What Emile Borel, the Great, Famous, French Mathematician, Missed by Skipping Part II of the a Treatise on Probability in His 1924 Review of that Book: The Impact of the Ignorance of Part II of the a Treatise on Probability on Modern Assessments of J M Keynes's Logical Theory of Probability

Abstract

In his review of J M Keynes’s A Treatise on Probability in 1924, Emile Borel made an extraordinary admission. He stated that he was not able to follow what Keynes was doing in Part II of the A Treatise on Probability. Borel’s review, then, as he admitted, is based on an assessment that is, at best, incomplete. Borel further admitted that he understood that this was the most important part of the book for Keynes. This additional frank admission on Borel’s part simply means that Borel’s review, which primarily criticized Keynes’s major claim, that all probabilities can’t be represented by a single numeral between 0 and 1, needed to have been supplemented by some other reviewer who did, in fact, cover Part II of the A Treatise on Probability if a correct overview of Keynes’s book was to be made available for logicians, statisticians, and decision theorists so that they could assess the importance of this book from a historical point of view.

Unfortunately, there was no such reviewer, except for the second, camouflaged review of Part II of the A Treatise on Probability done in 1934 by Edwin B Wilson, who deliberately chose to cover up this review by titling it “A Problem of Boole”.

Wilson was attempting to cover up the self admitted deficiencies, openly acknowledged by Wilson in private correspondence with F Y Edgeworth, that existed in his first 1923 review of the A Treatise on Probability. Wilson did, however, correctly point out in this correspondence with Edgeworth that F Y Edgeworth was the one person who was the most qualified in the world to review Keynes’s A Treatise on Probability. That contention was correct. Edgeworth’s two reviews, together, are the best reviews of Keynes’s A Treatise on Probability ever made with the possible exception of Bertrand Russell. The only problem was that Edgeworth himself also admitted that he could not follow Part II of the A Treatise on Probability. de Finetti’s 1938 review, based on Borel’s review, also makes no mention of Part II of the A Treatise on Probability. I have already demonstrated that F. Ramsey, in 1922 and 1926, and R. Fisher, in 1923, also showed in their reviews that they had absolutely no understanding of Part II of the A Treatise on Probability”, in my recently published 2016 book, “Reviewing the Reviewers of Keynes’s A Treatise on Probability”, that was originally written in 1986.

The failure of academics, especially economists, philosophers, and historians, in the 20th and 21st centuries to read and understand Part II of the A Treatise on Probability, as well as to integrate Part II into an overall, objective assessment of what Keynes had done in the A Treatise on Probability, has led to a historically false assessment of the A Treatise on Probability and failure to comprehend the General Theory. This false assessment claims that Keynes’s logical probability theory is, at best, an ordinal theory that can’t be applied most of the time. Nothing could be further from the truth. Keynes’s theory in Part II is a non additive, non linear, interval valued theory firmly based on the upper and lower probability approach first derived mathematically by G Boole in 1854 in The Laws Of Thought. Boole, in 1854, and not Koopmans in 1940, was the first to provide the foundations in math and logic for an explicit upper and lower probabilities approach. Part II of the A Treatise on Probability is based on hard logical and mathematical analysis that follows directly from Boole. It is not based on mere “views”, “ideas”, “intuitions”, “some hints” or ”suggestions” about the possibility of devising a system of interval valued probability as, for example, argued by Kyburg (see references).

Keywords: de Finetti Keynes, Borel, Part II, Interval Valued Probability, Approximation

JEL Classification: B10, B12, B14, B16, B20, B22