Is there any proof that it [automating programming, mathematics, scientific method] is or isn't possible? Is there a way to prove or disprove it mathematically?
> Is there any proof that it [automating programming, mathematics, scientific method] is or isn't possible?
Decidability results can be useful, but note that it's certainly possible to do automated reasoning for undecidable theories. So decidability results are more useful for predicting the difficulty and determining appropriate approaches than for actually determining anything about the possibility of creating a useful system.
E.g. the halting problem is famously undecidable, but there's lot of impressive work on termination checking. Misinterpreting undecidable problems as "impossible problems" is a classic undergrad mistake.
> Is there a way to prove or disprove it mathematically?
No, not really, unless you can be very specific about the exact subproblem you're interested in. And even then, probably not.
Is there any proof that it [automating programming, mathematics, scientific method] is or isn't possible? Is there a way to prove or disprove it mathematically?