Hah, I'll let Decartes go (or condition him on a workable concept of existence -- but that's more of a spitball than the hardball I was going for).

But in answer to your non-contradiction question... I think I'd be epistemically entitled to just sneer and walk away. For one reason, again, if we're in any conventional (i.e. not paraconsistent) logic, admitting any contradiction entails that I can prove any proposition to be true. And, giggle giggle, that includes the proposition "the law of non-contradiction is true." (Isn't logic a beautiful thing?) So if this mathematician thinks s/he can argue me into accepting the negation of the law of non-contradiction, and takes the further step of asserting any statement whatsoever to which it purportedly applies (i.e. some P, for which P&~P, such as the whiteness of snow), then lo and behold, I get the law of non-contradiction right back.

I suppose if we wanted to split hairs, we could say that one can deny the law of non-contradiction without further asserting an actual statement to which that denial applies -- i.e. ~(~(P&~P)) doesn't have to entail the existence of a statement P which is both true and false ((竏パ)Np, where N stands for "is true and not true?" Abusing notation? Never!) But then what would be the point of denying the law?

(That being said, what I'd actually do is stop long enough to listen to the argument -- but I don't think that commits me to changing my zero probability. I'd listen to the argument solely in order to refute it.)

As for the very tiny credence in the negation of the law of non-contradiction (let's just call it NNC), I wonder what the point would be, if it wouldn't have any effect on any reasoning process EXCEPT that it would create weird glitches that you'd have to discard? It's as if you deliberately loosened one of the spark plugs in your engine.