This argument moves from the fact that he can doubt the existence of the material world, but cannot doubt the existence of himself as a thinking thing, to the conclusion that his thoughts belong to a nonspatial substance that is distinct from matter.

The argument is fallacious. It relies on conceivability based in ignorance. Descartes has not included anything in the argument to ward off the possibility that he, as a thinking thing, is in fact a complex material system. He has merely relied on the fact that he can doubt the existence of matter to conclude that matter is distinct from mind. This argument is clearly inconclusive. From the fact that the Joker cannot, at a certain moment, doubt the existence of Batman (because he is with him), but he can doubt the existence of Bruce Wayne (who might, for all the Joker knows, have been killed by the Joker's henchmen), it does not follow that Bruce Wayne is not Batman. In fact, he is Batman. The Joker is merely ignorant of that fact.