No, I’m not just writing this to pick on Dr. Richard Carrier (that’s just a perk; anybody who thinks they can resolve a ~300 year debate in the mathematics community in a book on historical methods is either just about the greatest mathematical mind the world has ever seen or doesn’t understand math very well but doesn’t let that stop him from some extravagant claims). It’s because I get to review research (something that I don’t do enough of here) while talking about fundamental tools and methods in the sciences. In fact, because I’ve reviewed at least one of Carrier’s works before, this review will be much more focused. It is the first of 2 or 3 criticisms I have with On the Historicity of Jesus, not counting any of the issues relating to historical Jesus research or even problems with Carrier’s historical analysis. You may well ask, “How can you review historical scholarship without touching on its historical analysis?” Boy, I am sure glad you asked. Dr. Carrier has been kind enough to base his entire approach (to all of historical research) on his misunderstanding of Bayes’ Theorem (BT), which he has supplemented with misuse of logic and a more general ignorance of statistics and probability. As a result, I need only show that the foundations of his entire argument (and method, as outlined in a sort of methodological prequel Proving History). And to do this, I get to introduce essential components of logic, probability, and Bayesian statistics/methods (though not all in this post).
Dr. Carrier’s probability space and real probability spaces:
Carrier has set himself up to be knocked down so thoroughly that, whereas normally I’d be saying “for example” or “to illustrate”, my first “example” is the foundation for his entire work. On p. 30 (whence comes the scanned image below), Carrier reiterates how he intends (and indeed already has) appropriately classed together all possible hypotheses regarding Jesus’ existence:
We are told the bottom two hypothesis classes are so improbable we can ignore them, and then the entirety of the book proceeds with his use of BT to evaluate which of the remaining two hypotheses is more likely (or how likely each is). But we already have a problem: probability space. A basic understanding of probability (or sample) space can go a long way (technically these are not the same thing, but for a basic understanding this doesn’t really matter), and it is critical to understanding everything from determining your chances at winning in poker to understanding the logic of various statistical tests. It is so critical that I took some time going through statistics texts I’ve taught or tutored with or simply reviewed to find a good, concise definition with a straightforward example. The following were scanned from Dekking, F. M., Kraaikamp, C., Lopuha, H. P., & Meester, L. E. (2005). A Modern Introduction to Probability and Statistics: Understanding Why and How. Springer.
2 quick examples:
Experiments have events AND outcomes? What?
I chose this source in part because it is imperfect in just the right way. Unfortunately, because a lot of developments in academia end up being used in contexts they were never intended for, often enough we wind up retaining terminology even when it is more confusing. For example, the word “experiments” here would be fine if it this weren’t an elementary statistics textbook or if sufficient context were given such that it is clear we don’t intend the word to mean what it usually does. Simply put, because we’re already using the word “event” to refer one or more outcomes in the probability/sample space, and we’re already using “outcomes” to refer collectively to all the elements of the set that is the probability space. To understand the difference, look at the elements in the set of outcomes in the second example. Now imagine if we were interested in knowing not only what month the person on the street’s birthday falls on, but also whether the month ended in the letters “ber”. The “event” corresponding to this latter question is a subset of the set of months: December, October, & November. Honestly, one reason it can be better to stick with the misleading word “experiment” is because “event” and “outcome” are both taken.
For most purposes you can get away without such distinctions as that between “event” and “outcome”, because most people aren’t that concerned with mathematical rigor. What is absolutely crucial, though, is ensuring that you haven’t doomed yourself from the get-go by describing the situation/experiment, sample space, events, or outcomes in ways that render statistical or probability analysis hopelessly flawed.
That’s where logic first comes in. Even though plenty of people take advanced probability courses (advanced as in undergraduate level and requiring calculus) they don’t usually need a course in mathematical/formal logic. That’s because it’s not hard for a probability textbook to cover how formal language required for probability must be of a certain nature that differs from “natural” language. Which brings us back to Carrier’s “hypotheses”.
Formal language: Counterintuitive yet clear?
In mathematics in general, including logic and probability, it is absolutely essential that when you use language (rather than mathematical symbols) you are unambiguous. This is why, for example, in logic and probability the word “or” is ALWAYS understood to be inclusive. Thus, the logically correct answer if one is giving directions and is asked “…and then do I go left or right” is “yes”. If one wishes to formulate the question such that the answer is helpful, then one has to ask it in weird ways such as “is it the case that I go right AND IF I were to go left, would that be correct?” Carrier doesn’t just use natural language rather than formal logic or language that lends itself to probabilistic analysis, he does worse. All four “classes” concern Jesus’ historicity. Now, granting Carrier’s concise but excellent explanation concerning what we mean by Jesus’ historicity (sorry, folks, but for that, buy the book!), the main question is clearly “was Jesus a historical person”? Either he was, or he wasn’t. Note that there are two little marks before two of the “h’s”. These are negation operators from logic. Carrier refers to his “parcel[ing] out the entire probability space”. The entire probability space is parceled out with the proposition “Jesus was historical”, and its negation (“it is not the case that Jesus was historical”).
Parceling out…well, something
This may seem trivial, but it isn’t. First, notice that even if we assume “mythical” is exactly the same as “not historical”, then the probability space is still divided in two. Whether or not Jesus was historicized or mythicized involve additional properties. If we are interested in the probability of Jesus’ historicity and another property such as “Jesus spoke Aramaic”, then this additional property can ONLY be true if Jesus was historical. The next problem, though, is that unlike “Jesus spoke Aramaic”, Carrier adds the technical terms “historicized” and “mythicized”. Logically, there are two possibilities. Either these properties are unrelated to an individual’s historicity, or they are related. If they are related, then a hypothesis such as “Jesus was a historical person mythicized” is already a problem for probability, because we are not only being asked to determine the probability that Jesus was historical, but ALSO that he was mythicized WITHOUT knowing how being mythicized effects the probability of being historical. The jargon makes this difficult, so let me use a simpler example. Imagine we wished to know the probability that “Joe is a wealthy person with a $600,000 salary.” If it is true that Joe is making that much money, this affects the probability that Joe is wealthy. Or consider “Joe is a wealthy person with billions of dollars”, an example in which the added property ENSURES the probability that Joe is wealthy.
Things get worse. Because Carrier doesn’t include hypotheses like “Jesus was a historical figure historicized” we must assume that either he has missed large “portions” of the probability space, or that there is a relation between being historicized and being historical such that one can’t be historical and be historicized (after all, it’s not as if Carrier is leaving out highly improbable hypotheses, because he tells us we can ignore half of the ones he provides). This means that EVERY SINGLE hypothesis is really a combination of hypotheses. In fact, as two of them actually use the English word for the negation operator (“not”), we’re asked not only to evaluate the probability of two statements about Jesus for each hypothesis, we have to consider the effect of the negation operators on two truth values.
Carrier’s problems with negation don’t end by including them in what are supposed to be propositions or hypotheses that correspond to a single outcome or event. When we use natural language, we often indicate that something is false by indicating an opposing or contradicting idea is true. We don’t do that in mathematics, or if we do it is done VERY carefully. The negation of the statement “All Californian residents who are named Joe are wealthy” is “there exists at least one Californian resident named Joe who isn’t wealthy.” The negation of “Jesus was a historical person mythicized” is “it is not the case that Jesus was a historical person mythicized” if we interpret the predicate to be “historical person mythicized”, which we can’t (another hypothesis has it following the negation operator). Really, we have the following two propositions as our first hypothesis: “Jesus was a historical person AND Jesus was mythicized.” Logically, this hypothesis comes out false if Jesus was not a historical person, if Jesus was mythicized, or both. Carrier’s conclusion is that this hypothesis is most likely false, which means that it is perfectly possible (even highly likely) for Jesus to have been historical so long as he wasn’t mythicized.
What’s to be done?
But what about the negation of this hypothesis? It’s not a negation at all, but another hypothesis that combines two propositions. We haven’t even gotten to the two hypotheses with logical operators in them, and already we’re in a mess. How do we dig ourselves out?
First, we recognize that either Jesus was historical, or he wasn’t. That’s how we “parcel out” the probability space, because together this hypothesis and its negation form a complete probability space and the two possibilities are mutually exclusive. We next get rid of the improper use of the negation operator. Finally, we determine (if we want to include them) how the properties “mythicized” and “historicized” belong in this probability space (i.e., if someone is “mythicized”, does that mean they must be historical or can they also be not historical?). After all, we’ve covered every possible outcome with “Jesus was historical” and “it is not the case that Jesus was historical,” so any extras have to fit in this probability space somewhere. This is also crucial, as we’ll see in part two, because Bayes’ Theorem doesn’t work unless we are dealing with every possible outcome and all outcomes are mutually exclusive. Not that it could work here anyway, but it’s important to understand why Carrier’s particular misuse fails.
1) Because Carrier has collapsed multiple propositions into single hypotheses, his conclusion that it is unlikely that “Jesus was a historical person mythicized” is really evaluating the probability of the “Jesus was a historical person AND Jesus was mythicized”, which can have a probability of 0 while the probability of “Jesus was a historical person” is 100%.
2) You can’t parcel out a probability space with elements that you’ve inexorably linked together.