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?
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.
On the other hand, automating all of programming is a pipe dream. If you could do that, you could also automate mathematics and by extension most of the sciences...