Can someone run me up to speed on exactly how information is preserved in the case of an object losing kinetic energy to friction and that energy being dissipated as heat? How is the information about the original velocity of the object preserved ?
As I undertand it, and IANAPhysicist, the information is there, even if it is unfeasible for us mere mortals to obtain it and to process it: the whole Universe is a recording machine of everything that has happened so far, otherwise, if whatever you postulate had not happened, or if it had happened in a different way, the Universe would be different too, and we could know it. So the premise is that you know exactly how and when the motion was transformed into heat by friction, so from that heat pattern, and the particles this friction has set in motion (which you also know exactly) you can infer the original motion. And from that you can infer what did set that motion in motion, and so on ad infinitum. It’s motions and causes for every single particle and all of it’s interactions all the way back to the Big Bang. Theoretically. Of course, to be able to compute all those gazillions of data you would need a computer more massive than the Universe you are analyzing, and more time than has ever elapsed. And that is without taking Heisenberg’s Uncertainty Principle into account.
On top of the uncertainty principle and the measurement constrains IRL, I am told physicists are worried about the informaton that goes into black holes: that seems to disappear and causes them sleepless nights because that should not be.
But at least according to theory, leaving aside the Uncertanty Principle and black holes, the Universe itself is proof of everything that ever was, so the information must be out there.
So to answer tho OP: with an all powerfull computer, i.e., the Universe itself, and knowledge of the position of every particle, which the Universe did have, even though we did not, because we did not measure them, you could have known were the hat was. In fact, the Universe did know at that very moment and it still has the data that stem from that fact, because that fact had consequences. Those data are part of the present Universe. The Universe, the most powerful computer imaginable, knew and still “knows”, somehow. But you or anybody else are never going to be able to calculate that.
Simple, isn’t it? 
And probably wrong on many levels, but that is the best I can do.
Well sure, if the computer is all powerful.
My simple understanding is that the balls are at rest but the energy of the balls were transferred into other things setting them in motion and causing heat etc.
If you could theoretically calculate back from billiard balls bouncing around to where they were a second before, then you could from all those other molecules as well. It’s just a lot more and smaller balls. Limitations discussed noted.
We assume it is.
The point of these questions is to explore the limits of the known laws of physics, and in particular explore the question of free will. It is an old argument. The OP came in with a clever recasting of the old question.
The existence of the mooted computer is just a modern way of expressing the question. It isn’t actually needed. But it is a useful way of expressing the constraints on information present. What matters is that the information present is sufficient to allow such a computation. Not that the computation take place. Current understanding of physical laws says there is sufficient information.
Layperson question - does saying if you (impossibly) know the position and velocity of every particle you know all the past imply there is only one possible path for all particles to get in a particular position and velocity?
In other words, is the universe like Sudoku, with only one solution? I don’t know. Maybe a very simple system could have more than one valid solution. It’s a moot point though, because not only can we not know the position and velocity of all of the particles in the universe, it is not even possible to know both the position and velocity of just one particle. Not even theoretically.
I would think a more complex system could have even more paths to getting there, just as a function of the number of possible interactions that could lead to the same set of position/velocity in the impossible snapshot? I mean, as a lower limit, if there’s just one particle, there is actually only one possible path it could take given its current p/v. I think ditto for a 2-particle universe. But thereafter, not so sure whether there’s more than one set of bounces between particles that could lead to state NOW. And the more possible interactions, the more the chance of there being multiple paths goes up, I think?
And I know it’s impossible, that’s why I qualified my original question with the parenthetical “impossibly”.
There’re two impossibles. One, it’s impossible to know the position and velocity of every particle because there are too many of them. Two, it is fundamentally impossible to know the position and velocity of any particle. I wasn’t sure which you were referring to.
The second, I’m well aware of the UP.
If a single particle can only have one possible past, can’t a universe of particles be thought of as a collection of single particles, each with only one possible past?
Maybe.
In a universe of clockwork and bouncing billiard balls, there is one answer. This is a deep question, as it suggests that the universe does not have a sense of direction of time. It also demands pre-determinism of human action. Which is where the OP’s question comes in. People get very worried about the implications of pre-determinism. It has all manner of unpleasant issues for some. Others are quite comfortable about it.
In a universe of quantum mechanics and evolving probability functions, maybe there isn’t one single answer. Herein lies the nub of the problem. QM can, in some interpretations, provide a way for a non-deterministic choice to be made. At it simplest, this choice is perfectly random - God playing dice. But there are more than a few people who saw this gap as a place they could insert consciousness and also free will. Roger Penrose was not the first, but is currently probably the most prominent. There are of course other interpretations of QM. Some are deterministic.
This is the point about the OP’s question. It isn’t about whether it is physically possible to record the state of the universe exactly, or whether it is physically possible to compute the progress of the universe from that state (either forward or backwards.) It is a way of stating a thought experiment that allows us to reason about the existence of free will, and by extension moral choice.
In particular the question hinges on a single moral choice. That choice is the one made by John Wilkes Booth. In our universe Booth made the choice to murder Lincoln. In the moments after the time at we are targeting calculation of the position of Lincoln’s hat, Booth decides to pull the trigger. So there is a moral choice being made that appears in the timeline we are computing over. Is this choice one of free will, or was Booth’s action predestined?
We take the current state of the universe and calculate backwards. At a critical point we hit the state space of Booth’s brain just as he has made the choice to commit murder. We calculate back to the moment before that choice. The core question is - can we perform that explicit calculation step? If we can, we can then continue on to work out where Lincoln’s hat was.
But what are the implications of being able to calculate the state space of a human just prior to making a moral free will choice knowing what the outcome of that choice was? Is this a way of proving free will does not exist? It would seem it does. If we can compute where Lincoln’s hat is, we have computed Booth’s moral choice in reverse, and thus it was captured in the physical laws we calculated - which leaves no room for free will.
So, it appears we have clever argument about free will, one that is built entirely on events that have already happened and thus in some sense set in stone. There is a hidden idea that the universe evolves like a tree, with choices causing an expanding set of possible states (cf many worlds) and thus that there is only one possible pre-choice state that we compute back to. If we claim there is only one such state, the thought experiment might be able to step around questions of QM in the evolution of the state going forward. That perhaps provides a clever way of making a more powerful argument about free will than the more usual hand wringing about calculating the future.
Well that is disputed. There are positions such as “compatibilism” that would disagree with that claim.
I’ve made my views on free will clear before, so I’ll avoid bringing that here. For the sake of this thread, it suffices to say that that claim is disputed.
I’m not sure I am understanding this. Presumptively a particle has had only one past. That does not mean that there are not any infinite number of pasts that possibly could have resulted in the particle at that specific position and velocity in space time. Any specific particle possibly could have has many other paths to that point.
I’m not sure I am either.
Assuming that the laws of physics work the same in either time direction, forwards or backwards, which is our understanding. If we think of a billiard table, working forwards in time, if we hit the cue ball, there is only one possible future. The ball will hit other balls which will hit other balls, all in a predictable way. Working backwards in time, the same would be true, the present would lead to only one possible past for the system. Ergo, there is only one possible past for a system of particles / billiard balls.
I’m just musing here, if anyone has counter ideas, I’m all ears.
See this portion of @Francis_Vaughan’s post: the base is stochastic froth.
Though I never have bought the argument that randomness or the impossibility of prediction in any way implies Free Will, or even opens up the possibility of it. A game in which the outcome is completely determined by random chance does not imply that my Will impacted the outcome.
@Francis_Vaughan your moral choice question is essentially a rephrasing of The Ghost in the Machine. If there is a Ghost in the Machine, not resultant of it, then it is making that moral choice which cannot be calculated even if all physical processes could be. If our experienced consciousness is instead the result of physical processes then they in that hypothetical could be.
Exactly so.
I’m not taking any position on the question. This is GQ anyway. But I was seeking to outline the underpinning ideas that seemed to the the basis for the OP’s question.
IMHO everyone should at least be familiar with the basic ideas, even if they don’t have a personal stance. If ever there was a question for which there is currently no factual answer, this would have to be it. These questions go back a very long way. In Christian theology at least to Thomas Aquinas.
Personally I regard these questions as interesting, and worth understanding, but impossible to take a stance on. We neither have a complete physics or much understanding of cognition at all. There are quite a few caveats in the arguments, and more than enough wiggle room to take any stance one likes. Compatibilism is an easy answer but not exactly satisfactory. But I’d have to put myself there for want of any better pigeon hole.
For the very very little it is worth I go with Free Will as a necessary illusion, an illusion required as part of our sentience. But one can also argue that all of experience is an illusion created by our brains as interpretations of reality filtered for that which is most likely salient. We experience choice and in that sense it is real.
If I understand correctly that position is consistent with compatibilism.
The simple answer is yes: the fundamental laws of physics are time-reversal invariant, so in a ‘billiard ball’ universe, just as much as you only get one future given the present state, you only get one past given the present state. This is really just another way of saying that information is preserved: if you had the possibility of multiple possible pasts for a given present configuration, then some information about the past would have been irretrievably erased—because even given full information about the current state, you couldn’t exactly determine the state at some point in the past, and would need ‘extra information’ not contained in the current state to determine it.
This continues to be true in quantum mechanics. The dynamics given by the Schrödinger equation is reversible, such that given the present state of a system, you can evolve it uniquely backwards as well as forwards in time. Now, however, we have the complication of measurement: there, in general, after having measured the current state, you won’t have any access to the state prior to measurement (getting a result of ‘spin up’ won’t tell you if the state was ‘spin up’ before, or a superposition of ‘spin up’ and ‘spin down’). But things depend on the interpretation of QM here. Typically, one still expects the Schrödinger equation to be valid for the total system, i. e. the object and whatever does the measurement; so there, still, information conservation would hold, and dynamics are reversible. In particular, one then expects this to be the case for the universe in total. But things like objective collapse theories might paint a different picture.
—and the reason that’s only the simple answer is that you can, in a classical Newtonian context at least, construct pathological examples such as Norton’s Dome, which is usually taken to question determinism in a classical setting: a ball sitting on the apex of a dome slides down in an arbitrary direction. But the time-reversal of this is also possible: giving the particle, resting at the bottom of the dome, a push such that it comes to stop at the exact apex, where then the information from where the particle started would be lost.