28 March 2012

Thinking Broadly

I just read an article called “Overthinking”, which spawned some interesting discussion on Hacker News. I was rather disappointed in the article, because we do overthink, but not in exactly the way the author suggests. So here’s what I think I think—if I haven’t overthought it.

You cannot deliberately solve a problem without thinking about it. Even in the rare case that a solution arises from your subconscious for a problem you didn’t realise you had, your brain has been processing patterns and, as it were, thinking for you. This happens whether you like it or not—and for some reason it happens extraordinarily often in the shower.

Now, whenever we set out to solve a difficult problem, we use our powers of reasoning and intuition to guide us in the direction of a solution. Starting from the origin of a problem, we use thought to propel ourselves through the space of possible solutions in search of local maxima:

Overthinking is not thinking too much, because that’s simply not possible. Rather, to overthink is to doggedly pursue a single direction as far as possible before moving on, like the non-solution above. That direction might eventually lead to a good solution, or even the very best one. But there are solutions in other directions that require much less work to get to.

That’s why “overthinking” is often a waste of time: only when you know the direction to go should you put the blinders on and get to work. That’s essentially why I’m more of a starter than a finisher: I find it much easier to discover paths than to persist in treading them. But again, overthinking is not a measure of thinking too much or too little. It’s conducting a depth-first search when you ought to be searching breadth first.

Children are undoubtedly better than adults at avoiding overthinking, and the reason is simple. Without having had the chance to develop specialised knowledge, their solution spaces are relatively shallow and uniform:

Children don’t know as much as adults, so they give equal weight to most of the directions they could take when solving a problem, and thus are better at appearing observant and intuitive. We generally use intuition to decide our angle, and knowledge to go the distance—polar coordinates, literally. When you have knowledge in a specific area, you often discount alternative ways of thinking about a problem—in other words, if all you have is a hammer, everything looks like a nail.

But that’s not necessarily a bad thing. A lot of the time, your problem is in fact a nail, or something very like one, and your hammer will work just dandy. The idea that knowledge is the problem couldn’t be further from the truth. No, the real enemy is closed-mindedness.

Take this number puzzle from the article:

To an adult, puzzles involving numbers are almost always seen as mathematical. Through years of exposure to math problems involving numbers, we condition ourselves to solve numerical problems mathematically. Our solution space is specialised. We see a numbers problem, and we know with great certainty that it must be a math problem.

However, this cannot really be too mathematical a problem, since it “can be solved by preschool children in five to ten minutes”. To a person without significant mathematical experience, numbers are used primarily for counting. And when numbers and letters are both fairly new to you, you place them in basically the same category: visual figures. By thinking along these lines and keeping an open mind, anyone can arrive at the solution straight away.

I must admit that for this particular problem I have a bit of an unfair advantage. Since I have grapheme→colour synesthesia, I see figures with different features very differently from one another. Thus I can’t help but notice that the number on the right is just the count of enclosed loops in the figures on the left. The experience is hard to describe, but it’s for basically the same reason that a non-synesthete can’t help but notice the same fact in this image:

The author writes: “…to become truly cre­ative you need to be able to lib­er­ate your mind from the the shell of knowl­edge, edu­ca­tion and adul­ti­fi­ca­tion you have accu­mu­lated”. This is, I think, too narrow a view—there is nothing wrong with having knowledge and using it, nothing wrong with listening to your education. There is also nothing wrong with being an adult, but I’ll pass.

The key is to keep an open mind, to judge each new thing on its own terms. To relax and let our work tell us how it ought to be solved. And above all, to think broadly before thinking deeply.