Some analysts have proposed various concepts(!) of nonconceptual content:
The central idea behind the theory of nonconceptual mental content is that some mental states can represent the world even though the bearer of those mental states need not possess the concepts required to specify their content. This basic idea has been developed in different ways and applied to different categories of mental state. Not all of these developments and applications are consistent with each other, but each offers a challenge to the widely held view that the way a creature can represent the world is determined by its conceptual capacities.
Or there is G. Kreisel (who famously argued that, "CH is resolvable," is a sentence that is itself resolvable in second-order logic, albeit only so far as generally stated, not modulo any particular claimed solution), who speaks of "informal rigor":
Informal rigor makes the analysis precise to eliminate the doubtful properties of the intuitive notions. The principal emphasis is on intuitive notions, which do not occur in ordinary mathematical practice but leads to new axioms for current notions. The difference between familiar independence results and the independence of the continuum hypothesis is discussed in the chapter; the difference is formulated in terms of higher order consequence.
Hannah Arendt made much of the notion of dialogue-with-oneself, which in her theory of totalitarianism is linked to problems of solitude, isolation, and loneliness, along with the "hyperlogical" thinking of the totalitarian preacher (those who start from some premise A and deduce B from that, then C from B, and so on without disjunction, but only conjunction (or not even that but conditional reasoning divorced from reformulation in terms of a more ambiguating logical operator) down the whole "murderous alphabet").
Do these theories involve entire distinct languages in the way you propose? One way to look into this would be by constructing a language yourself. In constructing some L, can you avoid imposing or imparting mathematical/logical patterns on/with the more intuitive/perceptual reference expressions? Is, or at least can, poetry be thought of as a predominantly non-mathematized use of a given language?
Note that it is perhaps less "likely" that one can speak of strictly non-overlapping languages, but rather one could think of priorities on the intuitive-vs.-logical levels. One might think neither level is more important, or one might focus on either level first; but alternations like syntax/semantics, or intension/extension, are too elementary to be totally avoidable in one direction. To use poetry as an example again, as soon as one uses a meter or dedicated rhyme scheme (as for limericks or haikus), one will have "mathematized" a poem to some (significant) extent.
Or as a last example: in mathematics, given the relative scarcity of easily-typable symbols, it is common practice to use words, or at least parts of words, from natural language, to mark out various specified functions. For example, one common way to notate a particular species of forcing is to write, "Add(a, b)," to indicate that one is to add a-many forceable objects to (the powerset of) b. And outside of procedural notation, we will repurpose natural-language terms like mouse or skeleton, sometimes with clearer, sometimes with hazier, grounding in the natural use of said words.
