Tongue firmly in cheek:
Einstein apparently wrote
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
so statistics is the branch of maths that describes reality. ;o)
I'd say statistics is a branch of mathematics in the same way that logic is a branch of mathematics. It certainly includes an element of philosophy, but I don't think it is the only branch of mathematics where that is the case (see e.g. Morris Kline, "Mathematics - The Loss of Certainty", Oxford University Press, 1980).