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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."
Using mathematical tools from number theory and finite fields,
Applied Algebra: Codes, Ciphers, and Discrete Algorithms, Second
Edition presents practical methods for solving problems in data
security and data integrity. It is designed for an applied algebra
course for students who have had prior classes in abstract or
linear algebra. While the content has been reworked and improved,
this edition continues to cover many algorithms that arise in
cryptography and error-control codes. New to the Second Edition
Downloadable resources containing an interactive version of the
book that is powered by Scientific Notebook (R), a mathematical
word processor and easy-to-use computer algebra system New appendix
that reviews prerequisite topics in algebra and number theory
Double the number of exercises Instead of a general study on finite
groups, the book considers finite groups of permutations and
develops just enough of the theory of finite fields to facilitate
construction of the fields used for error-control codes and the
Advanced Encryption Standard. It also deals with integers and
polynomials. Explaining the mathematics as needed, this text
thoroughly explores how mathematical techniques can be used to
solve practical problems. About the AuthorsDarel W. Hardy is
Professor Emeritus in the Department of Mathematics at Colorado
State University. His research interests include applied algebra
and semigroups. Fred Richman is a professor in the Department of
Mathematical Sciences at Florida Atlantic University. His research
interests include Abelian group theory and constructive
mathematics. Carol L. Walker is Associate Dean Emeritus in the
Department of Mathematical Sciences at New Mexico State University.
Her research interests include Abelian group theory, applications
of homological algebra and category theory, and the mathematics of
fuzzy sets and fuzzy logic.
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.
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