The years of searching in the dark for a truth that one feels but cannot express, the intense desire and the alternations of confidence and misgiving until one breaks through to clarity and understanding, are known only to him who has experienced them himself.
We come now to the question: what is a priori certain or necessary, respectively in geometry (doctrine of space) or its foundations? Formerly we thought everything; nowadays we think nothing. Already the distance-concept is logically arbitrary; there need be no things that correspond to it, even approximately.
The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things.
Of all the intellectual faculties, judgment is the last to mature. A child under the age of fifteen should confine its attention either to subjects like mathematics, in which errors of judgment are impossible, or to subjects in which they are not very dangerous, like languages, natural science, history, etc.