But are people more likely to accept the axioms of Principia Mathematica (and the soundness of every logical step from page 1 to 300) than they are to accept the notion that 2 + 2 = 4 based on intuitive notions of what twoness, fourness and plusness are?
Of course they are more likely to believe their intuitions. They also believe that it makes no difference whether or not you swap doors in the Monty Hall problem, and don't believe that with only 23 people the odds of a shared birthday are more than 50%.
To some extent, there is the problem. People trust their intuitions, and their intuitions are often wrong. That's why for some things we need proper proofs.
But proofs always come back to axioms, and on what basis do we accept axioms? That they sound intuitively right. So we've just kicked the problem upstairs a bit, we can't avoid using our intuition.
Personally I'm more likely to believe 2 + 2 = 4, something I can easily check to my own satisfaction using four objects, than I am to believe the Axiom of Choice.
We still use our intuitions, but now everyone knows the starting set of assumptions.
As for the axioms in use, I think the big reasons they were chosen is: They lead to results we already wanted/proved to be true.
Another thing to keep in mind, not everyone works with the same sets of axioms. Which, as someone with a formalist[2] view on mathematics, I find interesting. For example, not everyone studying logic assumes the principle of the excluded middle[1]. One of the consequences of this is that you can no longer do proofs by contradiction.
The axiom of choice is another example of this where two groups of mathematicians accept it or not. I'm a formalist, so I don't have issues with this (as long as both sets of axioms are interesting and "intuitive"), other philosophies of maths might.
It comes down to levels of assurance. The rules of inference you are using are part of your design space. Mathematics at its core is about clever problem solving. When you encounter a problem you have to decide what you would accept as a proof, and this forces you to accept certain reasoning principles.
The caveat is that this is non-trivial and it's very easy to make people accept assumptions which are completely wrong. That's really the main reason to accept the axioms of set theory: People have been trying to poke holes in them for a hundred years and nobody has managed it yet. If you can use set theory (or something equiconsistent) to solve your problem, chances are that nobody will be able to call you out on a mistake.
Well, the whole idea is to pick simple axioms, so it's harder to get wrong. And also, every successful prediction that math makes based on the axioms, is in a sense a verification of them.
