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Full Description
A compact survey, at the elementary level, of some of the most important concepts of mathematics. Attention is paid to their technical features, historical development and broader philosophical significance. Each of the various branches of mathematics is discussed separately, but their interdependence is emphasised throughout. Certain topics - such as Greek mathematics, abstract algebra, set theory, geometry and the philosophy of mathematics - are discussed in detail. Appendices outline from scratch the proofs of two of the most celebrated limitative results of mathematics: the insolubility of the problem of doubling the cube and trisecting an arbitrary angle, and the Gödel incompleteness theorems. Additional appendices contain brief accounts of smooth infinitesimal analysis - a new approach to the use of infinitesimals in the calculus - and of the philosophical thought of the great 20th century mathematician Hermann Weyl.
Readership: Students and teachers of mathematics, science and philosophy. The greater part of the book can be read and enjoyed by anyone possessing a good high school mathematics background.
Contents
1 Numerals and Notation.- 2 The Mathematics of Ancient Greece.- 3 The Development of The Number Concept.- The Theory of Numbers.- Perfect Numbers.- Prime Numbers.- Sums of Powers.- Fermat?s Last Theorem.- The Number ?.- What are Numbers?.- 4 The Evolution of Algebra, I.- Greek Algebra.- Chinese Algebra.- Hindu Algebra.- Arabic Algebra.- Algebra in Europe.- The Solution of the General Equation of Degrees 3 and 4.- The Algebraic Insolubility of the General Equation of Degree Greater than 4.- Early Abstract Algebra.- 5 The Evolution of Algebra, II.- Hamilton and Quaternions.- Grassmann?s "Calculus of Extension".- Finite Dimensional Linear Algebras.- Matrices.- Lie Algebras.- 6 The Evolution of Algebra, III.- Algebraic Numbers and Ideals.- Abstract Algebra.- Groups.- Rings and Fields.- Ordered Sets.- Lattices and Boolean Algebras.- Category Theory.- 7 The Development of Geometry, I.- Coordinate/Algebraic/Analytic Geometry.- Algebraic Curves.- Cubic Curves.- Geometric Construction Problems.- Higher Dimensional Spaces.- Noneuclidean Geometry.- 8 The Development of Geometry, II.- Projective Geometry.- Differential Geometry.- The Theory of Surfaces.- Riemann?s Conception of Geometry.- Topology.- Combinatorial Topology.- Point-set topology.- 9 The Calculus and Mathematical Analysis.- The Origins and Basic Notions of The Calculus.- Mathematical Analysis.- Infinite Series.- Differential Equations.- Complex Analysis.- 10 The Continuous and The Discrete.- 11 The Mathematics of The Infinite.- 12 The Philosophy of Mathematics.- Classical Views on the Nature of Mathematics.- Logicism.- Formalism.- Intuitionism.- Appendix 1 The Insolubility of Some Geometric Construction Problems.- Appendix 2 The GÖdel Incompleteness Theorems.- Appendix 3 The Calculus in Smooth InfinitesimalAnalysis.- Appendix 4 The Philosophical Thought of A Great Mathematician: Hermann Weyl.- Index of Names.- Index of Terms.
