Hume, Bayes, and the Criteria For Miracles
Trading off false-positives and false-negatives
Hume has dominated how we think about miracles for more than 250 years. His popularity has not waned among atheists in our time. For example, Bart Ehrman takes a Humean approach to Biblical miracle claims.[1] Cambridge University philosopher Arif Ahmed’s famous debate on Jesus’ resurrection with Gary Habermas essentially became a debate about Hume.[2] His influence is also seen in atheist YouTubers like Paulogia[3] and Matt Dillahunty.[4]
Christian responses are sometimes dismissive of Hume. But grappling with him can help better understand why many atheists reject miracles.
Some aspects of his work are a helpful corrective. We want to minimize ‘false positives’ — believing miracles based on insufficient data. Yet, Hume’s criteria are vulnerable to ‘false negatives’ — denying miracles despite sufficient data.
In statistics, we are constantly trading off these risks. If our criteria are too restrictive, we risk false negatives. But overly liberal thresholds can lead to false positives. Bayes’ rule has for centuries helped us with this dilemma.
Hume’s criteria for accepting a miracle
Hume’s famous maxim is reasonable:
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.[5]
Atheist philosopher JH Sobel[6] interpreted this to mean the prior probability of a miracle (A) must be greater than the probability of testimony (a) about a miracle (e.g. Jesus’s followers claiming that his tomb was empty) when the event didn’t actually happen (~A):
Sobel showed these criteria are consistent with Bayes’ rule:
· when the probability of testimony is not impossible (p (a) > 0) and
· a miracle is more likely than not based on this testimony p(A|a > 0.5) then
· probability of miracle must be greater than the probability that the testimony is false: p (A) > p (a|~A).
Hume on miracles: possible or impossible?
Hume’s position on the possibility of miracles is challenging and much debated by scholars. But clearly he considered all miracle claims in history were based on insufficient evidence:
For first, there is not to be found, in all history, any miracle attested by a sufficient number of men, of such unquestioned good-sense, education, and learning, as to secure us against all delusion in themselves; of such undoubted integrity, as to place them beyond all suspicion of any design to deceive others; of such credit and reputation in the eyes of mankind, as to have a great deal to lose in case of their being detected in any falsehood; and at the same time, attesting facts performed in such a public manner and in so celebrated a part of the world, as to render the detection unavoidable: All which circumstances are requisite to give us a full assurance in the testimony of men.[7]
This has led some to think Hume assumed miracles were impossible. Or, at least, sufficient evidence for them was not possible. But he denies both:
…there may possibly be miracles, or violations of the usual course of nature, of such a kind as to admit of proof from human testimony.[8]
A better route is to assess whether his argument is internally consistent. Hume claims all miracles have shown to be false. Or, at least, something rational people should not believe. Which makes him sound like a rationalist. But, as an empiricist, he is keen to show sufficient evidence is at least possible.
Should we accept Hume’s theoretical miracle?
Hume’s theoretical example, is that in 1600 there were eight days of total darkness:
Thus, suppose, all authors, in all languages, agree, that, from the first of January 1600, there was a total darkness over the whole earth for eight days: suppose that the tradition of this extraordinary event is still strong and lively among the people: that all travellers, who return from foreign countries, bring us accounts of the same tradition, without the least variation or contradiction: it is evident, that our present philosophers, instead of doubting the fact, ought to receive it as certain, and ought to search for the causes whence it might be derived.[9]
He proposed the following testimonial data were sufficient to believe this event happened:
· all authors, in all languages, agree that it happened
· strong tradition remains till Hume’s day (148 years later)
· All travellers to foreign countries return testifying that other countries have the same tradition without variation or contradiction
We should not assume the possibility that God caused this miracle. So the following data are available for the prior for eight days of darkness, p(A):
· Approximately 45 eight-day periods in a year (365/8)
· Astronomical records going back approximately 2,600 years (since 6th century BCE)
· Approximately 100,000 eight day periods (45x2600) where we have astronomical data
Hume argued there was insufficient evidence for all miracle claims in his day. It is likely he would have concluded the same about miracle claims in the proceeding 250 or so years following Of Miracles. Therefore a prior probability of 1/100,000, p(A)=0.00001 seems a fair estimate.
I also assume the testimonial evidence proposed by Hume is strong — 100% certainty. If the miracle happened we would expect to see that type of testimonial evidence, p(a|A)=1.
The probability of observing this evidence is also unlikely if there wasn’t a miracle. But how unlikely? One of the benefits of Bayes’ rule is that multiple independent testimonies cumulatively increase certainty. One eccentric author might claim to have witnessed a strange event. But multiple independent authors, in multiple languages, on the same event is less likely.
There is an important trade-off. Certainty cumulatively increases with consistent testimony from many independent witnesses. But unanimity of agreement may also cumulatively increase suspicion of collusion. For example, legal scholar Thomas Starkie pointed out we expect minor variations in testimony:
It so rarely happens that witnesses of the same transaction perfectly and entirely agree in all points connected with it, that an entire and complete coincidence in every particular, so far from strengthening their credit, not unfrequently engenders a suspicion of practice and concert.[10]
Additionally, humans are often contrarian, we rarely agree 100% on anything. Philosophers have debated whether we can assume the law of non-contradiction is true.[11] People have written books denying the holocaust, the ellipsoid shape of the earth, and the moon landings. Events with strong evidence.
Could the absolute consistency in Hume’s example give us pause for doubt? People with power able to delete dissenting voices? We cannot rule out the possibility, but it seems unlikely. Who would gain from doing this? We don’t know enough from Hume’s illustration to tell.
Giving him the benefit of the doubt, we assume it extremely unlikely to observe this evidence, if the event didn’t happen, perhaps 1/1000, p (a|~A) = 0.001.
Plugging these values into Bayes’ rule leads to a posterior probability, p(A|a)= 0.01 (1/100).[12] Hume’s maxim leads to the same conclusion. The prior probability, p(A)=0.00001, is less than the probability of observing the testimonial evidence if no miracle occurred, (a|~A=0.001). We should not believe the miracle.
This suggests application of Hume’s criteria leads to false negatives. If his best example of sufficient evidence leads to a judgement of ‘extremely unlikely’ — then it is unsurprising he considered all other testimony insufficient.
A neutral definition of miracles?
Where does Hume’s example go wrong? The evidence is strong, we can’t increase p(a|A), since it is as certain as can be. It doesn’t seem possible to reduce p(a|~A) any lower than 0.001, as there is a limit to how much certainty can accumulate before raising suspicions of collusion.
So the prior probability seems to be the only parameter we can modify. Hume’s definition of miracles may lead to an over-inflated prior:
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.[13]
Hume suggested the laws of nature and miracles are competing explanations. An early example of methodological naturalism. Gregory Dawes, an atheist philosopher, notes that the term ‘methodological’ can be misleading:
After all, what does this so-called “methodological naturalism” entail? It demands that we investigate the world etsi Deus non daretur: as if there were no supernatural causes. And this means investigating the world as though ontological or metaphysical naturalism — the belief that there are no supernatural causes — were true. It follows that their naturalism is not “merely” methodological: it adopts, at least for the purposes of explanation, a working ontology, a set of assumptions about what kinds of entities are likely to exist.[14]
So a methodological naturalist estimates the prior probability of a miracle based on how often we experience the laws of nature to be fixed. For example, the laws of nature in our universe determine that when we die our bodies decay. There are no known ‘natural’ exceptions to this law. So miracles are virtually impossible.
Although “acting as if metaphysical naturalism is true” is technically provisional, it is unclear whether such an assumption can be revised in practice. For example, we’ve seen with Hume’s example, not even the strongest evidence can overcome the prior.
Are laws of nature and miracles necessarily in conflict?
Peter Harrison (a historian of science) argued methodological naturalist definitions imply:
…an unproblematic distinction can be drawn between ‘natural’ and ‘supernatural’, and that this distinction was routinely operative in the history of science. This turns out to be mistaken.[15]
To rescue Hume’s empirical argument, we need to investigate alternatives to methodological naturalism. For example, Christian theists like John Polkinghorne, formerly professor of mathematical physics at Cambridge University, consider the laws of nature reflect God’s constant activity in the world. Therefore, these laws:
…are not the grain against which a wonder-working deity occasionally acts, but their regularities are the pale reflection of the faithfulness of the Creator.[16]
On this definition, the laws of nature reflect God’s regular activity upholding the universe. Since these laws depend on God, he is free to change how these work on isolated occasions. If there is a God, it is at least possible, he could choose to depart from his usual operation of the regularities of the universe.
In evaluating the evidence for miracles, we must trade-off two opposing risks: false positives and false negatives. Hume is right, we can be too credulous about miracles sometimes. But his approach risks errors in the opposite direction. A prior taking into account uncertainty about the truth of metaphysical naturalism provides the best balance for limiting false positives and false negatives.
Notes
[1] Ehrman B. How Jesus Became God. Harper One; 2014
[2] Ahmed A, Habermas G. Did Jesus Bodily Rise From the Dead?
https://www.youtube.com/watch?v=Mg7rYJxHA4Y
[3] Paulogia, McLatchie J. Did Jesus Rise From the Dead?
https://www.youtube.com/watch?v=T3WgGV1njJU
[4] Dillahunty M. Atheist Debates — Thoughts on Hume’s axioms for today.
https://www.youtube.com/watch?v=j3NdzLCNI-E
[5] Hume D. An Enquiry Concerning Human Understanding; 1748
[6] Sobel JH. Hume’s Theorem on Testimony Sufficient to Establish a Miracle. Philosophical Quarterly 1991; 41:229–237.
[7] Hume, ibid.
[8] Hume, ibid.
[9] Hume, ibid
[10] Cited in McGrew T, McGrew L. The Argument from Miracles:
A Cumulative Case for the Resurrection of Jesus of Nazareth. In W Lane Craig, JP Moreland (Eds.), The Blackwell Companion to Natural Theology. Blackwell; 2009.
https://www.lydiamcgrew.com/Resurrectionarticlesinglefile.pdf
[11] Feyerabend P. Science in a Free Society. Verso Books; 1978.
[12] p(A|a) = p(A) x p(a|A)/ p(A) x p(a|A) + p(~A) x p(a|~A)
[13] Hume, ibid.
[14] Dawes GW. In defense of naturalism. International Journal for Philosophy of Religion 2011; 70 (1):3–25.
[15] Harrison P. Naturalism and the success of science. Religious Studies 2020; 56: 274–291.
[16] Polkinghorne J. The Faith of a Physicist: Reflections of a Bottom-up Thinker. Augsburg Fortress; 1996.
