TITLE: How is a system of axioms different from a system of beliefs?
QUESTION [100 upvotes]: Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?

REPLY [3 votes]: While I agree that faith is not required to accept the rules of certain game in order to play that game, I am pretty sure that, if the rules of the game, and the consequences of playing by those rules, were entirely arbitrary, a lot of mathematicians wouldn't be able to earn a living by doing mathematics.
For ex. according to some recent stats, mathematician is one of the most employable occupations in the U.S., mostly in finance industry. If that doesn't indicate that the rules have some validity in the real world, I don't know what does?!
Would physicists use mathematics if it were completely incapable to describe what they are theorizing and measuring? 
There is evidence that axioms may be true, a lot of evidence.
EDIT:
I have found the following article by R.W.Hamming very illuminating:
The Unreasonable Effectiveness of Mathematics 
Some quotations:

The Postulates of Mathematics Were Not on the Stone Tablets that Moses Brought Down from Mt. Sinai.

also

The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way. For over thirty years I have been making the remark that if you came into my office and showed me a proof that Cauchy's theorem was false I would be very interested, but I believe that in the final analysis we would alter the assumptions until the theorem was true. 

Amusingly, he warns in the introduction:

I am well aware that much of what I say, especially about the nature of mathematics, will annoy many mathematicians. My experimental approach is quite foreign to their mentality and preconceived beliefs