Why would the ball slide down the dome? If it is at the apex, exactly, then it should stay there.
Edit: I see from the link that this is indeed a valid solution to the setup of the problem.
That’s one solution of the setup, but in a sense, the position of the particle is infinitely instable, so that it requires no force at all to be pushed away from it—and you can just use the time-reversal argument to see that this must be a possible solution to the system: if you can push the particle up, and have it come to rest, and stay there for an indefinite amount of time, then there must also be a solution where the particle stays there for an arbitrary amount of time, and then slides down, as a time-reversal of the former.
I think the “all-powerful computer” part is cheating, in a sense. If a computer sufficiently powerful to compute the interactions of every single fragment of energy and matter would require more resources than the universe as a whole consists of, you’ve got an untenable hypothetical, one whose existence violates the very laws it’s supposed to be attending to. It would be one thing if it were merely “well, we don’t have the technology at the moment”, but I think it’s more of an intrinsic law, so it becomes a bit like saying “what if we ignore Heisenberg, Gödel, and Einstein and an all-powerful God really existed, how would that be?” – I mean, isn’t it?
It depends on what you intend the question to be. You could use the computer as merely a framing device, effectively asking whether there’s enough information in the present state of the universe to reconstruct the placement of Abraham Lincoln’s hat at the time of his assassination—that’s a wholly unambiguous question for which such a computer, or the task of actually reconstructing that placement given said information, need not actually be feasible.
I read that that explanation of the Norton’s Dome hypothetical and what am I missing? To me that reads as saying that the simple answer is no. There is no way to know even in this Newtonian hypothetical whether the ball would stay there forever or at some arbitrary point, unable to be predicted, roll down in some arbitrary direction that cannot be predicted. Each are mathematically valid sets of solutions. We do not know where or when or if that ball will roll and we cannot know which path it took to get there.
Precisely as you point out: in that example once at rest on the top “the information from where the particle started would be lost”.
It is the only way to understand the hypothetical. If there was an entity that was outside of the universe and nature (i.e. a supernatural entity) that could measure every single aspect of every single particle of the universe at one single point in time and had calculating power greater than of the entire universe (i.e. all powerful), could it calculate backwards to what was (and therefore by extension what will be)? Or would that God still themself be limited to rolling the dice at every move as part of every calculation?
If so then a time traveller heading backwards could end in an infinite number of possible pasts that had resulted in their present, just as much as an infinite number of possible futures exist …
The hypothetical doesn’t need a computer. In the past the question has been posed using all powerful beings and probably other mechanisms. It is just a way of placing the core issue in easier to understand terms. In the modern world we talk of computers. 100 years ago the same question would be posed in a different setting. But what matters doesn’t change. What matters is the conservation of information, deterministic physical laws, and the philosophical implications.
One might try some form of wiggle room that try to invalidate the conclusions because the calculation cannot actually be performed. Maybe try to shoehorn an argument that the universe is somehow special wrt free will.
Of course if everything is just a simulation, all bets are off.
I don’t know anything about physics, quantum mechanics, or computer engineering, but I looked it up, and according to this article,
When Lincoln was shot, the hat was on the floor beside his chair.
Well, it’s a hypothetical thought experiment. It may be pointing out that information can be lost or it may be pointing out a flaw in the mathematical description of Newtonian mechanics. Maybe information in such a system can be lost but it is so improbable that as a practical matter it isn’t lost. For one thing I don’t see that a subatomic Norton Dome can exist so it’s not going to factor in to calculating the movement of subatomic particles.
Oh I get that. My confusion was and is over how that hypothetical was being used to support the simple answer yes that there is but one path possible forward or back. I am assuming that I am not comprehending some aspect of the point being made but cannot figure out what I am not understanding. It seems to me to be a thought experiment in favor of indeterminism more than determinism.
It was a counterpoint. He said the simple answer is yes but you can device edge cases that bring the simple answer into question. The Norton Dome is a complication, not support for the simple answer.
Thank you. That’s what I was not comprehending.
As a side question, is it possible to simulate a universe of size X, using something that is smaller (let’s say in terms of fundamental particles) than X?
It seems like the answer is going to be ‘not in all cases, therefore no’ because of the whole Shannon thing about data compression and coding.
This hardly begins to answer your question, but imagine a universe that has even one real number as a parameter somewhere. That is already an infinite amount of information, so merely running it through “zip” is unlikely to help very much.
It’s trivial to simulate all possible universes on your laptop, provided you can keep it running long enough. The history of any given universe up to some point in time can be encoded within a finite string of bits. Generally, a typical string of bits can’t be significantly compressed, because it will be random, and hence, there won’t be any regularities to take advantage of to perform the compression. But all of them simultaneously can be compressed down to pretty much nothing at all: just a program that computes all possible outputs, which can be done in an interleaving sort of way. Fix an enumeration of possible outputs, then, first, compute the first bit of the first output, then the second bit of the first, then the first bit of the second, then the third bit of the first, the second bit of the second, the first bit of the third, and so on. Every bit of every possible output will eventually be computed—which will include every history of every possible universe up to any given point in time.
The reason things like that work is that information (algorithmic information in this case) isn’t additive; a collection of things can have less information than any of the things within it. The entirety of the natural numbers can be described using a few, short axioms; but a general set of natural numbers won’t be describable in a way that’s significantly shorter than listing all its members.
Would the computer runtime be shorter than the life of the universe? If not then it’s not really smaller than the universe.