of new technical advances and applications. It brings together 21 contributions from today's leading authors on intuitionism.