Last week, in the first installment of this essay we examined the application of probability analysis to the existence or non-existence of God with reference to Pascal’s Wager. This week, in part two, we will examine a more exotic application of probability theory to theological questions.
We will examine, in particular, Bayesian theory – named for the 18th century mathematician Thomas Bayes who invented it. While it is well-known that Bayesian theory now lies at the heart of modern probability theory, and thus is foundational for modern statistical theory, computer science, and even artificial intelligence, here we examine the curious, and surprising fact that Bayes actually created his theory to disprove David Hume’s arguments against miracles.
David Hume by Allan Ramsay (1754)
The Preacher Who Used Math to Defend Christianity
Some time in the 1740s or 1750s a Presbyterian minister by the name of Thomas Bayes sketched out an interesting mathematical argument and put it back in his desk. It stayed in that desk until after Bayes died in 1761. In 1763, Bayes’ assistant and protégé Richard Price, a radical nonconformist minister, finally published the paper through the British Royal Society. The result has since become known as Bayes’ Theorem, and it is one of the most important findings of statistical science — indeed, as we have noted, it’s one of the core engines of contemporary Artificial Intelligence.
What is fascinating about Bayes’ Theorem, however, is that those scientists, engineers, and computer programmers that use it every day seem mostly unaware of why it was invented.
Reverend Bayes never saw himself as coming up with a tool for sophisticated engineering, rather he saw the argument that he sketched out as proof that the miracles of the Bible – especially of the New Testament – could easily have taken place. In this regard, Bayes came up with his Theorem to counter the arguments of sceptics like the Scottish philosopher David Hume. If this were not interesting enough, it should be noted that contemporary philosophers of science – even those who are not themselves religious – recognise that Bayes and Price had the better of the argument. In fact, they tore Hume’s infamous work discrediting miracles – a work that is still regularly cited today – to pieces.
Sadly, the debate that took place in England in the 18th century on miracles is now largely forgotten. This is strange as it is something that people still talk about extensively. Recalling my own childhood and early adolescence, perhaps some of the earliest ontological questions that I discussed with others were about whether miracles were possible, whether ghosts might exist, and whether there might be a supernatural. No doubt that many readers will relate to this as such discussions are extremely commonplace and have never lost their fascination. The fact that supposedly ‘educated’ people no longer address such questions simply shows that the quality of education has fallen in recent years. Sceptical people have not been taught to debate such matters rationally, rather they have been taught to summarily dismiss them as being silly – a sort of intellectual class prejudice, of sorts. This degradation of the minds of the educated in our societies seems to overlap with a general deterioration of education – especially non-science and mathematics education – into imposing liberal dogmatism of the shallowest sort onto people and assuring them that they need not know anything else. This ‘education’ is typically supplemented with popular and semi-popular books that anyone technically trained in philosophical questions would find boorish and ridiculous.
Let us take a trip even further back to a time, then, when education was more robust. The terms of the debate around miracles were laid out by John Locke in his late-17th century work Essay Concerning Human Understanding. Here Locke distinguishes between two types of evidence. One he called “common observation in like cases”. We might think here of the famous example of the sun rising. We “commonly observe” the sun rising, and it is always a “like case”. These sorts of observations allow us to establish lawlike relationships between various events. When we hit the ball, the ball moves in a certain direction at a certain speed reliably. Sciences like physics then create general formulae to describe these “common observation in like cases”. The second type of evidence that Locke discusses is “particular testimonies”. These testimonies are particular in so far as they do not fit in with the general trends of “common observation in like cases”. Here we might think of a person or persons saying that they woke up one morning in a location where the sun has reliably risen since anyone can rememberand the sun did not rise. Or that they hit a ball, and the ball simply disappeared without a trace. Such instances rub against the “common observation in like cases” and so rub against scientific laws in general. These instances we deem miraculous.
Hume’s entire chapter entitled ‘Of Miracles’ in his 1748 book An Enquiry Concerning Human Understanding is an attempt to deny the validity of “particular testimonies” and thereby disprove the existence of miracles. While Hume’s chapter has proved very popular since he wrote it – and is very likely at the root of the snobbery that is currently passed off as education on the matter – recent analytical examinations have shown it to be greatly wanting. Recent analyses of the chapter show that it does not contain much in the way of strong arguments. Rather it contains a series of somewhat flimsy arguments that are tossed together with rhetorical flourish and cynical sneering to give the illusion that something has been demonstrated. Here is a classic quote from Hume’s chapter:
A miracle is a violation of the laws of nature; and as a firm and unalterable experience has established these laws, the proof against a miracle, from the very nature of the fact, is as entire as any argument from experience can possibly be imagined.
Time and again, many of Hume’s arguments appear tautological – of the form: “miracles do not happen because the laws of nature always hold, and the laws of nature always hold because miracles do not happen”. We might isolate something resembling a positive statement if we phrase Hume’s argument in terms of the particular and the universal: “what most people perceive with regularity most of the time, all people perceive with regularity all of the time”. But this sentence is obviously incorrect: just because most people perceive something regularly most of the time does not mean that all people perceive that regularity all the time. Perhaps it is best to focus in on Hume’s argument against the testimony that miracles have taken place (even though this argument ultimately rests on the aforementioned tautological arguments):
That no testimony is sufficient to establish a miracle, unless the testimony be of such a kind that its falsehood would be more miraculous, than the fact which it endeavours to establish.
But even here it is not clear what Hume really means. When he says that the falsehood of a statement would be “more miraculous” than the statement that something miraculous happened, he seems to be alluding to some sort of scale of miraculousness. Yet a miracle is a miracle – it occurs when the laws of nature are violated. It is hard to see how there is any sort of scale here. Perhaps we might say that a testimony of a miracle might be believed if its contrary implied that multiple miracles occurred, but this seems different to what Hume is trying to convey. In truth, Hume seems to be dressing up a very simple statement in more complex terms to hide the fact that the statement is simple and not all that compelling. That simple statement runs something like this: “miracles are events that occur in contrast to what we would typically expect from experience; because of this, when a miracle is reported we have to be extra cautious about the credibility of the testimony.” Of course, this is no different from the position of, say, the Catholic Church when it tries to verify any given miracle. This really is the best that Hume has to offer.
Thomas Bayes’ Miraculous Calculator
It is here where Bayes enters the picture and with him probabilities. Bayes wants to discuss the weight of testimony as evidence, and he wants to use probability analysis to do so. Here he is explicitly following Keynes: Bayes is not actually interested in the probability of whether a miracle took place – that is impossible to tell because the miracle has happened and will not happen again – but rather the probability that someone’s statement that the miracle took place is accurate. We are not here discussing the probability of events, but rather the probability of arguments – the difference is subtle, but extremely important. What Bayes’ Theorem shows is that while one report of a highly improbable event might signify a very low probability of that event having actually occurred, multiple independent reports of a highly improbable event greatly increase the probability that this event occurred – despite the event itself being improbable.
This can be shown mathematically. But first let us indicate how it fits with common sense. If a single person tells you that he was in a room the previous day when a chair started floating in thin air you would immediately think that he was either lying or delusional. But if he then introduces you to ten other people who report the same event, at the very least that makes you reconsider whether it may have happened. The probability that they all experienced the same delusion at the same time is intuitively implausible. The probability that they are colluding and lying to fool you is admittedly higher – but the probability that eleven people are lying is intuitively lower than the probability that one person is lying. Let us turn briefly to the mathematics plugged into Bayes’ Theorem, although we will explain what is happening in the equation for those unfamiliar with it. Here is the equation as laid out by John Earman in the paper referred to earlier:
We need not read the whole equation. Just focus on the p and the q in the right-hand lower corner. The p and q are simply as follows:
p = the probability that the testimony of the miracle proves the miracle.
q = the probability that the testimony of the miracle does not prove the miracle.
Here is Earman’s summary of the argument when it is plugged into Bayes’ Theorem:
The witnesses may be very fallible in the sense that q can be as close to [certainty] as you like. But as long as they are minimally reliable in the sense that p > q, it follows from that the posterior probability of [a miracle having occurred] can be pushed as close to [certainty] as you like by a sufficiently large cloud of such fallible but minimally reliable witnesses provided.
Stated differently: unless you can argue, on its own terms, that the witnesses are more likely to be either lying or delusional than not (i.e. q > p) then it follows that even if what they are saying seems highly improbable, the more independent witnesses that claim that the miracle occurred, the more probable that the miracle in fact occurred – and indeed, with a certain number of independent witnesses, the probability that a miracle did in fact occur becomes very close to being a certainty and because the number of witnesses is subject to an exponent (see the ‘N’ term in the bottom right of the equation) as the number of witnesses rises linearly, the probability of the miracle being true rises exponentially. This is what Baye’s Theorem shows – and, unlike in the case of Hume’s rambling chapter, it shows it with precision and certainty. We can disagree with the data being inputted – for example, we can say that the witnesses are not independent, and they are colluding – but we cannot disagree with the nature of the argument Bayes is putting forward. Here is a framework that not only shows how you might assess miracles, but most importantly, here is a framework that shows definitively that miracles can indeed exist.
Today Bayes’ refutation of Hume’s arguments against miracles is being used to produce technology that materialists and atheists gawk at as being quasi-miraculous. People speak in hushed tones about Artificial Intelligence and attribute metaphysical properties to a technology based on statistical correlation and probability calculus that it in no way deserves. Is this a coincidence? It is hard to think that that it is.
Bayes’ Theorem has an air of magic about it, as can be attested to by anyone who has studied it. Collecting evidence and feeding it into Bayes Theorem produces conclusions that are impossible to reach by simply holding the evidence in one’s mind. When we program a computer with the Theorem it can be used to recognise our face and unlock our electronic devices, or it can recognise our voice and convert this into text, or it can translate one language into another. These are not miracles, of course, and when one fully understands the technology, it is revealed to be no more than impressive engineering. Bayes’ Theorem does not truly produce miracles. But as we have seen, it establishes perfectly well that miracles are within the realm of the possible. We would be better off considering deeply the implications of the Theorem itself than getting distracted with the impressive gadgetry that it has been used to produce. It makes for a far more compelling philosophical argument.