There’s a piece I think you’re missing with respect to maps/territory and math, which is what I’ll call the correspondence between the map and the territory. I’m surprised I haven’t this discussed on LR.
When you hold a literal map, there’s almost always only one correct way to hold it: North is North, you are here. But there are often multiple ways to hold a metaphorical map, at least if the map is math. To describe how to hold a map, you would say which features on the map correspond to which features in the territory. For example:
For a literal map, a correspondence would be fully described (I think) by (i) where you currently are on the map, (ii) which way is up, and (iii) what the scale of the map is. And also, if it’s not clear, what the marks on the map are trying to represent (e.g. “those are contour lines” or “that’s a badly drawn tree, sorry” or “no that sea serpent on that old map of the sea is just decoration”). This correspondence is almost always unique.
For the Addition map, the features on the map are (i) numbers and (ii) plus, so a correspondence has to say (i) what a number such as 2 means and (ii) what addition means. For example, you could measure fuel efficiency either in miles per gallon or gallons per mile. This gives two different correspondences between “addition on the positive reals” and “fuel efficiencies”, but “+” in the two correspondences means very different things. And this is just for fuel efficiency; there are a lot of correspondences of the Addition map.
The Sleeping Beauty paradox is a paradoxical because it describes an unusual situation in which there are two different but perfectly accurate correspondences between probability theory and the (same) situation.
Even Logic has multiple correspondences. ”∀x.ϕ and “∃x.ϕ” mean in various correspondences: (i) ”ϕ holds for every x in this model” and ”ϕ holds for some x in this model”; or (ii) “I win the two-player game in which I want to make ϕ be true and you get to pick the value of x right now” and “I win the two-player game in which I want to make ϕ be true and I get the pick the value of x right now”; or (iii) Something about senders and receivers in the pi-calculus.
Maybe “correspondence” should be “interpretation”? Surely someone has talked about this, formally even, but I haven’t seen it.
There’s a piece I think you’re missing with respect to maps/territory and math, which is what I’ll call the correspondence between the map and the territory. I’m surprised I haven’t this discussed on LR.
When you hold a literal map, there’s almost always only one correct way to hold it: North is North, you are here. But there are often multiple ways to hold a metaphorical map, at least if the map is math. To describe how to hold a map, you would say which features on the map correspond to which features in the territory. For example:
For a literal map, a correspondence would be fully described (I think) by (i) where you currently are on the map, (ii) which way is up, and (iii) what the scale of the map is. And also, if it’s not clear, what the marks on the map are trying to represent (e.g. “those are contour lines” or “that’s a badly drawn tree, sorry” or “no that sea serpent on that old map of the sea is just decoration”). This correspondence is almost always unique.
For the Addition map, the features on the map are (i) numbers and (ii) plus, so a correspondence has to say (i) what a number such as 2 means and (ii) what addition means. For example, you could measure fuel efficiency either in miles per gallon or gallons per mile. This gives two different correspondences between “addition on the positive reals” and “fuel efficiencies”, but “+” in the two correspondences means very different things. And this is just for fuel efficiency; there are a lot of correspondences of the Addition map.
The Sleeping Beauty paradox is a paradoxical because it describes an unusual situation in which there are two different but perfectly accurate correspondences between probability theory and the (same) situation.
Even Logic has multiple correspondences. ”∀x.ϕ and “∃x.ϕ” mean in various correspondences: (i) ”ϕ holds for every x in this model” and ”ϕ holds for some x in this model”; or (ii) “I win the two-player game in which I want to make ϕ be true and you get to pick the value of x right now” and “I win the two-player game in which I want to make ϕ be true and I get the pick the value of x right now”; or (iii) Something about senders and receivers in the pi-calculus.
Maybe “correspondence” should be “interpretation”? Surely someone has talked about this, formally even, but I haven’t seen it.