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At the same time however, all the talk about proof made me eager for another set of proofs...Maybe a proof that the chance of a quarter landing heads is 1 out of 2. Sure, flip the coin a few billion times and report back to me, but for any given trial, how do we know the true probability?
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If you want to prove that the area of a rectangle is width times height, you can do this with pure mathematics. But if you want to prove that the area of Colorado is 104185 square miles, you cannot. To prove the latter statement, you must specify what you mean by "the area of Colorado", you must show that this object satisfies (at least approximately) the properties of a rectangle, and then you must take some measurements.
The same is true of probabilistic statements. If you want to prove abstract theorems in probability, such as the law of large numbers or Bayes' theorem, you can do that with pure math. But if you want to prove a probabilistic statement about something outside of mathematics, such as coins or gods, the "proof" must also lie, in part, outside of mathematics.
The first hurdle to overcome is that you must specify exactly what you mean when you say "the probability of a quarter landing heads is 1/2". In this thread alone, you can see the shadow of different interpretations lurking behind some of the responses. (ChrisV seems to be adopting either the frequency or propensity interpretation, while Kimbell is adopting the logical interpretation.)
This post explains four of the most common interpretations of statements such as this.