Remark: my question deals more particularly with the axioms of set theory, arithmetic, probability theory, etc. I think the status of the axioms in geometry is clearer.
The French fictitious mathematician Bourbaki writes somewhere ( " The Architecture of Mathematics " in Jean-François Le Lionnais, Great Currents Of Mathemaatical Thought ) that the approach of the mathematician is comparable to the way experimental science proceeds.
The physicist makes observations and looks for the best explanation: he adopts as his theory the hypothesis ( or set of hypotheses) from which these observations can be deduced at minimuml cost.
If the comparison holds, the " observations" would, in mathematics, be some pieces of mathematical knowledge the mathematician wants to " secure" or justify , and the axioms would be the best available explanation. For example, the mathematician first wants addition to be commutative, multiplication to distribute over addition, etc., and after that, he seeks hypothesis or axiom(s) from which this desired results could follow.
My question is : is this view of axiomatizing in mathematics correct? and could this conception of axiomatizing be helpfull to correct the feeling of gratuitousness or arbitrariness of axioms?
There are some similarities and some differences.
Marcus du Sautoy once attempted in a documentary to distinguish mathematics from empirical sciences by arguing that, whereas Einstein refuted Newtonian mechanics, the truths of Euclidean geometry stand forever. My issue with him saying this is that both theories have the following similarities, which through sleight of hand he tries to pretend is a difference: the fact that their axioms imply specific theorems cannot be overturned, and their theorems have been empirically shown to fail as descriptions of our world. (The universe's geometry appears to be Riemannian, not Euclidean.)
But what do mathematicians and scientists do when such empirical failures occur? Scientists look for new theories that don't have the same flaws, while retaining old ones in their domains of applicability and for teaching purposes. Mathematicians keep using old theories with the same aplomb as before. They don't need their theories to describe specific things in Nature, because such descriptions are appended to "look at this maths for its own sake", which is all that maths requires. That Euclidean geometry can't serve astronomers well any more doesn't detract from its "default" status in ordinary mathematics. (If you post a problem about a triangle on here, without saying the geometry is non-Euclidean, no-one will even mention the unspoken assumption that the geometry is Euclidean.)
Where mathematicians do reject a system of axioms is when they lead to logical contradictions. That's why set theorists go nowhere near unrestricted comprehension. The response to Russell's paradox seems straightforward now, but a lot of systems of axioms got debated in its immediate aftermath. If we ever find a contradiction in ZFC, we'll have to start over again.