Title
The Foundations of Mathematics,New
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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 firstyear 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 epsilondelta 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.
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- Q: How many pages does this book have? A: This book has four hundred thirty-two pages. It offers a comprehensive exploration of mathematics transitioning from school to university.
- Q: What is the size of the book? A: The book measures eight point four inches in length, one inch in width, and five point four inches in height. These dimensions make it a manageable size for reading.
- Q: What is the binding type of this book? A: This book is bound in paperback. This makes it lightweight and easy to carry around.
- Q: Who is the author of this book? A: The author of this book is Ian Stewart. He is well-known for his contributions to mathematics and educational literature.
- Q: What is the main focus of this book? A: The main focus is on the foundations of mathematics. It helps students transition from intuitive to formal mathematical methods.
- Q: Is this book suitable for beginners in mathematics? A: Yes, this book is suitable for beginners. It is designed to help first-year university students grasp foundational concepts.
- Q: How can this book help students with formal mathematics? A: This book provides insights into the motivations behind abstract mathematics. It suggests ways to understand formal ideas through familiar school experiences.
- Q: Does this book include practical examples? A: Yes, it includes practical examples and a new chapter on the theory of groups. This enhances understanding of abstract concepts.
- Q: What age group is this book intended for? A: This book is intended for university students, typically aged eighteen and above. It's aimed at those transitioning from high school to higher education.
- Q: How should I store this book? A: Store this book in a cool, dry place, upright. This helps maintain its condition and prevents damage to the binding.
- Q: Can I clean the pages of the book? A: No, you should not clean the pages of the book. To avoid damage, simply keep it away from moisture.
- Q: What if the book arrives damaged? A: If the book arrives damaged, you can initiate a return. Check the retailer's return policy for specific instructions.
- Q: Is there a warranty for this book? A: No, there is no warranty for this book. However, returns for damaged copies are generally allowed.
- Q: What if I have questions while reading this book? A: If you have questions while reading, consider researching additional resources or seeking help from a mathematics instructor.
- Q: Is this book aligned with current university curricula? A: Yes, this book is aligned with current university mathematics curricula. It addresses common gaps in understanding for students.
- Q: Does this book cover advanced mathematical concepts? A: Yes, the book covers advanced concepts such as infinite cardinal numbers and larger ordered fields. It bridges intuitive and formal mathematics.