system
Given the following post, respond as if you're commenting on Reddit:
human
Why is it important to "prove" mathematical conjectures/hypotheses?

For example, the Rieman hypothesis. What is the benefit to proving it over simply assuming it is true/false and seeing what the consequences of that are?

After all, mathematics rests upon any number of axioms and posulates that are accepted as true, and new branches of math can arise from flipping that around (e.g. discarding the parallel postulate).

Obviously, there may be novel math involved in solving these theorems, so this question is about the value inherent in the solution existing, independent of the how it was discovered.
gpt
Based on our assumptions, how will we know if it is true if we don't have a proof? We can't just add a new axiom that says the Riemann Hypothesis is true or false, because the axioms we have now already determine if it is true or false. A collection of axioms must be consistent with each other, that is, we are not able to prove true=false using them. If the Riemann Hypothesis is false and we assume that it's true, then we get true=false. A proof is the only way to know if it is true.

That being said, it has a $1,000,000 reward because we expect it's proof to give us new math. Though, it is a common assumption in Number Theory to prove things under the assumption that it's true, if it is false then thousands of results will be affected.