The incompleteness theorem isn’t an argument against formalism, it’s just a bound on what you can do with it (you can’t use one set of axioms and expect to prove everything). That’s ok, because you are,
1. Not required to prove everything which is true,
2. Not limited to one set of axioms.
The nice thing about formalism is that you can use a computer program to check whether a proof is sound or not. Isn’t that nice? It has applications in e.g. computer science and engineering, where you can prove that a program does not crash, or prove that a traffic light never shows green in conflicting directions, or prove that a real-time kernel always provides enough computational time for your rocket computer to steer.
Working math, there are more and more complex proofs that cannot reasonably be checked by humans. Having a computer check them is a nice alternative.
1. Not required to prove everything which is true, 2. Not limited to one set of axioms.
The nice thing about formalism is that you can use a computer program to check whether a proof is sound or not. Isn’t that nice? It has applications in e.g. computer science and engineering, where you can prove that a program does not crash, or prove that a traffic light never shows green in conflicting directions, or prove that a real-time kernel always provides enough computational time for your rocket computer to steer.
Working math, there are more and more complex proofs that cannot reasonably be checked by humans. Having a computer check them is a nice alternative.