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why does the existence of such problems disprove the existence of a mathematical foundation? A well-founded mathematical foundation would prove/predict/explain why such problems don't "fit" with the "structure of NNs" with precise lower/upper bounds. Anything that works, and especially everything that doesn't work, must have an explanation. God doesn't play dice.

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Maybe there's a reason. But I don't think you will ever be able to "prove" it. In any kind of formal way that would be satisfying to mathematicians.

Solomonoff induction is the best attempt to try to formalize machine learning. And in theory any machine learning algorithm that works, works because it approximates Solomonoff induction somehow. But proving anything approximates Solomonoff induction is absurdly difficult or impossible. Because it's incomputable and involves the search space of all possible computer programs.

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