Art of the Intelligible

• 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.

'... impressively broad and far-ranging... written in an accessible and engaging style... The appendix on Godel's theorems...is clear and easy to follow... a clear and straightforward survey of the conceptual development of mathematics. It is a wonderful addition to the literature...provides an accessible introduction to the subject matter. I recommend it to anyone who has an interest in mathematics and its development.' Philosophia Mathematica, 8:3 (2000)