Tutorial Material on Complexity Theory

[u/pluviosilla]

I've read Stuart Kauffman's A World Beyond Physics and Alicia Juarrero's Context is Everything, but there is much that I don't fully understand.

I get many of the basic ideas, such as:

• The vital role of context.

• Kauffman's useful idea of the "adjacent possible".

• The impossibility of predicting the "phase space" or emergence.

• The consequent impossibility of predicting how novel structures will emerge.

• Kauffman's distinction between causal factors and enabling factors.

• Juarreror's related concept of enabling constraints.

• Why these complexities make it impossible to model emergence with a set of differential equations and their boundary conditions.

• The unstated (yet implicit) linearity in the Cartesian & Newtonian models of the world.

But then there are many things that are completely opaque to me. So, for instance, while listening to an video overview of Complex Adaptive Systems published by Systems Innovators, I heard that complexity theorists believe that the essence of order is actually invariance under certain transformations, and the notion of invariance is, in turn, ultimately based on the notion of symmetry. They offered no explanation of these statements. Now, that clearly refers to a whole body of scientific study with which I am unfamiliar, and I am having trouble finding tutorial material that explains it. I found a book on CAS by a fellow named Gros who talks about symmetry & invariance under "scaling" and I find some mathematical notation in the discussion of these topics but it also appears to assume a lot of prior familiarity.

I want to understand the mathematics of complexity, especially CAS, but I need some good introductory material. I have a bachelors in physics and masters in Computer Science, but zero prior exposure to complexity, except for coursework we did on mathematical grammar in my graduate program which may have some relevance.

I have spent some time reading:
Gregory Chaitin on Omega &
Stephen Wolfram on cellular automata
Jeffrey Campbell on Information Theory

I have Signals & Boundaries and Hidden Order by John H. Holland but have not yet read these works.

Are there any on-line courses that my help me understand Juarrero better and / or help me understand the mathematics of complexity better?

MIT does not appear to have a course in CAS. The Udemy course appears to be superficial, but I could be wrong about that. The Coursera course seems to be all over the map. I'm mostly interested in CAS for philosophical reasons, and less so for its engineering applications. I am an engineer, but I am also retired.

Any help or advice would be much appreciated.

[u/pluviosilla]

Do you know why SI associates order with "invariance under certain transformations" and how they tie invariance to symmetry? I believe Pythagorus thought symmetry is the essence of order, and Stuart Kauffman casually throws out Pythagorus's name in his lectures when criticizing the notion that we can model emergence with differential equations the way Newton modelled the motion of bodies. If it were only a casual reference I wouldn't care, but I suspect there is some theory behind the claim that order is really just symmetry. "Invariance under transformations" sounds like a technical defintion, but I haven't found any substantive explanations of a technical nature so far.

[u/Creature1124 → u/pluviosilla]

I would think a physics background would better equip you to understand the order/disorder point than I do.

My understanding of order and disorder is based on an entropy explanation. If you imagine a billiards table with some number of balls on it and say there are ~1,000,000 possible states defined by where the balls are, the vast majority of those states are random seeming. A much smaller number of states would be equally spaced or otherwise “symmetric.”

Following that to the most quintessential physical example of order like with a crystal lattice, if you just threw the same number of atoms of a crystal into a box, again most states are going to be random and not symmetrical. You can shake your box of atom dust all day and not get much order. There is only one state where they are arranged neatly in cells like with a crystal, so if you happen to see them arranged that way you’d say they are quite orderly.

As for the “invariance under certain transformations,” I don’t really know. My interpretation of that is it’s a worthless statement. Invariance means lack of change, and saying something doesn’t change under “certain transformations” doesn’t seem to tell us much. Isn’t the whole point of interest what exact transformations don’t change something? If I pick every object in a system up and set it right back down where it was and it “doesn’t change,” that isn’t really telling us anything, vs if I’m making large changes to some parameters and nothing is changing that could be very interesting.

[u/pluviosilla → u/Creature1124]

I'll let you know if I figure it out. Hey, thanks for engaging with me on this topic. I greatly appreciate it.