I think there’s a practically infinite number of edge cases. For a system to run the algorithm, it would have to perform a sequence of operations on natural numbers. If we simplify this a bit, we could just look at the values of the variables in the program (like a, x, y; I don’t actually know the algorithm, I’m just looking at the pseudo-code on Wikipedia). If the algorithm is running, then each variable goes through a particular sequence, so we could just use this as a criterion and say the system runs the algorithm iff one of these particular sequences is instantiated.
Even in this simplified setting, figuring this out requires a mapping of physical states to numbers. If you start agreeing on a fixed mapping (like with a computer, we agree that this set of voltages at this location corresponds to the numbers 0-255), then that’s possible to verify. But in general you don’t know, which means you have to check whether there exists at least one mapping that does represent these sequences. Considered very literally, this is probably always true since you could have really absurd and discontinuous mappings (if this pebble here has mass between 0.5g and 0.51g it represents the number 723; if it’s between 0.51g and 0.52g it represents 911...) -- actually you have infinitely many mappings even after you agree on how the system partitions into objects, which is also debatable.
So without any assumptions, you start with a completely intractable problem and then have to figure out how to deal with this (… allow only reasonable mappings? but what’s reasonable? …), which in practice doesn’t seem like something anyone has been able to do. So even if I just show you a bunch of sand trickling through someone’s hands, it’s already a hard problem to prove that this doesn’t represent the Miller-Rabin test algorithm. It probably represents some sequence of numbers in a not too absurd way. There are some philosophers who have just bitten the bullet and concluded that any and all physical systems compute, which is called panpcomputationalism. The only actually formal rule for figuring out a mapping that I know is from IIT, which is famously hated on LW (and also has conclusions that most people find absurd, such that digital computers can’t be conscious at all bc the criterion ends up caring more about the hardware than the software). The thing is that most of the particles inside a computer don’t actually change all that much depending on the program, it’s really only a few specific locations where electrons move around, which is enough if we decide that our mapping only cares about those locations but not so much if you start with a rule applicable to arbitrary physical systems.
That all said, I think illusionists have the pretty easy out of just saying that computation is frame-dependent, i.e., that the answer to “what is this system computing” depends on the frame of reference, specifically the mapping from physical states to mathematical objects. It’s really only a problem you must solve if you both think that (a) consciousness is well-defined, frame-invariant, camp #2 style, etc., and also (b) the consciousness of a system depends on what it computes.
I think there’s a practically infinite number of edge cases. For a system to run the algorithm, it would have to perform a sequence of operations on natural numbers. If we simplify this a bit, we could just look at the values of the variables in the program (like
a,x,y; I don’t actually know the algorithm, I’m just looking at the pseudo-code on Wikipedia). If the algorithm is running, then each variable goes through a particular sequence, so we could just use this as a criterion and say the system runs the algorithm iff one of these particular sequences is instantiated.Even in this simplified setting, figuring this out requires a mapping of physical states to numbers. If you start agreeing on a fixed mapping (like with a computer, we agree that this set of voltages at this location corresponds to the numbers 0-255), then that’s possible to verify. But in general you don’t know, which means you have to check whether there exists at least one mapping that does represent these sequences. Considered very literally, this is probably always true since you could have really absurd and discontinuous mappings (if this pebble here has mass between 0.5g and 0.51g it represents the number 723; if it’s between 0.51g and 0.52g it represents 911...) -- actually you have infinitely many mappings even after you agree on how the system partitions into objects, which is also debatable.
So without any assumptions, you start with a completely intractable problem and then have to figure out how to deal with this (… allow only reasonable mappings? but what’s reasonable? …), which in practice doesn’t seem like something anyone has been able to do. So even if I just show you a bunch of sand trickling through someone’s hands, it’s already a hard problem to prove that this doesn’t represent the Miller-Rabin test algorithm. It probably represents some sequence of numbers in a not too absurd way. There are some philosophers who have just bitten the bullet and concluded that any and all physical systems compute, which is called panpcomputationalism. The only actually formal rule for figuring out a mapping that I know is from IIT, which is famously hated on LW (and also has conclusions that most people find absurd, such that digital computers can’t be conscious at all bc the criterion ends up caring more about the hardware than the software). The thing is that most of the particles inside a computer don’t actually change all that much depending on the program, it’s really only a few specific locations where electrons move around, which is enough if we decide that our mapping only cares about those locations but not so much if you start with a rule applicable to arbitrary physical systems.
That all said, I think illusionists have the pretty easy out of just saying that computation is frame-dependent, i.e., that the answer to “what is this system computing” depends on the frame of reference, specifically the mapping from physical states to mathematical objects. It’s really only a problem you must solve if you both think that (a) consciousness is well-defined, frame-invariant, camp #2 style, etc., and also (b) the consciousness of a system depends on what it computes.