The Art of the Intelligible: An Elementary Survey of Mathematics in its Conceptual Development
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.
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Higher Dimensional Spaces
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addition algebraic integers algebraic numbers angle arbitrary Archimedes arithmetic assertion Boolean algebra calculus called cardinal number circle coefficients commutative complex numbers concept cone construction coordinates corresponding cube cubic cubic equation curve decimal Dedekind defined definition denote derived divisor doubling the cube elements equation of degree Euclidean geometry Euclidean tools Euler example extensive quantities fact Fermat finite follows formula function geometry given Greek idea ideal identity infinite infinitesimal integers intersection ISBN known lattice line segments linear logical mathematical mathematician matrix means morphism multiplication natural numbers noneuclidean geometry objects obtained octonions operation pair parallel postulate partially ordered sets permutation plane polygon polynomial postulate prime numbers problem proof quantity quaternions rational numbers real numbers represented result ring roots satisfies shows smooth infinitesimal analysis solution solving space square straight line subset surface symbol tangent theorem theory topological triangle unit vector