*The mystery of their intentions lies only in their terms* - Thomas Vaughan

The terms "complexity" and "chaos" are often times used to describe something that is not understand. But that has nothing to do with the "understandability" of the thing being described. The same is true for the next generation of anything - Management 3.0 and Project Management 2.0 are two examples based on the terms "complexity" and "chaos." Neither of which are based on the foundations of the actual science of complex systems. Instead are based on the "populist" notions without the benefit of the underlying mathematics of how complex large scale systems actually work.

So when there is a quote like "it takes complexity to handle complexity," it's not exactly clear what kind of complexity is being talked about.

One of the most pervasive an least well-defined concepts today is the notion of a "system." In a vague. personal way, almost everyone who thinks about it can visualize what a system is. Sometimes they can even verbalize this image, "elements" interacting in a "complicated" manner in order to achieve some fuzzy "objective."

Sometimes two separate descriptions actually overlap within enough in common to have a meaningful discussion or collaboration. But more often huge gaps appear - intellectual, professional, and cultural. Gaps too great, communications breaks down despite good intentions and all the parties.

**Mathematical Descriptions of Systems**

There are many different descriptions of systems and their complexity. They include:

- Compactness - a mathematical description can eleminate fuzzy or ill-thought-out notions that may not be apparent in a lengthy verbal description.
- Clarity - by associating a symbol in the mathematical description with a known aspect of the process being studied, clear interrelationships between the various process variables can be revealed.
- Computability - once the mathematical description is agreed to, this description can be manipulated using the rules of logic with nontrival results. This model provides the basis for various computational studies as well, with the hope of a predictive outcome.

**Schools of System Descriptions**

The principle types of mathematical descriptions of systems include:

- Internal descriptions - since the time of Newton, these are standard mathematical descriptions of a dynamic process using differential equations. These describe the time evolution of a system in tyerms of their variables - position, velocity, temperature, etc. The key here is these descriptions are
*local*. The equations describe the process in the local neighborhood of the current state. While this works in the realm of physics - for the most part - it does not work so well in areas like social science or economics. Before Newton the concept of*local*was unheard of. Physical processes were dominated by Aristotelian views "*in the order of Nature the State is prior to the household or individual. For the whole must needs be prior to its parts.*" Modern physics does exactly the opposite. The whole is explained in terms of the elementary (local) parts. - External descriptions - this is the common system found in experimental sciences - the input-output relationships. This description is the opposite of the
*local*description, since all the local information is masked "inside" the system and only the Inputs and Outputs are visible to the observer. - Finite State descriptions - are a class of system analyzed by algebraic means. The number of dimensions in the system is replaced by the number of "states" in the system. The states are finite and are processed by a "finite state machine."
- Potential and Entropy Functions - are an alternative to the internal and external descriptions.

**Complexity**

One of the most overloaded terms in software development is "complexity." This term is used in a variety of ways, "complex systems," "degrees of complexity," "complex problems," ad nausium. But there is usually very little to indicate the authors really has any true understanding of the principles of complex systems beyond repeating the words from populist books and magazine articles. By populist I mean texts that have abstracted and obfuscated all the mathematical underpinings of "complex systems" theory.

Like the concept of time everyone has an opinion of what it means, until it is necessary to actually define it in the absence of analogies and words. There is much folklore around "complex systems," and "complex adaptive systems." Even famous names mentioned in support of this folklore.