The Recovery of Case

I t is April 1977. Noam Chomsky and Howard Lasnik are about to publish an important essay in linguistics.1 Having seen and studied the preprint of “Filters and Control,” Jean-Roger Vergnaud wrote to its authors.2 He had “some ideas to communicate.” Chomsky and Lasnik were unable to incorporate those ideas in their essay. Time was short; the mail, slow. They did something better. They incorporated them into their work.

It is not a sentence that suggests very much, but it is a grammatical English sentence. This is something that English speakers recognize at once, and recognize without effort. Sentences such as 1 may be embedded in still other sentences:

and so on ad infinitum? An allusion to infinity suggests an obvious question: how could infinitely many sentences be encompassed by the human brain, which, like the human liver, is blunt in its boundaries? In the first half of the twentieth century, Alonzo Church, Kurt Gödel, Emil Post, and Alan Turing created in the theory of recursive functions a mathematical scheme commensurate with the question’s intellectual dignity. The theory is one of the glories of twentieth-century mathematics.3 The factorial function n!, to take a simple example, is defined over the numbers n = 0, 1, 2, 3, …. Its domain and range are infinite. Two clauses are required to subordinate the infinite to finite control. The base case is defined outright: 0! = 1; and, thereafter, (n + 1)! = (n + 1)n! If the functions inherent in a natural language are recursive, the language that contains them comprises infinitely many sentences.

Sentences used in the ordinary give-and-take of things are, of course, limited in their length. Henry James could not have constructed a thousand-word sentence without writing it down or suffering a stroke. Nor is recursion needed to convey the shock of the new. Four plain-spoken words are quite enough: Please welcome President Trump. Prefacing 1d, on the other hand, with yet another iteration of Ralph believes, is no improvement on the original. Quite the contrary. It is a deprovement, like one hundred rounds of “For He’s a Jolly Good Fellow.” If sentences in English can be new without recursion, they can also be recursive without being new. The rules of grammar establish only that natural languages are infinite. Why they are as they are, no one knows. The same displacement of attention is at work in arithmetic. For anyone unaccountably persuaded that thirty-eight is the largest natural number, the rules of arithmetic say otherwise. The rules, note. The argument needs no further steps.

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