When you try to model a natural phenomenon, you inevitably drop out some of the phenomenon as irrelevant; that’s the nature of modeling. You can’t then complain that the model doesn’t capture something that it wasn’t intended to capture.

Lt’s take snow flakes as an example. When snowflakes form, they start from a seed. When a crystal bond is formed, some heat is released, inhibiting neighboring molecules from attaching. So, make a rule that says a cell will only be filled in if ___, and you get a snowflake form. You can make predictions from this such as: Big snowflakes will have holes in them from where arms collide; that turns out to be true. The model works. But it won’t answer a question like how far an arm will grow at a particular temperature. [I don’t know why.]

So, how does one assess models? The best models are ones where you put a little in and get a lot out. A bad model gets more complex because you have to keep adding new considerations until you’re putting in more than you get out.

Models used to be mechanistic: you push A and B moves; there’s a small chain of inference. In another form of modeling, about 300 years old, we use equations to model systems. That’s a much more abstract form of modeling. But some of the equations can be very hard. E.g., you can explain snowflake generation via partial differential equations, but they’re very hard to solve. But NKS adds a new type of model: a simple set of rules producing complex phenomena.

It’s not proper to object that snowflakes aren’t made of CA cells because CA is a model of snowflakes. We don’t think that the earth is solving a differential equation when it moves through space. Differential equations are an abstraction. Similarly, a CA model of a snowflake is an abstract representation of how snowflakes work.

Randomness in models

We see examples of randomness in the natural world, e.g., fluid motion. Where does the randomness come from? There are three possible origins:

1. Classically, randomness comes from external perturbation, e.g., a boat being kicked around by the randomness of the ocean’s surface. Randomness of this sort: Brownian motion and some electronic noise. The randomness you get out isn’t part of the system you’re studying.

2. Chaos theory points to systems in which the initial conditions are random, e.g., a toin coss or the spin of a wheel. The three-body problem in gravity was one of the first cases of this studied: a change in a billionth of a degree results in hugely different results. There’s some effect from the outside that causes the initial conditions to be random. The randomness doesn’t come from the system we’re modeling.

3. You can get randomness without going outside the system. E.g., Rule 110 is intrinsically random.

Constraints

In traditional mathematical models, one can have an equation that is a constraint on the system that solves the equation. Typical example is a boundary problem. [Suddenly over my head. Prepare for vagueness. ] Constraint-based models don’t tell you how to fill in the constraints to solve the problem. His example of a constraint-based model is the question of what the closest packing of circles is. This is very hard to solve if the circles are different sizes. [I’ve lost the point. Damn not-knkowing-math-iness!]

Biology wants to know where the complexity of organisms come from. Initially, Wolfram assumed it was a different class of phenomenon because the biological systems adapt and change over time. But he’s concluded that adaption and evolution isn’t the issue. What forms of explanation should we give for biological systems? Will simple rules do or do we need much more complicate rules? Maybe biology is in fact sampling really simple programs. So, does the complexity come as a response to the visual system of predators [he seems to be thinking about patterns of fur] or does it come from simple programs? We think it has to have a complex explanation (evolution) because it’s complex. Nah, says Wolfram. [Evolutionists don’t necessarily think that pigmentation patterns have been “carefully tuned” by natural selection. The question is whether we can get past pigmentation and get to flight or sight or kidneys.] Natural selection is good at progressively shortening or lengthening bones, but it’s not good at creating complicated things. Natural selection actually simplifies things, not makes them more complicated. We see this in technology where a form of natural selection makes stuff simpler, e.g., Fedex bills have gotten simpler. Natural selection operates well where you can make small changes and not have them be disastrous, just as engineering does.

His model explains how sea shells are formed. If you exclude ridiculous shapes — e.g., ones that leave no room for the animal — all of the ones his model draws are found in nature. So, you don’t need natural selection to explain them. Likewise for the shapes of leaves.

How do you find a model? If you think it’s a CA sort of thing, you can just match ’em up. But it can be really hard to go from a natural phenomenon to its model. It is an unsolvable computational problem in the general case. So what do you do in practice? First, you can use Wolfram’s Atlas of Simple Programs and see if you recognize one. [The mug shot approach.]

The other thing you can do is search through all possible models of some particular kind. This seems crazy because if you were to search all possible equations, you’d never find it. But because you’re looking for simple rules, it can work.

Categories: misc dw