prevalent, however increasing its weighting suggests that we think it is ‘more correct’, which is surely a problematic concept in mathematics in and of itself. In any case, the fatal flaw of the meta-average is that it is always susceptible to revision unless we can somehow prove that our methods form an exhaustive set. Can we ever be certain that there is no risk of a novel method being found that yields yet another different solution, thus changing the meta-average? Another solution was proposed by Edwin T. Jaynes in his 1973 paper “The Well-Posed Problem”. He says the issue arises because “we have not been reading out all that is implied by the statement of the problem; the things left unspecified must be taken into account just as carefully as the ones that are specified”. He demonstrates how this idea can help us choose between the three possible answers to Bertrand’s paradox as well as place restrictions on which methods can be applied to similar problems. The key is that the three methods discussed by Bertrand give rise to different probability distributions of chord lengths (or distributions of positions of midpoints, since a chord is uniquely defined by its midpoint). It seems obvious that we need to know “which probability distribution describes our state of knowledge when the only information available is that given in the statement of the problem”, but Jaynes’ genius insight is to realise that “if we start with the assumption that Bertrand’s problem has a definite solution in spite of the many things left unspecified, then the statement of the problem automatically implies certain invariance properties, which in no way depend on our intuitive judgments.”
Basically, because the problem does not state the orientation, size or position of the circle, then the correct method must be general enough to give the same answer if these parameters are altered – we need to be able to arrive at the same probability regardless of rotations, enlargements or changing position. We say the solution must be rotation invariant, scale invariant and translation invariant as these are left unspecified in the statement of the problem. Jaynes formulates these ideas mathematically and demonstrates that the probability distribution of chords must be of a certain form in order to meet these three criteria, which is enough to eliminate two of the three possible solutions. His method does not
necessarily show us which distribution is correct, but allows us to eliminate distributions that are definitely wrong (because they violate the indifference criteria). We can also demonstrate this visually by graphing the distributions of chords and their midpoints when generated “randomly” according to each of the three methods. After adjusting for the fact that the centre of the circle is the only midpoint that does not uniquely define a chord (an infinite number of diameters share this midpoint), we see that method 2 is the only one that is both scale invariant and translation invariant. The distribution looks the same if we change the size or location of the circle. Method 3 is only scale invariant and method 1 is neither. This confirms the result from Jaynes’ calculations – that 2 out of the 3 methods lead to chord distributions that are not invariant in the desired ways and therefore must be rejected. Fortunately in this case this leaves only one candidate.
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