In response to a student who complained that he didn't understand quantum mechanics, Von Neumann is supposed to have answered: nobody understands it. You just get used to it. Of course, quantum physicists have a set of mathematical tools that -- though they do not allow one to visualize what is going on in the subatomic world, at least allow one to make predictions that coincide with measurements -- and so they have something to work with which helps in "getting used to it".
Believers, in response to arguments against religion from non-believers, claim that such arguments do not take mystical reality into account -- that God, or more generally, the transcendent, is real, but ordinary language is incapable of dealing with it. The non-believer responds with charges of obscurantism, that the believer is evading the issue by taking refuge in nonsense.
But there are two issues here. The first is whether or not there is anything that is real but where all attempts at description -- staying within the confines of common sense language and Aristotelian logic -- fail. The second is what we do about it. I would argue, first, that the subatomic world is such a reality, though that alone does not grant license to the believer to believe (in God or whatever). To the objection that there is a mathematical language (which of course obeys Aristotelian logic -- the laws of identity, contradiction, and the excluded middle) for quantum physics, I repeat that this language does not describe the reality -- it just allows the physicist to make predictions. Hence there are a multitude of interpretations of that world, all of which are metaphysical positions, not scientific.
But, secondly, I would argue that there is an even more obvious reality that qualifies, namely plain, ordinary, everyday consciousness. The reason it qualifies is that the "now" is not an instant -- a point on time's continuum, but instead is extended over a small stretch of time (and space). Because the now is extended, I don't see any way that it could emerge from a strictly spatiotemporal process. In a spatiotemporal process every event is separated in time and/or space from every other event. Consciousness, on the other hand, puts together zillions of these separated events to form the "now". Within the "now" is the experience of time passing, but how is that possible? Consciousness somehow connects those zillions of events into one flowing whole, while within a strictly spatiotemporal process there is no way for events to aggregate as experience of anything larger than a single event. As I see it, this means that consciousness transcends time (and space), and so cannot itself be a consequence of a spatiotemporal process.
Granted, the argument in the preceding paragraph is no more than arm-waving. But there is enough of a mystery to consciousness that it leads a diehard materialist like Colin McGinn to assert that he doesn't expect there ever to be an explanation of consciousness, and another (David Chalmers) to hypothesize what he calls "naturalist dualism" to account for consciousness. What I propose instead is to assert the reality of the non-spatiotemporal (which in theological language is called the eternal -- not to be confused with time everlasting). What if the reason that quantum reality defies comprehension is that it too is non-spatiotemporal? That would "explain" how an unobserved electron could be in a superposition of states, that the position/momentum uncertainty is there simply because -- unobserved -- quantum particles are simply not at definite spatiotemporal locations, because at that level there is no space and time. And, of course, it would "account for" the non-locality observed in the Aspect experiments. But note that I put the words "explain" and "account for" in scare quotes, because appealing to non-spatiotemporal reality is not an understandable answer. But the point is that if one buys into this line of argumentation, then one should not expect one. Yet something definite has been argued for: that there is a reality for which our ordinary language fails.
What clinches the argument for me -- and is the reason I became religious -- is that mystics have been saying for millenia that fundamental reality is not spatiotemporal. And they have said so, or so they claim, by virtue of knowledge (of "experiencing" non-spatiotemporal reality), not by metaphysical guesswork. Should we believe them? Given the argumentation above I have no problem believing them. But it should be pointed out that mystics also say something else, that just arguing from consciousness and/or quantum physics does not, and that is that the eternal is not merely real, but also Good, and that it is possible to realize that Goodness. It is that addition that turns all this from metaphysical speculation to religion.
This, then, is my answer to the first issue: there is a reality that defies common sense language, and why it must be dealt with. Still to come: how to forge a language to deal with it.