Concept Car

According to Carl Benjamin Boyer, in the introduction to his The History of the Calculus and its Conceptual Development, concepts in calculus do not refer to perceptions. As long as the concepts are useful and mutually compatible, they are accepted on their own. For example, the concepts of the derivative and the integral are not considered to refer to spatial or temporal perceptions of the external world of experience. Neither are they related in any way to mysterious limits in which quantities are on the verge of nascence or evanescence, that is, coming into or going out of appearance or existence. The abstract concepts are now considered to be totally autonomous, even though they originated from the process of abstracting or taking away qualities from perceptions until only the common, essential attributes remained.
Concepts in Empirical Investigations
Concepts, as abstract units of meaning, play a key role in the development and testing of theories. For example, a simple relational hypothesis can be viewed as either a conceptual hypothesis (where the abstract concepts form the meaning) or an operationalized hypothesis, which is situated in the real world by rules of interpretation. For example, take the simple hypothesis Education increases Income. The abstract notion of education and income (concepts) could have many meanings. A conceptual hypothesis cannot be tested. They need to be converted into operational hypothesis or the abstract meaning of education must be derived or operationalized to something in the real world that can be measured. Education could be measured by “years of school completed” or “highest degree completed” etc. Income could be measured by “hourly rate of pay” or “yearly salary” etc. The system of concepts or conceptual framework can take on many levels of complexity. When the conceptual framework is very complex and incorporates causality or explanation they are generally referred to as a theory. Noted philosopher of science Carl Gustav Hempel says this more eloquently “An adequate empirical interpretation turns a theoretical system into a testable theory: The hypothesis whose constituent terms have been interpreted become capable of test by reference to observable phenomena. Frequently the interpreted hypothesis will be derivative hypotheses of the theory; but their confirmation or disconfirmation by empirical data will then immediately strengthen or weaken also the primitive hypotheses from which they were derived.”