TY - BOOK AU - Stillwell,John AU - Stillwell,John TI - The Story of Proof: Logic and the History of Mathematics SN - 9780691234373 PY - 2022///] CY - Princeton, NJ PB - Princeton University Press, KW - SCIENCE / History KW - bisacsh KW - MATHEMATICS / History & Philosophy KW - MATHEMATICS / Logic KW - Theorem KW - Axiom KW - Natural number KW - Computation KW - Geometry KW - Real number KW - Mathematics KW - Peano axioms KW - Predicate logic KW - Summation KW - Equation KW - Rule of inference KW - Well-order KW - Pythagorean theorem KW - Proof theory KW - Subset KW - Continuous function (set theory) KW - Gentzen's consistency proof KW - Zorn's lemma KW - Truth value KW - Computable function KW - Direct proof KW - Algorithm KW - Axiom of choice KW - Set theory KW - Turing machine KW - Determinant KW - Mathematical induction KW - Prime number KW - Special case KW - Playfair's axiom KW - Countable set KW - Extreme value theorem KW - Rational number KW - Credential KW - Addition KW - Mathematician KW - Fundamental theorem KW - Quaternion KW - Desargues's theorem KW - Permutation KW - Number theory KW - Commutative property KW - Intuitionism KW - Inference KW - Infimum and supremum KW - Self-reference KW - Prime factor KW - Calculation KW - Analogy KW - Analysis KW - Associative property KW - Recursively enumerable set KW - Dedekind cut KW - Hypothesis KW - Prediction KW - Logical connective KW - Intermediate value theorem KW - Aleph number KW - Total order KW - Constructive analysis KW - Reason KW - Infinitesimal KW - Identifiability KW - Power set KW - Hypotenuse KW - Logic KW - Proof by infinite descent KW - Satisfiability KW - Quantity N1 - Frontmatter --; 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; restricted access N2 - How the concept of proof has enabled the creation of mathematical knowledgeThe 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 UR - https://www.degruyterbrill.com/isbn/9780691234373 UR - https://www.degruyterbrill.com/document/cover/isbn/9780691234373/original ER -