But the real answer to Bertrand’s paradox is that the word ‘random’ is not meaningful without any knowledge of the structure of your sampling process.
Indeed, this is a fully general feature of attempting to know what’s true. What we’d like to do is create an objective map the territory, but any map we construct is contingent on large numbers of non-objective assumptions that hinge on everything from sampling, selection, and attention processes to what categories have been historically useful to include in our ontology. This limitation stands in the way of many clever arguments that would seek to do an end-run around uncertainty, but ultimately fail because they smuggle in unjustified assumptions (cf. anthropic arguments).
Suppose that someone chose the chord, generated random numbers uniformly in , then told us that such numbers are less than the radius chosen. How well will we be able to identify the radius, if I had a prior that it was chosen via Method 1, you had a prior that it was chosen via Method 2, Sam had the prior that it was chosen via Method 3?
Sure, within a fixed framework, we get convergence, even when using different methods. But this is possible because concepts like “the radius of the chord” are already agreed on. The paradox isn’t about not knowing which method generated the chord, it’s that the question itself is underspecified and can’t be fixed without making a choice (introducing contingency) because we lack access to a view from nowhere to adjudicate it.
Indeed, this is a fully general feature of attempting to know what’s true. What we’d like to do is create an objective map the territory, but any map we construct is contingent on large numbers of non-objective assumptions that hinge on everything from sampling, selection, and attention processes to what categories have been historically useful to include in our ontology. This limitation stands in the way of many clever arguments that would seek to do an end-run around uncertainty, but ultimately fail because they smuggle in unjustified assumptions (cf. anthropic arguments).
Suppose that someone chose the chord, generated
Answer: we ALL will be able to identify it as
Sure, within a fixed framework, we get convergence, even when using different methods. But this is possible because concepts like “the radius of the chord” are already agreed on. The paradox isn’t about not knowing which method generated the chord, it’s that the question itself is underspecified and can’t be fixed without making a choice (introducing contingency) because we lack access to a view from nowhere to adjudicate it.