An Episodic History of Mathematics: Mathematical Culture Through Problem Solving

By Steven G. Krantz

Catalog Code: EHM
Print ISBN: 978-0-88385-766-3
Electronic ISBN: 978-1-61444-605-7
360 pp., Hardbound, 2010
List Price: $68.95 Member Price:$55.95
Series: MAA Textbooks

An Episodic History of Mathematics delivers a series of snapshots of mathematics and mathematicians from ancient times to the twentieth century. Giving readers a sense of mathematical culture and history, the book also acquaints readers with the nature and techniques of mathematics via exercises.

The book introduces the genesis of key mathematical concepts. For example, while Krantz does not get into the intricate mathematical details of Andrew Wiles's proof of Fermat's Last Theorem, he does describe some of the streams of thought that posed the problem and led to its solution. The focus in this text, moreover, is on doing’getting involved with the mathematics and solving problems. Every chapter ends with a detailed problem set that will provide students with avenues for exploration and entry into the subject.

• Recounts the history of mathematics.

• Offers broad coverage of the various schools of mathematical thought to give readers a wider understanding of mathematics.

• Includes exercises to help readers engage with the text and gain a deeper understanding of the material.

Preface
1. The Ancient Greeks and the Foundations of Mathematics
2. Zeno's Paradox and the Concept of Limit
3. The Mystical Mathematics of Hypatia
4. The Islamic World and the Development of Algebra
5. Cardano, Abel, Galois, and the Solving of Equations
6. René Descartes and the Idea of Coordinates
7. Pierre de Fermat and the Invention of Differential Calculus
8. The Great Isaac Newton
9. The Complex Numbers and the Fundamental Theorem of Algebra
10. Carl Friedrich Gauss: The Prince of Mathematics
11. Sophie Germain and the Attack on Fermat's Last Problem
12. Cauchy and the Foundations of Analysis
13. The Prime Numbers
14. Dirichlet and How to Count
15. Bernhard Riemann and the Geometry of Surfaces
16. Georg Cantor and the Orders of Infinity
17. The Number Systems
18. Henri Poincaré, Child Phenomenon
19. Sonya Kovalevskaya and the Mathematics of Mechanics
20. Emmy Noether and Algebra
21. Methods of Proof
22. Alan Turing and Cryptography
Bibliography
Index

Excerpt: Henri Poincaré, Child Phenomenon (p. 291-292)

It is not generally well known that Poincaré discovered special relativity just about the same time as Einstein. He gave a lecture on the subject at Washington University in St. Louis on the occasion of the 1904 World's Fair, a full year before Einstein's ideas appeared in print. In fact Poincaré and Einstein had a considerable rivalry in this matter, and they never acknowledged each other's work. Poincaré's ideas appear in a journal called The Monist, and they bear a remarkable similarity to textbook treatments of relativity that we see today.

Poincaré is arguably the father of topology (popularly known as "rubber sheet geometry") and also of the currently very active area of dynamical systems. He made decisive contributions to differential equations, to geometry, to complex analysis, and to many other central parts of mathematics.