On Problem Solving

Welcome back to our regularly scheduled programming.

Recently, I was rereading an old, old post over at the GoodMath blog (http://scientopia.org/blogs/goodmath/) dealing with the Pumping Lemma and regular languages (http://scientopia.org/blogs/goodmath/2011/05/01/what-if-its-not-regular-pump-it/). I had read the post in the past, but really didn't understand it. I was overwhelmed by the notation and when I get overwhelmed, it's easy for me to skip past important parts. During that reading, it was frustrating and I didn't feel like I was smart enough to know what he was talking about, and that really made me annoyed. But I didn't really feel like there was anything I could do.

Today, I reread the post and even at a glance, I understood it much deeper than ever before. What had changed? There are a few things. This post is not about what changed, but rather how I went about figuring out how to solve problems of a much deeper kind from the normal bugs and defects I fix every day. My hope is that you will come to understand this process of discovery as well, and will benefit by applying it to your everyday lives too.

First off, what I noticed today is that instead of getting frustrated by notation, I simply saw the notation for what it really was: A tool for condensing a larger piece of information into a smaller one. Because of this, I knew that all I really had to do was remember what the notation stood for when reading the rest of the post in order to understand what it all meant; in a sense, I was compiling the meaning for each piece of notation into the new parts of the post and translating it thus.

Secondly, I broke everything down into idea-sized chunks and ensured that I knew the ideas in the order presented before moving on to the next idea. In a way, I view this as a form of ratiocination, or the art of thinking precisely. Always know that at any point in a particular paper or article or book or codebase that you know what exactly is happening at that point in time, and understand the inputs and the outputs at each level. Without that information - without even some trivial piece of context - it's possible that a wrench is thrown into the works, rendering any solution you come up with invalid.

With these two tools combined - ensuring that you know precisely the meaning of the symbols involved and breaking down the problem into smaller parts that you can prove connect to each other at points - I became intelligent today. I've never truly felt that way before. I'd felt the process of discovery as the most beautiful process on earth, but I never really felt like I was actually intelligent, because I didn't know to do the above.

I feel like heretofore, I had been simply taking it for granted that I understood what was happening, and, without trying, would know what the problem was at every point in time. This couldn't be further from the truth. What I realized, ultimately, I think, is that I cannot rely on intuition when trying to solve problems. I should instead always break it down with the steps above, and only then can I really understand and approach a problem intelligently.

I think that's a real issue for others as well. I think people think that an answer that is pulled more or less out of their asses is good enough, because hey, they're always the smartest person in the room anyway, right? That's just not how it works. The people you see and know to be really bright people, they don't get there by not working. The truly intelligent ones work hard to keep being good at what they're good at, and continue learning day by day, pushing themselves for bigger, better things. They're not simply complacent with what they know. And they approach problems in ways that work for them, which is what I really want this post to be about: Approach problems in a way that works for you and resolves to correct solutions, and push yourself. I know I have started doing that, and I'll never look back.