True. It is not logically implied from him being right that he had good reason to believe he was right. However, I think it is very strong evidence. Fair warning: I am very new to using Bayes’ Theorem, so please make sure to be highly critical of my math, and tell me what, if anything, I’m doing wrong.
First, we must assess the prior probability of Einstein having sufficient evidence, given that he thought he was correct. How often do modern scientists come up with theories that are quickly falsified? Let’s be pessimistic and assign 0.001 prior probability for Einstein making his claim with sufficient evidence. That is, only 1 in 1000 credible scientists who publish theories come up with theories that aren’t easily falsified.
What is the probability of him being correct if he had sufficient evidence? Well, if we say that having sufficient evidence means having evidence such that your prediction has P>0.95, then, if someone has sufficient evidence, their prediction must have P>0.95. Let’s assign a probability of 0.95.
What is the probability of him being correct if he had insufficient evidence? To be strictly logical about this, we would need to take this probability as 0.95 as well, to avoid a false dichotomy. It is not true that either Einstein had p=0.95 worth of evidence or he had no evidence at all. If we say that he necessarily has p > 0.95 given that he has sufficient evidence, we’d have to say that anything under p=0.95 is insufficient evidence; in which case, to be pessimistic, we’d have to assign the probability of him being correct given insufficient evidence to be infinitesimally less than 0.95. This would result in a likelihood ratio of approximately 1. However, this is only the case if we view “insufficient evidence” and “sufficient evidence” to be distinguished by a sharp point on the real number line. This would contradict common sense; we don’t say that p=95 is sufficient but p=94999999 is insufficient. It’s a gradient. So we should choose a number that is definitely insufficient evidence. What you claimed was that “He could have been a lucky draw from the same process”, so let’s go with sufficient evidence being p>0.95 and insufficient being no evidence at all: a guess. With no evidence the probability would be 1/n, n being the number of competing hypotheses. Let’s go with Eliezer’s number of 1⁄100,000,000.
Plugging these numbers into Bayes’ Theorem, we get a posterior probability of roughly 0.999 that Einstein had sufficient evidence to support his belief. Note that this number is larger than 0.95, our previously assumed standard for sufficient evidence. If we instead use 0.99 as our standard, Bayes’ Theorem spits out a posterior probability of roughly 0.999 regardless.
In either case, Einstein being correct about his theory gives us more than enough evidence to conclude that he had sufficient evidence to make the claim.
Is that true? Does the discovery of a non-mysterious explanation serve as negative evidence of a mysterious one? If by “mysterious” we mean “not strongly predicting any outcomes” then when a theory that is strongly predictive is discovered, the available evidence will shift probability from the mysterious explanation to the predictive one. And I do think that’s what you mean by “mysterious”, so that’s a good observation.
However, I’m not sure that this is the case with Elan vital. I’m not sure that Elan vital was weakened by virtue of being nonpredictive and thereby losing some of its credibility to a more predictive explanation. It may also be the case that Elan vital predicted something which was contradicted by the synthesis of urea. If this was the case, then it is inaccurate to say that Elan vital served only as a curiosity-stopper or was a fake explanation. It would, in fact, be a perfectly fine explanation.
So what was the theory of Elan vital, and did it prohibit urea synthesis? The theory was that the matter of living bodies has a property that non-living bodies can’t have. This is clearly a predictive theory. It prohibits any living matter from being identical to non-living matter because if two things are identical then there aren’t any properties that one has that the other doesn’t.
The synthesis of urea showed that at least one type of biological matter could be produced using purely mechanical means. Is this evidence that the synthetic urea and the natural urea are identical? No. One could postulate that there was a non-material difference between them. However, this would make that difference impossible to observe, and thus make the theory nonpredictive.
According to your post:
The fact that this synthesis served as negative evidence implies that the proponents of Elan vital believed that the difference between biological matter and non-biological matter would be observable. Therefore, the theory of Elan vital, as portrayed in your post, was predictive. Admittedly, not very predictive, but also not merely a curiosity stopper. Rather, it was a sufficiently vague theory to reflect the lack of evidence that they had at the time.
On the other hand, it’s a self-contradictory theory. If they thought the difference would be observable, they must have also thought that it was physical. Yet they thought that the processes of life could not be explained with physical interactions. Yet they clearly thought that those processes were caused by the physical structure of biological matter. How can something be caused entirely by physical matter but not be explainable by a physical explanation?
In conclusion, there are valid criticisms of Elan vital, as portrayed by your post, but not the ones you leveled against it.