This text is the translation of the last synthesis that Noam Chomsky made of his theory and conception developed over half a century. In the intention of the main author, it is a synthesis that: a) reaffirms many of the fundamental elements that – in the broad or contextual framework – the author has well established over time; b) specifies things and aspects that we believe have not been accurately and correctly understood; c) improves the results obtained as his ideational construction grows. No matter how much effort someone with good will might make to look at the writing and no matter how much help the author might have received from collaborators and from the co-author-disciple, for someone neutral this text does not complete and especially does not add (elements or perspectives that had not already been given the importance that they proved to have). It repeats and consolidates the already known idea that, from the beginning, Noam Chomsky’s conception was correct in relation to the state of affairs of processes and things, while at the same time being directed in relation to its evolution (the imperfections being due to the general context given by the level of knowledge, the dominant concerns over time, etc. – hence the inherent hesitations of the stage). Naturally, the former are so, the fact itself being universally valid in time and space. It also shows that the elements apparently not included at the beginning (especially regarding the origin of human language and the importance of the biological factor) are in fact pioneering actions of the author. The fact is well known to many followers of Chomsky (excellent operators of his methods and tools, attentive to the surface of the ideational conception, even if not always to the deep ideational and conceptual essence). * The translators’ effort was not motivated and sustained by their linguistic opinions, but by an authentic fuel, namely the desire to make available to the public a work that synthesizes received knowledge in linguistics and a presentation of the original conception of one of the important authors for the history of 20th-century linguistics.

The syntactic Merge operation of the Minimalist Program in linguistics can be described mathematically in terms of Hopf algebras, with a formalism similar to the one arising in the physics of renormalization. This mathematical formulation of Merge has good descriptive power, as phenomena empirically observed in linguistics can be justified from simple mathematical arguments. It also provides a possible mathematical model for externalization and for the role of syntactic parameters.

This paper suggests that the study of language may be approaching a new era in which we can envision, and sometimes attain, genuine explanations for basic features of language and answers as to why language should have these modules and not others. This Strong Minimalist Thesis that UG provides genuine explanations may now be a realistic objective, with some promising initial steps even if the goal is still remote.

In this paper we compare some old formulations of Minimalism, in particular Stabler's computational minimalism, and Chomsky's new formulation of Merge and Minimalism, from the point of view of their mathematical description in terms of Hopf algebras. We show that the newer formulation has a clear advantage purely in terms of the underlying mathematical structure. More precisely, in the case of Stabler's computational minimalism, External Merge can be described in terms of a partially defined operated algebra with binary operation, while Internal Merge determines a system of right-ideal coideals of the Loday-Ronco Hopf algebra and corresponding right-module coalgebra quotients. This mathematical structure shows that Internal and External Merge have significantly different roles in the old formulations of Minimalism, and they are more difficult to reconcile as facets of a single algebraic operation, as desirable linguistically. On the other hand, we show that the newer formulation of Minimalism naturally carries a Hopf algebra structure where Internal and External Merge directly arise from the same operation. We also compare, at the level of algebraic properties, the externalization model of the new Minimalism with proposals for assignments of planar embeddings based on heads of trees.

We extend our formulation of Merge and Minimalism in terms of Hopf algebras to an algebraic model of a syntactic-semantic interface. We show that methods adopted in the formulation of renormalization (extraction of meaningful physical values) in theoretical physics are relevant to describe the extraction of meaning from syntactic expressions. We show how this formulation relates to computational models of semantics and we answer some recent controversies about implications for generative linguistics of the current functioning of large language models.