My motivation for those comments was observing that you don’t need the measure except for worlds in E, those that are compatible with observation.
Say you have in the original “possible worlds” both some that are computable (e.g. Turing machine outputs) and some that are not (continuous, real-valued space and time coordinates). Now, suppose it’s possible to distinguish among those two, and one of your observations eliminates one possibility (e.g., it’s certainly not computable). Then you can certainly use a measure that only makes sense for the remaining one (computable things).
There might be other tricks. Suppose again, as above, you have two classes of possible worlds, and you have an idea how to assign measures within each class but not for the whole “possible worlds” set. Now, if you do the rest of the calculations in both classes, you’ll obtain two “probabilities”, one for each class of possible worlds. If the two agree on a value (within “measurement error”), you’re set, use that. If they don’t, then it means you have a way to test those two worlds.
For the time being, it seems more worthwhile to describe the mental operation of assigning subjective probabilities (and thus understanding possible worlds in this sense) and avoid conflating rationality with physics.
I certainly didn’t intend the schema as an actual algorithm. I’m not sure if your comment about subjective probabilities means. (I can parse it both as “this is a potentially useful model of what brains ‘have in mind’ when thinking of probabilities, but not really useful for computation”, and as “this is not a useful model in general, try to come up with one based on how brains assign probability”.)
What is interesting for me is that I couldn’t find any model of probability that doesn’t match that schema after formalizing it. So I conjectured that any formalization of probability, if general enough* to apply to real life, will be an instance of it. Of course, it may just be that I’m not imaginative enough, or maybe I’m just the guy with a new hammer seeing nails everywhere.
(*: by this, I mean not just claiming, say, that coins fall with equal probability on each side. You can do a lot of probability calculation with that, but it’s not useful at all for which alien team wins the game, what face a dice falls on, or even how real coins work.)
I’m really curious to see a model of probability that seems reasonable and general, and that I can’t reduce to the shape above.
My motivation for those comments was observing that you don’t need the measure except for worlds in E, those that are compatible with observation.
Say you have in the original “possible worlds” both some that are computable (e.g. Turing machine outputs) and some that are not (continuous, real-valued space and time coordinates). Now, suppose it’s possible to distinguish among those two, and one of your observations eliminates one possibility (e.g., it’s certainly not computable). Then you can certainly use a measure that only makes sense for the remaining one (computable things).
There might be other tricks. Suppose again, as above, you have two classes of possible worlds, and you have an idea how to assign measures within each class but not for the whole “possible worlds” set. Now, if you do the rest of the calculations in both classes, you’ll obtain two “probabilities”, one for each class of possible worlds. If the two agree on a value (within “measurement error”), you’re set, use that. If they don’t, then it means you have a way to test those two worlds.
I certainly didn’t intend the schema as an actual algorithm. I’m not sure if your comment about subjective probabilities means. (I can parse it both as “this is a potentially useful model of what brains ‘have in mind’ when thinking of probabilities, but not really useful for computation”, and as “this is not a useful model in general, try to come up with one based on how brains assign probability”.)
What is interesting for me is that I couldn’t find any model of probability that doesn’t match that schema after formalizing it. So I conjectured that any formalization of probability, if general enough* to apply to real life, will be an instance of it. Of course, it may just be that I’m not imaginative enough, or maybe I’m just the guy with a new hammer seeing nails everywhere.
(*: by this, I mean not just claiming, say, that coins fall with equal probability on each side. You can do a lot of probability calculation with that, but it’s not useful at all for which alien team wins the game, what face a dice falls on, or even how real coins work.)
I’m really curious to see a model of probability that seems reasonable and general, and that I can’t reduce to the shape above.