Carrier, Richard. Proving History: Bayes’s Theorem and the Quest for the Historical Jesus. Prometheus Books, 2012.

This is a review of Carrier’s book from purely a mathematical perspective, the historical merit has been reviewed elsewhere. Given the primary audience of this blog, and the book, however, I will review the mathematics in fairly non-technical terms, though I will assume some knowledge of probability theory.

Proving History is the first book of a pair on the topic of whether there was a historical figure of Jesus, written by independent scholar Carrier (who describes himself as a historian and philosopher), funded philanthropically by members of an online atheist group. It sets up a thesis based on using probability theory to reason about historical evidence. In particular, Carrier focuses on what he calls Bayes’s Theorem as the fundamental underlying process of doing history.

I will be unable to deal with every mathematical problem in the book in a short review, so will limit myself to issues arising from the first mathematical chapter. The issues below are not mitigated later in the book.

Chapter three introduces Bayes’s theorem (BT) as

[sic] which is a confusing and unnecessarily complex form of the formula more easily stated as

While Carrier devotes a lot of ink to describing the terms of his long-form BT, he nowhere attempts to describe what Bayes’s Theorem is doing. Why are we dividing probabilities? What does his denominator represent? He comes perilously close in chapter six (“Hard Stuff”), when talking about reference classes (which are quite closely related to the meaning of the denominator), but doesn’t try to bring his audience to the level of competency to do anything more than take his word for these mathematical assertions.

So this unduly complex version of BT serves as a kind a magic black box:

“the theorem was discovered in the late eighteenth century and has since been formally proved, mathematically and logically, so we know its conclusions are always necessarily true—if its premises are true”

In this Carrier allows himself to sidestep the question whether these necessarily true conclusions are meaningful in a particular domain. A discussion both awkward for his approach, and one surely that would have been more conspicuously missing if he’d have described why BT is the way it is.

The addition of background knowledge (b in his formula) in every term (the probability theory equivalent of writing x1 or +0) is highly idiosyncratic, though I’ve seen William Lane Craig use the same trick. Footnote 10 states that he made the choice so as to make explicit that background knowledge is always present. Clearly his audience can’t be expected to remember this basic tenet of probability theory.

Carrier correctly states that he is allowed to divide content between evidence and background knowledge any way he chooses, provided he is consistent. But then fails to do so throughout the book. For example on page 51 is an explanation of a ‘prior’ probability which explicitly includes the evidence in the prior, and therefore presumably in the background knowledge (emphasis original):

“the measure of how ‘typical’ our proposed explanations is a measure of how often that kind of evidence has that kind of explanation. Formally this is called the prior”

Going on to say (emphasis original):

For example, if someone claims they were struck by lightening five times … the prior probabilty they are telling the truth is not the probability of being struck by lightening five times, but the probability that someone in general who claims such a thing would be telling the truth.

This is not wrong, per se, but highly bizarre. One can certainly bundle the claim and the event like that, but if you do so Bayes’s Theorem cannot be used to calculate the probability that the claim is true based on that evidence. The quote is valid, but highly misleading in a book which is seeking to examine the historicity of documentary claims.

The final problem I want to focus on in chapter three, is the claim of BT special status. Carrier asserts it is both necessary and sufficient for any probabilistic reasoning about evidence. This is indicative of a confusion of nomenclature that permeates the book, at times he uses Bayes’s Theorem to mean probabilistic reasoning generally, then switches to using his idiosyncratic equation form (as if his claims about the former, therefore lead one to the latter necessarily), and at other times uses it as a stand in for Bayesian reasoning (to which I’ll return below). If he had started from the definition of conditional probability:

he might have noticed that Bayes Theorem is merely an application of basic high-school algebra to the definition, and there are many other such applications which would not give BT, but would be equally valid. Thus statements such as

“any historical reasoning that cannot be validly described by Bayes’s Theorem is itself invalid”

(which he claims he will show in the following chapter, but does not credibly do so) are laughable if understood to mean

but have been argued for (though by no means to universal acceptance) if taken to mean the Bayesian interpretation of probabilities.

Carrier joins that latter debate too, in what he describes as a “cheeky” unification of Bayesian and Frequentist interpretations, but what reads as a misunderstanding of what the differences between Bayesian and Frequentist statistics are. Describing what this means is beyond my scope here, but I raise it because it is illustrative of a tone of arrogance and condescension that I consistently perceived throughout the book. To use the word “cheeky” to describe his “solution” of this important problem in mathematical philosophy, suggests he is aware of his hubris. Perhaps cheeky indicates that his preposterous claim was made in jest. But given the lack of mathematical care demonstrated in the rest of the book, to me it came off as indicative of a Dunning-Kruger effect around mathematics.

I had many other problems with the mathematics presented in the book, I felt there were severe errors with his arguments a fortiori (i.e. a kind of reasoning from inequalities — the probability is no greater than X); and his set-theoretic treatment of reference classes was likewise muddled (though in the latter case it coincidentally did not seem to result in incorrect conclusions). But in the interest of space, the above discussion gives a flavour of the issues I found throughout.

Conclusion

Outside the chapters on the mathematics, I enjoyed the book, and found it entertaining to consider some of the historical content in mathematical terms. I strongly support mathematical literacy in the arts. History and biblical criticism would be better if historians had a better understanding of probability (among other topics: I do not think the lack of such knowledge is an important weakness in the field).

I am also rather sympathetic to many of Carrier’s opinions, and therefore predisposed towards his conclusions. So while I consistently despaired of his claims to have shown his results mathematically, I agree with some of the conclusions, and I think that gestalts in favour of those conclusions can be supported by probability theory.

But ultimately I think the book is disingenuous. It doesn’t read as a mathematical treatment of the subject, and I can’t help but think that Carrier is using Bayes’s Theorem in much the same way that apologists such as William Lane Craig use it: to give their arguments a veneer of scientific rigour that they hope cannot be challenged by their generally more math-phobic peers. To enter an argument against the overwhelming scholarly consensus with “but I have math on my side, math that has been proven, proven!” seems transparent to me, more so when the quality of the math provided in no way matches the bombast.

I suspect this book was always designed to preach to the choir, and will not make much impact in scholarly circles. I hope it doesn’t become a blueprint for other similar scholarship, despite agreeing with many of its conclusions.