We document the influence on programming language semantics of the Platonism/formalism divide in the philosophy of mathematics.

Sections 1 through 4 define, in the usual inductive style, various classes of object including one which is called the “combinatory terms of polymorphic type”. Section 5 defines a reduction relation on these terms. Section 6 shows that the weak normalizability of the combinatory terms of polymorphic type entails the weak normalizability of the lambda terms of polymorphic type. The entailment is not vacuous, because the combinatory terms of polymorphic type are indeed weakly normalizable, as is proven in Sect. 7 (...) using Tait and Girard’s computability predicates. The remainder of the paper is devoted to arguing that interesting consequences would follow from the existence of an “ordinally informative” proof, i.e. one which uses transfinite induction over recursive ordinal numbers and otherwise finitary methods, of normalizability for as large a class of the terms as possible. (shrink)

Cooper storage is a widespread technique for associating sentences with their meanings, used in diverse linguistic and computational linguistic traditions. This paper encodes the data structures and operations of cooper storage in the simply typed linear \-calculus, revealing the rich categorical structure of a graded applicative functor. In the case of finite cooper storage, which corresponds to ideas in current transformational approaches to syntax, the semantic interpretation function can be given as a linear homomorphism acting on a regular set of (...) trees, and thus generation can be done in polynomial time. (shrink)