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Sam Harris (a famous atheist) argues in an interview with Cenk Uygur that the probability of Jesus Christ coming back to Jackson County, Missouri, USA is less likely than the probability of Jesus Christ coming back anywhere on Earth. He says that it's a "mathematically true point. This is just probability theory...you're going to hear from a bunch of mathematicians who are going to insist that you grant that."

However, Cenk Uygur (also an atheist) argues that since Jesus Christ is not coming back at all (as the whole thing is "totally untrue"), then speaking about probabilities makes as much sense as dividing by zero.

Can mathematics (probability theory) be applied to a future event that might be impossible, specifically the Second Coming of Jesus Christ? If so, under what conditions/assumptions, would Sam Harris be correct?

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Steve McQueen
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    Cenk's point is correct if there literally is no probability of Jesus returning, giving both possibilities a probability of $0$ (though this does not mean that it is nonsensical to speak about), or if the only possible place Jesus could return to is Jackson County. If we acknowledge some nonzero probability that Jesus returns, and say that it is possible for him to return to other places (given that he returns), then Sam Harris is correct. – Eff Oct 27 '14 at 11:48
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    Applying probability theory or statistics (which deal with the study of repeated events or behaviour) to singularities makes little logical sense. – Lucian Oct 27 '14 at 12:23
  • @Lucian there is more than one interpretation of probability. You are using the "probability as long-run frequency" definition. Look up subjective probability and Bayesian data analysis...both of which can speak meaningfully about singular events. –  Oct 27 '14 at 12:27
  • @Eupraxis1981: True, but one might then wonder what the Bayesian probability of the Big Bang ever having happened is. But that is more an observation about the way in which ideologically driven propaganda uses, misuses, and abuses objective scientific notions. – Lucian Oct 27 '14 at 12:52
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    @Lucian I agree that you cannot verify subjective probabilities or credences, and I really only think about probability in a frequentist sense myself for that very reason. However, I just wanted to point out that logically, you can attach a probability to a singular event and discuss it, just like you can come up with axioms and derive "theorems" from them. Of course, you don't have to agree with the axioms... –  Oct 27 '14 at 12:55
  • I watched that interview a few days ago, and was in total agreement with Sam Harris on his probability argument. I'm quite amused this question found its way to this forum, @SteveMcQueen are you per chance trying to settle some internet dispute? – Olivier Bégassat Oct 28 '14 at 01:51

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If $A$ is a subevent of $B$, then $P(A) \leq P(B)$.

Edit: As 5xum points out, Harris says that $A$ is less likely than $B$, which literally means that $P(A) < P(B)$. This is, strictly speaking, incorrect, as it does not account for the possibility that $P(B \setminus A) = 0$. He cannot conclude that $A$ is less likely than $B$, only that $A$ is no more likely than $B$.

Michael Joyce
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    True, but sayint "$A$ is less likely than $B$" implies that $P(A) < P(B)$. – 5xum Oct 27 '14 at 11:30
  • Yes, there is unfortunately an asymmetry in the English language where it is much easier to say $P(A) < P(B)$ concisely than it is to say $P(A) \leq P(B)$. Thus, if Harris is attempting to speak precisely, then he is incorrect in his assertion. Editing my answer to reflect your observation. – Michael Joyce Oct 27 '14 at 11:36
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    I would say that the easiest way to say $P(A)\leq P(B)$ is to say "$B$ is at least as likely to occur as $A$". – 5xum Oct 27 '14 at 11:37
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    Right, and that is more complicated to parse than "$B$ is more likely than $A$". In my experience, in conversational speech, few people distinguish between the different meanings of the two expressions! And hence the silliness that apparently ensued in this conversation is not uncommon. – Michael Joyce Oct 27 '14 at 11:39
  • If you are going to say something is a "mathematically true point," the onus is on you to state the point precisely (or else you will never make tenure). – Tahlor May 26 '21 at 02:50
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There are many possible probability measures that could allow us to meaningfully compare probabilities of probability zero events, using something like the Radon-Nikodym derivative. (After a little reading, I have found that there is a general statistical construction called the conditional probability distribution that fits the bill nicely.)

To take the first example that popped into my head, suppose that $P(x)$ is a continuous probability distribution function on $\mathbb{R}_{\geq 0}$ such that $\int_{0}^\epsilon P(x) dx$ is the probability that somebody will appear with genetic distance at most $\epsilon$ from Jesus. Then (by our assumptions about the distribution) there must be zero probability that Jesus Himself will appear.

But nevertheless, we could have $P(0) > 0$ (and I would guess that we do, given my many encounters with people similar to Jesus). Given some other distribution $Q$ (like the weighted distribution function for Jackson County), we could imagine calculating that $P(0) >> Q(0)$ very explicitly.

This is a meaningful way to numerically compare the relative probabilities of the various impossible appearances of Jesus, so Cenk Uygur has clearly jumped to conclusions, most likely from an inadequate foundation in Lebesgue integration and measure theory.

His "like dividing by zero" comment further suggests that he has never even heard of the Dirac delta measure! I find his willingness to comment on such sensitive mathematical matters an appalling abuse of the interviewer's chair, and I insist that he recuse himself from the public stage until he has done something to address these foundational statistical deficiencies.

At my university, even first-year graduate students in statistics should be able to compare impossible events with ease. There is really no excuse for these standards, especially from celebrities.

Andrew Dudzik
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I side with Cenk Uygur here, Sam Harris' argument is not good math.

Harris is probably referring to the fact that P(A ∧ B) ≤ P(A), but that rule is only relevant if you already have a probabilistic model for your system.

However, here they are talking about which probabilistic model is the right one. Harris is succumbing to the issue Laplace tried to describe in his "sunrise problem". If you assume a probabilistic model where there is a constant probability p that the sun rises each day, you can use Bayes theorem (as Laplace did) combined with the observation that the sun rose every day you are alive (10000 days, say) to show that the probability that the sun will NOT rise tomorrow is 1/10001. However, this is clearly nonsense, as we have a much better physical model of the solar system from which we know that the probability that the sun will rise is pretty much 100%.

By Harris' logic, the 'uniform probability' model is more likely because it simply predicts whether or not the sun will rise. In contrast our solar system model not only predicts whether it will rise, but also the precise time at which it will rise, and is thus is less believable. This illustrates that the fact that a model makes more precise predictions does not make it more wrong.

Really, the issue Sam and Cenk are discussing is a "model selection" problem, specifically for the three models of Atheism, Christianity and Mormonism. Model selection is not math, it is philosophy, as you can read in the link. Sam's position is not mathematically justified.

ealloc
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  • It's true that Sam's point is (almost) mathematically true as others have suggested, but as you suggest, not a particularly good argument. By the same logic, P(no god) ≤ P( god ∪ no god); the way Sam is using this "mathematically true point" would suggest his atheism is necessarily inferior to agnosticism. – Tahlor May 26 '21 at 03:10
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  1. There is no way to prove that God* does not exist with 100% certainty.
  2. Thus, the existence of God is possible and has a non-zero probability, however small.
  3. Then it follows that the event of God bringing back Jesus Christ anywhere on Earth is also possible and has a non-zero probability.
  4. Therefore, the probability of God bringing back Jesus Christ to a specific location on Earth (Jackson County, Missouri) has a smaller probability than God bringing back Jesus Christ to any possible location on Earth.
  5. So, Sam Harris is correct.

*An omniscient, omnipotent, omnipresent, omni-everything God, of course

jacobernelson
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    Uh, does not God "bringing back Jesus" mean you are assuming a Christian God? Also, you can prove an omnipotent God can't exist. Can She create a rock so heavy she can't lift it? – Cheerful Parsnip Oct 28 '14 at 02:01
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    @GrumpyParsnip I don't think dei could produce a rock that dei couldn't lift deiself, but dei may ask deis deity friends to make one for deis. – Olivier Bégassat Oct 28 '14 at 02:14
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    I couldn't understand that very well, but if a friend makes such a rock, that implies the original deity is not omnipotent. – Cheerful Parsnip Oct 28 '14 at 10:16