Why Mereological Reductionism Fails Universals — Epoche C1
The position under attack, correctly identified Mereological nominalism is the proposal that a property such as redness is nothing but the single scattered object composed of all red things, and that for something to be red is for it to be a part of that object. The proposal is worth taking seriously because, if it worked, it would settle one of metaphysics' longest disputes at a stroke; this essay argues that it does not work, and that the reasons it fails are precise enough to state as arguments rather than as misgivings. Begin with the vocabulary, since the argument turns on it. Mereology is the formal theory of the part-whole relation, given its standard modern axiomatisation by Henry Leonard and Nelson Goodman in 1940 under the name 'the calculus of individuals' and developed by Goodman in The Structure of Appearance (1951) as a nominalist's alternative to set theory — nominalist because it recognises only individuals and their aggregates, never abstract entities. A fusion (or mereological sum) of some objects is the object that has all of them as parts and no part disjoint from all of them; the fusion of two shoes is a scattered object weighing what the pair weighs. Classical mereology adds two principles that matter here. Unrestricted composition says any objects whatever have a fusion, however scattered or gerrymandered. Extensionality — what the earlier version of this essay called the identity of parts thesis — says that objects with exactly the same proper parts are identical, so a fusion is fixed entirely by what goes into it. Peter Simons's Parts (1987) surveys the system and the disputes about these axioms. The nominalist analysis can now be stated exactly. Where $F$ is a predicate and $\mathrm{Fu}(F)$ the fusion of all the things that satisfy it, $$a \text{ is } F \quad \Longleftrightarrow \quad a \text{ is a part of } \mathrm{Fu}(F)$$ and the property $F$-ness is identified with $\mathrm{Fu}(F)$ itself. The attraction is genuine parsimony: no abstract objects, no sets, and identity conditions supplied by extensionality rather than by a further primitive. Its ambition is to displace realism about universals — the view, defended in the modern literature above all by David Armstrong, that in addition to particulars there exist repeatable entities, universals, which are wholly present in each of their instances, and that a particular's being red consists in its instantiating the universal redness. What Lewis actually held Before the analysis is attacked, an attribution must be withdrawn. The earlier version of this essay presented mereological nominalism as David Lewis's position and described an 'identity of parts thesis' by which he reduced universals to sums of instances. Lewis held no such view, and saying so matters, because his actual view is immune to the objection the earlier version pressed hardest. Lewis was a class nominalist. In On the Plurality of Worlds (1986) he identifies a property not with a fusion but with a set — and not the set of its actual instances but the set of all its actual and possible instances, drawn from the whole of modal space. His mereological work in Parts of Classes (1991) concerns the internal structure of classes, arguing that a class is the fusion of the singletons of its members; it is a thesis about set theory, not a reduction of properties to heaps of concrete things. The difference is not verbal. The earlier version objected that if being a unicorn is the sum of all unicorns, and there are none, then the property fails to exist though we can plainly think about it. Against mereological nominalism this objection lands, since the fusion of nothing is nothing. Against Lewis it does not, because the set of all possible unicorns is not empty: it contains the unicorns of other possible worlds, which for Lewis are as real as the inhabitants of this one. Modal realism is a heavy price, and one may decline to pay it, but the objection as stated does not reach the view. What Lewis did concede is more useful to the essay's thesis than the misattribution was. In 'New Work for a Theory of Universals' (1983) he argues that identifying properties with sets yields far too many of them — one for every arbitrary collection, most of them gerrymandered and useless — so that any adequate theory needs a distinction between natural properties and the rest, to do the work of grounding similarity, causal powers, and the reference of our predicates. That distinction, he argues, must be bought somewhere: by accepting Armstrong's universals, by accepting tropes, or by taking naturalness as an unanalysed primitive. Lewis takes the last option and says so. The point for present purposes is that even the most sophisticated extensional account concedes that extensions alone do not deliver what a theory of properties is wanted for. The dispute is not dissolved by reduction; it is relocated into the choice of primitive. Three arguments that do defeat the mereological analysis With the target correctly identified, the case against it can be made. Three arguments are available, and they are of increasing strength. First, the fusion has the wrong parts. Armstrong's objection in Universals and Scientific Realism (1978) is that the biconditional above fails from right to left. Take $F$ to be 'is a cat'. The fusion of all cats has as parts not only the cats but every part of every cat: tails, whiskers, individual cells. A cat's tail is therefore a part of $\mathrm{Fu}(\text{cat})$, but a tail is not a cat. The same holds for 'white': the fusion of all white things contains, as parts, molecules and sub-microscopic regions that are not white at all, since whiteness is not inherited by arbitrary parts. The analysis therefore over-generates massively, and the obvious repair — restricting the biconditional to parts of the right size or kind — cannot be made in mereological vocabulary, since mereology knows nothing of kinds. The repair would have to appeal to the very notion be