The Foundations of Mathematics by Ian Stewart

The Foundations of Mathematics

byIan Stewart, David Tall

Paperback | April 22, 2015

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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 generalconcepts, 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 ofschool 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 isexemplified 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 "nonstandardanalysis", 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.

About The Author

Ian Stewart is Emeritus Professor of Mathematics at the University of Warwick. He remains an active research mathematician and is a Fellow of the Royal Society. Famed for his popular science writing and broadcasting, for which he is the recipient of numerous awards, his bestselling books include: Does God Play Dice?, Nature's Numbers,...
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Details & Specs

Title:The Foundations of MathematicsFormat:PaperbackDimensions:416 pages, 8.5 × 5.43 × 0.96 inPublished:April 22, 2015Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:019870643X

ISBN - 13:9780198706434

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Table of Contents

I: The Intuitive Background1. Mathematical Thinking2. Number SystemsII: The Beginnings of Formalisation3. Sets4. Relations5. FunctionsIII: The Development of Axiomatic Systems8. Natural Numbers and Proof by Induction9. Real Numbers10. Real Numbers as a Complete Ordered Field11. Complex Numbers and BeyondIV: Using Axiomatic Systems12. Axiomatic Structures and Structure Theorems13. Permutations and Groups14. Infinite Cardinal Numbers15. InfinitesimalsV: Strengthening the Foundations16. Axioms for Set Theory

Editorial Reviews

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