Missing data

A book review gives an opportunity to its author to express an idea without supplying any justification - the necessary brevity of a review provides a good excuse. The result is that an idea expressed by the reviewer may be a dead end - it cannot be further discussed because there is no foundation for any further discussion. Readers who like the general tone of the review will applaud the reviewer, and those who do not like the general tone will ignore the specific idea. I will illustrate the above with three examples coming from three reviews of my book, in chronological order.

The first version of the review of my book by Christian Robert, posted on Arxiv on January 28, 2010, contains the following remarks (I could not find these remarks in the second version of the review). First of all, Robert quotes the first sentence of my book: “This book is about one of the greatest intellectual failures of the twentieth century - several unsuccessful attempts to construct a scientific theory of probability.” Later he writes "[...] I can readily think of more pressing intellectual failures in the twentieth century." Robert does not say what failures made his list. Communism and fascism come to my mind but I would label them "genocidal ideologies" rather than "intellectual failures". I am as critical of Robert's review as he is of my book but I am genuinely curious about what he (and other people) consider to be the greatest intellectual failures of the twentieth century.

Andrew Gelman published a review of my book in Bayesian Analysis (2010) vol. 5, pp. 229-232. He quotes my claim that "Similarly, the 'Bayesian statistics' shares nothing in common with the 'subjective philosophy of probability." Then he writes "It's true that in our book (Gelman et al. 2003), we emphasize that Bayesian data analysis does not rely on subjective probability, but ... `nothing in common'? That's a bit strong." So Gelman thinks that the Bayesian statistics and the subjective philosophy of probability have something in common. Indeed, they do. Both fields assume that probabilities are numbers with values between 0 and 1. I am sure that one can find some other common trivialities. I would be curious to know whether Gelman thinks that the two fields have non-trivial elements in common and if so, what they are. I cannot learn this from his review.

Finally, I will discuss a puzzling remark in the Math Review of my book (MR2510150) by A.I. Dale. He quotes, among other things, one of my laws: "(L5) An event has probability 0 if and only if it cannot occur. An event has probability 1 if and only if it must occur." Then he says "[...] one might strain at the gnat of the ‘only if’ in (L5) [...]". In other words, Dale is not supportive of the following implication "If an event has probability 0 then it cannot occur." I have to guess what his doubts are. A standard philosophical objection to the implication in question is that if we choose a real number from the interval [0,1] in a random and uniform way then for every fixed real number x in this interval, the probability of choosing x is zero; but some real number will be chosen. I discuss a similar example on page 50 of my book. This philosophical problem is related to the fact that probability is NOT uncountably additive. I do not know if Dale's remark about the "gnat" refers to this type of philosophical problem, whether he is dissatisfied with my analysis and whether he knows of a better philosophical treatment of this problem.