> And no practical amount of shaking the jar will ever return the marbles to a high order state again.
This can't be true. Take the case where there's only 2 marbles, it's likely that at some point that shaking them will return them to the original state. For higher numbers of marbles the probability becomes lower and lower, but is never 0.
> For higher numbers of marbles the probability becomes lower and lower, but is never 0.
Exactly. It doesn’t take a very large jar to make a return to an original state practically impossible. But even for two marbles, shaking them to return them to their original locations costs a fair amount of organized energy.
And that is organized chemical energy in your muscles, which you are dissipating as disorganized heat energy due to your efforts, increasing the total disorganization.
And the marbles are microscopically wearing down, shedding compounds. The jar is also wearing down. Both becoming less organized.
So even with two marbles, overall entropy continues increasing.
We can locally reverse entropy, only at the cost of actually increasing entropy in total.
- Organizing your day takes effort and uses up useful organized energy.
- Computation uses up useful organized energy.
- Your cells, maintaining their metabolism, uses up useful organized energy.
With enough marbles, the discrete distribution behaves like a continuous distribution. The point probability of a continuous distribution is infinitely improbable -- in other words it is 0.
Any point is infinitely improbable when sampling with infinite precision, only because this calculation is not computable.
Even if the point probability is 0 or approaching zero very fast, some event occurs with probability 1.
The probability that life would evolve on a blue planet that its anthropomorphic inhabitants will come to call Gaia or Earth, has as a star Sol, in this corner of the Milky Way, in this particular local group and so on is zero even if we limit ourselves to just the observable universe.
At the same time the probability that life will exist at some planet at some solar system at some galaxy at some local group and so on, is practically 1.
That is to say, depending on how you categorise and count said marbles, the reordering may occur.
While any individual marble will not be at its place with P=1. You can still end up in a situation where the marbles are ordered in layers.
The whole thing regarding entropy is concerned with a closed system. You can very much exchange energy to decrease entropy, but said exchange is a) leaky, and b) implies that the system is not isolated.
A probability of zero does not mean that the event is impossible. When we refer to probabilities we are talking about probability densities. Infinite precision is a useful modeling tool in the same way that approximations of pi are useful.
I think I was misunderstood. My comment regarding infinite precision was mostly aiming to argue that any single point has probability zero but an event still occurs.
This was then used again when mentioning earth to argue that depending on the definition of "order" here:
> And no practical amount of shaking the jar will ever return the marbles to a high order state again.
It is entirely possible to reach a state of higher order if you don't require that individual marbles don't hold the same arrangement relative to adjacent marbles, meaning that any permutation of marbles with identical color is acceptable.
This happens because the event you are asking for is a very large subset.
Define "large", at least in an estimate. Dividing two astronomically large numbers is not easily estimated. And define "entirely possible". Yes, could happen once, maybe, in the entire Universe is entirely possible. Anything is "entirely possible" because humans are not omniscient, so we have to start from Bayesian priors.
In physics, nothing can be measured exactly, so exact events are in fact impossible. Measurements are approximate, and thus have finite extent, and thus can have nonzero probability.
This whole thread is silly, The 2nd law is about probabilities which are asymptotically 0, so small in practice that we could never do experiments to frequentist probability empirically, using "jars of marbles" with more than 100 marbles", let alone trillions of trillions of particles.
That is a great point. The laws of thermodynamics exist as a logical consequence to fundamental statistical principles. It is statistical mechanics that gives rise to entropy, not the other way around. (hence the beautifully elegant equation S = Kb*ln(Omega) where Omega is the number of microstates)
I realize that programmers are not always good scientists. Yet some claims are absurd in ways that I now believe this phenomenon necessitates rigorous study.
Mathematics applies insofar as it can model the subject under discussion. It's a great tool for, dare I say, 99.999999...% of practical problems. But mathematics is an imperfect model of objective reality that cannot resolve metaphysical problems like Zeno's paradoxes [0]. Any discussion of retro-causality is inherently one of the philosophy of space and time, which is the domain of metaphysics, not mathematics, so we can't necessarily apply familiar mathematical lemmas to resolve the problems it creates.
I'm neither a mathematician nor a physicist, so I'm not nearly informed enough to elaborate on the subject properly, but I would note that the axiom of choice is not proven, and while you can mathematically divide a continuum into infinitesimally small sets, you cannot do the same to physical matter. From my understanding, many of the contradictions between classical and quantum physics arise at this boundary between the discrete and the continuous, where classical physics generally assumes continuity while quantum physics is constructed around discrete quantization mostly independent of time.
Again, I'm not a physicist, but I think it's telling that the validity of continuum mechanics [0] depends on a model and multiple assumptions. I have no trouble agreeing with your original statement, "with enough marbles, the discrete distribution behaves like a continuous distribution," when speaking mathematically, but mathematics by its nature is an idealized model of the world - I'm not willing to accept that it's objectively representative of physical reality. The discussion we're having here is one of metaphysics, so it feels a bit like the height of hubris to use mathematics as the tool for describing objective reality, because metaphysically, we cannot say that mathematics is anything other than a tool we've constructed for approximating the model of the world as we understand it. Considering the subject of this post is about retrocausality, we're already throwing out some pretty wild ideas, so I think it's a bit hubristic to dismiss them by citing a branch of mathematics that assumes the existence of countably infinite sets [1].
This whole thread hinges on your not giving credence to the word "practically" , as in "approximately". No countably infinity needed, and Continuum Hypothesis isn't relevant at all in any way, as it is aboit uncountable infinity. Until you explain what you are trying to say, you aren't saying anything meaningful. The one showing hubris here is you, tossing around technical terms don't admit you don't understand.
Indeed, the original claim treats mathematics as a modeling tool, not philosophy. Nevertheless, it is my belief that untestable hypotheses about True Reality (should it even exist) are not debatable without causing migraines of epic proportions. There's no physics without mathematics -- as physicists say, shut up and calculate.
> The one showing hubris here is you, tossing around technical terms don't admit you don't understand.
Hence the disclaimer. I just think this stuff is fascinating. If you're interested, this is the article that originally got me thinking about these ideas as they relate to the flow of time: "Does Time Really Flow? New Clues Come From a Century-Old Approach to Math" [0]. It's effectively an appeal to intuitionist mathematics and a rejection of constructivism, where - simplified - every particle since the big bang can be assigned some "number" that describes its position on its world line, with a precision that is finite, but always increasing - and it's this increase in precision from which emerges the "flow" of time. If you ctrl+f the article for "continuum," you'll see some better constructed arguments for what I'm trying to argue, e.g.:
> Moreover, the continuum can’t be cleanly divided into two parts consisting of all numbers less than ½ and all those greater than or equal to ½. “If you try to cut the continuum in half, this number x is going to stick to the knife, and it won’t be on the left or on the right,” said Posy. “The continuum is viscous; it’s sticky.”
The impression I'm left with is that the difference between constructivist vs. intuitionist mathematics (which rejects the law of the excluded middle) has some parallels to the difference between eternalism (there is a past, present and future) vs. presentism (there is only the present).
As a layman, I just can't help but notice the same patterns come up again and again when looking at unresolved problems between classical and quantum mechanics, namely those that lie at the boundary between the continuous and discrete. FWIW, I asked ChatGPT for some unresolved mathematical problems that might relate to this, and it came up with: The Continuum Hypothesis, The Axiom of Choice, and The Riemann Hypothesis, all of which it admitted are possibly impossible to prove because they're effectively unfalsifiable. (I also showed it this entire thread, and it agreed with your critique of my arguments, but also that every argument in the thread is unfalsifiable - and then it pointed me to the Boltzmann Brain paradox).
The number of possible states asymptotically approaches infinity, so we can model it as such. You can get a more "accurate" model using more math (with measure theory), but the terms will coalesce such that they are negligible.
you're correct. In stats/thermo class there's a commonly taught "what's the probability of all the molecules of air in a room spontaneously moving to <extremely small location>". The problem is that shoving all those molecules into a tiny location woudl increase the pressure tremendously, sending all the particles in directions that would eventually return to a uniform distribution.
IIUC my professor right the probability is non zero but is practically impossible for a large number of incompressible spheres.
This can't be true. Take the case where there's only 2 marbles, it's likely that at some point that shaking them will return them to the original state. For higher numbers of marbles the probability becomes lower and lower, but is never 0.