Math Review

A.I. Dale wrote a Math Review (MR2510150) of my book. I find his style somewhat strange and I do not think that the aphorisms quoted by Dale contribute much to our understanding of probability or to the discussion
initiated in my book (see On aphorisms). However, I consider the review to be fair and reasonably accurate. I have multiple complaints about only one paragraph in Dale's review. The paragraph in question says "[Burdzy] gives a system of five sufficient (and necessary) axioms-none excitatory-ranging from ‘(L1) Probabilities are numbers between 0 and 1, assigned to events, whose outcomes may be unknown’ to ‘(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.’ Though one might strain at the gnat of the ‘only if’ in (L5), it is perhaps more difficult to swallow the camel of deductions Burdzy draws from his system."

(i) I made a considerable effort to draw a distinction between axioms and scientific laws, see Sections 11.4 and 11.5 of my book. Calling my laws "axioms" is misleading, unless the reviewer wants to say that he disagrees with my justification for calling (L1)-(L5) "laws" (I have no reason to think that he does).

(ii) I argue on pages 38-39 of my book that "Some widely used procedures for assigning probabilities are not formalized within (L1)-(L5)." I discuss a possible sixth law (L6) in Section 3.6.1. I consider (L1)-(L5) "the best compromise [...] between accuracy, objectivity, brevity, and utility in description of the past situations involving uncertainty" (see page 43 of my book). Hence, I do not understand why the reviewer said that (L1)-(L5) are "sufficient (and necessary)" - I do not recall saying this anywhere in the book. I do argue in Section 3.6, titled "Moderation is golden", that (L1)-(L5) are a good starter kit, not too small and not too large. This is very far from saying that (L1)-(L5) are "sufficient and necessary".

(iii) I have already expressed my surprise at and my lack of understanding of the "gnat" in another entry in this blog (see Missing data).

(iv) The referee finds it "difficult to swallow the camel of deductions Burdzy draws from his system." I am surprised that he found any deductions at all. The book contains a substantial chapter devoted to the discussion of the five laws but "discussion" is not the same as "deductions". It is possible that what the referee means by "deductions" is what I call "discussion". But this is more than just a linguistic matter. Kolmogorov's axioms are a starting point for mathematical deductions. The subjective philosophy of probability is sometimes presented as a formal system (due to von Neumann and Morgenstern) in which one adopts axioms concerning rational decision making which are followed by deductions that lead to the representation of rational decisions as maximizers of (some) expected value of (some) utility. The reader of Dale's review may be under impression, after reading the word "deductions", that my laws are similar to the Kolmogorov theory or the von Neumann-Morgenstern theory. Laws (L1)-(L5) are supposed to have the same scientific character as Newton's laws of motion or laws of thermodynamics, not any deductive system.