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Several strikingly different modes of formal theorizing about human language syntax arose during the 20th century. The one that clearly dominates today, generative grammar, emerged a century ago out of work on mathematicizing logical proof. Mathematically well-developed alternatives exist, but have attracted little interest in the last 70 years. I briefly sketch this history, and then discuss several fundamental but often ignored syntactic properties of human languages that bear on the appropriateness of generative grammatical theorizing. I stress two theoretical desiderata: (i) that syntactic constraints should be stated in a way that is not dependent on lexical facts, and (ii) that human languages should not be represented as sets of determinate size. The latter topic connects up with some work done here at UZH. It also bears on the counterproductive and needlessly choleric dispute about Pirahã syntax, which dissolves away under the view I advocate.
