In most materialist theories (and non-materialist theories that wish to fit experimental evidence) it is supposed that somehow mental states or properties supervene on physical ones. Not all collections of physical properties have such supervening mental properties, but in the cases where the mental does supervene on the physical it does so necessarily (meaning that there are no physically possible worlds where that specific mental property or properties does not supervene on that collection of physical properties). This then raises the question: how do these mental properties causally affect the world. We know from experience that the mental does play a causal role, but supervenience seems to leave no room for it.
In this picture the Ms are mental properties and Ps the physical properties. The arrows upwards from the Ps to the Ms represent the supervenience relation, and the black horizontal arrow represents physical causation. The red and blue arrows are both possibilities for real mental causation, but in both cases it would seem physical causation and the supervenience relation have already determined matters. Let me simply assert that the red arrows are not possible causal links, and if you want the details as to why I suggest you read Jaegwon Kim’s excellent book Mind in a Physical World.
One way out of this dilemma is instantiation. Instead of appealing to a generic notion of supervenience we instead say that the physical properties instantiate mental properties. A visual representation of what instantiation is like is depicted below:
In this picture the physical properties, the Ps, instantiate a mental property M. The same mental property may be instantiated by different collections of physical properties (represented by the different colors of the Ps), and if a collection of physical properties instantiates some mental properties there are no physically possible worlds in which it doesn’t instantiate those mental properties (as under supervenience). Strictly speaking we might say that under this view the mental properties, as independent entities, don’t exist. However, they are certainly valid descriptions of similarities between different physical realizers, and so they do exist as valid abstractions. Under this view mental causation doesn’t pose a significant problem.
Because the mental properties are no longer something over and above the physical mental causation simply is physical causation. As the he mental is in a sense “made of” the physical it has causal powers because its physical realizers have causal powers, and there is no longer a need for additional mental causation.
There is a second, less talked about, problem with mental causation under a materialist view of the mind. The problem arises because we think that there can be many different physical realizers of a mental state. Because their physical properties are different these realizers will be the cause of different subsequent collections of physical properties, which is as we expect. However, there is no guarantee, under many materialist theories, that these subsequent physical properties will realize the same mental properties. This situation is depicted below.
If this is true then it is hard to say how there could be mental laws at all. Given any mental state there could be an infinite number of possible successor states, which would make the mental realm definitely strange, in a way that we should find unacceptable (because it conflicts with experience). What we need to do is find a materialist theory that permits one mental state to be followed only by some limited subset of the possible mental states under normal circumstances. One solution is functionalism, which states that a collection of physical properties realizes mental properties only when it has the correct kind of causal powers, specifically those that result in the right kind of subsequent mental states, basically building the requirement that some kind of regularities in the succession of mental states right in. Another approach is to look at the whole system, extended through time, as what can be conscious. The fact that one state is succeeded by another then becomes an additional property that can determine if the system is instantiating mental properties, again making only certain sequences of mental states possible if mental properties are to be instantiated at all.