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.
|Publisher:||Oxford University Press|
|Product dimensions:||5.40(w) x 8.40(h) x 1.00(d)|
About the Author
Ian Stewart, Emeritus Professor, University of Warwick,David Tall, Emeritus Professor, University of Warwick
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, and Professor Stewart's Cabinet of Mathematical Curiosities. He also co-authored The Science of Discworld series with Terry Pratchett and Jack Cohen
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).
Table of Contents
I: The Intuitive Background
1. Mathematical Thinking
2. Number Systems
II: The Beginnings of Formalisation
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
V: Strengthening the Foundations
16. Axioms for Set Theory