The function of logic in mathematics is critical rather than constructive. - George Frederick James Temple

Mathematics has also been developed as a philosophy, in the sense in which this term is defined by A.N. Whitehead as 'the endeavor to frame a coherent, logical and necessary system of general ideas in terms of which every element of our experience can be interpreted'. Substitute 'mathematics' for 'experience' and we have an admirable description of its speculative and philosophic development. ...Philosophy of mathematics... has its paradoxes and antimonies, and also diverse schools of thought...

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