Happy to see this made it to the front page. I submitted it because I'm hoping someone here will be able to understand it well enough to explain its significance. Is this Big News or a parlor trick? If it really is an example of an uncomputable physical process, that would seem to me to be earth-shattering, and yet this result does not appear to have made a splash. It is based on a construction of J. Myhill of a continuous recursive (and hence computable) function whose derivative is not computable. The proof is non-constructive, and it assumes the existence of a partially decidable set called A which somehow (this is the part I don't quite understand) results in a computable function despite the fact that its derivative can't be computable because that would imply A is decidable. Any clues would be much appreciated.
[UPDATE] Let me put this more succinctly: how is it possible that there can be a continuous computable function whose derivative is uncomputable? What exactly is it about this function that makes numerical differentiation fail?
[UPDATE2] Here is a link to the Myhill paper for easy reference.
[UPDATE] Let me put this more succinctly: how is it possible that there can be a continuous computable function whose derivative is uncomputable? What exactly is it about this function that makes numerical differentiation fail?
[UPDATE2] Here is a link to the Myhill paper for easy reference.
https://projecteuclid.org/download/pdf_1/euclid.mmj/10290006...
It's remarkably short, only two pages.