Introduction to Mathematical Thinking
内容简介
Many students encounter difficulty going from high school math to college-level mathematics. Even if they do well at math in school, most students are knocked off course for a while by the shift in emphasis from the K-12 focus on mastering procedures to the “mathematical thinking” characteristic of much university mathematics. Though the majority survive the transition, many do not. This short book is written to help them make that crucial shift.
Mathematicial thinking is not the same as “doing math” — unless you are a professional mathematician. For most people, “doing math” means the application of procedures and symbolic manipulations. Mathematicial thinking, in contrast, is what the name reflects, a specific way of thinking about things in the world that humans have developed over three thousand years. It does not have to be about mathematics at all, which means that many people can benefits from learning this powerful way of thinking, not just mathematicians and scientists.
You won't learn much new mathematics from this book. But by acquiring the ability to think like a mathematician, you will be able to master new mathematics quickly, whenever you need it.
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作者简介
Dr. Keith Devlin is a mathematician at Stanford University in California, where he is Executive Director of the university's H-STAR institute. He is a World Economic Forum Fellow and a Fellow of the American Association for the Advancement of Science. His current research is focused on the use of different media to teach and communicate mathematics to diverse audiences. He also works on the design of information/reasoning systems for intelligence analysis. Other research interests include: theory of information, models of reasoning, applications of mathematical techniques in the study of communication, and mathematical cognition. He has written 31 books and over 80 published research articles. His books have won a number of prizes, including the Pythagoras Prize, the Peano Prize, the Carl Sagan Award, and the Joint Policy Board for Mathematics Communications Award. In 2003, he was recognized by the California State Assembly for his "innovative work and longtime service in the field of mathematics and its relation to logic and linguistics." He is "the Math Guy" on National Public Radio.
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目录
Preface
What this book is about
1 What is mathematics?
1.1 More than arithmetic
1.2 Mathematical notation
1.3 Modern college-level mathematics
1.4 Why do you have to learn this stuff?
2 Getting precise about language
2.1 Mathematical statements
2.2 The logical combinators and, or, and not
2.3 Implication
2.4 Quantifiers
3 Proofs
3.1 What is a proof?
3.2 Proof by contradiction
3.3 Proving conditionals
3.4 Proving quantified statements
3.5 Induction proofs
4 Proving results about numbers
4.1 The integers
4.2 The real numbers
4.3 Completeness
4.4 Sequences
APPENDIX: Set theory
Index
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