Thursday, March 30, 2006

Sparse Properties

I've always found it incredibly implausible to think that properties like "being coloured if red", or even various non-trivial ones like "having four legs", should belong to our fundamental ontology. They're just obviously not fundamental components of reality. So I rejected the theory of universals, for this bloated ontology and the ludicrous claim that all objects are partly composed of indefinitely many trivial properties (e.g. "being coloured if red"). And note that trope theory faces the same problem. But the situation might improve if we distinguish between the "sparse" and "abundant" properties, holding that universals consistute only the former. As Lewis writes:
[U]niversals or tropes are credible only if they are sparse. It is quite easy to believe that a point particle divides into a few non-spatiotemporal parts in such a way that one of them gives the particle its charge, another gives it its mass, and so on. But it is just absurd to think that a thing has (recurring or non-recurring) non-spatiotemporal parts for all its countless abundant properties! And it is little better to think that a thing has a different non-spatiotemporal part for each one of its properties that we might ever mention or quantify over.

-- David Lewis, On the Plurality of Worlds, pp.66-67.

The sparse properties are those "perfectly natural" properties which a completed physics could uncover. They "carve [nature] at the joints" (p.60). These could plausibly be fundamental constitutents of reality. Not so for the "abundant" properties, however. Those are better understood as nominalistic constructions, perhaps as the sets of all their instances (across all possible worlds). Lewis suggests (p.56) that this construction captures one conception of the 'property' role, but for those that want to distinguish necessarily co-extensive properties (e.g. triangularity and trilaterality), we can also construct structured properties to satisfy them.

Pretty cool, really. I'm happy with the idea of sparse tropes and constructed abundant properties. Seems sensible enough, which is more than can be said of the alternatives I'd previously come across!



  1. First, it should be noted that most advocates of universals have wanted sparse universals (the issue is discussed in Plato's "Parmenides", and may well have come up earlier). Perhaps the best known modern advocate of universals, David Armstrong, has always insisted on their sparseness.

    However, while sparse properties may have appealed to the two most prominent philosophical Davids of recent times, I don't think they're tenable. We have no good reason to believe there are any fundamental or perfectly natural properties. At least, that's one of the conclusions of my (sadly as yet unfinished) dissertation; for published discussion of some of the issues which lead me to that conclusion, you might check out Jonathan Schaffer's stuff on infinite complexity.

  2. One other interesting question (at least to me) regarding sparse properties is whether there is a single set of sparse properties sufficient for explanation.

    That is let us say hypothetically there is some Set S1 such that the members of S1 are the properties such that they are all "orthagonal" and yet describe all properties in our universe. Is it the case that there is no Set S2 with a different set of properties that does the same?

    To make the mathematical analogy, consider the axis in Cartesian space {x,y}. Clearly there are infinite other orthagonal axis which could be used instead. Say {theta, r}. If sparse properties are equivalent to these orthagonal axis, does the same thing occur.

    The second thing to wonder is whether they really need to be orthogonal. Perhaps there is some property in a sparse set that isn't fully orthagonal to some other property, but is needed to explain all other properties. Would that still count as sparse even if it is a minimal property set.

    OK, maybe this is only of interest to me. But way back when I first got into thinking about properties in college it always seemed a big issue. One reason I wonder whether sparse properties are workable.


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