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Philosophy of Numbers

Philosophy of Numbers
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Item Code: IDI811
Author: Jayant Burde
Publisher: Munshiram Manoharlal Publishers Pvt. Ltd.
Language: English
Edition: 2007
ISBN: 8121511763
Pages: 262
Cover: Hardcover
Other Details: 8.8" X 5.6
weight of book: 467 gms

From The Jacket

This book is about numbers and so many questions relating to them. What is the nature of numbers are they discovered or invented? What is mystical about them? Mathematicians develop a hierarchy of numbers in which mysterious dichotomies appear. For example, the integer 5 is not the same as the rational 5 which in turn is different from the real 5. The author explains how this conceptual maze does not affect the laypersons' arithmetic. He also discusses such fascinating topics as primes, perfect numbers, inaccessible numbers and many other unsolved problems relating to the treacherous terrain of infinity, which have baffled mathematicians and philosophers alike.

Jayant Burde has academic/professional qualifications in mathematics, physics, haw and banking. His published papers contain mathematical models in finance and organizational structure. He is also the author of the book Rituals, Mantras and Science.

Preface

This book is about numbers. It tries to answer many questions relating to them. What is the nature of number? Are they discovered or invented? Are they eternal or immortal? What are finite and infinite numbers? What is mystical about them?

It shows, how starting with a collection of objects a hierarchy of numbers-the natural numbers, integers, rationals, irrationals, real numbers and complex numbers can be constructed. It also discusses such topics as primes, perfect numbers, infinite numbers, cryptopgraphy, Fibonacci numbers, the Fermat's last theorem and also many unsolved problems relating to numbers.

The only prerequisite for understanding this book is the knowledge of high school mathematics and a desire to know more about numbers. All mathematical concepts which the common reader is not likely to be familiar are explained at the appropriate stages. Since this is not a treatise on mathematics most theorems are stated without proof.

Most of these results can be appreciated intuitively except those which baffle even mathematicians. I hope, the many illustrations which I have included will enable the reader to have an insight which a rigorous formal treatment cannot provide.

Though meant for a general audience, the students of mathematics and philosophy should also find the book interesting. Bertrand Russell often reprimanded philosophers for not learning sufficient mathematics. A similar criticism can be leveled against mathematicians who feel that philosophy is either irrelevant or far beneath their intellectual level.

With their superior logical mind mathematicians can create countless systems which are self-consistent. Most of these systems are, however, drab unless they have a philosophical dimensions. If mathematicians care to develop a philosophical approach to their problems, they will find a new depth which will enable them to unify and solve many problems which appear disparate as well as intractable. What's more, they will find, to their surprise, tha philosophy can provide a link between their abstruse world of symbols and the world in which common people live.

There are a few books on the philosophy of mathematics, but I have not come across a treatise which exclusively deals with numbers. I hope this book will fill this void.

 

  Preface ix
  Acknowledgements xi
1 The Numbers We Know 1
2 The Natural Numbers-The Serial Concept 10
3 Sets and Classes 14
4 The Algebra of Sets 21
5 Relations 27
6 The Natural Numbers and Logic 35
7 Mapping 42
8 The Natural Numbers and Mapping 50
9 Abstract Algebra 56
10 Order 66
11 Finite and Infinite Numbers 72
12 Relation Number and Structure 79
13 The Integers and Rationals as Relations 83
14 The Integers as Classes 87
15 Rational Numbers as Classes 94
16 Real Numbers-Dedekind's Theory 98
17 Sequences 103
18 The Real Numbers 110
19 The Real Numbers and Infinite Continuum 119
20 The Complex Numbers 128
21 Infinite Cardinals and Ordinals 135
22 The Elusive Primes 143
23 Numbers and Secret Communications 152
24 The Mystical Perfect Numbers 163
25 The Fermat's Last Theorem 171
26 Some Intriguing Numbers 180
27 The Grammar of Numbers 187
28 The Treacherous Infinity 199
29 Wrestling with Numbers 207
30 Numbers: Reality and Mysticism 222
31 The Nature of Numbers 233
  Bibliography 247
  Index 249

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