I posted the first version of my rebuttal to Tim and Lydia McGrews' argument for the Resurrection last year, and it was more difficult to understand and more heavily mathematical than was necessary. This was unfortunate, for I believe that it is still the Internet's only rebuttal that engages the math head on. Many people have noted ways the argument is “obviously” invalid, and my first impression was exactly the same. But the “obviously” invalid step isn't shoved under the rug – the McGrews give specific reasons in defense of this step. In my opinion, it is not at all obvious what in particular is incorrect about their defense.

Although, there is a limit to how simple math can be made. The McGrews' argument uses Bayes factors, and so neither their argument nor my rebuttal can be understood without some knowledge of probability. This second try should be simple enough that if you understand Bayes' Theorem, you should be able to understand this post.

The McGrews' Main Point

Tim and Lydia McGrew have written a chapter in The Blackwell Companion to Natural Theology titled The argument from miracles: a cumulative case for the resurrection of Jesus of Nazareth. Their argument needs to be understood in the context of the standard argument for the Resurrection based on the disciples' testimony and death:

Claim 1: The disciples believed that they saw Jesus after rising from the dead, and they believed with enough sincerity to die for this belief.

Claim 2: Based on these beliefs, it is probable that Jesus rose from the dead.

This is not the McGrews' primary argument in the chapter. They are not making a full argument for the Resurrection. Their primary claim is not even a full defense of Claim 2, although it comes very close.

Let R be the Resurrection of Jesus, and let P, D, and W be the events that each of Paul, the disciples, and the women at the tomb claimed to have seen Jesus, and in many of these cases, died for this belief. The McGrews' primary claim is that P & D & W together provide a Bayes factor of 10^44 in support of R over ~R. Within this post, I am rebutting exactly one thing: their primary claim.

Edited to add: I need to be very specific about the sort of death it takes to qualify as D. What if the disciples died for their belief in a moralistic religion based on Jesus, but not the Resurrection in particular? What if Jewish leaders in general were rounded up and killed, and the disciples qualified as leaders? What if they didn't have the ability to recant? In this case, lying disciples dying for their faith is plausible. If these possibilities still count as D, then Claim 2 is weaker. If these don't count as D, then D is less likely and Claim 1 is weaker. I'm defining D to be the event that they died for their belief in the Resurrection in particular, and they had the ability to save their lives by recanting. I'm defining it this way to make the McGrews' argument stronger and show that their fundamental argument is wrong regardless of details like this.

The McGrews' Argument

There are 13 disciples in the argument (the twelve minus Judas plus Matthias plus James the brother of Jesus.) Under the hypothesis ~R, the probability that, say Matthias would persevere as a Christian is about 1000 times smaller than that probability that he would do the same under R. From this it follows that for each disciple and for Paul, we have a Bayes factor supporting R over ~R of 1000. They estimate the factor for W to be 100.

First, suppose these are independent. If so, the cumulative Bayes factor is found by multiplication, which gives 10^3 * 10^(3 * 13) * 10^2 = 10^44. This would be strong enough to overcome a prior probability on R as extraordinarily small as 10^-40, and make R 1000 times as likely as ~R. (Of course, they aren't independent, and this is what makes the argument “obviously” wrong.)

The fact that the events are not independent is recognized by the McGrews and responded to on pages 40-46. While dependence could lead to overestimating the factor, it could go the other way too. While it's possible that killing one martyr encourages the others, the more likely effect is that it scares off other people, who now realize that their life is in danger. So while the McGrews recognize that these aren't independent, their claim is that factoring in the dependence makes the Bayes factor even larger.

Rebuttals I'm Omitting

The bulk of the factor comes from the 10^39 factor for D, and so I will focus my rebuttal on that point and make no further mention of P or W.

One could argue that D is not true. This completely fails to rebut the McGrews' argument. They are defending Claim 2, and changing the subject to Claim 1 does not rebut Claim 2.

One could argue that the factor of 1000 used for each disciple is too large. Most rebuttals used against Claim 2 in the standard apologetic argument fall in this category. While these are important rebuttals, they are ineffective against the McGrews' version. Suppose the correct factor is 30 per disciple – then the Bayes factor for D is still over 10^19. Clearly, either the McGrews' argument is mostly correct, or there is a much bigger error somewhere else.

The primary reason the calculated value is so big is that they are multiplying the factors together. I will show why this is incorrect.

A Telekinetic Digression

I'm going to start with a related (and fictional) story that more cleanly illustrates the McGrews' mistake. As a digression from my digression, the reason mathematicians and economists often make up unrealistic and fictional scenarios is that they are instructive. Clearly reasoning through simplified examples is an essential prerequisite to reasoning through the more complicated and more realistic scenarios.

While at a carnival, I found a traveling circus performer who claimed to be able to control the flipping of coins with his telekinetic powers. He wrote down a sequence of three heads or tails and gave me the piece of paper. Next, I took a coin out of my pocket, and I flipped the coins myself while he started very intently at the process. To my surprise, all three predictions were correct. What should I make of this?

First, I will use the incorrect argument employed by the McGrews:

I can't think of a plausible way the performer could have known the result in advance, and controlling a coin that I will provide and I will flip is very difficult. But a very plausible idea comes to mind: maybe this is just a probabilistic trick. The idea behind the trick is that while it totally flops 7 out of 8 times, 1 out of 8 audiences are dazzled. The performer hopes to earn sufficient tips from the hapless few who happen to see the trick work. Or maybe he has a hidden video camera, and sells the recordings of the trick working.

Before flipping any coins, I thought that the odds that this was a probabilistic trick relative to telekinesis were 1,000,000,000 : 1. Maybe this is the wrong number – I don't care. This example is about what to do with the numbers, not about which input numbers are correct. If telekinesis were being used, I would expect every flip to be called correctly. Each flip gives a Bayes factor of 2 supporting telekinesis over luck: this value is computed via P(correct prediction | telekinesis) / P(correct prediction | luck) = 1 / (1/2) = 2. Three flips give a Bayes factor of 8 in support of telekinesis. So now the odds are 125,000,000 : 1. I continue to accept the usual laws of physics.

But then I pressed the performer, and in violation of the usual practice of magicians, he agreed to perform the trick as many times as I wanted. To my skeptical shock and dismay, he called 150 coin flips in a row! The cumulative Bayes factor supporting telekinesis over luck is now 2^150. This is about 10^45, which means that odds of a probabilistic trick to telekinesis are now 1 : 10^36. Have you spotted the mathematical error? I hope not, for I haven't made it yet. So far, all of my statements have been completely true.

And so I conclude that it is highly probable that the performer has telekinetic powers. Now there's the mistake. Although it should be obvious that something is wrong with allowing every talented illusionist to convince you of the paranormal, it's far less obvious what in particular is wrong with the argument.

But to explain why the inference is a mistake, let me go back to the start and name the possibilities more explicitly:

A: There was no illusion and no magic. He got lucky.

B: There was an illusion, or some other scientific means of controlling the coins.

C: It was his telekinetic powers.

This time I will not bury possibility B. While I can't think of a plausible way for B to work, I can think of some implausible ones. Maybe his assistant will sneak a magnetized coin in my pocket and will be using a hidden electromagnet to make it land properly. Maybe the first toss will be probabilistic, and then he will find a way to swap the coin out after it's out of my pocket. Maybe he writes out eight predictions, and finds a way to swap the pieces of paper. However, I know that the trick is rarely this complicated, and that these wild guesses are very likely to be wrong. (Alternatively, B can be thought of as the possibility that it's an illusion using a mechanism that I can't think of.) I would guess that A is 100 times as likely as B. Before flipping any coins, I would expect the odds of A, B, and C to be about 1,000,000,000 : 10,000,000 : 1.

Just as before, each correct call gives a Bayes factor of 2 supporting C over A. However, the same factor supports B over A, which provides us with no information in helping us decide between B and C. After the first three coin flips, the odds of A, B, and C are now 125,000,000 : 10,000,000 : 1.

After ten coin flips, the odds of A, B, and C are 1,000,000 : 10,000,000 : 1. At this point, I'm pretty close to convinced that there is a trick, and that the trick isn't probabilistic. (Actually, the trick could be partially probabilistic, but most of what's going on is something else.) So at this point, I think it is likely that the performer will call my coin tosses indefinitely.

When he does so, the odds of A, B, and C end up at 10^-36 : 10,000,000 : 1. As I claimed, it's actually true that telekinesis is more likely than luck at this point. Telekinesis really is supported over random chance by a massive factor. However, a known (or unknown) mechanism is also supported over random chance by a similarly massive factor. The result of these two is that the known (or unknown) mechanism goes from implausible to a virtual certainty, while telekinesis only goes from very, very, very unlikely to very, very unlikely.

Here's the general set-up of the mistake. Start with three possibilities where the first is likely, the second is unlikely, and the third is astronomically unlikely. Next, show the second possibility to be unlikely, and ignore it beyond this point. Next, reveal evidence that absolutely buries any shred of reasonableness in the first possibility. If you continue to ignore the (initially) unlikely possibility, only the astronomically unlikely option remains.

The next question is how to measure the degree to which evidence for telekinesis has been provided. I'm not asking for a number. What do we measure to determine the strength of the evidence? The answer is the obvious one. The strength of the evidence is measured as the initial degree of certainty that a non-probabilistic solution is impossible. I don't know how to compute an actual number for the strength of this evidence. But I do know how not to: 2^(number of flips).

On to the Resurrection

With the telekinetic coin flipper in mind, most of what needs to be done to refute the McGrews' argument is to label the relevant events. As would be expected, the flaw starts with the independence assumption. Although, I hasten to add that it's not really an assumption. What I really mean is that the flaw is in their justification for why this assumption doesn't mess up the calculation.

If Jesus didn't rise from the dead, the disciples' behavior would certainly influence each other. It's possible that circumstances would cause their behavior to be negatively correlated. It's also possible that circumstances would cause their behavior to be positively correlated. I suppose the McGrews and I agree so far.

I will divide the possibilities as:

A: Jesus didn't rise from the dead, and the disciples' reactions were close to uncorrelated or negatively correlated.

B: Jesus didn't rise from the dead, and the disciples' reactions were strongly positively correlated.

The McGrews go on to argue that A is much more likely than B. I don't know if I agree, although their argument does not work either way. They write: “If their belief that Christ was raised from the dead was false, either they had good reasons to believe it or they did not. The analogy of their belief to the subjective enthusiasm of religious zealots assumes that they did not. But their actual actions would be highly improbable under this condition.” Well, how improbable is it? Is it one in 100? One in a billion? We will see that justifying a Bayes factor of 10^39 for D requires justifying a similarly astronomical improbability of B. The McGrews do not attempt to quantify “highly improbable.”

I'll go with one in a billion as the probability that the disciples' behavior was strongly correlated. This includes the naturalistic explanations that have been suggested, and it includes the explanations that we haven't thought of. The McGrews hypothetically suggested prior odds of R as 1 in 10^40. I'm leaving out W & P, and so I will already include their factors of 100 and 1000 by thinking through the implications of the prior odds of R being 1 in 10^35. I have no reason to think any of these numbers are reasonable – my topic is what should be done with the input numbers, not what the input numbers are.

From here, the argument proceeds in much the same way as the telekinesis argument. The odds of A, B, and R start at about 10^9 : 1 : 10^-26.

The death of the first disciple is a 1 in 1000 surprise to both A and B, while R saw it coming. This changes the odds to about 10^9 : 1 : 10^-23. Note that the odds of the Resurrection went up by a thousand due to the first disciple – this much of the McGrews' argument is true.

But the death of the second disciple is very different, and the odds start acting like they did with telekinesis. Hypothesis A is shocked by the second death, B isn't all that surprised, and R knew it was coming. If the disciples are bound to act the same way and disciple 1 willingly died, then disciple 2 was reasonably likely to willingly die too. The effect is that the ratios P(A)/P(B) and P(A)/P(R) are drastically reduced, while P(B)/P(R) does not change much. (How much it changes depends on the precise meaning of “strongly positively correlated.”) Suppose that under B, after the first death the probability that the second disciple will die is about 1/2. Just as before, R is supported over A by a Bayes factor of 1000. However, R is supported over B by a Bayes factor of only P(second martyrdom | R & first martyrdom) / P(second martyrdom | B & first martydom) = 1 / (1/2) = 2.

After two disciples, the odds of A, B, and R are about 2*10^6 : 1 : 2*10^-23. (The math: Because R is supported by a factors of 1000 and 2 over A and B respectively, this means B is supported by a factor of 500 over A. Thus, I divided the number for A by 500, left the number for B the same, and multiplied the number for C by 2.)

The final odds of A, B, and R will be about 4*10^-24 : 1 : 4*10^-20. The Resurrection is as it started – drastically implausible. (The math: the last eleven disciples give a factor of 2^11 = 2*10^3 supporting R over B, and a factor of 500^11 = 5*10^29 supporting B over A.) It is true that R ends up more plausible than A. This fact is also completely irrelevant.

The final question is what to measure to determine the degree to which the Resurrection has been supported. The first relevant number is the odds that the first disciple would die for his faith. The second relevant number is the odds that their choices were strongly positively correlated. The third relevant number is just how strong this correlation would be.

We have returned full circle. These are the same questions that must be answered to assess the strength of the standard argument for the Resurrection based on the disciples' testimony and death. I have not shown the standard argument to be invalid, as that was not my goal. What I have shown is this the McGrews' Bayes factor of 10^3 * 10^(3 * 13) * 10^2 = 10^44 is of absolutely no use in evaluating the argument for the Resurrection.

The Second Problem

There is a second problem with the McGrews' use of math in the argument, which is essentially the first problem in a different context. Until now, I've considered the question “If D is mathematically certain, how does this affect the probability of R?” Except this really isn't relevant, except as a means to finding the answer to the correct hypothetical: “If conservative Christians are correct, and the most reasonable explanation of the data is D, how does this affect the probability of R?” Quantifying “most reasonable” will put an upper limit on the Bayes factor supporting R.

Suppose that the data is overwhelming, and the odds of D are 10^9 : 1. Suppose further, that the McGrews are correct and D supports R over ~R by a factor of 10^39. As before, suppose the prior odds against R are 10^35 to 1.

A: The disciple's died for their false belief in Jesus

B: The disciple's didn't die for a belief in Jesus

R: The disciple's died for their true belief in Jesus

The odds of A, B, and R start at 10^35 : 10^26 : 1. The McGrews' argument gives a factor of 10^39 supporting R over A and supporting B over A. However, the McGrews' argument does not give any information helping one choose between B and R. The odds of A, B, and R end up at 10^-4 : 10^26 : 1. R has been supported by 10^9, which is the number in the initial odds of D.

So even if the McGrews' argument gives a valid conclusion when taking D as a mathematical certainty, the way to measure to degree to which the Resurrection has been supported is to look at the chance of ~D. The factor 10^39 is again of no use in evaluating the strength of the argument for the Resurrection.