The Induction Book

Every mathematician and student of mathematics needs a familiarity with mathematical induction. This volume provides advanced undergraduates and graduate students with an introduction and a thorough exposure to these proof techniques. 2017 edition.

The fundamental burden of a theory of inductive inference is to determine which are the good inductive inferences or relations of inductive support and why it is that they are so. The traditional approach is modeled on that taken in accounts of deductive inference. It seeks universally applicable schemas or rules or a single formal device, such as the probability calculus. After millennia of halting efforts, none of these approaches has been unequivocally successful and debates between approaches persist. The Material Theory of Induction identifies the source of these enduring problems in the assumption taken at the outset: that inductive inference can be accommodated by a single formal account with universal applicability. Instead, it argues that that there is no single, universally applicable formal account. Rather, each domain has an inductive logic native to it.The content of that logic and where it can be applied are determined by the facts prevailing in that domain. Paying close attention to how inductive inference is conducted in science and copiously illustrated with real-world examples, The Material Theory of Induction will initiate a new tradition in the analysis of inductive inference.

When is it better to induce labour than to let a woman's body or baby decide the best time for birth? What are the pros and cons of waiting and of being induced? What about after the due date? When the baby is thought to be bigger than average? When the woman is older? If she had IVF? Or when her waters have broken earlier than usual? Induction of labour is an increasingly common recommendation and more and more women find themselves having to decide whether to let their body and baby go into labour spontaneously or agree to medical intervention. This book explains the process of induction of labour and shares information from research studies, debates and women's, midwives' and doctors' experiences to help women and families become more informed and make the decision that is right for them.

According to the great mathematician Paul Erdös, God maintains perfect mathematical proofs in The Book. This book presents the authors candidates for such "perfect proofs," those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems from number theory, geometry, analysis, combinatorics, and graph theory. As a result, this book will be fun reading for anyone with an interest in mathematics.

Handbook of Mathematical Induction: Theory and Applications shows how to find and write proofs via mathematical induction. This comprehensive book covers the theory, the structure of the written proof, all standard exercises, and hundreds of application examples from nearly every area of mathematics. In the first part of the book, the author discusses different inductive techniques, including well-ordered sets, basic mathematical induction, strong induction, double induction, infinite descent, downward induction, and several variants. He then introduces ordinals and cardinals, transfinite induction, the axiom of choice, Zorn's lemma, empirical induction, and fallacies and induction. He also explains how to write inductive proofs. The next part contains more than 750 exercises that highlight the levels of difficulty of an inductive proof, the variety of inductive techniques available, and the scope of results provable by mathematical induction. Each self-contained chapter in this section includes the necessary definitions, theory, and notation and covers a range of theorems and problems, from fundamental to very specialized. The final part presents either solutions or hints to the exercises. Slightly longer than what is found in most texts, these solutions provide complete details for every step of the problem-solving process.

This book serves as a very good resource and teaching material for anyone who wants to discover the beauty of Induction and its applications, from novice mathematicians to Olympiad-driven students and professors teaching undergraduate courses. The authors explore 10 different areas of mathematics, including topics that are not usually discussed in an Olympiad-oriented book on the subject. Induction is one of the most important techniques used in competitions and its applications permeate almost every area of mathematics.

Practical reasoning is not just a matter of determining how to get what you want, but of working out what to want in the first place. In Practical Induction Elijah Millgram argues that experience plays a central role in this process of deciding what is or is not important or worth pursuing. He takes aim at instrumentalism, a view predominant among philosophers today, which holds that the goals of practical reasoning are basic in the sense that they are given by desires that are not themselves the product of practical reasoning. The view Millgram defends is "practical induction," a method of reasoning from experience similar to theoretical induction. What are the practical observations that teach us what to want? Millgram suggests they are pleasant and unpleasant experiences on the basis of which we form practical judgments about particular cases. By generalizing from these judgments--that is, by practical induction--we rationally arrive at our views about what matters. Learning new priorities from experience is necessary if we are to function in a world of ever-changing circumstances. And we need to be able to learn both from our own and from others' experience. It is this, Millgram contends, that explains the cognitive importance of both our capacity for pain and pleasure and our capacity for love. Pleasure's role in cognition is not that of a goal but that of a guide. Love's role in cognition derives from its relation to our trusting the testimony of others about what does and does not matter and about what merits our desire. Itself a pleasure to read, this book is full of inventive arguments and conveys Millgram's bold thesis with elegance and force. It will alter the direction of current debates on practical reasoning.

Discusses the importance of training, supporting, and retaining new teachers, presents a step-by-step process for structuring an induction program, and features a list of replicable induction programs.