Bertrand's Paradox and the Principle of Indifference

Abstract

The principle of indifference is supposed to suffice for the rational assignation of probabilities to possibilities. Bertrand advances a probability problem, now known as his paradox, to which the principle is supposed to apply; yet, just because the problem is ill-posed in a technical sense, applying it leads to a contradiction. Examining an ambiguity in the notion of an ill-posed problem shows that there are precisely two strategies for resolving the paradox: the distinction strategy and the well-posing strategy. The main contenders for resolving the paradox, Marinoff and Jaynes, offer solutions which exemplify these two strategies. I show that Marinoff's attempt at the distinction strategy fails, and I offer a general refutation of this strategy. The situation for the well-posing strategy is more complex. Careful formulation of the paradox within measure theory shows that one of Bertrand's original three options can be ruled out but also shows that piecemeal attempts at the well-posing strategy will not succeed. What is required is an appeal to general principle. I show that Jaynes's use of such a principle, the symmetry requirement, fails to resolve the paradox; that a notion of metaindifference also fails; and that, while the well-posing strategy may not be conclusively refutable, there is no reason to think that it can succeed. So the current situation is this. The failure of Marinoff's and Jaynes's solutions means that the paradox remains unresolved, and of the only two strategies for resolution, one is refuted and we have no reason to think the other will succeed. Consequently, Bertrand's paradox continues to stand in refutation of the principle of indifference.