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The Foundations of Mathematics, by Ian Stewart, David Tall
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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 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.
- Sales Rank: #77292 in Books
- Published on: 2015-05-01
- Released on: 2015-05-01
- Original language: English
- Number of items: 1
- Dimensions: 5.40" h x 1.00" w x 8.40" l, 1.30 pounds
- Binding: Paperback
- 432 pages
Review
"In their writing of Foundations of Mathematics, the authors motivate, encourage and teach their readers. If I were to offer a future course in foundations, I would certainly use this book and highly recommend it to others, both as a reference and as a text." --Frank Swetz, MAA Reviews
"The writing is both rigorous and thorough, and the authors use compact presentations to support their explanations and proofs. Highly recommended." --CHOICE
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).
Most helpful customer reviews
12 of 13 people found the following review helpful.
Comprehensive and clear
By James Woodward
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.
6 of 6 people found the following review helpful.
Excellent Book for the High School Mathsters
By Dr. Charles Saunders
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.
3 of 3 people found the following review helpful.
essential
By Laura
This book is a perfect introduction to the subject; the language is clear, direct and the content is self-explanatory. A great suport for beginners.
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