Abstraction

"I discovered the works of Euler and my perception of the nature of mathematics underwent a dramatic transformation. I was de-Bourbakized, stopped believing in sets, and was expelled from the Cantorian paradise. I still believe in abstraction, but now I know that one ends with abstraction, not starts with it. I learned that one has to adapt abstractions to reality and not the other way around. Mathematics stopped being a science of theories but reappeared to me as a science of numbers and shapes."

"The abstraction is often the most definite form for the intangible thing in myself that I can only clarify in paint."

"Words are a medium that reduces reality to abstraction for transmission to our reason, and in their power to corrode reality inevitably lurks the danger that the words will be corroded too."

"Denotation by means of sounds and markings is a remarkable abstraction. Three letters designate God for me; several lines a million things. How easy becomes the manipulation of the universe here, how evident the concentration of the intellectual world! Language is the dynamics of the spiritual realm. One word of command moves armies; the word Liberty entire nations."

"Everything abstract is ultimately part of the concrete. Everything inanimate finally serves the living. That is why every activity dealing in abstraction stands in ultimate service to a living whole."

"For me, abstraction is real, probably more real than nature. I'll go further and say that abstraction is nearer my heart. I prefer to see with closed eyes."

"Abstract terms (however useful they may be in argument) should be discarded in meditation, and the mind should be fixed on the particular and the concrete, that is, on the things themselves."

"An abstraction is one thing that represents several real things equally well."

"A major danger in using highly abstractive methods in political philosophy is that one will succeed merely in generalizing one’s own local prejudices and repackaging them as demands of reason. The study of history can help to counteract this natural human bias."

"The reliance on the abstract is thus not a result of an over-estimation but rather of an insight into the limited powers of our reason. It is the over-estimation of the powers of reason which leads to the revolt against the submission to abstract rules. Constructivist rationalism rejects the demand for this discipline of reason because it deceives itself that reason can directly master all the particulars; and it is thereby led to a preference for the concrete over the abstract, the particular over the general, because its adherents do not realize how much they thereby limit the span of true control by reason. The hubris of reason manifests itself in those who believe that they can dispense with abstraction and achieve a full mastery of the concrete and thus positively master the social process."