Metric Space

Abstract

Abstraction and idealization are two major techniques to allow building simplified mathematical models in the sciences. Using a continuous function to represent discrete population values is an instance of idealization. Idealization helps to ignore and smooth over insignificant details, and therefore it simplifies the models, but it produces models that do not exactly represent real things, especially when idealization to infinity and continuity is used. Therefore, the applicability of idealization is not logically transparent, because one cannot straightforwardly translate the mathematical premises about an idealized model into literally true realistic assertions about finite real things without modifying the logical structures of those mathematical premises. Abstraction does not have this problem. Abstraction means presenting concepts, thoughts and proofs in some highly schematic and abstract format, and then they can be instantiated into more and more concrete concepts, thoughts and proofs in a few stages, with their logical structures preserved. The resulted thoughts and proofs can become much more complex than the original ones. That is how abstraction helps to simplify the presentation of a theory.