7 Perdurance
Perdurance is parallel to spatial extension. For a road to be spatially extended is for the road to consist of different spatial parts, one for each spatial region it occupies. Perdurance is similar. For a person to be temporally extended is for the individual to consist of different temporal parts, one for each time at which the individual exists.
7.1 Temporal Parts
One potential objection concerns the concept of a temporal part. We are completely familiar with the relation a part like the torso of a statue bears to a whole like the statue, but that is spatial part. The torso is a spatial part of the statue, and it is in fact part of it at a given part. But what exactly is for something to be a temporal part of something else?
7.1.1 Parts and Wholes
It will be helpful to fix some terminology. Talk of parts and wholes is often regimented and systematized by the axioms of a formal theory called mereology. We may take as primitive a predicate for the part to whole relation:
\[ x \leq y \tag{$x$ is a part of $y$} \]
Some definitions:
\[ \begin{array}{lll} x \circ y & := & \exists z (z \leq x \wedge z \leq y) & \text{($x$ overlaps $y$)}\\ x Fu S & := & \forall z (z \leq x \leftrightarrow \exists y (y \in S \wedge z \circ y)) & \text{($x$ is a fusion of $S$)} \end{array} \]
One axiomatization of Classical Mereology in terms of part is found in (Lewis 1991) and consists of three simple axioms:
7.1.2 Classical Mereology
\[ \begin{array}{llll} \text{Transitivity} & & (x \leq y \wedge y \leq z) \to x \leq z\\ \text{Unrestricted Fusion} & & \forall S \ \exists x \ xFu S\\ \text{Unique Fusion} & & \forall S \ \forall x \forall y \ ((xFu S \wedge y Fu S) \to x = y)\\ \end{array} \]
One axiom states that fusion is transitive and the other two claim that each set has one, and only one fusion. Several important consequences are that the relation of part to whole is reflexive and anti-symmetric:
\[ \begin{array}{lll} \text{Reflexivity} & & x \leq y \\ \text{Anti-symmetry} & & x \leq y \wedge y \leq x \to x = y \\ \end{array} \]
Notice that this is an atemporal relation of part to whole. That is, a relation an object \(x\) bears to an object \(y\), regardless of time. We are in a position to use that relation as a primitive in order to define what is for something to be a temporal part of something else:
\(x\) is a temporal part of \(y\) at time (or time interval) \(t\) if and only if
- \(x\) is part of \(y\).
- \(x\) exists at \(t\), and only at \(t\).
- \(x\) overlaps every part of \(y\) that exists at \(t\).
Your temporal parts are supposed to be parts of you with a certain temporal extent or location. It is not that they are part of you at one time but not at another. This is, on the face of it, very different from spatial part: one object may be part of another at one time and not at another. We understand what it is for an object to be part of another at a given time, but it is not clear we have room for a simple two-place relation of part to whole. Or at least it is not clear how it relates to the more familiar three-place relation: \(x\) is part of \(y\) at \(t\).
One way for the perdurantist to address the issue is to present the view in terms of a temporally relativized relation of part to whole at a time. (Wasserman 2017) begins with two primitive predicates:
\[ x \ \text{is a part of} \ y \ \text{at} \ t, \]
and
\[ x \ \text{exists at} \ t. \]
Both are familiar relations. Ordinary objects change their parts. The windshield that used to be part of my car before its replacement is now no longer a part of my cart. On the other hand, objects exist at some time intervals and not others.
The first task now is to explain what is for a time interval to be time span of a given object:
a time interval \(t\) is the time span of \(x\) if, and only if:
\(x\) exists at \(t\).
\(x\) exists at every subinterval of \(t\).
\(x\) does not exist at intervals wholly distinct from \(t\).
Two intervals are wholly distinct if they have no times or time intervals in common. For the definition of temporal part, we introduce one more defined predicate: \(x\) overlaps \(y\) at \(t\).
\(x\) overlaps \(y\) at \(t\) iff there is some \(z\), which is both a part of \(x\) and \(y\) at \(t\).
We are now in position to deliver a tentative definition of temporal part in terms of the primitive relations isolated above:
\(x\) is a temporal part of \(y\) at \(t\) if, and only if:
\(t\) is the time span of \(x\).
\(x\) is part of \(y\) at \(t\).
\(x\) overlaps at \(t\) every part of \(y\) at \(t\).
Temporal parts at a given time or time interval \(t\) are material objects. The first clause makes sure these material objects have the appropriate temporal extent; the second clause requires the objects in question to be genuine parts of the temporally extended object. Finally, the last clause ensures that the temporal part overlaps parts of the object at the relevant time.
The definition is parallel to the definition of instantaneous temporal part (Sider 2003) provides in p. 59. The main difference is that the definition above generalizes to non-instantaneous time intervals, e.g., my temporal part at the extended interval from 8:30am to 9am.
This matters because there are models of time without instants. Instead, we find only extended temporal intervals, which may approximate what would have been an instant as closely as one may want.
Perdurantism presupposes an ontological thesis according to which spatiotemporally extended objects have temporal parts at every interval at which they exist.
Necessarily, each spatiotemporal object has a temporal part at every time interval at which it exists.
Notice that as stated, four-dimensionalism is merely an ontological thesis. It posits the existence of temporal parts for spatiotemporal objects, but it does not by itself address the question of persistence: nowhere in the formulation of four-dimensionalism it is said that material objects persist through time by virtue of having different temporal parts at different times at which they exist. This suggests to (Wasserman 2017) that perdurantism should be formulated differently:
Objects persist through time by having different temporal parts at different times at which they exist.
Perdurance presupposes four-dimensionalism but it provides an explicit answer to the question of persistence.
7.2 How to Argue for Perdurantism
The combination of four-dimensionalism and perdurantism against the backdrop of the B-theory of time is often offered as a package deal, which promises to deliver elegant solutions to a wide family of puzzles in metaphysics:
Change
Some objects exemplify change in intrinsic properties. A candle may be 7-inches long at one time and 5-inches long at another. Whether the candle is 5- or 7-inches long is not a matter of how the candle is related to something else. Yet, nothing can be both 5- and 7-inches long.
The Puzzles of Coincidence
The statue is, on the face of it, something over and above the portion of clay of which it is made. But both statue and clay are material objects. No two different material objects can occupy the same space at the same time, yet the statue and the clay appear to do just that.
Fission
What it makes a person at one time to be the same as another at an earlier time is that there is a chain of psychological continuity and connectedness between one and the other stage. But we can imagine cases in which one and the same stage is psychologically continuous and connected to two different person stages in the future. On the other hand, since the future person stages are different from each other, the earlier stage cannot be identical to both.
7.3 Time Travel and Perdurance
Does time travel present special problems for perdurance?
Self-visitation is a staple of time travel fiction. Consider Tim, the time traveler who is born in 1980 and decides to step into the time machine once he turns fifty. Tim turns the dial and finally emerges in his college room in the year 2000. We may even imagine that the fifty-year old Tim comes to warn twenty-year old Tim about his future.
On the face of it, the fifty- and the twenty-year old are two different people:
- Fifty-year old Tim is wholly gray-haired.
- Twenty-year old Tim is wholly brown-haired.
- Nothing is both wholly gray- and wholly brown-haired.
But if this is in fact a case of self-visitation, then there is exactly one person in the room, who is wholly gray-haired and wholly brown-haired.
How is perdurance supposed to help with this scenario? One may be tempted to respond that there are two different person stages at the time \(t\) of self-visitation:
- Tim has a fifty-year old temporal part at \(t\)
- Tim has a twenty-year old temporal part at \(t\)
Furthermore, one may be tempted to add that the temporal parts in question differ in spatial location at \(t\). Unfortunately, that is not what follows from the account of temporal part with which we started. Instead:
- Tim has a spatially disconnected temporal part at \(t\), one that is scattered in the room.
But then the question arises whether Tim is wholly gray-haired or wholly brown-haired at \(t\), and it looks like the answer should be neither: Tim’s temporal part at \(t\) does not exemplify the property of being wholly gray-haired or the property of being wholly brown-haired.