What does it mean for something to behave lawfully?
Roughly, we ought to be able to discover some mathematics which correctly predict the thing’s behavior.
An example of something with lawful behavior is the position over time of a rock dropped from the top of a building. If the rock has mass m, and its initial position is i, then its position at time t is p = i + 0.5G * t^2.
It is clear that pretty much and for the most part, physical things have lawful behavior.
What does it mean for something to behave randomly?
Roughly, it has no discernible pattern in its behavior. It has an even probability distribution over all possible outcomes. An example of something with random behavior is a toss of a fair six-sided die.
Some things have behavior which involves a mixture of lawfulness and randomness. For instance, suppose we toss the six-sided die several times, producing a series of numbers d1, d2, d3, …, and then define a function f(n) whose value is the square of d1 + d2 + … + dn. f(n)’s behavior involves a mixture of lawfulness and randomness.
Could there be a kind of behavior which is neither lawful nor random, nor a mixture of the two? What would that look like?
Such a thing would have behavior which was not evenly distributed over all possible outcomes. But it would also have the property that we could not discover any mathematics which would correctly predict its behavior.
Can we imagine an example of this? Well, let’s suppose that we have a black box, which constantly outputs a string of ones and zeroes. It can read our thoughts, and so it knows everything that we think about it.
The black box’s output always follows some rule. But, every time we figure out what the rule is, it changes the rule.
Suppose that we discover that the black box changes the rule according to some meta-rule. Then it would have to be the case that, when we figure out the meta-rule, the meta-rule changes.
We would need to say the same thing about any meta-meta-rule, meta-meta-meta-rule, etc.
So this black either has an infinite series of rules, or at some point the series of rules stops. Let us return to the simple case where there is no meta-rule.
In this case, there is simply a Creative Void which can come up with arbitrary mathematical rules. The rules of the black box look like this:
1. Output ones and zeroes according to the current rule.
2. If the humans have figured out the rule, ask the Void for a new rule.
So we can imagine our black box in two ways. It could have an infinitely complex rule set; or, it could have a finitely complex rule set, and a Creative Void which changes the rules sometimes.
It is an important point that we could not distinguish empirically between these two cases.
Whether or not such a box could exist in our universe, we can imagine, in thought experiment, what it would be like to interact with such a box. We have the simple empirical observation that this box never does what we expect it to.
We have clarified the idea of something whose behavior is neither lawful nor random. Let us call it “magical.”
The black box in the thought experiment is magical. But is there anything magical in our universe? It is possible that humans are such a thing. What observations would make us think this?
Suppose we that construct a mathematical model of human psychology which correctly predicts everybody’s behavior in every case we have observed so far. Then we tell somebody, “our model predicts that you will do x.” Out of sheer capriciousness, they decide to do y instead. If humans are magical, then we would expect that this could happen.