The Emergence of Number

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World Scientific, 1987 - Mathematics - 222 pages
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This book presents detailed studies of the development of three kinds of number. In the first part the development of the natural numbers from Stone-Age times right up to the present day is examined not only from the point of view of pure history but also taking into account archaeological, anthropological and linguistic evidence. The dramatic change caused by the introduction of logical theories of number in the 19th century is also treated and this part ends with a non-technical account of the very latest developments in the area of Gödel's theorem. The second part is concerned with the development of complex numbers and tries to answer the question as to why complex numbers were not introduced before the 16th century and then, by looking at the original materials, shows how they were introduced as a pragmatic device which was only subsequently shown to be theoretically justifiable. The third part concerns the real numbers and examines the distinction that the Greeks made between number and magnitude. It then traces the gradual development of a theory of real numbers up to the precise formulations in the nineteeth century. The importance of the Greek distinction between the number line and the geometric line is brought into sharp focus.This is an new edition of the book which first appeared privately published in 1980 and is now out of print. Substantial revisions have been made throughout the text, incorporating new material which has recently come to light and correcting a few relatively minor errors. The third part on real numbers has been very extensively revised and indeed the last chapter has been almost completely rewritten. Many revisions are the results of comments from earlier readers of the book.
 

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Contents

Preface to the First Edition via Prologue
1
Genesis 1 The start of the search
4
Palaeolithic counting
5
Is counting instinctive?
6
The abstraction of number
11
The development of counting
12
Concrete counting
14
A primitive system
17
Frege
51
Conclusion
55
Complex Numbers
56
Latency 1 The Babylonians
58
Euclid
60
Hindu Algebra
64
AlKhwanzmi
65
Diophantos
70

The smallest numbers
20
The limits of counting
22
Imprecise limits
25
Unending repetition
27
Historic Times 1 The Greeks
30
Nicomachus
34
The Dark Ages
36
Renaissance
39
The Seventeenth Century
43
Dedekind and Peano
46
Omar Khayyam
73
The Early Italians
76
Bologna
81
Revelation
88
Real Numbers
109
The Totality of Real Numbers
133
Epilogue
154
References
198
Index
216
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