Turning Points in the History of Mathematics

✍ Scribed by Hardy Grant, Israel Kleiner

• Publisher: Birkhäuser
• Year: 2016
• Tongue: English
• Leaves: 112
• Series: Compact textbooks in mathematics
• Edition: 1st ed.
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-ix
Axiomatics—Euclid’s and Hilbert’s: From Material to Formal....Pages 1-8
Solution by Radicals of the Cubic: From Equations to Groups and from Real to Complex Numbers....Pages 9-18
Analytic Geometry: From the Marriage of Two Fields to the Birth of a Third....Pages 19-25
Probability: From Games of Chance to an Abstract Theory....Pages 27-35
Calculus: From Tangents and Areas to Derivatives and Integrals....Pages 37-47
Gaussian Integers: From Arithmetic to Arithmetics....Pages 49-56
Noneuclidean Geometry: From One Geometry to Many....Pages 57-66
Hypercomplex Numbers: From Algebra to Algebras....Pages 67-73
The Infinite: From Potential to Actual....Pages 75-83
Philosophy of Mathematics: From Hilbert to Gödel....Pages 85-93
Some Further Turning Points....Pages 95-103
Back Matter....Pages 105-109

✦ Subjects

Mathématiques -- Histoire;Mathematics -- History;Mathematics