[dekhn]

The halting problem isn't scientific, though. It's entirely mathematical (mathematics and science are typically treated as distinct domains, see https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf for some discussion). We do not know if there is a scientific way to make a machine which exceeds the ability of a Turing machine, see for example this paragraph from the wikipedia page on the halting problem:

"It is an open question whether there can be actual deterministic physical processes that, in the long run, elude simulation by a Turing machine, and in particular whether any such hypothetical process could usefully be harnessed in the form of a calculating machine (a hypercomputer) that could solve the halting problem for a Turing machine amongst other things. It is also an open question whether any such unknown physical processes are involved in the working of the human brain, and whether humans can solve the halting problem"

One of the most important things to recognize about science is that we rarely, if ever, work with absolutely well-determined systems with analytically solvable equiations. INstead, we work almost untirely with underdetermined systems with only approximate methods, and while somewhat unsatisfying, those methods are almost always a more efficient way to make falsifiable hypotheses and run experiments. I don't think anybody ever truly makes a falsifiable hypothesis- in the sense of Descartes' great deceiver, we can't truly know for certain what the underlying state of the system was.

[lisper]

> The halting problem isn't scientific, though. It's entirely mathematical

No, it isn't. The halting problem arises out of a mathematical model of a physical system. We don't know for certain that it's impossible to build an oracle for the halting problem, just as we don't know for certain that it's impossible to do an end-run around the Second Law. But the evidence in both cases is (IMHO) equally compelling.

[dekhn]

Everything written about halting machines presumes a mathematical system, not a physical system. It makes statements about model systems, some of which have physical counterparts. THe whole point of a turing machine is that it's abstract, not made of tape or transistors or anything else.

The halting problem is not really widely discussed in physics journals. They care much more about the physical limits of actual computing. If you have good examples of physics people talking about the halting problem in a non-theoretical and distant way, I'm happy to see it. But from waht I can tell, physicists are not concerned with uncomputable functions.

[lisper]

> The halting problem is not really widely discussed in physics journals.

So? Golf clubs aren't widely discussed in physics journals either but they are physical systems nonetheless. You can't draw valid conclusions about what is not a physical system based on what is absent from physics journals.