We are 100% agreed in the value of math for proving stuff. It's the same power that's found in the true engineering fields -- one can design a product and know exactly how it will perform without spending a penny making stuff.Let's try another example: exponentially fitted methods for the Black Scholes PDE and other linear convection-diffusion-reaction PDE are stable for any values of drift and diffusion. Standard FDM fail when convection dominance kicks. We can prove this without having to write a single line of code.
Now, has AI classified input types/categories so that having done that you know what to expect and which methods work? The bespoke arx files seem to suggest not. DL is only a few years old so miracles take longer. Be carefui with hype.
BTW here is a great example of constructivist mathematics
https://en.wikipedia.org/wiki/Banach_fi ... nt_theorem
Maybe this is an eye-opener
https://en.wikipedia.org/wiki/Construct ... thematics)
Why beat one's head against a mathematical wall of trying to find a proof of a solution if said proof does not exist?
It's the future of humanity Jim what's at stake here: Newton did it, Stokes did it, so AI should do it.
We are agreed that AI does not have that. And we all agree that it would be really really good if it did. I think the show-stopper disagreement is in whether it's possible and tractable.
Note: there's another failure mode for math that is potentially latent in this problem. Even if one finds a proof that predicts which input systems are learnable by which AI methods, there's no guarantee that the proof encodes a simple calculation of existence or robustness. In seeking a proof of existence of a solution, we assume that the proof exists and that the phase space for well-behaved and ill-behaved systems is simple and easily calculated. Calculating whether a given AI method will work in a given context may well be harder than just running the AI method (Wolfram's point about irreducible complexity).
As much as I truly love the deductive power of math and engineering, I also know that sometimes trial-and-error is the best and cheapest method.