As an aspiring mathematician, I often get asked, "Lindsay, why do you like math? Isn't math just equations and numbers that you have to memorize to pass an exam?" I am then forced into defense mode in which I must defend the subject that is so near and dear to my heart. However, this is such a difficult thing to do. Sure, the quadratic equation and triangle congruence rules that we had to memorize in high school fit under the umbrella of "Mathematics," but that's not all that it includes.
It is my understanding that mathematics is a collection of tools that we use to quantify and describe the world around us. We use mathematics very similarly to how we use language. Using language, we can identify objects, convey ideas, and argue. Math can be used in the exact same way when communicating scientific ideas, defining mathematical objects, and proving theorems. The most interesting relationship between language and mathematics is that both can be utilized to describe events and objects that do not exist in the physical universe. Language can be used to create poetry which describes abstract emotions or fictional events. Similarly, mathematics can be used to describe things like how a 2-dimensional circle would look if it were projected on a 1-sided Klein bottle (like what Dr. William Dickinson at GVSU researches.) I think that, at its very root, mathematics is a form of communication.
A very large turning point for math was when mathematical concepts could be argued and verified through what we all now recognize as a proof. This began in the field of geometry and this method of logical argument showed that math concepts were concrete and could be verified. The conceptual topics that were being proved early in mathematical history were rooted in applications. Another key turning point for math was when observable events and objects were abstracted into general cases. This pushed the field of mathematics from simply being computation and quantification into a field that could describe both concrete and abstract situations.