Chaos: Good For More Than Absolutely Nothing

Originally Posted by Meridyth Mascio and Tom Plick

Chaos, War, and Disarmament

Debate has raged over whether wars are due mainly to long-term causes (oppression, economic depression, nationalistic zealotry) or whether they only find their genesis in freak events (the murder of a ruler, the sinking of a fleet). Many believe the former; among these people was Lewis Richardson, who, in the 1930s, developed a simple model of an arms race between two countries. He related the size of a nation's stockpile to its willingness to go to war, for two reasons: Firstly, arms could be easily qualified, whereas the "feelings" of a nation were not easily transformed into numbers. Secondly, the larger a nation's stockpile, the greater its chance of winning a war, and thus, the greater its chance of starting a war. (When each nation's stockpile is sufficiently large, the system becomes crisis-unstable: each nation wants to attack the other(s) first, lest the other(s) make the first blow.)

Richardson's model was the first attempt at modeling an arms race mathematically. Many analysts point out the inapplicability of Richardson's model to modern conflicts - the model mishandles many situations, because of its simplicity and the assumptions made to secure that simplicity.

Others after him, including Alvin Saperstein, took Richardson's idea and elaborated on it, in an attempt to model reality more accurately. Saperstein modeled an arms race between three nations under two systems - in one, the nations were allowed to ally as they saw fit, so that a superpower might find the other two nations allied against it. In the other system, the nations were not permitted to ally, and thus were at odds with both of the other nations. Saperstein's model was complex, and non-linear; recall that non-linearity breeds chaos.

Saperstein showed the system permitting alliances to be stable - in time, he says, each country's stockpile will dwindle to zero. Saperstein uses this observation in support of encouraging cooperation between countries instead of competition.

More interesting than the alliance system is the system that disallows alliances.
In the independent-nations scenario, several outcomes are possible, depending on the initial stockpiles of the nations and the "fear and loathing" values between the nations, plus many other parameters defined by Saperstein.

1. Strong stability: Each nation's stockpile decreases to zero (that is, in the limit, not necessarily in any finite amount of time).
   
2. Weak stability: Each nation's stockpile decreases to some non-zero amount.
   
3. Weak chaos: The system has a strange attractor, but its basin of attraction is small, and so the system is still largely crisis-stable.
   
4. Strong chaos: The system has a strange attractor, whose basin encompasses the entire system.
   

According to Saperstein, the strange attractor signals crisis instability, since once on a strange attractor, what happens in the long-term is anyone's guess. The way to maintain peace, he says, is to avoid the appearance of the strange attractor. To him, knowing chaos is useful for avoiding it in world affairs.

You can see our Powerpoint presentation again here.

Notes on Universality and on Colors

Originally Posted by Tom Plick

Even after Feigenbaum discovered the constant that bears his name, I find it amazing that he could use it to predict the bifurcations of all sorts of different functions - in our experiments, the numbers did not converge quickly, and so there was likely a lot of error involved there.

We talked a lot about the similarities between different types of physical phenomena - for instance, between phase transitions and the onset of turbulence - and we said how the mathematics at the boundary is similar in both cases. This may be, but even if the equations for the two phenomena match up, I still don't see how the wide-reaching concept of universality can pair them. Phase transitions involve changes among the solid, liquid, and gaseous states of matter; turbulence involves a multiplicity of frequencies in an oscillating fluid, resulting in "turbulent or not turbulent". I see no way to parallel these two problems, between three states and two states.

I agree a lot with Goethe's views on color, in particular that perception of color is subjective. I read an article two months ago, talking about linguists' analysis of color words in different languages. In ancient societies, some say, the people had fewer words for colors - the Greeks had only identified a handful. It is interesting to think that perhaps, back then, they actually couldn't see as many colors as we can today. That would be very subjective, no?

Information about the linguistic analysis of color words can be found at the Straight Dope; there is also a nice chart here.

Pondering Life...A Reaction to Non Serviam

Originally Posted by Meridyth Mascio

In the personal struggle to determine whether my mind has been created to think mathematically, I have started to look at my general feelings of disdain for computers. In actuality, I realize their usefulness, but I struggle with my one-on-one daily encounters with them. It has always bothered me that if a program stopped running or an error occurred, then I could not obtain a direct response to my questioning what had caused the error. I think that I now see that I find it unnatural to communicate with this entity as it does not possess consciousness (a term described frequently in Non Serviam).

A computer does not perceive the contradictions that qualify humanity. Logic is not the only component of a person; beings are diverse. As mathematics underlies the structure of the world, the "chaotic" element are the perspectives of human beings. We see things as we perceive them, and so the world is defined intrinsically to each individual as such. However, are these decisions and perspectives "programmed" into us? If so, can is it possible for us to ever replicate this program?

So many other questions remain in my mind from this reading: Why have we been "created"? What is the general purpose of humanity? What is my individual purpose? Are humans guided by some unknowing force? How do religious beliefs factor into our existence?

I had never before considered the idea of an "intermediate" God, where the one who has directly created us has been created by yet another "higher" power. Is this chain infinite? Can a process then ever be broken down to one?...

These questions plague my mind. I feel as though humans are trapped in the midst of incomprehensible infinities, never able to grasp the full extent of everything in the universe and yet not ever quite able to break down one situation, one entity completely (some sort of fractal nature).

What I do realize is that the scope of the world is beyond me; more specifically, I think that the reality of one's individuality is beyond any limited human perspective.

Journal about Fractals

Originally Posted by John Sehi

When I first thought of dimensions, I thought about the usual 1-D, 2-D, and 3-D. I did not examine it farther, did not think that there would be a need for any more thought. A fractal dimension was something I didn't know about. I didn't really question their existence, but more of something that's not talked about. Since learning about the Middle Thirds Cantor Set in which is less than one dimension and the Sierpinski Triangle that is between 1-D and 2-D, this just fascinated me. Especially the Menger Sponge which has infinite area but no volume.

I am surprised at what can be done with fractals. There are so many possibilities with fractals that it is sometimes hard to comprehend all that can be accomplished with mathematics.

God

Originally Posted by Matt Venanzi

It never ceases to amaze me, even before "Chaos & Fractals," how random, yet logical, strange, yet sensible, intricate, yet explicable the world is. Could God just be watching us discover every little intricacy he has made about the world, like a proud parent, or an accomplished inventor, marveling at their "creation"? What, then, is still left to be discovered? And I mean that, not in that we have discovered it all, but in that we have barely touched the tip of the iceberg!

Is it laughable, then, that we think we are so far "advanced"?

But how could God, Him Herself, have done all of this?
It just seems like too much! Then again, is it all relative?
Do we have the capacity to fathom all that there is?
Is that "capacity" expanding ~ with each generation"? "Evolutionarily"?
Do some have more "capacity" than others?

This all seems so esoteric, yet comprehensible-----
I guess its just another one of those woes of God...
I'll keep trying...

Is God a Mathematician???

Originally Posted by Rachel Hensey

Answer: NO.

It is true that God created the universe which is perfect in every way. The universe is without a doubt mathematically correct. Many people would argue that because God is the creator and the universe is mathematically correct then God is a mathematician. However, the definition of a mathematician is someone who studies how things work. Since God is all knowing and already knows how everything works, there is no need for him to study. There is nothing for him to solve. Thus, God is above mathematics. He is NOT a mathematician.

Chaos Is Everywhere

Originally Posted by Pat Rafferty

Click here for interactive NASA site

It is incredible to see the wide variety of subjects that chaos has an affect on. Things that could not normally be understood are simplified under the study of chaos and fractals. For example, the giant red spot of Jupiter is explained as a calmness that can be found within the disorder of the gaseous system. This perfectly explains the idea of chaos, a structured order found within seeming randomness. Chaos theory even has conceptions on other areas such as weather prediction, population growth, and turbulence.

On Non Serviam and Personoids

Originally Posted by Sean Houlihan

The concept of "personoids" from Non Servium is really interesting. What does this mean about God, life, etc? Could we be living in a fractal-like, self-similar world? Could our decisions be governed by mathematics?

In essence I hope nope. The reality portrayed in this work is a little scary. I would hope certain aspects of the fractal nature of personoids is not present in reality.

Notes on Two Bradbury Stories

Originally posted by Tom Plick

At the end of "A Sound of Thunder," I was amazed at the slightness of the changes that occurred because of Eckels' killing the butterfly. The English language was spelled differently, but still pronounced the same way. Everyone who had existed "before" the trip still existed afterward. (Think of that old conceit in the movies, where someone's parents never meet, and so the person is never born.) It boggles my mind that such small changes are all that take place over thousands of millennia.

I read another of Bradbury's stories in grade school, entitled "All Summer In a Day." It is about a group of kids in an underground society on Venus; they live underground to avoid the constant rains. The rain only stops every seven years, for an hour at a time. The kids in this one class are skeptical about the outdoors, except for a frail little girl named Margot, who has known the wonders of nature and misses them dearly. When the teacher prepares the kids to go outside during the brief window, the kids hatch a plan against Margot.

I won't spoil it for you - you can read the story by clicking here. (It's about four pages.) I will tell you that years later, I am still turning it over in my head, pondering its true meaning.