"We may come back to the more general question, what one should consider as the characteristic features of such a closed system of axioms and definitions. Perhaps the most important feature is the possibility of finding a consistent mathematical representation for it. This representation must guarantee that the system does not contain contradictions. Then the system must be suited to describe a wide field of experience. The great variety of phenomena in the field should correspond to the great number of solutions to the equations in the mathematical representation. The limitations of the field can generally not be derived from the concepts. The concepts are not sharply defined in their relation to nature, in spite of the sharp definition of their possible connections. The limitation will therefore be found from experience, from the fact that the concepts do not allow a complete description of the observed phenomena."
A concept is a fundamental unit of cognition that classifies entities and encodes shared features. Concepts make it possible to form and combine ideas, draw inferences, and refer to external objects. They act as the meanings of words and play a central role in many cognitive processes, including perception, memory, and reasoning. Researchers distinguish different types of concepts based on their i
"The aim of science is... a comprehension, as complete as possible, of the connection between the sense experiences in their totality, and... the accomplishment of this aim by the use of the minimum of primary concepts and relations."
"All concepts, even those which are closest to experience, are from the point of view of logic freely chosen conventions."
"Concepts can only acquire content when they are connected, however indirectly, with sensible experience. But no logical investigation can reveal this connection; it can only be experienced. ...this connection ...determines the cognitive value of systems of concepts."
"Kant says that concept without percepts are empty, percepts without concepts are blind; it would perhaps be truer to say that concepts and percepts are inseparable; and if torn asunder, they are nothing."
"... mathematics is the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts. Mathematics would soon run out of interesting theorems if these had to be formulated in terms of the concepts which already appear in the axioms. Furthermore, whereas it is unquestionably true that the concepts of elementary mathematics and particularly elementary geometry were formulated to describe entities which are directly suggested by the actual world, the same does not seem to be true of the more advanced concepts, in particular the concepts which play such an important role in physics."
"There are three aspects of thinking as more or less complete stages of it. These are, conception, judgment, and reasoning. They are not to be considered three distinct acts; not even three successive stages. No one of them could occur without each of the others."