23 Quotes by Carl B. Boyer

The Greek thinkers was no way of bridging the gap between the rectilinear and the curvilinear which would at the same time satisfy their strict demands of mathematical rigor and appeal to the clear evidence of sensory experience.

A quantity is something or nothing: if it is something, it has not yet vanished; if it is nothing, it has literally vanished. The supposition that there is an intermediate state between these two is a chimera.D'Alembert

Now we can see what makes mathematics unique. Only in mathematics is there no significant correction – only extension. Once the Greeks had developed the deductive method, they were correct in what they did, correct for all time. Euclid was incomplete and his work has been extended enormously, but it has not had to be corrected. His theorems are, every one of them, valid to this day.

Berkeley was unable to appreciate that mathematics was not concerned with a world of “real” sense impressions. In much the same manner today some philosophers criticize the mathematical conceptions of infinity and continuum, failing to realize that since mathematics deals with relations rather than with physical existence, its criterion of truth is inner consistency rather than plausibility in the light of sense perception of intuition.

Definitions of number, as given by several later mathematicians, make the limit of an infinite sequence identical with the sequence itself. Under this view, the question as to whether the variable reaches its limit is without logical meaning. Thus the infinite sequence .9, .99, .999,... is the number one, and the question, “Does it ever reach one?” is an attempt to give a metaphysical argument which shall satisfy intuition.