"Using your logic we would have to observe every number between 1 and 10200 before we could trust that 10200 can actually be relied on. For all we know, integers follow different rules above 10199."
Your analogy does not apply.
The basics premise of mathematics is based on definition, not theory. You are confusing definitions and theories.
Something which is defined is ALWAYS constant with no chance of changing. A theory is based on observations, and is only valid as long as no one produces an exception to the data which spawned the theory.
Such as a specific religious belief?
A theory is based on observations, and is only valid as long as no one produces an exception to the data which spawned the theory.
Sounds like science to me. What's the problem?