Neuroclassicism is the view that cognition is explained by “classical” computing mechanisms in the nervous system that exhibit a clear demarcation between processing machinery and read–write memory. The psychologist C. R. Gallistel has mounted a sophisticated defense of neuroclassicism by drawing from ethology and computability theory to argue that animal brains necessarily contain read–write memory mechanisms. This argument threatens to undermine the “connectionist” orthodoxy in contemporary neuroscience, which does not seem to recognize any such mechanisms. In this paper I argue that the neuroclassicist critique rests on a misunderstanding of how computability theory constrains theorizing about natural computing mechanisms.