The Story of Proof : Logic and the History of Mathematics

**How the concept of proof** **has enabled the creation of mathematical knowledge** __The Story of Proof__ investigates the evolution of the concept of proof—one of the most significant and defining features of mathematical thought—through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge. Stillwell begins with Euclid and his influence on the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as “infinitesimal algebra,” and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved. Shedding light on the workings of mathematics at its most fundamental levels, __The Story of Proof__ offers a compelling new perspective on the field’s power and progress. "The proof is one of the most significant and defining features of mathematical thought. Many of the most significant advances in mathematics have been advances in the concept of proof, and in many ways, the history of mathematics can be viewed as a history of the proof. Yet proof itself is not considered an interesting topic by many mathematicians. In the United States, proof is not deemed an essential part of mathematics education until the upper-undergraduate level. In The Story of Proof, John Stillwell argues that the proof is an important object of study in itself. He provides insight into the history, theory, and practice of proof in order to shed light on the way that mathematics works at its most fundamental levels. The book begins with the proof of one of mathematics' most foundational theorems, the Pythagorean Theorem. The book then proceeds to look at the role of proof in the various fields of mathematics, including geometry and number theory, algebra, algebraic geometry, and calculus. By examining the history of the proof, Stillwell sheds new light on age-old questions in mathematics, including how logic, computation, and abstraction are connected with one another and with the rest of mathematics"-- Provided by publisher Contents Preface CHAPTER 1 Before Euclid CHAPTER 2 Euclid CHAPTER 3 After Euclid CHAPTER 4 Algebra CHAPTER 5 Algebraic Geometry CHAPTER 6 Calculus CHAPTER 7 Number Theory CHAPTER 8 The Fundamental Theorem of Algebra CHAPTER 9 Non-Euclidean Geometry CHAPTER 10 Topology CHAPTER 11 Arithmetization CHAPTER 12 Set Theory CHAPTER 13 Axioms for Numbers, Geometry, and Sets CHAPTER 14 The Axiom of Choice CHAPTER 15 Logic and Computation CHAPTER 16 Incompleteness Bibliography Index