Mathematicians worry about proofs and the intrinsic value of something as elusive as 'understanding'. They are deeply ingrained in the study, deeply concerned with anything effecting the field. Yet they're still emotional beings looking for beauty and meaning in life that might come from an understanding how the universe works purely from a math perspective. I'm glad Mathematicians exist, I certainly can't do that type of work.
And I trust their results: technology wouldn't be possible without advancing our understanding of the world in various fields, including math.
Your idea sounds great for the Mathematicians.
There's a more pragmatic view though, and unrelated to proofs themselves: does understanding a proof help us to advance Humanity in some way?
Do we have better lives afterwards? What if we give up understanding proofs and focus only on results.
In other words, if an AI solves a problem for you, but you don't understand how it works, should you continue building anything on top?
I suppose the results are truly what matter. If AI solved cancer, disease, anything that lowers quality of life, but you have no idea how it did it: is that good enough?
Your second approach seems good to help figuring out results from both theory and application of math to solve problems.
But also, what if there is no true beauty in Math, the way Dirac and Einstein wanted?
What if these AI brute force proofs are all that's left?
> If AI solved cancer, disease, anything that lowers quality of life, but you have no idea how it did it: is that good enough?
A lot of medicine is already like this. Shown to work in clinical trials, no complete end to end mechanism understood. They still get approved if the empirical results are strong.
I'll answer one of your points partially: if AI builds a better sorting algorithm and proves its performance characteristics, it's useful. I'd be able to use it to make my programs faster even if I don't/couldn't understand it.
It would be a bit disappointing but still useful and make humanity slightly better.
It's interesting to measure how much we believe in something, and how much trust we've lended in order to have a working model of our reality: enough understanding for us to get around, move about, and be content.
I'm sure you would only trust the improved 'blackbox' AI sorting algorithm after it has been proved out through benchmarks. Once you've seen better, repeatable numbers: your trust would rise and eventually you'd feel confident enough to use the blackbox in other areas of your application. You'd build on top of the trust you lended to the blackbox. And you would continue measuring yourself as you build out, making sure you trust the foundation as you go.
A proper engineering mindset if you ask me, but it's only useful in the physical world when solving physical problems.
The Mathematicians build 'castles in the sky' with vast equations that link up together in shapes that make sense. There's trust being lent to the linking as you go. How do you validate these 'castles in the sky'?
Through understanding. But then, how much understanding is needed? This is where Theory meets Application: and the Article is purely in the Theory territory. Your measure is purely in the Application territory.