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Consider Jayne's solution to the Bertrand paradox using the principle of indifference. Why doesn't a similar argument apply to the Borel-Kolmogorov paradox?

Is there something wrong with arguing that since the problem does not specify an orientation for the sphere, rotating the sphere should not affect the resulting distribution arrived at by the chosen limiting process?

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Given that this is a non-mathematical argument, you can always use it! And equally always find someone arguing against it...! – Xi'an Jul 26 '12 at 7:45
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Also I do not thing Jaynes' argument closes the debate about Bertrand paradox: there are an infinite number of ways of physically drawing lines at random, as discussed in this post of mine. – Xi'an Jul 26 '12 at 7:50
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Did you notice how that Wikipedia article actually quotes Jaynes on the B-K paradox? "… the term 'great circle' is ambiguous until we specify what limiting operation is to produce it. The intuitive symmetry argument presupposes the equatorial limit; yet one eating slices of an orange might presuppose the other." It seems to me this answers your question. – whuber♦ Jul 26 '12 at 12:41
@whuber: I took that to mean that the question-asker had to specify the limiting process. I didn't think it meant that the principle of indifference could be used to force a unique choice in the limiting process. Is that how you see the statement? – Neil G Jul 26 '12 at 16:54
I'm mainly looking for insight into when the principle of indifference can be applied and how to apply it. I have not seen any applications except the Bertrand paradox, and it still seems pretty mystical to me. – Neil G Jul 26 '12 at 16:57
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I don't see the point of the "principal of indifference". The Wikipedia articles answer is better: "Probabilities may not be well defined if the mechanism or method that produces the random variable is not clearly defined." In other words, without even restricting ourselves to questions of probability, "An ambiguously-posed question does not have a single unambigous answer."

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Thanks for your answer. Did you read Jaynes' defence of the principle of indifference? E. Jaynes, “Where do we stand on Maximum Entropy?,” R. Levine and M. Tribus, Eds. The MIT Press, 1979, pp. 15–118. – Neil G Aug 6 '12 at 16:00

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