Provides a comprehensive overview of the major turning points in the history of mathematics, from Ancient Greece to the present
Substantial reference lists offer suggestions for resources to learn more about the topics discussed
Problems and projects are included in each chapter to extend and increase understanding of the material for students
Ideal resource for students and teachers of the history of mathematics
This book explores some of the major turning points in the history of mathematics, ranging from ancient Greece to the present, demonstrating the drama that has often been a part of its evolution. Studying these breakthroughs, transitions, and revolutions, their stumbling-blocks and their triumphs, can help illuminate the importance of the history of mathematics for its teaching, learning, and appreciation.
Some of the turning points considered are the rise of the axiomatic method (most famously in Euclid), and the subsequent major changes in it (for example, by David Hilbert); the “wedding,” via analytic geometry, of algebra and geometry; the “taming” of the infinitely small and the infinitely large; the passages from algebra to algebras, from geometry to geometries, and from arithmetic to arithmetics; and the revolutions in the late nineteenth and early twentieth centuries that resulted from Georg Cantor’s creation of transfinite set theory. The origin of each turning point is discussed, along with the mathematicians involved and some of the mathematics that resulted. Problems and projects are included in each chapter to extend and increase understanding of the material. Substantial reference lists are also provided.
Turning Points in the History of Mathematics will be a valuable resource for teachers of, and students in, courses in mathematics or its history. The book should also be of interest to anyone with a background in mathematics who wishes to
learn more about the important moments in its development.
Topics
History of Mathematics
Mathematics Education
Mathematics in the Humanities and Social Sciences
Geometry
Algebra
Author(s): Hardy Grant, Israel Kleiner
Series: Compact Textbooks in Mathematics
Edition: 1
Publisher: Birkhäuser
Year: 2016
Language: English
Pages: C,IX,109
Tags: Математика;История математики;
Chapter 1 Axiomatics-Euclid’s and Hilbert’s: From Material to Formal
1.1 Euclid’s Elements
1.2 Hilbert’s Foundations of Geometry
1.3 The Modern Axiomatic Method
1.4 Ancient vs. Modern Axiomatics
References
Further Reading
Chapter 2 Solution by Radicals of the Cubic: From Equations to Groups and from Real to Complex Numbers
2.1 Introduction
2.2 Cubic and Quartic Equations
2.3 Beyond the Quartic: Lagrange
2.4 Ruffini, Abel, Galois
2.5 Complex Numbers: Birth
2.6 Growth
2.7 Maturity
References
Further Reading
Chapter 3 Analytic Geometry: From the Marriage of Two Fields to the Birth of a Third
3.1 Introduction
3.2 Descartes
3.3 Fermat
3.4 Descartes’ and Fermat’s Works from a Modern Perspective
3.5 The Significance of Analytic Geometry
References
Further Reading
Chapter 4 Probability: From Games of Chance to an Abstract Theory
4.1 The Pascal–Fermat Correspondence
4.2 Huygens: The First Book on Probability
4.3 Jakob Bernoulli’s Ars Conjectandi (The Art of Conjecturing)
4.4 De Moivre’s The Doctrine of Chances
4.5 Laplace’s Théorie Analytique des Probabilités
4.6 Philosophy of Probability
4.7 Probability as an Axiomatic Theory
4.8 Conclusion
References
Further Reading
Chapter 5 Calculus: From Tangents and Areas to Derivatives and Integrals
5.1 Introduction
5.2 Seventeenth-Century Predecessors of Newton and Leibniz
5.3 Newton and Leibniz: The Inventors of Calculus
5.4 The Eighteenth Century: Euler
5.5 A Look Ahead: Foundations
References
Further Reading
Chapter 6 Gaussian Integers: From Arithmetic to Arithmetics
6.1 Introduction
6.2 Ancient Times
6.3 Fermat
6.4 Euler and the Bachet Equation x2.+.2.=.y3
6.5 Reciprocity Laws, Fermat’s Last Theorem, Factorization of Ideals
6.6 Conclusion
References
Further Reading
Chapter 7 Noneuclidean Geometry: From One Geometry to Many
7.1 Introduction
7.2 Euclidean Geometry
7.3 Attempts to Prove the Fifth Postulate
7.4 The Discovery (Invention) of Noneuclidean Geometry
7.5 Some Implications of the Creation of Noneuclidean Geometry
References
Further Reading
Chapter 8 Hypercomplex Numbers: From Algebra to Algebras
8.1 Introduction
8.2 Hamilton and Complex Numbers
8.3 The Quaternions
8.4 Beyond the Quaternions
References
Further Reading
Chapter 9 The Infinite: From Potential to Actual
9.1 The Greeks
9.2 Before Cantor
9.3 Cantor
9.4 Paradoxes Lost
9.5 Denumerable (Countable) Infinity
9.6 Paradoxes Regained
9.7 Arithmetic
9.8 Two Major Problems
9.9 Conclusion
References
Further Reading
Chapter 10 Philosophy of Mathematics: From Hilbert to Gödel
10.1 Introduction
10.2 Logicism
10.3 Formalism
10.4 Gödel’s Incompleteness Theorems
10.5 Mathematics and Faith
10.6 Intuitionism
10.7 Nonconstructive Proofs
10.8 Conclusion
References
Further Reading
Chapter 11 Some Further Turning Points
11.1 Notation: From Rhetorical to Symbolic
11.2 Space Dimensions: From 3 to n (n..3)
11.3 Pathological Functions: From Calculus to Analysis
11.4 The Nature of Proof: From Axiom-Based to Computer-Assisted
11.5 Experimental Mathematics: From Humans to Machines
References
Further Reading
Index
