I have, for quite a while now, advocated statistical inference as a solution to the infamous problem of induction. The problem of induction is this: we’ve seen, say, the sun rise again and again. But how do we justify the inference from “the sun has always risen in the past” to the conclusion “the sun will probably rise tomorrow”?
Well, if we’ve taken a look at so many days and have seen the sun rise, it’s fairly straightforward inference to the conclusion that the sun rises on at least most days (it would be tremendously improbable for the sun to rise on, say ten percent of all 24-hour periods but for us to somehow have seen the sun always come up). If the sun rises on at least most days, that dictates that it will probably rise on some particular day (like tomorrow).
However, something that’s bugged me about this solution for quite a while is this: what if the world around us was structured in such a way that before we were born the sun did not rise and after we die it will never rise again? Of course, no one actually believes this is the case. However, coming up with a sound, non-inductive refutation of this scenario is quite difficult. You can’t, for example, say that you disbelieve this because other people tell you that the sun rose before your birth, because testimony itself seems to have no justification apart from inductive reasoning, and so the problem rears its ugly head again.
Here is a solution I have been thinking about. If I draw 50 gumballs out of a gumball machine and all of them are pink, it could logically be the case that only the 50 gumballs closest to the bottom of the machine are pink. However, if most of the gumballs were some other color, there would be far more possible ways to distribute those gumballs that would have me getting at least one gumball of another color. If we assume that all possible gumball arrangements have an equal probability because we have no reason to prefer one over the other, then this means that evidence of 50 pink gumballs effectively falsifies the hypothesis that most of the gumballs are some other color (since it is very improbable that the 50 pink ones would have all been at the bottom, that is but one possible arrangement out of many, and most other arrangements would lead to a different result).
Of course, my solution depends upon treating all possibilities as equally likely because of the fact that we have no a priori way to ajudicate between them. This is called the principle of indifference and it is very contentious among philosophers, I explain one of the reasons for the hostility here. I feel that work needs to be done on this principle: there should be, if you like, a “User’s Guide to the Principle of Indifference” that can give us a way to use it consistently (there are often many ways one could apply the principle and get conflicting results) and in a way that will avoid paradoxes typically associated with it.
