> How does a human generate proofs, given no algorithm exists to solve the halting problem? To be fair, this has some trivial answers -- despite being a fascin

> How does a human generate proofs, given no algorithm exists to solve the halting problem? To be fair, this has some trivial answers -- despite being a fascinating question. For example, given a formal proof verification system (e.g. Lean), you can search for proofs (either by brute force or hopefully using heuristics) and if a proof exists, you will find it after some finite time. The problem is that sometimes no proof exists. You may be thinking of getting "smart", and also trying to prove no proof exists in parallel. But that's just another proof, such that this recursion tower (proof that (proof that (... proof doesn't exist)) doesn't resolve. But I think overall this mostly means you should be able to accept non-existence or arbitrary proving times of proofs (maybe also a sociological problem for mathematicians!). That's of course how human mathematics works: no one now if any famous theorem (e.g. Riemann hypothesis) will be proven within some time frame, or maybe never.