The claim that language is in principle inaccessible to consciousness may look counterintuitive but is not as challenging as finding an answer to the subsequent question of why that must be the case -- if language is a function that is in the service of consciousness and we cannot imagine why language would have existed at all without the existence of consciousness. On the one hand, language is the cognitive capacity that seems best fit to support consciousness in its monitoring and control functions; on the other hand, language learning (learning the rules of one's own language), language structure and language processing turn out upon closer scrutiny to be in principle inaccessible to consciousness. I present a set of arguments in favour of the thesis that language is in principle inaccessible to consciousness on the basis of a set of asymmetries between sentence structure and the structure of consciousness. If the thesis in question is on the right track, we have to face two basic problems. The first deals with linguistics method(s), namely how can we study a very complex mental phenomenon like language if it is not available to introspection? The second problem is related to the question put in the title of this article. The suggested answer is along the lines that inaccessibility of language to consciousness enables a cognitive architecture that can run a Cartesian theatre.