Well, firstly there are "important numbers" much bigger than the range we've empirically tested, so our empirical results aren't actually good enough. If there

Well, firstly there are "important numbers" much bigger than the range we've empirically tested, so our empirical results aren't actually good enough. If there was some deep relationship with number theory, maybe it just happens that a number in this range is the first to break the pattern, which could be a deep and beautiful result. But also, it's just the nature of mathematics, proving what is true is just as important (if not more so) than knowing what is true. From a practical perspective, doing mathematics, you often don't really grok why something is true until you prove it. From a philosophical perspective, there's not much in the universe we can know for sure (as you mention with the gravitational constant), but a mathematical proof we know for sure. That's the beauty of it. if X is true and the implication (X => Y) is true, then it must be that Y is true. By definition of a logical implication. Add in a whole bunch of axioms and suddenly you can take these simple logical atoms and build them into a field that spans the working tools of every engineer and beautiful arcane subjects like set theory. And every single thing we prove, we know to be true, as we build it from logical atoms that can be traced back to definitions and axioms. It "could be" that for certain types of matter or at certain distances, the gravitational constant changes. It cannot be that there exists a bijection between the natural numbers and the real numbers under ZFC. Cantor's diagonalization argument *proves* it.