The way I use “extensibility” here is between two different models of reality, and just means that one can be obtained from the other merely by adding details to it without removing any parts of it. In this case I’m considering two models, both with abstractions such as the idea that “fish” exist as distinct parts of the universe, have definite “weights” that can be “measured”, and so on.
One model is more abstract: there is a “population weight distribution” from which fish weights at some particular time are randomly drawn. This distribution has some free parameters, affected by the history of the tank.
One model is more fine-grained: there are a bunch of individual fish, each with their own weights, presumably determined by their own individual life circumstances. The concept of “population weight distribution” does not exist in the finer-grained model at all. There is no “abstract” population apart from the actual population of 100 fish in the tank.
So yes, in that sense the “population mean” variable does not directly represent anything in the physical world (or at least our finer-grained model of it). This does not make it useless. Its presence in the more abstract model allows us to make predictions about other tanks that we have not yet observed, and the finer-grained model does not.
The way I use “extensibility” here is between two different models of reality, and just means that one can be obtained from the other merely by adding details to it without removing any parts of it. In this case I’m considering two models, both with abstractions such as the idea that “fish” exist as distinct parts of the universe, have definite “weights” that can be “measured”, and so on.
One model is more abstract: there is a “population weight distribution” from which fish weights at some particular time are randomly drawn. This distribution has some free parameters, affected by the history of the tank.
One model is more fine-grained: there are a bunch of individual fish, each with their own weights, presumably determined by their own individual life circumstances. The concept of “population weight distribution” does not exist in the finer-grained model at all. There is no “abstract” population apart from the actual population of 100 fish in the tank.
So yes, in that sense the “population mean” variable does not directly represent anything in the physical world (or at least our finer-grained model of it). This does not make it useless. Its presence in the more abstract model allows us to make predictions about other tanks that we have not yet observed, and the finer-grained model does not.