Supervenience seems to be neither an explanation nor a description. It is a mirage in a desert of ignorance.
Well to be fair, we might speak of supervenience in terms of description in that it describes a relation. But that alone seems not all that interesting. Most will probably agree that the physical and mental are related. I've been knocked out before, it seems my thinking was knocked out right along with it. But, to say that the mental supervenes on the physical seems to favor a certain kind of relation. The relation here is asymmetrical. It works one way from the physical to the mental so that the mental is dependent on the physical. Now at this point not a whole lot has been described. Certainly how this relation takes place is a great mystery. So, we don't have an explanation either.
What is interesting is that it favors the physical and so the materialists go home and laugh and feel satisfied. Be that as it may, there is the possibility of a symmetrical relation where the relata are somehow mutually interdependent. Then, there are those who favor the relation in the opposite direction. What is the genetic code but information? Of course folks like Stephen Myer and Thomas Nagel get run out of town for messing with the direction of the sacred relation as it has been handed down. All that to say, even if we allow it as a description it isn't describing all that much in terms of a mechanism or a process.
And of course the flip-side of supervenience is reduction. If B supervenes on A, then B can be reduced to A. Or can it? It is difficult to show how reduction works (or would work). We will not go into the various kinds of reductionism available. And I won't bore you with bridge laws or with a long discussion of the Multiple Realization Thesis. What I will do is introduce you (in case the two of you have not yet met) to C.D. Broad and an observation he made long ago when reductionism and emergentism (the forerunner of supervenience) were still...emerging.
In his The Mind and It's Place In Nature (copy found here), first published in 1925, C.D. Broad points out the properties of some wholes are not predictable simply by knowing the properties of their constituents in isolation. I can know all the properties of oxygen in isolation and all the properties of hydrogen in isolation, but I would still not be able to predict the properties of water.
"Oxygen has certain properties and Hydrogen has certain other properties. They combine to form water, and the proportions in which they do this are fixed. Nothing that we know about Oxygen by itself or in its combinations with anything but Hydrogen would give us the least reason to suppose that it would combine with Hydrogen at all. Nothing that we know about Hydrogen by itself or in its combinations with anything but Oxygen would give us the least reason to expect that it would combine with Oxygen at all. And most of the chemical and physical properties of water have no known connexion, either quantitative or qualitative, with those of Oxygen and Hydrogen. Here we have a clear instance of a case where, so far as we can tell, the properties of a whole composed of two constituents could not have been predicted from a knowledge of the properties of these constituents taken separately, or from this combined with a knowledge of the properties of other wholes which contain these constituents. Let us sum up the conclusions which may be reached from these examples before going further. It is clear that in no case could the behaviour of a whole composed of certain constituents be predicted merely from a knowledge of the properties of these constituents, taken separately, and of their proportions and arrangements in the particular complex under consideration. Whenever this seems to be possible it is because we are using a suppressed premise which is so familiar that it has escaped our notice" (C.D. Broad; The Mind and It's Place In Nature, Routledge 2013, p.63).None of this proves anything except our ignorance. To date, Broad's observation still holds. The great reductive hope at this point is a reduction of everything to physics. But, in good ole Broadian fashion, it seems unthinkable that we would ever have guessed (much less predicted) the properties of our everyday world merely by knowing the properties observed at the quantum level (in isolation or in relation). And to date, the "hard problem of consciousness" still reigns supreme. And unfortunately, psychology is still a legitimate science (or at least they are still handing out degrees for it). So what are we doing? It seems to me this whole project, as stalled as it may be, is simply a constructed haven for physicalists and materialists. That is not an indictment so much as an observation. Maybe there are no better alternatives, but I fear we will never know at this rate.
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