Why does your reasoning not apply to the coin toss? What’s the mathematical property of the motion of the coin that motion in my system does not possess?
The coin toss is (or we could imagine it to be) a deterministic system whose outcomes are entirely dependent on its initial states. So if we want to talk about probability of an outcome, we need first of all to talk about the probability of an initial state. The initial states come from outside the system. They are not supplied from within the system of the coin toss. Tossing the coin does not produce its own initial state. The initial states are supplied by the environment in which the experiment is conducted, i.e., our world, combined with the way in which the coin toss is realized (i.e., two systems can be mathematically equivalent but might be realized differently, which can affect the probabilities of their initial states). When you presented your toy model, you did not say how it would be realized in our world. I took you to be describing a self-contained toy universe.
What is “intrinsic geometry” when translated into math? (Is it just symmetry?
You can’t have symmetry without geometry in which to find the symmetry. By intrinsic geometry I mean geometry implied by the physical laws. I don’t have any general definition of this, I simply have an example: our own universe has a geometry, and its geometry is implied by its laws. If you don’t understand what I’m talking about I can explain with a thought experiment. Suppose that you encounter Flatland with its flatlanders. Some of them are triangles, etc. Suppose you grab this flatland and you stretch it out, so that everything becomes extremely elongated in one direction. But suppose that the physical laws of flatland accommodate this change so that nobody who lives on flatland notices that anything has changed. You look at something that to you looks like an extremely elongated ellipse, but it thinks it is a perfect circle, because when it regards itself through the lens of its own laws of physics, what it sees is a perfect circle. I would say that Flatland has an “intrinsic geometry” and that, in Flatland’s intrinsic geometry, the occupant is a perfect circle.
Your toy model, considered as a self-contained universe, does not seem to have an intrinsic geometry. However, I don’t have any general idea of what it takes for a universe to have a geometry.
Can you give an example of a system equipped with an intrinsic geometry (and therefore an “objective probability”) where symmetry doesn’t play a role?
I’m not sure that I can, because I think that symmetry is pretty powerful stuff. A lot of things that don’t on the surface seem to have anything to do with symmetry, can be expressed in terms of symmetries.
This will be my last comment on this. I’m breaking two of Robin’s rules—too many comments and too long.
Why does your reasoning not apply to the coin toss? What’s the mathematical property of the motion of the coin that motion in my system does not possess?
The coin toss is (or we could imagine it to be) a deterministic system whose outcomes are entirely dependent on its initial states. So if we want to talk about probability of an outcome, we need first of all to talk about the probability of an initial state. The initial states come from outside the system. They are not supplied from within the system of the coin toss. Tossing the coin does not produce its own initial state. The initial states are supplied by the environment in which the experiment is conducted, i.e., our world, combined with the way in which the coin toss is realized (i.e., two systems can be mathematically equivalent but might be realized differently, which can affect the probabilities of their initial states). When you presented your toy model, you did not say how it would be realized in our world. I took you to be describing a self-contained toy universe.
What is “intrinsic geometry” when translated into math? (Is it just symmetry?
You can’t have symmetry without geometry in which to find the symmetry. By intrinsic geometry I mean geometry implied by the physical laws. I don’t have any general definition of this, I simply have an example: our own universe has a geometry, and its geometry is implied by its laws. If you don’t understand what I’m talking about I can explain with a thought experiment. Suppose that you encounter Flatland with its flatlanders. Some of them are triangles, etc. Suppose you grab this flatland and you stretch it out, so that everything becomes extremely elongated in one direction. But suppose that the physical laws of flatland accommodate this change so that nobody who lives on flatland notices that anything has changed. You look at something that to you looks like an extremely elongated ellipse, but it thinks it is a perfect circle, because when it regards itself through the lens of its own laws of physics, what it sees is a perfect circle. I would say that Flatland has an “intrinsic geometry” and that, in Flatland’s intrinsic geometry, the occupant is a perfect circle.
Your toy model, considered as a self-contained universe, does not seem to have an intrinsic geometry. However, I don’t have any general idea of what it takes for a universe to have a geometry.
Can you give an example of a system equipped with an intrinsic geometry (and therefore an “objective probability”) where symmetry doesn’t play a role?
I’m not sure that I can, because I think that symmetry is pretty powerful stuff. A lot of things that don’t on the surface seem to have anything to do with symmetry, can be expressed in terms of symmetries.
This will be my last comment on this. I’m breaking two of Robin’s rules—too many comments and too long.