The enigma of the Emergence of Natural Languages, coupled or not with the closely related problem of their Evolution is perceived today as one of the most important scientific problems. The purpose of the present study is actually to outline such a solution to our problem which is epistemologically consonant with the Big Bang solution of the problem of the Emergence of the Universe}. Such an outline, however, becomes articulable, understandable, and workable only in a drastically extended epistemic and scientific oecumene, where known and habitual approaches to the problem, both theoretical and experimental, become distant, isolated, even if to some degree still hospitable conceptual and methodological islands. The guiding light of our inquiry will be Eugene Paul Wigner's metaphor of ``the unreasonable effectiveness of mathematics in natural sciences'', i.e., the steadily evolving before our eyes, since at least XVIIth century, ``the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics''. Kurt Goedel's incompleteness and undecidability theory will be our guardian discerner against logical fallacies of otherwise apparently plausible explanations. John Bell's ``unspeakableness'' and the commonplace counterintuitive character of quantum phenomena will be our encouragers. And the radical novelty of the introduced here and adapted to our purposes Big Bang epistemological paradigm will be an appropriate, even if probably shocking response to our equally shocking discovery in the oldest among well preserved linguistic fossils of perfect mathematical structures outdoing the best artifactual Assemblers.