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...

Genius at Play by Siobhan Roberts is a biography revolving around the life and works of John Conway, most notoriously (much to his distaste) the creator of The Game of Life (the computer game, not the board game!). The structure of the book keeps your attention throughout as it switches between the author narrating pieces of Conway’s life according to different acquaintances’ recounting, interviewing Conway about different memories (of the few he seems to have), and reviewing pieces of Conway’s work. Jumping from works ranging from knot theory to a proof of the free will of humanity, Siobhan highlights how Conway’s heavy focus on games and the mathematics behind them seemed to fuel his work. In what seems to be a strange turn for most of the famous mathematicians, much of Conway’s work and understanding of mathematics seems to stem from him having a physical grasp of what he is working with (as demonstrated by his parallax contraption to view the world in the 4 th ...

Who was Archimedes? To begin, Archimedes was born in 287 BC in Syracuse, Italy. He was a Mathematician, a Physicist, an Inventor, and an Astronomer. Considered not only one of the greatest Mathematicians of his time, but of all time, his contributions to the field were many. He gave a numerical approximation of pi, found an equation for the circumference of a circle, found an equation for the area of a circle, developed the Archimedean property, found the volume of a sphere, and even proposed an estimate of the number of grains of sand that would fill the universe - and these were only his contributions to Mathematics.  Archimedes was born in Syracuse, studied in Alexandria, and then returned to Syracuse to continue his work. He died around the age of 75 during the Second Punic War. Syracuse held against the siege of the Romans for 2 years before finally falling, and Archimedes played no small role in that. Due to his contributions, the Roman commander had great respect for...

A person on the street might say that Math is...   My professor for my Math capstone class asked us what is Math? Some responses were that Math is problem solving, it is anything to do with numbers, or, as I originally thought, a way to model what we see around us in a methodical manner (you can see my physics background show here). After some further thought on the topic, however, I'm changing my view a bit.   Three brothers inherit 35 camels...        In a slightly modified version of the well known Arabic Algebra problem, our professor told us that there are three brother's whose father left a will telling them how to divide their camels. The father said the oldest should take half of the camels, the middle brother should take one third, and the youngest brother should take one ninth. Faced with the issue of all of the fractions not resulting in whole numbers, the brothers turn to a Mathematician riding by on camel back with his companion to try...