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I must admit to some amount of silliness – the first thought I had upon stumbling onto LessWrong, some time ago, was: “wait, if probability does not exist in the territory, and we want to optimize the map to fit the territory, then shouldn’t we construct non-probabilistic maps?” Indeed, if we actually wanted our map to fit the territory, then we would not allow it to contain uncertainty – better some small chance of having the right map, then no chance, right? Of course, in actuality, we don’t believe that (p with x probability) with probability 1. We do not distribute our probability-mass over actual states of reality, but rather, over models of reality; over maps, if you will! I find it helpful to visualize two levels of belief: on the first level, we have an infinite number of non-probabilistic maps, one of which is entirely correct and approximates the territory as well as a map possibly can. On the second level, we have a meta-map, which is the one we update; it consists of probability distributions over the level-one maps. What are we actually optimizing the level-two map for, though? I find it misleading to talk of “fitting the territory”; after all, our goal is to keep a meta-map that best reflects the state of the data we have access to. We alter our beliefs based (hopefully!) on evidence, knowing full well that this will not lead us to a perfect picture of reality, and that a probabilistic map can never reflect the territory.
I rather like this way of thinking. Clever intuition pump.
Hmmm, I guess we're optimizing out meta-map to produce accurate maps. It's mental cartography, I guess. I like that name for it.
So, Occam's Razor and formal logic are great tools of philosophical cartographers. Scientists sometimes need a sharper instrument, so they crafted Solomonoff induction and Bayes' theorem.
Formal logic being a special case of Bayesian updating, where only p=0 and p=1 values are allowed. There are third alternat... (read more)