Using the tools of mathematics and logic in order to formalize natural languages is an idea already well understood by Leibniz. However, it is only after Ferdinand de Saussure that such tools started to be used consistently by linguists so to have new insights on natural languages. A recent proposal, rooted on the structural syntax by Lucien Tesnière, formalized natural languages using modern tools of constructive mathematics, in particular categories and topos theory. This proposal is called Constructive Adpositional Grammars (CxAdGrams) and it is already tested in psychological discourse analysis and second language acquisition in a primary schools. The talk wants to give an overview on the CxAdGrams, putting it into relation with Natural Logic, in order to explore further directions of research.