You are here

Mathematics: The Man-Made Universe: An Introduction to the Spirit of Mathematics

Sherman K. Stein
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Underwood Dudley
, on

Sherman K. Stein received his Ph.D. from Columbia University in 1953 and spent almost all his academic career at the University of California, Davis, where he attained emeritus status in 1993. He has written several books, texts and otherwise; this is his first, originally published in 1963.

It may seem to be a bit late to be getting around to reviewing a book approaching its fiftieth year of life, but it may be worthwhile to take a look at a book that has gone through three editions with its original publisher, W. H. Freeman & Co., growing all the while (1963, 316 pp.; 1969, 415 pp.; 1976, 573 pp.), had a reissue by McGraw-Hill in 1994, and two printings, 1999 and 2010, with Dover Publications. The page count is now 575. Few books, mathematical or not, survive so long.

It’s a text for a course for students outside the mathematical mainstream. Such courses have their ups and downs, and the 60s were one of the up times. Many schools offered courses with titles like “Mathematics for the Liberal Arts” and quite a few texts were available for them. My school had one of the courses and used a text by Richman, Walker, and Wisner that is also still in print; I taught the course a few times and it went over fairly well. In the 70s, it disappeared, either because requirements for students changed or the department thought that its time could be better spent elsewhere, I forget which. Thus do ups and downs go: up, and then down.

As such texts do, it considers a variety of topics, related only by being mathematical, interesting, and within the grasp of students knowing little mathematics, as one can see from the table of contents.

The writing is clear and its pace is appropriate. There are many exercises, their number at the end of chapters 2, 3, and 4 being 57, 66, and 71 (including a picture of a slide rule), many of which have answers at the back of the book. The text’s emphasis is always on the mathematics. Problem 11 in chapter 2 starts “Prove that” and there are many calls for proofs throughout. That is the way things were in the 60s. I think it was a good way, but tastes change. I took a quick look around the internet for the contents of current liberal-arts mathematics courses and found quite a few including topics like “financial mathematics” or “consumer mathematics”. It’s also a good thing to know about interest rates, but a choice has to be made.

The book’s age shows in some of its topics. Dissecting a square or a rectangle into different-sized subsquares or subrectangles was more fashionable then than it is now, and I expect that you could think of a few topics that have arisen since 1963 that could replace the two chapters they take up. Even if you can’t there is plenty of other material. Or, you could always do more problems. There are plenty to go around. has a review of the book by a student, who gives it a lowly two stars. Even so, he or she says “If I were reading this book for my own enjoyment or in conjunction with a more formal text then it is an interesting and informative read ... even enjoyable at times.” Even enjoyable! High praise indeed. If you are looking for a text for a course for which this could be a text, give it consideration. Texts don’t last fifty years for nothing.

Woody Dudley once wrote a text and is proud that it is still going, though it’s a comparative youngster at the age of 42.

Map; Guide; Preface
1. Questions on weighing
Weighing with a two-pan balance and two measures—Problems raised—Their algebraic phrasing
2. The primes
The Greek prime-manufacturing machine—Gaps between primes—Average gap and 1/1 + 1/2 + 1/3 + . . . + 1/N—Twin primes
3. The Fundamental Theorem of Arithmetic
Special natural numbers—Every special number is prime—"Unique factorization" and "every prime is special" compared—Euclidean algorithm—Every prime number is special—The concealed theorem
4. Rationals and Irrationals
The Pythagorean Theorem-—he square root of 2—Natural numbers whose square root is irrational—Rational numbers and repeating decimals
5. Tiling
The rationals and tiling a rectangle with equal squares—Tiles of various shapes—use of algebra—Filling a box with cubes
6. Tiling and electricity
Current—The role of the rationals—Applications to tiling—Isomorphic structures
7. The highway inspector and the salesman
A problem in topology—Routes passing once over each section of highway—Routes passing once through each town
8. Memory Wheels
A problem raised by an ancient word—Overlapping n-tuplets—Solution—History and applications
9. The Representation of numbers
Representing natural numbers—The decimal system (base ten)—Base two—Base three—Representing numbers between 0 and 1—Arithmetic in base three—The Egyptian system—The decimal system and the metric system
10. Congruence
Two integers congruent modulo a natural number—Relation to earlier chapters—Congruence and remainders—Properties of congruence—Casting out nines—Theorems for later use
11. Strange algebras
Miniature algebras—Tables satisfying rules—Commutative and idempotent tables—Associativity and parentheses—Groups
12. Orthogonal tables
Problem of the 36 officers—Some experiments—A conjecture generalized—Its fate—Tournaments—Application to magic squares
13. Chance
Probability—Dice—The multiplication rule—The addition rule—The subtraction rule—Roulette—Expectation—Odds—Baseball—Risk in making decisions
14. The fifteen puzzle
The fifteen puzzle—A problem in switching cords—Even and odd arrangements—Explanation of the Fifteen puzzle—Clockwise and counterclockwise
15. Map coloring
The two-color theorem—Two three-color theorems—The five-color theorem—The four-color conjecture
16. Types of numbers
Equations—Roots—Arithmetic of polynomials—Algebraic and transcendental numbers—Root r and factor X—r—Complex numbers—Complex numbers applied to alternating current—The limits of number systems
17. Construction by straightedge and compass
Bisection of line segment-Bisection of angle-Trisection of line segment—Trisection of 90° angle—Construction of regular pentagon—Impossibility of constructing regular 9-gon and trisecting 60°
18. Infinite sets
A conversation from the year 1638—Sets and one-to-one correspondence—Contrast of the finite with the infinite—Three letters of Cantor—Cantor's Theorem—Existence of transcendentals
19. A general view
The branches of mathematics—Topology and set theory as geometries—The four "shadow" geometries—Combinatorics—Algebra—Analysis—Probability—Types of proof—Cohen's theorem—Truth and proof—Gödel's theorem
Appendix A. Review of arithmetic
A quick tour of the basic ideas of arithmetic
Appendix B. Writing mathematics
Some words of advice and caution
Appendix C. The rudiments of algebra
A review of algebra, which is reduced to eleven rules
Appendix D. Teaching mathematics
Suggestions to prospective and practicing teachers
Appendix E. The geometric and harmonic series
Their properties—Applications of geometric series to probability
Appendix F. Space of any dimension
Definition of space of any dimension
Appendix G. Update
Answers and comments for selected exercises