First, I would have assumed that you understood I was excluding the 1353-1384 range rather than the 1262-1312 range.
On the contrary, I regret that I am not always able to read your mind and thereby determine which of the many ways to misunderstand statistical analysis you will favor us with on any given day. Obviously despite my last message, your mystification over multimodal distributions continues undeterred. You are right to expect radiocarbon dates to be normally distributed. However, when those dates are calibrated to calendar dates—accounting for the varying amount of
14C in the atmosphere throughout history—the result is not normally distributed. The resulting distribution is instead multimodal. It doesn't matter which confidence band you "object" to; they all belong to the same distribution. The different modes aren't severable.
If you have a pair of 2σ confidence bands—let's say 100-200 CE and 350-500 CE—arising from a calibration, you can identify them on the x-axis of a probability mass function as
But the correct way to interpret their multimodal distribution is not
Instead, it's
Multimodal distributions are the norm for calibrated calendar dates in radiocarbon dating. Attached you can see the radiocarbon dating report for several samples form an excavation for engineering purposes I was involved with in Farmington, Utah a few decades ago—roughly the same era as the shroud test. Note how most of the dozen-ish samples sent for dating have multimodally-distributed calendar dates. Note also the notation about possibly having to compensate for the increase in
14C as the result of above-ground nuclear weapons testing in my state back in the day.
This was in the early 1990s, using a much later revision of the software than was available in 1988. As such, we had the luxury of weighting estimates giving the likely amplitude of each mode in the distribution. Leese, writing in 1986, notes that at that time there was no good agreement on how to reckon the contribution of each mode to the overall distribution, as is given in our report. Hence no such balancing was attempted in the shroud dating, which remained scientifically conservative on that point. Leese notes, however, that various forms of estimation were possible—some empirical, some purely probabilistic. These are referred to in our report as Method A, Method B, and so on.
What you need to come to terms with is that we can tell that you're bluffing your way along in a discussion with people who actually have to do this for their jobs and who know the science and the math inside and out. You don't know radiocarbon dating. You don't know statistics. Yet you're arrogantly convinced every single day that you've found some smoking gun that proves the shroud dating is "obviously" wrong. As I said before, you either need to take some serious time off and grapple with the actual complexity of the sciences you're Googling your way through, or admit the obvious—that you're in over your head.
If it is documented historically to have appeared in 1356, then yes the radiocarbon date is incompatible with dating by historiographical methods.
It's hard to know where to begin to tell you everything that's wrong with this statement.
First, and most obvious: "incompatible" is not a thing here. That a true value will gravitate toward one mode or another in a multimodal distribution is the intended behavior of the distribution—and, in fact, of
any distribution.
Second, the modes are not somehow mistakes or missteps. They are simply reflections of a thoroughly irreconcilable ambiguity in the underlying phenomenon. Something with a certain amount of
14C can either be a young object that hasn't undergone as much radioisotopic decay, or it may be an old object that was around when there was a lot of
14C around. That the math gives you two or more answers doesn't mean someone messed up and needs to be chastised for it. Just because some functions in math, for example, have more than one zero—and that only one of them may be the right answer to the problem—doesn't mean that algebra is trying to hide something.
Third, that the shroud is documented to have been witnessed in a certain year doesn't preclude that it didn't exist much earlier. More importantly, radiocarbon dating tells you when the flax was harvested, not when the thread was spun, not when the cloth was woven, not when the image was created, and not when it was finally written about by someone whose writings persisted. You're trying to narrow the goalposts in a particularly ridiculous way. It's not valid history, it's not valid science, and it's not valid statistics.
Seems this paper agrees with excluding the 1352-1384 range.
No, you're just projecting your confusion onto Christen. Exclusion isn't a thing here. There is no concept in the interpretation of multimodal distributions for "excluding" a mode. There is only the concept of the most likely value gravitating toward one or the other of the available modes. That other factors may contribute in your estimation of where the true value most likely lies within your multimodal distribution does not make the radiocarbon methods or the statistical methods somehow inaccurate, or indicate that they were somehow done incorrectly. Nor does it provide any justification for misstating the statistical outcome by omitting a mode that you think (for other reasons) shouldn't apply. It would have been highly dishonest for Damon et al. to
not report a confidence band that was generated by the calibration.
I referred to Christen to establish two points that were material to my rebuttal of Casabianca. First, to dispel the notion that there is a single, God-given statistical model that definitively answers a given question. Hence, there will never be a "slam-dunk" based on statistical analysis, so stop looking for one and stop believing others when they desperately tell you they have one. Second, statistical norms are not applied as inflexibly as you are trying to do—neither generally in radiocarbon dating nor specifically in the case of the shroud. You get points for at least seeking out and reading what I cite to. But you lose all those points and more when it appears your interest in the material rises no higher than mining it for the next day's desperate ploy for yet another poorly-informed, slam-dunk smoking gun. Try to imagine a reality in which all the world's scientists are not conspiring to hide the truth about the shroud of Turin.