Proof or dare

 Originally published on Language and Philosophy, August 27, 2010

While commenting on David B. Hart’s “Believe it or not” at First Things, I learned that many outside the field of linguistics and philosophy (at least, outside analytic philosophy) seem to view linguistics as just a ‘soft’ social science. I’m moved to correct:

Current syntax began with computer theory, a form of math — pure math, strictly speaking. It originated in cryptology, but as developed in computer theory, it’s all abstract elements and abstract structures replete with proofs, either by logical deduction or mathematical induction on those elements.

When I studied CT, it was presented exactly as that: pure math, including proofs for every single result. The prof — this was a while ago — had no idea Chomsky was a linguist, but knew him as a mathematician, because Chomsky had proved several important mathematical results: Chomsky Normal Form (that any multiple expansion can be reduced to equivalent set of binary expansions) and the Chomsky Hierarchy of Languages (abstract languages, not human languages). Syntax is still using that mathematical model, though much augmented and much more speculative.

Current phonology, Optimality Theory, is a highly mathematical form of complex flow chart and bears a close relationship with Game Theory.

The area I work in, semantics, is in large part applied mathematical logic — the Dutch flatly consider it little more than applied mathematical logic. Quantifiers, negation, modals, all are rendered in mathematical models including set theory, lattices, lambda abstraction, Curry functions, modal logic, with results expressed in monotonicity, scope relations, scales; and in some cases semantics has shown important results for mathematical logic itself. Montague’s work belongs among the work of Gödel and Tarski. Kratzer’s work brings logic much closer to reality. But mostly it’s a bunch of mathematicians and logicians analyzing language.

Learnability is a mathematical problem. Computability — how much space your brain can contain, assuming it’s got a finite capacity — is a mathematical problem. And Diver’s followers treat the whole field as information theory, which is entirely math.