This is a mathematical coming-of-age book, for students on the cusp, who are maturing into mathematicians, aspiring to communicate mathematical truths to other mathematicians in the currency of mathematics, which is: proof. This is a book for students who are learning—perhaps for the first time in a serious way—how to write a mathematical proof. I hope to show how a mathematician makes an argument establishing a mathematical truth.
Proofs tell us not only that a mathematical statement is true, but also why it is true, and they communicate this truth. The best proofs give us insight into the nature of mathematical reality. They lead us to those sublime yet elusive Aha! moments, a joyous experience for any mathematician, occurring when a previously opaque, confounding issue becomes transparent and our mathematical gaze suddenly penetrates completely through it, grasping it all in one take. So let us learn together how to write proofs well, producing clear and correct mathematical arguments that logically establish their conclusions, with whatever insight and elegance we can muster. We shall do so in the context of the diverse mathematical topics that I have gathered together here in this book for the purpose.

lgrsnf/Hamkins J. Proof and the Art of Mathematics 2020.pdf

lgli/Hamkins J. Proof and the Art of Mathematics 2020.pdf

Hamkins, Joel David

AAAI Press

United States, United States of America

Cambridge, Massachusetts, 2020

{"isbns":["0262539799","9780262539791"],"last_page":232,"publisher":"MIT Press"}

An introduction to writing proofs, presented through compelling mathematical statements with interesting elementary proofs.This book offers an introduction to the art and craft of proof writing. The author, a leading research mathematician, presents a series of engaging and compelling mathematical statements with interesting elementary proofs. These proofs capture a wide range of topics, including number theory, combinatorics, graph theory, the theory of games, geometry, infinity, order theory, and real analysis. The goal is to show students and aspiring mathematicians how to write proofs with elegance and precision. The book is organized around mathematically rich topics (rather than methods of proof), allowing students to learn to write proofs with material that is itself intrinsically interesting. Students will find the early chapters the easiest. Chapter 4 explains the method of mathematical induction, which is used in many arguments throughout the book. Later chapters offer chapter-length developments of major theorems, and the final chapters are more abstract. The book is generously illustrated; an extended chapter on proofs-without-words shows the power of figures and diagrams to communicate mathematical ideas—but also acknowledges the dangers of such an approach. Each chapter includes exercises, and sample answers are provided at the end of the book.

"This book offers an introduction to the art and craft of proof writing. The author, a leading research mathematician, presents a series of engaging and compelling mathematical statements with interesting elementary proofs. These proofs capture a wide range of topics, including number theory, combinatorics, graph theory, the theory of games, geometry, infinity, order theory, and real analysis. The goal is to show students and aspiring mathematicians how to write proofs with elegance and precision.The book is organized around mathematically rich topics (rather than methods of proof), allowing students to learn to write proofs with material that is itself intrinsically interesting. Students will find the early chapters the easiest. Chapter 4 explains the method of mathematical induction, which is used in many arguments throughout the book. Later chapters offer chapter-length developments of major theorems, and the final chapters are more abstract. The book is generously illustrated; an extended chapter on proofs-without-words shows the power of figures and diagrams to communicate mathematical ideas—but also acknowledges the dangers of such an approach. Each chapter includes exercises, and sample answers are provided at the end of the book."-- Résumé de l'éditeur

An introduction to writing proofs, presented through compelling mathematical statements with interesting elementary proofs.
This book offers an introduction to the art and craft of proof-writing. The author, a leading research mathematician, presents a series of engaging and compelling mathematical statements with interesting elementary proofs. These proofs capture a wide range of topics, including number theory, combinatorics, graph theory, the theory of games, geometry, infinity, order theory, and real analysis. The goal is to show students and aspiring mathematicians how to write proofs with elegance and precision.

A classical beginning -- Multiple proof -- Number theory -- Mathematical induction -- Discrete mathematics -- Proofs without words -- Theory of games -- Pick's theorem -- Lattice-point polygons -- Polygonal dissection -- Functions and relations -- Graph theory -- Infinity -- Order theory -- Real analysis

"A textbook for students who are learning how to write a mathematical proof, a validation of the truth of a mathematical statement"-- Provided by publisher

2021-10-24