Turns out they don’t support in-determinism either.
I suggest you would do well to always interpret such posts as an invitation to respond. It would save you lots of typing.
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Both Newton’s and Schrödinger’s equations can have conditions under which solutions are not unique.
Few such cases for Newton’s are ever actually discussed – many physicists never even hear about them – though they are no less realistic than the ones that do have unique solutions. There is at most an argument to be had, that since Newtonian mechanics do not apply on the molecular scale anymore, the precise initial conditions are never truly met in nature, so it’s not worth worrying about.
For Schrödinger’s, it is the minority of conditions that actually yield a unique solution, and they are often highly artificial ones, like enforcing that one is looking for one particular state, instead of just solving the equation. Even then it’s not on the face of it obvious that the solution would be unique in both time directions, but at least such problems are being regularly considered.
Of course, if the uncommon set of non-deterministic problems of classical mechanics is insufficient to make the case that classical physics is non-deterministic, we can always introduce thermodynamics, or any of the classical field theories. Off the top of my head I’m not sure how much introducing quantum thermodynamics or field theory would shift quantum physics towards or away from determinism, though…
@AlanFox ,
One could say that an AI-driven robot assistant CHOOSES the best tool for a worker to use for a particular project.
But was the choice a “free” choice? Or was the choice made actually determined by all sorts of predefined parameters? Would 1,000,000 highly trained humans facing a similar choice ALSO have selected the same tool 99.9999% of the time?
If the robot or humans had the ability to choose outside these bounds and parameters - - could we trust them at all? I think not!
Not to worry! I’m confident the Terminator hypothesis is fiction. Robots (should they become self-sustaining) cannot conceive of or construct entities more complex than themselves. Intelligence can only advance (under selection) at an evolutionary rate.
Why not? What would prevent it?
I think you misunderstand my point.
The AI uses 50 factors to make its choice. The human version also uses 50 factors.
The 50 factors produce only 873 possible answers.
If a human started to produce a few dozen answers that were not part of thd valid set, we would send him to a doctor or a therapist. If the AI did the same, it would be labeled broken.
Just to make sure I understand; is the statement here that we can conceive of a particular state of a system under Newtonian physics, for example, where multiple distinct prior states could give rise to that later state?
It’s rather what would be the outcome if it were possible. It would have happened resulting in civilisations of god-like entities having the power to rule the universe. Where are they?
The statement is even more general than that here:
- Yes, one can construct a problem where multiple distinct states evolve into a single one over time.
- One can even construct a problem that meets such a final state after a finite time.
- One can also construct a problem where a single initial state can give rise to more than one distinct later states.
To be fair, I was “only” thinking of the last point when I made the statement. But you are right. Technically, if two or more states collapse into one, it becomes impossible to retrodict the past states definitively. So a system where that can happen (and they are entirely permitted within Newtonian mechanics) technically also counts as a non-deterministic system.
At my age, it happens more and more.
You’re right. I don’t follow.
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Would it? Why? What would facilitate this apparent inevitability?
I don’t know. Maybe they know better than to waste their time zooming to the other end of the galaxy. Or maybe they don’t exist yet. Or maybe they are destroyed already by any number of ways that could happen. Or maybe it is not the case that a series of robots who can conceive of or construct things more complex than themselves would result in anything like an interstellar or intergalactic civilization. It’s not like we have any evidence to suggest this should happen, or do we?
In the scenario presented, whether a human or a machine, if a chooser CHOOSES incorrectly - - it is called either crazy or broken.
I’m a determinist, but open to compatibilism (so a ‘soft’ determinist it seems). I do believe that I have a choice, but that (apart from a degree of randomness) my choices are predetermined, by my existing mental state (tastes, mood, outlook on life, etc), and external circumstances. The biggest question in my mind is the balance between determinism and randomness – is for example my tendency to mischoose due to absentmindedness (misreading the label, etc) or tunnel-vision, a form of determinism or randomness?
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Now Not sure that I trust them now. 
It’s true that we don’t see Von Neumann Replicators, but any high intelligence is going to be very careful about turning loose self-replicating machines. They could have other reasons for not being seen. Maybe we just aren’t interesting enough yet.
If they are actually God-like, which I think I unlikely, then they don’t need to see us or us them.
Hmm. Though, seriously, the speed of light is a safety barrier for us. Humanity has (or should have) many more immediate concerns.
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I’m having a difficult time seeing how this is possible using Newtonian physics if you’re not smuggling in uncertainty to begin with.
Are you saying there is a system, for example a box containing a bunch of gas particles where each particle has a definite position and momentum, and as this system evolves over time there will come a point where a gas particle collides with another and where this collision permits multiple distinct solutions?
Hmm I’m not sure if one could devise an n-body interaction potential that would do that also. I was thinking of something much simpler, like Norton’s Dome:
Consider a particle in a potential given by V(r)=-\left[1-\left(1-r\right)^{\frac23}\right]^{\frac32}, where 0\leq r<1 is the distance from the origin. We can get Newton’s equations of motion by partially derivating the Hamiltonian with respect to each spatial coordinate (here we are looking at a two-dimensional system, best parametrized by aforementioned distance r and a polar angle 0\leq\theta<2\pi with respect to some arbitrary reference axis).
For the specific initial condition where the particle is stationary at the origin at some initial time t_0 there exist infinitely many distinct solutions to these equations of motion. One solution is that the particle remains atop the dome described by the potential. However, for every t>t_0 there also exists a solution where the particle remains stationary until that t, at which time it will begin moving in an arbitrary direction \theta. The trajectory will be fully determined for all times after the motion begins, but if or when that happens, or in which direction is not determined for this problem.
Newtons axioms place no restriction on the shape of domes one is allowed to suggest for treatment by the theory. One can come up with excuses like whether a dome shaped perfectly like this could possibly exist, or whether a particle could possibly be at perfect rest and a perfectly defined position (especially in light of the uncertainty principle), but these criticisms are external to the theory, and seldom come up to dismiss the deterministic solutions to, say, the spherical, or the parabolic dome. As it stands, nothing about this example violates Newtonian mechanics. The only thing it violates is the conjecture that Newtonian mechanics is deterministic.
Why would it do that if it is at rest?
I was going to be humorous and suggest that you’d have to ask the particle, commenting that Newtonian mechanics is not a philosophy concerned with why things behave as they do. But I can give a (maybe?) more satisfactory answer than that.
Since we are dealing with a second order ordinary differential equation, it should suffice to specify initial value of the function and initial value of its derivative to formulate a complete set of initial conditions (this is known as Cauchy boundary condition). Because Newton’s theory is all about forces, the second derivatives of the trajectory, a particle’s state is generally fully described by its position and momentum. However, merely the specification of an initial value problem is not a guarantee that a unique solution exists. For that, the Hamiltonian has to be Lipschitz-continuous in position and momentum, otherwise there may still be a unique solution, or there may be no solution at all, or there may be more than one solution. In this case, the Hamiltonian is not Lipschitz-continuous at r=0 and \vec p=0, and there exists, as a matter of fact, more than one solution to this problem. This is a consequence of the problem’s own properties, and it is mathematically impossible to non-arbitrarily eliminate any of the solutions.
That doesn’t answer why any particular trajectory would be assumed, but then that’s kind of the point: It is not determined by the given rules and initial state, there is nothing to cause any of the possible trajectories to be assumed in place of another. The only reason the deterministically fixed trajectory in a deterministic problem is assumed is precisely the fact that it is the only one that can be. Here, all we can expect is that one of the possible solutions will be realized, but we cannot predict it, even fully knowing the system’s initial state and forces at all times and locations.
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