Inferring Probabilities From Symmetries
Michael Strevens · Noûs · 1998
Consider a perfectly symmetrical dodecahedron ~see Figure 1!.Paint one face red, and it may be used as a die.What is the probability that such a die, when tossed, lands red face up?Without having touched, or even laid eyes on, such an object, it is possible to give a firm answer: the probability is one in twelve.Find such a die, and experience will bear out this assertion.The red face will land uppermost roughly one twelfth of the time.We have made, apparently, an a priori inference to a fact about the world.How is it that we are able to infer the probability with such success?The conventional answer to this question is that we invoke a rule called the Principle of Indifference ~or the Principle of Insufficient Reason!. 1 This rule tells us that, in the absence of any known reason to assign two events differing probabilities, they ought to be assigned the same probability.Applied to the dodecahedral die, the principle allows us to reason as follows.The die may land with any of twelve faces uppermost.We know nothing that distinguishes these twelve possibilities, so we must assign each an equal probability.Since the sum of these probabilities must equal one ~some face must be uppermost!, each possibility should be assigned a probability of one twelfth.This explanation of our probabilistic prowess cannot be right.The Principle of Indifference is not even the right kind of rule to explain our successful inference.It purports to tell us which probabilities are rational given our ignorance, helping us to do the right thing under adverse epistemic circumstances.But our achievement is not mere rationality; it is truth.We infer the correct physical probability for the event in question.This suggests that our inference is not based on ignorance at all, and thus not based on a principle of "indifference" or "insufficient reason".Two arguments support this suggestion.First, it is surely the case that we can never reliably get from ignorance to truth, because the nature of the world is independent of our epistemic deficits.The fact we do not know anything about A does not constrain the way things are with A.