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 <em>scratch</em> the proofs of two of the most celebrated <em>limitative</em> 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. <br/> <em>Readership:</em> 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.