A formal system is a structure in logic that consists of basic symbols and a set of rules for manipulating these symbols. These symbols are “basic” because they have no inherent meaning. Sets of symbols are said to be “well-formed” if their structure follows the rules laid out by these…

Brownian Motion is the name for any object which experiences small, random fluctuations. It was first described by Robert Brown in the 19th century when he observed small pollen particles moving in a jittery fashion. …

It is a common intuition that performing an experiment more times will lead to more accurate results. Formally, this intuition is known as the Law of Large Numbers. However, this was not always common knowledge. Why did we learn to think this way? …

Statistics is full of different ways to analyze data sets. Often times, there is a larger population that we can only take a sample of due to limitations. We use this sample to extract as much information about the overall set as we can. Two of our most important tools…

Probability theory is full of surprising results. Often times, these results go against our nature yet reveal key aspects of how the world works. My personal favorite, first introduced to me by the Feynman Lectures, is the random walk. …

If you look close enough, our sky is just a bunch of particles bouncing off each other. These particles all follow predictable patterns of motion. We can easily picture a point mass moving through the air and calculate how much kinetic energy it has. This particle will have a few…

You may have heard of Freeman Dyson, a 20th-century mathematician who made enormous progress in quantum electrodynamics, along with a variety of other fields such a number theory. He worked extensively with Richard Feynman, translating his thoughts into mathematical equations. Dyson is also known for his big ideas about the…

One approach when studying the history of mathematics is to view it as a series of unification. That is taking seemingly unrelated concepts and connecting them in novel ways. Some examples include Galois Theory (fields and groups), Set Theory (numbers and logic), and Analytical Geometry (algebra and geometry). I hopefully…