Turning Points in the History of Mathematics

  • Hardy Grant
  • Israel Kleiner

Part of the Compact Textbooks in Mathematics book series (CTM)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Hardy Grant, Israel Kleiner
    Pages 27-35
  3. Hardy Grant, Israel Kleiner
    Pages 37-47
  4. Hardy Grant, Israel Kleiner
    Pages 49-56
  5. Hardy Grant, Israel Kleiner
    Pages 57-66
  6. Hardy Grant, Israel Kleiner
    Pages 67-73
  7. Hardy Grant, Israel Kleiner
    Pages 75-83
  8. Hardy Grant, Israel Kleiner
    Pages 85-93
  9. Hardy Grant, Israel Kleiner
    Pages 95-103
  10. Back Matter
    Pages 105-109

About this book


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.


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

Authors and affiliations

  • Hardy Grant
    • 1
  • Israel Kleiner
    • 2
  1. 1.York UniversityWoodlawnCanada
  2. 2.MathematicsYork UniversityTorontoCanada

Bibliographic information

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