Earlier quoted context omitted.
"Formal mathematics has no concept of absolute truth" This is false. The axioms don't have to be true. You can still talk about their implications in absolute terms. "Assuming a=0 implies a=0" is absolutely true. Regardless of whether a actually is 0.
"The axioms don't have to be true." No, not at all. Axioms are better thought of as universally accepted truths which everything else depends on. https://en.wikipedia.org/wiki/Logical_atomism
It's obvious when applying math to real world. If you're talking about objects on Earth surface, for example, at first Euclid's geometry will get you good results, because objects you're working with can be assumed to fulfill it's axioms. However, when your scale gets bigger, you'll have apply a more complicated geometry apparatus.
Easiest analogy about axioms is interface in software engineering: you don't care what the object really is, but as long as it shows certain properties, you can prove theorems about it. Which will be true for any objects with these properties, and as true as exactly these properties are fulfilled by the real life object.