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I found the following statement attributed to Richard Dawkins:

"A miracle is something that happens, but which is exceedingly surprising. If a marble statue of the Virgin Mary suddenly waved its hand at us we should treat it as a miracle, because all our experience and knowledge tells us that marble doesn’t behave like that"

"In the case of the marble statue, molecules in solid marble are continually jostling against one another in random directions. The jostlings of the different molecules cancel one another out, so that the whole hand of the statue stays still. But if, by sheer coincidence, all the molecules just happened to move in the same direction at the same moment, the hand would move. If they then all reversed direction at the same moment the hand would move back. In this way it is possible for a marble statue to wave at us. It could happen. The odds against such a coincidence are unimaginably great but they are not incalculably great. A physicist colleague has kindly calculated them for me. The number is so large that the entire age of the universe so far is too short a time to write out all the noughts! It is theoretically possible for a cow to jump over the moon with something like the same improbability. The conclusion to this part of the argument is that we can calculate our way into regions of miraculous improbability far greater than we can imagine as plausible"

Is this valid and if so what would the probability and mechanics behind both the statue waving and the cow jumping of the moon ?

Edit#1: I forgot to also ask if the "physics" mentioned in the article I linked denying the possibility of Dawkins' example is right?

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    $\begingroup$ It's valid, but any calculation of the probability would be somewhat of a wild guess. A really really really small number. $\endgroup$ – Aaron Stevens Jul 22 at 22:26
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    $\begingroup$ I feel that when the probability of something happening passes below some very small but non-zero value, then it becomes misleading to use phrases such as "it could happen". In plain English, such events could not happen. Note also that Dawkins has picked for his illustration a rather meaningless event, one for which we have no reason to think that special considerations might apply. $\endgroup$ – Andrew Steane Jul 23 at 0:01
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To get a feel for this kind of (im)probability, consider the simplified case where molecules either move left or right with 50% chance. A marble statue hand is maybe 1 kg, the molar weight of marble is conveniently nearly exactly 100 g/mole, implying 10 moles, e.g. $10N_A=6.02214076\times 10^{24}$ molecules. The chance that they would all be moving to the left simultaneously is 1 in $2^{10 N_A}=2^{6.02214076\times 10^{24}}=10^{1.812845\times10^{24}}$.

Molecules typically change direction fast, so we get a lot of "trials" every second, but we do not have to calculate how many there are since the above probability is so small than even if they tried once every Planck time it would be far, far longer than the expected time till proton decay or the end of the black hole era.

One can try to refine the calculation with more directions, not every molecule moving in the same direction and so on. But the answer is still the same. This is why thermodynamics can rely on statistical mechanics: since there are so many molecules, macroscopic averaged properties behave in very lawful ways, and fluctuations tend to be tiny.

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    $\begingroup$ Not just that such a probability is ridiculously small, it is so small that the probability of the whole of physics, on the basis of which we are doing the calculations, being wrong is likely to exceed it. $\endgroup$ – Dvij Mankad Jul 23 at 1:38
  • $\begingroup$ @FeynmansOutforGrumpyCat - Yes! Kind of. I actually have a paper on the problems of reasoning about this kind of tiny probabilities arxiv.org/abs/0810.5515 If I see a waving statue the probability of me hallucinating is vastly higher than it actually happening (even a believer should think this, since miracles are presumably more rare than madness), so I have a good reason to discount it. However, physics being wrong means a lot of independent checks (every boiler, every microchip, every argument) being wrong at the same time - and that can perhaps be made less probable. $\endgroup$ – Anders Sandberg Jul 23 at 9:04
  • $\begingroup$ Thank you for the link! Looks like an interesting read :) $\endgroup$ – Dvij Mankad Jul 23 at 11:37
  • $\begingroup$ It occurred to me that there are numerous ways that a statue could wave without breaking physical laws; e.g. a focused gravity wave; an incident intense particle beam that caused a change in elastic properties of a small region; etc. All highly improbable but a lot more likely than coincidental movement of a mole of molecules. By ignoring the possibility of correlated and cumulative processes Dawkins is here committing a classic error, the same one made by "intelligent design" arguments in biology. $\endgroup$ – Andrew Steane Jul 24 at 19:33
  • $\begingroup$ @AndrewSteane - Not really. When you allow the hypothesis space to become big enough it becomes meaningless. After all, you could apply the same thinking to religious miracles and claim that they just happen because of thermodynamic miracles and hence do not give any religious evidence. Sure, you might meet an angel giving an eloquent speech, but it could be random thermal motion of air and light. The key thing is that the probability of that is very low - alternative hypotheses, including actual angels and hallucinations - are far more likely. $\endgroup$ – Anders Sandberg Jul 25 at 8:44
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Such a thing as a statue waving its arm is indeed physically possible but gigantically improbable. However, some analogous things are physically impossible. The moon, for example, cannot suddenly leap out of orbit by itself. A pebble floating freely in space cannot suddenly take off in an arbitrary direction. Both of those would violate conservation of momentum. A statue could in principle wave its arm without violating conservation laws because the arm is attached to the rest of the statue. The arm might move left; the statue would shift to the right, and momentum would be conserved. The probability calculated by @AndersSandberg is smaller than can be imagined, but the actual probability is probably even vastly smaller, because of conservation laws.

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    $\begingroup$ what about the other example of the cow, is it possible or does it violate conservation of momentum? $\endgroup$ – c k Jul 23 at 5:04
  • $\begingroup$ It depends on where the cow starts its jump. If it starts on the Earth's surface, or even in mid-air, the Earth can carry the backward momentum. However, the cow couldn't immediately accelerate to the necessary velocity or it would be smashed (and then incinerated by atmospheric friction); so it would need help from molecules in the air to push it along, gradually building up its speed in the right direction. All together, probably even less probable than a marble statue jumping off the ground and doing a back-flip. $\endgroup$ – S. McGrew Jul 23 at 5:11
  • $\begingroup$ Or the ground could improbably launch the cow into orbit. $\endgroup$ – Aaron Stevens Jul 23 at 23:00
  • $\begingroup$ The extreme value of these improbabilities is so extreme, it makes me wonder if there is a minimum nonzero improbability, vaguely analogous to Planck length. Probably not-- $\endgroup$ – S. McGrew Jul 24 at 0:40

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