"Imagine a world in which logic, programming, and mathematics are unified, in which every proof corresponds to a program, every program to a mapping, every mapping to a proof!"
The thing I never understood about statements of this sort is that in my understanding of model theory, Godel's theorem and so-forth, a proof is a rare thing. Most of the true statements in a given model don't have proofs. Any consistent proof system admits an uncountable sets of independent theorems and so-forth.
While Goedel's first incompleteness theorem indeed shows that there will always be true statements that don't follow from a given set of (computable) axioms, this is almost never a problem in practise. It is hard to find natural examples of such statements. Almost every mathematical statement (or its negation) you or I can come up with is a consequence of the axioms of ZFC set theory, or whatever other foundation you prefer.
Bob is coming from the perspective of Brouwerian Intuitionism, and so he is not really that interested in a closed "formal system". So, "proofs" in the setting that Bob cares about are not derivations in a formal system, so Gödel's result doesn't really apply.
In bob's setting, the computation system is not determined or fixed. It is open-ended...
In intuitionistic mathematics, we of course accept that there is no Turing computable halting oracle, but we do not rule out the possibility that there is some other effective oracle that can decide halting. (This is in contrast to recursive mathematics / Russian constructivism)
Unfortunately, that isn't a world in which you could connect a USB camera to your laptop and just have it work after the system finds the right driver.
The thing I never understood about statements of this sort is that in my understanding of model theory, Godel's theorem and so-forth, a proof is a rare thing. Most of the true statements in a given model don't have proofs. Any consistent proof system admits an uncountable sets of independent theorems and so-forth.