  The existence of a non-algorithmic side of the mind, conjectured by Penrose
on the basis of G\"odel's first incompleteness theorem, is investigated here in
terms of a quantum metalanguage. We suggest that, besides human ordinary
thought, which can be formalized in a computable, logical language, there is
another important kind of human thought, which is Turing-non-computable. This
is methatought, the process of thinking about ordinary thought. Metathought can
be formalized as a metalanguage, which speaks about and controls the logical
language of ordinary thought. Ordinary thought has two computational modes, the
quantum mode and the classical mode, the latter deriving from decoherence of
the former. In order to control the logical language of the quantum mode, one
needs to introduce a quantum metalanguage, which in turn requires a quantum
version of Tarski Convention T.
