Varieties Of Constructive Mathematics
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Varieties Of Constructive Mathematics
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Author : Douglas Bridges
language : en
Publisher: Cambridge University Press
Release Date : 1987-04-24
Varieties Of Constructive Mathematics written by Douglas Bridges and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1987-04-24 with Mathematics categories.
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.
Varieties Of Constructive Mathematics
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Author : Douglas S. Bridges
language : en
Publisher:
Release Date : 2014-05-14
Varieties Of Constructive Mathematics written by Douglas S. Bridges and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-05-14 with MATHEMATICS categories.
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.
A Course In Constructive Algebra
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Author : Ray Mines
language : en
Publisher: Springer Science & Business Media
Release Date : 1987-12-18
A Course In Constructive Algebra written by Ray Mines and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1987-12-18 with Mathematics categories.
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.
Constructive Analysis
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Author : E. Bishop
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Constructive Analysis written by E. Bishop and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
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.
Mathematics As A Constructive Activity
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Author : Anne Watson
language : en
Publisher: Routledge
Release Date : 2006-04-21
Mathematics As A Constructive Activity written by Anne Watson and has been published by Routledge this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-04-21 with Education categories.
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.
Foundations Of Constructive Probability Theory
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Author : Yuen-Kwok Chan
language : en
Publisher: Cambridge University Press
Release Date : 2021-05-27
Foundations Of Constructive Probability Theory written by Yuen-Kwok Chan and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-05-27 with Mathematics categories.
Using Bishop's work on constructive analysis as a framework, this monograph gives a systematic, detailed and general constructive theory of probability theory and stochastic processes. It is the first extended account of this theory: almost all of the constructive existence and continuity theorems that permeate the book are original. It also contains results and methods hitherto unknown in the constructive and nonconstructive settings. The text features logic only in the common sense and, beyond a certain mathematical maturity, requires no prior training in either constructive mathematics or probability theory. It will thus be accessible and of interest, both to probabilists interested in the foundations of their speciality and to constructive mathematicians who wish to see Bishop's theory applied to a particular field.
Essays In Constructive Mathematics
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Author : Harold M. Edwards
language : en
Publisher: Springer Nature
Release Date : 2022-09-29
Essays In Constructive Mathematics written by Harold M. Edwards and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-09-29 with Mathematics categories.
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
Foundations Of Constructive Analysis
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Author : Errett Bishop
language : en
Publisher: Ishi Press
Release Date : 2012-07
Foundations Of Constructive Analysis written by Errett Bishop and has been published by Ishi Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-07 with Mathematics categories.
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.
Handbook Of Constructive Mathematics
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Author : Douglas Bridges
language : en
Publisher: Cambridge University Press
Release Date : 2023-05-11
Handbook Of Constructive Mathematics written by Douglas Bridges and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-05-11 with Mathematics categories.
Constructive mathematics – mathematics in which 'there exists' always means 'we can construct' – is enjoying a renaissance. fifty years on from Bishop's groundbreaking account of constructive analysis, constructive mathematics has spread out to touch almost all areas of mathematics and to have profound influence in theoretical computer science. This handbook gives the most complete overview of modern constructive mathematics, with contributions from leading specialists surveying the subject's myriad aspects. Major themes include: constructive algebra and geometry, constructive analysis, constructive topology, constructive logic and foundations of mathematics, and computational aspects of constructive mathematics. A series of introductory chapters provides graduate students and other newcomers to the subject with foundations for the surveys that follow. Edited by four of the most eminent experts in the field, this is an indispensable reference for constructive mathematicians and a fascinating vista of modern constructivism for the increasing number of researchers interested in constructive approaches.
Constructive Real Analysis
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Author : Allen A. Goldstein
language : en
Publisher: Courier Corporation
Release Date : 2012-01-01
Constructive Real Analysis written by Allen A. Goldstein and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-01-01 with Mathematics categories.
This text introduces students of mathematics, science, and technology to the methods of applied functional analysis and applied convexity. The three-part treatment consists of roots and extremal problems, constraints, and infinite dimensional problems. Topics include iterations and fixed points, metric spaces, nonlinear programming, polyhedral convex programming, linear spaces and convex sets, and applications to integral equations. 1967 edition.