Euler's formula evaluated when x = pi: The five most important numbers in math

What's the Context? To begin, let's frame the question in which we are exploring these ways of thinking. What is the nature of mathematics? Is it something that exists regardless of human existence (discovered), or is it something which humans formulated (invented)? This is a question that has been tossed around for centuries (if you couldn't tell by the name of Platonism), and there still isn't one widely accepted answer. The goal here is to present two of the most prevalent fields of thought, my current thinking, and short-comings of all three.   Allegory of the Cave One of the more well known thought experiments is Plato's Allegory of the Cave. In this particular conceptualization, there is a group of prisoners which are restrained such that the only interaction they have with the outside world is to see a flat 2D shadow of the things that lie behind them. Plato considered mathematics to be no different than this. He believed there to be a "realm of...

But wait... I thought this was Mathematics, not Physics! Before I begin with the topic of String Theory, I want to touch on one of the prevalent ideas of the course: what is mathematics? Although I still haven't quite pinned down an exact definition for myself so far this semester, I would say that if we look to the past, one of the defining characteristics thus far of many mathematicians' work was an attempt at describing the world around them. String theory is another one of these attempts, and therefore rest easy knowing that I'm not terribly off topic with this brief introduction of String theory  (link to "Making sense of String theory", a presentation by Brian Greene).   Newton, Einstein, and Kaluza walk into a bar... There are several advancements in describing the world around us which led to the development of String theory in the first place. Beginning back in the time of Newton, the idea of gravity was developed, and a mathematical model was ma...