The philosopher Immanuel Kant writes in the popular introduction to his philosophy: "There is no single book about metaphysics like we have in mathernatics....
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Résumé
The philosopher Immanuel Kant writes in the popular introduction to his philosophy: "There is no single book about metaphysics like we have in mathernatics. 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 its most parts, 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 toward 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 accompanying section, the reader will find items of general interest for mathematics, such as the question of parallels, squaring the circle, problem and theory, what rigor 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.
Sommaire
General Historical Remarks
The Contents of the Elements
The Origin of Mathematics 1: The Testimony of Eudemus
Euclid Book I
The Origin of Mathematics 2: Parallels and Axioms
The Origin of Mathematics 3: Pythagoras of Samos
Euclid Book II
The Origin of Mathematics 4: Squaring the Circle
Euclid Book III
The Origin of Mathematics 5: Problems and Theories
Euclid Book IV
The Origin of Mathematics 6: The Birth of Rigor
The Origin of Mathematics 7: Polygons After Euclid
Euclid Book V
Euclid Book VI
The Origin of Mathematics 8: Be Wise, Generalize
Euclid Book VII
The Origin of Mathematics 9: Nicomachus and Diophantus
Euclid Book VIII
The Origin of Mathematics 10: Tools and Theorems
Euclid Book IX
The Origin of Mathematics 11: Math Is Beautiful
Euclid Book X
The Origin of Mathematics 12: Incommensurability and Irrationality
Euclid Book XI
The Origin of Mathematics 13: The Rôle of Definitions
Euclid Book XII
The Origin of Mathematics 14: The Taming of the Infinite
Euclid Book XIII
The Origin of Mathematics 15: Symmetry Through the Ages
The Origin of Mathematics 16: The Origin of the Elements.