Proof And Knowledge In Mathematics

Distinguished contributors tackle the main problem that arizes when considering an epistemology for mathematics, the nature and sources of mathematical justification.

In the four decades since Imre Lakatos declared mathematics a "quasi-empirical science," increasing attention has been paid to the process of proof and argumentation in the field -- a development paralleled by the rise of computer technology and the mounting interest in the logical underpinnings of mathematics. Explanantion and Proof in Mathematics assembles perspectives from mathematics education and from the philosophy and history of mathematics to strengthen mutual awareness and share recent findings and advances in their interrelated fields. With examples ranging from the geometrists of the 17th century and ancient Chinese algorithms to cognitive psychology and current educational practice, contributors explore the role of refutation in generating proofs, the varied links between experiment and deduction, the use of diagrammatic thinking in addition to pure logic, and the uses of proof in mathematics education (including a critique of "authoritative" versus "authoritarian" teaching styles). A sampling of the coverage: The conjoint origins of proof and theoretical physics in ancient Greece. Proof as bearers of mathematical knowledge. Bridging knowing and proving in mathematical reasoning. The role of mathematics in long-term cognitive development of reasoning. Proof as experiment in the work of Wittgenstein. Relationships between mathematical proof, problem-solving, and explanation. Explanation and Proof in Mathematics is certain to attract a wide range of readers, including mathematicians, mathematics education professionals, researchers, students, and philosophers and historians of mathematics.

Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.

How to write mathematical proofs, shown in fully-worked out examples. This is a companion volume Joel Hamkins's Proof and the Art of Mathematics, providing fully worked-out solutions to all of the odd-numbered exercises as well as a few of the even-numbered exercises. In many cases, the solutions go beyond the exercise question itself to the natural extensions of the ideas, helping readers learn how to approach a mathematical investigation. As Hamkins asks, "Once you have solved a problem, why not push the ideas harder to see what further you can prove with them?" These solutions offer readers examples of how to write a mathematical proofs. The mathematical development of this text follows the main book, with the same chapter topics in the same order, and all theorem and exercise numbers in this text refer to the corresponding statements of the main text.

This volume examines appropriate axioms for mathematics to prove particular theorems in core areas.

Requiring only a basic background in plane geometry and elementary algebra, this classic poses 28 problems that introduce the fundamental ideas that make mathematics truly exciting. "Excellent . . . a thoroughly enjoyable sampler of fascinating mathematical problems and their solutions"—Science Magazine.

Focuses on essential knowledge for teachers about proof and the process of proving. It is organised around five big ideas, supported by multiple smaller, interconnected ideas-essential understandings. Taking you beyond a simple introduction to proof and the activities involved in proving, the book will broaden and deepen your mathematical understanding of one of the most challenging topics for students...and teachers.

Learn how to develop your reasoning skills and how to writewell-reasoned proofs Learning to Reason shows you how to use the basic elements ofmathematical language to develop highly sophisticated, logicalreasoning skills. You'll get clear, concise, easy-to-followinstructions on the process of writing proofs, including thenecessary reasoning techniques and syntax for constructingwell-written arguments. Through in-depth coverage of logic, sets,and relations, Learning to Reason offers a meaningful, integratedview of modern mathematics, cuts through confusing terms and ideas,and provides a much-needed bridge to advanced work in mathematicsas well as computer science. Original, inspiring, and designed formaximum comprehension, this remarkable book: * Clearly explains how to write compound sentences in equivalentforms and use them in valid arguments * Presents simple techniques on how to structure your thinking andwriting to form well-reasoned proofs * Reinforces these techniques through a survey of sets--thebuilding blocks of mathematics * Examines the fundamental types of relations, which is "where theaction is" in mathematics * Provides relevant examples and class-tested exercises designed tomaximize the learning experience * Includes a mind-building game/exercise space atwww.wiley.com/products/subject/mathematics/

The aim of this book is to help students write mathematics better. Throughout it are large exercise sets well-integrated with the text and varying appropriately from easy to hard. Basic issues are treated, and attention is given to small issues like not placing a mathematical symbol directly after a punctuation mark. And it provides many examples of what students should think and what they should write and how these two are often not the same.