Advanced Mathematics

6.1Symmetries and Algebraic Systems

Purpose of Section

To introduce the idea of a symmetry of an object in the plane which will act as an introduction to the study of the algebraic group.

6.1.1 Abstraction and Abstract Algebra

The ability to think abstractly is a unique feature of human thought, a capacity not shared by “lower forms” of living creatures.¹ The power to capture the essence of what we experience is so engrained in our mental processes that we never give it a second thought. If the human mind did not have the capability to abstract commonalities in daily living, we would be living in a different world. Imagine lacking the capacity to grasp the “essence” of what makes up a chair. We would be forced to call every chair by a different name in order to communicate to others what we are referring to. The statement “the chair in the living room” would have no meaning unless we knew exactly what chair was being mentioned. Parents point to a picture of a dog in a picture book and tell their one‐year‐old infant, “dog,” and it is a proud moment for the parent when the child sees a strange dog in the yard and says, “dog!”

The concept of number is a crowning achievement of human's ability to abstract the essence of size of sets. It is not necessary to talk about “three people,” “three days,” “three dogs,” and so on. We have abstracted among those things the commonality of threeness, so there is no need to say “three goats plus five goats is eight goats,” or “three ...