Isomorphism: Abstract and Concrete Representations

Abstract

Looking at concrete representations of mathematical problems from an isomorphic perspective, this article suggests that every concrete representation of a mathematical concept is understood by reference to an underlying abstract representation in the mind of the comprehender. The complex form of every abstract representation of a problem is created by the gradual development of its elementary form. Throughout the process of cognitive development, new features are added to the elementary form of abstract representation, which leads to gradual formation of a fully developed abstract representation in the mind. Every developed abstract representation of a problem is the underlying source for understanding an infinite number of concrete isomorphic representations. Deep or abstract representations of a problem are shared by the concrete realizations or concrete forms of that problem. In other words, concrete representations of a problem are the realizations of a single abstract representation. This discussion is extended to mind-brain relationship and the possible isomorphism that could exist between mind and brain.