The Foundations of Mathematics

By David Tall and Ian Stewart

Book Overview

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.

• Format: Paperback
• Language: English
• ISBN: 019870643X
• ISBN13: 9780198706434
• Release Date: May 2015
• Publisher: Oxford University Press
• Length: 416 Pages
• Weight: 1.14 lbs.
• Dimensions: 0.9" x 5.4" x 8.3"

Customer Reviews

Comprehensive and clear

Stewart and Tall have written some excellent mathematical texts, and this book is one of them. It deals with the basics of mathematics and treats them in a clear but thorough way. Although it is probably not essential, it is a good accompaniment to anybody starting a mathematics degree. They take the approach that one should already be familiar with an idea (via examples) before seeing its definition. This works well, although sometimes the effect is that the pace is slow and the exposition is drawn out.

Excellent Book for the High School Mathsters

This book was written for those inquisitive/advanced high school students (6th formers in England) to give them an introduction to the number systems, formalization in mathematics and the concept of proof. Stewart is well-known for his clear writing and this book is an excellent example.