> which is that mathematicians compute /all sorts/ of uncomputable systems, with no apparent restriction
You are flattering us, but that's not actually true. We already have lots of trouble with proving (which is the same as computing, essentially) many computable things as it is. It is only occasionally that we can make progress on some uncomputable problem, and it's always because of making use of some properties very specific to said problem. The point of Rice theorem is that there's no universal method to solve any problem without any concern for the guts of that problem, and that's true: when mathematicians or computer scientists are proving that some computer program halts, they don't apply the same general machine, but instead they try to figure out some "tricks" based on the particular situation at hand, which will not generalize to every conceivable problem.
You are flattering us, but that's not actually true. We already have lots of trouble with proving (which is the same as computing, essentially) many computable things as it is. It is only occasionally that we can make progress on some uncomputable problem, and it's always because of making use of some properties very specific to said problem. The point of Rice theorem is that there's no universal method to solve any problem without any concern for the guts of that problem, and that's true: when mathematicians or computer scientists are proving that some computer program halts, they don't apply the same general machine, but instead they try to figure out some "tricks" based on the particular situation at hand, which will not generalize to every conceivable problem.