The Foundations of Mathematics

Description

The transition from school 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. This book bridges the divide.

• 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 has been completely rewritten and significantly expanded.
• A brand new Part IV of the book on Using Axiomatic Systems contains three further new chapters on · Axiomatic Structures and Structure Theorems · Permutations and Groups · Infinitesimals

About the Author(s)

Ian Stewart, Emeritus Professor, University of Warwick, and 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
3:Sets
4:Relations
5:Functions
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
15:Infinitesimals
V: Strengthening the Foundations
16:Axioms for Set Theory

Reviews

"The writing is both rigorous and thorough, and the authors use compact presentations to support their explanations and proofs. Highly recommended." - N. W. Schillow, CHOICE