Creating a new theory is not like destroying an old barn and erecting a skyscraper in its place. It is rather like climbing a mountain, gaining new and wider views, discovering unexpected connections between our starting points and its rich environment. But the point from which we started out still exists and can be seen, although it appears smaller and forms a tiny part of our broad view gained by the mastery of the obstacles on our adventurous way up.
When I examine myself and my methods of thought, I come to the conclusion that the gift of fantasy has meant more to me than any talent for abstract, positive knowledge.
It is important for the common good to foster individuality: for only the individual can produce the new ideas which the community needs for its continuous improvement and
requirements - indeed, to avoid sterility and petrification
All these constructions and the laws connecting them can be arrived at by the principle of looking for the mathematically simplest concepts and the link between them.
The best that Gauss has given us was likewise an exclusive production. If he had not created his geometry of surfaces, which served Riemann as a basis, it is scarcely conceivable that anyone else would have discovered it. I do not hesitate to confess that to a certain extent a similar pleasure may be found by absorbing ourselves in questions of pure geometry.
The most valuable thing a teacher can impart to children is not knowledge and understanding per se but a longing for knowledge and understanding, and an appreciation for intellectual values, whether they be artistic, scientific, or moral. It is the supreme art of the teacher to awaken joy in creative expression and knowledge. Most teachers waste their time by asking questions that are intended to discover what a pupil does not know, whereas the true art of questioning is to discover what the pupil does know or is capable of knowing.