The Foundations of Mathematics

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Ian Stewart (Emeritus Professor, Emeritus Professor, University of Warwick) David Tall (Emeritus Professor, Emeritus Professor, University of Warwick)

The Foundations of Mathematics

Ian Stewart (Emeritus Professor, Emeritus Professor, University of Warwick), David Tall (Emeritus Professor, Emeritus Professor, University of Warwick)

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English

Oxford University Press
19 March 2015

Mathematics & Sciences; Mathematical logic; Set theory; Number systems; Educational: Mathematics & numeracy

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.

By: Ian Stewart (Emeritus Professor Emeritus Professor University of Warwick), David Tall (Emeritus Professor, Emeritus Professor, University of Warwick)
Imprint: Oxford University Press
Country of Publication: United Kingdom
Edition: 2nd Revised edition
Dimensions: Height: 214mm, Width: 141mm, Spine: 24mm
Weight: 508g
ISBN: 9780198706434
ISBN 10: 019870643X
Pages: 432
Publication Date: 19 March 2015
Audience: College/higher education , A / AS level
Format: Paperback
Publisher's Status: Active

I: THE INTUITIVE BACKGROUND ; 1. Mathematical Thinking ; 2. Number Systems ; II: THE BEGINNINGS OF FORMALISATION ; 3. Sets ; 4. Relations ; 5. Functions ; III: THE DEVELOPMENT OF AXIOMATIC SYSTEMS ; 8. Natural Numbers and Proof by Induction ; 9. Real Numbers ; 10. Real Numbers as a Complete Ordered Field ; 11. Complex Numbers and Beyond ; IV: USING AXIOMATIC SYSTEMS ; 12. Axiomatic Structures and Structure Theorems ; 13. Permutations and Groups ; 14. Infinite Cardinal Numbers ; 15. Infinitesimals ; V: STRENGTHENING THE FOUNDATIONS ; 16. Axioms for Set Theory

David Tall is Emeritus Professor of Mathematical Thinking at the University of Warwick. Internationally known for his contributions to mathematics education, his most recent book is How Humans Learn to Think Mathematically (2013).

Reviews for The Foundations of Mathematics

Review from previous edition 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