Matrix syntax is a formal model of syntactic relations, based on a conservative and a radical assumption. The conservative assumption dates back to antiquity: that the fundamental divide in human language is between nouns and verbs, which are “conceptually at right angles” (as different as substantive words can be). The radical assumption is that such a conceptual orthogonality could be treated as a formal orthogonality in a vector space, with all its consequences. Under certain simple assumptions (placing the orthogonal attributes in the diagonal of a 2x2 square matrix), the resulting mathematical structure resembles some aspects of quantum mechanics, as stated in the structure of a group that surprisingly includes Pauli's. Perhaps even more surprising is the fact that Matrix Syntax allows us to simply describe a number of language phenomena that are otherwise either stipulated or difficult to explain. These include conditions of basic semantic selection (for verbs, prepositions, etc.) and, more generally, so-called chain conditions, as arising in passive voice, relative clauses, etc. It is curious how sentences can be simply modelled as vectors in a Hilbert space with a tensor product structure, built from the 2x2 matrices belonging to a specific group.