How do we know that the universe isn't sitting on the back of a giant turtle?
Most of the math that we use is self-consistent, and our proof system is, as far as we can tell, relatively consistent with the known properties of the universe.
What I'm getting at is that no, we don't actually KNOW with 100% certainty that, for example, certain proven divergent series continue to diverge at infinity, if your standard for proof requires trial of an infinite span of numbers; but such speculation is so far out of the realm of what is known and "proven" according to our [mostly] self and externally consistent set of knowledge[1] that it is more or less in the realm of pseudoscience/metaphysics.
Most of the math that we use is self-consistent, and our proof system is, as far as we can tell, relatively consistent with the known properties of the universe.
What I'm getting at is that no, we don't actually KNOW with 100% certainty that, for example, certain proven divergent series continue to diverge at infinity, if your standard for proof requires trial of an infinite span of numbers; but such speculation is so far out of the realm of what is known and "proven" according to our [mostly] self and externally consistent set of knowledge[1] that it is more or less in the realm of pseudoscience/metaphysics.
1. Excluding the Incompleteness Theorem