The presentation of mathematics in schools should be psychological and not systematic. The teacher, so to speak, should be a diplomat. He must take account of the psychic processes in the boy in order to grip his interest, and he will succeed only if he presents things in a form intuitively comprehensible. A more abstract presentation is only possible in the upper classes.
Every one who understands the subject will agree that even the basis on which the scientific explanation of nature rests is intelligible only to those who have learned at least the elements of the differential and integral calculus, as well as analytical geometry.
If the activity of a science can be supplied by a machine, that science cannot amount to much, so it is said; and hence it deserves a subordinate place. The answer to such arguments, however, is that the mathematician, even when he is himself operating with numbers and formulas, is by no means an inferior counterpart of the errorless machine, "thoughtless thinker" of Thomas; but rather, he sets for himself his problems with definite, interesting, and valuable ends in view, and he carries them to solution in appropriate and original manner. He turns over to the machine only certain operations which occur frequently in the same way, and it is precisely the mathematician -- one must not forget this -- who invented the machine for his own relief, and who, for his own intelligent ends, designates the tasks which it shall perform.
Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions.
Projective geometry has opened up for us with the greatest facility new territories in our science, and has rightly been called the royal road to our particular field of knowledge.
The definition of e is usually, in imitation of the French models, placed at the very beginning of the great text books of analysis, and entirely unmotivated, whereby the really valuable element is missed, the one which mediates the understanding, namely, an explanation of why precisely this remarkable limit is used as a base and why the resulting logarithms are called natural.
The greatest mathematicians, as Archimedes, Newton, and Gauss,
always united theory and applications in equal measure.
Regarding the fundamental investigations of mathematics, there is no final ending ... no first beginning.
Thus, in a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs.