Mathematical holism has a number of counter-intuitive ontological consequences, which result from there being no way to discriminate in terms of ontological status among parts -- none is more 'real' than another; they are all equally real; they are all equally parts of the world.
One rather strange result: there are 'too many' parts and they don't have the structure we expect. For example, partitions don't have unique complements, whereas we usually feel that a part has a unique environment ("everything else") which together with that part constitutes the whole. Technically, the set of all partitions constitutes a semi-matroid lattice which is not uniquely complemented and is not distributive, whereas our natural (pre-theoretic) concept of parts and wholes is structured as a Boolean lattice which is distributive and has unique complements. I cannot yet give an account which explains just how this gap arises, but an intriguing possibility is that it is entirely phenomenological -- that is, a necessary structural feature of awareness perhaps arising from a kind of 'lateral inhibition' which prevents us from maintaining simultaneously multiple and conflicting representations of the same thing -- you can't see the Neckar cube from both perspectives at once.
The characteristic cliché of holism is that "the whole is greater than the sum of its parts." In the current theory, we can understand this more precisely.
On the one hand, the whole is exactly the "sum", that is, the lattice-theoretic join, of all its parts. For the parts are partially ordered by inclusion, and for any two parts there is a unique largest part which includes them both -- this is their sum or join. The sum of all the parts of the world is the maximal partition I , which contains them all -- but this is just the finest partitioning of S, which is the partition of singletons: {{s}: s is a World state}. (In what follows, we exclude I as a proper part of the World, since it is isomorphic to the set of World states.)
On the other hand, there is a sense in which the cliché says something important, for there is not just one collection of (atomic) parts whose sum is the whole, as we might expect from the usual mereology in which wholes decompose into a list of independent atoms. For the atoms of the World are not independent (if they were, then they would constitute a Boolean algebra and complementation would be unique [Roosen-Runge, 1967]); as a result, the World can be taken as a sum of atomic (and hence disjoint) parts in more than one way -- the whole is more than a single sum of its parts.
There is a more difficult twist we can give to the cliché -- difficult in the sense that it seems hard to state clearly. We are used to thinking of a sum of a collection as something that is determined by that collection as "seen from the outside"; to form a set ( a union or sum) from atomic objects a and b, we do not need to know anything more than the labels in order to describe or identify the sum as {a, b}, and there are no two other atomic objects which sum just to that, so the identity criterion for the sum object is based on just the identification of the parts or summands; the internal structure of the parts is not involved in defining their sum. This works for part-whole structures which have the structure of Boolean algebras, as do most of such structures in the phenomenal world. But it fails for the parts of the World as defined here. In the lattice of partitions, the sum of two partitions can only be identified from the internal structure of the partitions -- for World parts, that means in terms of the sets of World states which constitute the states of the parts.
What part of the world is the sum of a and b? We can give it a name a + b, but to identify it, we need to know how many states it has and how the states of a and b are determined by its states. Suppose a has states A1 and A2; b has states B1, B2, and B3. From this alone, we can conclude very little about a + b. It may have as many as 6 states or as few as 3 and there are several possible ways in which the states of a and b might be determined by the states of a + b. In this sense, the whole is not simply determined by listing the parts of which it is a sum.
Only if the internal structure is made visible -- if, for example, a = {{a,b}, {c, d, e}} and b = {{a}, {e}, {b, c, d}}, -- only then is the sum a + b completely determined; in this case, as {{a}, {b}, {c, d}, {e}}.
But the structure which determines the sum is the structure of the part states as constituted by World states, and this is inaccessible from within the World, that is, inacessible to World parts. No part less than the whole can fully represent the whole, so no proper part of the World can ever completely reveal the way it depends on the states of the World. Thus, from within the World, the parts of the World do not reveal enough to completely determine their sums, even less that sum which is the whole.