No comment at this time on Kolmogorov, but the Poincaré example I found particularly sketchy. Clearly Bertillon and arguably also Poincaré were not trying to use mathematics to find the right answer, they had a right answer in mind and were trying to convince others of it. (Lies, damned lies, and statistics, as they say.)

In any case Poincaré was not putting forth a mathematical argument for innonce so much as destroying Bertillon's mathematical argument for guilt. Are we really sure he advocated the use of subjective priors here? (And again, the controversial point is not Bayes' Theorem itself, but the priors). So I would like to see more details on this.