Mathematics for the Nonmathematician

by Morris Kline

Overview

Practical, scientific, philosophical, and artistic problems have caused men to investigate mathematics. But there is one other motive which is as strong as any of these — the search for beauty. Mathematics is an art, and as such affords the pleasures which all the arts afford." In this erudite, entertaining college-level text, Morris Kline, Professor Emeritus of Mathematics at New York University, provides the liberal arts student with a detailed treatment of mathematics in a cultural and historical context. The book can also act as a self-study vehicle for advanced high school students and laymen.
Professor Kline begins with an overview, tracing the development of mathematics to the ancient Greeks, and following its evolution through the Middle Ages and the Renaissance to the present day. Subsequent chapters focus on specific subject areas, such as "Logic and Mathematics," "Number: The Fundamental Concept," "Parametric Equations and Curvilinear Motion," "The Differential Calculus," and "The Theory of Probability." Each of these sections offers a step-by-step explanation of concepts and then tests the student's understanding with exercises and problems. At the same time, these concepts are linked to pure and applied science, engineering, philosophy, the social sciences or even the arts.
In one section, Professor Kline discusses non-Euclidean geometry, ranking it with evolution as one of the "two concepts which have most profoundly revolutionized our intellectual development since the nineteenth century." His lucid treatment of this difficult subject starts in the 1800s with the pioneering work of Gauss, Lobachevsky, Bolyai and Riemann, and moves forward to the theory of relativity, explaining the mathematical, scientific and philosophical aspects of this pivotal breakthrough. Mathematics for the Nonmathematician exemplifies Morris Kline's rare ability to simplify complex subjects for the nonspecialist.

Product Details

ISBN-13:	9780486248233
Publisher:	Dover Publications
Publication date:	02/01/1985
Series:	Dover Books on Mathematics
Pages:	672
Sales rank:	186,755
Product dimensions:	5.50(w) x 8.50(h) x (d)

About the Author

Morris Kline: Mathematics for the Masses
Morris Kline (1908–1992) had a strong and forceful personality which he brought both to his position as Professor at New York University from 1952 until his retirement in 1975, and to his role as the driving force behind Dover's mathematics reprint program for even longer, from the 1950s until just a few years before his death. Professor Kline was the main reviewer of books in mathematics during those years, filling many file drawers with incisive, perceptive, and always handwritten comments and recommendations, pro or con. It was inevitable that he would imbue the Dover math program ― which he did so much to launch ― with his personal point of view that what mattered most was the quality of the books that were selected for reprinting and the point of view that stressed the importance of applications and the usefulness of mathematics. He urged that books should concentrate on demonstrating how mathematics could be used to solve problems in the real world, not solely for the creation of intellectual structures of theoretical interest to mathematicians only.

Morris Kline was the author or editor of more than a dozen books, including Mathematics in Western Culture (Oxford, 1953), Mathematics: The Loss of Certainty (Oxford, 1980), and Mathematics and the Search for Knowledge (Oxford, 1985). His Calculus, An Intuitive and Physical Approach, first published in 1967 and reprinted by Dover in 1998, remains a widely used text, especially by readers interested in taking on the sometimes daunting task of studying the subject on their own. His 1985 Dover book, Mathematics for the Nonmathematician could reasonably be regarded as the ultimate math for liberal arts text and may have reached more readers over its long life than any other similarly directed text.

In the Author's Own Words:
"Mathematics is the key to understanding and mastering our physical, social and biological worlds."

"Logic is the art of going wrong with confidence."

"Statistics: the mathematical theory of ignorance."

"A proof tells us where to concentrate our doubts." ― Morris Kline

Table of Contents

1 Why Mathematics?
2 A Historical Orientation
2-1 Introduction
2-2 Mathematics in early civilizations
2-3 The classical Greek period
2-4 The Alexandrian Greek period
2-5 The Hindus and Arabs
2-6 Early and medieval Europe
2-7 The Renaissance
2-8 Developments from 1550 to 1800
2-9 Developments from 1800 to the present
2-10 The human aspect of mathematics
3 Logic and Mathematics
3-1 Introduction
3-2 The concepts of mathematics
3-3 Idealization
3-4 Methods of reasoning
3-5 Mathematical proof
3-6 Axioms and definitions
3-7 The creation of mathematics
4 Number: the Fundamental Concept
4-1 Introduction
4-2 Whole numbers and fractions
4-3 Irrational numbers
4-4 Negative numbers
4-5 The axioms concerning numbers
* 4-6 Applications of the number system
5 "Algebra, the Higher Arithmetic"
5-1 Introduction
5-2 The language of algebra
5-3 Exponents
5-4 Algebraic transformations
5-5 Equations involving unknowns
5-6 The general second-degree equation
* 5-7 The history of equations of higher degree
6 The Nature and Uses of Euclidean Geometry
6-1 The beginnings of geometry
6-2 The content of Euclidean geometry
6-3 Some mundane uses of Euclidean geometry
* 6-4 Euclidean geometry and the study of light
6-5 Conic sections
* 6-6 Conic sections and light
* 6-7 The cultural influence of Euclidean geometry
7 Charting the Earth and Heavens
7-1 The Alexandrian world
7-2 Basic concepts of trigonometry
7-3 Some mundane uses of trigonometric ratios
* 7-4 Charting the earth
* 7-5 Charting the heavens
* 7-6 Further progress in the study of light
8 The Mathematical Order of Nature
8-1 The Greek concept of nature
8-2 Pre-Greek and Greek views of nature
8-3 Greek astronomical theories
8-4 The evidence for the mathematical design of nature
8-5 The destruction of the Greek world
* 9 The Awakening of Europe
9-1 The medieval civilization of Europe
9-2 Mathematics in the medieval period
9-3 Revolutionary influences in Europe
9-4 New doctrines of the Renaissance
9-5 The religious motivation in the study of nature
* 10 Mathematics and Painting in the Renaissance
10-1 Introduction
10-2 Gropings toward a scientific system of perspective
10-3 Realism leads to mathematics
10-4 The basic idea of mathematical perspective
10-5 Some mathematical theorems on perspective drawing
10-6 Renaissance paintings employing mathematical perspective
10-7 Other values of mathematical perspective
11 Projective Geometry
11-1 The problem suggested by projection and section
11-2 The work of Desargues
11-3 The work of Pascal
11-4 The principle of duality
11-5 The relationship between projective and Euclidean geometries
12 Coordinate Geometry
12-1 Descartes and Fermat
12-2 The need for new methods in geometry
12-3 The concepts of equation and curve
12-4 The parabola
12-5 Finding a curve from its equation
12-6 The ellipse
* 12-7 The equations of surfaces
* 12-8 Four-dimensional geometry
12-9 Summary
13 The Simplest Formulas in Action
13-1 Mastery of nature
13-2 The search for scientific method
13-3 The scientific method of Galileo
13-4 Functions and formulas
13-5 The formulas describing the motion of dropped objects
13-6 The formulas describing the motion of objects thrown downward
13-7 Formulas for the motion of bodies projected upward
14 Parametric Equations and Curvillinear Motion
14-1 Introduction
14-2 The concept of parametric equations
14-3 The motion of a projectile dropped from an airplane
14-4 The motion of projectiles launched by cannons
* 14-5 The motion of projectiles fired at an arbitrary angle
14-6 Summary
15 The Application of Formulas to Gravitation
15-1 The revolution in astronomy
15-2 The objections to a heliocentric theory
15-3 The arguments for the heliocentric theory
15-4 The problem of relating earthly and heavenly motions
15-5 A sketch of Newton's life
15-6 Newton's key idea
15-7 Mass and weight
15-8 The law of gravitation
15-9 Further discussion of mass and weight
15-10 Some deductions from the law of gravitation
* 15-11 The rotation of the earth
* 15-12 Gravitation and the Keplerian laws
* 15-13 Implications of the theory of gravitation
* 16 The Differential Calculus
16-1 Introduction
16-2 The problem leading to the calculus
16-3 The concept of instantaneous rate of change
16-4 The concept of instantaneous speed
16-5 The method of increments
16-6 The method of increments applied to general functions
16-7 The geometrical meaning of the derivative
16-8 The maximum and minimum values of functions
* 17 The Integral Calculus
17-1 Differential and integral calculus compared
17-2 Finding the formula from the given rate of change
17-3 Applications to problems of motion
17-4 Areas obtained by integration
17-5 The calculation of work
17-6 The calculation of escape velocity
17-7 The integral as the limit of a sum
17-8 Some relevant history of the limit concept
17-9 The Age of Reason
18 Trigonometric Functions and Oscillatory Motion
18-1 Introduction
18-2 The motion of a bob on a spring
18-3 The sinusoidal functions
18-4 Acceleration in sinusoidal motion
18-5 The mathematical analysis of the motion of the bob
18-6 Summary
* 19 The Trigonometric Analysis of Musical Sounds
19-1 Introduction
19-2 The nature of simple sounds
19-3 The method of addition of ordinates
19-4 The analysis of complex sounds
19-5 Subjective properties of musical sounds
20 Non-Euclidean Geometries and Their Significance
20-1 Introduction
20-2 The historical background
20-3 The mathematical content of Gauss's non-Euclidean geometry
20-4 Riemann's non-Euclidean geometry
20-5 The applicability of non-Euclidean geometry
20-6 The applicability of non-Euclidean geometry under a new interpretation of line
20-7 Non-Euclidean geometry and the nature of mathematics
20-8 The implications of non-Euclidean geometry for other branches of our culture
21 Arithmetics and Their Algebras
21-1 Introduction
21-2 The applicability of the real number system
21-3 Baseball arithmetic
21-4 Modular arithmetics and their algebras
21-5 The algebra of sets
21-6 Mathematics and models
* 22 The Statistical Approach to the Social and Biological Sciences
22-1 Introduction
22-2 A brief historical review
22-3 Averages
22-4 Dispersion
22-5 The graph and normal curve
22-6 Fitting a formula to data
22-7 Correlation
22-8 Cautions concerning the uses of statistics
* 23 The Theory of Probability
23-1 Introduction
23-2 Probability for equally likely outcomes
23-3 Probability as relative frequency
23-4 Probability in continuous variation
23-5 Binomial distributions
23-6 The problems of sampling
24 The Nature and Values of Mathem
24-4 The aesthetic and intellectual values
24-5 Mathematics and rationalism
24-6 The limitations of mathematics
Table of Trigonometric Ratios
Answers to Selected and Review Exercises
Additional Answers and Solutions
Index