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Abstract

  1. Top of page
  2. Abstract
  3. Attrition of believability
  4. The vanishing of historical probability
  5. References

A friend of Isaac Newton asked how strongly we should believe ancient history. His calculations gave a date for the end of the world. Julian Champkin looks at a little-known foreshadower of statistical thinking.

The world will end in the year of our Lord 3150. Belief will have all but vanished from the earth; and the Second Coming will be nigh.

Mock not. The date was worked out by John Craig, occasionally spelled Craige, mathematician and friend of Isaac Newton. He did the calculation in the year 1696; and if the statistics behind his calculation were not totally sound, they were certainly pioneering. And they were based on premises that were at least reasonable, and that few would argue with today.

How do we judge the believability of events that we have not witnessed ourselves? This was Craig's theme; and he applied rational judgements to it to work out the date of the end of the world. His contemporaries mocked him; but he has aroused the admiration of no less a historian of statistics than Stephen Stigler.

The probability of a long-ago event having happened as described to us depends, among other things, on the reliability of those who witnessed it and described it. The longer ago the event is supposed to have happened, the less certain we can be about it. Those would seem reasonably uncontentious statements. Memories fade, eye-witnesses grow old and die; the past becomes another country.

A first-hand report is better than one at second- or third-hand. A recent witness is better than one from long ago, and a live witness is better – or potentially more believable – than a dead one. That, too, seems reasonable. We can cross-examine a live witness, and gain an impression from his or her answers and demeanour of how likely it is that he or she is telling the truth. More: if the witness died within living memory there may be people still around who knew him or her, and whose judgement we in turn may form judgements upon. If the witness died long ago, all we may have is written records, perhaps copied out several times in succession before they reach us. Texts become progressively more garbled and corrupt the more often they are copied (true at least in the age before digital printing; and quite possibly true even in the age after it). Documents copied laboriously by monks or scribes gain errors; a copy of a copy of a copy will have error upon error upon error. Classical scholars spend lifetimes analysing fourth- or fifth-generation manuscripts to try to work out what exactly the ancient poet they are studying actually wrote. The decay is even faster of course if the story is passed down orally; the game of Chinese whispers is based on exactly that.

The more witnesses there are to an event, the more certain we can be about it. But the longer ago it happened, the less certain we can be

These were the considerations to which Craig applied his reasoning. Just for starters he reckoned that a written record from a primary historian was worth ten times as much as one passed on by word of mouth. Even a written account lost 10 per cent of its credibility every time it was copied. He went on to deduce remarkable things.

Craig was born around 1663 in Scotland, at Hoddam in Dumfries where his father was a priest. Little is known of his life. He graduated in 1687 from the University of Edinburgh; while an undergraduate he had journeyed to Cambridge to meet Newton, who read one of his papers and allowed Craig to read one of his own. After Edinburgh he became a clergyman, with parishes in London and in Wiltshire, and became a canon of Salisbury Cathedral, but was friendly with Halley and De Moivre and corresponded with Scottish mathematicians such Maclaurin. After Newton's death he strongly defended him against allegations of religious unorthodoxy (though Newton certainly was religiously unorthodox). He published eight papers in the Philosophical Transactions of the Royal Society and in 1711 was elected a fellow.

This dual career as churchman and mathematician was not unusual for the time and seems particularly relevant for statisticians. Thomas Bayes, after all, was a Nonconformist minister, and is held by some to have devised his theorem as an attempt to prove the existence of God (see Significance, February 2013).

Similar considerations moved Craig. He published several mathematical books. Despite his friendship with Newton he seems to have been the first to use Leibnitz's dx/dy notation for calculus in British print – though he reverted to Newton's fluxions notation in a later book – and he was first also with the ∫ integration symbol in print. Even so, such fame as he has rests on one rather peculiar work. Theologiae Christianae Principia Mathematica [Mathematical Principles of Christian Theology] was written in 1696 and published three years later1. It attempts to work out the date of the Second Coming of Christ; and it does so by analysing changing probabilities of belief as evidence for those beliefs gets less.

He was laughed at, then and later. Augustus De Morgan in 1837 called it “a very silly attempt to apply numerical reasoning to historical evidence”. “An insane parody of Newton's Principia” was an 1830 view2. And, from no less an authority than Laplace, “His analysis is as mistaken as his hypothesis about the duration of the world is bizarre”.

But Stephen Stigler reckons this to be unjust. He claims Craig as an original. What he was groping towards, he says, was log-likelihood ratios; Stigler's rehabilitation of John Craig came in his paper “John Craig and the probability of history: from the death of Christ to the birth of Laplace”3, which I have drawn on heavily here.

Attrition of believability

  1. Top of page
  2. Abstract
  3. Attrition of believability
  4. The vanishing of historical probability
  5. References

We should note at the outset that Craig's definition of probabilities are not ours. He lived before the mathematicisation of the term. “Probability is the appearance of agreement or of disagreement of two ideas through arguments whose conclusion is not fixed, or at least is not perceived to be so”, wrote Craig. Stigler describes Craig's probability as “Not modern probability, but the amount of change (in a vaguely-understood likelihood) due to evidence or data”. It is about changes in the weight of evidence. The evidence deteriorates over time; so, therefore, does the likelihood of a reported event having actually happened. There are two sides to every story; Craig's probability is the balance between the two sides.

“Suspicion of historical probability is the application of the mind to the contradictory sides of an historical event.” At what rate does suspicion increase? That was the key to all his thinking.

The rate of increase of suspicion grows linearly with time; suspicion itself, therefore, increases quadratically, as the square of time. Here his parallels with Newton's inverse-square law of gravity are clear. From that insane, or not-soinsane, parody he produced theorems – see the box. And from the theorems he deduced equations that describe the decay of belief.

What he wanted to discover was the date, far into the future (he hoped), when suspicion of the Gospel story had grown, and belief in it correspondingly diminished, to the point of vanishing. At that date, he believed, Christ would come again.

The vanishing of historical probability

  1. Top of page
  2. Abstract
  3. Attrition of believability
  4. The vanishing of historical probability
  5. References

The passing of time and generations, the corruption of manuscripts, the exaggerations and omissions in stories handed down – all these mean there comes a point when the degree of probability of a story decreases to near zero. For modern man, the story of, say, Noah's flood falls into this category. We have the biblical account – but we know very well, most of us, that the raven, the dove, and the grounding of the Ark on Mount Ararat, might not be historically true. On one hand, ancient sources such as the Epic of Gilgamesh from Mesopotamia, and a 4000-year-old Babylonian clay tablet4 speak of a great flood around the Tigris and the Euphrates; some modern geological evidence from the Caspian, on the other hand, hints at an inundation there. Those who were around at the time knew all about the flood; their oral testimonies were not written down, on clay or in the Bible, until many hundreds of year later, by which time they had already become embellished myth. Our degree of belief in a flood or floods is moderately great. We are much less sure about where or when it happened; our belief in animals going in two by two has been reduced to practically zero.

Craig's Theorems

John Craig's theorems about historical probability are:

  • Theorem I: “Any history (not contradictory) confirmed by the testimony of one first [i.e. primary] witness has a certain degree of probability.”
  • Theorem II: “Historical probability increases in proportion to the number of primary witnesses.” The more original sources there are to a story the better. Every modern journalist knows that a single-source story needs confirmation.
  • Theorem III: “Suspicions of historical probability transmitted through single successive witnesses (other things being equal) increase in proportion to the numbers of witnesses through whom the history is handed down.” A first-hand witness is preferable to a secondhand report; and a third-hand report is worth even less.
  • Theorem IV: “Suspicions of historical probability transmitted through any period of time (other things being equal) increase in double proportion to the times taken from the beginning of the history.” The older a story, the less we are likely to believe it; and the degree of disbelief increases as the square of the time elapsed.
  • Theorem V: “Historical probability transmitted by one historian, and through only one series of witnesses, although it continually decreases, nevertheless in no given time utterly vanishes.” But it may decrease to very close to zero.

From these theorems he produced a formula for the degree of belief we are likely to give to any historical account of a past event:

  • display math

It takes into account the number of primary witnesses, the number of stages of transmission that have occurred before their account has reached us, and the additional effect of time. The terms are explained in the text.

Craig was not much concerned with Noah's flood; he was deeply concerned with the story of Christ. In particular, he was concerned with the Second Coming. Many at the time were saying that it would happen very soon. Might they be right? Luke 18:8 gave a clue: “Nevertheless when the Son of Man cometh, shall He find faith on the earth?” The date when faith disappeared from the earth would set a bound upon the date of the Second Coming.

Craig wanted to find a date by which faith would be all but absent – that is, the believability of the Gospel stories would have sunk to zero, or to imperceptibility.

As we have seen, the degree of believability of an event depends on the number of people who witnessed and told of it. There were of course for Craig four primary witnesses to the Gospel events: the four authors of the Gospels, Matthew, Mark, Luke and John. He assumed they wrote down their accounts immediately – which made their four testimonies equivalent to that of 40 witnesses who tell their tale by word of mouth.

He had his formula for the decay of belief, given in the box:

  • display math(1)

Here c is the number of primary witnesses to an event, 4 in this case. n is the number of copyings of their original accounts before the version that we have today. For the Bible he reckoned this to be one every 200 years – so by the year 1696 it had been through 8.48 successive copyists. T is the number of years that have elapsed since the event. And t is a time unit of 50 years.

He also needed values for the constants z, f and k. z is the believability of a single first-hand record written down at the time. As we have seen, Craig took it to be ten times the believability of an oral account, which he called x; so z = 10x. He gave the other constants also in terms of x. f is the suspicion arising at each recopying. He reckoned that 100 recopyings are necessary to erode the testimony of one eyewitness to imperceptibility, so f = –x/100. And k he set also at k = –x/100: the suspicion arising by delaying the telling of the story by 50 years is equivalent to that of one recopying.

So the formula becomes:

  • display math(2)

T, for Craig, was the year in which he was writing, 1696. Which gave:

  • display math(3)

And that in turn tells us that P = 28x. That is, the Gospel story initially, at the time of Christ, had a believability equivalent to that of 40 independent eyewitnesses passing their testimony by word of mouth; after 1696 years its believability had decayed to be worth that of 28 eyewitnesses. Craig calculated further. He could extrapolate into the future. The historical probability of this now-written testimony would not vanish entirely (or have the believability of a tale with less than one witness to it) until ad 3150.

That was when faith would have departed from the earth and Christ would return. The Second Coming, said Craig, was not imminent. It would not occur for another 1454 years.

As we have seen, Craig was mocked. This seems unfair. If his subject-matter seems strange to us, remember that Archbishop James Ussher had not long before applied far more basic mathematics, as well as considerable scholarship, to the Bible to calculate the date of the Creation (as Sunday, October 23rd, 4004 bc, near the autumnal equinox), and was widely respected for it; Craig's attempt, to date the end of the world rather than its beginning, was far less simplistic.

His neglect by statisticians is also unfair. As Stigler points out, Craig was effectively operating de novo. There was no reason for him to adopt the modern definition of probability, since De Moivre and others had yet to define it. Furthermore, he was faced with an important, and an extraordinarily difficult, question. He had attempted, as his detractors had not, to use covariates, of time and of manuscript distance, to find his answer. Reliability of evidence is surely a statistical matter, and he was one of the first to grapple with it, and to try to quantify it. Craig's treatise is a remarkable early example of the application of mathematical statistics to a problem in social science, says Stigler – and one that shows him to have been operating intuitively as a highly sophisticated twentieth-century statistician.

Towards the end of his life Craig went to London in the hope that his mathematical abilities would be noticed. It does not seem to have happened. He died in 1731 in High Holborn, and was buried on October 14th in the churchyard of St James's, Clerkenwell.

References

  1. Top of page
  2. Abstract
  3. Attrition of believability
  4. The vanishing of historical probability
  5. References
  • 1
    Craig, J. (1699) Theologiae Christianae Principia Mathematica, London: Timothy Child. (Reprinted in 1755 with commentary by Daniel Titius in Leipzig in 1755. A reprint of the Latin original of most of the 1699 material up to the end of Chapter 2, with facing English translation, appeared in 1964 as: Craig's rules of historical evidence, in History and Theory: Studies in the Philosophy of History, Beiheft 4. The Hague: Mouton.)
  • 2
    Todhunter, I. (1865) A History of the Mathematical Theory of Probability. London: Macmillan. (Reprinted in 1965 by Chelsea, New York.)
  • 3
    Stigler, S. M. (1986) John Craig and the probability of history: from the death of Christ to the birth of Laplace. Journal of the American Statistical Association, 81, 879887.
  • 4
    Finkel, I. (2014) The Ark before Noah: Decoding the Story of the Flood. London: Hodder & Stoughton.