> It's not surprising that a complex limit-based process like taking a derivative would be also be non-computable.
The reason it was surprising to me is that it would naively imply that an analog computer could compute something that a TM could not. If that were true, it would be Big News. But the construction of a computable function whose derivative is uncomputable is such that this function cannot be physically realized because you run into quantum effects.
Nice, thanks for the link. I knew that "printable" and "computable" reals were equivalent logically but it didn't occur to me that the equivalence itself would be uncomputable! This is exactly what I mean about a minefield of subtlety haha.
But to my point, I don't even think the examples in the OP paper are that convincing to me. Sure, you can construct examples of circuits with computable inputs and non-computable outputs, but this basically comes down to fractal-like driving behaviour that can't be approximated at any scale correctly. It's an artifact of the mathematics, which lets you do this kind of thing to an input signal, but at that point (my belief is that) you've left the realm of physics/reality and are now chasing pure logical implications.
It's like extrapolating from the edge of a map and concluding that there's no ocean because your original map had no water in it. Actually, you just needed a different map.
The reason it was surprising to me is that it would naively imply that an analog computer could compute something that a TM could not. If that were true, it would be Big News. But the construction of a computable function whose derivative is uncomputable is such that this function cannot be physically realized because you run into quantum effects.
Also, you might find this interesting:
https://www.sciencedirect.com/science/article/pii/S030439752...