1) If Alice and Bob observe the system in your first example, and Alice decides to keep track precisely of X’s possile states while Bob just says “2-8”, the entropy of X+Y is 2 bits for Alice and 2.8 for Bob. Isn’t entropy a property of the system, not the observer? (This is the problem with “subjectivity”: of course knowledge is physical, it’s just that it depends on the observer and the observed system instead of just the system.)
2) If Alice knows all the molecules’ positions and velocities, a thermometer will still display the same number; if she calculates the average speed of the molecules, she will find this same number; if she sticks her finger in the water at a random moment, she should expect to feel the same thing Bob, who just knows the water’s temparature, does. How is the water colder? Admittedly, Alice could make it colder (and extract electricity), but she doesn’t have to.
You lost me there.
1) If Alice and Bob observe the system in your first example, and Alice decides to keep track precisely of X’s possile states while Bob just says “2-8”, the entropy of X+Y is 2 bits for Alice and 2.8 for Bob. Isn’t entropy a property of the system, not the observer? (This is the problem with “subjectivity”: of course knowledge is physical, it’s just that it depends on the observer and the observed system instead of just the system.)
2) If Alice knows all the molecules’ positions and velocities, a thermometer will still display the same number; if she calculates the average speed of the molecules, she will find this same number; if she sticks her finger in the water at a random moment, she should expect to feel the same thing Bob, who just knows the water’s temparature, does. How is the water colder? Admittedly, Alice could make it colder (and extract electricity), but she doesn’t have to.