I'm interested in smoothing out the spaghetti-mess language that us humans use to try to logically formulate exactly what is really being said. Not as in a logic class, where language is converted to logic symbols, but to understand certain words better. Words with extremely ambiguous meaning: love, empathy, and respect for example. In this case, the word "complexity". There are instances where "complexity" would not work in a sentence but "difficulty" would work. So there is a segregation there. In other instances, "complexity" and "difficulty" are interchangeable.
I use the word "complexity" all the time, and so at some level I do understand it, yet logically I cannot give an exact definition of "complexity". Dictionary.com says that "complex" is something consisting of "interwoven parts". "Interwoven" is used as a metaphor there. No definition can be accurately defined using metaphors. Complexity is not made of any type of fabric, so it therefore cannot literally consist of "interwoven parts". The definition also says that complexity is interconnected parts. What is the difference between "interconnected parts" and "connected parts"? There is no difference that I know of. The word "interconnected" seems like nothing more than a longer version of "connected". So I believe the dictionary means that complexity is only "connected parts". But if complexity is "connected parts", then how are some things more complicated than others? Either something is connected to another thing or it isn't, right? Or maybe there is more than one connection with two given parts, so it is more complicated? Surely "connected parts" satisfies many uses of the word "complex", but I don't see how it satisfies all uses. You havn't defined a word until you satisfy all of the common usage for the word. A lot of people would think of "math" when they think of "complicated"... I do. I suppose you could refer to each step in a math problem as a part. And I suppose the more steps there are in a math problem, the more complex the problem is. So maybe the definition does really apply to nearly every instance of the word usage. I believe "complex" and "complicated" are the same thing. However, I don't believe complex and difficult is the same thing.
Is a problem with four steps twice as complicated as a math problem with two steps? I would guess so. In that way, complexity can be mathematically calculated. How complex something is would be a counting problem. It would be interesting if that were the case, as if someone asked you: "how complicated is the problem", perhaps you could respond: The problem is 8 complexity. Perhaps you can mathematically calculate exactly how complex something is by counting the number of connections of one part to another.