I wonder to what degree this is an artifact of the definition of "computable real" in common use (originally sure to Turing I believe), which is a Turing machine that outputs successive digits of your real number.
The point, I think, is that this gives you a good model for doing mathematics - it's a simple definition, and it allows you to talk about real numbers to arbitrary precision in arbitrary computational problems. But the downside is it's a minefield of subtle logical issues - addition is non-computable! So is the order relation, and equality, and a host of other basic stuff. It's not surprising that a complex limit-based process like taking a derivative would be also be non-computable.
On the other hand physicists want to model reality, which is a totally different set of constraints. I suspect that, as mathematicians, we've taken a logical construction that maps reasonably well onto the universe in some cases (real numbers) and stretched it far beyond its applicability there, because it leads to interesting mathematics. Physicists, if they ever find the cracks, will just come up with a better abstraction and move on.
> 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 point, I think, is that this gives you a good model for doing mathematics - it's a simple definition, and it allows you to talk about real numbers to arbitrary precision in arbitrary computational problems. But the downside is it's a minefield of subtle logical issues - addition is non-computable! So is the order relation, and equality, and a host of other basic stuff. It's not surprising that a complex limit-based process like taking a derivative would be also be non-computable.
On the other hand physicists want to model reality, which is a totally different set of constraints. I suspect that, as mathematicians, we've taken a logical construction that maps reasonably well onto the universe in some cases (real numbers) and stretched it far beyond its applicability there, because it leads to interesting mathematics. Physicists, if they ever find the cracks, will just come up with a better abstraction and move on.
Anyway, not sure if I'm making sense here.