Of course they matter. We're discussing arguments that apply in the real world, not in math theory.
In math, if you have this axiom:
f(x) > 5 for all x >= 20
you are not then allowed to change it with a later axiom
except when x is divisible by 240
However, in real life this happens a lot. I have a company that is taxed a fixed amount per year... except for the years where I make over 100k euros, in which case things become quite complicated. If I omit the second part (which is not impossible, given that I never made over 100k euros a year with that company), and suddenly get a big payout from someone, the result will be very different from my initial estimation - as sound as it was WITHOUT that additional axiom / assumption / rule.
In math, if you have this axiom:
you are not then allowed to change it with a later axiom However, in real life this happens a lot. I have a company that is taxed a fixed amount per year... except for the years where I make over 100k euros, in which case things become quite complicated. If I omit the second part (which is not impossible, given that I never made over 100k euros a year with that company), and suddenly get a big payout from someone, the result will be very different from my initial estimation - as sound as it was WITHOUT that additional axiom / assumption / rule.