Was the Resurrection a Conspiracy? A New Mathematical Approach

1 Introduction

This article introduces a new method for evaluating the sincerity of the testimony of miracles. While others have emphasized the relevance of the number of those claiming to be witnesses of a miracle, how long the witnesses maintain their testimony has not yet been adequately appreciated. Building on David Grime’s mathematical model for evaluating conspiracy theories,^[1] I develop a model that incorporates both how many witnesses and how long they maintain their testimony to evaluate the sincerity of their testimony. I then apply the model to two case studies: Hume’s hypothetical example of the death and recovery of the Queen of England, and the resurrection of Jesus. This analysis demonstrates that, contrary to Hume’s assertion, many witnesses who maintained their testimony for many years provide compelling evidence of the sincerity of their testimony.

It has been widely recognized that the number of witnesses is relevant to the credibility of the testimony of a miracle.^[2] The more witnesses there are that testify to the resurrection, the less likely it is that they are making it up. Suppose, for example, the probability that one witness would fabricate the testimony of a miracle is 50%. If there were two independent witnesses, then the probability that they would both fabricate the testimony of a miracle is 25%. If there were three witnesses, then the probability would be 12.5% and so on. The more witnesses there are, then, the more quickly it becomes implausible that the witnesses of the resurrection were conspiring.

What has not yet been adequately appreciated, though, is that how long the witnesses assert their testimony is also relevant to the probability that the miracle occurred. Being a witness of the resurrection was not a one-time event. If the probability that a witness would intentionally deceive others about a miracle on one occasion were 50%, and the probability of deceiving others on a second occasion is equally likely, then the probability of deceiving others given two opportunities would be only 25%. For three opportunities, the probability would be only 12.5% and so on. Just as the number of witnesses matters for the probability that the testimony of a miracle is an intentional act of deception, how long the supposed conspiracy to deceive others goes on also matters. The main point of this article, then, is to show that the testimony of a miracle is very unlikely to be a conspiracy if there are many witnesses over many years.

The aforementioned examples are an oversimplification of the math involved (if only it were so easy!), but they illustrate an important point: the more opportunities there are for failure, the more likely it is that a conspiracy will fail. Further, these opportunities for failure increase with the number of witnesses and the length of time that they maintain their testimony. Therefore, both the number of witnesses and duration of their testimony are relevant to evaluating the sincerity of the testimony of miracles. The model I develop in this article is unique in that it accounts for both the number of witnesses and the number of years they maintain their testimony.

In this article, then, I first explain how Grimes uses his model to debunk prominent conspiracies and develop a similar model for evaluating the sincerity of the testimony of miracles (Section 1). Next, I show how this model can be applied to particular cases. I begin with Hume’s example of Queen Elizabeth dying and reviving 1 month later (Section 2), and then I show how the model can be applied to the testimony of the resurrection and some of the other miracles of Jesus (Section 3 and 4). Finally, I conclude that the model developed here provides a new and useful tool for evaluating the sincerity of the testimony of miracles, and, as an upshot, prove that it is very unlikely that the testimony of the resurrection was a conspiracy.

2 A Strategy for Debunking Conspiracies

Grimes develops a mathematical model to debunk large-scale conspiracy theories.^[3] He assumes that there is a small but nonzero probability that any one co-conspirator would accidentally or intentionally make a confession that reveals the conspiracy. If so, then the probability of a confession would increase with a greater number of witnesses and more time. He then argues that if the model predicts that confession would be very likely, but there has been no confession, then this is evidence that it was not a conspiracy.

Let’s consider a simple example to show how the probability of revealing a conspiracy grows over time. Suppose Mary, Peter, and John are part of a secret club, and they each have a 10% chance of leaking the club’s secret in a given year. We want to know the probability that at least one of them would reveal their secret club. Put differently, we want to know the probability that Mary or Peter or John make a confession that reveals their secret club. Since they each have a 10% chance of revealing the secret club, we might expect there to be a 30% chance that at least one of them will reveal the secret. But this is not quite correct. More than one of them could confess. We are interested in the probability that at least one person confesses, which could include more than one person. The overlapping possibilities of more than confession can be seen clearly in a Venn diagram (Figure 1). If we simply add the probability of each person confessing, we will be double counting those overlapping areas where more than one person confesses. In reality, then, if they each have a 30% chance of confessing, then the combined probability that at least one of them would confess is a little less than 30%.

Figure 1

Venn diagram. The overlapping probabilities that Mary, Peter, or John reveals the secret club.

We need to count each area of the Venn diagram once, but no more than once. So, to avoid double-counting, we take the sum (30%) and subtract the probability that both Mary and Peter confess, and that both Mary and John confess, and that Peter and John confess. However, when we do so, we subtract the middle most area (the overlapping area where Mary, Peter, and John all confess) three times, and thus, it is not counted. Finally, then, we should add the probability of all of them confessing to the total. Now, every area of the Venn diagram will be counted once, and none will be counted twice. Thus, we can calculate the combined probability that at least one of them will reveal the secret club as follows:

The probability of leaking the secret in year 1

Thus, the probability that either Mary or Peter or John reveals the secret club in the first year is 27.1%.

What, then, is the probability that someone would leak the secret over time? Assuming that the club has the same number of members in year 2, and the probability that any one member would reveal the secret in year 2 remains 10%, then the probability that at least one person would reveal the secret in year 2 is again 27.1%. Since there is about a 27% chance of a confession each year for two years, we might estimate the probability to be 27% + 27% = 54%, but again this estimate double-counts the possibility that someone leaks in both years. To be more precise, then, we should subtract the probability that one leak occurs in both years. We can calculate the probability as follows:

The probability of leaking the secret in 2 years

We can see that the probability that at least one of the members would reveal the secret club in 2 years is about 47%, a little shy of 50%.

We could repeatedly apply the same method of arithmetic to calculate the probability that someone would reveal the secret club in 3 years, or more. Likewise, as members of the club increase or decrease, we could factor that into our calculations. However, when we add more years or club members, the arithmetic quickly becomes unwieldy. Fortunately, we can use different formula to derive the same results.

A probability mass function (PMF) allows us to calculate the probability of an outcome given number of trials. For example, if we flip a coin 10 times, the binomial PMF can tell us how likely it is that the coin will land on heads 4 times (20%), or 5 times (25%), etc. Similarly, we can use the binomial PMF to calculate the probability of c confessions in a given number of trials. The binomial PMF for c confessions in a year given n witnesses can be stated as follows:

Binomial PMF

Here, c is the number of confessions, n is the number of conspirators, and p is the probability of each conspirator making a confession that undermines the conspiracy. If c = 0, then this formula simplifies to ( 1 − p ) n , and it tells us the probability that no one would confess in a year. In this case, the number of “trials” in each year is determined by the number of club members (n). If there were a 10% chance that each person would confess, then the binomial PMF entails that the probability that there would be no confessions is 72.9%. But this calculation is for only one year.

To calculate the total that there would be no confession over several years, we would multiply the probability there would be no confession in one year by the number of years. For 2 years, the probability of no confession would be 72.9% × 72.9% = 53.1%. For 5 years, we would multiply 72.9% five times: i.e., 72.9% × 72.9% × 72.9% × 72.9% × 72.9%, which is equivalent to 97.5%. So, to get an accurate prediction that there would be no confession over time, we raise the formula for each year by t : i.e., ( ( 1 − p ) n ) t , which is equivalent to ( 1 − p ) n t .

Finally, we can use the compliment rule to obtain the probability that at least one witness would confess at time t by subtracting ( 1 − p ) n t from 1 as follows:

Miracle Conspiracy Model

I call this formula the Miracle Conspiracy Model because I use it to evaluate the theory that the testimony of a miracle is a conspiracy, or a deliberate attempt to deceive others into believing a miracle occurred when those testifying know this to be false.

If we apply this model to the secret club, we can see how the probability that there would be at least one confession grows over time (Figure 2). In Figure 2, we can see that the probability that there would be at least one confession that is inversely proportional to the probability that there is no confession. This means that as the probability of no confession decreases, the probability of at least one confession increases. In this case, the secret club is not likely to be revealed in its first 2 years (with less than a 50% chance), but by 10 years, it is overwhelmingly likely (greater than 95%) that someone will confess and reveal the secret.

Figure 2

Secret Club of 3 (10%). The decreasing probability that no one reveals the secret club (assuming there is a 10% chance per club member per year), and the inversely proportional probability that at least one person reveals the secret.

The probability that they can keep the club secret depends on several variables. First, the number of members: the more members there are, the harder it is to keep the secret. Second, how long they try to keep it secret: the longer they try to keep the secret, the more opportunities there are for someone to intentionally or accidentally let the secret slip that they have a club. These two variables determine the number of “trials” there are (this is the nt part of the formula), and the more trials there are, the more likely it is that someone will make a confession. Finally, the probability that there will be at least one confession depends on how likely each person is to confess in a given year (represented by variable p).

We can see how each variable changes the overall probability of at least one confession by varying them individually. For example, here, we assumed that there was a 10% chance that Peter would reveal the secret club, and the same for John and Mary. Let’s compare that with an individual probability of only 1% (Figure 3).

Figure 3

Secret Club of 3 (10% vs. 1%). The increasing probability that at least one person reveals the secret (assuming there is a 10% or 1% chance per club member per year).

If the individual probability of a confession is 10% per club member per year, then the overall probability that someone will make a confession that reveals their secret club after 10 years is about 95%, but if the individual probability of a confession is only 1%, then the overall probability is only 28%.

Alternatively, let’s keep the individual probability of a confession fixed but vary the number of members of the club. Suppose the individual probability of a confession (p) is fixed at 10% per member per year. Now compare the overall probability of a confession if there were 3 members of the club versus 10 members (Figure 4). A 10-member club has a 95% chance of being revealed after only 3 years, whereas a 3-member club does not reach a 95% chance of exposure until 10 years. We can see, then, that the more club members there are, the faster the secret club would be expected to be revealed.

Figure 4

Secret Club (3 vs. 10 Members). The increasing probability that at least one person reveals the secret (assuming there is a 10% chance per club member per year, and 3 or 10 members).

Grimes argues that it is highly unlikely for a large number of conspirators to maintain a conspiracy over a long period. His model shows that, as the number of conspirators increases, and the longer they try to keep it secret, the more likely there will be at least one confession. A conspiracy can only succeed if it remains secret. Conversely, if one of the conspirators makes a confession that reveals the truth and undermines the plausibility of the conspiracy, then the conspiracy fails. When the probability of at least one confession reaches 95%, then we may say that the conspiracy theory would be “expected to fail.”^[4]

Grimes argues that if the probability that there would be at least one confession is high enough, and yet there is no such confession, this is evidence against the conspiracy theory. The logic behind this may seem counterintuitive, but it parallels how hypotheses are tested in science. Suppose a scientific theory makes a prediction with high probability. If the prediction is correct, this is evidence for the theory, but if it is wrong, then this is evidence against the theory. For example, if an astronomer’s theory predicts that a planet will pass in front of a star with 95% probability, and it does not, then this is evidence against the theory. Grimes uses this same logic to test conspiracies. In this framework, the conspiracy theory functions as the null hypothesis, and he recommends rejecting it when the probability of no confessions becomes sufficiently low, mirroring the logic of classical hypothesis testing. If a conspiracy like the moon landing hoax were real, for instance, then it would be overwhelmingly likely (greater than 95%) that someone at NASA would have made a confession that reveals the conspiracy. But no one has made a confession that reveals a conspiracy. So, because this is contrary to what we would expect if the theory were true, the fact that none of the supposed conspirators have confessed is evidence against the conspiracy theory.

Now imagine we suspect that Mary, Peter, and John have excluded us from their secret club, but after 10 years, none of them has ever confirmed its existence. According to Grimes’s method, we would be in a position to reject the hypothesis that they have a secret club if no one has confessed, and the probability of at least one confession, assuming the club exists, would be 95% or greater. For example, if each of the 3-member club has a 10% chance per year of revealing the secret, the combined probability of at least one confession over 10 years would exceed 95%, and we could therefore reject the secret club hypothesis (Figure 3). However, if the individual probability were only 1%, we would not reach that threshold; in that case, given the decision rule we have adopted, we would not be in a position to reject the secret club hypothesis (Figure 3).

Although I follow Grimes’ general approach in this article, there is one significant difference: Grimes derives his Large Conspiracy Model from the Poisson PMF, whereas I derived my Miracle Conspiracy Model from the binomial PMF. The Poisson PMF can be stated as follows:

Poisson PMF

In this formula, λ is the average rate of confessions over a continuous interval, and here, it is equal to 1 − ( 1 − p ) n (Grimes 2016: 4). Grimes then derives the following model:

Large Conspiracy Model

Grimes’s model tells us the probability of at least one confession over a continuous interval of time, assuming the events happen independently and at a constant average rate.

There are several reasons that I use a different model than Grimes. First, Grimes is interested in large-scale conspiracies (such as the conspiracy that the NASA moon landing was a hoax) that have potentially thousands of conspirators. The Poisson PMF works well for modeling rare events over a large number of trials, making it a good fit for the conspiracies Grimes discusses. However, when the number of trials is relatively small, the Poisson approximation becomes less accurate. Since I am interested in evaluating the testimony of miracles, which typically have a small number of witnesses, the binomial PMF is a better fit for modeling these cases.

Second, the Poisson PMF is often used to approximate the probability of rare events within an indefinitely large number of independent trials over a continuous interval. For example, it can be used to estimate the number of emails you receive in an hour, or to calculate the probability that more cars arrive to turn left than can complete their turn while the light is green. Similarly, Grimes’s Large Conspiracy Model estimates the probability of at least one confession based on an expected rate of confessions (represented by the variable λ ). This approach is useful when the exact number of conspirators or the precise duration of the conspiracy is unknown. In contrast, my Miracle Conspiracy Model easily and explicitly calculates the number of trials using the formula n × t (i.e., the number of witnesses multiplied by the number of years). This discrete approach is preferable when modeling a known, small, and fixed number of trials.

Finally, I base my Miracle Conspiracy Model on the binomial PMF, rather than the Poisson PMF, for pedagogical clarity. Nonspecialists may be unfamiliar with the mathematical constant e in the Poisson PMF. Also, calculating the number of trials is less intuitive when using a rate-based model. The binomial PMF avoids both of these potential sources of confusion: the formula does not include e, and others who are interested in using the model can easily calculate the number of trials by multiplying the number of witnesses by the number of years.

While there are some differences between Grimes’s Large Conspiracy Model and my Miracle Conspiracy Model, there are also important similarities. According to the Poisson limit theorem, the results of the Poisson PMF are an accurate approximation of the binomial PMF for a large number of trials. So, the results of the model will be similar.

Another important similarity is that both models assume that confessions are independent. Independence is assumed in two ways. First, the confession of one conspirator is independent of any other conspirator. Second, the model also assumes that the probability of a confession in one year is independent of any other year. Although the actual probability of a confession will vary from person to person and from situation to situation, the probability assumed in the model can be taken as an average across persons and time. This simplifying assumption is necessary for us to use a PMF model to predict the probability that the conspiracy would fail.

Assuming the independence of a confession may be a controversial simplification, but it is a defensible one. When the police arrest two people who they suspect committed a crime together, they immediately separate the two suspects. The suspects are then interrogated separately. The police are hoping for one of two outcomes: at least one of the suspects might intentionally confess to committing the crime; or, alternatively, at least one of the suspects may unintentionally make a confession that undermines the alibis of the suspects. When suspects are separated in this way, they must individually decide whether to intentionally confess or not, and if they are trying to maintain their alibi then they must try to avoid unintentionally saying something that would undermine their alibi. Both of these outcomes are realistic possibilities, and the more witnesses there are, and the longer they are interrogated, the more chances there are that they would make a confession that exposes the truth.

Similarly, Grimes argues that his model can be applied to large conspiracies. Each conspirator individually decides whether to confess, and if not, they must maintain the conspiracy without unintentionally revealing the truth. It is reasonable to assume the chances that one conspirator would accidentally reveal the truth is independent of another accidentally doing so. Even if the independence assumption used in the model is an oversimplification, it is similar enough to real-world decision making that the model can provide a useful approximation of the probability that there would be at least one confession.

The witnesses of the resurrection, and other miracles of Jesus, repeated their testimony of the miracle for many years. They would have been asked about it on separate occasions and under varying conditions. In some cases, they were threatened with severe punishment for asserting their testimony of seeing the resurrected Jesus.^[5] So, there are many opportunities for the witnesses to recant their testimony, and even if they do not intentionally reveal the conspiracy, there remains a nonzero chance that they will accidentally reveal it. Given these additional opportunities for the miracle conspiracy to fail, then, the chances of undermining the conspiracy grow over time.

As with all models, the Miracle Conspiracy Model abstracts from the full complexity of human psychology and social dynamics to analyze specific structural features. Mathematical models do not aim to capture every detail of reality; rather, they illuminate patterns that we otherwise would not see. So, although a model does not account for every relevant factor, it can nonetheless clarify the relevance of some of the important variables. In this case, the model demonstrates that the number of witnesses and the duration of their testimony are structurally relevant to the plausibility of a conspiracy.

The main goal of this article is to introduce this method for evaluating the sincerity of the testimony of witnesses of a miracle. Just as Grimes uses the lack of a confession by a conspirator as evidence to reject a conspiracy theory, we can use the lack of a confession by the witnesses of a miracle as evidence to reject the hypothesis that the testimony of the witnesses of a miracle is insincere, rejecting the theory that the witnesses are conspiring to deceive us rather than report what they believe to be true. In what follows, I show how the method can be applied to the testimony of miracles.

3 Hume’s Queen Elizabeth Conspiracy

David Hume argues that even if it is unlikely that multiple witnesses of a miracle are conspiring, it is even more unlikely that the miracle occurred:

When anyone tells me, that he saw a dead man restored to life, I immediately consider with myself, whether it be more probable, that this person should either deceive or be deceived, or that …[it] should really have happened.^[6]

And then he adds that a miracle is always less likely than deception:

[…]there is not found, in all history, any miracle attested by a sufficient number of men, of such unquestioned good-sense, education, and learning, …as to place them beyond all suspicion of any design to deceive others[…]^[7]

It is strange, a judicious reader is apt to say, upon the perusal of these wonderful historians, that such prodigious events never happen in our days. But it is nothing strange, I hope, that men should lie in all ages. …Be assured, that those renowned lies, which have spread and flourished…arose from like beginnings[…]^[8]

Hume implies that a miracle is extremely unlikely and that a conspiracy is not unlikely.

Nor does it matter, on Hume’s view, whether there are a large number of credible witnesses to the miracle. He imagines all of the court of Queen Elizabeth testifying to her death and miraculous return to life. He insists that, notwithstanding the large number of witnesses, it is still more likely to be deception than a genuine miracle:

But suppose, that all the historians who treat of England, should agree, that, on the first of January 1600, Queen Elizabeth died; that both before and after her death she was seen by her physicians and the whole court, as is usual with persons of her rank…and that, after being interred a month, she again appeared, resumed the throne, and governed England for three years: I must confess that I should be surprised at the concurrence of so many odd circumstances, but should not have the least inclination to believe so miraculous an event. I should not doubt of her pretended death, and of those other public circumstances that followed it: I should only assert it to have been pretended, and that it neither was, nor possibly could be real. You would in vain object to me the difficulty, and almost impossibility of deceiving the world in an affair of such consequence; …but…I should rather believe the most extraordinary events to arise from their concurrence, than admit of so signal a violation of the laws of nature.^[9]

Hume insists that, in such a case, the death and miraculous recovery were only “pretended, and that it neither was, nor possibly could be real.” But is it plausible to insist that all of these witnesses are conspiring to deceive others?

Let’s put this conspiracy hypothesis to the test. We can apply the Miracle Conspiracy Model to Hume’s Queen Elizabeth case by estimating the number of conspirators (n) and the probability a conspirator would confess in a given year (c). Since the case is hypothetical, we can just stipulate estimates without having to justify them. It will also be instructive to make multiple estimates so that we can see how the probability of a confession changes with different estimates.

First, let’s say there were 100 witnesses (so, n = 100) and then vary the average probability that, if they were conspiring, one of the witnesses would confess in a given year (p). In the previous section, we saw how it is very unlikely that 3 club members could keep their club secret for 10 years if they each had a 10% chance of revealing their secret per year. Here, there are many more witnesses. If they each had a 10% chance of revealing the conspiracy, it would be expected to fail almost immediately, having a greater than 95% chance of failure in 4 months. So, instead, let us compare the individual probability of a confession is 1% and 0.1% (Figure 5). Figure 5 shows that the conspiracy is overwhelmingly likely to fail 3 years if there were 100 conspirators and the probability of a confession for each conspirator was 1% per year.

Figure 5

Queen Elizabeth Conspiracy (1% vs 0.1%). The increasing probability that if there were 100 witnesses (and a probability of 1% or 0.1% per witness per year), there would be at least one confession that undermines the Queen Elizabeth conspiracy.

If there are significantly more witnesses, then the individual probability of a confession could be much lower, and yet it would be likely that there would be at least one confession that undermines the conspiracy. Since we already know that a conspiracy of 100 witnesses would be likely to fail if each witness had a 1% chance of confessing, let’s instead consider a much lower probability: say 0.1%. We can then compare the probability the conspiracy would fail if there were 100 or 200 witnesses (Figure 6). If there were 200 witnesses, and each had a 0.1% chance of making a confession that reveals the conspiracy, then the conspiracy would be 95% likely to fail in 15 years. If there were only 100 witnesses, then it would take twice as long (30 years) to be 95% likely to fail.

Figure 6

Queen Elizabeth Conspiracy (100 vs 200). The increasing probability that if there were 100 or 200 witnesses (and a 0.1% probability of a confession per witness per year), there would be at least one confession that undermines the Queen Elizabeth conspiracy.

From these examples, we learn several important lessons about the tenability of a miracle conspiracy. First, the probability that at least one witness would confess is sensitive to the number of conspirators: increasing the number of conspirators significantly reduces the time needed to reach a high probability of at least one confession. Second, changes in the average probability that conspirator would confess can drastically alter the overall probability that there would be at least one confession: a higher average probability of a confession leads to a much higher overall probability of at least one confession within a shorter time frame. Finally, and most importantly, Hume is wrong to suggest that a conspiracy to deceive others about a miracle is not unlikely. To the contrary, when the probability of a confession is not extremely low, the overall probability of at least one confession increases significantly over time, especially with a larger number of conspirators. Therefore, the probability of a confession can become very likely within a reasonable period of time, which implies a correspondingly low probability of the miracle conspiracy hypothesis given that there were no confessions.

In this section, I have used Hume’s fictional example to show how the lack of a confession from a large number of witnesses of a miracle can be strong evidence against the hypothesis that their testimony of the miracle is a conspiracy. But this is a purely hypothetical example. It has not yet been shown that we can reach the same conclusion in any actual case. Let us turn our attention, then, to the testimony of the resurrection of Jesus.

4 The Number of Witnesses

After the death of Jesus, some of his disciples claimed to discover his tomb empty, and others testified that they saw him alive. In response, some Jewish leaders accused the disciples of stealing Jesus’s body, implying that their claims of seeing the resurrected Jesus were fabricated. The Gospel of Matthew (c. 75–85 AD) reports that “this saying is commonly reported among the Jews until this day.”^[10] If the disciples had stolen Jesus’s body but claimed to have seen him alive, their testimony would therefore constitute a conspiracy.

However, the Miracle Conspiracy Model can show that the resurrection conspiracy is untenable. To apply the model to the resurrection, we first need to make approximate but realistic estimates about the number of potential conspirators (n) and the probability of a confession per person, per year (p). We can then use the models to predict how long it would take for a confession that undermines the conspiracy to be overwhelmingly likely. In this section, I estimate the number of potential conspirators (n), and in the next section, I estimate the probability, per witness per year, that a witness would make a confession that undermines the conspiracy, and, finally, I will use these estimates to show that the testimony of the resurrection is very unlikely to be a conspiracy. Let’s start, then, by estimating the number of witnesses.

Multiple early sources attest that the disciples claimed to have seen the resurrected Jesus. This is widely regarded as a historical fact. Even the most skeptical scholars admit this. For example, the participants in the Jesus Seminar (as it is called) reject 84% of the gospel narratives as unhistorical, yet they accept as “probably reliable” that several of the disciples claimed to see the resurrected Jesus.^[11] More recently, Bart Ehrman, who doubts the reality of the resurrection, nonetheless acknowledges “that some of Jesus’s disciples claimed that they saw him alive after he had died.”^[12] Scholars across the ideological spectrum agree. In a meta-analysis of approximately 4,500 scholarly publications on the resurrection (from 1975 to 2024), there is “virtually unanimous” agreement the disciples of Jesus “reported experiences…of the risen Jesus” and this “proclamation of Jesus’s resurrection and appearances took place very early, soon after the experiences themselves.”^[13] Of course, not all scholars agree that the resurrection actually occurred, only that the disciples said that it did.

It should be granted, then, that there were many witnesses of the resurrection. But how many witnesses were there? Scholars disagree. Fortunately, we do not need to resolve this disagreement here. Instead, we can use multiple estimates to evaluate the resurrection conspiracy. One strategy, sometimes called the “minimal facts” approach,^[14] seeks to identify the claims for which there is strong historical evidence and which nearly all scholars agree. We can use this approach to identify the minimum plausible estimate of the number of witnesses of the resurrection. Another strategy, which has been called the “maximum data” approach,^[15] first argues that the New Testament is largely reliable and then uses the New Testament accounts of the resurrection as historical evidence, even if that evidence is contested by skeptical scholars.^[16] We can use the maximal data approach to make an estimate on the high range of plausibility.

First, let’s use the minimal facts approach to identify a low-end estimate of the number of witnesses. The earliest and most historically significant evidence about the many witnesses of the resurrection is First Corinthians. There is no real doubt that Paul wrote this letter, and, moreover, there is good reason to believe that Paul’s list of witnesses comes from an even earlier oral tradition.^[17] Here is his list:

[…]he [Jesus] was seen of Cephas [Peter], then of the twelve: After that, he was seen of above five hundred brethren at once; of whom the greater part remain unto this present, but some are fallen asleep. After that, he was seen of James; then of all the apostles. And last of all he was seen of me also[…]^[18]

The appearance of Jesus to more than 500+ witnesses is not one of the minimal facts, so let’s set it aside for now. Instead, let’s focus on the others in his list.

Paul’s list of witnesses

This list includes the eyewitness testimony of Paul himself,^[19] and Paul reports meeting Peter and James in Jerusalem 3 years after his conversion,^[20] and he met with them again, with John and other disciples, 14 years after his conversion.^[21] Since Paul knew many of the witnesses personally, he was in a very good position to confirm that there were at least these 13 witnesses of the resurrection. For this reason, “a nearly unanimous consensus of modern scholars” accept that these 13 witnesses claimed to see the resurrected Jesus.^[22]

But even a minimum of 13 is too modest. All four gospels report that multiple women went to the tomb of Jesus and found it empty; there, an angel told them that the “he is risen.”^[23] Each of the gospels provides a slightly different list of which women were at the empty tomb:

The female witnesses of the empty tomb

Mark, Luke, and John each affirm that there were at least 3 women at the empty tomb, and Matthew affirms that there were at least 2 (but does not assert that there were only 2). The primary reason for accepting this account as historical is that if the story of the empty tomb was fabricated, the authors of the gospels would have likely made the witnesses male disciples. Instead, they are women, who at the time were not seen as credible witnesses. As Michael Licona persuasively argues, though, “the reason for the report’s lack of credibility in the first century is a reason for its credibility in the twenty first.” He adds:

Accordingly, the most plausible explanation for the inclusion of women witnesses in the resurrection narratives is that the remembrance of the tradition was so strong and widespread that it had to be included.^[24]

In Habermas’s survey of the literature, he reports that in the last 50 years about 75–80% of the scholars have agreed that some of the female disciples found the tomb empty, and the consensus view is that the women themselves are the source of this tradition.^[25] Given that each of the accounts has multiple women, and several of them are specified by name, let’s say that there were at least 3 women witnesses of the resurrection. According to the minimal facts approach, then, the minimum plausible estimate of the number of witnesses is 16.

The minimal facts estimate of the number of witnesses

Alternatively, we can use the maximal data approach to estimate the number of witnesses. This method argues, on historical grounds, that the New Testament is mostly reliable, and then identifies the number of witnesses with the number of people described to have seen the resurrected Jesus in the New Testament. For example, Craig Keener argues that “external sources regularly confirm most of [Luke’s] information that can be tested,” and he concludes that most “scholarship today…finds his story largely reliable.”^[26] Since Luke gives us accurate information about the times and places he was writing about, Keener suggests that we can also trust his reports about what the apostles were doing and saying after the death of Jesus. Similarly, Craig Blomberg argues the New Testament is mostly reliable and then he argues that we should therefore trust the New Testament’s report that there were many witnesses of the resurrection.^[27]

On this approach, there is good reason to believe that there were more than 3 women at the empty tomb. Mark and Luke agree that Mary Magdalene, another Mary, and one other woman (Salome or Johanna) were present at the empty tomb. If we assume that both Mark and Luke are right, then Salome and Johanna would both be present, making 5 women. Further, Luke’s names 3 women at the tomb (Mary Magdalene, another Mary, and Johanna) and, in addition, says there were “other women” (plural) there; this again implies that there were at least 5 women at the tomb.

Luke also describes other resurrection appearances. He describes an appearance of Jesus to 2 disciples on the road to Emmaus,^[28] and he implies that Matthias and Joseph, the 2 candidates to replace Judas as an apostle, were both at the ascension of Jesus.^[29] Finally, he recounts how Stephen reported seeing the resurrected Jesus just before being stoned to death.^[30] This adds 5 more witnesses.

Given these reports in the New Testament, the maximal data approach allows us to make a reasonable estimate of 23 witnesses.

The maximal data estimate of the number of witnesses

The list of witnesses of the resurrected Jesus could have been many times longer. For example, he told the disciples he would meet them in Galilee, but we do not have a report of that meeting, and so we cannot say how many people were there. (Is that Paul’s 500+ witnesses?) Luke also reports that Jesus remained with the disciples for 40 days, but, again, we do not know who claimed to see Jesus during that time. So, setting aside Paul’s 500+ witnesses, even if we grant that the New Testament is reliable, we do not have good textual evidence for more than 23 witnesses.

Finally, let’s include Paul’s assertion that there were 500 witnesses of the resurrected Jesus as another separate estimate.^[31]

In summary, we have considered three ways to estimate the number of witnesses of the resurrection.

Estimates of the number of witnesses

Below, I will use these estimates to calculate the probability that, if they were conspiring, at least one of these witnesses would have made a confession that undermines the plausibility of the resurrection.

However, the number of witnesses of a miracle diminishes over time. This raises questions about how a diminishing number of witnesses might affect the probability that the resurrection conspiracy will eventually fail. Grimes uses a Gompertzian survival function to estimate the decline in the number of conspirators over time due to natural mortality, but his estimates are based on modern life expectancy data and would not accurately model mortality rates of the witnesses of the resurrection. A better method here would be to use Robert McIver’s estimates. He uses what is known about the population and lifespans of those living in the Roman Empire in the first century AD to “estimate the number of eyewitnesses of Jesus’ ministry who would still be alive at different periods after the resurrection.”^[32] He estimates that 62,755 people saw Jesus during his public ministry, and then he estimates what percent of these witnesses would be alive 5, 10, and 15 years, etc., after the death of Jesus (Table 1).

Although there is reliable historical evidence about the lifespans of some of the witnesses (e.g., Peter, Paul, and James), the historical record is silent or unclear about the others. For example, we do not know what happened to the women at the tomb, the two disciples on the road to Emmaus, or other lesser-known figures after the gospel narratives end. In my view, Luke implies that most of the apostles (other than James) were still alive at the time of the Jerusalem Council in 49 AD,^[33] and Paul suggests that they were still alive when he wrote First Corinthians.^[34] However, to keep the analysis focused on the model rather than this historical reconstruction, I will not pursue that here. Instead, for simplicity, I will assume that the lifespans of all the witnesses follow the cohort data reported by McIver.

If we assume that the witnesses follow the expected lifespans reported by McIver, then we can estimate how many of the witnesses of the resurrection remained alive over time (Table 2).

In Section 1, we assumed that the number of witnesses remained the same over time, so we calculated the probability of a confession by multiplying the number of witnesses (n) by the number of years that they maintained their testimony (t). However, we are now assuming that the number of witnesses of the resurrection changes over time. In that case, the number of “trials” is equal to the sum of the number of witnesses each year (Table 2). Using N to represent the total number of trials, we can restate the Miracle Conspiracy Model as follows:

Miracle Conspiracy Model

In this way, the model can accommodate changes in the number of witnesses over time.

The goal of this section has been to estimate the number of witnesses so that it can be used in the Miracle Conspiracy Model. Although there is some disagreement about the number of witnesses, we can avoid settling the issue by providing multiple estimates. According to the minimal facts method, the minimum plausible number of witnesses would be 16. According to the maximal data method, the number of witnesses would be at least 23. Finally, Paul reports that there were 500+ witnesses of Jesus at once.

Even after identifying the total number of witnesses, one remaining challenge has been to estimate how long these disciples maintained their testimony of the resurrection. Here, I used McIver’s general population data to estimate how many of the witnesses would still be alive (Table 2). We can now use these estimates to determine the number of “trials” there were, meaning the number of witnesses per year that could have revealed the conspiracy.

If others are dissatisfied with these estimates, then they can provide their own. Yet, the overall point of the model will be substantiated in any case. The point of the model is to show that both the number of witnesses and the length of time they maintained their testimony are relevant to the probability that they were conspiring. Even if other estimates are used, the model will still be useful for seeing how much these variables matter for the overall plausibility of the resurrection conspiracy. Further, on any reasonable estimate of the number of witnesses, we will obtain approximately the same results, vindicating the overall conclusion that the resurrection of Jesus is unlikely to be a conspiracy.

5 The Individual Probability of a Confession

The Miracle Conspiracy Model requires, in addition to an estimate of the number of witnesses (n), an estimate of the individual probability that, if they were conspiring, one of the witnesses would intentionally or accidentally make a confession that undermines the plausibility of the conspiracy (p). In this section, I suggest a range of plausible estimates.

The standard for what counts as a “confession” in this context is rather low. Grimes defines a confession as a statement by a conspirator that actually convinces people to reject the conspiracy. By contrast, my operational definition of a “confession” leaves open the possibility that a conspirator might confess the truth, but others do not believe it. For example, if Mary Magdalene lets it slip that the dead body of Jesus was removed from the tomb and buried elsewhere, this would count as a “confession” in the relevant sense, even if no one was convinced to reject the disciples’ testimony of the resurrection. Thus, for my purposes, any statement by a witness that should convince us that the testimony of a miracle is a conspiracy will count as a confession, regardless of whether it convinces others.

Grimes estimates the probability of a confession by surveying a few examples of exposed conspiracies. Grimes estimates how long the conspiracy went on before it was revealed, and how many people knew of the conspiracy and nonetheless allowed the public to be misled. He then concludes that the probability of a confession can be as low as 0.0004.^[35] As we have seen (in Section 1), even if the probability of a confession is only 0.0004%, it is still overwhelmingly likely that the moon landing conspiracy would have failed, and likewise for other prominent conspiracy theories he discusses.

However, Grimes’s estimate is much too low for the resurrection conspiracy. One reason why his estimate is too low is that we have different standards for what counts as a conspiracy. For Grimes, a statement counts as a “confession” only if it persuades people not to believe the conspiracy. But, obviously, no statement like that has been made; there are still billions of Christians! On my criterion, any historically reliable report of a witness saying something that, if true, should convince us the resurrection was a conspiracy counts as a confession. We do not have any record of that kind of statement either, but the issue is how likely it would be that we would have such a statement if the resurrection was a conspiracy.

A second reason Grimes’ estimate is too low is that the background conditions of the would-be conspirators are very different. On Grimes’ analysis, if a mid-level manager at NASA knew that they did not land on the moon but remained silent, then this silence is taken as being complicit in the conspiracy. But it is much easier to keep the conspiracy a secret if the participants remain silent. This was decidedly not the case for the witnesses of the resurrection; they went around talking about it! One of the minimal facts that almost every scholar agrees to is that many of the disciples went around telling people they had seen the resurrected Jesus.^[36] Others knew that they claimed to be witnesses, and both believers and skeptics likely asked the witnesses about their experiences.^[37] What are the chances that, if the witnesses were conspiring, they would (either intentionally or accidentally) make a confession that undermines the plausibility of their conspiracy? Is it less than one-hundred-thousandth of a percent? No! Intuitively, it should be much higher than that.

As a way to estimate the probability that a witness would confess in a given year, consider an analogy to the secret club. Suppose we suspect that Mary, Peter, and John have a secret club. Further, we frequently hear them talking about how they spend time together, and we pointedly ask them on multiple occasions whether they have such a club. Focusing on the individual risk, how likely is it that any one of them would reveal the secret in a given year? Let’s grant that each member is very unlikely to let their secret slip. Yet, the more the talk about it, and the more are asked about it by others, the harder it becomes to keep their club a secret. So, we should avoid assuming that the individual annual probability of confession is so low that it’s practically zero. As a conservative working estimate, we might say that each member has a 1% chance per year of revealing the club.

For similar reasons, the individual probability that one of the witnesses to the resurrection would confess should be not lower than 1%. These witnesses frequently spoke about their experiences in public, and likely in private as well, repeating their testimony to other believers, potential converts, and even skeptics hostile to Christianity.^[38] Each conversation posed a small but real risk of disclosure. Given this context, it is implausible to assume that the individual annual risk of confession was practically zero. For the purposes of the Miracle Conspiracy Model, I will assume that p, the annual probability of a confession per witness, is 1%.

Some may object that even if one of the witnesses did confess, we would not know about it, and so the probability of a recorded confession should be lower. However, Christian authors unanimously assert that the apostles and other disciples continued to affirm the resurrection of Jesus.^[39] Further, if there were evidence to the contrary, this would have been used by the opponents of Christianity. Michael Licona persuasively argues:

We must also keep in mind that there is an absence of any hints that any of the Twelve (other than Judas) had recanted or walked away from the Christian community. If the news had spread that one or more of the original disciples recanted, we would expect … that a recantation by any of the disciples would have provided much ammunition for Christian opponents like Celsus and Lucian[…]^[40]

If Peter or another well-known witness made a public confession that undermined the plausibility of the resurrection, this would not have gone unnoticed and unreported. So, given the evidence we have, we can be confident that none of these witnesses made a confession that undermined the conspiracy.

It might be true, though, that a confession by one of the less well-known disciples among the 500+ would be less likely to be recorded. So, let’s say that the annual probability of a confession for each of these witnesses could be as low as 0.1 or 0.01%. But for the well-known witnesses, I will assume that the probability is not lower than 1%.

Again, some may disagree with these estimates. If so, they are free (even encouraged!) to test the resurrection conspiracy hypothesis using alternative values for p. The Miracle Conspiracy Model is designed to accommodate a range of inputs. However, based on the considerations outlined earlier, I will proceed using an annual individual confession probability of 1%.

6 Debunking the Resurrection Conspiracy

Now that we have estimates for the number of conspirators and the probability of a confession, we can use these estimates as inputs in the model. The output of the model is the overall probability that at least one witness would make a confession that undermines the conspiracy. Following Grimes, if the probability that there would be at least one confession reaches as high as 95%, then we can take this as evidence against the hypothesis that the testimony of the resurrection is a conspiracy.

Given how many conspirators there were, and how long they maintained their testimony, it is overwhelmingly likely that at least one of the witnesses would have made a confession that undermines the plausibility of the conspiracy (Figure 7). A brief look at Figure 7 and two conclusions immediately become apparent. First, having more witnesses raises the probability that there would be a confession. Individually, each witness is unlikely to confess. Collectively, though, the more witnesses there are the higher the probability that at least one of them is to confess. Second, the probability that there would be at least one confession grows over time. Just as adding more witnesses raises the probability of a confession, even so adding more time raises the probability of a confession.

Figure 7

The Resurrection (16 vs 23 Witnesses). The increasing probability that if there were 16 or 23 witnesses, there would be at least one confession that undermines the resurrection conspiracy.

More precisely, Figure 7 shows how long it would take for it to become overwhelmingly likely that there would be at least one confession and so the conspiracy would be expected to fail. If there were originally 23 witnesses, and the probability of any one witness confession was only 1%, then it would take 17 years for there to be a 95% probability of at least one confession. If there were 16 witnesses, then it would take 28 years for it to be overwhelmingly likely that there would be at least one confession. In either case, the conspiracy would be expected to fail within a generation, and within the expected lifetimes of many of the witnesses. Given that it is so unlikely that they could successfully maintain the conspiracy for that long, it is instead very likely that they were not conspiring.

Notice the cumulative probability of at least one confession never decreases. In fact, if the number of witnesses remained the same over time, then the probability that at least one of them will eventually confess continues to rise, approaching 100%.^[41] In reality, however, the number of witnesses diminishes over time. As the number of witnesses decreases, the probability of a confession per year declines. Yet the cumulative probability of at least one confession continues to increase.

For example, consider what happens to the secret club of Mary, Peter, and John when one of them leaves the group. We are assuming that each member of the club presents a risk of exposing the club each year. Since there are 3 members of the club in year 1, there are 3 “trials,” of 3 opportunities for club members to make a confession that reveals the club or not. In year 2, there are still 3 members. So, in the first 2 years, there are a total of 6 trials (Table 3). However, let’s assume that in year 3 John is no longer able to reveal the secret club. (Perhaps John died, or, less morbidly, moved far away and so can no longer tell any of his old friends about the secret club.) In year 3, then, we add only 2 trials, bringing the total number of trials to 8. The total number of trials has increased, but, since there are fewer club members, it has increased at a slower rate (Table 3).

This simplified example illustrates two important points. First, when the number of witnesses decreases over time, the annual probability of a confession declines. Second, even though the annual probability of a confession declines, the cumulative probability of at least one confession still increases over time.

A similar dynamic applies to the resurrection witnesses. As the number of living witnesses declines, the per-year probability of a confession also diminishes. However, as long as even some witnesses remain alive and continue to affirm their testimony, the number of opportunities for a confession (the “trials”) continues to increase. And as the number of trials grows, so does the cumulative probability that at least one confession will occur. Once all witnesses have died, the cumulative probability no longer increases, but it also doesn’t decrease. This is why the line graphs showing the cumulative probability of a confession flatten out over time, but do not go down.

Next, let’s consider Paul’s assertion that Jesus appeared to more than 500 witnesses at once.^[42] If each of the 500 witnesses have a 1% chance of revealing the conspiracy in a given year, then the probability that at least one of them would confess after only 1 year is 99%! Let’s suppose, then, that the probability is much lower: say, 0.1% or 0.01% (Figure 8). With so many witnesses, even if the individual probability of a confession is only 0.1%, the resurrection conspiracy would be expected to fail in 6.5 years. If the probability that each witness would confess were as low as 0.01%; however, it would not be expected to fail within 30 years. In the latter case, we would not be in a position to use the fact that there has not been a confession as grounds for rejecting the conspiracy.

Figure 8

The Resurrection (500 Witnesses). The increasing probability that if there were 500 witnesses, there would be at least one confession that undermines the resurrection conspiracy.

More realistically, though, Paul’s 500 witnesses would include the 16 witnesses identified in our low-end estimate (Section 3). The probability that one of the 500 witnesses would confess, then, can be no lower than the probability that one of these 16 witnesses confessed. If we assume that each of the well-known witnesses have a 1% chance of making a confession that reveals the conspiracy, and, in addition, there are 500 other witnesses each with a 0.01% chance of making a confession that undermines the conspiracy, then the conspiracy would be expected to fail even faster (Figure 9).^[43]

Figure 9

The Resurrection (16 and 500 Witnesses). The increasing probability that either one of the 16 witnesses or one of the other 500 witnesses would make a confession that undermines the resurrection conspiracy.

If there were only 16 witnesses, the conspiracy would be expected to fail in 28 years (Figure 7). However, if in addition to the 16 witnesses there were 500 more, then even if the 500 witnesses each have a very low probability of making a confession that undermines the conspiracy, the conspiracy would still be likely to fail in 5 years (if p = 0.1%) or 18 years (if p = 0.01%).

I have argued that, under a wide range of estimates, the resurrection conspiracy would be expected to fail within the lifetimes of the witnesses. However, the failure of a conspiracy is not inevitable. By the point of comparison, let’s consider two other miracles by Jesus. When Jesus healed the daughter of Jarius, there were only 6 people present.^[44] If these 6 witnesses were conspiring, and the probability of a confession is 1% per person, per year, then the conspiracy would not be expected to fail within 25 years (Figure 10). A conspiracy with so few might well succeed. Alternatively, there were 5,000 witnesses when Jesus fed the crowd. But among so many witnesses, even if one confessed that the feeding was not miraculous, it may be unlikely that this confession would be believed. So, let’s suppose the probability that a witness could make a confession that would undermine the plausibility of this conspiracy is only 0.0004% (Grimes’ estimated value for p). In that case, the conspiracy might well concede because the probability of a confession is so low (again, see Figure 10). These examples show that not all conspiracies would be expected to fail. The conspiracies that are expected to fail are those that have a large number of witnesses and who maintain their testimony over a long period of time.

Figure 10

Conspiracy of Other Miracles of Jesus. If the healing of Jairus’s daughter or the healing of the blind man were a conspiracy, then the probability of at least one confession would increase only modestly over time.

The miracle conspiracy model vindicates the sincerity of the witnesses given a wide range of reasonable estimates, but this outcome is not inevitable. If there were fewer witnesses, or they were much less likely to confess, then the lack of a confession would not provide strong grounds for rejecting the resurrection conspiracy. For example, if the healing of the daughter of Jarius or the feeding of the crowd of 5,000 were conspiracies, then the miracle conspiracy model does not predict that these conspiracies would fail. By contrast, if the resurrection of Jesus were a conspiracy, then given the number of witnesses, the probability of a confession, and how long they maintained their testimony, the conspiracy would be overwhelmingly likely to fail. This implies that there is a correspondingly low probability that the testimony of the resurrection is a conspiracy, or an intentional deception of the witnesses. Given our evidence, then, it is very likely that the testimony of the resurrection is sincere.

7 Conclusion

One goal of this article has been to introduce a new method for evaluating the sincerity of the testimony. The miracle conspiracy model uses the number of witnesses and the probability of a confession to predict how long it would take for it to become very likely that at least one of the witnesses, if they were conspiring, would make a confession that undermines the conspiracy. If the probability of at least one confession by t years is greater than 95%, then this implies that the probability that there would be no confession by t is less than 5%. So, if a miracle conspiracy is very unlikely, less than 5%, then we can use that as evidence to reject the conspiracy hypothesis.

However, if the model does not show that the conspiracy would be likely to fail, this does not mean that we should accept the conspiracy hypothesis. In hypothesis testing in the social sciences, “failing to reject” a hypothesis is not the same as accepting it. If the evidence is unlikely given the hypothesis, then this counts as evidence against it. Typically, we reject a hypothesis if the probability of the evidence given the hypothesis is below a certain threshold, usually 5 or 10%. However, if the probability does not fall below this threshold, that does not mean we accept the hypothesis. For example, even if the healing of the daughter of Jairus were a conspiracy, the model shows a 78% probability of at least one confession in 25 years. We should not use this evidence to accept the hypothesis. Instead, we should conclude only that we do not have enough evidence to reject the hypothesis.

A second goal of this article has been to show that the testimony of the resurrection is very unlikely to have been a conspiracy. I have shown how, given that many witnesses of the resurrection maintained their testimony for many years, it is very unlikely that the witnesses of the resurrection were conspiring to deceive us about the resurrection of Jesus, and thus, it is very likely that the witnesses of the resurrection sincerely believed their testimony. This is an important step, though not the only step, in showing that the testimony of the disciples is true.

The next step in an argument for the resurrection of Jesus is to argue that the sincere testimony of the witnesses was unlikely to be mistaken. That the witnesses were somehow honestly mistaken likewise strains credulity. The witnesses report that Jesus provided them with “many infallible proofs” of his resurrection.^[45] As William Paley points out:

It was not one person but many, who saw him; they saw him not only separately but together, not only by night but by day, not at a distance but near, not once but several times; they not only saw him, but touched him, conversed with him, ate with him, examined his person to satisfy their doubts.^[46]

Given the abundance of empirical evidence of the resurrected Jesus provided to the witnesses, their testimony would not be the result of perceptual error. So, their testimony is either deception or they were telling the truth. And it wasn’t deception.