Mathematics in 10 Lessons: The Grand Tour - Softcover

King, Jerry

Softcover

ISBN 10: 1591026865 ISBN 13: 9781591026860

Verlag: G38-Prometheus, 2009

Inhaltsangabe

Many of us trained mainly in the humanities and liberal arts may respect mathematics as an essential scientific discipline, but have done very little mathematics and often feel intimidated by its rigors. If you’ve ever wondered what mathematicians mean by the aesthetic elegance of their subject, here is your chance to experience firsthand mathematics’ intellectual pleasures.
Martin Gardner, in his review of Jerry King’s The Art of Mathematics, praised King:

"Creative mathematicians seldom write for outsiders, but when they do, they usually do it well. Jerry King, a professor at Lehigh University, is no exception."

For his new book, Jerry P. King has designed a grand tour of mathematics in ten essential lessons for the general reader who wants to know how mathematics is done. Almost no prior mathematical knowledge is assumed and through lively exposition and lucid explanations real mathematics is made not only palatable, but even enjoyable to the uninitiated.

Professor King begins by establishing two key points: first, all mathematics flows from a few fundamental principles. Second, aesthetic considerations provide both the motivation for mathematics research and the standards for evaluating that research. The book is structured so that the reader gradually builds up an ever-greater skill set as each lesson is mastered.

The essential concepts introduced include symbolic logic, infinity, rational numbers, number theory, real and imaginary numbers, function, probability, calculus, and the building of mathematical models in applied mathematics. Throughout his exposition, King provides brief historical digressions, which highlight key developments made by the giants in the field of mathematics.

Eloquently written and clearly presented, Mathematics in 10 Lessons will inspire the reader to go on to learn more and will instill a true appreciation for mathematics as both an art and a science.

Über diesen Titel

Über die Autorin bzw. den Autor

Jerry P. King, PhD (Bethlehem, PA), is professor emeritus of mathematics and dean emeritus of the graduate school at Lehigh University, where he spent forty-five years teaching mathematics and was awarded the university’s two most prestigious teaching awards. He is the author of the critically acclaimed The Art of Mathematics and numerous professional papers and reviews.

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MATHEMATICS in 10 LESSONS

The Grand TourBy JERRY P. KING

Prometheus Books

Copyright © 2009 Jerry P. King
All right reserved.
ISBN: 978-1-59102-686-0

Contents

ACKNOWLEDGMENTS...............................................7HOW TO READ THIS BOOK.........................................9INTRODUCTION..................................................11Chapter 1. Truth and Beauty...................................19Chapter 2. One, Two, Three, ..., Infinity.....................51Chapter 3. Beyond Counting....................................81Chapter 4. Number Theory......................................137Chapter 5. Numbers Real and Imaginary.........................187Chapter 6. Number Machines....................................219Chapter 7. Probability........................................241Chapter 8. Calculus...........................................293Chapter 9. Patterns and Paradox...............................351Chapter 10. Summing Up........................................369NOTES.........................................................381KEY TO SYMBOLS................................................385INDEX.........................................................391

Chapter One

TRUTH AND BEAUTY

Like Macbeth's air-drawn dagger, mathematics lives in the mind. Mathematical objects-things like numbers, equations, matrices-are abstract and imagined. They do not belong to the real world. Certainly, mathematical symbols can be written on pages and printed in books that are indeed real-world objects. But the subject itself comes from somewhere deep in the mind. Mathematics is made of pure thought, as are air-drawn daggers. It is not made of ink. Mathematicians make it out of airy nothing. And, as poets do, they give it form:

... as imagination bodies forth The forms of things unknown, the poet's pen Turns them to shapes and gives to airy nothing A local habitation and a name.

As I write I have an illustration before me-perhaps by N. C. Wyeth-of a unicorn standing with his spiraled horn in the air and pale moonlight on his silver flanks. Beyond him lies water and stone towers that shine above the trees like Camelot. It is a lovely picture and in it the silver creature seems truly fabled. But it is only a picture. Yet it is as close as any of us will come to unicorns.

Go look. Search the dark forest. Stand still as a tree. Wait beside the bright pool until the moon wanes and you turn cold as stone. You will see no unicorns. Unicorns do not live on this earth. Neither does mathematics.

As you search, you may find a ledger left behind in some abandoned campsite. On the first page you may see faded mathematical symbols including, let's say, the number 6. But you cannot conclude from this that "sixes" exist in the real world. To do so would be analogous to concluding that unicorns exist because N. C. Wyeth once drew a picture of one of them. What you have found, on the old ledger, is a picture of "the number six"-a symbol that represents a mathematical idea. All those who know elementary mathematics share this idea. It connotes to them other commonly shared ideas.

For example, they know that "six" is the name of the natural number (natural numbers are the numbers: 1, 2, 3, 4, 5, 6, 7, ...) that follows five and precedes seven. Also, they are aware of certain arithmetical properties of 6 such that 6 = 1 + 2 + 3, and 6 = 2 3. (Here the "dot" represents multiplication. This is one of the symbols we will use for this operation. Nowhere in this book do we indicate multiplication with the "X" which is commonly used in elementary school. "X" looks too much like "x," which we will use for something else.)

Thus, the number six is an idea that stands apart from the world of reality, as do unicorns or daggers made of air. Moreover, we can extend this discussion from the number six to any mathematical notion. Each mathematical notion-no matter whether it is as simple as that of a positive integer or as complicated as a topological space-is an abstraction and lives in the world of ideas.

As we proceed, it will become increasingly important to formalize this distinction. Think of the two worlds-the world of reality and the world of mathematics-as existing side by side. They can be represented schematically in various ways. Figure 1 shows each world as a simple rectangle, the real world on the left and the mathematical world on the right. Real-world objects-the things that live there-are just what you think: poems and people, palaces and plasterboard, all those things you can see or touch. Real things.

The other, the mathematical world, contains only ideas. But they are special ideas and they have names like number or function or inequality. These are mathematical ideas and they have properties that they inherit from simpler mathematical notions and they can be manipulated and extended-according to the rules of mathematics-to produce more complicated ideas. Part of our purpose in this book is to come to grips with the mathematical world and to understand some of the fundamental ideas that live in it. This will come as our story unfolds. What we want to understand now is the existence of the mathematical world and its separateness from the world of reality.

It is in this sense of separateness, of abstraction, that mathematics is similar to Wyeth's unicorn or to Macbeth's dagger of the mind. All are mere constructs, composed of nothing but pieces of pure thought. But there is a significant sense in which mathematics differs from the others. The dagger comes from the heat-oppressed brain of a man who would be king and who stands on the bloody edge of murder. The unicorn derives from Dark Age myth and from the tales of magic told on long nights around leaping fires. Each has its place in literature and in the unfolding of civilization. But neither is necessary. The world would be pretty much as it is if neither the dagger nor the unicorn had been conceived-not as rich a world, but still the world. Malory and Shakespeare are important to me and I prefer a world containing all their ideas. I do not like to imagine a world without the concepts of unicorn or that of Macbeth's dagger. But I can imagine such a world. And I know it would be not much changed.

On the other hand, it is inconceivable to imagine a world without mathematics. (Without mathematics there would exist no science beyond mere description and categorization. The world would be absent the great predictive and explanatory powers provided by mathematics. No part of such a world could be free of opinion, of dogma, or of wishful thinking.) The two worlds of figure 1 are fundamentally entwined like the branches of adjacent vines. They are essential for one another. The real world gives us people who create mathematics. The mathematical world gives us truth.

CREATION

A lot has been slipped into the final two sentences of the above. I've mentioned the two philosophically fundamental concepts of creation and truth. We need to take a closer look at each of them.

As we have seen, the people who pursue research in mathematics are called mathematicians. And when they engage in research they refer to the process as "doing mathematics." To "do mathematics" means to set down on paper new mathematics: mathematics that has heretofore not existed. Ordinarily, this new mathematics becomes the content of a research...

Details zur Ausgabe

• Verlag: G38-Prometheus
• Erscheinungsdatum: 2009
• Sprache: Englisch
• ISBN 10: 1591026865
• ISBN 13: 9781591026860
• Einband: Taschenbuch
• Anzahl der Seiten: 394
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Simon and Schuster Netherlands BV; info@simonandschuster.nl; ; Herculesplein 96; Utrecht; 3584 AA; Niederlande