Includes bibliographical references (pages 383-385) and index.

The transition from school to university mathematics is seldom straightforward. Students face a schism 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. This 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 proofs lead to amazing new ways of defining, proving, visualising, and symbolising mathematics beyond previous expectations. -- Back cover.

Part I. The intuitive background -- 1. Mathematical thinking -- 2. Number systems -- Part II. The beginnings of formalisation -- 3. Sets -- 4. Relations -- 5. Functions -- 6. Mathematical logic -- 7. Mathematical proof -- Part 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 -- Part IV. Using axiomatic systems -- 12. Axiomatic systems, structure theorems, and flexible thinking -- 13. Permutations and groups -- 14. Cardinal numbers -- 15. Infinitesimals -- Part V. Strengthening the foundations -- 16. Axioms for set theory.

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