A Statistical Analysis of the Hallucination Hypothesis Used to Explain the Resurrection of Christ

Abstract

Given the centrality of the resurrection to the Christian faith, the post-crucifixion appearances of Jesus to his disciples continue to be a central topic in historical inquiry regarding the origins of the Christian faith. While a number of hypotheses have been proposed for these post-crucifixion appearances, a leading naturalistic explanation suggests that these appearances are best explained by grief-induced bereavement hallucinations. Although scholars acknowledge that such hallucinations are somewhat unlikely, prior works have not provided a quantitative analysis of the hallucination hypothesis, so the question remains: Just how improbable is the hallucination hypothesis? This paper presents a statistical analysis to address this question for some of the hallucination scenarios proposed by scholars to show that the probabilities are extremely low, even given mitigating circumstances.

1. Introduction

The traditional Christian faith hinges on the question of whether or not Christ was raised from the dead—not metaphorically or mystically, but as an objective event that occurred in space and time. The apostle Paul put it this way: “For if the dead are not raised, not even Christ has been raised. And if Christ has not been raised, your faith is futile and you are still in your sins” (1 Corinthians 15:16–17, ESV). Given the centrality of the resurrection to Christianity, and given that the resurrection appearances as reported in the New Testament constitute one of the key supports for demonstrating the historicity of the resurrection, it should come as no surprise that challenges to the Christian faith often center around discrediting the resurrection appearances. Two of the most popular naturalistic explanations for the post-resurrection appearances include the legend hypothesis (the resurrection accounts are largely legendary) and the hallucination hypothesis (the resurrection accounts are due to hallucinations, either alone or in conjunction with other psychological phenomena). The goal of this paper is to address the hallucination hypothesis, including some of the significant variants, via a probabilistic analysis based on published statistical data.

The hallucination theory has a long history, going back at least as far as the second century, when the Roman philosopher Celsus attributed the postmortem appearances to hallucinations (see The True Word, c. 177, cited by Origen in Book II, Chapter 60 of Contra Celsum, c. 248). The hallucination theory experienced a revival in the Victorian era when it was popularized by David Strauss in A New Life of Jesus, published in 1879. Over the past three decades, a number of scholars have proposed that the resurrection appearances can be explained by hallucinations, or other similar psychological phenomena (Habermas 2001), including Michael Goulder (Goulder 1996), Jack Kent (Kent 1999), Dale Allison (Allison 2005), and Gerd Lüdemann (Lüdemann 2004, 2009). More recently, this trend has continued with Carrier (Carrier 2014), Eisenberg (Eisenberg 2016), and Smith (Smith 2020). One of the common themes of these scholars is the performance of psychological analysis on the disciples in order to conclude that the resurrection appearances can be attributed to stress-induced hallucinations triggered by factors such as guilt and bereavement. These conclusions are often justified by citing relevant hallucination studies, such as the study by Dewi Rees documenting a post-bereavement hallucination rate of nearly 50% (Rees 1971). Another common theme is to invoke multiple hypotheses, such as an initial guilt and grief-induced hallucination experienced by Peter that triggers a chain reaction in the disciples (Lüdemann 2004), a combination of hallucinations and collective delusions (Smith 2020, 2023), or hallucinations plus an improbable sequence of events (Eisenberg 2016).

There are, however, a number of problems with these theories: (1) Psychological analysis is difficult even when the patient is present to support detailed evaluation, and extreme caution must be exercised when dealing with ancient historical figures. As pointed out by Bergeron (a medical doctor) and Habermas, hallucination theories are “primarily proposed by non-medical writers”, and, as a result, “the analysis of potential medical causes for these hallucinatory symptoms is generally flawed and often absent” (Bergeron and Habermas 2015). (2) The statistics from the studies have been misapplied—in particular, the actual rate for either visual or auditory hallucinations is much lower than the overall rate, which is dominated by a feeling of the presence of the deceased (Bergeron and Habermas 2015). (3) Even if all the disciples individually experienced auditory and/or visual hallucinations (a highly unlikely scenario, as we will see later), statistical analysis indicates that simultaneous hallucinations (i.e., time-coincident hallucinations by different individuals) are extremely improbable. Since the first two issues have been addressed in the recent literature, we will focus on the third problem, which has not been addressed from a statistical perspective in the literature. Although we will discuss some of the multi-factor hypotheses, some are not amenable to statistical analysis, and will not be addressed in detail in this work. Furthermore, although it is possible that the statistics from modern hallucination studies are not necessarily applicable to ancient civilizations, we will still use these statistics since we lack any other statistics.

The contributions of this work include a probabilistic analysis that considers the following: (1) Hallucination statistics, including the probability of an individual experiencing hallucinations tied to an event such as the death of Jesus. (2) The duration, frequency, and mode of hallucinations, given that an individual is experiencing hallucinations. (3) The probability of the existence of a messianic or similar figure, who would be a candidate for belief in a miraculous resurrection, at any given point in time. (4) The probability of such a belief arising not just at one particular time and location (namely, Palestine during the 30s), but at any other time or location across all of recorded history given a conservative estimate of the total number of messianic figures. This last factor is important because even if a given particular event is unlikely, given a large enough population pool, an event can become likely. For example, if the probability of an individual winning the lottery is roughly one in a million, the probability that someone among a pool of 100 million people will win the lottery is quite high. Thus, this work considers whether or not the total population pool across all of recorded history is large enough that a belief in a resurrected messianic figure is likely to have arisen based on hallucinations (or similar psychological phenomena) experienced by a group of disciples.

The remainder of this paper is organized as follows. Section 2 summarizes some of the key resurrection appearances as described in the New Testament. Section 3 discusses the results from the hallucination literature, including some of the key studies often cited by hallucination proponents, and summarizes the facts and the parameters upon which the statistical analysis will be based. Section 3 also discusses messianic-like candidates for resurrection appearances, along with the relevant multiple opportunities across history for the possibility of an unlikely set of convincing hallucinations leading to a belief in resurrection. Next, Section 4 presents the mathematics derived from statistics and combinatorics theory that will be used to support the statistical analysis. Section 5 revisits the New Testament appearances to apply the probability analysis to these appearances. Section 6 evaluates some of the leading variants of the hallucination hypothesis. Section 7 presents an overall evaluation of the hallucination hypothesis, and then the paper concludes with Section 8.

2. Reported Resurrection Appearances in the New Testament

This section summarizes some of the resurrection appearances that are relevant to the analysis of the hallucination theory. The purpose of this section is not to demonstrate the accuracy of the biblical accounts, but to summarize the appearances that are typically discussed by scholars who propose the hallucination theory. Some scholars accept many of the appearance reports, while other scholars discount some of the appearance reports as non-historical. Later we will assess the probabilities associated with these reported appearances, where possible.

2.1. Indivudial Appearances

• Peter: Based on 1 Cor. 15:5a and Luke 24:34, Peter saw the resurrected Jesus separately from the other disciples. Per Lüdemann, there were at least two individual appearances to Peter: in addition to the appearance in 1 Cor. 15:5a, there is a separate appearance to Peter implied by Luke 24:34 when the Eleven proclaim that “It is true! The Lord has risen and has appeared to Simon” (Lüdemann 2004).
• James, the bother of Jesus: In addition to Paul’s testimony in 1 Cor. 15:7 that James saw the resurrected Jesus, we have corroborating data, including the fact that James was skeptical of Jesus prior to the crucifixion, a finding based on passages such as John 7:5 and Mark 3:21, but was somehow transformed into a leader in the early church (Acts 12:17, 15:13; Gal. 1:19), and was executed by the religious authorities as the leader of the church in Jerusalem according to Josephus, Antiquities1 (Licona 2010).
• Eleven disciples: According to Acts 1:1–26, in addition to accounts such as Luke 24:33–51, the apostles, including all 11 remaining disciples, thought that they saw and heard the resurrected Jesus.
• Other early witnesses: Based on Acts 1:21–26, there were a number of additional disciples besides the Eleven who saw the resurrected Jesus. The fact that two were nominated (“Joseph called Barsabbas (also known as Justus) and Matthias”) would suggest that the group of witnesses was considerably larger. In this paper, we will consider the case that there was a total group of 72 who were not as close to Jesus, and that at least 10 of the 72 were experiencing hallucinations at some point. The value of 10 is a minimum value consistent with the situation in Acts 1, while from Luke 10:1 we can infer that 72 is a reasonable value for the number of disciples not as close to Jesus.

2.2. Group Appearances

We now summarize some of the key group appearances reported in the New Testament.

• Two disciples traveling to Emmaus: Luke 24:13–32 describes an appearance by Jesus to Cleopas and another disciple which took place over an extended period of time.
• Group of disciples in Jerusalem: Luke 24:33–51 describes how Jesus appeared to the “Eleven and those with them”, in addition to Cleopas and the other disciple who were traveling to Emmaus, corresponding to a minimum group size of 15 witnesses. This group appearance is supported by 1 Cor. 15:5b, which is part of an early tradition that Paul received when he was in Jerusalem, and is much too early to be legendary, per Lüdemann, who dates 1 Cor. 15:5 to 3 years after the crucifixion when Paul was converted near Damascus circa 33 (Lüdemann 2009). In addition, John 20 describes two group appearances to the disciples in Jerusalem—one without Thomas and one with Thomas. According to Allison, the John 20 account provides an at least partially independent confirmation of the other accounts (Allison 2005). We will focus on 11 as the most conservative value.
• Group of disciples in Galilee: Matthew 28:16–20 describes the appearance of Jesus to the eleven disciples in Galilee. The Mark 16:7 account, in which the angel tells the women to “go, tell his disciples and Peter, ‘He is going ahead of you into Galilee. There you will see him, just as he told you’”, provides early evidence supporting a group appearance in Galilee (Lüdemann 2009). John 21:1–14 also describes a group appearance in Galilee, although in this passage only seven disciples are specifically mentioned.
• Women: All four Gospel accounts are in agreement that a number of women from Galilee were at the crucifixion (Mark 15:40–41; Matthew 27:55; Luke 23:49,55; and John 19:25), that women were the first to see the empty tomb with the stone already rolled away, that one or two angels (or messengers) were at the tomb, and that the women learned that Jesus had risen (whether announced by the angels, or by Jesus himself as described in John). Mark 16:1 lists Mary Magdalene, Mary the mother of James, and Salome; Matthew 28:1 lists Mary Magdalene and the “other Mary”, which based on 27:55, is the mother of James and Joses; Luke 24:10 lists Mary Magdalene, Joanna, and Mary the mother of James, in addition to the “other women”; John 20:1–2 only lists Mary Magdalene by name, but implies that there were other women by the expression “we don’t know where they have put him”. Luke implies that there were at least five women who saw a messenger who proclaimed that Jesus was risen, which is supported in the broad details by the other Gospels. In addition, women were apparently with the disciples in some of the group appearances.
• Paul and at least two companions: The appearance of Jesus to Paul recorded in 1 Cor. 15:8, and corroborated by numerous other passages in Paul’s letters, is described in more detail in Acts. Acts 9:1–19, along with Acts 22 and Acts 26, describes the conversion of Paul while he was enroute to Damascus to persecute the Jewish Christians. Paul’s companions saw a bright light and heard a sound, although they did not see or hear anything specifically distinguishable. Virtually all scholars accept that Paul believed he had an encounter with the risen Jesus; the only question is whether this was an objective or subjective experience, and if subjective, how best to account for it (vision or hallucination or both, given the experiences of Paul’s companions).

3. Hallucination Data and Statistics

This section presents the studies and data used in the analysis.

3.1. Definitions

The American Psychological Association defines a hallucination to be a “false sensory perception that has a compelling sense of reality despite the absence of an external stimulus” (American Psychological Association n.d.). A similar definition by Tien is an “occurrence of visual, auditory, olfactory, or somatic sensations without external stimuli” (Tien 1991). Another type of experience that is sometimes used in the hallucination hypothesis is a sense of presence of the deceased, a phenomenon which is reported in numerous post-bereavement hallucination studies (Castelnovo et al. 2015).

Visual hallucinations can come in numerous forms, “ranging from simple fleeting shapes or lights” to fully formed “hallucinations of meaningful percepts, such as people, faces, animals, or objects” (Collerton et al. 2023). Note that while fully formed hallucinations are relevant to the hallucination hypothesis, simple hallucinations consisting of fleeting shapes, lights, or sounds are not relevant. While there are a number of different psychological models for hallucinations, we will employ the pragmatic approach that Collerton et al. used for visual hallucinations (and extended to other hallucinations), which is that a fully formed visual hallucination consists of seeing a recognizable object or person when there is nothing there to see (Collerton et al. 2023). Finally, we note that hallucinations are individual rather than shared experiences.

Although delusions are often associated with hallucinations, they are a different psychological phenomenon, and most often involve “paranoia, reference, and grandeur”, per a recent large online survey (Linszen et al. 2022). A number of the hallucination-hypothesis variants propose group delusions (also referred to as mass or collective delusions), which are delusions shared by a group of individuals. However, possibly due to the rarity and complex nature of this phenomenon, there are no significant statistical studies characterizing the prevalence of group delusions. Thus, this paper will not attempt any statistical analysis of group delusions, although later we will observe that some of the group delusions proposed as analogies to the New Testament resurrection appearances do not fit the data.

Altered States of Consciousness (ASCs) are similar to hallucinations, except that the experience does not seem as real, and they are more frequent when falling asleep or waking up (Rabeyron and Loose 2015). Note that, per the pragmatic definition we have adopted, we will not distinguish between ASCs and hallucinations, a choice which is consistent with some studies in which ASCs appear to have been lumped in with hallucinations, as discussed in the following.

3.2. Hallucination Rates in the General Population

Researchers have conducted a number of studies aiming to measure the percentage of the population that experiences hallucinations. Tien evaluated two large-scale studies, including a study published in 1894 and conducted by Sidgewick, with a group of 17,000 adults, and the other conducted by the Epidemiologic Catchment Area (ECA) Program between 1980 and 1984, and involving 18,572 people (Tien 1991). The Sidgewick study indicates that 8.4% of the general population claims to have experienced visual hallucinations, 3.3% have experienced auditory hallucinations, and 0.9% have experienced tactile hallucinations, with overall positive responses, including all categories, of 9% among men and 12% among women. Tien noted that the ECA study showed similar overall rates, although the proportion of visual hallucinations in Sidgewick was higher. Tien suggests that this may be because the Sidgewick study included hallucinations that were likely sleep-related (this would include ASCs), since the mind is much more suggestive during these times, leading to more apparent hallucinations.2 Waters et al. report a visual hallucination rate in the general population of 6% when excluding drugs and physical illness, and an overall rate of just over 7% when drugs and illness are not necessarily excluded, based on taking a weighted mean of earlier large-scale studies (including the Tien study) (Waters et al. 2014). They also report that visual and auditory hallucinations often occur together. Johns and Van Os report an annual audio or visual hallucination rate of 4% among a white population group, 9.8% among a Caribbean group, and 2.3% among a South Asian group; of the affirmative respondents, about one-third to one-fourth (1.2%, 2.8%, and 0.6% total percentage, respectively) reported having, at least once in their life, an auditory hallucination involving partial or complete sentences (Johns and Van Os 2001).

These studies, spanning about a century in time and covering multiple cultural groups, indicate that a fraction of the population ranging from 2% to 10% report having experienced audio, visual, or combined audio–visual hallucinations. Note that these reported hallucination rates relative to the general population also include hallucinations affecting those who have experienced bereavement, which we will discuss next.

3.3. Post-Bereveament Hallucination Rates

The hallucination rate for post-bereavement is considerably higher, with reported rates in various clinical studies ranging from 30% to 60%, per Castelnovo et al. (Castelnovo et al. 2015). However, this apparently high hallucination rate is deceptive, because in many of these studies the hallucination rate is dominated by reports of a feeling of the presence of the deceased, while reported visual and/or auditory hallucination rates are far lower. For example, in a study by Rees involving bereavement after the death of a spouse, the reported total hallucination rate was about 47%; however, by category, 39% felt the presence of the deceased, while 14% experienced visual hallucinations, 13% experienced auditory hallucinations, 12% of reports involved speaking to the deceased, and 3% involved being touched by the deceased (Rees 1971). Thus, although the reported total hallucination rate was almost 50%, both the auditory and the visual hallucination rate were much lower. In addition, Rees observed that older patients are much more likely to experience hallucinations than are younger patients (about 22% for those under 40 years old, and about 50% for those over 60). In addition, the hallucination rate varied with years of marriage as well, with a 30% total rate for those married under 10 years, and up to almost 60% for those married over 40 years. Finally, Rees noted that although hallucinations tend to go away with time, for some, the hallucinations can continue for years. Thus, the data from the Rees study are consistent with the Castelnovo results, but provide more insight into the nature and distribution of the bereavement-related hallucinations. Rees also found that the post-bereavement hallucination rate did not vary by cultural group in his study, although Rees does point out that a Japanese study reported considerably higher rates of feeling the presence of the deceased among widows living in Tokyo.

In a study involving bereavement in a wide range of relationships (spouse/partner, parents, children, siblings, friends, and other family) in a variety of circumstances, including violent death by murder, Simon reports auditory and visual hallucination rates of about 4% within the general study group, and about 10% for the subset of the group that was diagnosed with “complicated grief” (Simon et al. 2011). This study is useful since the bereavement parameters (the mixture of relationships, and including consideration of violent death) are more in line with the situation of Jesus and his followers. In another study focusing on post-bereavement hallucinations among elderly people, Grimby reports that the most common hallucination was a feeling of presence, followed by speaking to the deceased, and hearing, seeing, and being touched by the deceased (Grimby 1993). Furthermore, Grimby provides data consistent with Rees’ statement, showing that hallucinations in this population group tend to decrease over time, with feeling of presence decreasing by about 20% over the first year, and hearing-related phenomena decreasing by about 80% over the first year.

3.4. Hallucination Characteristics

Johns and Van Os note that auditory and visual hallucinations, for persons who do not need care, are generally of brief duration (Johns and Van Os 2001). Waters et al. report that non-clinical auditory and visual hallucinations are generally of brief duration, and even among psychosis cases, visual hallucinations typically last “several seconds to several minutes” (Waters et al. 2014). Anecdotal hallucination accounts described by Shuchter (Shuchter 1986) as well as by Allison (Allison 2005) confirm the brief, transitory, and relatively infrequent nature of hallucinations among those who experience hallucinations. Sommer et al., however, report a much higher frequency and much longer duration, with an estimated 10–15% of the general population experiencing an average frequency of 3.6 auditory hallucinations per week, lasting an average of 128 s3 (Sommer et al. 2010). Given that this type of study is prone to self-selection bias (Elston 2021), we must treat their results with caution (Sommer et al. discuss limitations given their subject-selection methodology). Nonetheless, these parameters can be used to form an upper bound for the statistical analysis, as we will see later.

In addition to typically being relatively short in duration, hallucinations are also characterized as individual events rather than shared events. While some researchers suggest that group hallucinations might not be that uncommon based on first-hand accounts of people seeing the same apparition, they also acknowledge the lack of support for this finding in the psychology literature (e.g., Allison 2005). Using the keywords “group hallucination” or “collective hallucination” to search Google Scholar reveals that the vast majority of works mentioning either of these two expressions fall into one of four categories: (1) works that argue for group hallucinations in order to account for religious phenomena such as the resurrection appearances; (2) works that argue against group hallucinations in order to account for such religious phenomena; (3) paranormal and occultic works; and (4) reviews about fictional literary sources that involve group hallucinations, such as Hamlet. Apparently, no peer-reviewed psychology works provide any statistics or case-study data on group hallucinations. Some advocates of the hallucination hypothesis, such as Lüdemann, have proposed that one disciple, such as Peter, experienced a hallucination, which then caused a chain reaction of mass hysteria, leading the other disciples to also see the resurrected Jesus, similar to the manner in which some of the crusaders involved in the 1099 conquest of Jerusalem reported seeing St. George above the city walls (Lüdemann 2004). However, such examples of mass delusion or hysteria do not apply to most of the resurrection appearances, which occurred with small groups of disciples, and with auditory and visual interaction in close quarters. The exception is the appearance to five hundred of the disciples at one time, which was recorded in Paul’s first letter to the church at Corinth, which this work will not consider in its statistical analysis since mass hysteria or delusion cannot be ruled out in this case. As an alternative to mass delusion, we can consider the possibility that a first hallucination caused a chain reaction of hallucinations among the disciples. Although there is insufficient data to support the chain-reaction effect for hallucinations, even if we allow for such a scenario, there is still the problem of simultaneous hallucinations, which will be statistically analyzed later.

3.5. Messianic Candidates for Resurrection Appearances

When evaluating the hallucination hypothesis, we need to consider both the probability of a particular candidate event for resurrection appearances, such as the crucifixion of Christ, and the total number of events that qualify as candidates for resurrection appearances throughout history. Any given resurrection candidate may have an extremely low probability when accounting for post-mortem appearances via hallucinations, but with multiple opportunities throughout history, the cumulative probability will be higher, and must be assessed.

However, not every individual who has died will qualify as a candidate for resurrection appearances. For example, Kent argues that the context of Jesus as a major prophet or messianic figure is an important consideration in any analysis of the resurrection appearances (Kent 1999). Michael Licona makes a similar point regarding the identification of a miracle such as the resurrection: “We may recognize that an event is a miracle when the event (1) is extremely unlikely to have occurred given the circumstances and/or natural law and (2) occurs in an environment or context charged with religious significance” (Licona 2010). For example, Bar Kokhba, who led the Jewish revolt against Rome in 132 as a messianic figure would qualify, as well the Roman emperors, who, from the time of Augustus, were routinely worshipped as divine in portions of the empire.

To assess the number of resurrection candidates throughout history, we need to estimate the percentage of the total population who are significant prophetic or messianic figures, or similarly significant charismatic leaders, and how many people have lived during the historical era. One approach to estimating the percentage of resurrection candidates among the general population is to consider first-century Jewish messianic contenders, which will yield a statistically conservative estimate, given the political turmoil and messianic expectations in the region around Palestine during this time period (Licona 2010). In the context of this analysis, a “conservative” estimate means one that tends to favor the hallucination hypothesis. Per Ingolfsland, there were likely six messianic contenders in first-century Palestine, including Jesus and five additional figures mentioned by New Testament scholar JD Crossan (Judas in Galilee, Simon in Perea, Athronges in Judea, Manahem son of Judas the Galilean, and Simon son of Gioras) (Ingolfsland 2002). Per Taylor, the population of this region was probably at least a few million (Taylor 2021), corresponding to a total population pool across the century of at least 10 million, given the ancient typical life expectancies and age distribution described by Scheidel (Scheidel 2001). As another example, in a time and region with similar political and religious instability, per Goldberg, there were three known messianic contenders in 19th century Yemen (Goldberg 1995), within a population pool similar to the region around first-century Palestine. Based on these examples, a rate of one messianic contender per million would be conservative when computing upper bounds for the probability of simultaneous resurrection hallucinations in the context of a messiah-like figure.

Next, we consider the total population pool of adults throughout history. This can be estimated by integrating U.S. Census Bureau estimates for population by time period from 10,000 BC through 20254, then dividing by a conservative average life span of 20 years to yield an estimate of about 68 billion people who have lived during this time period (if we consider the increasing average life span in recent decades, then the total population pool over history is significantly smaller). This includes children, so the adult population pool is much smaller. This paper will assume an extremely conservative value for the total population pool, specifically, 100 billion (again, conservative in the sense that a larger population pool throughout history provides more opportunities for simultaneous hallucinations).

Therefore, given an average rate of one messianic-like resurrection candidate per million people born, and an upper bound for the total historical population determination of 100 billion, then the upper bound for the total number of potential resurrection candidates throughout all of recorded history is 100,000. Thus, any set of hallucinations centered on a potential candidate for resurrection claims will have under 100,000 opportunities to occur, and the evaluation of the hallucination hypothesis will conservatively allow for 100,000 opportunities.

3.6. Hallucination Analysis Parameters

This section describes the key hallucination parameters that will be used in the statistical analysis. This paper assumes that modern bereavement hallucination studies apply cross-culturally and cross-temporally; this is for two reasons: (1) those who propose the hallucination theory generally make this assumption, and (2) the studies report roughly similar hallucination rates for different cultural groups. Therefore, given this assumption, and with an additional margin to be conservative, this analysis will use a post-bereavement audio–visual hallucination (AVH) rate of 15% for those closest to Jesus, and a rate of 10% for those not as close to Jesus.

Furthermore, we will consider this hallucination rate as being constant over a 2 month (60 day) period, which is conservative compared to the traditional 40-day period of appearances immediately following the crucifixion (Habermas 2001; Smith 2020). For hallucination frequency and duration among those experiencing hallucinations, the analysis will assume a rate of one hallucination per month lasting one minute in duration, which corresponds to two per two-month period. Thus, we will assume, as a conservative value, that 15% of Jesus’ family and closest followers would experience an average of two combined audio–visual hallucinations lasting one minute during the 60-day period following the crucifixion.

Finally, as discussed earlier, we will assume the existence of 100,000 messianic candidates across history whose deaths would provide opportunities for simultaneous post-death resurrection appearances that could lead to a belief in a risen messiah.

The hallucination analysis parameters are summarized in

Table 1. Audio–visual hallucination (AVH) analysis parameters.

Parameter	Value
Probability that a close disciple of Jesus will experience AVH	15%
Probability that other disciples of Jesus will experience AVH	10%
AV hallucination rate if a disciple is experiencing AVH	2 per 60 days
AVH duration per hallucination	1 min
Period of time after crucifixion to consider AV hallucinations	60 days
Number of messianic resurrection candidates throughout history	100,000

, below.

4. Probability Theory Used in Hallucination Analysis

This section discusses the probability theory that will be employed in the hallucination analysis. This includes the probability of individual appearances to a subset of the disciples (i.e., appearances not at the same time) as well as the probability of simultaneous hallucinations among a subset of the disciples.

4.1. Probability of Individual Appearances

Let P_H represent the probability that an individual will experience audio–visual hallucinations. For post-bereavement periods, we saw that P_H ≈ 0.04 (or as high as 0.10 for complicated grief) (Simon et al. 2011), or about 0.14 for visual or audio individually (combined audio–visual would be lower) (Rees 1971). As discussed earlier, this analysis will use P_H = 0.10 for disciples not very close to the deceased (the 72 disciples sent out in Luke 10:1 for example), and P_H = 0.15 for disciples and individuals very close to the deceased (11 disciples and Jesus’ family, in this case).

Given P_H, the probability P_H:M,N that at least M people out of a group of N individuals will experience hallucinations can be found by summing the right tail of the binomial probability distribution, starting at m = M and continuing through N:

$$P_{H:M,N}=\sum _{m=M}^N\left(\begin{array}{c}N\\ m\end{array}\right){\left(P_H\right)}^m{(1-P_H)}^{N-m} where \left(\begin{array}{c}N\\ m\end{array}\right)\equiv {\displaystyle \frac{M!}{m!(N-m)!}}$$

(1)

If we consider the case that everyone in a group will experience hallucinations at some point in time, then (1) simplifies to P_H:N,N = (P_H)^N for a group of size N:

$$P_{H:N,N}={\left(P_H\right)}^N$$

(2)

Thus, Loke observes that the probability of all 11 disciples experiencing a visual hallucination is (0.14)¹¹, based on the 14% rate from Rees that Loke uses (Loke and Meader 2024). Note that P_H:M,N does not represent the probability of a group of disciples having simultaneous hallucinations. Rather, it represents the much higher probability that a group of disciples will experience hallucinations at some point in time. To determine simultaneous hallucinations among a group, we consider two scenarios: (a) everyone in that group is in the subset of individuals who will experience at least one hallucination at some point in time, and (b) everyone in that group is experiencing hallucinations at the same point in time. Loke, however, computes the probability for the first condition (a) only, and does not consider the intermittent nature of these hallucinations, a factor which would make condition (b) very unlikely, so the actual probability of all 11 disciples simultaneously experiencing a hallucination of any sort is far lower than (0.14)¹¹, as we will see in the next section.

4.2. Probability of Simultaneous Appearances

We will now consider the conditional probability P_SH:K|M,N that at least M disciples of a group size of N will experience simultaneous hallucinations, given that all N disciples are experiencing hallucinations. The probability P_SH:K|M,N is computed for an average rate of P_K hallucinations per unit of time for a total of K units of time. To illustrate what K represents, if we have a group of individuals experiencing hallucinations at an average rate of once per month, and if each hallucination is 1 min in duration, then we can approximate the total number of candidate time slots for time-coincident hallucinations as 30 days/month * 16 h/day * 60 min/hour = 28,800 min/month. Here we assume 16 h/day represents the typical time the disciples are awake and potentially in a group. In this case, K = 28,800, and P_K = 1/28,800. If we consider a 2-month period with same rate, then K = 57600, while P_K remains at 1/28,800. It can be shown that the probability P_SH:K|M,N is approximately given by the following expression:

$$P_{SH:K|M,N}\approx K\sum _{m=M}^N\left(\begin{array}{c}N\\ m\end{array}\right){\left(P_K\right)}^m{(1-P_K)}^{N-m}$$

(3)

Note that the basic form of (3) is similar to (1), but with P_H replaced by P_K, and a multiplying factor of K. To understand how to interpret the probability P_K, we note that if an individual is experiencing hallucinations with an average duration of 1 min at an average rate of L hallucinations per K minutes, then the individual probability of having a hallucination during any given minute is P_K = L/K. Thus, the probability of M of N disciples having a hallucination during any given minute, given that they are all experiencing hallucinations, follows (1), but with P_H replaced by P_K. The probability over a total span of K minutes is multiplied by K to account for multiple opportunities. Although this is an approximation that tends to yield probabilities that are slightly too high (i.e., favoring the hallucination hypothesis), it is accurate for P_SH:K|M,N << 1.0 (i.e., much less than 1.0, or in this context, less than 0.01), which is typically the case in simultaneous hallucination analysis. Also, note that P_SH:K|M,N in (3) is a conditional probability conditioned on the subset of M or more simultaneous hallucinations occurring among a set of N disciples who are all experiencing hallucinations at the given rate and duration. The final probability is lower, as discussed next, if we consider the fact that only a fraction of the disciples would be expected to experience any hallucinations at all.

To determine the final probability P_SH:K,M,N of M simultaneous hallucinations out of a total of N people, with P_H being the probability of any given disciple experiencing hallucinations, and P_k the probability of a hallucinating disciple having a hallucination in a specific time slot k out of the total of K time opportunities, we need at least M of the N to be hallucinating, and we need to have at least M time-coincident hallucinations among the subset of M or more that are experiencing hallucinations. Combining (1) and (3) across each of the M-out-of-N possibilities yields

$$P_{SH:K,M,N}\approx K\sum _{m=M}^N\left(\begin{array}{c}N\\ m\end{array}\right){\left(P_{Hk}\right)}^m{\left(1-P_{Hk}\right)}^{N-m} where P_{Hk}=(P_H)(P_k)$$

(4)

Thus, returning to Loke’s calculation, the probability of all 11 disciples experiencing at least one simultaneous visual hallucination lasting one minute at some point during a 2-month period is not 0.14¹¹ ≈ 4 × 10⁻¹⁰ (equivalent to 1 out of 2.5 billion as expressed by Loke), but is given by a much lower probability: P_{SH:57,600,11,11} ≈ 3 × 10⁻⁵⁴. Note that the dominant factor is the low probability of simultaneous hallucinations, so even under the unlikely scenario that all disciples were assumed to be experiencing hallucinations (100% probability versus 14% probability for any given disciple), the probability would be given by (3), or 7 × 10⁻⁴⁵.

Finally, as discussed earlier, this paper will consider not just this one single case in Palestine with Jesus, but all possible messianic contenders throughout all of history:

$$P_{SH:K,M,N,R}\approx R\left(P_{SH:K,M,N}\right)$$

(5)

where P_SH:K,M,N,R is the total probability that M disciples out of a group of N total disciples of a messianic figure across all of history will experience simultaneous resurrection hallucinations during the first two months after death. Here, R represents the total number of resurrection candidates across history. Using the conservative estimate of 100,000 as discussed earlier, this yields a total historical probability for the same-sized group with R = 100,000 = 1 × 10⁵. For Loke’s case of a group of 11 disciples, this yields a probability value of 3 × 10⁻⁴⁹ instead of 3 × 10⁻⁵⁴. Thus, although considering all opportunities across history for hallucinations among the disciples of a messianic figure leading to a resurrection belief greatly increases the probability, the total probability is still infinitesimally small for the case of 11 out of 11 disciples. Again, this is an approximation, but it is accurate as long as the total probability is <<1.0, which is the case here.

The next section returns to the appearances reported in the New Testament to support later analysis considering specific hallucination proposals.

5. Probability of New Testament-Reported Resurrection Appearances

This section includes examples of probabilities for some specific cases reported.

5.1. Indivudial Appearances

• Peter: Although Peter’s witness has significant explanatory power according to Licona (Licona 2010), this work will not attempt to perform a statistical analysis for a single individual.
• James, the bother of Jesus: As with Peter, this work will not perform statistical analysis for a single individual, despite the strong explanatory power provided by the case of James.
• Eleven disciples: Even without accounting for group appearances (discussed later), all eleven disciples hallucinating individually is a highly improbable event. Applying (1) with this work’s conservative statistical analysis AVH rate of 15% among disciples close to Jesus, we determine that P_H:11,11 = (0.15)¹¹ ≈ 1 × 10⁻⁹.
• Other early witnesses: Applying (1) to the case that 10 of the 72 disciples not close to Jesus experience hallucinations, we determine that P_H:10,72 = 0.18, where the probability that these disciples not as close to Jesus will experience post-bereavement audio–visual hallucinations is given by P_H = 0.1, as discussed earlier. Although this event by itself is not that improbable, this probability is independent of the probability that all 11 disciples hallucinated, and the joint probability is therefore lower.

5.2. Group Appearances

We now consider the probabilities of simultaneous hallucinations over a 2-month period following Jesus’ death, using the hallucination analysis parameters.

• Two disciples traveling to Emmaus: To provide a conservative upper bound approximation of the probability of a hallucination, we will not consider the reported long duration of both seeing and talking, which, if this is a hallucination, is much longer than the nominal 1 min duration, and is far less likely than a single 1 min time-coincident hallucination. Thus, for this case, we need the probability of 2 simultaneous hallucinations among a group of 72 disciples with P_H = 0.1 for this group and K = 57,600 as a conservative upper bound: P_{SH:57,600,2,72} ≈ 0.0018. Note that in this case, we can double-check the approximate probability via a different method. If P_H = 0.1 for this group, then the expected number of disciples experiencing hallucinations out of this group is 10%, or approximately seven. Given a group of seven experiencing hallucinations, then the probability of two simultaneous hallucinations among this group of 7 is P_H:57,600,2,7 ≈ 0.0015, which is close to the more accurate estimated probability of 0.0018.
• Group of disciples in Jerusalem: Given the unlikely situation that all 11 disciples are experiencing post-bereavement hallucinations, the probability that all 11 will have a simultaneous hallucination during the first two months after the crucifixion, assuming they are experiencing one hallucination per month lasting one minute on average, is given by P_{SH:57,600,11,11} ≈ 7 × 10⁻⁴⁵, where we use P_H = 1.0 for the case that all 11 disciples are experiencing hallucinations. With a more realistic value of P_H = 0.15, we determine that P_{SH:57,600,11,11} ≈ 6 × 10⁻⁵⁴.
• Group of disciples in Galilee: The probability associated with 11 of 11 is the same as in Jerusalem. If we assume that 11 are present, but only a simple majority of 6 have a simultaneous group audio–visual hallucination, then the probability is given by P_{H:57,600,6,11} ≈ 5 × 10⁻²⁰. Again, this assumes that all 11 disciples are experiencing hallucinations at some point. With a more realistic value of P_H = 0.15, we determine that P_{SH:57,600,6,11} ≈ 6 × 10⁻²⁵.
• Women: Although the appearances to the women are significant, the New Testament does not provide enough information to perform a proper statistical analysis, so we will not include the women, although inclusion of appearances to the women further weakens the hallucination case.
• Paul and at least two companions: Although Paul and his companions experiencing a hallucination relating to Jesus is an unlikely event, meaningful statistical analysis is problematic due to the many unknown factors.

6. Analysis of Selected Hallucination Theories

In this section, we cover some of the significant hallucination theories that have been proposed, including theories that accept virtually all of the reported resurrection appearances and theories that discount most of the resurrection appearances.

6.1. Kent

Kent suggests that the experiences by Mary Magdalene and the disciples were “very normal grief-related illusions”, and he cites Rees in support of this theory (Kent 1999). Kent also performs a psychological analysis of Jesus’ followers, and concludes that prior to the crucifixion, they experienced a “transference” of themselves to Jesus, which, along with normal bereavement hallucinations after the crucifixion, resulted in a transformation in their lives.5 Although Kent cites Rees, his discussion potentially suggests that all 11 close disciples are experiencing hallucinations. Although this is extremely unlikely, for Kent, we will use P_H = 100% for the 11 disciples rather than the more realistic P_H = 0.15. Surprisingly, Kent does not appear to discount any of the resurrection appearances as summarized in the Gospel accounts, except that regarding the group appearance to the 11 disciples in Galilee, Kent concludes that not all agreed about seeing Jesus; this is on the basis of Matthew’s comment in 28:17: “When they saw him, they worshiped him; but some doubted” (Kent 1999). Although doubting does not imply not seeing (for example, when Jesus showed them his hands and his feet, “they still did not believe it because of joy and amazement” (Luke 24:41)), we will analyze Kent with the assumption that a simple majority of 6 out of 11 disciples saw Jesus in the group setting in Galilee. Therefore, for Kent, we have the following probabilities to multiply together:

• Individual, other early witnesses: With P_H = 0.10, we determine that P_H:10,72 = 0.18;
• Group, two disciples traveling to Emmaus: With P_H = 0.10, we determine that P_H:57,600,2,7 ≈ 0.0015;
• Group, 11 disciples in Jerusalem: With P_H = 1.0, we determine that P_{SH:57,600,11,11} ≈ 7 × 10⁻⁴⁵;
• Group, 6 of 11 disciples in Galilee: With P_H = 1.0, P_{H:57,600,6,11} ≈ 5 × 10⁻²⁰.

Multiplying together, we calculate that the joint probability that these resurrection appearances are accounted for by hallucinations is about 1 × 10⁻⁶⁷, which after accounting for 100,000 historical opportunities, yields 1 × 10⁻⁶² as the integrated probability across history. Clearly, an explanation based on ordinary post-bereavement hallucinations, as proposed by Kent, is not compatible with the available data, even if we allow for the unlikely case that all 11 of the closest disciples were experiencing hallucinations.

6.2. Lüdemann

Lüdemann is one of the more pessimistic of the New Testament scholars, and discounts a number of the resurrection appearances. For example, Lüdemann claims that the group appearance of Luke 24:36-43 is the result of “the imagination of the second-generation Christian community”, that Joanna of Luke 8:3 and 24:10 is “probably a redactional addition”, and that it is evident that John 21 was created by later editors (Lüdemann 2004). Although he originally considered it as “all but certain” that James saw Jesus after the crucifixion, Lüdemann later said that the appearance to James of 1 Cor. 15:7 is “a legitimizing formula without any basis in history” (Lüdemann 2004).

However, despite discounting many of the resurrection appearances, Lüdemann still accepts a historical core, and says that “No doubt Jesus ‘appeared’ to Peter and then to the Twelve” (Lüdemann 2004). As noted earlier, Lüdemann also considers the appearance to the two disciples on the road to Emmaus to be reliable. However, Lüdemann also claims that only Peter had independent hallucinations, and that his hallucinations caused a chain reaction in which the other disciples started hallucinating. Therefore, in line with Lüdemann, we will only consider one of the group-of-11 appearances, and, although not supported by any large-scale hallucination studies, we will allow, per Lüdemann, that all the disciples were hallucinating (P_H = 100% rather than 15%). We thus have the following probabilities to multiply together:

• Group, two disciples traveling to Emmaus: With P_H = 0.10, we determine that P_H:57,600,2,7 ≈ 0.0015;
• Group, 11 disciples in Jerusalem: With P_H = 1.0, we determine that P_{SH:57,600,11,11} ≈ 7 × 10⁻⁴⁵.

Multiplying together, we calculate that the joint probability that these resurrection appearances are accounted for by hallucinations is 1 × 10⁻⁴⁷. When multiplying by 100,000 messianic candidates throughout history, we determine 1 × 10⁻⁴² to be the final probability of an occurrence of Lüdemann’s hallucination hypothesis at any point in history. Again, this is an infinitesimally small probability.

6.3. Goulder

Goulder suggests that, based on his psychological analysis, Peter had a hallucination that was triggered by cognitive dissonance, guilt, and bereavement, which then precipitated hallucinations by the other disciples, who were also suffering loss, leading to a “collective delusion” (Goulder 1996).6 Although Goulder considers some portions of the Gospel accounts to be unreliable, he does not appear to object to the resurrection appearances as described in the Gospels, Acts, and 1 Cor. 15. Thus, the probability of Goulder’s hypothesis falls somewhere between those of Kent and Lüdemann.

6.4. Eisenberg

Eisenberg proposes a variant that includes a sequence of events that starts with Jesus surviving his crucifixion for a short while and concludes with the disciples having feelings of present hallucinations; this sequence of events, when combined with the presence of a stranger in settings such as the group of disciples recorded in Luke 24:33–51, resulted in audio–visual hallucinations (Eisenberg 2016). Eisenberg’s proposal is problematic for a number of reasons, including (1) the mistaken identity of a stranger for Jesus is not a hallucination; and (2) whether hallucination or other psychological phenomena, there are no studies that provide empirical support or data for this proposal. Therefore, Eisenberg’s variant of the hallucination hypothesis is not statistically testable, so it is outside the scope of this paper.

6.5. Smith

Smith proposes that a combination of hallucinations and collective delusion is the best explanation for the resurrection appearances (Smith 2023). In support of collective delusions, Smith references historical mass delusions such as the “Bigfoot” sightings in South Dakota in 1997 (Smith 2020). Although this type of collective delusion is not amenable to statistical analysis due to a lack of data, and is thus outside the scope of this paper, we will observe that these Bigfoot sightings are not relevant since these sightings did not involve small groups of people engaged in conversations with Bigfoot. Smith also cites the Rees study in support of a variant similar to Lüdemann’s, in which Peter, driven by guilt, hallucinates the risen Jesus, leading to a type of mass hysteria during which all the disciples begin to experience hallucinations of Jesus (Smith 2020). Unlike the Bigfoot sightings, Smith’s variant that aligns with Lüdemann’s is testable, and was shown to have an infinitesimally small probability, as discussed earlier.7

6.6. Parameter Sensitivity Analysis

We will perform a brief parameter sensitivity analysis to show that even with parameters that are much more optimistic, the hallucination hypothesis still leads to infinitesimally small probabilities, even when starting from Lüdemann’s pessimistic position. If the number of messianic resurrection candidates is assumed to be 1000 times higher (i.e., one out of a thousand instead of one out of a million), then Lüdemann’s scenario has a probability that is 1000 times higher: 1 × 10⁻³⁹ as the final probability of Lüdemann’s hallucination hypothesis occurring at any point in history. If we also assume that the hallucinations are taking place over a year-long period, we calculate a probability for Lüdemann that is 6 times higher, or 6 × 10⁻³⁹ as a total probability for all of history. Alternatively, we can consider a much higher rate and duration of audio–visual hallucinations by using the audio hallucination statistics from Sommer (Sommer et al. 2010); with 3.6 hallucinations per week lasting an average of 128 s, then K = 28,800 for a 2-month period consisting of 2 min time opportunities instead of 1 min time opportunities, and P_K ≈ 1 × 10⁻³. Given these values, the hallucination probability for a group of 11 disciples is ≈ 4 × 10⁻²⁹, which corresponds to ≈ 4 × 10⁻²⁴ as a total probability throughout all of history. While far higher than the other probability bounds we have discussed, this is still an infinitesimally small probability, and it is based on a study that, per its own admission, is limited by the selection bias introduced by its subject-selection methodology, and is, furthermore, focused on audio hallucinations versus full audio–visual hallucinations.

6.7. Interpreting Hallucination-Hypothesis Probabilities

To help the reader visualize how small these probabilities are, a 1 m diameter column of sand going from the earth to the sun can contain about 1 × 10²⁰ grains of sand, assuming a typical grain size of 1 mm. Thus, a probability of 1 × 10⁻²⁰, which represents an event far more likely than any of the cases discussed, corresponds to the probability of randomly selecting a single diamond the size of a grain of sand hidden in a column of sand 1 m wide stretching from the earth to the sun. Or, if we consider the probability that six of eleven disciples will experience simultaneous hallucinations at any point in history, assuming all eleven are experiencing hallucinations, then P_{H:57,600,6,11,R} ≈ 5 × 10⁻²⁰ × 100,000 = 5 × 10⁻¹⁵. This probability is roughly equivalent to the probability of finding a single diamond grain hidden in a 1 m wide column of sand stretching across the width of China. This is also roughly the same probability of winning a lottery jackpot from a pool of 50 million entries every second without pausing for over 2 years.

7. Evaluation of the Hallucination Hypothesis

We have employed statistical analysis to formally demonstrate that hallucinations cannot account for key resurrection appearances described in the New Testament that are often accepted by hallucination proponents, including the event involving the group of disciples described by Paul in 1 Cor. 15:5b, which, as we noted earlier, even Lüdemann accepts as being an early report dating to within a few years of the crucifixion. The analysis in this work is based on reported hallucination statistics from the same psychology literature typically cited by the hallucination proponents. Although some hallucination variants lack sufficient data to support statistical analysis, in all cases in which the hypothesis is amenable to statistical analysis, the results show that the probability of a modestly sized group comprising 11 disciples of a messianic figure experiencing time-coincident hallucinations that could lead to a belief in resurrection is virtually zero, even when considering all of history. Sensitivity analysis allowing for highly optimistic parameters still leads to infinitesimally small probabilities. Based on these results, we conclude that the hallucination hypothesis should be rejected.

This raises a question—are naturalistic explanations preferable over supernatural explanations? Some of the hallucination proponents that we have examined have demonstrated an extremely strong bias against any form of supernatural explanation. For example, Kent makes his bias clear when he states that “There never was nor will there ever be a resurrection of a body if that body was in fact dead” (Kent 1999). Similarly, Lüdemann refers to the idea of the resurrection as “absurd” (Lüdemann 2004). Goulder considers the appearances as “evidence for grief and not for resurrection”, without even considering the possibility that they might really be evidence for the resurrection (Goulder 1996). While this analysis does not show that the hallucination hypothesis is logically impossible, it does show that it is not supported by any available statistical data.

The limitations of this work include the following considerations: (1) This paper relies on published hallucination statistics, which, by necessity, are all from the modern era. Such statistics must be used with caution due to dissimilarities with the ancient New Testament culture, as noted by O’Collins (O’Collins 2011). With this caution in mind, the advantage of this approach is that using modern studies allows direct evaluation of contemporary hallucination hypotheses, which often rely on modern studies. (2) In close relation to the first limitation, even within modern studies, the statistical values are subject to uncertainty. In order to address both of these limitations as much as possible, this paper uses conservative estimates for statistical parameters to yield probabilities that err on the side of the hallucination hypothesis, and also considers the sensitivity of the results to parameter variations within a wide range of values. (3) This paper does not evaluate alternatives to the hallucination hypothesis to account for the belief in the resurrection.

8. Conclusions

This paper examines the hypothesis that the resurrection appearances are best accounted for by hallucinations, and demonstrates that this hypothesis is extremely improbable, based on a statistical analysis that is significantly more detailed than those performed to date, including those advanced by any proponents of the hallucination theory. In particular, statistical analysis shows that the various proposed hallucination hypotheses all lead to probabilities so low that they would normally be deemed a miracle, even when using the proponent’s assumptions. Thus, the hallucination hypothesis should be rejected as an explanatory mechanism used to account for the New Testament-reported resurrection appearances.