For a mathematician or scientist, a natural step after identifying something important is to attempt to quantify it, in the hopes of determining some of its properties. Us humans actually have a decent intuitive sense of different quantities of complexity. For example, we could all agree that Mozart's 40th is more complex than Twinkle Twinkle Little Star, or that sovling a crossword puzzle is more complex than tying a shoe. Other comparisons are less clear: Is a horse a more complex animal than a lion? Is China's economy more complex than India's?

Complexity researchers have identified several different ways that complexity can be quantified. These measures roughly fall into three categories:

- Variety - the complexity of an object can be quantified in terms of the number of actions it can take or the number of states in which it can exist.
- Descriptive complexity - the complexity of an object can be quantified in terms of the length of the shortest complete description of that object
- Algorithmic complexity - the complexity of a process can be quantified in terms of the number of steps or amount of time required to complete that process.

This approximateness is a problem for many scientists, who are used to dealing with the exact. How can we apply our analytical tools to a quantity which can never be precisely measured?

The way forward, in my opinion, is as follows. We (complex systems researchers) will investigate abstract models in which complexity can be mathematically quantified. The goal of investigating such models will be to discover laws of complexity which may carry over to the real world. At the same time, these laws must be checked against real-world experiment and observation. Because of the semi-fuzzy nature of complexity, the laws we discover will likely not be quantitative (e.g. F=ma or e=mc^2), but qualitative (e.g. "energy is conserved.")

In the near future, we will investigate two examples of such laws: Occam's Razor (the simplest explanation is the likeliest) and Ashby's Law (an organism must be as complex as its environment.) In the meantime, can anyone think of other qualitative laws of complexity/complex systems?

Sorry this isn't answering the question you asked, but I just wanted to say I just started reading your blog and I love it so far. I'm just a high school student in calculus right now, but you do a good job of explaining it to someone not on the bleeding edge of the field without it seeming like it's being dumbed down.

ReplyDeleteThat's great! That's exactly what I was hoping this blog could be. Keep reading!

ReplyDeleteYeah, I'd like to second that emotion...your voice is very similar to Hofstadter's in

ReplyDeleteGEB, just like I thought it would be!Awww, I love you guys!

ReplyDeleteYou're off to a good start, that's fer sure! :-)

ReplyDeleteOn the subject of math and complex-systems blogs: Cosma Shalizi writes interesting and informative things, but doesn't allow comments. The math-oriented blogs I read most frequently are, in no particular order, God Plays Dice; The Unapologetic Mathematician; Good Math, Bad Math; and Shtetl-Optimized.

It is great to spot your blog. I am in the search of how I can do what you do.

ReplyDeleteGreat! I love it.