In Four-Dimensionalism, Ted Sider gives the following definition of a temporal part (p. 59):
x is an instantaneous temporal part of y at instant t =df (1) x exists at t, but only at, t; (2) x is part of y at t; and (3) x overlaps at t everything that is part of y at t.
I am a little puzzled about how to apply this definition to temporal parts of time travelers. Following Lewis, let’s distinguish external time (physical, objective) and personal time (subjective, “inner”). Intuitively, I think that different instants of personal time correspond to different temporal parts. I will explain why I believe that this intuition is incompatible with Sider’s definition of a temporal part, and it would be great if someone could tell me whether I am right.
Consider the case of Takouhi the time traveler. Let external time and personal time be represented as extT_ and persT_, respectively. At extT0 and persT0, Takouhi is in Ann Arbor. In the future, extT1 and persT1, she gets in the time machine to travel back to extT0 but to Chicago. To be specific: at extT0 and persT2, Takouhi is in Chicago. Now let TAKOUHI be the spatiotemporally extended “worm” of her whole existence. Given that I intuitively think that different instants of personal time correspond to different temporal parts, I also think that there are two temporal parts of TAKOUHI at extT0. Call them takouhi-annarbor and takouhi-chicago, corresponding to her at persT0 and persT2 respectively. (By the way, I find the “worm” picture to be somewhat suggestive of this intuition.)
But if Sider’s definition of a temporal part were right, then my intuition must be wrong. For suppose, in accordance with my intuition, that takouhi-annarbor and takouhi-chicago are both temporal parts of TAKOUHI at extT0. Then by clause (3) of Sider’s definition, takouhi-annarbor would overlap everything that is part of TAKOUHI at extT0. However, it is false that takouhi-annarbor overlaps everything that is part of TAKOUHI at extT0 because it does not overlap takouhi-chicago.
Hence, according to Sider’s definition of a temporal part, takouhi-annarbor and takouhi-chicago cannot be temporal parts of TAKOUHI at extT0. They are instead merely parts of TAKOUHI. In fact, the temporal part of TAKOUHI at extT0 is the mereological sum of takouhi-annarbor and takouhi-chicago. A more general consequence is that, contrary to my intuition, a temporal part can correspond to more than one instant of personal time.
By the way, I am not really raising an objection against four-dimensionalism. This result about time travelers and their temporal parts is unintuitive, but I don’t think it hurts the theory all that much. My main goal is just to make sure that my understanding is correct. So, dear metaphysicians, help!
Update 10/03/08: I think I now understand better what is going on. My intuition in the case is driven by considerations about what Sider calls continuants—“the referents of ordinary terms, members of ordinary domains of quantification, subjects of ordinary predications, and so on” (p. 60). Sider is quite explicit that four-dimensionalism is an ontological thesis that does not decide one way or another what continuants are. This clarification suggests a new understanding of my result about time travelers and their temporal parts: temporal parts cannot be what continuants are. To look ahead (p. 188-208), the notion of continuants is used to capture mental and linguistic content. Thus, an initially plausible way that ontology relates to content is ruled out by my result. To say what continuants are, we instead need conceptual resources beyond ones given by the four-dimensionalist ontology. Update 04/02/09: There is a related question for stage-theory. What distinguishes stage-theory from Lewis’s worm-theory is precisely what they designate as continuants (in most cases). Stage-theory says that ordinary terms like “persons” refer to stages (in most cases). However, if I am right that temporal parts cannot be what continuants are, then either (a) stages are not temporal parts or (b) stages are not continuants. Both, prima facie, seem to be a problem for stage-theory. The question, then, is whether it would be less costly to deny (a) or (b); or alternatively, reject the time-traveler case altogether.
Update 05/09/09: I think my worry is actually already addressed by Sider in a footnote. On p. 101, footnote 30, he writes,
According to the stage view, there are actually two person involved, since each person stage is a person. One of these can say that he will be the other; the other can say that he was the first, provided the stage-theoretic truth condition for `x will be F’ is modified to read: `x has a counterpart in his personal future that is F’.
How satisfying that response is depends on whether one thinks the proposed modification to the temporal counterpart relationship is appropriate or not. To answer that question, more work needs to be done in clarifying what personal time is, and how it relates to the persistence of objects.