Developments In Nonstandard Mathematics

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Developments in Nonstandard Mathematics

Author : Nigel J Cutland
Publisher : CRC Press
ISBN 13 : 1000724646
Total Pages : 278 pages
Book Rating : 4.0/5 (7 download)

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Book Synopsis Developments in Nonstandard Mathematics by : Nigel J Cutland

Download or read book Developments in Nonstandard Mathematics written by Nigel J Cutland and published by CRC Press. This book was released on 2020-01-30 with total page 278 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains expository papers and articles reporting on recent research by leading world experts in nonstandard mathematics, arising from the International Colloquium on Nonstandard Mathematics held at the University of Aveiro, Portugal in July 1994. Nonstandard mathematics originated with Abraham Robinson, and the body of ideas that have developed from this theory of nonstandard analysis now vastly extends Robinson's work with infinitesimals. The range of applications includes measure and probability theory, stochastic analysis, differential equations, generalised functions, mathematical physics and differential geometry, moreover, the theory has implicaitons for the teaching of calculus and analysis. This volume contains papers touching on all of the abovbe topics, as well as a biographical note about Abraham Robinson based on the opening address given by W.A>J> Luxemburg - who knew Robinson - to the Aveiro conference which marked the 20th anniversary of Robinson's death. This book will be of particular interest to students and researchers in nonstandard analysis, measure theory, generalised functions and mathematical physics.

Non-standard Analysis

Author : Abraham Robinson
Publisher : Princeton University Press
ISBN 13 : 1400884225
Total Pages : 315 pages
Book Rating : 4.4/5 (8 download)

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Book Synopsis Non-standard Analysis by : Abraham Robinson

Download or read book Non-standard Analysis written by Abraham Robinson and published by Princeton University Press. This book was released on 2016-08-11 with total page 315 pages. Available in PDF, EPUB and Kindle. Book excerpt: Considered by many to be Abraham Robinson's magnum opus, this book offers an explanation of the development and applications of non-standard analysis by the mathematician who founded the subject. Non-standard analysis grew out of Robinson's attempt to resolve the contradictions posed by infinitesimals within calculus. He introduced this new subject in a seminar at Princeton in 1960, and it remains as controversial today as it was then. This paperback reprint of the 1974 revised edition is indispensable reading for anyone interested in non-standard analysis. It treats in rich detail many areas of application, including topology, functions of a real variable, functions of a complex variable, and normed linear spaces, together with problems of boundary layer flow of viscous fluids and rederivations of Saint-Venant's hypothesis concerning the distribution of stresses in an elastic body.

Nonstandard Methods in Stochastic Analysis and Mathematical Physics

Author : Sergio Albeverio
Publisher : Courier Dover Publications
ISBN 13 : 0486468992
Total Pages : 529 pages
Book Rating : 4.4/5 (864 download)

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Book Synopsis Nonstandard Methods in Stochastic Analysis and Mathematical Physics by : Sergio Albeverio

Download or read book Nonstandard Methods in Stochastic Analysis and Mathematical Physics written by Sergio Albeverio and published by Courier Dover Publications. This book was released on 2009-02-26 with total page 529 pages. Available in PDF, EPUB and Kindle. Book excerpt: Two-part treatment begins with a self-contained introduction to the subject, followed by applications to stochastic analysis and mathematical physics. "A welcome addition." — Bulletin of the American Mathematical Society. 1986 edition.

Lectures on the Hyperreals

Author : Robert Goldblatt
Publisher : Springer Science & Business Media
ISBN 13 : 1461206154
Total Pages : 292 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Lectures on the Hyperreals by : Robert Goldblatt

Download or read book Lectures on the Hyperreals written by Robert Goldblatt and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introduction to nonstandard analysis based on a course given by the author. It is suitable for beginning graduates or upper undergraduates, or for self-study by anyone familiar with elementary real analysis. It presents nonstandard analysis not just as a theory about infinitely small and large numbers, but as a radically different way of viewing many standard mathematical concepts and constructions. It is a source of new ideas, objects and proofs, and a wealth of powerful new principles of reasoning. The book begins with the ultrapower construction of hyperreal number systems, and proceeds to develop one-variable calculus, analysis and topology from the nonstandard perspective. It then sets out the theory of enlargements of fragments of the mathematical universe, providing a foundation for the full-scale development of the nonstandard methodology. The final chapters apply this to a number of topics, including Loeb measure theory and its relation to Lebesgue measure on the real line. Highlights include an early introduction of the ideas of internal, external and hyperfinite sets, and a more axiomatic set-theoretic approach to enlargements than is usual.

Nonstandard Analysis, Axiomatically

Author : Vladimir Kanovei
Publisher : Springer Science & Business Media
ISBN 13 : 366208998X
Total Pages : 421 pages
Book Rating : 4.6/5 (62 download)

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Book Synopsis Nonstandard Analysis, Axiomatically by : Vladimir Kanovei

Download or read book Nonstandard Analysis, Axiomatically written by Vladimir Kanovei and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 421 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the aftermath of the discoveries in foundations of mathematiC's there was surprisingly little effect on mathematics as a whole. If one looks at stan dard textbooks in different mathematical disciplines, especially those closer to what is referred to as applied mathematics, there is little trace of those developments outside of mathematical logic and model theory. But it seems fair to say that there is a widespread conviction that the principles embodied in the Zermelo - Fraenkel theory with Choice (ZFC) are a correct description of the set theoretic underpinnings of mathematics. In most textbooks of the kind referred to above, there is, of course, no discussion of these matters, and set theory is assumed informally, although more advanced principles like Choice or sometimes Replacement are often mentioned explicitly. This implicitly fixes a point of view of the mathemat ical universe which is at odds with the results in foundations. For example most mathematicians still take it for granted that the real number system is uniquely determined up to isomorphism, which is a correct point of view as long as one does not accept to look at "unnatural" interpretations of the membership relation.

Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory

Author : Mauro Di Nasso
Publisher : Springer
ISBN 13 : 3030179567
Total Pages : 211 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory by : Mauro Di Nasso

Download or read book Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory written by Mauro Di Nasso and published by Springer. This book was released on 2019-05-23 with total page 211 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of this monograph is to give an accessible introduction to nonstandard methods and their applications, with an emphasis on combinatorics and Ramsey theory. It includes both new nonstandard proofs of classical results and recent developments initially obtained in the nonstandard setting. This makes it the first combinatorics-focused account of nonstandard methods to be aimed at a general (graduate-level) mathematical audience. This book will provide a natural starting point for researchers interested in approaching the rapidly growing literature on combinatorial results obtained via nonstandard methods. The primary audience consists of graduate students and specialists in logic and combinatorics who wish to pursue research at the interface between these areas.

Nonstandard Models of Arithmetic and Set Theory

Author : Ali Enayat
Publisher : American Mathematical Soc.
ISBN 13 : 0821835351
Total Pages : 184 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Nonstandard Models of Arithmetic and Set Theory by : Ali Enayat

Download or read book Nonstandard Models of Arithmetic and Set Theory written by Ali Enayat and published by American Mathematical Soc.. This book was released on 2004 with total page 184 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the proceedings of the AMS special session on nonstandard models of arithmetic and set theory held at the Joint Mathematics Meetings in Baltimore (MD). The volume opens with an essay from Haim Gaifman that probes the concept of non-standardness in mathematics and provides a fascinating mix of historical and philosophical insights into the nature of nonstandard mathematical structures. In particular, Gaifman compares and contrasts the discovery of nonstandard models with other key mathematical innovations, such as the introduction of various number systems, the modern concept of function, and non-Euclidean geometries. Other articles in the book present results related to nonstandard models in arithmetic and set theory, including a survey of known results on the Turing upper bounds of arithmetic sets and functions. The volume is suitable for graduate students and research mathematicians interested in logic, especially model theory.

Nonstandard Methods in Fixed Point Theory

Author : Asuman G. Aksoy
Publisher : Springer Science & Business Media
ISBN 13 : 1461234441
Total Pages : 149 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Nonstandard Methods in Fixed Point Theory by : Asuman G. Aksoy

Download or read book Nonstandard Methods in Fixed Point Theory written by Asuman G. Aksoy and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 149 pages. Available in PDF, EPUB and Kindle. Book excerpt: A unified account of the major new developments inspired by Maurey's application of Banach space ultraproducts to the fixed point theory for non-expansive mappings is given in this text. The first third of the book is devoted to laying a careful foundation for the actual fixed point theoretic results which follow. Set theoretic and Banach space ultraproducts constructions are studied in detail in the second part of the book, while the remainder of the book gives an introduction to the classical fixed point theory in addition to a discussion of normal structure. This is the first book which studies classical fixed point theory for non-expansive maps in the view of non-standard methods.

Lectures on the Philosophy of Mathematics

Author : Joel David Hamkins
Publisher : MIT Press
ISBN 13 : 0262542234
Total Pages : 350 pages
Book Rating : 4.2/5 (625 download)

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Book Synopsis Lectures on the Philosophy of Mathematics by : Joel David Hamkins

Download or read book Lectures on the Philosophy of Mathematics written by Joel David Hamkins and published by MIT Press. This book was released on 2021-03-09 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice. In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations.

Nonstandard Methods and Applications in Mathematics

Author : Nigel J. Cutland
Publisher : Cambridge University Press
ISBN 13 : 1108621295
Total Pages : 260 pages
Book Rating : 4.1/5 (86 download)

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Book Synopsis Nonstandard Methods and Applications in Mathematics by : Nigel J. Cutland

Download or read book Nonstandard Methods and Applications in Mathematics written by Nigel J. Cutland and published by Cambridge University Press. This book was released on 2017-03-30 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the twenty-fifth publication in the Lecture Notes in Logic series, grew from a conference on Nonstandard Methods and Applications in Mathematics held in Pisa, Italy from 12–16 June, 2002. It contains ten peer-reviewed papers that aim to provide something more timely than a textbook, but less ephemeral than a conventional proceedings. Nonstandard analysis is one of the great achievements of modern applied mathematical logic. These articles consider the foundations of the subject, as well as its applications to pure and applied mathematics and mathematics education.

Real Analysis Through Modern Infinitesimals

Author : Nader Vakil
Publisher : Cambridge University Press
ISBN 13 : 1107002028
Total Pages : 587 pages
Book Rating : 4.1/5 (7 download)

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Book Synopsis Real Analysis Through Modern Infinitesimals by : Nader Vakil

Download or read book Real Analysis Through Modern Infinitesimals written by Nader Vakil and published by Cambridge University Press. This book was released on 2011-02-17 with total page 587 pages. Available in PDF, EPUB and Kindle. Book excerpt: A coherent, self-contained treatment of the central topics of real analysis employing modern infinitesimals.

How To Measure The Infinite: Mathematics With Infinite And Infinitesimal Numbers

Author : Vieri Benci
Publisher : World Scientific
ISBN 13 : 9813276606
Total Pages : 346 pages
Book Rating : 4.8/5 (132 download)

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Book Synopsis How To Measure The Infinite: Mathematics With Infinite And Infinitesimal Numbers by : Vieri Benci

Download or read book How To Measure The Infinite: Mathematics With Infinite And Infinitesimal Numbers written by Vieri Benci and published by World Scientific. This book was released on 2019-02-19 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: 'This text shows that the study of the almost-forgotten, non-Archimedean mathematics deserves to be utilized more intently in a variety of fields within the larger domain of applied mathematics.'CHOICEThis book contains an original introduction to the use of infinitesimal and infinite numbers, namely, the Alpha-Theory, which can be considered as an alternative approach to nonstandard analysis.The basic principles are presented in an elementary way by using the ordinary language of mathematics; this is to be contrasted with other presentations of nonstandard analysis where technical notions from logic are required since the beginning. Some applications are included and aimed at showing the power of the theory.The book also provides a comprehensive exposition of the Theory of Numerosity, a new way of counting (countable) infinite sets that maintains the ancient Euclid's Principle: 'The whole is larger than its parts'. The book is organized into five parts: Alpha-Calculus, Alpha-Theory, Applications, Foundations, and Numerosity Theory.

Elementary Calculus

Author : H. Jerome Keisler
Publisher : Orange Groove Books
ISBN 13 : 9781616100315
Total Pages : 992 pages
Book Rating : 4.1/5 (3 download)

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Book Synopsis Elementary Calculus by : H. Jerome Keisler

Download or read book Elementary Calculus written by H. Jerome Keisler and published by Orange Groove Books. This book was released on 2009-09-01 with total page 992 pages. Available in PDF, EPUB and Kindle. Book excerpt:

The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics

Author : John L. Bell
Publisher : Springer Nature
ISBN 13 : 3030187071
Total Pages : 320 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics by : John L. Bell

Download or read book The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics written by John L. Bell and published by Springer Nature. This book was released on 2019-09-09 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal.

Mathematics and Its History

Author : John Stillwell
Publisher : Springer Nature
ISBN 13 : 3030551938
Total Pages : 400 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Mathematics and Its History by : John Stillwell

Download or read book Mathematics and Its History written by John Stillwell and published by Springer Nature. This book was released on 2020-11-07 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook provides a unified and concise exploration of undergraduate mathematics by approaching the subject through its history. Readers will discover the rich tapestry of ideas behind familiar topics from the undergraduate curriculum, such as calculus, algebra, topology, and more. Featuring historical episodes ranging from the Ancient Greeks to Fermat and Descartes, this volume offers a glimpse into the broader context in which these ideas developed, revealing unexpected connections that make this ideal for a senior capstone course. The presentation of previous versions has been refined by omitting the less mainstream topics and inserting new connecting material, allowing instructors to cover the book in a one-semester course. This condensed edition prioritizes succinctness and cohesiveness, and there is a greater emphasis on visual clarity, featuring full color images and high quality 3D models. As in previous editions, a wide array of mathematical topics are covered, from geometry to computation; however, biographical sketches have been omitted. Mathematics and Its History: A Concise Edition is an essential resource for courses or reading programs on the history of mathematics. Knowledge of basic calculus, algebra, geometry, topology, and set theory is assumed. From reviews of previous editions: “Mathematics and Its History is a joy to read. The writing is clear, concise and inviting. The style is very different from a traditional text. I found myself picking it up to read at the expense of my usual late evening thriller or detective novel.... The author has done a wonderful job of tying together the dominant themes of undergraduate mathematics.” Richard J. Wilders, MAA, on the Third Edition "The book...is presented in a lively style without unnecessary detail. It is very stimulating and will be appreciated not only by students. Much attention is paid to problems and to the development of mathematics before the end of the nineteenth century.... This book brings to the non-specialist interested in mathematics many interesting results. It can be recommended for seminars and will be enjoyed by the broad mathematical community." European Mathematical Society, on the Second Edition

Harvey Friedman's Research on the Foundations of Mathematics

Author : L.A. Harrington
Publisher : Elsevier
ISBN 13 : 9780080960401
Total Pages : 407 pages
Book Rating : 4.9/5 (64 download)

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Book Synopsis Harvey Friedman's Research on the Foundations of Mathematics by : L.A. Harrington

Download or read book Harvey Friedman's Research on the Foundations of Mathematics written by L.A. Harrington and published by Elsevier. This book was released on 1985-11-01 with total page 407 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume discusses various aspects of Harvey Friedman's research in the foundations of mathematics over the past fifteen years. It should appeal to a wide audience of mathematicians, computer scientists, and mathematically oriented philosophers.

Infinitesimal Calculus

Author : James M. Henle
Publisher : Courier Corporation
ISBN 13 : 0486151018
Total Pages : 146 pages
Book Rating : 4.4/5 (861 download)

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Book Synopsis Infinitesimal Calculus by : James M. Henle

Download or read book Infinitesimal Calculus written by James M. Henle and published by Courier Corporation. This book was released on 2014-01-15 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introducing calculus at the basic level, this text covers hyperreal numbers and hyperreal line, continuous functions, integral and differential calculus, fundamental theorem, infinite sequences and series, infinite polynomials, more. 1979 edition.