Really? Thanks, Art'. I'll take that under consideration.
...since you ask...
vonNeumann's Definition of Randomness:
(Fanfare in brazen picolo trumpet in the key of C ....)
As my great uncle said "Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin.''
Hehehehh! No, no, no. I'm not going to pretend to be an expert on this. I'm going to set a greater example than that. Even though it is tempting to do as Cyborg points out that I do..."I take myself too damn seriously."
Suffice to say that the subject of randomness is much deeper than I perceive most posters here give it credit. This OP asks a good question yet we can only poke at it.
I do have my opinions
du jour , tentative and subject to change. I'll offer up some tidbits:
1. Quantum randomness is the only true randomness in our physical universe, the mechanism of which is apparently completely hidden from us.
2. Algorithmic Complexity as described by Gregory Chaitin, sheds new light on randomness from a meta-mathematical perspective. Chaitin says,
"There's only one definition of randomness (divided into the finite and the infinite case for technical reasons): something is random if it is algorithmically incompressible or irreducible. More precisely, a member of a set of objects is random if it has the highest complexity that is possible within this set. In other words, the random objects in a set are those that have the highest complexity. Applied to the set of all n-bit strings this gives one of our definitions, applied to infinite binary sequences this gives our second definition. "
So Art', my answer is that my opinion on randomness is that Chaitin makes more sense on the essense of randomness than anything else I am currently aware of. It seems to me he has encapsulated what transpired from Hilbert->Goedel->Turing.
Since I believe quantum randomness is the only true physical randomness, and since mathematics models quantum mechanics even while we cannot see how quantum mechanics works, I believe therefore the explanation for randomness is best to be found in pure mathematics.