The Importance of Mathematics

This post is a change of pace and the mathematics included will be minimal. Before continuing with the first of multiple posts on set theory, I would like to explain the importance of mathematics and why you should study it (i.e., why you should continue to follow my blog!).

Since the dawn of mathematics, it has been used as a tool for understanding the natural world. Newton developed differential calculus in order to explain motion of moving bodies, differential equations have been used to study motion, population growth, and decay rates of materials, partial differential equations were used to study heat conduction in metals, the way waves propagate in media and the study of quantum mechanics. Mathematics is everywhere: in the fan circling above your head, in the satellites orbiting the Earth providing you with GPS navigation, in the warming body that we call the Sun. Each of these three topics will be treated in a physics blog I intend to create later down the road.

When grinding through a mathematics class, it is easy to miss the forest despite all of the trees. Sometimes you may feel inadequate as you are learning a new topic. It is imperative that you remind yourself that the world's best men had difficulty developing many topics you learn as an undergraduate. Mathematics was not built in a day. Newton and Leibniz developed calculus in the 1600s, but at the time calculus was considered mysterious. Newton's fluxions - as they were called - became what we call differentials (extremely small quantities); even the great Bishop Berkeley called these fluxions "ghosts of departed quantities" and many others felt similarly. Calculus was heavily used by mathematical physicists (called natural philosophers at the time), but some true mathematicians considered calculus incomplete or possibly incorrect. It was not until the 1800s that Weierstrass gave us our current definition of what a limit is. As you can see, it was not an easy task even for the best mathematicians at the time to make calculus a rigorous topic in its own right. The moral of the story is that in your courses you are learning potentially hundreds of years of mathematics in a very short period of time, so do not be discouraged.

Having said that, mathematics becomes more abstract from here. Mathematics is wonderful because it has the ability to explain, but is not limited to, the natural world. Some people believe all of mathematics will find its way into explaining the world around us and it certainly is true that much of mathematics has. I am unsure of whether or not all branches of mathematics will find its way into explaining the natural world, but it is an exciting concept!

In the near future, I will make a post concerning abstractions found in mathematics. The idea of a factorial can be extended, the ideas surrounding vectors (as you probably used in physics in high school) can be generalized and so can the concept of a length. You might ask: why generalize these ideas? At first generalization can seem unwarranted or perhaps even unnecessary, but many times these "unnecessary" generalizations are used when modeling something in the real world. The abstraction of vectors into vector spaces and length is hugely important in quantum mechanics (as will be covered in my physics blog).

On that note, I end this post. The upcoming topic of set theory is very rich and full of intrigue and mystery, concreteness and abstraction. Some of the arguments may be difficult to internalize at first, but I will try to make them as clear and understandable as possible. And in case I don't see you, good afternoon, good evening and goodnight.