Numbers, Sets and Axioms

The Apparatus of Mathematics

Author: A. G. Hamilton

Publisher: Cambridge University Press

ISBN: 9780521287616

Category: Mathematics

Page: 255

View: 3958

Following the success of Logic for Mathematicians, Dr Hamilton has written a text for mathematicians and students of mathematics that contains a description and discussion of the fundamental conceptual and formal apparatus upon which modern pure mathematics relies. The author's intention is to remove some of the mystery that surrounds the foundations of mathematics. He emphasises the intuitive basis of mathematics; the basic notions are numbers and sets and they are considered both informally and formally. The role of axiom systems is part of the discussion but their limitations are pointed out. Formal set theory has its place in the book but Dr Hamilton recognises that this is a part of mathematics and not the basis on which it rests. Throughout, the abstract ideas are liberally illustrated by examples so this account should be well-suited, both specifically as a course text and, more broadly, as background reading. The reader is presumed to have some mathematical experience but no knowledge of mathematical logic is required.

The Foundations of Mathematics

Author: Ian Stewart,David Orme Tall

Publisher: Oxford University Press on Demand

ISBN: 9780198531654

Category: Fiction

Page: 263

View: 1961

"There are many textbooks available for a so-called transition course from calculus to abstract mathematics. I have taught this course several times and always find it problematic. The Foundations of Mathematics (Stewart and Tall) is a horse of a different color. The writing is excellent and there is actually some useful mathematics. I definitely like this book."--The Bulletin of Mathematics Books

The Art of Proof

Basic Training for Deeper Mathematics

Author: Matthias Beck,Ross Geoghegan

Publisher: Springer Science & Business Media

ISBN: 9781441970237

Category: Mathematics

Page: 182

View: 9971

The Art of Proof is designed for a one-semester or two-quarter course. A typical student will have studied calculus (perhaps also linear algebra) with reasonable success. With an artful mixture of chatty style and interesting examples, the student's previous intuitive knowledge is placed on solid intellectual ground. The topics covered include: integers, induction, algorithms, real numbers, rational numbers, modular arithmetic, limits, and uncountable sets. Methods, such as axiom, theorem and proof, are taught while discussing the mathematics rather than in abstract isolation. The book ends with short essays on further topics suitable for seminar-style presentation by small teams of students, either in class or in a mathematics club setting. These include: continuity, cryptography, groups, complex numbers, ordinal number, and generating functions.

Axiomatic Set Theory

Author: Paul Bernays

Publisher: Courier Corporation

ISBN: 0486666379

Category: Mathematics

Page: 227

View: 2648

A monograph containing a historical introduction by A. A. Fraenkel to the original Zermelo-Fraenkel form of set-theoretic axiomatics, and Paul Bernays’ independent presentation of a formal system of axiomatic set theory. No special knowledge of set thory and its axiomatics is required. With indexes of authors, symbols and matters, a list of axioms and an extensive bibliography.

How to Prove It

A Structured Approach

Author: Daniel J. Velleman

Publisher: Cambridge University Press

ISBN: 9780521675994

Category: Mathematics

Page: 384

View: 4096

Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.

Logic for Mathematicians

Author: A. G. Hamilton

Publisher: Cambridge University Press

ISBN: 9780521368650

Category: Mathematics

Page: 228

View: 8688

This is an introductory textbook which is designed to be useful not only to intending logicians but also to mathematicians in general.

Sets for Mathematics

Author: F. William Lawvere,Robert Rosebrugh

Publisher: Cambridge University Press

ISBN: 9780521010603

Category: Mathematics

Page: 261

View: 7970

In this book, first published in 2003, categorical algebra is used to build a foundation for the study of geometry, analysis, and algebra.

Linear Algebra: Volume 2

An Introduction with Concurrent Examples

Author: A. G. Hamilton

Publisher: Cambridge University Press

ISBN: 9780521310420

Category: Mathematics

Page: 328

View: 8063

Emphasis is placed on applications in preference to more theoretical aspects throughout this readable introduction to linear algebra for specialists as well as non-specialists. An expanded version of A First Course in Linear Algebra.

Simon Stevin

Author: Simon Stevin

Publisher: N.A


Category: Mathematics

Page: N.A

View: 3710

Mathematics and Logic

Author: Mark Kac,Stanislaw M. Ulam

Publisher: Courier Corporation

ISBN: 0486670856

Category: Philosophy

Page: 170

View: 4708

Fascinating study of the origin and nature of mathematical thought, including relation of mathematics and science, 20th-century developments, impact of computers, and more.Includes 34 illustrations. 1968 edition."

Introduction to Logic

and to the Methodology of Deductive Sciences

Author: Alfred Tarski

Publisher: Courier Corporation

ISBN: 0486318893

Category: Mathematics

Page: 272

View: 7472

This classic undergraduate treatment examines the deductive method in its first part and explores applications of logic and methodology in constructing mathematical theories in its second part. Exercises appear throughout.

Great Events from History

The 20th century, 1901-1940

Author: Robert F. Gorman

Publisher: Salem PressInc


Category: History

Page: 3453

View: 3976

Presents essays arranged in chronological order on key world events that occurred in such areas as politics, science, medicine, communications, literature, music, philosophy, and international affairs during the first forty years of the twentieth century.

The Foundations of Mathematics

Author: Ian Stewart,David Tall

Publisher: Oxford University Press, USA

ISBN: 019870643X

Category: Mathematics

Page: 432

View: 972

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.


Author: N.A

Publisher: N.A


Category: Academic libraries

Page: N.A

View: 4147