Notation

One aspect of modern mathematics that is obvious to even the casual observer is the use of abstract notations: algebraic expressions, complicated-looking formulas, and geometric diagrams.

Different aspects of reality require different forms of description. For example, the most appropriate way to study the layout of the land or to describe to someone how to find their way around a strange town is to draw a map. Text is far less appropriate. Analogously, annotated line drawings (i.e. blueprints) are most appropriate for representing the construction of a building. And musical notation is most appropriate for representing music on paper. 

In the case of various kinds of abstract, formal patterns and abstract structures, the most appropriate means of description and analysis is mathematics, using mathematical notations, concepts, and procedures. The mathematicians’ reliance on abstract notation is a reflection of the abstract nature of the patterns they study. 

Mathematical notation is no more mathematics than musical notation is music!

A page of sheet music represents a piece of music. The music itself is what you get when the notes on the page are sung or performed on a musical instrument. It is in its performance that the music comes alive and becomes part of our experience. The music exists not on the printed page but in our minds. The same is true for mathematics. The symbols on a page are just a representation of the mathematics. When read by a competent performer (in this case, someone trained in mathematics), the symbols on the printed page come alive—the mathematics lives and breathes in the mind of the reader like some abstract symphony.