Agnes Scott College

Edna E. Kramer

The Nature and Growth of Modern Mathematics
Hawthorn Books, Inc., 1970

Cover Page


The Nature and Growth of Modern Mathematics races the development of the most important mathematical concepts from their inception to their present formulation. Although chief emphasis is placed on the explanation of mathematical ideas, nevertheless mathematical content, history, lore, and biography are integrated in order to offer an overall, unified picture of the mother science. The work presents a discussion of major notions and the general settings in which they were conceived, with particular attention to the lives and thoughts of some of the most creative mathematical innovators. It provides a guide to what is still important in classical mathematics, as well as an introduction to many significant recent developments.

Answers to questions like the following are simple and will be found in this book:

Proceeding from illustrative instances to general purposes, let us state that the author's objectives are:

1. To survey the entire field of mathematics, with emphasis on twentieth-century ideas.

2. To furnish the type of exposition that should make it possible for a layman who is educated by not a specialist in mathematics to gain insight into the manifold aspects of modern mathematics, including its essential relationship to all areas of scientific thought. This objective was formulated because the "new" mathematics which has become the vogue in our schools is not really new, and those who week popular treatment of contemporary mathematics can find only occasional superficial articles in periodicals. Books offering fuller exposition, including the author's Main Stream of Mathematics, have generally terminated with material from the early decades of the present century. In the potentially democratic world which men of good will envision, the man in the street must be entitled to more mathematical stimulation than the puzzle column in a Sunday newspaper, an occasional profile of a Nobel prize winner, an enigmatic summary of some recent discovery in applied mathematics—whether that man is an engineer in a remote village in India who is seeking to fill loopholes in his mathematical knowledge, a retired physician anxious to convince himself he is a mathematician manqué, a high school senior inquest of a research topic for a science talent contest, a nun whose objective is to inspire her students with an account of the accomplishment of women mathematicians, or a stockbroker eager to indulge in some pure mathematics. (Such are some of the individuals who have corresponded with the author in re mathematical information.)

3. To stress "human interest" and thereby to reveal mathematics as a living, growing endeavor, holding a strong place in man's culture. This aim was conceived because there is danger to the humanities in the present educational crash programs designed to produce a large number of mathematicians, physical scientists, engineers, and technical workers. Our times make such programs a necessity, and the leaders who suggest the concentrated curricula or write the texts are rendering an inestimable service. Although these men understand the value of the pure mathematical content of the courses of study they prescribe, can the same be said of the students or of the teachers in elementary and secondary schools, or of the general public? They must have the opportunity to realize that there is more purpose to the "new" mathematics than recounting the tale of Little Red Riding Hood in terms of set theory or computer language. Thus, part of the third objective of the present work is to supply material which can serve as a cultural background or supplement for all those who are receiving rapid, concentrated exposure to recent advanced mathematical concepts, without any opportunity to examine the origins or gradual historical development of such idea. Hence, although designed for the layman, this book would be helpful in courses in the history, philosophy, or fundamental concepts of mathematics.

Table of Contents

  1. From Babylonian Beginnings to Digital Computers
  2. Mathematical Method and Main Streams Are Launched
  3. Mathematical Reasoning from Eudoxus to Lobachevsky
  4. Algebra from Hypatia to Hamilton
  5. Equations, Human and Inhuman
  6. A Universal Language
  7. Forefathers of Modern Mathematics and Their Legacy
  8. A Calculus for Heaven and Earth
  9. Determinism and Its Creators
  10. The Elements of Strategy in War and Peace
  11. Probabilistic Models, Great Expectations, and Randomized Strategies
  12. General Games and Statistical Decision Theory
  13. From Dice to Quantum Theory and Quality Control
  14. Realm of Random Variables
  15. Demons, Energy, Maxwell, and Gibbs
  16. Sweet Manuscript of Youth
  17. The Unification of Geometry
  18. A Special Group and Its Application
  19. Geometry for Universe-Builders
  20. Post-Relativity Geometry
  21. East Meets West in the Higher Arithmetic
  22. The Reformation of Analysis
  23. Royal Roads to Functional Analysis
  24. Infinite Hierarchy
  25. Angelic Geometry
  26. The Leonardos of Modern Mathematics
  27. Twentieth-Century Vistas—Analysis
  28. Twentieth-Century Vistas—Algebra
  29. Twentieth-Century Vistas—Logic and Foundations
  30. Retrospect and Prospect