Mathematical Thought Quotes

Mathematical discoveries, small or great are never born of spontaneous generation.

Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.

The history of mathematics, lacking the guidance of philosophy, [is] blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the history of mathematics, is empty.

Another characteristic of mathematical thought is that it can have no success where it cannot generalize.

Number, place, and combination . . . the three intersecting but distinct spheres of thought to which all mathematical ideas admit of being referred.

Mathematicians may flatter themselves that they possess new ideas which mere human language is as yet unable to express. Let them make the effort to express these ideas in appropriate words without the aid of symbols, and if they succeed they will not only lay us laymen under a lasting obligation, but, we venture to say, they will find themselves very much enlightened during the process, and will even be doubtful whether the ideas as expressed in symbols had ever quite found their way out of the equations into their minds.

Among mathematicians in general, three main categories may be distinguished; and perhaps the names logicians, formalists, and intuitionists may serve to characterize them.