The Open AI solution (see previous post) to the Clay Navier-Stokes problems sends shock waves into the world of analytical (pure) mathematics. Is the proof correct? Can correctness be checked by human mathematicians or only formally by AI itself? What if AI says the proof is correct. Will AI then get the prize?
If so the traditional split of mathematics into analytical (formulas) mathematics and numerical mathematics (number crunching), will no longer be functional. An AI proof is the result of an ultimately computational process and of course the same is true for a numerical solution.
Traditionally, analytical mathematics has been associated with generality by offering proofs of existence of solutions (but not their values) for general data, while numerical solutions have been particular for each choice of data. Thus analytical-general and numerical-particular.
But OpenAI offers a single counterexample to existence, not generality but extreme particularity, while numerical solution to Navier-Stokes equations for almost any data offers generality.
We see that the distinction between analytical and numerical mathematics with computational AI gets blurred, which can be seen as a lift for numerical mathematics traditionally viewed as lower level. Mathematics is fundamentally computational.
It will be interesting to see the effects of the shock waves now sweeping over the field of mathematics, including choice of topics and education. Leibniz would have been thrilled to experience this development which he prepared 350 years ago.
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