How is a system of axioms different from a system of beliefs? 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?

The best example is the Euclidean geometry and non-Eucliedean geometry. The first four postulates are common to these geometries and are rather elegant


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*A straight line segment can be drawn joining any two points

*Any straight line segment can be extended indefinitely in a straight line

*Given any straight line segment, a circle can be drawn having the segment as radius and one end point as center.

*All right angles are congruent.


The fifth one did not share their grace
(5) If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
This postulate is not used in the first twenty-eight proposition of Euclid's Elements and it is a rather ugly duckling.
Euclid thought of this postulate to be inferior to the other four and certainly Euclid would have find it far preferable to prove this ugly duckling rather than to have to assume it. But he found no proof and therefore adopted it.
Over the centuries untold numbers of people gave untold years of their lives in attempting to prove that the fifth postulate was itself part of the four-postulate geometry.
Saccheri assumed the fifth postulate is opposite (he hoped he will find a contradiction) and worked with that as his fifth postulate and he got the hyperbolic geometry.
Non-Euclidean geometry was discovered in 1823 simultaneously, in one of those inexplicable coincidences by Janos Bolyai and Nikolay Lobachevskiy.
The meanings of "point", "line" and so on can be determined by the sets of theorems in which they occur and there are many geometries that have the first four axioms and they are all internally consistent but only Euclidean geometry is consistent with the external world.
This is from chapter IV, Consistency, Completeness and Geometry of Godel, Escher, Bach: an Eternal Golden Braid by Douglas H. Hofstadter
