"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' . 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."

“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.”

“What would geometry be without Gauss, mathematical logic without Boole, algebra without Hamilton, analysis without Cauchy?”

“The history of mathematics throws little light on the psychology of mathematical invention.”

“For the great majority of mathematicians, mathematics is... a whole world of invention and discovery—an art. The construction of a new theorem, the intuition of some new principle, or the creation of a new branch of mathematics is the triumph of the creative imagination of the mathematician, which can be compared to that of a poet, the painter and the sculptor.”

“The professional mathematician can scarcely avoid specialization and needs to transcend his private interests and take a wide synoptic view of the whole landscape of contemporary mathematics. His scientific colleagues are continually seeking enlightenment on the relevance of mathematical abstractions. The undergraduate needs a guidebook to the topography of the immense and expanding world of mathematics. There seems to be only one way to satisfy these varied interests... a concise historical account of the main currents... Only by a study of the development of mathematics can its contemporary significance be understood.”

“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.”