Note on the meaning of “complexity”
In an email discussion I am involved in the question was rasied if “Complexity” Theory was related to “Complex Numbers”.
Normally when the various books about complexity talk about it they are not specifically thinking about complex or imaginary numbers. They are talking about a determinate set of things which is not just complicated but complex. But “complexity” has the other meaning of imaginary numbers which is usually not meant.
Here is the problem. When “complex” or “imaginary” numbers were discovered they did not know what to do with them and so they just gave them random names that signified that they were hard to talk about and think about. They were “imaginary” rather than real. They were “complex” rather than just complicated or simple. But it turns out that these names although just meant to distinguish them from what we were used to in terms of mathematics were apropos in many ways to the nature of the numbers themselves and the fact that they were useful for describing physical phenomena meant we started thinking about the phenomena that they described along those lines such that there was a feed back loop between how we conceived them and what they ended up signifying and how we understood the phenomena to which they were applied.
If complexification means “complex or imaginary numbers” and phenomena that lends itself to being described by that kind of mathematics then this changes what is normally meant by complex dynamics or complex systems. Complex as it is used normally for systems means we don’t understand them and they are so difficult to understand that we will never untangle what is going on in the phenomena so we need to consider them as a whole rather than in pieces. While complicated means that it is like a mechanism but it is just a rather large and difficult to understand mechanism but ultimately it can be unentangled and understood determinately and discreetly if we were to take the time and effort to do so. Complexity means that we do not believe that there is enough time, effort, or computing power to understand the system discretely due to the entangled and enmeshed and ensnared nature of its relations.
Let me give an example I just learned about. Groupoids. 3-element groupoids have been completely determined 3-ELEMENT GROUPOIDS. But 4-element groupoids are seen as beyond the possibilities of computation. And what is more distrurbing is that there appears to be some of them which are themselves not finite at the 4 element level and so not only would it take too long to compute them given current computing power, even if we could there might be anomalous 4 element groupoids that themselves were nonfinite, which is very strange indeed. 2 elment groupoids are determinate. 3-groupoids are complicated but still discrete. 4 element groupoids are complex because even if we could compute them all, i.e. we had much faster computers, we would still find groupoids among them that were not determinate in the finitary sense. So given this mathematical example it becomes clear what the difference between determinateness, complicatedness and complexity is and the “imaginary or complex” numbers play no role here in this case.
However, if we take complexification to mean that we have somehow crossed the threshold into a phenomena that needs the “complex and imaginary” numbers to sort out and describe, then we have entered a whole different kind of complexity which is the realm that I call Speical Systems Theory.
That is a realm in which the Hypercomplex Algebras provide the principle of organization. And I don’t believe that is what most people mean when they talk about complex systems, but it is definitely a variety of complexity but only one kind among others. It is an important kind because it is determinate but entangled in a way that cannot be unentangled. The entanglement is determined by the complex numbers and the higher hypercomplex numbers which are Quaternions and Octonions and even Sedenions. Basically what I have done in Special Systems Theory is work out the Systems Theoretic meaning of these mathematical structures for this kind of Complexity. For further information see my electronic book athttp://works.bepress.com/kent_palmer on Reflexive Autopoietic Dissipative Special Systems Theory. I think there are many kinds of complexity and this is an important one which is actually based on complex numbers, but there are other types of indeterminacy, entanglement as well and I think that when people are thinking of complex systems they are not specifically thinking of systems driven by the hypercomplex numbers for their organization.
It is unfortunate that there is a confusion in the meanings of the word “complexity”. But it gives me a way to explain what I have been doing all these years, which is to explore the kind of complex system that is explicitly organized by hypercomplex algebras which is an esoteric type of complexity but I would argue a very important one because it is what makes Life, Consciousnes, and the Social possible. It is the kind of Complexity referred to by R. Rosen in Life Itself even though he did not realize it at the time. But the loops of entailment that he talks about as making biology possible as a science are a key part of the structure of hypercomplex algebras. Recently I presented my ideas on General Schemas Theory to the Page on incose.org working group on Systems Science. Just let me within that context that Special Systems Theory is also a ‘schema’, however it is a special set of sub-schemas that is between the System and the Meta-system and is partially Systemic and partially Meta-systemic. General Schemas Theory is the background for the exploration of Special Systems which are Holonomic in the sense of the term coined by Koestler.
See Emergence and Complexity in relation to General Schemas Theory
Meta-systems, Complexity and Emergence .ppt
For General Schemas Theory see Final thirteenth version for CSER Presentationand Final version E of General Schemas Theory short paper and Longer Explanation of Research Program
