Correction of worked example of dependence factor in undesigned coincidence article

Part of being a conscientious scholar is admitting one's earlier mistakes. While it can be tempting to toss earlier mistakes into a memory hole and hope nobody notices them, it's something I try to avoid. Another temptation can be retconning what one said so as to imply that one meant something that one can still maintain. I don't suffer from the latter temptation, but I do sometimes experience the former temptation. I try hard to resist both.

Let it be said here at the outset that the correction here does not mean that "undesigned coincidences are not a thing," "undesigned coincidences have been refuted," "professional publication on undesigned coincidences is fraudulent and needs to be retracted" or anything of the kind. Skeptics, I'm lookin' at you; if you try to use this post for that purpose, you're making a mistake. Also, even if you don't slog through this whole post, please read the last paragraph.

In 2020 an article (by me) on the probabilistic analysis of undesigned coincidences was published in the high-level journal Erkenntnis. The accepted manuscript version of that article can be found here, and it now has a dated cover sheet with a short update note linking back, in turn to this blog post for further details.

Recently I have been reviewing that article while preparing for a talk on the mathematics of undesigned coincidences. I've realized that in the appendix to that article, where I give some illustrative worked examples, Example 3 has a couple of errors. Here are the errors and the corrected work:

The first (and greater) error is that, when discussing dependence under H, I treat the fact that, when we are considering the conditional probability given H, we are taking H to be true, as if it confers additional reason to take the witness to be reliable. This is incorrect, since this is a mere conditional probability and does not actually involve receiving further independent evidence that H is in fact true.

The second error is a sheer matter of incorrect calculation using the model. While I intended (and even emphasized) that the subhypothesis of H that I dubbed H', and its negation, should screen off testimony 1 from testimony 2, under the assumption of H, I did not in fact build that screening off into the model correctly. This error was related to my not properly taking account of the fact that, in this example, I am creating dependence under H by stipulating that a portion of the probability space under H actually gives each of the individual items of testimony higher probability than the rest of the space under H. This differs from the nature of the dependence under ~H modeled in the previous two examples; the relevant subhypotheses there directly hypothesize collusion or copying and make no difference to the probability of an individual item.

Here is a corrected worked example, modeling a dovetailing undesigned coincidence:

H is the umbrella hypothesis that some event took place. (In the text of the article the event is a picnic.)

Let H' be the subhypothesis that the event took place and also that some further fact about the event is true (in the paper, this was that a picnic took place and also that a Joyce scholar was present at the picnic).

Let P(H'|H) = .05

Suppose that what each of the witnesses says confirms not just H but also H'.

Contrary to what I did in the published paper, the credibility of the witnesses 1 and 2 should be kept at a Bayes factor of 10/1 (as it already was) for their information that confirms H', even though this is a detail of their testimony that H, since no independent information (aside from their testimonies) confirming H is being modeled.

For the sake of not complicating the model yet further, suppose that each of their testimonies confirms H' to the same extent.

Both testimonies are screened off from one another given H' and given its negation, modulo H, but they are not screened off from one another by H itself, because they both confirm H'. The probability of the conjunction of the testimonies is higher given H' than given H and than given (H & ~H'), because the probability of the individual items is higher given H' than given (H & ~H'). This produces the fruitful dependence that actually creates a kind of "bonus" probabilistic boost for H from the conjunction of these testimonies.

Let P(T2|H & ~H') = .02

And the same for T1.

Let P(T2|H & H') = .2

And the same for T1.

Therefore, the overall probability of T1 individually and T2 individually given H is

(.95)(.02) + (.05)(.2) = .019 + .01 = .029

(It might be possible to set all the probabilities just right so that this comes out as .03 as it previously did in example 3 in the published paper, but I did not keep trying all possibilities, and .029 was the closest I was able to get in the numbers I attempted, while keeping everything else as it should be—for example, the BF between H' and ~H' as 10/1. Since this is similar to the other worked examples where the individual probability of each item of testimony given H is .03, and since this is merely an illustrative example, I decided to go with this modeling. Nothing rides on having the total individual probability given H be .03 rather than .029, as long as everything else is done accurately within the assumptions stated.)

Let the individual, independent Bayes factors between H and ~H be 10/1 as in the other appendix examples, so that P(T1|H)/P(T1|~H) = .029/.0029 and the same for T2, so that treated as independent the joint BF would be 100/1, as in the other examples.

Confirmation of H' within H by a 10/1 internal Bayes factor:

Since P(T1|H & H') = .2 and P(T1|H & ~H') = .02, under H, T1 confirms H' by a factor of 10/1, and so does T2.

By the odds form

P(H'|H)/P(~H'|H) x P(T1|H' & H)/P(T1|~H' & H) = P(H'|T1 & H)/P(~H'|T1 & H)

.05/.95 x 10/1 = .5/.95

To turn these odds (under H, giving us the probability of H' given either of the testimonies and H) into a probability,

.5/(.95 + .5) ≈ .3448

So given H and either one of the testimonies (say, T1), the probability of subhypothesis H' is approximately .3448 and the probability of its negation is approximately .6552.

We have stipulated that, under H, both H' and ~H' screen off T1 from T2 and that they each have the same probabilities (.2 and .02 respectively) given (H & H') and (H & ~H'). Given such screening, the conditional probabilities of .2 and .02 remain the same.

 Therefore, the probability of T2 given (H & T1) =

 (.3448)(.2) + (.6552)(.02) = .06896 + .013104 = .082064

 To calculate the fruitful dependence factor between T1 and T2 under H:

 P(T1|H) x P(T2|H & T1)/P(T1|H) x P(T2|H) = (by cross canceling P(T1|H) on the top and bottom)

 P(T2|H & T1)/P(T2|H) =

 .082064/.029 ≈ 2.82979/1

 This, then, is the correction factor by which T1 confirms T2 given H—which will be helpful to the confirmation given by the conjunction (T1 & T2) to H.

This is almost exactly half of the size of the dependence factor of 5.617 given in the published paper in the worked Example 3, modeling dovetailing details.

In the text of the paper other than the appendix, the closest that I come to stating the erroneous extra boost of the witness's credibility is this sentence on p. 21:

Given that P (asserted by Source 1) is true, there is some reason to think that Source 1 is truthfully reporting about that day, including the extra detail about the Joyce scholar's presence.

 This should be reworded along these lines:

 Since Source 1 has, as stipulated, some credibility for what he attests, this credibility can be applied to the extra detail about the Joyce scholar's presence.

Concerning the correction factor for dependence under ~H, I argued in the appendix section on Example 3 that dependence on ~H (which weakens the case for H) should not be any higher than in example 2 (varied but not dovetailing details), where it is 2.33 to 1 against H, and plausibly should be lower.

I stand by that. In that calculation, the only dependence arose from modeling two subhypotheses under ~H of PC (partial/possible copying) and Cr (crafty copying) as having a relative advantage in predicting the conjunction (T1 & T2) over complete independence modulo ~H. If that relative advantage is kept proportionally identical in Example 3 (the one I am recalculating), then of course that dependence factor will be exactly the same.

If the dependence given ~H is kept the same as in example 2, on the new calculation the approximately 2.82979 dependence given H more than offsets the approximately 2.33 dependence given ~H, though obviously not as much as the dependence factor of twice that much in the worked example in the paper.

I then pointed out that arguably in the dovetailing case hyper-crafty collusion/copying would be necessary so as to produce the dovetailing between the details with the appearance of casualness/undesignedness. In other words, partial/possible copying would do an even worse job at uniting the testimonies, as would crafty copying. (Full copying of course is just out of the picture here, as it is in example 2.) I suggested that we should even reasonably consider eliminating a dependence factor under ~H altogether in the case of dovetailing details, since in order to have T1 and T2 as described here we have to have not just variation but extra craftiness to think of varying in a way that is both unobvious and dovetailing, which might simply be overlooked by an audience and which colluders are unlikely to think of. And it's even less likely than non-dovetailing variation to come about by partially copying the content while also varying in some unthinking fashion.

As stated in the paper, if dependence given ~H is eliminated due to these considerations, then the dependence given H is all gain for the confirmation of H from the conjunction of the testimonies, over and above the 100/1 Bayes factor treating them as independent.

It is also worth emphasizing that these numbers are merely illustrative. If the two witnesses have even higher credibility in the first place, that will affect this type of example particularly strongly, since the witness's existing credibility should apply to the subhypothesis H'.

Entanglement concerns don't rule out use of all granted propositions

 It was recently suggested to me that, if my concerns about epistemic entanglement in the minimalist arguments for the resurrection were legitimate, they should preclude ever stating that any opponent acknowledges or grants anything that one is going to use in an argument with a conclusion that the opponent doesn't grant. This was supposed to be a reductio of my concerns on the assumption that it's obviously sometimes okay to state that an opponent grants something that one wants to use.

 To recap, I object to the use in a resurrection argument of the statement that the vast majority of scholars grant that the disciples had experiences which they believed were appearances of Jesus risen. I also object to the use of particular quotations mentioning appearances by specific mainstream scholars (like Gerd Ludemann, Norman Perrin, and Dale Allison) in the same way. My reason for the objection is that it equivocates on the term "appearances." What is needed for the argument is that the disciples at least claimed that they had experiences of a particular phenomenological kind. It seemed to them that Jesus was with them on earth in 3D, in groups, able to eat and be touched, etc. But these scholars don't grant that they had experiences of that kind. Please note this isn't just an objection on the grounds that some of these scholars don't really think Jesus was risen. I'm objecting on the grounds that the phenomenological nature of the experiences they think occurred was very different from what is actually needed for the argument. E.g. Perrin seems to think the disciples merely had an experience of theological enlightenment, which he calls a "vision," but doesn't seem to think even had a visual component. All three of these scholars think that the group bodily "seemings"—Jesus eating, having long conversations—were not something the disciples experienced. The experiences rather were fleeting at most, and the physical manifestations to groups were made up later.

 When one says that someone grants appearance experiences and then uses this "fact" later in the argument in a way that the person does not grant, this is at best a fallacy of equivocation. Moreover, if the person literally means something by talking of the appearance experiences that would be evidence against one's conclusion, one is using something that is evidence against one's conclusion as if it is evidence for it.

 The example given as an alleged reductio of my concerns was "Richard Dawkins concedes that there are gaps in the fossil record" in the course of an argument for intelligent design in biology—a conclusion that of course Dawkins does not agree with. 

But it is not true that any use of an opponent's concession runs into an entanglement problem. I may do a video on this some time, but for now: Entanglement is a specific type of issue in which the very meaning of an opponent's admission (and/or the reasoning he uses to reach it) is such that, if that meaning were clearly understood, the proposition in question would (if true) either be quite worthless to one's conclusion or even tell against the conclusion for which someone wants to use it.

 I've given multiple examples: E.g. Turning "A is from Georgia" into "A is from the United States" in the service of an argument for "A is from Massachusetts." Turning "The number in the box is 20" into "The number in the box is less than 21" in the service of an argument for "The number in the box is 5." Here is a video I did on entanglement. This can also be thought of as using the popular fallacy of equivocation, if that is an easier way to think of it.

In the case of the minimalist argument, this involves turning what would be accurately stated as "The disciples had only very brief group appearance experiences of Jesus in which he definitely did not eat and at most spoke only briefly" into "The disciples had appearance experiences of Jesus" in the service of an argument for "Jesus rose bodily from the dead." In some cases the person in question (e.g., Bart Ehrman) doesn't grant group appearances at all.

 In the case of Dawkins, as far as I can tell, there is no necessary entanglement issue with using "Dawkins acknowledges gaps in the fossil record" in the service of an intelligent design conclusion. It would be a good idea, for the sake of clarity, to state immediately in the context that Dawkins thinks that such "gaps" occur millions of years ago and over periods of millions of years. Hence a term like "suddenness" if Dawkins uses it cannot be used to support a young earth. But there are intelligent design arguments (about, say, the Cambrian explosion) that are not arguments for a young earth. It would also be good to point out that Dawkins makes much of what he considers the similarly abrupt appearance of flatworms as a sort of reductio of the use of gaps in the fossil record to argue for design. This, however, would be solely for the purpose of avoiding all appearance of taking Dawkins out of context, not because Dawkins is literally acknowledging gaps in a sense of that term that is incompatible with or tells against a design conclusion. And whether or not it is useful for supporting a design conclusion is something that can be debated explicity with followers of Dawkins without any equivocation or "fuzzifying" of the meaning of Dawkins's statement about gaps in the record.

It is possible to envisage entangled uses of Dawkins' admission that could be made. For example, if a YEC attempted to use Dawkins's admission in a way that implied (or had to be taken to mean) that Dawkins acknowledges that fossil gaps appear over a much shorter period of time.

Or, to change the actual situation, suppose that Dawkins had acknowledged only very small gaps, where small is understood in terms of phenotypical appearance. This is not actually the case. It seems that he acknowledges what appear to be "large" gaps in terms of appearance, but he thinks these do not have the significance that creationists attribute to them. If Dawkins meant only small gaps and one used the statement that "Dawkins acknowledges gaps in the fossil record" in an argument that requires large phenotypical gaps, while giving the impression that Dawkins acknowledges what one needs for the argument, this would run into an entanglement problem. But this doesn't appear to be what is going on.

Another example of an unentangled granting is Bart Ehrman's defense of the existence of Jesus against Jesus mythers. It's quite easy to use a definite description of "Jesus" that is reasonable and understandable by both Christians and non-Christians. E.g., "A Jewish teacher who lived during the reign of Tiberius Caesar, was known as 'Jesus of Nazareth', and was crucified by order of Pontius Pilate." One can state correctly, without any sliding or equivocation, that Bart Ehrman acknowledges that Jesus existed. Even Ehrman's arguments for the existence of Jesus are such that Christians can also use them—e.g., that Paul said in a letter that he met Jesus' brother, and that Paul appears in his letters to think of Jesus as a real, historical person. 

Ehrman doesn't even seem to think of this as something he "grants" or "admits" in the sense that he does so reluctantly. He's quite overt and even emphatic in rejecting Jesus mythicism. But insofar as one thinks of Ehrman as a "hostile" source (to Christianity), one might wish to use some sentence like, "Even Bart Ehrman thinks that Jesus existed." There doesn't seem to be any entanglement here. Entanglement would arise if one implied that Bart Ehrman thinks that Jesus existed in such a sense that "Jesus" is defined as including his actually working miracles. Entanglement would also arise if one seemed to be implying that Bart Ehrman understands "Jesus" to mean, among other things, "A person who claimed to be God," while in fact, Ehrman strongly denies that Jesus claimed to be God. But I don't know of anyone who does that.

 When checking for entanglement, two things to check for are term meaning and the actual arguments that are used by the person admitting the proposition in question to support that proposition. On this basis, as well, Dawkins appears to be acknowledging apparent gaps in the fossil record for the same reason (namely, simply that in some cases we don't have a smooth set of fossils showing a lot of transitional forms) that a proponent of ID would use.

Another hypothetical example of entanglement (which perhaps someone out there gets snared in) would be if a person arguing for a young universe used the premise, "Such-and-such a scientist acknowledges that the universe had a beginning" when the scientist in question believes that the universe had a beginning only as the conclusion of arguments supporting a Big Bang occurring many millions of years ago. In that case the entanglement occurs at the level of the reasons for which the scientist grants that the universe had a beginning. If the scientist didn't believe that the universe is very old, he would not be able to use these scientific reasons to conclude that it had a beginning, but the hypothetical reasoner here wants to use his "acknowledgement" in the service of an argument that the universe is very young.

 In the minimalist case it is often assumed that, if some skeptic or liberal scholar "admits" that the disciples had appearance experiences, he must be doing so on the basis of strong arguments that are the same as arguments that the Christian would use. It is sometimes left vague what exactly this shared argument is supposed to be; usually we just find some sentence like "The underlying data are so strong that the vast majority of skeptics acknowledge these facts."

 But perhaps the shared argument is supposed to be "multiple, independent attestation to group appearances." Gary Habermas has stated, incorrectly, that the Jesus Seminar admits multiple, independent attestation to group appearances. This is not true; as I've argued elsewhere, the pages that Habermas cites from the Jesus Seminar are merely lists of ancient documents that mention group appearances. In context it's very clear that the Jesus Seminar is not treating these as independent of one another. Bart Ehrman has been explicit that he doesn't grant that there were group appearances (though he says that if there were, they were like "seeing at a distance"). Yet Habermas has boasted in a video that he can argue strongly for the resurrection using only "Bart Ehrman's facts."

Entanglement is a tricky epistemological subject. But no, avoiding entanglement does not make it impossible ever to use something that an opponent admits in the service of one's own argument. One just has to be very careful when doing so.

Probability and the non-partition

 Statistician Brian Blais has attempted to criticize Tim's and my use of a full partition between R and ~R (the resurrection and no resurrection) in evaluating the evidence for the resurrection. He has called this "inflating the Bayes factor with nothing." I intend to do a video series on my Youtube channel on various skeptical attempts to avoid using a partition, which is the actually confusing approach to modeling the impact of evidence. That series will talk about other skeptics who have tried something similar and what's epistemologically wrong with doing so.

Meanwhile, I thought I'd rescue from obscurity in a comments thread some comments I made in immediate response when Blais posted his criticisms. These are a little unorganized, but here they are for now.

There actually is no problem with this analysis (using a full partition), significant or otherwise. You simply prefer, for unknown reasons, to analyze an empirical inference in what to me seems like a confusingly roundabout way, in which one evaluates the specific evidence set E for some salient hypothesis H (e.g., testimony to some event, alleged video of that event, or what-not) by comparing the prior probability of H to the prior probability of some highly gerrymandered theory, ~H1, which is a subhypothesis of ~H specifically generated to give probability to the entirety of the set E that is equal to its conditional probability on H. 

It is not even clear that there always exists such a ~H1 (unless one simply considers a non-explanatory declaration "~H and E, somehow" to be a hypothesis), and such a procedure is absolutely riddled with opportunities for epistemic confusion. In contrast, the odds form of Bayes' Theorem, so far from being misleading, yields the correct answer even in your "exercise for the reader" and does so with ease.

 There you create a distribution in which the prior odds are 99/1 against H and the Bayes factor is 39/1 in favor of H from some evidence or set of evidence E. The use of the odds form shows immediately that of course E increases the probability of H but is not sufficient to swamp the prior odds. We have there posterior odds of 39/99 (H over ~H), and an extremely simple calculation (39/(99 + 39)) yields the posterior probability of H as approximately .28.

I have actually seen a professional probability theorist confused by not using a partition in evaluating the evidence for miracles, to the point that he switched between a naturalistic hypothesis which was the conjunction of a set of theories specifically evolved to account for the evidence for the resurrection (and frankly, even so it didn't do a good job!) and the individual conjuncts, or naturalism more broadly. In other words, he treated the statement that a miracle has a low prior probability as equivalent to saying that any naturalistic hypothesis, however convoluted and consisting of however many independent parts conjoined together, evolved to account for the evidence for the miracle, will always have a higher prior probability than that of the miracle. 

Needless to say, "Naturalism" is not the same thing as "this incredibly specific conjunctive naturalistic hypothesis that we dreamed up in a blatantly ad hoc manner so as to try to account for the evidence for this specific miracle." I believe that the failure to think in terms of partitions is at least partly to blame for this sort of error, which is rife.

Finally, I stress that I deem the full partition Bayes factor analysis to be the most natural way to conceive of evidence for any specific event for which there is specific evidence. When we say, "The evidence for a moon landing is very strong," a very natural way to think of that is as a reference to the specific evidence (videos, interviews, etc.) that man actually landed on the moon, not to more general evidence such as a knowledge of the state of technology at the time and whether it would have been good enough for a moon landing. The latter is relevant to the prior. The former is the specific evidence for the event. 

I submit that considering a Bayes factor using a partition makes a lot more sense than instead saying, "How does the prior probability of a moon landing compare to the prior probability of no moon landing plus a conspiracy hypothesis specifically gerrymandered to account for this whole set of evidence that we have?" If there actually were a subhypothesis of the negation that gave identical probability to the whole of the specific set E to that given by H (and I question that), and if one were able to think about it with perfect clarity (also unlikely) and deduct from the prior of the subhypothesis all that needed to be deducted for the various ad hoc embellishments one would need to make, the final result would be the same. But I can't imagine why anyone would try to go about it in such a way, and I submit that if we were considering a variety of mundane events for which we have specific evidence, no one would think of preferring such a method.

One more point: There is a longstanding and lively debate on the best way to measure evidential force. Various methods are suggested, such as measuring the difference between the prior and the posterior of H--P(H|E) - P(H). Another proposed measure, sometimes called the r measure, is P(H|E)/P(H), and there are more candidates. Tim and I favor the likelihood ratio, sometimes called the L measure, which is P(E|H)/P(E|~H). Sometimes probability theorists use the log of the L measure. 

In any event, I don't know of anyone who suggests that we should measure the force of evidence in the round-the-barn fashion that Brian is suggesting: "Take a highly ad hoc version of ~H, assuming that you can find one, that gives precisely the same conditional probability to the set of evidence E that H gives to E, while leaving the probability of E given all the rest of the partition as 0, then somehow or other calculate the prior probability of this subhypothesis ~H1, then calculate the posteriors of H and ~H as constituting a new partition in the same ratio as the priors of H and ~H1. Good luck with that."

 The whole point of finding a good measure of the force of specific evidence E is to figure out the most intuitive way to measure what E actually does for or against H qua evidence, not to try as hard as possible to make it look like E isn't doing anything. Part of the reason we favor the L measure (an extremely well-known measure, favored by plenty of people who wouldn't be caught dead being Christians) is because it separates out the power of the specific evidence so cleanly from separate considerations that feed into the priors. In any event, our use of the L measure has precisely nothing to do with weird religious apologists engaging in some sort of probabilistic skulduggery.

The Denarius Objection

I've been recently asked twice about an objection to the historicity of the story of the denarius and rendering tax to Caesar in the Synoptic Gospels, so I wrote up some comments. These comments are in rough form, even more so than usual for a blog post. I was asked to publish them publicly (having first written them for an email correspondence and then for a private Facebook group) and have decided to do so in the most efficient way possible rather than taking a lot of time to polish before publishing.

In the first instance obviously the objection is to the historicity of the entire incident. That is the thesis of Christopher Zeichmann, the author of the article. He thinks Mark made the story up to encourage Christians to pay the fiscus Iudaicus levied later by Vespasian. I'm told that it's also being used in skeptical circles to argue against Matthean authorship, since if the incident is unhistorical, a tax collector would have known this and wouldn't have repeated the story. As I pointed out to the correspondent, that a Gospel author made up a whole story and presented it as historical in order to induce Christians in his own time to cooperate smoothly with the Romans is of course the bigger claim.

Comments on Christopher B. Zeichmann, "The Date of Mark's Gospel Apart from the Temple and Rumors of War: The Taxation Episode (12:13-17) as Evidence.” The Catholic Biblical Quarterly 79:3 (2017), pp. 422-437. This article can be read online for free even if you have no institutional affiliation, using your Google profile.

The objection is based in the first instance on the fact that archaeologists found denarii in coin hoards, even finding denarii that were minted prior to the Jewish War, but found relatively few, fewer of them than at later periods, and this is used to argue that the episode in Mark (and hence Matthew and Luke) is non-historical (pp. 428-429). Talk about an argument from silence. It would be somewhat weak even if they had found none, since often things aren't found later in archaeological digs but nonetheless existed. But in this case the argument for such a strong conclusion is based merely on their finding relatively few. Zeichmann concludes that denarii were "extremely rare" in Judea and Galilee before the Jewish War, but this is itself a dubious extrapolation from coin hoards. He notes that some of those found were not in hoards but in individual findings of unknown date (of their having been dropped or buried there, presumably). He makes a big deal out of the fact that of those hoards we have found, where the hoard itself is believed to have been deposited before the war (it’s unclear how they know that), none contain denarii. In other words, denarii that were minted prior to the Jewish War were found either individually or in hoards that were (archaeologists think) deposited qua hoards in their current locations after the Jewish War. 

It's somewhat opaque as to how this supports the conclusion that they were "extremely rare" before the war. He seems to be assuming that if they were not "extremely rare" more of them would have been found in coin hoards and in particular that they would have been found in hoards where the hoards themselves can be independently confirmed to have been deposited prior to the Jewish War. This seems like an extremely shaky argument.

Zeichmann also argues (p. 431) that, since he was a Galilean, Jesus wouldn't have been interested, knowledgeable, or even asked about a tax levied in Judea (as the Gospel portrays). Jesus was being asked this question as a rabbi. (Of course in the story his questioners are trying to entrap him with the question.) Zeichmann says, merely on the basis that Jesus was asked this question by enemies in Judea, that Mark seems to be assuming that Galileans and Judeans were paying "the same taxes" to Rome. Rabbis were asked for Jewish legal rulings. There was plenty of travel between Galilee and Judea, and a rabbi who was often in Jerusalem (as Jesus was) could have been asked to make a statement in Jerusalem about a tax that Jews didn’t like, that was levied in Judea, regardless of whether he was originally raised in Galilee.

Just one page earlier (p. 430) Zeichmann assumes that Galilean taxation was relevant to the region as a whole. He refers to corn stored in Galilee to pay taxes to Caesar (per Josephus) and uses this to argue that all taxes to Rome were paid in kind prior to the Jewish War (see below for a response to this). In other words, at that point, Zeichmann is arguing inter alia that Judean taxes to Rome would only have been paid in kind in Jesus' time from Josephus' record of corn being kept to pay taxes in Galilee several decades later. But at the same time (p. 431) he argues that it was "impossible" (his word) that Galilean and Judean taxes to Caesar were the same during Jesus' lifetime, extrapolating that they couldn't even be sufficiently similar for Jesus to be asked about Judean taxes, because the administration of Judea and Galilee under Rome was not unified until A.D. 44. This is inconsistent. Either Galilean taxation is relevant to Judean taxation in Jesus’ time or it isn’t. Zeichmann treats its relevance in opposite ways depending on which position seems to further the argument against the historicity of the incident.

The payment in kind argument is very problematic. Zeichmann argues from the fact that Josephus mentions corn being kept for taxation purposes. Josephus also mentions the difficulty of paying tax to Rome if land remained unsown. Zeichmann also notes an absence (outside of the Gospels) of explicit reference to paying tax to Rome in the form of coins prior to the Jewish War (except in the Agrippa II incident, see below). He argues from this absence to the conclusion that tax to Rome was collected only in kind prior to the Jewish War “except in emergencies.” But this certainly does not follow. In an agricultural region, of course unsown lands could lead to an inability to pay taxes, even if at least some people sold their crops and paid tax in coins. Agriculture is going to be important to prosperity or poverty in the region. And the fact that corn was kept for taxation purposes shortly before the Jewish War merely shows that taxes could be rendered in kind, not that they always were.

Zeichmann's very strong use of the argument from silence is also evident here: 

In fact, no monetary capitation taxes are known at all in the southern Levant before the war, much less any paid in denarii or equivalent coinage (e.g. didrachm). In short, there was no κήνσος that a resident paid to Καίσαρ in δηνάριον or any other coin for that matter at the time. Whatever tax Mark had in mind, it did not exist during the life of Jesus. (p. 431)

Notice the seamless movement from "no monetary capitation taxes are known [in non-biblical sources]" before the war to the dogmatic statement that there was no such tax, a claim which Zeichmann continues to use throughout the rest of the article.

Most strikingly, Zeichmann argues that before the Jewish War, taxes were collected in coin only "in emergencies" from an incident Josephus tells about in A.D. 66, during the time of Agrippa II, when the taxes to Rome had remained unpaid for a while as a form of protest against the rule of Florus. Agrippa II hastily collected the taxes in coin in order to ward off war with Rome. In this incident Josephus (Jewish War II.17) says that they "soon got together forty talents" (!!) This shows not only that the taxes could be collected in coin but also that coins were widely enough available to collect a large amount in back taxes hastily in coin. That this was "only" done due to an "emergency" is not indicated at all in the Josephus passage. It is just Zeichmann's ad hoc construct to explain away the passage, which is otherwise a counterexample to his sweeping generalization that taxes to Rome were paid only in kind, not in coins. 

The rapid collection of forty talents in money for back taxes is especially notable in view of Zeichmann's insinuation on p. 431 that there were few coins in general used in the region prior to the war. 

Indeed, the economy of the southern Levant was not fully monetized during Jesus' time; the widespread use and circulation of coins in rural Judea was facilitated decades later by the postwar military occupation. This fact alone would have rendered any monetary collection an inefficient means of taxation. 

One wonders why "fully monetized," however that is measured, is the relevant standard. I suspect Zeichmann is assuming that any tax must be collected in only one way. In any event, the Agrippa incident shows that it is not true that collecting taxes in coinage was necessarily inefficient, due to the rarity of coins, since coinage must have been sufficiently available for that collection to take place in a short time.

The general idea that coinage was little used (and by the way, why are we suddenly focusing on rural Judea only?) is also contrary to other evidence about both denarii and drachmas, an approximate monetary equivalent of the denarius. Tobit 5:15 indicates that a drachma for a day's wage was reasonable, as does a parable of Jesus in Matthew 20:2, where the word "denarius" is used for a daily wage for agricultural workers. This parable occurs only in Matthew. And Mishna Bekharat 8.7 indicates that, as in the incident of the fish and the stater in Matthew 17, the Temple tax was two drachmas—i.e., ½ of a shekel. Jesus' parable of the talents in Matthew 25 also makes use of coinage. There is also a story in Mark 12 of people dropping money into the Temple treasury, clearly in coins.

There are in fact numerous references specifically to denarii throughout the Gospels, including in texts apparently independent of Mark. Not only does the parable of the vineyard workers refer to a single denarius as a daily wage; other references refer to denarii as if they are a natural way for people to think about how much something costs, how much something is worth, or how much is owed. Matthew 18:28, in the parable of the unforgiving servant (found only in Matthew), measures the amount owed in denarii. Luke 7:41-42 also uses denarii in the brief parable that Jesus tells to the Pharisee who judges the sinful woman. Jesus asks whether a man who is forgiven a debt of five hundred denarii will not love the one who forgives more than a man who is forgiven a debt of fifty. This story is unique to Luke. Mark 6:37 and John 6:7 both use denarii to describe the disciples' appalled statement about how hard it would be to buy bread for the crowd at the feeding of the five thousand. Mark 14:5 attributes a reference to denarii to those who criticize the woman who anoints Jesus' feet, referring to how much could have been given to the poor if the ointment were sold. Are we to believe that these varied apolitical biblical references to denarii were all made up for some obscure reasons? The motive Zeichmann attributes to Mark cannot apply to the parables, nor would it apply in these other contexts, including other contexts in Mark. Is it not more reasonable to think that Jesus (even on a skeptical view a master teacher) made use of coinage in his parables which his audience would be able to relate to? And should we not consider seriously the possibility that people in Jesus' time and place thought of denarii more often than Zeichmann's thesis would indicate?

The references both inside and outside the Gospels present a picture of a more "monetized" world in Jesus' place and time than Zeichmann portrays.

Returning to the story of the denarius and taxes, it seems that Zeichmann is imposing on the story in Mark (which others are then extending to Matthew) the idea that, according to the story, denarii must have been a very common coin in Judea and that the tax in question was always paid in denarii. Although the numerous references elsewhere in the Gospels do call into question the efficacy of the coin hoard method of estimating how common denarii were, in fact the story does not say either of these. It relies on the idea that Jesus was sufficiently familiar with a denarius to know what it would look like (hence his request to bring him one), that he could expect one to be available at the time of the conversation, and that the tax was at least sometimes paid in the form of this coin. It also assumes that Jesus’ audience knew this, though of course they could simply look at the denarius once it was brought to him and describe it. Nothing about the relative rarity of denarii in coin hoards prior to the Jewish War makes any of this impossible or even particularly improbable, especially since this same archaeological work confirms that such coins were in existence in the region prior to the Jewish War.

The exact wording of the story in Matthew 22:19 is slightly different: Jesus says, “Show me the coin for the tax.” But since this is the exact same incident as what is found in Mark 12:15, where Jesus says, “Bring me a denarius,” there is no reason to place weight on the exact Matthean wording. We need not think that the story (in either Gospel) Jesus is saying that the tax was per se paid using this coin, only that it could be. (Zeichmann says nothing about the Matthean wording. I'm bringing it up here to ward off a possible criticism.)

Zeichmann rightly warns (p. 431) against circularity between NT scholars and classicists, in which classicists cite Mark as a source concerning Roman taxation in the region and NT scholars then cite classicists as though the specific practices of taxation mentioned in the story were confirmed independently of Mark, though this is not the case. Of course we should make every attempt to avoid this sort of distributed circularity.

On the other hand, any classicists who cite the passage in Mark as historically relevant are entirely right to treat Mark as a 1st-century source on the subject in his own right. Why not? The sort of a priori history on display in this article is the dubious historical practice; the use of Mark as a source on coinage and taxation in the time of Christ is not.

In the final section (pp. 432-436), Zeichmann argues that the incident arises from the later fiscus Iudaicus imposed by Vespasian and is anachronistically inserted here. His suggestion is that some members of Mark's audience were Jews and/or were regarded as Jews by Rome and hence were being made to pay the tax, and that they were wondering if they should do so. The suggestion is that Mark projected a very similar tax back into Jesus' own time and placed words into the mouth of Jesus in order to reassure such readers that it was permissible to pay the fiscus Iudaicus. But in this section he repeats the arguments from earlier sections of the article, assuming their conclusions to be correct (e.g., that no earlier tax to Rome was collected in coins) and then using this to boost the claim that the incident arises from the sitz im Leben of Mark's audience. 

The only new argument in this section (p. 434) is the forceless point that "census" and "denarius" are transliterated into Greek from Latin. Zeichmann vaguely relates this to "perceived Romanness" in Mark and its relationship to Roman occupation of Judea. But as already pointed out, "denarius" is also used in widely varied stories in both Mark and other Gospels where the contexts have nothing to do with taxation or anything else particularly Roman. These widespread references actually put pressure on the whole argument of the paper that denarii were very rare before the Jewish War. In this last section Zeichmann seems to be arguing that the Latin transliterations of "census" and "denarius" would be likely to be picked up from places where a "Roman garrison was stationed" and where "Roman law was practiced" (p. 435) and that such situations would be more common in postwar Israel. This is part of his attempt to argue that Mark was written after the Jewish War. He even goes so far as to use the transliterated word "centurion" as part of this same argument (footnote 31, p. 435), which is truly bizarre, since as a matter of independent history we know that there were centurions in Israel at the time of Jesus.

Zeichmann points out that the fiscus Iudaicus had similar features to the tax that seems to be envisaged in the story in Mark--levied by census (a poll tax), collected in coin, and paid to the emperor. But that the fiscus Iudaicus had these three features does not make such a Markan invention a better explanation of the passage than the truth of Mark’s story. More than one tax could easily have these three broad features. So the observation that the fiscus Iudaicus had these features and the spinning out of a political motive on Mark's part doesn't constitute an independent argument that the story is unhistorical. It seems forceful only if we assume or at least take it as quite probable, based on the earlier arguments, that no tax to Rome was paid in denarii in the time of Jesus.