Multiple realizability is the thesis that the same kind of mental states may be manifested by systems with very different physical constitutions. Putnam (1967) supposed it to be “overwhelmingly probable” that there exists some psychological property with different physical realizations in different creatures. Cao will argue that this empirical bet is far less favorable than it might initially have seemed, because function constrains possible physical realizers -- and more complex integrated functions impose more constraints. Cao suggests that if we take on board the richer picture of neural and brain function that neuroscience has been uncovering over the past 40 years, in which all sorts of brain activities beyond electrical signaling play crucial roles in the functional architecture of the system, the multiple realizability of a sophisticated and flexible cognitive system will seem much less obvious.