Theoretically, there are two forms of future prediction. Interactive prediction and non-interactive prediction. A non-interactive predictor makes predictions about some world, but doesn’t share the predictions with that world. An example would be if I’m watching a play and try to predict the next scene in my head. The actors don’t know what prediction I make, so it can’t influence them. An interactive predictor, on the other hand, broadcasts his prediction into the world whose future he’s trying to nail.
The thing about interactive predictions is, as soon as you make one, it can influence the very future you’re trying to nail down.
Suppose I notice that every day my neighbor checks his mail at exactly 9am. I would be a fool to bet him that tomorrow he’ll check his mail at 9am. He’d accept and then intentionally alter his pattern to collect on the bet. The prediction fails. (But if I’d kept it to myself, making it a non-interactive prediction, it probably would’ve come true)
In the non-interactive case, assuming a deterministic universe, the configuration of the universe is simply a function of time. For each value of time, there’s a single fixed configuration of the universe, and reading the future is as easy as reading the values of this function. But put the crystal ball inside the world, where the world’s inhabitants can actually see it, and suddenly things get more complicated. The configuration of the universe at time T is no longer a function of just T, but also of whatever predictions were made before time T.
It isn’t that the interactive case is harder and the non-interactive case is easier. That’s an oversimplification. It really depends on the particular conjecture.
I predict that someone will read this sentence.
If I hadn’t written this article, the above assertion would be false. The mere act of my making it, made it true. It truly is a self-fulfilling prophecy. And even if, against all odds, it doesn’t come true, then I’m ok, nobody will ever know I failed
The configuration of the universe at time T is a function of two variables: T itself, and whether or not I introduced the assertion “someone will read this sentence” into the universe. If T=The year 2100, and “Did I introduce the assertion”=TRUE, then by 2100, someone will have read the sentence. This is mathematically tautological because if nobody reads it, then it won’t be introduced. The very word “introduce” here means somebody reads it.
Now suppose I start publishing prophecies about the value of GOOG stock. Truth is, the price will actually depend on my predictions. If my initial guesses are exactly right, eventually people are going to start noticing and taking it seriously. If my winning streak goes on long enough, people are going to start buying and selling according to what I predict. There’s a real possibility this will cause the prices to deviate from my projections. For example, if, before I came along, prices were about to stagnate and stay fixed for a month before suddenly collapsing– and I predict as much– then stockholders will try to get out before the fall. But by getting out, they’ll cause the fall to come sooner than I predicted.
If I want to make accurate assertions about GOOG stock forever, it’s not enough to calculate the stock prices in a closed universe. I must take into account how people will react to my prophecies– which reactions will become more dramatic the longer my winning streak gets– and adjust accordingly. If my only goal is to correctly make predictions, then I must find a fixed point, a price such that, by predicting this price, I make this price become true.