Mathematics
The Four Color Theorem: If you were to draw any number of lines on a piece of paper, no matter the configuration, you can always color the regions with only 4 colors, such that no two of the same colors touch. This isn’t too surprising, and in fact cartographers have known about this fact for centuries. Yet it is so complex that it was only proven to be true in 1976 with the aid of computers.
As a 12 year old, my computer programming tutor, a physicist by trade, introduced me to the Four Color Theorem. As a child I thought of math as just increasingly advanced arithmetic, and was curious about how an idea like the Four Color Theorem could even be mathematically thought of. Coloring maps is far different from arithmetic after all. Seeing my curiosity, my tutor continued to introduce me to more complex mathematics.
Let’s play a game. Pick any whole number, such as 2, 17, 34, etc., and if it is even then divide it by 2. If it is odd multiply it by 3 and add 1. I bet that if you repeat that procedure enough times, you will eventually reach 1. This bet is called the Collatz Conjecture.
This seemingly simple idea of eventually reaching 1 turns out to be so complex that to this day, it is still unproven. In fact it isn’t known if it is even true. We just haven’t found a number that disproves it.
I first heard of this problem in sophomore year of high school and it quickly occupied much of my time. When I had free periods between classes I would scribble the many connected sequences of numbers down on the whiteboards of the math tutoring center. I learned to read academic math papers in an attempt to find some pattern in this a-mazing sequence. When I couldn’t sleep I continued trying to find ideas in it, struggling to keep enough numbers in my head at once. I never got anywhere close to a proof.
What are the rules of math? What reason does 1 + 1 have to equal 2? The answer to this question, is one of the most beautiful ideas. Gödel’s incompleteness theorems are two realizations about the nature of mathematics and its provability. Consider a series of rules that the rest of mathematics could be based upon. The first theorem states that for any system of these rules, there will always be mathematical truths that are unprovable. The second theorem states that the system of rules cannot prove itself. This means that the things that we can have knowledge about are directly dependent on the rules we base them on. It means that nothing can be proven without the construction of arbitrary rules. The things that we take to be true are based on a simple set of beliefs that we have no reason to believe, and if we only were able to change our beliefs, we would find a whole new truth, a new reality.
https://commons.wikimedia.org/wiki/File:Four_Colour_Map_Example.svg
