I think we're at the point where most people at least realize that there is a point to studying math, as opposed to 3rd grade when people would whine to their teacher about how they were never going to use it. I mean, there's not really a point for everyone to learn math, but I would hope that most of my readership would already agree that math is a worthwhile endeavor for us as a species. However, one may have a hard time getting behind the practical relevance of the particular flavor of math that I study. If you're new to mathematics or come from an applied background, once I start in on the things that I'm going to talk about, you may have a hard time seeing the point of it all. I must admit, I can understand that feeling. So what I would like to do now is to show some of the motivation behind two of the topics that I'll be discussing and give some applications.
Abstract algebra is the study of algebraic structure. It is essentially taking a mathematical look into certain universes that may not seem mathematical at first. If you give me a bunch of various atoms, the set of all molecules that I can make with those atoms is an algebraic structure. Another algebraic structure is the set of all of the symmetries of a shape or an object. Even something like a map is algebraic - that is the set of all possible routes from one point to another point on a map is an algebraic structure. Below are some good examples of practical applications of abstract algebra. You'll notice that a lot of them deal with group theory, which is (currently) my primary area of interest.
Topology deals with the study of surfaces and/or objects with particular attention to their structure rather than their shape. When two structures are built the same way and it is possible to stretch, twist, or shape one into the other then they are considered topologically equivalent (the math term is homeomorphic) and they share a lot of very useful properties. Oftentimes if we have a very complicated structure that we know is equivalent to something simpler, we can "move" everything over to the simpler structure, do whatever work is needed, and then move back. Many of the practical applications of topology can be found here. However, some of the most important advances in topology have been using topological methods to prove theorems in other areas of math. Some examples of these things are shown below.
- Banach Fixed Point Theorem
- Hairy Ball Theorem (yes, that's actually what its called)
- Borsuk-Ulam Theorem
- Any subgroup of a free group is also free. This is a purely group-theoretic result but the simplest proof is one using topology
- Topological Combinatorics is a very useful area of mathematics that is a direct branch of topology
Aside from all of my supposed practical applications of theoretical mathematics, you may wonder why it is that I, myself, study math and love it so much. Its quite simple, really, and its because I think math is beautiful. (Does that make me shallow? maybe...) There are no opinions in mathematics, no grey areas, and no hypotheses. It is true, mathematicians sometimes conjecture about certain things that they think are true, but if they are to become accepted as fact it is only once an argument has been presented based on logic and truth - we don't use laboratories or experiments to determine truth to within an error percentage. Mathematics is the only science that can impart factual, universal truths on its students. Aside from that, there is no bound to mathematical knowledge - if there is a limit to mathematics, it is not in the math, but in the mathematician. Finally, math has its own spectacular elegance and satisfaction that is not something that can be explained, but must be experienced.
In my next post I plan on talking about a very important technique called mathematical induction. I know I said that I would stay away from proofs as much as possible and induction is a technique used to prove things, but it is necessary - induction is extremely useful.
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