An algorithm expresses the solution to a problem with the
elements of a given language, whatever language this is.
Malcolm has made the preposterous claim that "if you're thinking of an
algorithm you'll think of it in C terms".
The question of whether some "language" is needed (however abstract and
ill-specified) simply to *think* about algorithms, is one I am not
remotely qualified address. I could report what I feel is happening in
my head when I do such a thing, but will that help to arrive at a true
picture of the phenomenon? I doubt it.
Expressing an algorithm needs a notation, but that's as far as I'd be
prepared to go.