Ninety per cent of all the mathematics we know has been discovered (or invented), in the last hundred years... the advances made in each of some dozen directions are converging into one single discipline uniting algebra, topology and analysis.
During this century mathematics has been transformed...
The brilliant summaries by Bourbaki (1969) and Weyl (1951)... set a high standard of exposition, but... there is room for a history of mathematical ideas which will demand less mathematical expertise and offer a more detailed account of the motivation of research.
This book... is primarily and essentially an account of the discovery or invention of mathematical concepts, and the historical material is... divided and classified, neither chronologically nor biographically, but philosophically.
The subject matter of mathematics has increased so rapidly and extensively that there is some element of truth in maintaining that mathematics is not so much a subject as a way of studying any subject, not so much a science as a way of life. We turn, then, from the attempt to characterize the material object of mathematics to an attempt to determine its formal object, i.e., its methodology.
At each stage of in the advance of mathematical thought the outstanding characteristics are novelty and originality. That is why mathematics is such a delight to study, such a challenge to practise and such a puzzle to define.
The function of logic in mathematics is critical rather than constructive.
Logical analysis is indispensable for an examination of the strength of a mathematical structure, but it is useless for its conception and design. The great advances in mathematics have not been made by logic but by creative imagination.
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.
The 'language theory' is inadequate as a description of the nature of mathematics.
Mathematical activity has taken the forms of a science, a philosophy and an art.
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.
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.