The Achilles heel of philosophy of probability

The area of foundations of probability is awash with interesting ideas. The first memorable ideas were proposed at least three centuries ago by Bernoulli in relation to St. Petersburg paradox. The Achilles heel of foundations of probability is a construction of a complete philosophical theory of probability. A number of philosophers (Carnap, de Finetti, Gillies, Hacking, von Mises, to name just a few) constructed more or less complete and logically consistent philosophical theories of probability. When one reads any of these theories, the first impression is that of professionalism. The books are logical, the arguments are detailed and go in many directions, diverse aspects of the problem are considered, pro and con arguments are given, etc. But when you finish any of these books, you ask yourself whether you would use this theory as an explanation of the concept of probability in an undergraduate class. Or does this theory represent faithfully the science of statistics in its present shape? And the answer is invariably "no".

De Finetti and von Mises did not fail because they did not have innovative ideas. They did - the ideas of a "collective" and "consistency" in decision making were new. They did not fail because they did not use logic. They failed because they used logic in the strictest sense, like mathematicians. Just like mathematicians, they decided to accept logical conclusions, no matter how counterintuitive they were. There are at least two lessons from the failures of de Finetti and von Mises. One is that it is very hard to build a complete philosophical theory of probability. Another lesson is that once you arrive at conclusions that are clearly inconsistent with accepted beliefs and practices, it is a good idea to reexamine your assumptions. De Finetti and von Mises made the same mistake that Zeno made when he said that "Motion does not exist" over two thousand years ago. Some lessons are never learned.