Haynes is intuitively convincing, but intuition isn't everything. If you buy his interpretation of his desiderata and buy his assertions (like: plausibility is a real) then its convincing to believe that his argument leads to probability---though I think Cox's Theorem as presented by Jaynes has been disproved? I think it still holds in another form though.
The dogmatism is not that we should forsake math, but instead that we should accept any One True Math. History is littered with the broken careers of those who did so.
Disproved?! frantic search… Oh, you meant for infinite sets. Somehow I'm not surprised. When one can make a 1-1 correspondence between even numbers and natural numbers… Maybe that's one reason why Nick Bostrom says Infinite Ethics is hard? http://www.nickbostrom.com/ethics/infinite.html
Plausibility is a real… What else could it be? I still need something continuous, quantifiable, and totally ordered. Why, might one say? I'm not sure I can explain. Some things we just take for granted. Only a rock would take nothing for granted.
The dogmatism is not that we should forsake math, but instead that we should accept any One True Math. History is littered with the broken careers of those who did so.