The Story of Proof : Logic and the History of Mathematics /
Stillwell, John,
The Story of Proof : Logic and the History of Mathematics / John Stillwell, John Stillwell. - 1 online resource
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 http://purl.org/coar/access_right/c_16ec
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.
Mode of access: Internet via World Wide Web.
John Stillwell is emeritus professor of mathematics at the University of San Francisco. His many books include Elements of Mathematics and Reverse Mathematics (both Princeton).
In English.
9780691234373
10.1515/9780691234373 doi
SCIENCE / History.
MATHEMATICS / History & Philosophy.
MATHEMATICS / Logic.
Theorem Axiom Natural number Computation Geometry Real number Mathematics Peano axioms Predicate logic Summation Equation Rule of inference Well-order Pythagorean theorem Proof theory Subset Continuous function (set theory) Gentzen's consistency proof Zorn's lemma Truth value Computable function Direct proof Algorithm Axiom of choice Set theory Turing machine Determinant Mathematical induction Prime number Special case Playfair's axiom Countable set Extreme value theorem Rational number Credential Addition Mathematician Fundamental theorem Quaternion Desargues's theorem Permutation Number theory Commutative property Intuitionism Inference Infimum and supremum Self-reference Prime factor Calculation Analogy Analysis Associative property Recursively enumerable set Dedekind cut Hypothesis Prediction Logical connective Intermediate value theorem Aleph number Total order Constructive analysis Reason Infinitesimal Identifiability Power set Hypotenuse Logic Proof by infinite descent Satisfiability Quantity Theorem Axiom Natural number Computation Geometry Real number Mathematics Peano axioms Predicate logic Summation Equation Rule of inference Well-order Pythagorean theorem Proof theory Subset Continuous function (set theory) Gentzen's consistency proof Zorn's lemma Truth value Computable function Direct proof Algorithm Axiom of choice Set theory Turing machine Determinant Mathematical induction Prime number Special case Playfair's axiom Countable set Extreme value theorem Rational number Credential Addition Mathematician Fundamental theorem Quaternion Desargues's theorem Permutation Number theory Commutative property Intuitionism Inference Infimum and supremum Self-reference Prime factor Calculation Analogy Analysis Associative property Recursively enumerable set Dedekind cut Hypothesis Prediction Logical connective Intermediate value theorem Aleph number Total order Constructive analysis Reason Infinitesimal Identifiability Power set Hypotenuse Logic Proof by infinite descent Satisfiability Quantity
The Story of Proof : Logic and the History of Mathematics / John Stillwell, John Stillwell. - 1 online resource
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 http://purl.org/coar/access_right/c_16ec
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.
Mode of access: Internet via World Wide Web.
John Stillwell is emeritus professor of mathematics at the University of San Francisco. His many books include Elements of Mathematics and Reverse Mathematics (both Princeton).
In English.
9780691234373
10.1515/9780691234373 doi
SCIENCE / History.
MATHEMATICS / History & Philosophy.
MATHEMATICS / Logic.
Theorem Axiom Natural number Computation Geometry Real number Mathematics Peano axioms Predicate logic Summation Equation Rule of inference Well-order Pythagorean theorem Proof theory Subset Continuous function (set theory) Gentzen's consistency proof Zorn's lemma Truth value Computable function Direct proof Algorithm Axiom of choice Set theory Turing machine Determinant Mathematical induction Prime number Special case Playfair's axiom Countable set Extreme value theorem Rational number Credential Addition Mathematician Fundamental theorem Quaternion Desargues's theorem Permutation Number theory Commutative property Intuitionism Inference Infimum and supremum Self-reference Prime factor Calculation Analogy Analysis Associative property Recursively enumerable set Dedekind cut Hypothesis Prediction Logical connective Intermediate value theorem Aleph number Total order Constructive analysis Reason Infinitesimal Identifiability Power set Hypotenuse Logic Proof by infinite descent Satisfiability Quantity Theorem Axiom Natural number Computation Geometry Real number Mathematics Peano axioms Predicate logic Summation Equation Rule of inference Well-order Pythagorean theorem Proof theory Subset Continuous function (set theory) Gentzen's consistency proof Zorn's lemma Truth value Computable function Direct proof Algorithm Axiom of choice Set theory Turing machine Determinant Mathematical induction Prime number Special case Playfair's axiom Countable set Extreme value theorem Rational number Credential Addition Mathematician Fundamental theorem Quaternion Desargues's theorem Permutation Number theory Commutative property Intuitionism Inference Infimum and supremum Self-reference Prime factor Calculation Analogy Analysis Associative property Recursively enumerable set Dedekind cut Hypothesis Prediction Logical connective Intermediate value theorem Aleph number Total order Constructive analysis Reason Infinitesimal Identifiability Power set Hypotenuse Logic Proof by infinite descent Satisfiability Quantity