[…]

So certainly physics has not proven, and can never prove, that its theory of atomic decay is true.  The logical process is that if atoms decay randomly, then Poisson statistics will result.  Experiments show that Poisson statistics do indeed result, but logically this does not mean that atoms decay randomly.  Nevertheless, the way of science is that we do postulate that atoms decay randomly, until a new experiment calls this into question.  But no experiment ever has.  If this sounds like a reverse use of logic, then consider the same ideas for mechanics.  Ideas of gravity, mass and acceleration were originally produced by Newton through the same process: because they predicted planetary orbital periods that could be verified experimentally.  Because of this great success, expressions such as F = ma and F = GMm/r2 came to be canonical in physics.  The logic was indeed being used in reverse; but no one was surprised when, three centuries later, one of the moon astronauts dropped a feather and a hammer together in the moon's vacuum, and found that they both fell at the same rate (although it was still beautiful and dramatic to watch!).  That reverse logic had, after all, allowed him to get to the moon in the first place.  So this way of conducting science works very well.