The principal thesis for which I will argue is that: not only is Wittgenstein, as is too often thought, not putting forth some naturalistic explanation, or scientific theory of meaning, language or of intentional mental concepts. He wishes to persuade us of something much stronger: not only is he not trying to offer any such explanation in the face of philosophical problems of meaning and mind, but Wittgenstein's claim is that the desire for explanation in regard to understanding linguistic and mental concepts is itself the problem. In order to defend this thesis, however, there are three principal impediments which, I believe, serve to stand in the way of our being able to grasp the import of Wittgenstein's arguments, and I aim to clear these impediments away. The first is that the huge influence of Saul Kripke's celebrated reading is both a blessing and a curse for understanding the import of Wittgenstein's discussion. The second impediment which I attempt to clear away, is a characteristic tendency to receive Wittgenstein's insights within the very explanatory framework of thought which he is trying to train us out of; and hence not to receive Wittgenstein's insights at all. The third impediment is that, whilst John McDowell has grasped the import of Wittgenstein's insights into the desire for explanation of linguistic and mental concepts, his understanding of these insights remains obscured from the view of the vast majority of published commentators, who are distracted by an early published paper by McDowell which betrays a form of the very misunderstanding Wittgenstein is trying to warn us against. What has gone unremarked is that McDowell later explicitly repudiates his early reading, once he has grasped the import of Wittgenstein's discussion. By clearing these impediments away, my hope is that Wittgenstein's profound insights can be made more widely available to contemporary thought.