The Historical Importance of Keynes' A Treatise on Probability A visiting speaker in Economics recently happened to mention that John Maynard Keynes' A Treatise on Probability revolutionized probability theory. I have not heard any such claim before and it struck me as strange. The Wikipedia page contains some effusive praise from Russell but nothing specific. This leads me to ask:
1) Is this claim approximately true?
2) In what specific ways did it impact probability theory?
3) What are some specific citations which demonstrate this?
 A: If you have a copy of Kai Lai Chung's Elementary Probability Theory, you will find Keynes' picture next to that of Kolmogorov, Polya, Feller and other probability heavy weights. (I do not know if you would cosider this evidence.) According to my modest understanding, Kolmogorov formalized probability theory and practically re-wrote the field. As far as I know he provided the framework to study probability in a measure theoretic sense. Keynes was influential not in a mathematical sense, but in a philosophical sense. A situation that exemplifies his struggle between quantifiable probability and reality is that if you have a fair coin and toss it 20 times, say you get 15 heads in a row, it would be natural to argue the coin is not fair.
A: J M Keynes did not revolutionize probability theory.J M Keynes  revolutionized decision theory by showing that the mathematical laws of the calculus of probabilities were a special case requiring that the weight of the evidence,w,where w is defined on the unit interval[ 0,1],must equal ,approximate ,tend,or approach 1  in order for the probability calculus to be operationalized where 0 is less than or equal to w ,which is less than or equal to 1.The weight of the evidence ,w,measures the completeness of the relevant evidence upon which the probability estimates are based.w=1 requires that the sample space of all possible outcomes be specified  before any decision is made or,what amounts to the same thing,a specific ,unique probability distribution is known before any decision is made. A w <1 requires the use of interval estimates ,which contain an upper and lower bound.
The first economist to specify an interval estimate of probability was Adam Smith in the Wealth of Nations on pp.106-109 of the Modern Library(Cannan) edition See p.714 for Smith's clearcut statement that probabilities can't be estimated precisely  .
A: Thank you for your comment.Unfortunately, Kolmogorov's  axiom system assumes linearity and additivity. His precise or point estimate approach to probability gives correct answers only in those fields where the weight of the evidence, w, equals, approximates, or approaches 1 in the limit. It is in many of the areas of physics, chemistry, biology ,and engineering where the weight of the evidence is close to 1,where w, the weight of the evidence ,is defined on the unit interval between 0 and 1.w=1 means you are able to define a sample space of all possible outcomes before you begin your experiment or analysis.
Keynes's approach is based on Boole's indeterminate approach to probability and uses intervals, not point estimates ,because in many instances, especially in social science, liberal arts , economics, business, finance , and public policy, there is substantial missing evidence that will never be available to the decision maker when he must make a choice. Boole's system, Keynes's version, and Hailperin's 1986 linear programming approach ,would treat Kolmogorov's approach as a special case where w=1.
