In his most comprehensive book on the subject (1994), Roger Penrose provides arguments to demonstrate that there are aspects of human understanding which could not, in principle, be attained by any purely computational system. His central argument relies crucially on renowned theorems proven by Gödel and Turing. However, that key argument has been the subject of numerous trenchant critiques, which is unfortunate if one believes Penrose's conclusions to be plausible. In the present article, alternative arguments are offered in support of Penrose-like conclusions (although the present arguments differ markedly from his). It is argued here that a purely computational agent, which lacked conscious awareness, would be incapable of possessing crucial concepts and of understanding certain kinds of geometrically-based proofs.