Gödel’s Incompleteness Theorems

Godel’s Incompleteness Theorems are some of the most important results in logic, that I’ve used in many forms day to day. “any formal system (such as a system of the natural numbers), there are certain true statements about the system which cannot be proven by the system itself.” https://listverse.com/2013/05/05/10-coolest-mathematics-results/ This, although seemingly technical, means to solve a problem you need to look outside the problem itself. It isn’t necessarily the theorem itself, but the state of mind. We often think that we can find the problems and solutions of a problem on whether something is true or not, without looking past the system itself. It is like determining whether the legal system is just based only on the legal system.  Which leads us to the other Incompleteness Theorem, which basically says you can’t determine if something is consistent based on itself. This comes in handy whenever you are trying to figure out how to handle an issue.

The first step when presented with an issue is to find a good goal, or achievement you want to achieve, even if it’s broad, like, betterment for society, as long as there is an outside goal, you can work backwards to determine if you are considering everything.

The other great use, is that you can never fully figure out the impact of a system by looking at the system itself, consider iphones, while we may enjoy them, is our enjoyment worth the bad conditions of the people working to make them? And what can we do about it? Math is at its heart all about problem solving, and I feel like sometimes it gets undervalued to memorization of what dead people figured out and numbers, but really it is about solving problems given axioms. And this result tells us, we never have it all figured out. (and even if we thought we were at least on the right track, we wouldn’t know it)

Prisoner’s Dilemma

Despite our best efforts, it seems math is always there to thwart our brilliant ideas. Whether it be believing in lucky dice rolls or getting on a streak in blackjack, math is there to remind us, it’s all probability. A poor man once said… ” Always bet it all when you are on a winning streak”.

Despite math and logic, we are always looking for the psychological edge and interrogation is a prime example. Consider a scenario where there are two suspects who are believed to have worked together to commit a crime. A clever cop thinks, I know I’ll reward them for telling the truth. The cop individually goes to each suspect with the following…

If one confesses and the other doesn’t, the one who confesses gets off for free, while the other does 10 years. if both confess you each get 6 years, but if neither confess they each get only a year of jail. What would you do?

Believe it or not, you should confess every time. This is because no matter what the other person does, confessing lowers your jail time, and same with the other suspect. This means they will both get 6 years in jail if both individuals act rationally, and self-interested. Whereas cooperation would get them only 1 year!Image result for prisoner's dilemma

This is the so-called prisoner’s dilemma and despite the example being quite specific, it actually comes up quite a bit. It occurs whenever two parties acting bad, is better than a party acting good, while the other is bad. (think politics). This is an important concept which I try to consider whenever proposing an system for people to use, you need to make sure there is an incentive for cooperation rather than taking advantage of others.

This is the sort of thinking you need to have whenever you are in charge of competition, and recognizing little kinks in your proposal which allow for such stalemates can often make a seemingly good plan fail miserably, all because neither group wants to help the other. In many ways this is similar to the government shutdown, in that both groups don’t want to concede, and thus they both end up hurting each other rather than allowing productivity to happen. This is why I love math, because it shows up when you want it to and can often make things more robust, but at the cost of clarity and understanding unfortunately. That needs to change.

Pigeonhole Principle

Deconstruction

Math… equations, formulas, integrals, trigonometry. You’re not alone if some of those words give you cognitive dissonance. Unfortunately, math feels like the study of boredom rather than the study of patterns. That is not what math is. Math has had the terrible rap of being all about computation and memorization of complex formulas for solving problems. But why? I think that a lot of it has to do with the fact that it’s harder to teach somebody an idea than it is to teach somebody a formula to remember. But that’s not math. Anyone can remember a formula, but how many can derive it, or make it more general/better.

Construction

What is math? Unfortunately, math is tough to decipher. The reason I like it is not the reason other like it, which is not the reason society likes it. Often times it takes so much math for one to see its benefit that it’s not worth going that far into it past basic arithmetic, and maybe trigonometry. I want to build a different perception. I’m gonna talk about a part of math I really enjoy and tell you how it relates to everyday life, I hope it doesn’t get too boring, but it’s math so no promises!

Implementation

There are two people, (non-bald)in New York with the same number of hairs on their head.

How would one even go about proving such a thing, would I need to look at every person in New York and count each of the hairs on their head? How much time would that take? What if I miscount? What if midway through counting somebody grew hair/got a haircut? Who would actually let some random guy count the hairs on their head? Why does anybody care? All jokes aside, that last question is both the most and least important question on this list, most important for obvious reasons, but least important because it impedes our ability to solve this thought experiment, surprisingly, a majority of math is thought experiments which prepare you for real questions that people care about. Without too much commentary, onto the proof!

It is known(weirdly, that the maximum number of hairs on one’s head is roughly 500,000.) There are currently 8.5 million people in New York. So if everybody had a different number of hairs on their head, there would have to be somebody with 8.5 million hairs on their head, impossible!

Although the answer was simple and the question useless, it is the process and way of thinking outside the box which drew me to math, so despite its simplicity, the process is actually quite deep.

Introduction

Passion at first I thought of as a hobby, but as I struggled more and more I decided to extend this to include lifestyles. To be clear, I love math. Yet, today I feel as though one gets more relate-ability and humility tied to the mantra “I was always bad at math…” I swear this is the first, most common expression after people learn that I’m a math major. Usually followed by, “what exactly does a mathematician do?” But every time that mantra is uttered, I know that we’ve failed. Not because math didn’t appeal to that person, but because I’m willing to bet that person never even learned what math really is…