*Murky sees his students at a table in the coffee house and
joins them. *

Murky: Good evening!

De: Hi!

On: Hey! Murky: Not as glum as yesterday?

De: No, but still confused.

Murky: Maybe I should not have gotten onto the thermodynamics kick yet?

De: Well, that's sort of right. I'm still struggling with the broader notion of hard science co-opting complexity.

On: Yeah, tell us more about that! For instance, why has this happened so recently if the stuff was always out there?

Murky: Computers!

De: Computers have been around longer than the big surge in complexity research.

Murky: True enough, but I meant that computers are improving so rapidly. When Lorentz discovered chaotic dynamics in his differential equations for modeling weather, he was first inclined to believe it was his computer. Then he convinced himself that it was real. Then he had to convince OTHERS that it was real. This, at that time, meant eating up lots and lots of computer time. Now you can walk over to Barnes and Noble and get an inexpensive book called "The Complexity Lab" and with any new PC you can have Artificial Life, Cellular Automata, Genetic Algorithms, Fractals and Chaos to play with in your own home or office.

On: So what's the big deal? Isn't science supposed to be a question answering enterprise?

Murky: I used to have a plaque in my office that said this: THE LAW OF THE INSTRUMENT: "Give a small boy a hammer and he soon learns that the entire world needs pounding."

De: You mean that technology drives research? That's interesting. Sure, most equipment is expensive and it would be easy to be tempted to design research programs around some big lab toy that cost big bucks. But computers are cheap by comparison.

On: But humans are fond of sticking to familiar things especially if others reinforce that tendency. That's what fads are all about.

Murky: I want to make a point here, but also want to be fair. These "fads" are really bursts of creative effort which has a great deal of value. But like in Brian Arthur's "Lock in" theory of economics, they quickly take on a life of their own and become "established". Then they may actually stand in the way of new questions being asked. That's the nature of the co-optation I'm trying to describe. The other aspect of it has to do with egos, power, control, and all those other "non-scientific" mushy concepts (he laughs).

On: Let's see if I understand. Hey! This is beginning to be fun. Will someone please take my temperature? OK, complex things are now in the domain of hard science because the computer has evolved to a point where they can now handle this stuff?

De: So what's wrong with that?

Murky: On is right and there is nothing "wrong" as long as we don't let the law of the instrument dominate. For example, spending a lot of time on the computer might not be all that productive in complexity research if real complexity involves non-computable things!

De and ON in unison: Non- computable things?

De: So now you are into mysticism?

Murky: Well, will you believe it if I use the very ideas which physicists and others have used to put up computer models. That's not mysticism is it? You see, when you do what Lorentz did when he discovered chaos, you are forcing something that has a non-computable part to seem like it is computable. Notice how the language has evolved in such a way as to obfuscate this central problem. Have you heard of the "butterfly effect"?

De: Yes, you mean sensitivity to initial conditions.

On: How does that work?

Murky: It is the heart of the matter. What troubled Lorentz was that his computer kept giving him different answers for the same set of differential equations AND the same set of numbers for initial conditions!

De: That's not possible!

Murky: Right. He THOUGHT he was using the same numbers for initial conditions, but the way things got done on computers, there were subtle undetectable differences entered into the analysis. They were so small they shouldn't have mattered.

On: So how COULD they matter?

Murky: Bifurcations!

De: Aha! THERE's you non-computable component and it is in NO WAY mysterious! On: Speak for yourself!

De: I see it all now! Wow! This is neat. The computer is numerically calculation a solution to set of non-linear differential equations. We know that certain forms of these non-linearities do something remarkable, namely produce bifurcating solutions.

On: What is a bifurcation?

De: Ok. We calculate the trajectory of a cannon ball using the same sort of method, but due to the nature of the problem, we get a single curve for the trajectory and we can predict where the cannon ball lands. We try to do the same thing for the trajectory of the water molecules in the Benard system that Professor Schneider described and we get the bifurcations he mentioned. These are a different kind of solution to the differential equations....not a single, smooth curve, as a solution, but a solution that splits into TWO BRANCHES....a BIFURCATION. And THAT is why we will never explain the effect via differential equations! There IS a non-computable aspect to the situation. There is NOTHING in the math to tell us which of the branches to use.

On: I'm beginning to see. The fact that the solution BRANCHES, I mean bifurcates, sorry, means that at the branch point there is no way to determine which branch will actually be followed in any given case. Are there many of these?

Murky: In certain systems like chaotic systems and there are so many as to make the system extremely sensitive to initial conditions and also make it impossible to ever "curve fit" experimental data.

De: Right! In the "period doubling route to chaos" the first bifurcation leads to a periodic or oscillatory behavior. Then each branch bifurcates again and the period doubles. This happens over and over, hundreds of times. You can diagram it and at some point there have been so many bifurcations that the lines smear together.

On: Have I just been had? How can you do this stuff on a computer if it is non-computable? I mean how does the computer deal with bifurcations?

Murky: That's the rub! It deals with them "numerically" by some algorithm for finding the curve step by step. You can make the steps as small as you like, but you never really "see" the bifurcation point. You simply "jump over" it onto one or the other of the branches.

On: And what determines which branch?

Murky: That's as much a function of the numerical technique and how the computer expresses numbers as anything. But don't get hung up on that technical point. The real issue is that the computer is mimicking the solution by choosing a branch and thereby loosing the uncertainty at the branch point. Hence it can only give a particular case each time it goes through the process.

On: Hey! I don't need math to talk about this. This happens in Human endeavors all the time. Bifurcations are like decisions!

Murky: Now we've got the picture. The hard scientists take some small aspect of a widespread phenomenon, turn it into a somewhat obscure mathematical model, put it on a computer, and claim that they have discovered complexity! Is that or is it not co-optation?

On: Sounds like it to me! Time for my cyber-date. I'm otta here. Let's continue this. I like it!

De: You would! You'll take it as an excuse to be "mushy" for the rest of your life. And What's with this cyber-chick anyway? Can't you find someone to date in the real world? On looks at her carefully: Let's talk about that more later. Are you going to be around?

Murky: Time for me to go. Bye all (he leaves)

De: I have this new book "Turbulent Mirror" with me and I'll be right here reading.

On leaves with a big smile on his face.