Take as an example COnways game of life. It is perfectly predictable in the future - given current state, I can easily predict the state N steps in the future (I just run the simulation on my side computer). And clearly my side prediction will have no impact on how the game-of-life evolves. I can even insert my prediction into the game-of-life by changing the value of my cell (alive=watch TV, dead=listen to radio). Even if a "perverse" in-game character chooses to do the opposite of my prediction, it doesn't alter the fact that I can perfectly predict future of the game universe.

But if I am asked to make that prediction inside the game of life using a program embedded in the game-of-life, then that is a lot harder. It seems very unlikely that I could encode the state of the universe in a way that allows my prediction program to run in any way more efficiently than just letting the game-of-life play out. I'm pretty sure that the Halting problem could be used to demonstrate this.

So for me it makes a big difference whether you constrain the demon to "in universe" computation or "out of universe" computation.

First, that it is assumed that its answer has a physical representation in reality. The fact that solving a non-trivial maze from the inside is impossible without moving through the maze doesn't mean the maze cannot be solved without moving. It can be solved from above the maze. Similarly, assuming quantum mechanics is accurate in that every particle has a random nudge in its position and velocity, if we had access to the random generator, we could create that demon; otherwise, the solution would be expressed in terms of what that generator produces: a solution which doesn't have a physical representation, but is merely a probability distribution.

Second, there is this assumption that its answer be crystal clear for us to understand. Any amount of work with symbolic computation gives the intuition that sometimes, the answer isn't atomic. In that case, the demon's answer could be "if I say that you listen to the radio, what you will do is watch TV. Otherwise, you will listen to the radio".

But then there could be a Laplace's Demon+1 which sits above this all and knows precisely that Laplace's Demon+0 will render a response to that question and Demon+1 still can accurately predict the future.

I think the main premise rests on the same confusion about free will that shows up in Newcomb's paradox. Either there is free will that transcends anything's ability to predict, even in theory (perfect knowledge/measurement and infinite computation), or there is not. If there is not, then a demon/AI/God setting up a test/prediction based on your future decision may be possible, but your awareness of that fact will simply be an input into your future decision-making... an input that the demon/AI/God already accounted for when making the prediction. Just because something is recursive does not mean it's not deterministic. I happen to believe that Laplace's demon exists, it's called physics, but its state is the physical world, so it predicts things will happen at exactly the moment they happen (the prediction/computation being entropic decay); unfortunately that doesn't seem very useful for predictions in the practical sense.

If there is free will that transcends physics, then prediction is fundamentally impossible by something bound to physics—so it's a thought experiment where some things are bound to the laws of physics and others are not. What is the purpose of conducting thought experiments about the physical world, based on supernatural axioms? "Free will exists," the claim goes, and yet "here's a thing that can collapse free will and issue arbitrarily accurate predictions on human behavior." And then, "Let's see what happens when these axioms are applied to someone in the physical world choosing whether to try to make more money or less."

That was my first thought as well, but consider a chaotic system in a Newtonian universe. Laplace' demon must then have the capability to simulate the universe to be able to predict it. What chaos theory brings is that this simulation would require infinite precision to be done without diverging from reality at some point (and sooner rather than later as the error grows exponentially). Then you could use Cantor's diagonalization theorem to prove that no such simulator could exist, even for a finite system, because even a finite system would require an infinitely large state to represent it, and no finitely sized simulator could represent it.

There's Planck length and Planck time (entropic decay resolution?) to consider. If those are the basis for physical laws, isn't everything finite?

If the universe requires infinite computation to "go", it's doing that computation somehow, isn't it? If the universe at perceivable scale is not capable of infinite computation, but infinite computation is required, perhaps it's spawning infinite universes that do computations instantly with respect to our universe's arrow of time?

I don't understand the difference in computational requirements between predicting (in "advance") a future (entropic future) state of the universe, and predicting it JIT.

A functioning Newtonian universe must be capable of infinite computation, therefore Laplace's demon can exist (for that universe), even if only by being a copy of the universe, right?

1) a functioning Newtonian universe is impossible

2) functioning Newtonian universes are infinitely complex

The laws of physics already established 1, making 2 a moot point.

>>In some forms of quantum gravity, the Planck length is the length scale at which the structure of spacetime becomes dominated by quantum effects, and it is impossible to determine the difference between two locations less than one Planck length apart.

Chaos theory with regards to information doesn't take multiple observers into account - where the observers actually change the how the information is related and moves. If you can measure that, then it ceases to be chaotic. Whether that's possible (inferring intent), I am not sure.

> If there is free will that transcends physics, then prediction is fundamentally impossible by something bound to physics—so it's a thought experiment where some things are bound to the laws of physics and others are not. What is the purpose of conducting thought experiments about the physical world, based on supernatural axioms?

I think these theoretical suppositions are based more on a fear of something that people imagine to be real, and may be real. Instead of supernatural beings, suppose there are people that define the world, and people that live within the definition. The fear that the groups are mutually exclusive leads is directly analogous to the philosophical problem.

> "Free will exists," the claim goes, and yet "here's a thing that can collapse free will and issue arbitrarily accurate predictions on human behavior." And then, "Let's see what happens when these axioms are applied to someone in the physical world choosing whether to try to make more money or less."

The thing always depends on states of mind, explanations, the pretense of logic and maths, which ultimately convey themselves to me as rhetoric. There is mind, and there is the world.

Instead of considering discrete points in which perfect knowledge is known, I'm more concerned with rates of information flow.

    consider that there are only two options: (1) watch TV or (2) listen to the radio.

The demon might see that you will watch TV, but if it says that you will listen to the radio. If it sees both of those, even if it says something that becomes false, that doesn't mean the demon couldn't see the future or doesn't see the future. It merely means that the demon cannot accurately manifest the information.

Similarly, in the paradoxical question before, it makes the assumption that demon is honest; that it answering the question incorrectly is tantamount to it having predicted incorrectly. To me it seems clear that the constraint that the demon must be only honest and must transfer the knowledge into the system it is predicting the state of without changing the state is utterly silly.

The problem is the 'prediction loop', which has no clearly defined exit. There's no path that can be followed that will terminate it:

The demon understands every part of your brain, and knows that if it says that you're going to watch TV while you're determined to prove it wrong will make you listen to the radio instead. So it enters the next iteration of the loop. This cycle would continue for infinity, which is where we perceive a paradox.

There are several naive resolutions that come to my mind. Assuming that predicting the future takes some time, even if it's only a fraction of a second, will allow a way out. The Demon will simply enter the loop and exit it once the evening has passed and every possibility of 'predicting it' has vanished. Of course this could be considered cheating, it has never been established if the Demon's computations take time or not, but asking this question can be considered cheating as well, since it's probably impossible to have _only_ two possible things you could be doing. Which introduces the second solution.

The Demon is more intelligent than you are, and while you'd sworn to yourself that there were only two possible outcomes to this evening: doing the opposite of what the Demon predicts, the Demon sees beyond the 'two' possibilities and correctly predicts a third possibility that you didn't envision.

The last solution attacks the notion of free will, if the Demon can exactly predict the future than this implies that the universe is deterministic. So instead of having only two options, or a third invisible option, there is only one real option and the demon cannot be wrong or it cannot be right.

All of these 'solutions' either introduce a new or alter a previous assumption and are thus not really solutions to the question asked, as it stands I think the question is unanswerable and introducing new concepts and ideas is inevitable.

This third neither right nor wrong option reminds me of 'Mu' found in eastern philosophy and highlighted in Hofstadter's GEB and it forms another crucial part of self-reference and recursion as mentioned earlier in your comment.

I'm not sure it's even applicable to the universe we live in because of random events. Is the randomness we observe a result of a mechanical process, or is it inherently unpredictable? If the later, that would make a Demon inherently impossible. It could be we just don't have enough information to decipher the things that appear random to us. Or is randomness what happens when you try to apply Laplace's Demon. As you say, a Demon needs to be able to predict its own predictions. When such a self-referential loop occurs, then any possible event is a "correct" prediction, and that's what happens.

Or the "Star Trek" solution to the paradox: the demons starts repeating error in a high pitched mechanical voice and smoke pours out.

Alternatively, the demon could say one thing, and send a delayed timestamped message with the opposite.

The disproof assumes free will, but Laplace's universe assumes free will is an illusion. Devise an experiment to work out which is right -- a Laplace demon, if such a thing can exist, is probably a good experiment to test this.

However perhaps a device which could capture the state of a defined system, model it and monitor its boundary as a means of prediction would be possible? The question is could it be possible to capture the state of a system at a single instant and without altering the system?

http://dresdencodak.com/2009/02/16/exorcising-laplaces-demon...

laplace's demon is part of the universe.

to simulate the universe, laplace's demon must simulate itself.

assuming P!=NP, this self simulation cannot be more efficient than the "lower level" simulation.

so either the "self-simulation" hole will become increasingly deeper (to infinity), or the resources required to do the simulation at each level will exponentiate away.

proof 2;

the smallest space at which a single bit of information is storable is O(a subatomic particle). but that subatomic particle is defined by more than one bit of information.

therefore to have knowledge of the entire state of the universe at some time requires more matter than exists in the observable universe.

Physics oriented explanation