Springer | 2013 - reprint of the original 1st ed. 1999 edition | 350 páginas | 26 Mb
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The philosopher Immanuel Kant writes in the popular introduction to his philosophy: "There is no single book about metaphysics like we have in mathematics. If you want to know what mathematics is, just look at Euclid's Elements." (Prolegomena Paragraph 4) Even if the material covered by Euclid may be considered elementary for the most part, the way in which he presents essential features of mathematics in a much more general sense, has set the standards for more than 2000 years. He displays the axiomatic foundation of a mathematical theory and its conscious development towards the solution of a specific problem. We see how abstraction works and how it enforces the strictly deductive presentation of a theory. We learn what creative definitions are and how the conceptual grasp leads to the classification of the relevant objects. For each of Euclid's thirteen Books, the author has given a general description of the contents and structure of the Book, plus one or two sample proofs. In an appendix, the reader will find items of general interest for mathematics, such as the question of parallels, squaring the circle, problem and theory, what rigour is, the history of the platonic polyhedra, irrationals, the process of generalization, and more. This is a book for all lovers of mathematics with a solid background in high school geometry, from teachers and students to university professors. It is an attempt to understand the nature of mathematics from its most important early source.
Content:
Front Matter....Pages i-xvi
General Historical Remarks....Pages 1-2
The Contents of the Elements....Pages 3-10
The Origin of Mathematics 1: The Testimony of Eudemus....Pages 11-16
Euclid Book I: Basic Geometry....Pages 17-46
The Origin of Mathematics 2: Parallels and Axioms....Pages 47-50
The Origin of Mathematics 3: Pythagoras of Samos....Pages 51-60
Euclid Book II: The Geometry of Rectangles....Pages 61-71
The Origin of Mathematics 4: Squaring the Circle....Pages 73-78
Euclid Book III: About the Circle....Pages 79-91
The Origin of Mathematics 5: Problems and Theories....Pages 93-95
Euclid Book IV: Regular Polygons....Pages 97-107
The Origin of Mathematics 6: The Birth of Rigor....Pages 109-111
The Origin of Mathematics 7: Polygons After Euclid....Pages 113-120
Euclid Book V: The General Theory of Proportions....Pages 121-134
Euclid Book VI: Similarity Geometry....Pages 135-149
The Origin of Mathematics 8: Be Wise, Generalize....Pages 151-159
Euclid Book VII: Basic Arithmetic....Pages 161-182
The Origin of Mathematics 9: Nicomachus and Diophantus....Pages 183-191
Euclid Book VIII: Numbers in Continued Proportion, the Geometry of Numbers..
Pages 193-201
The Origin of Mathematics 10: Tools and Theorems....Pages 203-206
Euclid Book IX: Miscellaneous Topics from Arithmetic....Pages 207-211
The Origin of Mathematics 11: Math Is Beautiful....Pages 213-221
Euclid Book X: Incommensurable Magnitudes....Pages 223-228
The Origin of Mathematics 12: Incommensurability and Irrationality....Pages 229-253
Euclid Book XI: Solid Geometry....Pages 255-265
The Origin of Mathematics 13: The Role of Definitions....Pages 267-269
Euclid Book XII: Volumes by Limits....Pages 271-278
The Origin of Mathematics 14: The Taming of the Infinite....Pages 279-282
Euclid Book XIII: Regular Polyhedra....Pages 283-302
The Origin of Mathematics 15: Symmetry Through the Ages....Pages 303-316
The Origin of Mathematics 16: The Origin of the Elements....Pages 317-320
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