I’ll confess here that this post has nothing to do with AP Calculus AB or AP Calculus BC. It is geared towards people who have taken higher level courses centered around proofs (i.e. DE, CA, Analysis). But for those of you who see yourselves taking those higher level courses in the future, this might go to show you why math is bigger than just math (and maybe it’ll encourage you to continue your pursuits).Ever since I started learning Analysis in my junior year, I have always tried to apply ideas from Analysis to aspects of my everyday life. And the easiest application I saw fit was in arguments. With a free discussion environment in my math classes, I found myself often pondering how I could use abstract theorems to further a certain side of an argument, or even just to stimulate thought in the class. It wasn’t about being “right,” but rather just being able to apply a unique line of thought that would normally be tucked away in the minds of mathematicians. I saw this line of thought come to fruition in one discussion regarding the differences between judgments and opinions. My two major questions using this method were as follows (the first one is rather simple and the second is really just me running wild with an idea from analysis):

- Suppose that we define the set of judgments as J and the set of opinions as O. Is J a subset of O or is O a subset of J? Do your answers to these questions depend on which definitions you use for opinions and judgments? Note: This may be quite subjective based on your definitions but perhaps you should discuss with other people on what the difference is between an opinion and a judgment. How do they vary with time? Is evidence used in one and not the other?
- (Consider the Nested Interval Property). Having considered the sets J and O, now also consider the set E, of evidence based opinions. Denote E_k as a set of evidence based opinions with some quality of evidence relating to the index k such that as k increases, the evidence used to support the opinion is getting more and more refined (basically, the evidence is getting better and better as k increases). Now, suppose that E_{k+1} is a subset of E_k and so on and so forth. Is there an intersection for all E_k (as k goes to infinity) and if so, is this intersection what a fact is? Note: This was really me just expressing my creativity with the nested interval property, but there may be some validity to the sentiment.

For reference, the Nested Interval Property:

I think the two points I have brought up might make for some interesting discussions but more than just an interesting discussion, it goes to show how math, even abstract math, can extend beyond its preconceived boundaries. The power of math does not just rest in its applications to business or science, but rather in its ability to stimulate fresh thought. I’ll admit that I can’t pinpoint how I’ll be using this type of thought later in my life. But having such thought has made me realize the true amazing nature of mathematics and see the world in a different light. And I hope you can enjoy it in the same way too!Also, feel free to chime in with comments below.