A counterexample to Sider's argument for temporal parts

Sider famously argues for the existence of arbitrary temporal parts by arguing for the existence of arbitrary minimal diachronic fusions. A minimal diachronic fusion of a function f from times to non-empty sets of objects such that all the objects in f(t) exist at t is an object x that exists at all and only the times in the domain of f, and at any t in that domain is a fusion-at-t of all the objects in f(t). Here, an object x is a fusion-at-t of the objects in a set S provided that every member of S is a part of x at t, and every part of x at t overlaps at t some member at S.

The argument is a continuity argument starting with the “continuous series” assumption:

• if not every function f as above has a minimal diachronic fusion, there will be a continuous series of cases starting with one where there is a minimal diachronic fusion and ending where there isn’t one.

Then Sider argues that this results in an implausible abrupt cut-off in whether there is a minimal diachronic fusion.

But the continuous series assumption need not be true. Consider a world consisting of a single temporally extended simple, x, that exists at all times from 0 to 2. In this world, u is a part of v at t if and only if t is in the interval from 0 to 2 and u = v = x.

Now, let f₀ have an interval from 0 to 1 as its domain, and let f₀(t) = x for t in that interval. Then there is no minimal diachronic fusion of f₀, because the only object there is is x, and x exists at times outside the interval from 0 to 1. Next, let f₁ have the interval from 0 to 2 as its domain, and let f₁(t) = x for all x. Then x is a minimal diachronic fusion of f₁.

Is there a continuous series of cases between the two? The best candidate is to consider the functions f_u which has as its domain the interval from 0 to 1 + u, with f_u(t) = x for all x. It seems like this is a continuous series of cases. But it’s not. For in a continuous series of cases, nearby cases are very similar. But in this case, nearby cases are not very similar: for f_1 − δ for small positive δ differs very significantly from f₁, because the domain of f₁ has the special property that it includes all the times at which any of the objects in the range of f₁ exist, while the domain of f_1 − δ does not have that special property.

More generally, consider four-dimensional atomic worlds with temporally extended simples and classical atemporal mereology. The objects of these worlds are fusions of all non-empty sets of these temporally extended simples. A fusion of simples exists at all the times at which any of its simples do. Suppose further that x is a part of y at t provided that x exists at t, y exists at t, and all the simples that exist at t and are parts of x are parts of y. It is easy to verify that this parthood-at-t relation satisfies all the formal conditions in Sider’s footnote 10.

On this story, there will be minimal diachronic fusions only for functions f whose domain is a union of the sets of existence-times for simples. For instance, if all the simples come in and out of being at integer times, there will be minimal diachronic fusions only for functions f whose domain is a union of intervals of time with integer values.