Theory of Teaching Mathematics

Extension and abstraction without apparent direction or purpose is fundamental to the discipline. Applicability is not the reason we work, and plenty that is not applicable contributes to the beauty and magnificence of our subject.

The final truth about a phenomenon resides in the mathematical description of it; so long as there is no imperfection in this, our knowledge of the phenomenon is complete. We go beyond the mathematical formula at our own risk; we may find a model or a picture which helps us understand it, but we have no right to expect this, and our failure to find such a model or picture need not indicate that either our reasoning or our knowledge is at fault. The making of models or pictures to explain mathematical formulas and the phenomena they describe is not a step towards, but a step away from reality; it is like making a graven image of a spirit.

Their ideas seemed to him fruitful when he was reading or was himself seeking arguments to refute other theories, especially those of the materialists; but as soon as he began to read or sought for himself a solution of problems, the same thing always happened. As long as he followed the fixed definition of obscure words such as spirit, will, freedom, essence, purposely letting himself go into the snare of words the philosophers set for him, he seemed to comprehend something. But he had only to forget the artificial train of reasoning, and to turn from life itself to what had satisfied him while thinking in accordance with the fixed definitions, and all this artificial edifice fell to pieces at once like a house of cards, and it became clear that the edifice had been built up out of those transposed words, apart from anything in life.