When linguists talk mathematical logic

Given the importance of recursion in modern linguistics, there ought to be much to commend in Watumull et al.’s (2014) attempt to clarify what recursion is (or ought to be); I have trudged this very terrain myself, using some of the same sources, and in order to make similar points (e.g., Lobina, 2011, but especially in Lobina, 2012). However, there are so many issues with Watumull et al.’s own attempt that a proper response is in order. I will here limit myself to the following: (a) the characterization of recursion these authors offer is wholly mistaken, the unavoidable result of misunderstanding, misrepresenting, and misinterpreting the relevant literature from the formal sciences; and b) as a corrective, I provide a definition of recursion that stands on much firmer ground in order to then show how it relates to Chomsky’s introduction of recursive techniques into linguistics. Watumull et al. (WEA, from now on) base their definition on a quote of Gödel (1931) regarding what they call the “primitive notion of recursion” (p. 2), deriving therefrom three criterial properties of recursion: (a) the function must specify a finite sequence (Turing computability, WEA claim); (b) this function must be defined in terms of preceding functions, that is, it must be defined by induction, which WEA associate with strong generativity (i.e., the generation of ever more complex structure); and (c) this function may in fact just reduce to the successor function (that is, mathematical induction, which WEA associate with the unboundedness of a generative procedure). Unfortunately, this characterization of recursion is mistaken in both design and detail. To begin with, WEA don’t provide the full version of Gödel’s text, their ellipses omitting important material, as the full text demonstrably shows (I’ll be quoting from Davis (1965), which offers a different translation from the one WEA use, but this won’t affect my analysis):