What's all this whining about the environment? They're always talking about 'stop the clearcuts.' I mean do the math people. If we were out of trees then we wouldn't have any clearcuts to be complaining about now would we?
But the creative principle resides in mathematics. In a certain sense, therefore, I hold true that pure thought can grasp reality, as the ancients dreamed.
That Logic has advanced in this sure course, even from the earliest times, is apparent from the fact that, since Aristotle, it has been unable to advance a step, and thus to all appearance has reached its completion.
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.
There is an astonishing imagination, even in the science of mathematics. ... We repeat, there was far more imagination in the head of Archimedes than in that of Homer.
But there is another reason for the high repute of mathematics: it is mathematics that offers the exact natural sciences a certain measure of security which, without mathematics, they could not attain.
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.
We may discover resources on the moon or Mars that will boggle the imagination, that will test our limits to dream. And the fascination generated by further exploration will inspire our young people to study math, and science, and engineering and create a new generation of innovators and pioneers.