Probability of the Physical Resurrection

It is a Christian belief that Jesus came back from the dead (McGrew & McGrew, or as I will refer to the article, {\rm McGrew^2}). Some claim that the resurrection has significant historical evidence, great enough to compel a rational person to accept it as an historical fact. If something is an historical fact, then it happened physically, at some time and in some location. Maybe Jesus’s resurrection is not an historical fact, but rather a religious fact or some mystical experience. We can figure out the probability of it being an historical fact, but not of it being a mystical experience.

Let’s consider the physical resurrection. How unlikely is it? It’s hard to say with great precision, but an estimate can be made about how likely it is for a person to come back from the dead by known natural causes. It’s possible. A person is, physically speaking, the sum total of his atoms in a particular arrangement. Let’s say that each atom or molecule in this person’s body has nine indendent degrees of freedom (3 vibrational/velocity-components, 3 rotational, 3 translational). When he dies, let’s say some large fraction of his body is reduced to what can be approximated as an ideal gas. Let’s say 10^{26} molecules.

The simplest approximation to be made involves calculating the difference in probability function between an ideal gas and a person, and we can do this via Boltzmann’s entropy equation for entropy, considering the entropy of a dead person compared to the entropy of a live person:

S = k_B \log \Omega

Where \Omega is the number of microstates that correspond to the “macrostate” of “being alive” or “being dead”. For the ideal gas, we can use the Sackur-Tetrode equation, assuming the volume of, say, a sealed tomb near the first temple period. I’ll approximate the volume as {\rm 1000 \; m^3}. We also take for the internal energy 9/2 k_B T (assuming equipartition for the gas) and k_B is Boltzmann’s constant:

S_{\rm dead} = k_BN \log \Bigg[ \Big(\dfrac{V}{N}\Big)\Big(\dfrac{4 \pi m k_B T}{Nh^2}\Big)^{3/2} + \dfrac{5}{3}\Bigg]

where m = {\rm 3 \times 10^{-23} \; grams} is the average molecular mass, taken here to be the mass of a water molecule and h is Planck’s constant. We will assume that the temperature in the room is about 300 K. Our approximation for the entropy of a dead person is about S_{\rm dead} \approx 3 \times 10^{11} \; {\rm erg/K}. S_{\rm alive} will be crudely approximated by approximating the number of microstates that corresponds to “being alive” as \approx 1, and so S_{\rm alive} \rightarrow 0, at least compared to the value of S_{\rm dead}. So:

S_{\rm dead} - S_{\rm alive} = k_B \log \Big(\dfrac{\Omega_{\rm dead}}{\Omega_{\rm alive}}\Big)

And the probability is then:

P_{\rm resurrection} = \dfrac{\Omega_{\rm alive}}{\Omega_{\rm dead}} = \exp\big(-S_{\rm dead}/k_B\big) = 10^{-10^{27.4}}

But this is only for one attempt, one roll of the dice. We can roll the dice a whole bunch of times, over and over. Let’s say that the molecules completely re-arrange themselves over the average time it takes a molecule of the dead person to get from one end of the tomb to the other, assuming random diffusion. The diffusion equation is:

\dfrac{\partial \\\phi}{\partial t} = D \nabla^2 \phi

Which can be solved by taking the size of the room as the scale length, L and \tau is the time it takes for the system to get rearranged:

\dfrac{\phi}{\tau} = \dfrac{D\phi}{L^2}

And so for a diffusion coefficient for oygen-air D \approx 0.2 \; {\rm cm^2 s^{-1}}, the time-scale to reset is about \tau = L^2/D \sim 1 \; {\rm day}, or faster if warmer (although the entropy also increases in such a case).

In order for this to be likely to happen by the known laws of nature, we would have to wait 10^{26} times longer than the age of the universe.

Virtually any explanation is more likely than that Jesus came back from the dead through the known laws of nature. This means that, among other possibilities, there are either unknown laws of nature that make it more likely for people to return from the dead, or some supernatural force or entity has the capability and desire to raise at least one person from the dead. Since there is no good evidence* for either of these alternatives, it seems best to conclude that the resurrection of Jesus is at most a spiritual resurrection, and not a physical resurrection.

EDITED TO ADD:

We can do a very simplistic Bayesian analysis, given my prior and using the hand-wavy probabilities of {\rm McGrew^2}. We find that the probability for a miraculous resurrection ({\rm R}) vs. no miraculous resurrection ({\rm \lnot R}), given the facts of the matter, {\rm F}, and background information, {\rm \chi}:

{\rm P(R | F \& \chi)} = \dfrac{{\rm P(F \& \chi | R) \, P(R)}}{{\rm P(F \& \chi|R)P(R)+ P(F \& \chi|\lnot R)\,P(\lnot R)}} \approx 10^{44} \times 10^{-10^{27.4}}

\approx 10^{-10^{27.4}}

The probability for the resurrection is effectively completely unchanged by the available evidence, as unlikely as it is to have happened by chance.

*What does this “no good evidence” even mean?

All I mean by this is that I personally am not aware of any other value to set my priors, because I don’t know what other mechanism to use in order to estimate these priors. How likely should I think it is that someone resurrects before I look at the evidence? It seems best to compare to the physical principles that we know now, and treat the resurrection as a thermodynamic miracle that requires evidence. Even if someone thought that all physical events are enacted by God, this would suggest that God’s behavior is so orderly that him deviating from this order would be just as likely as a thermodynamic miracle.

The alternatives, new physics and divine free action, are difficult to incorporate into any Bayesian analysis. How do I include new physics? What processes should I invoke to estimate my probabilities? How do I include divine free action? By what method do I estimate the chance that God will decide to resurrect people? Stan says he came back from the dead and that he’s the son of God. Because of his special relationship with God, he claims it’s more likely he came back from the dead already. How do I determine the prior probability that Stan came back from the dead, before looking at the evidence? Do I assume his claim is true? If I assume it’s true, then why bother claiming that the resurrection is decisive evidence for the claim? If I assume at the beginning that his claim is likely false, then I’m back with the calculation above. Besides, why would God want to kill his own son? How am I going to psychoanalyze the Creator of the Universe?

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About שאול בן דניאל

I'm a dad, husband, scientist. I spend a lot of time wondering how life came about, and about how lightning works, and how these can be connected on planets outside our solar system.
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7 Responses to Probability of the Physical Resurrection

  1. steamchip says:

    “Since there is no good evidence for either of these alternatives, it seems best to conclude that the resurrection of Jesus is at most a spiritual resurrection, and not a physical resurrection.”

    The Shroud and its linked cloth Sudarium of Oviedo is believed by many to testify to an actual physical resurrection of Jesus the Christ.

    –Someone reproducing the Shroud with ancient or medieval even modern methods have thus far failed.
    –no one yet just discovered another burial cloth with a similar imprint. Of the billions of people died and among those an amount being wrapped up in a shroud and NO other cloth like it being found?!?
    –these things coming to pass would put the matter to rest in favor clever artist creation or by naturalistic processes .

    Shroud of Turin has been gone over by numerous scientists over many decades. Just when one of them declares the Shroud a fake, the methodology is shown to be flawed, erroneous or incomplete and they are back trying to solve the mystery of it.

    The account in the Bible of Thomas actually touching the Resurrected Jesus seems directly aimed at deflating the idea the event was “spiritual.” (John 20:24-29)

  2. And the odds of a low entropy universe (according to Roger Penrose) is 10^10^123 to one….. Yet here we are.

    • Good point, and one that is aptly handled with the probabilistic analysis. If new evidence was discovered such that the probability of no resurrection, given the new facts of the matter, was less than about 10^10^-27.4, then I would accept on the evidence that the resurrection happened. One way this could happen is if I could witness the resurrection by means of time travel.

  3. [—
    One way this could happen is if I could witness the resurrection by means of time travel.
    —]
    To think that someone ~2000 light years away could theoretically observe earth and see the resurrection or any other historical event is an amazing thing. In fact, when you think about it, every thing we do, good or bad, is captured in the cosmos forever. It can be chronicled by a spectator in the correct location, with the correct instruments, well after the events actually took place.

    • That’s right! But they would need a very good telescope. We can’t see the field of comets about 3/4 of a light year away, even with our best telescopes. But wow, what an amazing detection that would be.

  4. Sorry, but this is irrelevant claptrap. No one claims Jesus rose from the dead by natural means. Your glib, completely unsupported claim that “there’s no evidence” that a supernatural being has the power or will to raise someone from the dead (in the face of McGrew2’s article, among a lot of other fairly thorough and informed scholarship) counts for nothing.

    Here’s my article on the Prior Probability of the Resurrection. Thousands of reads, no serious rebuttal yet:

    http://christthetao.blogspot.com/2012/03/prior-probability-of-resurrection.html

    • Thanks for taking the time to comment on my blog. I read your argument for the priors and had trouble determining how you assigned any of the probabilities, especially for (2) whether God would resurrect someone. Let’s set the probability for God’s existence and for Jesus’s likeliness to be raised from the dead by God, if anyone is to be raised from the dead, to 1. Does God like raising people from the dead? Maybe God hates the idea. Maybe the chances for his raising people from the dead are zero. It might seem as though it would be a loving thing for God to raise Jesus from the dead. But God probably loved Jesus and let him die on a cross, even though he could have prevented it. It seems difficult to guess God’s actions even knowing his motives. I can’t always guess what my wife will do next, and she’s a human being that I intimately know. How can an ant hope to guess as the future actions of a human being? How can we hope to guess at the future actions of God?

      You say in one part (2e) that the resurrection would be “some ‘good disaster’ by which (to cite his friend C. S. Lewis) the ‘laws of nature would begin to work backwards,’ so that Entropy would not have the last word…” We can determine how likely God is to reverse entropy using the calculations I presented above. If God were given to reverse entropy more often in general, we should have seen this effect experimentally. The experimental evidence of how God deals with entropy is encapsulated in the estimate above.

      The reasonableness of a particular heuristic for assigning prior probabilities would seem to be determined by what sorts of conclusions that can be reached via consistent application of the heuristic. If the heuristic would result in the rejection of what you are confident is the case, or would result in acceptance of what you are confident is not the case, then the heuristic and examples both need to be examined more carefully.

      Your heuristic is vague, and it would seem as though you could assign any probabilities you like based on imputed motives and metaphysical speculation. Given the wide range of probabilities this might produce, you could end up accepting the veracity of psychic powers or the existence of ghosts or special creation accounts or a handful of other absurd claims. The heuristic seems as though it could overestimate the prior probability of unusual events.

      Does my heuristic fail in some obvious cases? Might it underestimate the prior probabilities? Can you think of any examples where applying physics to calculate the probability of the generic event in question leads to bad prior probabilities for historical events? Does it obviously underestimate prior probabilities or overestimate them? Can you think of any counter-example for the method I’ve employed here?

      Forget the controversial cases (like bodily resurrection!). Is there a non-controversial case where my analysis would vastly overestimate or underestimate the prior probabilities?

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