[Siegmund]

Those links don't seem to question the "purity of probability" - but point out situations where the standard set of assumptions that most the common probabilistic tools are based on might not apply.

Closely related is your other question, about whether mathematics Just Is, or is a way of explaining what we see. I am solidly of the view that mathematical structures have always been and will always be, independent of physical reality and whether any sentient beings are around to think about it. Almost nothing mathematical is stated in the form "this is true"; it is all carefully arranged: "if these first few facts are true, then so are these others." People are sloppy about listing their assumptions. And especially in probability and statistics, people are sloppy about making sure the assumptions are satisfied before they apply the tools to their data.

Re the Sklanksy question about whether God can find 3 integers such than x&3+y^3=z^3, that's as pointless a question as the old "can god create an immovable object?" question. No, not without changing the meaning of integer, +, ^, =, or 3, he can't; just as anything that doesn't move when a force is applied to it is something other than what we mean by 'object'.

If you allow tampering with the assumptions, it becomes easy to do, of course: for instance, assume that 0=12 (that is, define "=" to mean "congruent modulo 12" and 1^3+11^3=2^3+10^3=3^3+9^3=4^3+8^3=5&3+7^3=0^3.

All sorts of fun apparent paradoxes can arise if you slip a distribution whose moments are undefinded or a set that's not measurable into the standard probability tools. Says nothing about the tools' validity, only about how careful the user is.