What would geometry be without Gauss, mathematical logic without Boole, algebra without Hamilton, analysis without Cauchy? - George Frederick James Temple

From Pythagoras to Boethius, when pure mathematics consisted of arithmetic and geometry while applied mathematics consisted of music and astronomy, mathematics could be characterized as the deductive study of 'such abstractions as quantities and their consequences, namely figures and so forth' (Acquinas ca. 1260). But since the emergence of abstract algebra it has become increasingly difficult to formulate a definition to cover the whole of the rich, complex and expanding domain of mathematics.

Pure mathematics is much more than an armoury of tools and techniques for the applied mathematician. On the other hand, the pure mathematician has ever been grateful to applied mathematics for stimulus and inspiration. From the vibrations of the violin string they have drawn enchanting harmonies of Fourier Series, and to study the triode valve they have invented a whole theory of non-linear oscillations.

Most mathematicians are by nature Platonists who cheerfully, unreflectingly and habitually employ such loaded phrases as 'We assume there exists...' or 'Therefore there exists...' an entity with such and such characteristics. Challenged by the realist they would probably reply that since the truths of mathematics are absolute, universal and eternal it is hard indeed to deny them an existence independent of human intelligence.