Hawthorn Books, Inc., 1970

Preface

*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:

- Why should Pythagoras and his followers be credited with(or blamed for) some of the methodology of the "new" mathematics?
- How do modern algebras (plural) generalize the "common garden variety"?
- What single modern concept makes it possible to conceive in a nutshell of
*all*geometries plain and fancy—Euclidean, non-Euclidean, affine, projective, inversive, etc.? - What is the nature of the universal language initiated by a thirteenth-century Catalan mystic, actually formulated by Leibniz, and improved by Boole and De Morgan?
- How did Omar Khayyám solve certain cubic equations?
- What are the common features of any boy's "engagement problem," the geishas' pantomime of baseball, and modern engineering decisions?
- Who are the "Leonardos" of modern mathematics?
- How did Queen Dido set a precedent for mathematicians and physicists?
- Why should isomorphism, homomorphism, and homeomorphism be an intrinsic part of the vocabulary of every mathophile?
- How did Maxwell's "demon" make the irreversible reversible?
- Why did the mere matter of counting socks lead a millionaire mathematical genius to renounce mathematics in favor of finance?
- What are the beautiful "ideals" formulated by Richard Dedekind and advanced by Emmy Noether?

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**

- From Babylonian Beginnings to Digital Computers
- Mathematical Method and Main Streams Are Launched
- Mathematical Reasoning from Eudoxus to Lobachevsky
- Algebra from Hypatia to Hamilton
- Equations, Human and Inhuman
- A Universal Language
- Forefathers of Modern Mathematics and Their Legacy
- A Calculus for Heaven and Earth
- Determinism and Its Creators
- The Elements of Strategy in War and Peace
- Probabilistic Models, Great Expectations, and Randomized Strategies
- General Games and Statistical Decision Theory
- From Dice to Quantum Theory and Quality Control
- Realm of Random Variables
- Demons, Energy, Maxwell, and Gibbs
- Sweet Manuscript of Youth
- The Unification of Geometry
- A Special Group and Its Application
- Geometry for Universe-Builders
- Post-Relativity Geometry
- East Meets West in the Higher Arithmetic
- The Reformation of Analysis
- Royal Roads to Functional Analysis
- Infinite Hierarchy
- Angelic Geometry
- The Leonardos of Modern Mathematics
- Twentieth-Century Vistas—Analysis
- Twentieth-Century Vistas—Algebra
- Twentieth-Century Vistas—Logic and Foundations
- Retrospect and Prospect