Old and New Minimalism: a Hopf algebra comparison
Matilde Marcolli Robert C. Berwick Noam Chomsky
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.