One argument against hypercomputation
is that even if someone hands me a hypercomputer (that solves the halting
problem, say), I cannot verify that it really works as advertised from a
finite set of finite measurements. Without the ability to make a finite
verification, I can never really "know" that the hypercomputer is "really
solving" the halting problem.

The use of approximation as a method for dealing with complex problems is a fundamental research issue in Knowledge Representation. Using approximation in symbolic AI is not straightforward. Since many systems use some form of logic as representation, there is no obvious metric that tells us `how far' an approximate solution is from the correct solution.