Explicitly suggests to the student ways they can increase their understanding. Explains the motivation behind otherwise abstract foundational material in mathematics. Guides the reader from an informal to a formal, axiomatic approach. Extremely well-known bestselling authors. New edition of a widely used book. Highly illustrated. New to this edition The new edition of The Foundations of Mathematics is a major update, with four entirely new chapters and the remainder of the book updated as necessary. The original opening chapter on · Mathematical Thinking will be completely rewritten and significantly expanded. A brand new Part IV of the book on Using Axiomatic Systems will contain three further new chapters on · Axiomatic Structures and Structure Theorems · Permutations and Groups · Infinitesimals 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.
- ISBN: 978-0-19-870643-4
- Editorial: OXFORD UNIVERSITY PRESS.
- Encuadernacion: Rústica
- Páginas: 416
- Fecha Publicación: 01/03/2015
- Nº Volúmenes: 1
- Idioma: