Carrier’s “Proving History”, Chapter 3(a) — Review

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by Neil Godfrey

I have been studying the first half of Richard Carrier’s chapter 3, “Introducing Bayes’s Theorem”, in his recent book Proving History: Bayes’s Theorem and the Quest for the Historical Jesus. I mean studying. I want to be sure I fully understand the argument before tackling the second half of the chapter, headed Mechanics of Bayes’s Theorem, which promises to be “the most math-challenging section of the book” (p. 67). Maths is not my most outstanding strength so I want to be sure I get the basics clear before moving into those waters. I have come to a point where I can enjoy playing little mind-games with Bayes’ Theorem for the sake of reinforcing my understanding. Last night on the TV news was dramatic story of an unexpected resignation of a leading Australian political figure. So I found myself piecing all I heard, how I heard it and what I knew etc. into a Bayes’ equation and calculating the probability that the story was true. Kind of fun. At least for the moment before the novelty factor wears off.

Result: While I believe I can see Richard’s point some of my niggling questions have not yet gone away.

When did the sun go out?

Carrier begins by setting out our reasoning when we read in the Gospels that darkness covered the whole earth for three hours at the time of the crucifixion of Jesus. What he is seeking to do is to take readers through the processes they would undergo in order to conclude that such an event almost certainly never really happened.

To make the scenario work he posits at least a barely conceivable natural cause for the event: “a vast dense cloud of space-dust swiftly drifting through the plane of the solar system . . .” — Wouldn’t the Sun’s gravity prevent that? But I’m happy to go along with the exercise for sake of argument nonetheless.

The critical point for Carrier is that what would convince us that such an event really had happened in the past is if we could find records testifying of the event across all world cultures thousands of miles apart from Britain to China.

There could not fail to have been mention or discussion of such a remarkable and terrifying event across many of these cultures among their surviving textual traditions and materials. (p. 43).

The key point is that we know in advance that this is the evidence we would expect to find IF such an event had happened.

And if indeed that were the case, we would surely have adequate warrant to believe the sun was blotted out for three hours on the corroborated day . . .

What Carrier is preparing his readers for is to accept that reasoning about historical events is fundamentally similar to reasoning in the sciences. If such and such a hypothesis (or explanation) is true then we would predict (or expect) certain events (or evidence) to be manifest.

Then there is the converse. If such a hypothesis (explanation) were true, we would NOT expect to find a universal silence in the surviving records:

[A] single claim in a single religion repeated only in its own documents (and documents relying on those), is extraordinarily improbable — unless the event was entirely made up. . . . This is a slam-dunk Argument from Silence, establishing beyond any reasonable doubt the nonhistoricity of this solar event . . . (p. 44)

My niggling question:

I follow the reasoning. But in my mind, rather than taking me into the realm of mathematics, it all leads back to my own argument about how historians know anything at all about the persons and events of the past.

Common testimony across clearly independent cultures, and that is found within a range of genres, including “scientific” (by the standards of the day) treatises, philosophical discussions, diaries, official monuments, that are known for seriously conveying factual events, — it is this multiply and independently attested record that is our verification. Against this, all the claims that it really did happen derive from a single tradition and so are worthless when set against the unexpected and highly problematic silence of the wider record. This principle is at least as old as Albert Schweitzer:

In reality, however, these writers [those arguing for the historicity of Jesus against mythicists] are faced with the enormous problem that strictly speaking absolutely nothing can be proved by evidence from the past, but can only be shown to be more or less probable. Moreover, in the case of Jesus [or the sun going out for three hours at the crucifixion], the theoretical reservations are even greater because all the reports about him go back to the one source of tradition, early Christianity itself, and there are no data available in Jewish or Gentile secular history which could be used as controls. Thus the degree of certainty cannot even by raised so high as positive probability. (From page 402 of The Quest of the Historical Jesus, 2001, by Albert Schweitzer.)

But I know that there are biblical scholars who disagree with this, or at least who refuse to address the principle expressed here. The same principles are applied by the so-called “minimalists” in their study of the Old Testament and history of Israel. No matter, some of the same scholars dismiss them, too. Again we can find the principles explicitly set out in handbooks on basic historical method:

Historians can place trust in oral sources only to the extent that they can be verified by means of external evidence of another kind, such as archaeological, linguistic, or cultural. (p. 26, Howell and Prevenier)

In order for a source to be used as evidence in a historical argument, certain basic matters about its form and content must be settled. (That is, is it a mythical tale or a serious factual record? p. 43)

So when Bart Ehrman proclaims we have as many as seven independent or “partially independent” sources behind the Gospels, his words necessarily have to fall flat. Such “sources” alone tell us nothing about the historicity of they narrate. As Steven Carr recently noted, to suggest they do have such power is comparable to claiming that the sources behind the Old Testament (J, E, D, P) are evidence for the existence of Moses and Noah.

The question for me is why do biblical scholars who profess to be historians (and admittedly a few lazy historians in other areas) not follow such fundamental principles? Would it not save them the punishment of having to attend remedial mathematics classes?

Literary analysis and independent controls sound to me like solid and reliable methods for historians. Why don’t biblical scholars take them seriously?


From science to history

Historians need solid and reliable methods. Their arguments must be logically valid, and factually sound. Otherwise, they’re just composing fiction or pseudo-history. (p. 45)

I have had two different reactions to Carrier’s words above.

  1. As a stand-alone remark Carrier’s words strike me as jarring against the grain of everything that has been written about the nature (philosophy) of history since Leopold von Ranke. Von Ranke was famous for several lines, one of which was that history should tell it just like it happened — as if such an endeavour were at all possible. I suspect most historians would acknowledge that they place a certain spin, a certain ideology, a certain filter, on past events that necessarily has more to say about the present generation of scholars and readers than it does about the past generation of historical actors. But Carrier has made it clear he is writing this in the context of the quest for the historical Jesus, so I will cut him some slack here. Though I know I shouldn’t, given what I am just about to admit in the following insert:
    • Another niggling doubt: I personally don’t think “a quest for the historical Jesus” is an historical exercise at all. That quest necessarily begins with a presumption that there is an historical Jesus to be found, and that is hardly a “scientific” or valid methodology. Historians begin with a literary and theological Jesus. Since that is their starting point their logically valid procedure would be to explain that Jesus who is portrayed there in their evidence or documentation. Whether there was “an historical Jesus” or some other entity at the base of the evidence is what the historian should be interested in discovering.
  2. That’s my first reaction. But if you don’t like that one I have another. When it comes to “history wars” and ideological contentious questions over “what really happened” in the past then I can understand how any exercise that helps clear the air by cutting through the passion of belief and forcing professional noses into nothing but strictly sound methodologies is a good thing. I found myself at the pointy end of such an exercise when I was forced to acknowledge that I was being inconsistent with my arguments for historical methodologies in relation to Christian origins compared with how I treated the historical evidence relating to the European treatment of aboriginals in Australian history. (I posted on this at Ouch! My own beliefs undermined by my own historical principles!) Where the historical argument is over “what is the fact of the matter” and where the ideological and emotional stakes tend to stifle clear objective processes (e.g. what was the extent of the Holocaust? what was the real history of the sacred site at Ayodhya? and the Australian “history wars” example above) there is no doubt that the effort to think through all the evidence to accord with Bayesian thinking might be helpful.
    • Of course no method can be guaranteed to solve issues that revolve around ideological (or religious) commitment. One can anticipate the arguments for the ideological bias or use of Bayes’ theorem. Besides, my own change of mind was not a change of belief that certain events had happened in the past, but a reminder of acceptable and consistent methodology itself quite apart from Bayesian thinking. I suddenly had to face the fact that I could not support my beliefs about past events as objectively as I had thought I could. Maybe I should take the time to toy with Bayes’ theorem to see where my revised approach to the evidence might lead. There is certainly no reason to assume it will inevitably lead to the other side of politics being completely right.


Carrier believes all valid historical reasoning can be described by Bayes’ theorem.

Much has been written on the method and logic of historical argument. And yet, though note of it is aware of the fact, all of it could be reduced to a single conclusion: all valid historical reasoning is described by Bayes’s Theorem (or BT). (p. 45)

Now logically I have no problem whatever with Richard Carrier’s argument.

In simple terms, Bayes’s Theorem is a logical formula that deals with cases of empirical ambiguity, calculating how confident we can be in any particular conclusion, given what we know at the time. . . . [I]ts conclusions are always necessarily true — if its premises are true. By “premises” here I mean the probabilities we enter into the equation, which are essentially the premises in a logical argument. (p. 45)

When we have reasonable values [in the formula] . . . Bayes’ Theorem entails a particular conclusion as to how probable our theory is — given all that we know at that point in time, since a Bayesian result is a conditional probability. It’s conditional on current knowledge, which means if we discover new theories or facts, the conclusions might change. Bayes’s Theorem thus tells us what we are warranted in believing at any given time, fully acknowledging that this can change with new information. (p. 53)

I single out these passages because I see that at least one visitor to R. Joseph Hoffmann’s blog has, and with Hoffmann’s own expressed approval, ignorantly accused Carrier of a logical fallacy as if he is suggesting any old nonsense that is put into Bayesian terms is logically valid. Hoffmann does not like Carrier personally and appears to be quick to presume any fallacy even where there is none. Carrier is clear: the validity of the results hangs upon the validity and comprehensiveness of the inputs.

This is the same as saying that we must always consider our conclusions to be tentative. We must always be open to new evidence, and to discovering hitherto overlooked problems with our arguments. And this is all part of what Carrier says is built in to the thought processes of the Bayesian thinker.

Carrier argues that Bayes’ Theorem models everything that historians do or test against their hypotheses. It is for this reason that he recommends historians know more about it.

Carrier presents a strong case for accepting a certain similarity between the hard physical sciences and reasoning in historical inquiry. Compare geology. This is a study of the past processes in the earth’s formation, but it is also a science that works with predictions. It proposes hypotheses about what ancient evidence we can expect to find in unexamined areas of the earth and in the future. Out bushwalking in a mountain area and see some fossilized seashells? The hypothesis indicates to us that the earth we are walking upon was once below the sea.

All of these conclusions are theories: theories about how all the evidence came about that survives for us to see today. (p. 47)

I suspect Carrier takes us just one step too far when he goes on to say that these same theories entail predictions of what will happen in the future — which way a river will be found to be turning, etc. Is history really ever able to predict like this? Because Carrier writes, “History is the same.

But I mulled over Carrier’s next words for some time and I have finally (dinosaur thinker that I am) come to accept them as true:

The historian looks at all the evidence that exists now and asks what could have brought that evidence into existence. And tautologically speaking, what most likely brought it about is what most likely happened. [The historian] can then infer what other evidence could be found someday (whether finding it is at all likely or not), and what couldn’t, if her theory is true. (p. 47)

Carrier then addresses the “kinds of evidence” to be expected. That word “kinds” reminds me of the nature of the evidence we currently have for Jesus — letters, gospels and treatises. But what kinds of evidence are gospels? That is, what is their genre? To answer that means a literary analysis and a deep study of the theory of genre. (Bring in Bakhtin for starters.) Burridge’s theory-free, ankle-deep analysis of the genre of the Gospels that concluding the Gospels are “biographies” won’t do. And even when we do understand the kinds of evidence we are dealing with, we are still no closer to knowing the true motives or backgrounds of their authors. There are many variables here.

What finally convinced me to go along with Richard Carrier’s argument was when he pointed out the following equivalence:

[A]ctual predictions (such as that the content of Julius Caesar’s Civil War represents Caesar’s own personal efforts at political propaganda) . . . . follow from historical theories. This is disguised by the fact that these are more commonly called ‘explanations.’ But theories is what they are. (p. 48, my emphasis)

Okay. This addresses one of the concerns I raised in my first post in this series. Let’s continue to give Richard Carrier a fair hearing as I promised back then.

Carrier concludes this section of his chapter by proposing that theories in history are of two basic kinds:

  1. theories of evidence — how did the content of our literary documents come to be what it is and survive for us today
  2. theories of events — how real events in the past came about

Historians rarely realize the fact, but all sound history requires answering three difficult questions about any particular theory of evidence or events:

(1) If our theory is false, how would we know it? (e.g., what evidence might there then be or should there be?)

(2) What’s the difference between accidental agreement of the evidence with our theory, and an agreement produced by our theory actually being true — and how do we tell the two apart?

(3) How do we distinguish merely plausible theories from provable ones, or strongly proven theories from weakly proven ones?

In other words, when is the evidence clear or abundant enough to warrant believing our theory is actually true, and not just one possibility among many? (p. 48, my formatting)


What is Bayes’ theorem?

Richard Carrier acknowledges that the application of Bayes’ theorem to contemporary studies is not his own baby. Archaeologists, for example, are making sophisticated use of it. I myself recall (first?) encountering it in C. A. J. Coady’s chapter 10 of Testimony: A Philosophical Study, a book I probably read at the time I was engaged with Richard Bauckham’s Jesus and the Eyewitnesses. Carrier directs readers to several websites explaining it.

I won’t repeat the mathematical formula Carrier uses here. I would be certain to make it sound far more complicated than it really is.

Instead, here are a few of the positive or encouraging things that Carrier says are factored into a Bayesian formula. I add my own thoughts of how some of the principles might be applied.

  • The measure of how “typical” our proposed explanation is, is a measure of how often that kind of evidence (or that kind of event) has that kind of explanation (rather than some other).

To say that again:

  • In any kind of causal reasoning, [the measure of how typical our explanation is] does not measure how often such a thing happens, but how often such a thing happening is the explanation of that kind of evidence (rather than something else explaining that same evidence).
    • (That is, we are not measuring how often a person is resurrected from the dead, but how often a real resurrection is the explanation for a tale about an empty tomb)

I like repeating it and Carrier rightly repeats it because it keeps the focus on the evidence, or on the fact that we are discussing Gospel narratives. So often one reads biblical scholars naively reading a narrative as if it is a direct window to real events.

Then there’s another key point:

  • If our theory is true, then what sort of evidence do we expect, and how well does the evidence we actually have match that explanation?
  • How likely is the evidence if some other explanation is true — some explanation other than our own? To answer that question, we have to seriously look for, and seriously consider, alternative explanations to the evidence.
  • If all the evidence we can reasonably expect to have is someone’s word, that is exactly the same evidence we would expect to have whether they were lying or telling the truth.
    • The hermeneutic of charity (accepting someone’s word as true until we have reasons to doubt them) is just fine for neighbourhood and general social and personal cooperation, but it’s downright naive when applied to unknown authors.
  • The fact that a theory’s [explanation’s] prior probability is not an absolute probability, but a relative probability, is commonly overlooked yet this is one of the most important features of correct reasoning about a claim’s probability.

After explaining the workings of a simplified Bayesian formula (and that is the context of the above quotations) Carrier explores how this Bayesian reasoning can be applied to the problem presented at the beginning of the chapter: the explanation for the accounts of three hour darkness at the crucifixion.

Again I will not attempt to cover here the details of the logical steps but single out quotations that I found made the whole exercise pleasing to my tastes.

  • When the evidence really is good enough, even the incredibly improbable becomes likely.
  • Thus, even extremely low prior probabilities can be overcome with adequate evidence.
  • It’s not that anecdotal evidence is necessarily false. It’s just that it’s much too likely to be false.
  • Bayes’s Theorem is thus not an alien way of thinking. It’s just an exact model of how we always think (when we think correctly). Thus, when applied correctly, BT will not only represent correct thinking about any empirical claim; it will help us identify and expose incorrect thinking. Because the one thing Bayes’s Theorem adds to the mix is an exposure of all our assumptions and how our inferences derive from them. Instead of letting us get away with using vague verbiage about how likely or unlikely things are, Bayes’s Theorem forces us to identify exactly what we mean. It thus forces us to confront whether our reasoning is even sound.
    • Oh how often does one see in the writings of biblical scholars such meaningless copouts as “that’s not persuasive” or “this is more likely than that” etc! It is appalling! It is even more appalling when one hears a bible professor demanding his guild be treated with the same respect as those of the hard sciences. Respect will be earned to the extent that we can see something more professional than the level of reasoning one encounters everyday among the untrained. Knowledge and consensus opinion are not sufficient to impress. Valid methods of reasoning and argument are.

Okay, I have a natural love for any discussion that helps me better understand and practice clear thinking. While reading the above pages I was also wondering why the same principles could not be pointed out without any reference to mathematics and probability theory. One of my old and favourite books on historical methods, or at least on common errors found even in the works of some of the most renowned historians, is Historians’ Fallacies: Toward a Logic of Historical Thought by David Hackett Fischer. I picked it up cheap at a used book store and it has been a favourite of mine ever since.

What I’d love to see would be a comparable book picking out all the logical fallacies in the arguments of biblical scholars. Would not such a work, composed in a good-humoured way to sugar-coat the embarrassment many will no-doubt experience, be as effective in lifting the intellectual standards of many of these scholars? Maybe there should be an annual revision: updates of the worst or the most commonly boring logical howlers and their perpetrators.


Why Bayes’ Theorem

Carrier, however, is not arguing for such a book. At least through Bayes’ theorem he is attracting some attention for his arguments. He believes there are two main advantages of the Bayesian method:

  1. No one who accepts the validly stated premises can deny the conclusions;
  2. It forces us to consider what those premises really ought to be. That is, it pins down our subjective assumptions and makes them explicit and accessible to criticism.

With BT, instead of myopically working out how we can explain the evidence “with our theory,” we start instead by asking how antecedently likely our theory even is, and then we ask how probable all the evidence is on our theory (both the evidence we have, and the evidence we don’t) and how probable all that evidence would be on some other theory . . . . (p. 61)

Many will ask what maths has to do with history. Carrier answers this by arguing that whenever we say something is “more likely” or “most likely”, “plausible” or “implausible”, etc. we are in effect thinking mathematically. To say something is “probably true” is to mean it has a probability greater than 50%. We rarely think in absolutes when estimating probabilities. We order options in cascading levels of confidence.

My maths-challenged mind keeps asking: If I am really doing mathematical thinking behind all of my ordinary words then why do I find it so difficult to think mathematically? That’s not to say Carrier’s point is wrong. It probably isn’t. (A mathematical statement!) All those little neurons and synapses where our thinking takes place is surely at some basic level mechanical or operating on mathematical principles.

Carrier is a firm teacher, though. He will not accept the excuse that “math is hard”.

“It’s too hard” is not an argument we should ever hear from a professional historian — because mastering difficult methods is what separates professionals from amateurs. (p. 66)

I might in a future post present a few of my own little simplified exercises with Bayes’ theorem. I’m thinking of assessing the probability that our famous phrase about Paul meeting the brother of the Lord is best explained by Jesus having had a real brother, James. That’s just one. There are many more. Could be fun.

To be continued. . .

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8 thoughts on “Carrier’s “Proving History”, Chapter 3(a) — Review”

  1. Also “When the evidence really is good enough, even the incredibly probable becomes likely.”

    Didn’t you mean ‘incredibly improbable’? This is just application of the Sherlock Holmes ‘when you eliminate the impossible’ etc idea.

    I admit to being a bit lightweight mentally – advanced age makes it fatiguing to think really hard about difficult matters – but as you seem to indicate somewhat, all this ‘Bayesian’ stuff seems unnecessary. Bayes calculations properly refers to calculating the expection of B given C, both of which are subsets of a set A with given properties. It is purely a computational tool. In proper usage, you know things about A and C to arrive at the likelihood of B.

    Appropriating a purely computational tool into a historical investigation where the values (priors) are esimated rather than known doesn’t initially impress me. (I sure don’t mean to denigrate Carrier, his take down of Ehrman’s HuffPo piece was awsome.) I await hearing more of your impressions, since the ability to blow up the print size on the computer makes reading books a thing of my past (as also do cost considerations).

    1. I gather that Carrier’s point is not so much of using a “purely computational tool”. It’s more about knowing how probability works, since all proper logical thinking is proper probabilistic thinking. For example, Occam’s Razor is not just a logical argument, it’s a probabilistic argument. And why it makes so much sense becomes apparent when you think probabilistically. You don’t actually have to do computation at all, you just have to know the rule.

      Falsifiability, also, is a probabilistic argument… it’s not just a handy demarcation between “science” and “non-science”. Think about the theory of evolution and Christians’ response to it. Obviously, a god should have no reason to create humanity via such a horrible process like evolution, but liberal Christians say “well, god could have done it that way anyway”. On the other hand, a universe with no god can only have humans come about through evolution.

      If we compare the two hypotheses with Bayes’ theorem, we can see why an unfalsifiable hypothesis will always have less explanatory power than a falsifiable one without going through the whole computation.

      P(Christian God | Evolution) = P(Evolution | Christian God) * P(Christian God) / [P(Evolution | Christian God) * P(Christian God)] + [P(Evolution | No God) * P(No God)]

      When Christians say that god “could have” created humans via evolution, they are talking about the conditional probability P(Evolution | Christian God). But if humans came about by direct creation just like the Bible says, they wouldn’t claim that this is evidence against the Christian god so that has to be included in the conditional probability as well: P(No Evolution | Christian God). The two conditional probabilities, since they are each one half of the same coin, have to add up to 100% as a rule of probability; P(Evolution | Christian God) + P(No Evolution | Christian God) = 100%.

      On the other hand, P(Evolution | No God) we know is 100%. Thus without going through the entire computation of Bayes’ theorem, we know that P(Evolution | Christian God) has to be less than P(Evolution | No God). And also as a rule of probability, if one conditional probability is lower than the other, then the hypothesis with the higher conditional better explains the evidence. This situation with the conditional probabilities only came about because the Christian god is unfalsifiable. But if a historian isn’t thinking probabilistically, they might unwittingly posit an unfalsifiable hypothesis even if they have the best intentions and are thinking only logically.

  2. Thanks for the corrections.

    Social Sciences As SorceryI have in the past referred to another book I liked, Social Sciences As Sorcery by Andreski. Carrier would seem to come close to falling into this trap with his remark that professionals are set apart from amateurs by learning “difficult methods”.

    There is a difference, though, between difficult methods and explaining methods that are themselves simple enough to understand in a manner that makes them sound very difficult. The latter is a sign of the one who either does not understand the methods very clearly, or the sign of one who seeks to impress. In older days it was the use of Latin that set the professionals apart.

    At least Bayes’ theoerm does express genuine content, though. It is a cut above other jargon that is a smokescreen for vacuity.

  3. R. Joseph Hoffmann has responded to this @ http://rjosephhoffmann.wordpress.com/2012/03/25/4716/#comment-5258

    Here is his comment, though on the original site it comes complete with his nice shiny forehead branding:

    Vridar has a knack for using words like ignorantly to disguise his utter ignorance: there is nothing ignorant about accusing Carrier of having a crackpot theory made up of ersatz-logic pretending to be a logical knock down argument for his myth theory. Godfrey has become a cheerleader and postmaster for the mythtics, but hardly has anything worth contributing himself. This is what vetting and critique look like; it is what happens when NT scholars float ideas and theories. It is time for Carrier to respond to these criticisms with facts. He is frontloading assumptions into his Bayes machine and coming out with sausage. None of the assumptions as far as I can tell bear scrutiny–but we’ll get to that on this site in about a week…

    It’s made my day! 🙂

    1. This morning the Robert and Henrietta Campbell Professor of Religion, R. Joseph Hoffmann, called me “utterly ignorant”. [http://rjosephhoffmann.wordpress.com/2012/03/25/4716/#comment-5258]

      That was because I pointed out with quotations from Richard Carrier the falsity of a claim about Carrier’s argument on Hoffmann’s blog.

      This afternoon the Clarence L. Goodwin Chair in New Testament Language and Literature, James McGrath, called me “dishonest, uncomprehending and mad” [http://www.patheos.com/blogs/exploringourmatrix/2012/04/review-of-bart-ehrman-did-jesus-exist-part-two.html#comment-498538606]

      That was because I pointed to an article, and my discussion of it, by a Classicist, Professor John Moles (and an earlier article by Professor of Greek Studies, C. J. Mackie), that in my view abundantly demonstrates the plausibility of a divinity being assigned the otherwise common name of Jesus (or its Greek equivalent, Jason).

      Even Classicists can see through the fallacy on that one. You have surely read Classicists like John Moles and C. J. Mackie address that very question and conclude that Jason/Joshua/Jesus is THE MOST APT name for a new saviour/healing god!

      Check out “Greece & Rome”, vol. 48 no. 1, April 2001 (Mackie: The Earliest Jason: What’s in a name?” — this article takes on special significance when read alongside the next one . . . ) . . . that next one being John Moles’ more recent article, linked at Gospel Puns on the Name Above All Names and Creativity with the Name of Jesus the Healer in the Gospel of Mark.

      And remember back in the days when critical biblical scholarship was dominated by those far more extensively incisive French and Germans? 😉 — Even the anti-mythicists of those days could bring themselves to question the historical plausibility of the name of Jesus of Nazareth — Would the historical Jesus of Nazareth really have been named Jesus of Nazareth?

      I’m beginning to wonder if there’s a little intellectual insecurity in evidence or is it just pig-headed arrogance?

  4. 3(b) please! I’m sure that by now you’ve realized that the math (sorry, “the maths” for you) isn’t all that hard. It’s determining the priors / consequents that’s hard. And that’s on purpose.

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