Turning Points in 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

• Axiomatics
• History of Calculus
• History of Mathematics
• Mathematics Education
• Non-Euclidean Geometry
• Philosohpy of Mathematics

“This slim volume will be especially helpful to those teaching high school and junior college students the rudiments of selected ‘major turning points’ in the history of mathematics. It offers compact vignettes divided into ten basic topics: axiomatization, solutions of cubic equations, analytical geometry, probability, calculus, Gaussian integers, non-Euclidean geometries, hyper complex numbers, the infinite, and philosophy of mathematics from Hilbert to Gödel. … Thumbnail portraits and photographs of major mathematicians are also included.” (Joseph W. Dauben, Mathematical Reviews,May, 2017)

“Turning Points does provide a useful summary and outline of at least a portion of the subject, and also functions nicely as a way of helping to mentally organize the material. It contains a number of good quotes, and a decent selection of bibliographic references at the end of each chapter. There are also problems at the end of each chapter, generally calling for essay-type answers that should require the student to do further reading.” (Mark Hunacek, MAA Reviews, maa.org, June, 2016)

“Each chapter contains some problems and projects (they extend and increase the understanding of the material) as well as references and suggestions of further readings. A comprehensive index has been supplemented. The book can serve everyone interested in the historical development of mathematics – some mathematical background is of course required. It can serve teachers and students, can be used in courses in the history of mathematics as well as in courses in particular domains of mathematics … .” (Roman Murawski, zbMATH 1342.01005, 2016)

• York University, Woodlawn, Canada

  Hardy Grant

• Mathematics, York University, Toronto, Canada

  Israel Kleiner

Hardy Grant was, before his retirement, Associate Professor in the Department of Mathematics and Statistics, York University, Toronto, Canada

Israel Kleiner is Professor Emeritus, the Department of Mathematics and Statistics, York University, Toronto, Canada

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