The Heritage of Thales

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Springer Science & Business Media, 14 Δεκ 1998 - 331 σελίδες
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This is intended as a textbook on the history, philosophy and foundations of mathematics, primarily for students specializing in mathematics, but we also wish to welcome interested students from the sciences, humanities and education. We have attempted to give approximately equal treatment to the three subjects: history, philosophy and mathematics. History We must emphasize that this is not a scholarly account of the history of mathematics, but rather an attempt to teach some good mathematics in a historical context. Since neither of the authors is a professional historian, we have made liberal use of secondary sources. We have tried to give ref cited facts and opinions. However, considering that this text erences for developed by repeated revisions from lecture notes of two courses given by one of us over a 25 year period, some attributions may have been lost. We could not resist retelling some amusing anecdotes, even when we suspect that they have no proven historical basis. As to the mathematicians listed in our account, we admit to being colour and gender blind; we have not attempted a balanced distribution of the mathematicians listed to meet today's standards of political correctness. Philosophy Both authors having wide philosophical interests, this text contains perhaps more philosophical asides than other books on the history of mathematics. For example, we discuss the relevance to mathematics of the pre-Socratic philosophers and of Plato, Aristotle, Leibniz and Russell. We also have vi Preface presented some original insights.

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Περιεχόμενα

Introduction
1
History and Philosophy of Mathematics
5
Egyptian Mathematics
7
Scales of Notation
11
Prime Numbers
15
SumerianBabylonian Mathematics
21
More about Mesopotamian Mathematics
25
The Dawn of Greek Mathematics
29
The Law of Quadratic Reciprocity
169
Foundations of Mathematics
173
The Number System
175
Natural Numbers Peanos Approach
179
The Integers
183
The Rationals
187
The Real Numbers
191
Complex Numbers
195

Pythagoras and His School
33
Perfect Numbers
37
Regular Polyhedra
41
The Crisis of Incommensurables
47
From Heraclitus to Democritus
53
Mathematics in Athens
59
Plato and Aristotle on Mathematics
67
Constructions with Ruler and Compass
71
The Impossibility of Solving the Classical Problems
79
Euclid
83
NonEuclidean Geometry and Hilberts Axioms
89
Alexandria from 300 BC to 200 BC
93
Archimedes
97
Alexandria from 200 BC to 500 AD
103
Mathematics in China and India
111
Mathematics in Islamic Countries
117
New Beginnings in Europe
121
Mathematics in the Renaissance
125
The Cubic and Quartic Equations
133
Renaissance Mathematics Continued
139
The Seventeenth Century in France
145
The Seventeenth Century Continued
153
Leibniz
159
The Eighteenth Century
163
The Fundamental Theorem of Algebra
199
Quaternions
203
Quaternions Applied to Number Theory
207
Quaternions Applied to Physics
211
Quaternions in Quantum Mechanics
215
Cardinal Numbers
219
Cardinal Arithmetic
223
Continued Fractions
227
The Fundamental Theorem of Arithmetic
231
Linear Diophantine Equations
233
Quadratic Surds
237
Pythagorean Triangles and Fermats Last Theorem
241
What Is a Calculation?
245
Recursive and Recursively Enumerable Sets
251
Hilberts Tenth Problem
255
Lambda Calculus
259
Logic from Aristotle to Russell
265
Intuitionistic Propositional Calculus
271
How to Interpret Intuitionistic Logic
277
Intuitionistic Predicate Calculus
281
Intuitionistic Type Theory
285
Godels Theorems
289
Natural Transformations
303
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