Catholic Substackers are building a case for the truth of Catholicism, and they are building it for skeptics like you and me. Ethan Muse leads the charge, as he seeks to “present the miracle case for Catholicism in a way that is rigorous enough to persuade those that are strongly predisposed towards skepticism”, specifically by excluding non-Catholic skeptical explanations for claimed miracles using “multiple lines of evidence.” Muse’s most recent post in this project focused on Padre Pio, an Italian friar venerated by the Catholic Church, who per Wikipedia “was known throughout his life for numerous instances of supernatural phenomena”, including stigmata, miraculous healings, prophecy, and, most importantly for this post, biologically impossible fevers. Specifically, while mainstream medicine & biology tell us that at ~43°C, severe cellular damage is expected, and that by about 45°C protein structures break down and vital systems such as the brain and heart fail, Pio is claimed to have reached fevers as high as 52°C. That’s quite a claim!

As an atheist ‘strongly predisposed towards skepticism’, I am not keen to jump to theological conclusions; however, if it is true that this man survived with such high fevers, then biology and chemistry, as we know them, did not hold. While this wouldn’t necessarily make me a theist, let alone a Catholic, establishing that something like this likely happened would definitely challenge how I understand the world. So I take such claims seriously.

In this post, I'll be taking a Bayesian perspective to figure out whether someone like me should be convinced by Muse’s sources that Pio had biologically impossible fevers, which I name ‘half-boiling’ for brevity. I will start by suggesting my prior odds, based on a framework presented in my previous post, that a random person will get half-boiled. This is only the starting point for an investigation, which I then update given the evidence specific to this case. When considering the evidence, I will focus on the characteristics of the instruments used to measure Pio’s miraculous temperatures (mainly bath thermometers), and of the operators involved (the measurer & the measured). With this, I'll be able to decide whether the evidence provided should convince a skeptic that Pio was miraculously half-boiled.

2. Prior Costs

To consider Pio’s half-boiling from a Bayesian framework, we need to get a sense of the event’s prior odds. Previously, I suggested a practical approach for calculating the prior odds that a random person gets half-boiled as a function of two parameters: 1) how many people we are confident were never half-boiled, and 2) how confident we are that God exists and is in the business of half-boiling people. Based on these parameters, we can find an optimal rate of God’s ‘half-boiling’ tendency, the rate that makes the half-boiling hypothesis look as good as it possibly can given that we never observed it before. We then multiply this rate by our probability that ‘God exists and is in the business of half-boiling people’ to get a fairly miracle-friendly prior probability (because it gets an optimized rate for free). You can use my app to calculate your own prior.

Now for my prior: given biology & medicine, I think we can be confident that at least 40 billion people throughout human history reached adulthood without getting half-boiled. Furthermore, I am an atheist who finds the existence of God to be quite implausible, and I find the idea of God suspending biology & chemistry to half-boil people to be absurd. I look into my heart and, for the life of me, I can’t put a probability higher than 0.05 on this combination. With this input, I can calculate my prior odds: about 1 in 2.1 trillion. I can also convert these odds into a number of fair coin flips landing heads in a row in one attempt: beating odds of 1 in ~2.1 trillion is like getting 41 heads in a row.

The nice thing about the coin flips conversion is that it allows us to compare hypotheses in terms of a concrete currency with additive costs: before we see the specific evidence, the half-boiling hypothesis would cost 41 coins. Every observation that half-boiling explains better than the skeptical alternative costs coins for the skeptical hypothesis, and the question is whether the reports of Pio's temperature can cost it more than 41 coins. As Bayesian shoppers, we should prefer the cheaper and thus more likely hypothesis. So, is the evidence for half-boiling strong enough to cost a skeptical explanation more than 41 coins? Let’s find out!

3. The Instrument

In this section, I consider whether claims of Pio’s half-boiling can be explained by the use of poor instruments for measuring body temperature. In other words, I will try to assess whether, assuming honest measurement, the evidence we have could be reasonably explained by measurement error. This is an initial hurdle that many empirical claims, including miracle claims, must pass.

Measuring stuff is hard. Measuring stuff about humans is often extra hard because we’re a mess. While heat is well-understood physically, measuring human body temperature can be challenging. For example, a meta-analysis of multiple scientific studies looked at the ability of modern clinical axillary (armpit-based) thermometers used by healthcare professionals to detect fever, as measured via rectal thermometers (yikes). In appendix 2, they report that axillary thermometers used with 975 participants got things right in about 77.3% of cases, and that a fever reading corresponded with actual fever in 74.8% of cases. So measurement errors are a real issue for axillary thermometers.

Now, the claim is that Pio’s axillary temperatures reached very high levels on several occasions; mostly around 48°C, where one claim goes as far as 52°C. However, standard thermometers in 1920s Italy could not go further than around 42.5°C, an extreme but not outright biologically impossible temperature. Undeterred by technological limits, Pio’s contemporaries used bath thermometers, which go further than 42.5°C, to measure Pio’s axillary temperature. And so, our understanding of biology and chemistry hangs on the results of 1920s bath thermometers. Can they get the job done?

While Muse claims that Pio’s half-boiling axillary temperatures were observed “using many and various properly calibrated instruments”, I have to disagree with calling bath thermometers “properly calibrated” for human measurement. I looked for, and found exactly 0 data suggesting that bath thermometers, let alone those used in south Italy in the 1920s, can accurately assess human temperature. Simply because, as far as I can tell, the ability of bath thermometers to measure human temperature has not been studied. A normal person would stop here, but I want to get some sense of the evidence that bath thermometers can provide.

3.1. Measurement models

To reason about the evidence that bath thermometers may provide, we start by specifying two models representing the half-boiling and skeptical (no half-boiling) hypotheses. Normal thermometers back then were reasonably accurate, so the skeptical hypothesis can grant that Pio’s temperature reached 42.5°C, the highest point of a common thermometer and the edge of biological possibility. In contrast, the half-boiling hypothesis assumes that Pio was miraculously heated. This means that his expected temperature should have been at least 42.5°C, but how much higher? While God’s heating ability is unlimited, our measurement ability is limited. Some of the bath thermometers reported are described as having a range of up to 80°C, so let’s put God’s limit there1. Let’s also say that God has no preference between the different values, such that Pio’s expected temperature was equally likely to take on any value from 42.5°C to 80°C2.

Now that we have some sense of expected values, let’s push toward modeling the bath thermometers’ results. For this, we need to get a sense of measurement errors, but surprise surprise, there’s no data on this. Undeterred, I will extrapolate from the results of a 2015 meta-analysis of studies on the accuracy of modern peripheral thermometers studied in clinical settings. This gives us a standard deviation (SD) of about 0.74 for adult patients with fever, so let’s take 0.74 as a very optimistic starting estimate of measurement error. I will assume that measurement error is normally distributed and the same per hypothesis3, and that the measure can only go from 0°C to 80°C4. With these details, we can calculate & visualize how likely we are to measure 48°C once, per each model. This one observation costs the skeptical model about 35.5 coins, almost enough to overcome the prior odds! Does this mean that the measurement hurdle is easily passed? Not so fast.

Having measurement error represented by SD of 0.74 is critical for this comparison, and it’s an unrealistic best-case scenario which the skeptic shouldn’t grant. What if we assume that the imprecision of bath thermometers is 2-times worse than that of modern clinical thermometers? With SD of 1.48, a single observation of 48°C costs the skeptic 6.6 coins, far from enough to overcome the prior cost of 41 coins.

Why stop at 2-times imprecision? It may well be 3-times. With SD of 2.22, a single observation of 48°C is evidentially weak, costing the skeptic a meager 1.7 coins.

You might be thinking that I’m going too far here. Sure, we don’t have human-use data, but are bath thermometers really so bad? We should remember that the skeptical hypothesis has lots of coins to move around, and it should spend them wisely. How likely is it that 1920s bath thermometers are 3 or more times more imprecise than modern clinical thermometers? I don’t know! And since we don’t have a shred of data on human use, I don’t see why ‘buying’ this level of imprecision would cost the skeptical hypothesis more than 5 - 8 coins (odds ranging from 1-in-32 to 1-in-256), yet it almost neutralizes the evidential value of a single measurement by a bath thermometer. That’s money well-spent for a skeptical explanation.

But of course, we don’t have just one measurement, as Muse’s sources provide multiple reports of miraculous fevers. Can they be enough? Luckily, we can apply our models to finally answer the perennial question: How many bath thermometers does it take to prove a miracle?

Say that we’re dealing with a skeptical hypothesis that generously spent 8 coins to assume that bath thermometers are very bad at measuring human temperature (SD of 2.22). With 33 coins left to spend, we can ask: how many honest independent measurements with an average value of 48°C5 would cost the skeptic more than 33 coins? At long last, it’s time to answer the article’s main question: 9. It takes 9 honest, independent measurements using bath thermometers to prove a miracle.6 If you want to try different values, I made an online app for you.

3.2. How many measurements do we have?

Do we have the equivalent of 9 independent observations of half-boiling with a mean of 48? I consider a measurement to have confidently happened only when we have corroborated reports that identify who took it, with what instrument, on what occasion, and where the account is clearly distinct from another account that we already counted. Fairly minimal reporting standards for shaking the foundations of reality. If these conditions are unmet, the skeptical hypothesis can easily ‘pay’ to dismiss a report as hearsay, embellishment, or false memory.

First, let’s look at the eyewitness reports that Muse provides, i.e., people who say they personally took or saw the half-boiling measurements being done:

• We have Fr. Lorenzo, who in a sworn deposition reports measuring half-boiling twice and names several people involved, observing 48°C in an instrument brought by Dr. Festa and 45°C in an instrument that could go no further. Fr. Ignazio said in support that “A 48°C fever was observed”.

• We have Fr. D’Apolito, in his memoir published 68 years after the events, reporting that he observed 46/47°C “many times”. No corroboration or description of the measurement tools.

• We have Fr. Paolino, in his memoir published 61 years after the event, dramatically describing how he observed 52°C once. One advantage of this report is that its bath thermometer is said to be stored at the Padre Pio archives. However, Muse does not provide a chain of custody that confirms that this is indeed the thermometer used in 1917. So I'm not sure to what extent this is corroborated.

The quotes generally give little detail about the measurement & instrument conditions, the environment, all the people around, and the degree of supervision. With this level of reporting, maybe the skeptical hypothesis is over-paying for assuming that the measurements are awful. Next, let’s look at the non-eyewitness reports of half-boiling that Muse provides:

• Nino Salvaneschi reports that Doctors Merla and Giuva took measures twice, at 48°C and 50°C. Fr. Lorenzo reports that Dr. Merla and Dr. Gina (Francescantonio Giuva is likely a transcription error of Francesco Antonio Gina) were there when he took his two half-boiling measurements, so this report likely refers to the measurements taken by Lorenzo, which got inflated with retelling.

• We have an unnamed reporter reporting without any details that Professor Adelchi Fabrocino took a measure of 48°C. I would expect a professor from the 20th century to have published academic papers or books, or at least be commemorated by his university. But I can’t find anything about this person. I asked Claude to search for him in Italian, and it found nothing. The very existence of Professor Adelchi Fabrocino lacks corroboration (UPDATE7).

• Dr. Festa says that an unspecified city physician observed 48°C, and that Fr. Lorenzo observed 48°C and 48.5°C8. Although Festa never observed Pio’s temperatures go higher than 36.5°C, he wrote 18 years after the events that Pio’s miraculous half-boiling was “verified with complete certainty”, as he brought Pio’s convent “a thermometer of absolute precision” that Lorenzo reports using. Sadly, the technical details of this extraordinary invention, alongside any data on its accuracy for measuring human axillary temperatures, are forever lost to science. Hence, I treat it as no better than a bath thermometer.

This is a mess. 2 of 3 eyewitness accounts were published 60+ years after the event, show signs of embellishment, and don’t sit well with other reports. Fr. D’Apolito is the only one who reports that 46/47°C were observed “many times” while everyone else skips these values and are satisfied with only a few measures. On the other hand, Fr. Paolino’s dramatic report of a single 52°C measure is extreme even for the reports around Pio. Fr. Lorenzo’s case looks much better, being corroborated by Ignazio, Salvaneschi, & Festa. I’d say that these reports should make us confident in about 2-4 independent measurements at most. 2 from Lorenzo, maybe 1 from Paolino, and a big maybe for 1 from D’Apolito, but no more than that.

The non-eyewitnesses don’t fare well. Salvaneschi’s report of Dr. Merla’s and Giuva’s/Gina’s measurement seems to cover the same measurements that Lorenzo refers to. We don't get any details regarding Professor Adelchi Fabrocino’s measurement, and the report reaches us through an unnamed correspondent while no independent trace of this professor survives. Lastly, Dr. Festa’s mention of an unnamed city physician lacks corroboration and details, and Festa corroborates the observations that Fr. Lorenzo described. In my reading, these reports don’t add anything beyond corroborating Lorenzo’s 2 measurements.

While there are claims of many measurements being taken, I don’t think that a skeptic has much reason to grant anything close to the equivalent of 9 independent measurements. The skeptical hypothesis can easily spend a few coins, if it must, to assume that fewer than 9 measures were taken. Mainly by dismissing Fr. D’Apolito’s (although it can also easily grant this one9) uncorroborated and minimally-detailed account of taking many measurements, which was published 6 decades after the events.

Even if we grant that some legitimate independent measurements were taken, and generously assume that no contradictory/low measurements were ignored10, there’s too much uncertainty around the measurements’ imprecision, and of the number of independent measurements, to make a skeptical hypothesis cost more than the half-boiling hypothesis. Measurement error is the first hurdle that the claim of miraculous half-boiling has to pass, and it’s not doing well.

4. The Operator

The skeptical hypothesis I presented assumes that Pio reached 42.5°C multiple times. It does so because it can be costly to challenge standard mercury thermometers, and we have better reports of those reaching 42.5°C than of half-boiling. Furthermore, assuming the highest biologically possible value of ~42.5°C makes measurement error work better as an explanation for extreme readings. However, having to grant multiple cases of 42.5°C can be costly to the skeptical explanation; 42.5°C is freaky and extreme on its own. Can the skeptic make it cheaper? Yes, by going beyond the instrument, and looking at operator characteristics: of Pio and of those around him.

4.1. Necessity is the mother of invention

Growing up in a country with mandatory military service, I heard of many mysterious ailments that affect conscripted soldiers. Young and fit men with sudden high blood pressure, or who broke a bone overnight without signs of trauma, or, yes, suddenly showed unexplained fevers or wounds. Malingering, the faking or exaggeration of health problems to get a reward or to avoid an unwanted duty, is real and common. It has been recognized and studied with special focus on military settings.

Why do I mention malingering? Because Pio’s fevers were his first clear weird symptoms, and the only ones that started during his military service. Pio was conscripted to the Italian military in World War 1, and assigned as medical staff in a military hospital in Naples. However, due to health reasons, Pio was often taken off-duty and eventually discharged. Here’s how Dr. Festa characterized Pio’s military service: “it was quickly understood that, given the poor state of health he appeared to be in, it would be more useful to study him carefully than to use his labor in caring for the sick, in order to restore him, as much as possible, to a normal physical condition. An attempt was therefore made to begin a clinical study on him by methodically measuring his temperatures; but this had to be abandoned quickly because whenever a thermometer was applied to him, the mercury column would rise with such rapidity and force in the glass tube that it immediately burst. After repeating the attempt several times with the same result, the matter was dropped and arrangements were made for his permanent discharge.”

While I find that Festa is being credulous and/or fabulist about some details11, it seems that young Pio, like many other young conscripted men, suddenly showed mysterious medical symptoms during his mandatory military service, helping him to be frequently taken off-duty and eventually getting an early discharge. When a mysterious ailment gets a conscripted soldier consistently off-duty and eventually out of military service, we usually don’t call it divine intervention; we call it malingering.

Surely the military’s medical staff closely supervised Pio while measuring his temperature, so how could he inflate his axillary temperatures12? Easily. The biggest study of factitious fevers found that 9% of patients referred by doctors for prolonged fevers of unknown origin actually had factitious fevers. Their longest case of fever-faking lasted eight years until discovery. Doing something like Tummo meditation, which just involves breathing and mental imagery, can raise axillary temperatures by up to 2.2°C. Covering a sensor, or pressing it hard against the skin, can each increase measured skin temperatures by up to 1.3°C. We also have evidence that stress can increase body temperatures, sometimes leading to severe psychogenic fevers. And there are likely to be many more paths to temperature inflation that I’m not aware of13. So there’s nothing special about Pio combining a variety of techniques to inflate axillary measurements by ~3°C. With this, the skeptical hypothesis doesn’t require more than Pio having a few banal ~39.5°C temperatures that are inflated into ~42.5°C and get mismeasured.

Okay, inflating readings of axillary temperatures is nothing special, and wanting to be taken off military duty is a common motivation for conscripted soldiers. But Pio was not a Tummo practitioner, and how could he learn to inflate his temperatures? Note that Pio served as medical staff in the 10th Medical Corps unit. This is important, because paramedical personnel dominate adult samples of prolonged fever fakers, enabled by "their familiarity with hospital routine; their ready access to thermometers and drugs; and the special consideration given to them". Furthermore, Pio could have easily learned to inflate readings of axillary temperatures from soldiers that he met at medical ward and elsewhere. Read a bit about, or listen to soldiers who want to get off-duty. They have to get incredibly inventive to fool trained and suspicious medical personnel. I can personally attest to that, knowing men who, as mere teenagers who didn't even do well at school, fooled multiple specialized military doctors. Many soldiers, especially at times of war, feel that their lives are on the line. For many, malingering is a necessity, and necessity is the mother of invention.

Pio’s military service plausibly gave him an opportunity to learn how to inflate his temperatures and a motive to do so. Furthermore, the timing gives the skeptical hypothesis an explanatory edge, making temperature inflation very cheap to ‘buy’. Ask yourself: why did Pio’s unusual fevers start during his military service? If Pio was indeed inflating his temperatures, we would expect him to start doing it exactly during his military time. But from the miracle side, it’s a convenient coincidence that God started heating Pio up just in time to mess with his military service. So did Pio get off military duty and eventually discharged because of divine intervention, or perhaps, like so many other conscripted soldiers, he used mundane methods to help get him off duty and eventually discharged14?

4.2 A conspiracy?!

Muse rules-out that fraud took place, arguing that “If St. Pio’s extraordinary fevers were fraudulent then, unless there were a series of baffling conspiracies that implicated many and various clergymen and physicians over a span of decades, he would have had to repeatedly and reliably manipulate axillary temperature readings while under close supervision in diverse circumstances.”

It should be clear by now that Pio could have easily and repeatedly inflated axillary temperature readings. I’ll avoid repeating my earlier criticism of claims regarding “many and various clergymen and physicians over a span of decades”, and I don’t see anything remotely like “close supervision” in the reports of half-boiling. While some temperature inflation doesn’t require a conspiracy, It’s not a stretch to suspect that some of the key supernatural claims about Pio were fraudulently manufactured alongside with a few individuals. This was suspected by Dr. Agostino Gemelli. Gemelli was a Catholic friar, physician and psychologist, the first President of the Pontifical Academy of Sciences, and founder of the Catholic University of Milan. He personally investigated the Pio situation, and was considered the Vatican’s top expert on stigmata.

Stan Patton made a post where he introduces us to Gemelli. Patton’s post primarily draws from the book Padre Pio: Miracles and Politics in a Secular Age, by historian Professor Sergio Luzzatto. Briefly, Gemelli believed that Pio’s stigmata were chemically-inflicted, of the kind Gemelli had seen soldiers produce during WW115. Importantly16, Gemelli suspected that Pio was being manipulated by Pio’s adviser Father Benedetto, who Patton describes as “cultivating a community of escalating superstition”. Furthermore, Gemelli singled out a “ringleader” of Pio’s close circle. This was Emanuele Brunatto, a salesman of chemical products, a well-connected fascist propagandist, and a conman convicted of fraud & forgery who published an influential pro-fascist biography of Pio under a pseudonym. In other words, Gemelli believed that Pio fraudulently produced at least one supernatural claim, which the people closest to him were encouraging and promoting, while those surrounding this circle credulously believed what they were shown and told. I don’t know if Gemelli was exactly right; maybe not. But buying Gemelli’s characterization of this environment won’t cost much for the skeptical explanation, yet it provides another path to neutralize much of the evidentiary value of the reports.

Speaking of ‘baffling conspiracies’, it's worth asking how baffling this one would be. We are told that a hoax must implicate many and various clergymen and physicians over decades. But there’s no need to implicate more than Pio, a few people close to him, and some credulous men looking at thermometers. Dr. Gemelli lends credibility to the first part, and the reports from those surrounding Pio implicate their own credulity. For example, Fr. Paolino calls a mercury thermometer his "most authentic proof" which was "returned to me by the Padre after he had measured the degree of the fever." Dr. Festa, who measured Pio’s normal temperatures for days, reported "complete certainty" in biologically impossible measurements that he did not observe. Fr. D’Apolito tells us that he “had the good fortune and joy to see, to meet and to approach Padre Pio many times a day, to render him some service and to kiss his hand”, while telling next-to-nothing about the “many” measurements he took.17 We’ve seen far more baffling frauds than this one would be, inside and outside the Catholic Church, and much weirder cults popped-up. If a hoax happened here, then the credulous men looking at thermometers never stood a chance.

5. My Verdict

Overall, I find the evidential value of the reports regarding Pio’s fevers to be far from sufficient to convince a skeptic. The reports leave skeptical hypotheses with plenty of room to cheaply ‘buy’ various options that can neutralize the reports’ evidential value. If a skeptical explanation wisely allocates its probabilities across measurement error, temperature inflation, and fraud & credulity, it can easily provide a much more likely explanation than miraculous half-boiling. At least for someone with remotely skeptical tendencies.

Reaching the end of this post, I must admit to some frustration. In my view, much of the pro-miracle material flooding Substack is built on a sandcastle of data provided without a shred of skepticism, and yet boasts that all reasonable readers must be convinced. The lack of minimal critical engagement with reports and measurements plagues miracle arguments, and makes them inert for skeptics. What did I consider here? Measurement error, temperature inflation, fraud & credulity. Those are completely banal explanations! If the aim is to convince skeptics, then why do we get zero critical engagement with the measurements, zero critical consideration of Pio and his environment, and zero critical engagement with the reports? I might have made SILLY mistakes, and my reasoning might fall to OBVIOUS objections, but at least I tried. Isn't that what we're supposed to do? Try to critically evaluate ideas that we care about?

Suppose that one day the stars in the night sky aligned to read “In the beginning was the Word, and the Word was with God, and the Word was God” in a language that every human understands. Surely, this scenario would provide at least some evidence for the existence of God? If you agree, please nod your head.

This scenario is an example of ‘Awesome Theistic Evidence’ (ATE). These scenarios have been used to argue that ‘divine psychology’ objections to fine-tuning arguments go too far. In short, fine-tuning arguments look at the unexplained constants of established physical models and argue that they fall within a very narrow range of life-permitting values. This state of affairs, it is argued, shows that a life-permitting universe is very improbable if the universe was not intentionally designed for life. In contrast, the probability of God designing a life-permitting universe is not so low, and therefore the life-permitting constants provide strong evidence for there being a God.

‘Hold your horses’, respond skeptics. God could make any kind of life regardless of physical constants, and why would God be so interested in carbon-based biological life to begin with? To make life-permitting constants more probable under theism, we need to make quite a few assumptions regarding God’s goals & methods for achieving them. But this divine psychology does not share the anchors of any other types of psychology, such as biology, culture, empirical experience, or even physical limitations. This makes the probability of God doing something like fine-tuning physical constants inscrutable.

‘Your objection proves too much’ respond theists. If divine psychology is inscrutable, then observing an ATE would not provide any evidence for the existence of God. But an ATE would obviously provide some evidence for the existence of God. Remember nodding your head earlier? So clearly, divine psychology is not inscrutable.

ATE is a double-edged sword

Is divine psychology really constrained enough to make just a single case of Amazing Theistic Evidence (ATE) support theism, as the ATE response assumes? Upon over-analysis, this turns out to be trickier than it first seems, in a way that illustrates more general problems with theism as an explanatory framework. To show why, I will start by making a key assumption: ATEs could happen in many possible ways; there’s a ginormous number of ways that an all-powerful & all-knowing being could provide and combine Awesome Evidence for its existence.

This assumption can easily create problems for the ATE response. If God has great many possible ATEs at his disposal, and a tendency to provide ATEs, then it is quite strange that we only observe a single ATE. To illustrate how this becomes an issue for the ATE response, I'll use a simplified toy model.

Suppose that there are 1 billion different kinds of ATEs that God could provide. Since we’re dealing with an omnipotent being, there could be many more, but let’s start here. Next, let’s assume that the probability of each ATE on atheism is constant at 10^-300, the smallest number I can work with before running into numerical issues. This number is so vanishingly small that it makes the probabilities thrown around in fine-tuning arguments seem gigantic in comparison. Next, we need the probability of observing each ATE on theism.

As usual, God doesn’t tell us how many ATEs he plans to provide us with, so our divine psychology has a free parameter. This is ω (Omega), the probability of a random ATE occurring. For simplicity, I will model God’s ATE generation as a binomial process: each possible ATE may occur independently of any other with probability ω1. The higher ω is, the more ATEs we’d expect to observe proportionally to the number of possible ATEs.

Let’s say that ω can be anywhere from 1 to 10^-300; God might perform every possible ATE, or none of them. This reflects the constraint that each ATE should be at least as likely on theism as on atheism2, but it’s difficult to say in advance by how much. Now, since we assume that many ATEs are possible, a high ω should make us expect a high number of ATEs. Where in some cases, if ω is high enough, observing only 1 ATE would be even less likely on theism than on atheism.

I’ll visualize how ω shapes the expected number of ATEs given 1 billion possible ATEs. On the left plot, I set ω to 1 divided by 1 billion. In this setup, 0 or 1 ATEs are highly likely, and the probability of additional ATEs diminishes beyond 7. On the right plot, I set ω to 1 divided by 100 million, which leads us to expect multiple ATEs, while making 0 or even just 1 ATE quite unlikely.

With this setup, we can calculate how low ω has to be so that observing only 1 ATE would support theism; that is, which ω values make 1 ATE more likely under theism than under atheism when 1 billion ATEs are possible. After doing the math, we find that for only 1 ATE to support theism, ω has to be lower than 1 divided by 1.48 million. That’s a tiny range out of all possible ω values!3 But what independent reasons do we have for thinking that ω just happens to be so low? That is, without looking at the actual number of ATEs, why should we think that ω is so smol?

Turns out that having just 1 ATE support theism is not such an innocent assumption. For that to hold, ω, or God’s tendency to provide ATEs must be sufficiently low relative to the number of ATEs available to God. Otherwise, we would expect to observe much much more than just 1 ATE. But this constraint is not built into the ATE response itself, and it sits in tension with many common ways of thinking about divine action; where God is taken to reveal, guide, or intervene quite freely. Look at recent Substack discourse on miracles, ask religious theologians & philosophers about the presence of the Holy Spirit or about heaven, or, you know, read the bible. If there are good a-priori reasons for God to be shy about providing Awesome Evidence, many serious theistic thinkers and traditions don’t seem to accept them.

We can further use this setup to explore the implications of observing 0 ATE for the evidential power of fine-tuning. To make a connection between the two, let's assume that observing just 1 ATE would provide at least as much evidence for theism as fine-tuning does. After all, the probability of each ATE on atheism is as low as it gets, and we’re talking about clear & universally accessible linguistic communication formed from an otherwise well-understood system. This assumption enables us to ask the following question: for which values of ω can fine-tuning provide more evidence for theism than the absence of ATEs provides for atheism? In other words, for which ω values does evidence from 0 ATE out-perform evidence from 1 ATE and therefore evidence from fine-tuning?

This range of ω values is even narrower than the previous one; ω now has to be lower than about 1 divided by 2.96 million. Even if we insist that 1 ATE must support theism, evidence for theism from fine-tuning can only out-perform the evidence for atheism from 0 ATE in about 50% of ω values. So for fine-tuning to have any chance of providing more evidence for theism than 0 ATE provides for atheism, we have to even further constrain ω. Again, what independent justification do we have for doing that? Did our divine psychology just happen to be lucky?4

Remember that in the context of fine-tuning arguments, the divine psychology objection states that theism is not independently constrained enough to make fine-tuning more likely. The ATE response then contests that theism is constrained enough to make 1 ATE more likely, but that brings its own issue since 1) you need to fine-tune divine psychology itself for just 1 ATE to provide evidence for theism, and 2) if you want fine-tuning evidence to possibly beat evidence from 0 ATE, you need to even further finely-tune divine psychology. Upon over-analysis, the ATE response ends-up further illustrating theism’s lack of constraint instead of serving as a counter-example.

Objection!

There are many possible objections to the approach I've been taking here, but many of them are going to miss the point. I’ll focus on key objections that help sharpen the challenges I raise.

Objection 1: Aren’t you assuming way too many possible ATEs?

I might as well be lowballing it. We're dealing with an omnipotent being after all. But the main point is that to shrink the number of possible ATEs enough to diffuse the issues I raised, we need to place strict constraints on what God would/could do. For example, if we want 50% of ω values to allow 1 ATE to provide evidence for theism, we need to have only about 1,000 possible ATEs. Instead of restricting ω, we are now restricting the number of possible ATEs. But why?

Objection 2: You’re oversimplifying by modelling ATEs as independent events, but God has reasons that make them dependent. For example God cares about regularity in nature, so he wouldn’t just throw ATEs around.

Mere dependence embeds the issues I highlighted into a more complex model of divine action; it doesn’t make them go away. Observing 0 ATE will still provide evidence for atheism, which would still be amplified by the number of possible ATEs. Furthermore, observing only 1 ATE would, for many dependence structures, be even less likely on theism than my toy model suggests. Because if we ever stumble upon one ATE, many dependence structures would imply that we should expect more ATEs. Unless we are restricting divine psychology to a very specific type of dependence:

Objection 3: God would perform no more than 1 (or very few) ATE.

If we somehow know that God would perform no more than 1 ATE, the evidence from 0 ATE for atheism is minimized, and the evidence from 1 ATE for theism is maximized. But this approach has issues. First, why no more than 1? This is a convenient constraint which needs independent justification. Furthermore, this constraint has absurd consequences: if we really think that God outright refuses to perform more than 1 ATE, then multiple ATEs would not provide any further evidence for theism than just 1 ATE would. If it’s absurd to think that 1 ATE would provide no evidence for theism, then surely it’s much more absurd to think that millions of additional ATEs would provide no further evidence for theism!

Objection 4: God would find a sweet spot where there is strong but non-coercive evidence for theism. So we're not going to see many ATEs.

This approach has two main issues that reoccur in this context. First, while this view might be reasonable given the lack of ATEs, it’s not clear whether it can be established independently of us -not- observing ATEs. Second, if we take this view seriously, then it is not clear whether ATEs would provide evidence for theism, since many individuals might find them to be psychologically coercive. It's the same old problem in a different dressing: the constraints on divine psychology seem to be motivated by the lack of ATEs rather than established independently of it, and it’s not clear whether they actually enable 1 or more ATE to provide evidence for theism.

Now, I’m the first to admit that the approach I took here is simplified; I presented an illustrative toy model, not a true model5. But before you suggest that I neglected an alternative, check that it actually works! It won’t be easy to undermine my 2 key points: that divine psychology needs non-trivial constraints for only 1 ATE to provide evidence for theism, and needs even more constraints for fine-tuning to reliably beat 0 ATE. These are consequences of God’s unlimited power and inaccessible psychology, which lead to trouble when we're observing only 1 relevant event when many more are possible. Any modelling approach would need to severely constrain divine psychology to overcome these issues. Think I’m wrong? Show your work, expose its assumptions, and provide independent justification.

Conclusion

The ATE response is meant to show that divine psychology is not inscrutable, since God is predictable enough to make 1 ATE more likely than otherwise. But God’s omnipotence means that for that to be the case, we need to tightly constrain divine psychology without independent justification. Instead of defeating the divine psychology objection, the ATE response ends up illustrating it: God’s extreme power and inaccessible psychology make theism such a radically unconstrained explanation that even 1 case of Amazing Theistic Evidence may not support it. The ATE response needs a divine psychology that explains why the stars speak, yet remains unbothered by their silence. I don't think that it's lucky enough to have both.

To make things concrete, let’s consider an example from recent debates. In the 1920s, a priest’s body temperature was reportedly measured in the range of 48-52°C. Unfortunately, sustained human survival with body temperature above ~45°C is incompatible with established biology. Going over 40°C leads to heatstroke. At 43°C, severe cellular damage occurs; by ~45°C, protein structure breaks down and vital systems such as the brain and heart fail. This is textbook medicine & biology.

Because human survival is biologically unsustainable at body temperatures of 48-52°C, proponents of the miraculous argue that the priest’s alleged state of half-boiling can only be explained via divine intervention. Some even gesture at the language of probabilistic model comparison when making their case. Big mistake. Now, I happen know a thing or two about probabilistic model comparisons, so let’s see how miraculous explanations fare under this framework.

Comparing models

To make sense of the kind of evidence needed to establish a miraculous half-boiling, we start by specifying two models:

• Model 1: this kind of divine intervention DOES NOT happen. Perhaps God is not in the business of half-boiling people; maybe God is non-interventionist; or maybe God simply doesn’t exist.

• Model 2: God exists, this kind of divine intervention DOES happen, and the non-zero probability of it happening to a given person is ε (epsilon).

This framing immediately poses a problem: since God doesn’t tell us how often He plans to half-boil people, how are we supposed to know ε? Luckily, we have some data for the rescue, which we can analyze using Bayes theorem.

Suppose that we observed one-million people throughout their adult life, and are quite confident that none of them were ever half-boiled. The probability of each such observation is 1 according to Model 1, but it is 1 minus ε according to Model 2. As such, every case without divine intervention is more likely under Model 1, and provides some evidence against Model 2. However, the strength of this disconfirming evidence depends on how high ε is. This creates two opposing forces, and a corresponding dilemma for miraculous explanations:

• As ε increases, the data more strongly favors Model 1. This lowers our belief in Model 2, and should therefore increase skepticism that God half-boiled a given person.

• As ε decreases, the data more weakly favors Model 1. But God is also less likely to half-boil a person, and we should therefore increase skepticism that God half-boiled a given person.

In other words, our observations strongly disconfirm Model 2 if it assumes that these divine interventions are common, but only weakly disconfirms it if these divine interventions are super rare. Either way, without getting into theology or metaphysics, the result is that we should be skeptical that a given person experienced a divine intervention. But just how skeptical should we be? Well, here comes the fun part.

Optimizing the tradeoff

Once we define our models, specify our prior beliefs in Models 1 and 2, and gather data, we can ask: what value of ε gives the miracle hypothesis its best possible shot? Mathematically, we are looking for the ε 'sweet spot'. If the miracle hypothesis assumes that God half-boils people every Tuesday, the many cases of non-intervention destroy it. If it assumes that God only does it to one in a trillion guys, the probability that it happened to this guy gets too low. The optimal ε is where the miracle hypothesis is at its strongest.

By finding the optimal ε, we avoid arguments over what ε should be, and instead work with the highest prior probability that a given person was half-boiled. This is the most charitable approach we can take when evaluating miraculous explanations. If even this best-case probability is vanishingly small, we should demand proportionally strong evidence before accepting the miracle claim.

From here we'll be dealing with tiny probabilities, so I'll describe them via the metaphor of flipping a coin X times and getting heads every time (i.e. 0.5^X), and using odds.

Suppose that we are confident that 1 million people were never half-boiled, we never observed half-boiling, and we don’t have any preferences between Models 1 and 2. With this information, we can calculate the optimal ε value. Then, to get the best-case probability that the next person we consider gets this divine intervention, we multiply the probability that Model 2 is correct by the probability of intervention under it (ε). As I show below, it is equivalent to getting about 22 heads in a row, or odds of about 1 in 3.6 million. Is that how skeptical we should be that a given person was half-boiled?

Why restrict ourselves to only 1 million observations? We might be confident that no person alive today has ever been half-boiled, because biology is a thing. If we have 8 billion observations, the best-case probability that the next person we examine is half-boiled is about 35 heads in a row, or 1 in 28.7 billion.

Why stop at 8 billion? About 40 billion humans are estimated to have reached adulthood throughout history. Unless miracle proponents wish to argue that many of them were half-boiled, we can be confident that 40 billion humans were never half-boiled. Now the best-case probability that the next person we examine gets half-boiled is about 37 heads in a row, or 1 in 143.6 billion:

Shifting the priors

So far, we’ve been assuming equal prior probabilities for Models 1 and 2. But why should we? Many skeptics are atheists, or don’t believe that God is in the business of half-boiling people. If someone like that assigns Model 2 with prior probability of 0.1, the resulting probability corresponds to about 40 heads in a row, or odds of about 1 in a trillion:

What about someone who is near-certain that God exists and is in the half-boiling business? Let’s set the prior of Model 2 at 0.999. We still get a best-case of 33 heads in a row, or odds of 1 in 9 billion. Regardless of prior beliefs, having many cases without this kind of divine intervention should make us very doubtful that a random person is going to experience it.

If you are unhappy with the values I showed, then try it yourself! I made a simple interactive app that allows readers to play with this analysis. To use it, you can click on the image below, or follow the link:

https://asnetic.shinyapps.io/substack/

What’s in a number

Okay, so the massive preponderance of non-interventions should make us extremely skeptical that a given person has experienced a divine intervention like half-boiling. Even if one is .999 certain that God is into half-boiling people, the prior odds of this happening to a person are about 1 in 9 billion. It takes quite some evidence to overcome that! Working within the framework of probabilistic model comparison, a proponent of the miraculous could take 3 main response strategies to the dilemma I presented. I’ll outline them from the least to most promising.

Strategy 1: ‘This was not some random person, and God was particularly likely to intervene in this case’. The biggest problem with this response is that it needs to be substantiated; it’s easy to tell stories about God, but until independent substantiation is given, non-believers have no reason to accept that God was more or less likely to intervene in a given case. Furthermore, how confident should we be that an intervention is more likely for a particular person? And how much more likely is it? The large number of non-intervention cases present a massive probabilistic obstacle to surmount. Taking meaningful steps toward it with this approach requires prophetic powers when it comes to predicting God’s interventions; something that no non-believer is going to accept without receipts.

Strategy 2: ‘The evidence was good enough to overcome the prior probability’. This calls for contextualizing the numbers I presented: In areas such as medicine & behavioral science, the odds of a result being generated by chance alone need to be below 1 in 20 (~4.3 coin flips) for it to be given further consideration. In physics, the 5 sigma threshold required for discovering a new particle is about 1 in 3.5 million (~21 coin flips). These are peanuts compared to the evidential thresholds required for never-seen-before divine interventions such as half-boiling. To discover the Higgs Boson, humanity’s top physicists & engineers built a multi-billion dollar setup and collected massive amounts of precise data over several years. In contrast, the evidence for our half-boiled priest comes from “priests reading bath thermometers” in the 1920s. Peanuts indeed.

Strategy 3: 'These interventions happen quite often, and here's the evidence'. This is the most promising response, and in principle it could work if similar divine interventions are common enough. The obvious move is to point to a large body of independently reported miracles of similar nature and argue that the cumulative evidence challenges Model 1, and shows that ε is not low. An example could be the many reports of Marian apparitions, although camera recordings of modern apparitions raise serious concerns regarding the veracity of these reports.

To summarize, the probabilistic framework I outlined here allows us to set a charitable evidential threshold for one-off miracle claims using standard probabilistic reasoning. It does not assume naturalism, or theism, or any debated theology or metaphysics1. It simply shows the probabilistic consequences of having many weakly disconfirming observations. Meeting the resulting evidential standard for miraculous explanations is in principle possible; I outlined 3 strategies for doing so. Yet, I personally find that none of the cases made in the recent Miracle Discourse come remotely close to succeeding; they don’t even present evidential peanuts compared to the mountain they need.

But IBE cuts both ways. If it’s the right method for evaluating explanations of the universe, it’s also the right method for evaluating the methodology used by design arguments themselves. And when we turn IBE on design arguments, asking whether they are best understood as a productive knowledge-generating methodology or a degenerative one protecting a pre-established conclusion, the results are not flattering.

In what follows, I’ll argue that design arguments share three key structural methodological problems: weak models, old evidence, and understated evidence. These compound each other in ways that result in bias toward theism and against naturalism, while underestimating uncertainty regarding comparisons between the two. These problems, alongside the striking lack of new insights about nature despite nearly three decades of increasingly sophisticated argumentation, together suggest that modern design arguments are best understood as a degenerative research program protecting a pre-established conclusion.

Three methodological problems

1. Weak models

Design arguments are framed as comparisons between theism and naturalism. In practice, the comparison is much narrower: design arguments pit very specific and poorly substantiated versions of each against the other. On the naturalistic side, fine-tuning arguments typically assume something like a uniform probability distribution over a broad range of values, as if each value were equally likely. But this assumption isn’t derived from naturalism; neither the range nor the probability distribution are based on empirical data or established theories. It’s a convenient foil for theism to defeat. Instead of considering naturalism, design arguments examine a rigid and unlikely form of naturalism.

On the theistic side, the models and their predictions are constructed in a radically different way. Predicting the behavior of a single human is hard, but we can anchor expectations in shared biology, culture, experience, and physical limitations. Divine psychology has none of those anchors, yet design arguments ask us to assess which cosmic features a God would likely create. This requires assumptions regarding divine intent and how it translates into cosmic features. Since divine psychology cannot be independently corroborated, these assumptions are at high risk of being based on the observations that are meant to be explained.

This creates a telling asymmetry. Theistic models, backed by divine psychology, can flexibly accommodate a wide range of outcomes after-the-fact. Whereas specific naturalistic models are rigidly held to high scientific standards of justification before being allowed to favor some outcomes over others. The result is that a rigid, arguably unlikely naturalistic model gets compared against a flexible, arguably unlikely theistic one. This approach weakly translates into meaningful comparison between theism and naturalism, and I will argue that it is structurally biased in favor of theism.

2. Old evidence

When flexible and rigid models are compared on data they were both made to fit, model flexibility almost always wins. A flexible model can easily be adjusted after the fact to fit observations, regardless of its ability to explain them. This is a well-documented problem in model comparison, which applies directly to design arguments.

Design arguments begin with known observations, such as a life-permitting universe, and then ask which hypothesis best explains them. But the theistic model’s core advantage, its ability to easily accommodate a wide range of outcomes via divine psychology, is precisely what makes it suspect here. While theists argue that their model is derived from basic aspects of God, such as goodness or perfection, there’s a serious risk that their model isn’t predicting observations so much as accommodating them. The naturalistic model, meanwhile, is rigidly held to a stricter standard that penalizes it for not already having an ready-made, scientifically-established explanation.

It is well known in statistical modeling and machine learning that a flexible model evaluated against data it was made to fit will almost always look better than a rigid model. Even if the rigid model is the correct model. In other words, comparing between rigid and flexible models on old data creates a bias in favor of flexible models. I will visually illustrate the idea of overly-rigid, overly-flexible, and just right models, and how they translate into predictive performance. First, you can see below how different models fit ‘old data’, which was used for adjusting each models. At first glance, the most flexible model on the right seems to provide a very good fit for the data.

However, when we compare the models’ ability to predict new data, as visualized below, the most flexible model performs the worst. This is because its unconstrained nature allowed it to more flexibly accommodate the old data, giving it an advantage with old data, which was not maintained with the new data. We can see that visually, but also in terms of root mean squared error (RMSE) between observed and predicted values, which is highest (and therefore worst) for the most flexible model.

Okay, so there’s clearly a problem with comparing flexible and rigid models on old data; however, theists argue that their model of God wasn't adjusted to fit the data, and was instead derived independently from God's nature. But notice that theistic models seem only to ‘predict’ observations already in hand, and do not generate predictions about new ones. This convenient pattern is exactly what we'd expect from a model built around old data. Since we can't test theistic models with new data, we can instead ask a related question: do the assumptions used to explain a general fact also fit the finer-grained details of that same domain? A good explanation, which does more than accommodate the facts, should be able to do that.1

3. Understated evidence

If a model explains a general fact rather than accommodating it, we’d expect it to also handle the more specific facts in the same domain without requiring additional unevidenced assumptions. This gives us a useful test for design arguments. If they don’t generate new predictions, we can instead ask: do the assumptions used to explain some general feature of the world also fit the more specific, fine-grained details of that same domain?

They don’t.

Biological design arguments emphasize elegance and goodness in living systems, but biology is filled with poor design, evolutionary dead ends, parasites that blind children, and predators that eat prey alive. A consideration of the specific facts of biology shows a mix of elegance and waste, awe and horror. Is this what we’d expect from a God motivated by elegance and goodness?

Fine-tuning arguments highlight a life-permitting universe, but the universe is overwhelmingly hostile to life. Even on the tiny patch of spacetime hospitable for life, the air, water, and soil are full of invisible things trying to kill you, and the sun causes cancer. Is this what we’d expect from a God motivated by making a universe with life?

Psychophysical harmony arguments emphasize the alignment between conscious experience and physical reality, but human experience is riddled with phantom pain, irresistible delusions, optical illusions, and depraved desires. Is this what we’d expect from a God who intends conscious experience to align with physical reality?

Discoverability arguments emphasize nature’s intelligibility, but most humans throughout history have been deeply wrong about most aspects of reality, science is brutally hard to do well, and religious traditions disagree profoundly about God and the universe. Is this what we’d expect from a God motivated by making the world understandable by its inhabitants?

None of these understated evidence falsifies theism. In each case, the theistic model can be easily adjusted to accommodate these awkward details, and that’s precisely the problem. If your model constantly needs new un-evidenced assumptions to handle evidence in its own domain, it’s a strong sign that the model is not explaining the data and is instead absorbing it, one patch at a time.

What are design arguments doing?

Taken individually, each of these problems weakens design arguments. Together, they compound into something more serious, raising doubt about this whole endeavor.

Weak models mean the comparison was not between theism and naturalism, and was instead between a rigid unevidenced naturalistic model and an unevidenced theistic one free to accommodate whatever it needs to. Evaluation on old evidence then creates a bias in favor of flexible models, as models are only evaluated against observations they can be adjusted-for. And understated evidence reveals telltale signs of accommodation instead of explanation. The result is a methodology structurally biased toward a starting conclusion, because the combination of asymmetric model selection standards, evaluation on old evidence, and post-hoc accommodation creates a ratchet that only turns one way.

A knowledge generating research program would do more than accumulate arguments for the same conclusion that it started with. It would generate new questions, new mechanisms, and new predictions. Good explanatory systems suggest things we didn’t know before, and should now go check. They help us open up the world with new testable mechanisms and predictions rather than just accounting for what we already see. Despite nearly three decades of increasingly sophisticated design arguments, this hasn’t happened. The literature has grown; insights into nature haven’t. No new predictions, no new testable mechanisms, no better understanding of nature. Just more elaborate arguments for an answer in search of a question. These results, considered alongside the 3 issues I raised, are best explained as the output of a degenerative research program, as opposed to a knowledge-generating one. A research program that misuses IBE to reach a predetermined conclusion, as opposed to enabling the discovery of new truths and helping us make sense of our world.

Finishing note: repeated readers might have noticed a pattern in my posts, where I try to turn the methods of theistic arguments against these arguments. You might be wondering: is that my schtick? I do appreciate methods such as IBE and decision theory, and I also think that turning an argument’s methods against itself is a powerful argumentative move. But more fundamentally, these posts are a way for me to push back against misuse of these methods, and to encourage readers to see value in them. So yes, as long as legitimate methods are misused, I’ll do my part in calling that out.

Skeptics think they have the answers. “Static cling,” they say. “Dryer vents,” they protest. Or my personal favorite: “You misplaced them,” as if I am a simpleton. Some even point their fingers at my cat. But these explanations do not wash.

Let’s start with the so-called laws of physics. If socks could simply disappear into some unseen dimension of spacetime, then physics itself would collapse into absurdity. How do they get there? Why socks and not underwear? Why new socks more often than the old? If so-called scientists cannot explain this, then why should we trust their so-called equations?

Next, consider the speculative theory of chance. Skeptics expect us to believe that laundry chaos just sometimes produces odd outcomes. But chance cuts both ways. Why do we never find extra socks in the laundry? Where are all the extra socks that randomness owes me? The skeptics’ house of cards is collapsing under the weight of its absurdity.

As for human error, this is the weakest explanation of all. Are we to believe that trillions of people across the multiverse have all made the exact same mistake, always with socks? Humans make errors, but we are not that consistent. This pattern requires coordination. And coordination, on a cosmic scale, points to a grand design.

And what of my cat? Yes, Spinner sometimes paws at laundry. But this cannot explain why socks vanish in homes with dogs. Furthermore, hiding my socks is a risky and difficult business, which brings no benefits. If cats behaved this irrationally, they’d be long extinct! Another explanation bites the dust.

Where does this leave us? The skeptics’ basket is empty, and their excuses have dried out.

The cleanest explanation is the one skeptics refuse to consider: The Divine. Just like our Lord, socks vanish into the Heavens. This serves to remind us of His truth and His glory, while leaving enough space for genuine faith, love, and freedom. God has woven His intent into the very fabric of reality. Socks disappear by neither chance nor error, but because Providence is tugging at the threads of Creation.

Dismiss it if you wish. Call it coincidence. But the truth is out in the open, beckoning those with open hearts and minds: the laundry tells a story. We just have to listen. Doubt may stain the cloth of reason, yet it will not outlast the final rinse.

Instead of making one prediction and sharing it with all 160 recipients, I decided to make two predictions per match, and share only one of them per person. So, in match 1, 50% of emails will predict that team A will win, and 50% that team B will win. I will repeat this process over the 5 matches, resulting in 5 recipients who will receive perfect predictions. I will then email the lucky 5 fans, asking them for a fee in return for more predictions. The scam is visualized below:

We can understand why the 5 lucky basketball fans would feel inclined to pay for more predictions. 5 out of 5 is pretty impressive. However, the process by which I generated my predictions – what statisticians call the data generating process – does not involve any special ability to predict basketball matches. All that is needed for my perfect predictions is a selection effect that operates on my variety of guesses.

The miracle of selection

When we are trying to compare explanations for an event, we are in effect considering different data generating processes. Be they divine or more mundane. Therefore, we should consider the presence and explanatory role of selection effects. This is highly relevant to miracle claims, because a) there have been many miracle claims throughout history, and b) proponents of miracle claims tend to focus on a small subset of these claims. The questions, then, are what is selected-for, and whether this process could generate our observations.

Let’s consider the case of authenticated holy relics. There have been many fake relics throughout history, so people started using modern scientific tests to authenticate relics. We can grant that the testers conducting these tests are generally honest and capable. However, testers will sometimes report misleading findings. This could be due to technical issues such as sample contamination and measurement error, or bad practices ranging from selective reporting to outright fraud. Furthermore, holy relics authenticated by scientific tests are more likely to get attention than those disconfirmed by them. And voila! We have selection operating on natural variation, since we can expect mistaken authenticating tests to happen, and can expect them to get more attention.

Let’s assume the test authenticating a relic has been done perfectly. However, there are methods for faking this relic that would fool our test. If these methods are sufficiently difficult and rare, we might rhetorically ask ‘did the relic’s ancient forgers have modern tests in mind?’ Ignoring the obvious selection effects. People have faked many holy relics in many different methods, for different reasons, and under different constraints. As such, even strange and unusual forgeries exist. Since we selectively focus on relics that have been scientifically authenticated, we again have selection operating on natural variation. We can expect rare forgeries that can fool scientific tests to happen, and we can expect them to garner more attention than forgeries that cannot fool the tests. This requires neither miracles, nor time-traveling forgers.

We can also apply selection effects to eyewitness accounts of miraculous events. The main steps are: a) we have many false or mistaken miracle accounts by eyewitnesses, b) there is natural variation in how trustworthy eyewitnesses appear, and c) we selectively focus on accounts from seemingly trustworthy eyewitnesses. Again, we get selection acting on natural variation, resulting in incredible yet incorrect eyewitness accounts.

Comparing explanations

As I highlighted in my previous post, people can behave in very weird and unpredictable ways. Even the best established psychological rules have many exceptions. If you don’t believe that, then look up Florida Man. The natural weirdness and variation of human behavior, when combined with selection effects, can easily create the impression of miraculous-seeming events. Just like my random basketball guesses can selectively result in perfect predictions. When natural variation and selection effects apply, we will encounter extremely rare and seemingly difficult-to-explain events, even if no miracles apply. As such, how could a miraculous explanation ever overcome an explanation based on selection effects?

To overcome selection effects, competing explanations need to make sense of the specifics of the given case. When dealing with a holy relic, we can ask: why is this specific relic so unusual? If it’s about an experience, we can ask: why is this specific experience so unusual? These questions are not answered by selection effects, as selection explains why surprising cases appear to us, but not why this particular case occurred. The problem is that, from non-believers’ perspective, case-specifics are not answered by theological explanations either.

From an atheist’s perspective, it is reasonable to see theology as a “subject without an object”. Where theological assumptions are about as credible as reading tarot cards or sheep entrails. From this perspective, in rare cases where theological assumptions are clear enough to generate predictions, they are consistently wrong. More often, they are conjured to generate postdictions (‘predictions’ made after the fact) based on charismatic believers’ vibes, and made to fit the facts-to-be-explained. I am not trying to be demeaning; I just want to clarify that, for an atheist, theological explanations are vacuous.

So no, non-believers are not obligated to accept theological assumptions for explaining the specifics of an event, even if these assumptions make for nice stories. Until there are better ways to generate and evaluate theological assumptions, I don’t think that non-believers should accept miraculous explanations where observations could be explained by selection effects acting on natural human weirdness. One may disagree with this perspective, but it is difficult to deny that explanations involving selection acting on natural variation are incredibly powerful; often replacing theological explanations, and never being supplanted by one. Unless proponents of miracles seriously contend with the explanatory power of selection effects, and justify their theological assumptions, they should not expect non-believers to be convinced.

In this short post, I will provide an example of how psychological research operates under ideal conditions. Using that, I will make the case that psychological research relies on average behavioral tendencies, leaving considerable room for ‘psychologically implausible’ behavior by individuals. This means that even the best psychological research does not justify the ruling out of unusual individual behavior, leaving most folk-psychological attempts at doing so dead in the water.

The flanker effect

The flanker effect is one of the best-evidenced effects in psychology. Having been discovered in 1974, it has been studied, replicated, and extended in many ways since. The flanker effect can be studied in a simple setup, as depicted below, where participants are asked to press one key when the central arrow points left, and another key when the central arrow points right. Crucially, the central target is surrounded by ‘flankers’, or distractors, that sometimes match the central arrow (congruent trials), and sometimes mismatch it (incongruent trials). This results in a flanker task.

What happens when research participants complete a flanker task? Their responses are consistently slower and less accurate in incongruent trials compared to congruent trials. Does that mean that we can ‘rule-out’ a person -not- showing a flanker effect?

What the average means

Luckily, my own research can help answer this question. Having studied the flanker effect with more than 1,000 research participants and having published multiple scientific papers about it, I can confidently say that it is a real and robust psychological phenomenon. However, we absolutely cannot rule-out that a person did not fall for the flanker effect.

To understand how this could be, let’s consider how psychological1 research is done. In my case, I had many participants complete many trials of the flanker task. I then2 1) calculated each person’s mean response time per type of trial, 2) calculated the differences between each person’s mean per trial, and 3) calculated the mean of these differences & the uncertainty around it. The resulting flanker effects, based on 180,172 trials completed by 925 research participants, are visualized below.

As we see, a clear flanker effect emerges. The plot highlights the mean flanker effect of 57 milliseconds, which is surrounded by a 99% confidence interval3 ranging from 53 to 60. This effect is about 40 σ away from what we would expect by sampling error alone (5 σ is the standard of evidence for discovering a new particle in physics; behavioral & medical research usually uses 2 σ). The flanker effect is legit.

The flanker effect is one of the best-evidenced psychological phenomena. Nevertheless, 27, or 2.9% of participants did not show a flanker effect. So, would a psychological explanation that requires someone to -not- show the flanker effect be so implausible that we should rule it out in favor of a miraculous explanation? I don’t think so, and neither should you.

Zooming out

As you may have noted, the flanker effect relies on very simple behaviors, which are observed in very tightly controlled environments. I will leave it for you to imagine how much more difficult & uncertain it is to make psychological generalizations regarding complex behaviors. Especially those performed in messy day-to-day reality. If we cannot rule out psychologically implausible behavior in the flanker task, then what hope do we have for doing so with complex real-life scenarios?

My point in this post was to illustrate how psychological research relies on average behavioral tendencies, which leaves plenty of room for deviations by individuals. Psychological research can be very good at predicting average behavioral patterns; yet even at its best, it does not rule-out major deviations by individuals or small groups. Such that individual deviations from even the best-established psychological patterns are a serious possibility, and a common reality. If someone wants to argue that such a deviation is less plausible than a miracle, I wish them good luck. But if they simply ‘rule out’ psychologically unusual behavior, then I feel justified in ruling out the value of their opinion.

I divided this post into short parts. First, I will outline the concept of expected value (EV) and how it can be used to guide our actions. Then, I will consider what happens when we apply EV to infinite outcomes. Building on these initial outlines, I will argue that a Pascalian decision-theoretic analysis of infinite experiential outcomes is undermined by… a decision-theoretic analysis of this whole thing. So buckle up, because we’re going meta.

1. Expected value

A basic concept in decision theory is expected value (EV). To simplify things: EV considers the value of each possible consequence against its probability. Since you want to get the best consequences, you find which action has the highest EV, and go with it. Let’s make it concrete:

Imagine that I asked you to give me $10, with the promise that you’ll immediately get $20 back. If we apply a simple, EV-based analysis (ignoring messy psychological reality), it could look like this:

If you do not loan me $10, you are guaranteed to keep the money. So

meaning that you’ll always end up with $10.

Based on online reviews, you know that in 60% of cases you’ll get $20, while in 40% you will get nothing. Making

So, if you loan me $10, you can expect a higher value ($12) than what you can expect otherwise ($10). Since we are assuming that you seek value maximization and are impervious to messy psychological factors such as loss aversion, the rational action for you is to give me $10. Why, thank you.

The example is silly and simplistic, but the core idea of considering and balancing consequences by their probabilities is extremely powerful. It has been applied in areas ranging from Artificial Intelligence to medical decision-making and business management. EV is legit.

2. Infinite value?

We only considered finite consequences in our EV analysis. But what happens when dealing with infinite consequences? This is what Pascal’s argument touches on. The basic intuition is that as probabilities are between 0 and 1, and outcome values can be infinitely positive (∞) or negative (-∞), an action with the mere possibility of an infinite outcome dominates all other options. This infinity-dominance is expressed mathematically as:

as long as p, the probability of the outcome, is bigger than 0.

This has something going for it. I would avoid assigning a probability of exactly 0 to a proposition that does not clearly contradict itself. Why? Because 1) no evidence completely falsifies a proposition (Duhem-Quine etc.,), and 2) no evidence can increase a probability of 0. Simply put, evidence cannot lead you to assign a probability of exactly 0, and evidence cannot lead you out of a probability of 0. Personally, I’d avoid assigning probabilities that are not evidence-based and cannot be changed by evidence.

The preceding means that if Islamic theology promises heaven (∞ value) for belief, and hell (-∞ value) for disbelief, then I cannot assign a probability of exactly 0 to this theology. So I have to take the threat of infinite consequences seriously. Is it time to join a local mosque?

Not quite. Infinity is not a real number, so using it moves us beyond the realm of standard mathematical analysis. If we are applying non-standard analysis to outcomes, why can’t we do the same to their probabilities?

I’ll make a fairly intuitive claim: the prior (initial; before specific evidence) probability of a given outcome should be proportional to its magnitude. For example, the more money I propose in return for your $10, the more skeptical you should be. Returns of $10, $1,000, and trillion dollars should encounter increasing levels of skepticism. Simply put, the more I promise, the more difficult it is to deliver.

Let’s consider an example of keeping prior probabilities proportional to values. If going to heaven brings a value of X = ∞, I can assign the probability of going to heaven as inverse to the magnitude of its value (p = 1/X). Then, with some limits math we get:

Now, the probability takes an infinitesimally small yet non-0 value, leading to a finite value. Infinite possible outcomes no longer overwhelm our decision-theoretic analyses.

When working with decision theory, we need to decide how to assign prior probabilities based on the magnitudes of outcome values. Ignoring this is itself a decision, if a poorly-thought-out one. If we take seriously the idea that prior probabilities should reflect the size of possible outcomes, we arrive at two broad ways of assigning them. Specifically, when X stands for the outcome’s absolute value:

As we previously saw, Rule 11 neutralizes infinite but un-evidenced consequences. In contrast, Rule 2 lets infinite consequences overwhelm the EV because the expected value will practically always be infinite. While I think there are good epistemic reasons to prefer the first rule type, I find the decision-theoretic reasons for it to be decisive. Since Wager-style arguments require decision theory to guide actions, yet rely on Rule 2, they are undermined by the very framework they invoke.

3. Selecting assignment rules based on decision theory

The way in which prior probabilities correspond to outcomes’ magnitudes matters. I proposed two rules for doing so. If we take Rule 1, then the EV of actions that could have infinite consequences will be evidence-dependent. Whereas in Rule 2, the EV will be infinite regardless of evidence. Now, I will consider which rule should be applied in decision-theoretic contexts.

We divided decision-theory into two families, based on Rules 1 and 2. Now, let’s divide the world into two families where decision theories could be used.

World 1: where humans cannot have infinite experiential outcomes (suffering and bliss must be finite)

World 2: where humans can have infinite experiential outcomes (infinite suffering or bliss)

While I find World 1 to be much more likely than World 2, we don’t need to consider that here. All we need is to compare the EVs of using Rule 1 vs Rule 2 per World. This, I argue, shows that Rule 1 decision-theoretically dominates Rule 2.

3.1 World 1: humans cannot have infinite experiential outcomes

In this World, Rule 1 trivially dominates. Rule 1 ensures that decision-making remains sensitive to evidence and finite payoffs, whereas Rule 2 would only distort our reasoning.

3.2 World 2: humans can have infinite experiential outcomes

In this World, Rule 2 could in principle be applied without being misleading, but it offers no clear advantage over Rule 1. A proponent of wager-style arguments needs to at least show that Rule 2 dominates Rule 1 here. But I don’t think they can.

It's trivially easy to come-up with coherent propositions that involve infinite outcomes. We have examples of heaven & hell in competing religions such as Islam, Christianity, and their different denominations. We can also have atheistic versions. Maybe a non-minded force, like karma or AI, will infinitely reward atheists and punish theists? Since we can easily come up with not-impossible infinite EVs for opposing actions, Rule 2 leads to a broken decision theory. Our actions are insensitive to evidence, and the machinery of decision theory is paralyzed by conflicting actions involving infinite outcomes. While this makes me suspect that Rule 1 also dominates World 2, I don’t need more than parity to get the argument across.2

4. Applying decision theory to a decision theory

As I highlighted, the probability assignment rules that we use determine how infinite outcomes affect our decision-theoretic toolset, so we should be careful with them. We can pick rules that keep EVs finite and evidence-dependent, or rules that let them get infinite and insensitive to evidence. I argued that we have clear decision-theoretic reasons to pick rules that keep EVs finite and evidence-dependent. This means that Wager-style arguments rely on a defective probability assignment rule, and are therefore defeated by decision-theoretic considerations.

Now, some may not agree with my analysis, and that’s fine. The onus here is on arguing for the decision-theoretic benefits of probability assignment rules that let infinite outcomes dominate EVs. And that’s not something I’m willing to bet on.