The Heritage of ThalesSpringer Science & Business Media, 2012 M12 6 - 331 páginas 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. |
Contenido
3 | |
6 | |
7 | |
11 | |
12 | |
Prime Numbers | 15 |
SumerianBabylonian Mathematics 21 | 20 |
More about Mesopotamian Mathematics | 25 |
Quaternions Applied to Physics 211 | 210 |
Quaternions in Quantum Mechanics | 215 |
Cardinal Numbers 219 | 218 |
Cardinal Arithmetic | 223 |
Continued Fractions | 227 |
The Fundamental Theorem of Arithmetic | 231 |
Linear Diophantine Equations | 233 |
Quadratic Surds 237 | 236 |
The Dawn of Greek Mathematics | 29 |
Pythagoras and His School | 33 |
Perfect Numbers | 37 |
Regular Polyhedra | 41 |
The Crisis of Incommensurables | 47 |
From Heraclitus to Democritus 53 | 52 |
The Rationals 187 | 80 |
Alexandria from 300 BC to 200 | 94 |
Mathematics in China and India | 111 |
The Real Numbers | 191 |
Complex Numbers | 195 |
The Fundamental Theorem of Algebra 199 | 198 |
Quaternions | 203 |
Quaternions Applied to Number Theory | 207 |
Pythagorean Triangles and Fermats Last Theorem | 241 |
What Is a Calculation? | 245 |
Recursive and Recursively Enumerable Sets | 251 |
Hilberts Tenth Problem 255 | 254 |
Lambda Calculus | 259 |
Logic from Aristotle to Russell | 265 |
Intuitionistic Propositional Calculus 271 | 270 |
How to Interpret Intuitionistic Logic | 277 |
Intuitionistic Predicate Calculus 281 | 280 |
Intuitionistic Type Theory | 285 |
Gödels Theorems 289 | 288 |
Natural Transformations | 303 |
Renaissance Mathematics Continued | 319 |
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Alexandria algebra Archimedes arithmetic arrows assumed Axiom of Archimedes axioms Babylonians Brahmagupta calculate called Cardano Chapter circle commutativity complex numbers concrete category construct continued fraction cube cubic equation define Diophantine equation Diophantus divides divisor element equal Euclid Euclidean example Exercises expressed factor Fermat Fibonacci finite follows formula function functor geometry given Gödel's Gödel's Incompleteness Theorem Greek hence Hilbert infinite integer solutions integral domain intuitionistic Lagrange lambda calculus Leibniz Lemma linear logic mapping mathematical induction mathematicians matrix monoid multiplication natural numbers natural transformation number theory obtain pair perfect numbers polygon polynomial positive integers postulate prime numbers problem proof provable Prove Pythagoras Pythagorean triangles quaternion radius rational number real numbers recursively enumerable regular ruler and compass segment sides solve square roots straight line Suppose symbols Theorem