Biolinguistics is the interdisciplinary scientific study of the biological foundations, evolution, and genetic basis of human language. It treats language as an innate biological organ or faculty of the mind, rather than a cultural tool, and it challenges a behaviorist conception of human language acquisition as being based on stimulus-response associations. Extracting its most essential component, it takes seriously the idea that mathematical, algebraic models of language capture something natural about the world. The syntactic structure-building operation of MERGE is thought to offer the scientific community a "real joint of nature", "a (new) aspect of nature" (Mukherji 2010), not merely a formal artefact. This mathematical theory of language is then seen as being able to offer biologists, geneticists and neuroscientists clearer instructions for how to explore language. The argument of this chapter proceeds in four steps. First, I clarify the object of inquiry for biolinguistics: not speech, communication, or generic sequence processing, but the internal computational system that generates hierarchically structured expressions. Second, I argue that this formal characterization matters for evolutionary explanation, because different conceptions of syntax imply different standards of what must be explained. Third, I suggest that a sufficiently explicit algebraic account of syntax places non-trivial constraints on candidate neural mechanisms. Finally, I consider how recent neurocomputational work begins to transform these constraints into empirically tractable hypotheses, while also noting the speculative and revisable character of the present program.