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Author: Douglas Bridges Publisher: Cambridge University Press ISBN: 9780521318020 Category : Mathematics Languages : en Pages : 164
Book Description
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.
Author: Douglas Bridges Publisher: Cambridge University Press ISBN: 9780521318020 Category : Mathematics Languages : en Pages : 164
Book Description
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.
Author: M.J. Beeson Publisher: Springer Science & Business Media ISBN: 3642689523 Category : Mathematics Languages : en Pages : 484
Book Description
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.
Author: Douglas S. Bridges Publisher: ISBN: 9781107361331 Category : MATHEMATICS Languages : en Pages : 161
Book Description
This is an introduction to, and survey of, the constructive approaches to pure mathematics. The authors emphasise the viewpoint of Errett Bishop's school, but intuitionism. Russian constructivism and recursive analysis are also treated, with comparisons between the various approaches included where appropriate. Constructive mathematics is now enjoying a revival, with interest from not only logicans but also category theorists, recursive function theorists and theoretical computer scientists. This account for non-specialists in these and other disciplines.
Author: Giovanni Sommaruga Publisher: Springer Science & Business Media ISBN: 9400704313 Category : Mathematics Languages : en Pages : 312
Book Description
The book "Foundational Theories of Classical and Constructive Mathematics" is a book on the classical topic of foundations of mathematics. Its originality resides mainly in its treating at the same time foundations of classical and foundations of constructive mathematics. This confrontation of two kinds of foundations contributes to answering questions such as: Are foundations/foundational theories of classical mathematics of a different nature compared to those of constructive mathematics? Do they play the same role for the resp. mathematics? Are there connections between the two kinds of foundational theories? etc. The confrontation and comparison is often implicit and sometimes explicit. Its great advantage is to extend the traditional discussion of the foundations of mathematics and to render it at the same time more subtle and more differentiated. Another important aspect of the book is that some of its contributions are of a more philosophical, others of a more technical nature. This double face is emphasized, since foundations of mathematics is an eminent topic in the philosophy of mathematics: hence both sides of this discipline ought to be and are being paid due to.
Author: Harold M. Edwards Publisher: Springer Nature ISBN: 303098558X Category : Mathematics Languages : en Pages : 325
Book Description
Contents and treatment are fresh and very different from the standard treatments Presents a fully constructive version of what it means to do algebra The exposition is not only clear, it is friendly, philosophical, and considerate even to the most naive or inexperienced reader
Author: Ray Mines Publisher: Springer Science & Business Media ISBN: 1441986405 Category : Mathematics Languages : en Pages : 355
Book Description
The constructive approach to mathematics has enjoyed a renaissance, caused in large part by the appearance of Errett Bishop's book Foundations of constr"uctiue analysis in 1967, and by the subtle influences of the proliferation of powerful computers. Bishop demonstrated that pure mathematics can be developed from a constructive point of view while maintaining a continuity with classical terminology and spirit; much more of classical mathematics was preserved than had been thought possible, and no classically false theorems resulted, as had been the case in other constructive schools such as intuitionism and Russian constructivism. The computers created a widespread awareness of the intuitive notion of an effecti ve procedure, and of computation in principle, in addi tion to stimulating the study of constructive algebra for actual implementation, and from the point of view of recursive function theory. In analysis, constructive problems arise instantly because we must start with the real numbers, and there is no finite procedure for deciding whether two given real numbers are equal or not (the real numbers are not discrete) . The main thrust of constructive mathematics was in the direction of analysis, although several mathematicians, including Kronecker and van der waerden, made important contributions to construc tive algebra. Heyting, working in intuitionistic algebra, concentrated on issues raised by considering algebraic structures over the real numbers, and so developed a handmaiden'of analysis rather than a theory of discrete algebraic structures.
Author: Douglas S. Bridges Publisher: Springer Science & Business Media ISBN: 0387381473 Category : Mathematics Languages : en Pages : 227
Book Description
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.
Author: Anne Watson Publisher: Routledge ISBN: 1135630011 Category : Education Languages : en Pages : 407
Book Description
This book explains and demonstrates the teaching strategy of asking learners to construct their own examples of mathematical objects. The authors show that the creation of examples can involve transforming and reorganizing knowledge and that, although this is usually done by authors and teachers, if the responsibility for making examples is transferred to learners, their knowledge structures can be developed and extended. A multitude of examples to illustrate this is provided, spanning primary, secondary, and college levels. Readers are invited to learn from their own past experience augmented by tasks provided in the book, and are given direct experience of constructing examples through a collection of many tasks at many levels. Classroom stories show the practicalities of introducing such shifts in mathematics education. The authors examine how their approach relates to improving the learning of mathematics and raise future research questions. *Based on the authors' and others' theoretical and practical experience, the book includes a combination of exercises for the reader, practical applications for teaching, and solid scholarly grounding. *The ideas presented are generic in nature and thus applicable across every phase of mathematics teaching and learning. *Although the teaching methods offered are ones that engage learners imaginatively, these are also applied to traditional approaches to mathematics education; all tasks offered in the book are within conventional mathematics curriculum content. Mathematics as a Constructive Activity: Learners Generating Examples is intended for mathematics teacher educators, mathematics teachers, curriculum developers, task and test designers, and classroom researchers, and for use as a text in graduate-level mathematics education courses.
Author: E. Bishop Publisher: Springer Science & Business Media ISBN: 3642616674 Category : Mathematics Languages : en Pages : 490
Book Description
This work grew out of Errett Bishop's fundamental treatise 'Founda tions of Constructive Analysis' (FCA), which appeared in 1967 and which contained the bountiful harvest of a remarkably short period of research by its author. Truly, FCA was an exceptional book, not only because of the quantity of original material it contained, but also as a demonstration of the practicability of a program which most ma thematicians believed impossible to carry out. Errett's book went out of print shortly after its publication, and no second edition was produced by its publishers. Some years later, 'by a set of curious chances', it was agreed that a new edition of FCA would be published by Springer Verlag, the revision being carried out by me under Errett's supervision; at the same time, Errett gener ously insisted that I become a joint author. The revision turned out to be much more substantial than we had anticipated, and took longer than we would have wished. Indeed, tragically, Errett died before the work was completed. The present book is the result of our efforts. Although substantially based on FCA, it contains so much new material, and such full revision and expansion of the old, that it is essentially a new book. For this reason, and also to preserve the integrity of the original, I decided to give our joint work a title of its own. Most of the new material outside Chapter 5 originated with Errett.
Author: Douglas Bridges
Author: Douglas Bridges
Author: M.J. Beeson
Author: Douglas S. Bridges
Author: Giovanni Sommaruga
Author: Harold M. Edwards
Author: Ray Mines
Author: Douglas S. Bridges
Author: Anne Watson
Author: E. Bishop
Author: Yuen-Kwok Chan