In his wretched life of less than twenty-seven years Abel accomplished so much of the highest order that one of the leading mathematicians of the Nineteenth Century could say without exaggeration, "Abel has left mathematicians enough to keep them busy for five hundred years." Asked how he had done all this in the six or seven years of his working life, Abel replied, "By studying the masters, not the pupils."

Pick the assumptions to pieces till the stuff they are made of is exposed to plain view - this is the cardinal rule for understanding the basis of our beliefs.

If indeed, as Hilbert asserted, mathematics is a meaningless game played with meaningless marks on paper, the only mathematical experience to which we can refer is the making of marks on paper.

The longer mathematics lives the more abstract - and therefore, possibly also the more practical - it becomes.

The mistakes and unresolved difficulties of the past in mathematics have always been the opportunities of its future...

A number of aspects of mathematics are not much talked about in contemporary histories of mathematics. We have in mind business and commerce, war, number mysticism, astrology, and religion. In some instances, writers, hoping to assert for mathematics a noble parentage and a pure scientific experience, have turned away their eyes. Histories have been eager to put the case for science, but the Handmaiden of the Sciences has lived a far more raffish and interesting life than her historians allow.

If "Number rules the universe" as Pythagoras asserted, Number is merely our delegate to the throne, for we rule Number.