Mathematics has become so sophisticated that it is easy to lose touch with the mathematical realities out of which it arises. Many nowadays think of mathematics as complicated games with forests of symbols. Yet this is an image which could not have got a grip on the ancient mathematicians, like those of the Pythagorean brotherhood, who stood nearer to the roots of those forests. Now we are lost in the higher branches of mathematics, and we sometimes forget that all these growing branches are nourished ultimately by roots far below in down-to-earth reality.

The early Pythagoreans discovered things which are thus and so, and which could not have been otherwise. Yet although these things could not have been otherwise, the Pythagoreans who discovered them could have done otherwise. Hence the truths which the Pythagoreans discovered are things which would still have been so even if no one had ever discovered them.