Algebra, Number Theory and Mathematical Logic

Philosophical theories or ideas, as points of view, instruments of criticism, may help us to gather up what might otherwise pass unregarded by us. “Philosophy is the microscope of thought.” The theory or idea or system which requires of us the sacrifice of any part of this experience, in consideration of some interest into which we cannot enter, or some abstract theory we have not identified with ourselves, or what is only conventional, has no real claim upon us.

This fallacy [appeal to authority] is not in itself an error; it is impossible to learn much in today’s world without letting somebody else crunch the numbers and offer us explanations. And teachers are sources of necessary information. But how we choose our "authorities" and place a value on such information, is just another skill rarely taught in our education systems. It’s little wonder that to most folk, sound bites and talking heads are enough to count as experts. […] Teaching is reinforcing the appeal to authority, where anybody who seems more intelligent than you must ultimately be right. […] We educators must simply role-model critical thinking. […] Educators themselves have to be prepared to show that “evidence” and “answers” are two separate things by firmly believing that, themselves.

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.