Mathematics is formalised is to avoid this sort of philosophizing.
I used to think, for example, that the dirac delta function was mathematical fiction - a mathematical "hack". But then in an engineering control systems class, we did an experiment where we used a step function to approximate a dirac delta function. I could see the results both on the computer screen and in physical reality through a mass-spring-damper system. From that moment on I saw the dirac delta function in the same way that I see cosine/sine: The reason it works in math is because it has a basis in physical reality.
The lesson to be learned here, is that you don't know in advance whether something is obvious or not. To me, it doesn't make sense to decide whether the axiom of choice can be justified by looking at the axiom itself. You have to look at where it's used and required, and whether the proofs convey something that matches your physical intuition.
I used to think, for example, that the dirac delta function was mathematical fiction - a mathematical "hack". But then in an engineering control systems class, we did an experiment where we used a step function to approximate a dirac delta function. I could see the results both on the computer screen and in physical reality through a mass-spring-damper system. From that moment on I saw the dirac delta function in the same way that I see cosine/sine: The reason it works in math is because it has a basis in physical reality.
The lesson to be learned here, is that you don't know in advance whether something is obvious or not. To me, it doesn't make sense to decide whether the axiom of choice can be justified by looking at the axiom itself. You have to look at where it's used and required, and whether the proofs convey something that matches your physical intuition.