Written: Dec. 31, 2017
I just finished the first reading of Persi Diaconis' and Brian Skyrms' fascinating page turner, and hope to read it at least one more time.
I was particularly sympathetic to the efforts of Bruno de Finetti, described in Chapter 7, of trying to make everything as finite as possible. But even he had to resort to "countable unions", and to measure theory. It mentioned, in passing, that in fact, martingales over finite sample spaces suffice, and at the end of the day one can take the limit.
So for me, as a finitist, most of the "ideas" were interesting but not really necessary, "theology", how humans try to come to grips with that fictional notion of "infinity". One great idea suffices! Probability is Counting in a finite sample space, whose size depends on symbolic n. Then at the end, one can get the "continuous" limit, by taking a formal limit, essentially, replacing 1/n by 0. For example, here is a one-line Maple proof of the De-Moivre (local, i.e. stronger version) Central Limit Theorem (for a fair coin)
e-x2/2/(n Pi)1/2 + O(1/n1/2) .
So the 11th great idea about Chance is to use "symbol-crunching", and keep n symbolic, and take formal limits. For more on this approach, see here.
Even Chapter 8, on "Algorithmic Randomness" is a bit flawed. It uses the notion of Turing machine, that has "infinite tape". As we know, even the biggest computers, in our real world, have finite memory, so the right way to deal with it is to define the algorithmic complexity of a (finite!) sequence as the length of the shortest Turing machine that would generate it in time < T with memory < M, for parameters T and M.
I was also amused by the Chapter on Physical Chance, and all the (so far) unsuccessful attempts to solve the "Boltzmann Redux" (pp. 173 and 179) (it was heart-warming to find out that even Abel-prize winners make mistakes, and that the great mathematical physicist, Yasha Sinai, had to retract a claim that he solved the Boltzmann model for n hard spheres). I really liked the following quote (p. 179, 2nd paragraph)
"The problem remains as a mathematical problem, but it has receded from a central place in statistical mechanics"
and replaced (as Stephen Wolfram preaches) by simulations, that proves it, if not "rigorously", at least very convincingly. Besides, even Avogadro's number is a finite number, so any talk about "taking the limit as n goes to infinity" is a priori nonsense (of course, it could be made a posteriori meaningful by taking a "limit" in the sense of Maple, like in the above proof of CLT.)
The book is also full of shocking tidbits, for example that many (perhaps most) articles that appear in medical (and other!) journals, using traditional statistical methods are irreproducible (and hence garbage!). [ For much more detail on this disturbing fact, see Susan Holmes' fascinating article .]
Be it mathematics, and be it "theology", everything is very interesting. Go ahead and read "Ten Great Ideas about Chance"!