Multiple realizability is a notion that grew out of cognitive science in the 1950's. A computer software program can run on very different machines - as long as the program conforms to basic computing principles, such as those described by Alan Turing's Universal Computing Machine.
Likening the mind-brain relationship to that between software and hardware,
Hilary Putnam and Jerry Fodor generalized this relation to claim that the same mental state or property could be realized by different physical states.
This is reasonable. For example, information philosophy argues that the mind is in an important sense the pure information content in a brain. The information can be transmitted to other minds, by converting it to energy for communication, which can then embody the same information (perhaps with significant differences in the details).
Multiple realizability appears to attack the basic thesis of
mind-brain identity theory. But this is incorrect. The question is whether mind/software is something fundamentally different from brain/computer.
Multiple realizability led Putnam and Fodor to the notion of
functionalism, namely that the important things are the functions performed by the brain, not the particular physical implementation of those functions. This was something appreciated much earlier by Turing:
The fact that Babbage's Analytical Engine was to be entirely mechanical will help us rid ourselves of a superstition. Importance is often attached to the fact that modern digital computers are electrical, and the nervous system is also electrical. Since Babbage's machine was not electrical, and since all digital computers are in a sense equivalent, we see that this use of electricity cannot be of theoretical importance... If we wish to find such similarities we should look rather for mathematical analogies of function.
("Computing machinery and intelligence." In The Essential Turing (1950) p. 446.)
Multiple realizability was used to argue against
reduction, the idea that mental states can be reduced to brain states, which in turn are reducible to biological states, and those reducible to physics and chemistry.
Jaegwon Kim has argued that since the physical world is "causally closed," the idea of mental states that are not reducible to brain states (so called non-reductive physicalism) is impossible. By "causally closed," Kim means that
deterministic physical laws governing the motions of atoms and molecules at the base level are sufficient to determine all the higher "
supervening" biological and mental levels.