For those who object to the use of sets and sets of sets, mathematical holism is readily translatable into the nominalist language of the calculus of individuals described by Nelson Goodman [1966]. In this language, neither classes nor sets exist -- only individuals and n-place predicates applied to individuals.
The calculus assumes that a primitive relation overlaps is applicable to some pairs of individuals, and this relation is used to define a partial ordering among individuals defined by
x <= y if for all z, z overlaps x implies z overlaps y.
A key postulate of the calculus ([Goodman, p. 51]) is that every two individuals have a sum which is itself an individual. It is these sums which must supply whatever essential content there is in the now-eliminated concept of a class.
The sum x+y of two individuals x and y is defined to be the smallest individual which overlaps both x and y.
The individuals in the World can be identified with states of affairs, or equivalently, the states of parts (now no longer considered as sets of World states but rather as sums of World states.) We no longer distinguish between s and {s}; states of the World and states of parts are on the same ontological level -- they are both just individuals, so whereas before we distinguished between World states and states of affairs, now one term will do for both.
As a formal convenience, we add to the system of states as individuals the "universal" individual I which overlaps every individual.
Two part states overlap if there is a state of the World in which they co-occur, otherwise they are disjoint. One state s is smaller than another state t if t holds whenever s holds. Every state s has a unique 'negate' -s, such that s + -s = I, and there is no x which is both <= s and <= - s, (so s and - s are disjoint.)
The atomic states--those which contain no smaller states-- are just the individual states of the World. Although these atomic states are the same kind of entity as states of affairs, distinguished only by their minimal character, that distinction is important for holism; the key idea is that the holding or not holding of a state of affairs is always relative to some (atomic) World state. To assert that a state of affairs A holds is to assert that it holds in this World state, call it s, and the condition for the truth of that is expressed simply in the nominalist formalism by the condition that s <= A.
Just as in the set-theoretic formalism, there are two equivalent ways to represent parts (either as equivalence relations or as partitions), so also in the 'calculus of individuals', a part is representable in two distinct ways: either as an equivalence relation on atomic states, or. more simply, as a special kind of monadic (1-place ) predicate on states in general:
A monadic predicate P is a part of the World if P s and P t imply that s and t are disjoint, and the sum of all the states for which P holds is I.
The states of a part P are just the states s for which P s holds.
Thus, part which we usually think of as noun-like 'objects' or 'things' are here represented by adjective-like predicates, and what we normally think of as adjective-like states or characteristics of things are here the noun-like individuals.
The largest part (the one corresponding to "the whole World") is that predicate W which satisfies
W s iff s is atomic.If we adopt the view that parts are equivalence relations on the atomic states, then things are a bit more complicated--the states of a part R are then those individuals X which satisfy the constraint:
If x and y are atoms, then x + y <= X iff x R y.
The largest part W is the identity relation on atomic states.