The Atlas of Reality
Page 89
24.2.2 Substratism vs. Classical Genidentity Theory
In light of the small worlds objections lodged in the last section, Replacementists might consider taking on board a metaphysically fundamental genidentity relation. On this account, although only instantaneous objects are metaphysically fundamental, there is a fundamental relation of genidentity that holds between certain slices and not others, in such a way that persisting things exist when and only when there is a genidentity-unified spacetime worm. This is just Classical Genidentity Theory.
Is there a real difference between Substratism with its fundamental persisting things and Classical Genidentity Theory with its fundamental genidentity relation? There does seem to be some difference between the two, since the latter takes genidentity to be a metaphysically primitive relation, while the former defines genidentity in terms of the more fundamental relation of identity. Suppose that the genidentity relation is necessarily reflexive, symmetric, and transitive (what mathematicians call an ‘equivalence relation’):
Def D24.9.1 Reflexivity. A relation R is reflexive if and only if for every x, Rxx.
Def D24.9.2 Symmetry. A relation R is symmetric if and only if for every x and y, if Rxy then Ryx. [= Def D10.2]
Def D24.9.3 Transitivity. A relation R is transitive if and only if for every x, y, and z, if Rxy and Ryz, then Rxz.
Genidentity an Equivalence Relation. Genidentity is reflexive, symmetric, and transitive.
Identity is clearly reflexive, symmetric, and transitive, since everything is identical to itself, if x is identical to y then y is identical to x, and if x = y and y = z then x = z. This is a theorem of logic because it follows from Leibniz's Law, which says that when x and y are identical, we can substitute one of x's names for one of y's names in a truth and still obtain a truth as a result. If x = y and y = z, then we can replace ‘y’ with ‘x’ in the second sentence, obtaining the further truth that x = z.
Given the definition of genidentity and the transitivity of identity, the hypothesis that genidentity is an equivalence relation would follow logically from the hypothesis that each time-slice is a temporal part of only one persisting thing. This is No Temporal Coincidence:
24.4T No Temporal Coincidence. Necessarily, no instantaneous thing is a time-slice of more than one persisting thing.
24.4A Temporal Coincidence. It is possible for an instantaneous thing to be a time-slice of two distinct persisting things.
In a world with temporal coincidence, there could be two distinct spacetime worms, each corresponding to the career of a persisting thing, that perfectly coincide at an instant of time. The contrary thesis, No Temporal Coincidence, rules out such criss-crossing of spacetime worms that correspond to persisting things. Let's prove that No Temporal Coincidence entails the transitivity of genidentity, if we assume that genidentity is definable in terms of identity:
Assume No Temporal Coincidence.
Assume that x and y are genidentical and that y and z are genidentical.
By the definition of genidentity, x and y are time-slices of some persisting thing A, and y and z are time-slices of some persisting thing B.
No Temporal Coincidence entails that A and B are identical, since y is a time-slice of each of them.
Consequently, x and z are both time-slices of A. (This step tacitly assumes that identity is eternal: that if A and B are identical at the time of y's occurrence, then they were also identical at the time of x's occurrence. We'll take up this issue in the next subsection.)
Thus, x and z are genidentical.
Thus, Substratists have an explanation for the reflexivity, symmetry, and transitivity of genidentity, something that Classical Genidentity Theorists have to posit as a brute metaphysical necessity. This provides the Substatists with an advantage in terms of Ockham's Razor (PMeth 1.2).
However, is it really the case that genidentity is transitive? If genidentity weren't transitive, then this failure of transitivity would give a decisive advantage to Classical Genidentity Theory, since Substratists are forced to take definable genidentity to be transitive.
There are a number of puzzling cases in which genidentity seems to be non-transitive, including possible cases of fusion and fission. Suppose an amoeba A divides into two “daughters”, B and C. It seems natural to think that A is the same amoeba as B, and also the same amoeba as C, even though obviously B and C are different amoeba.
24.2.2.1 The eternity of identity and distinctness.
However, the very cases that raise questions about the transitivity of genidentity could be used to raise questions about the eternity of identity, which is needed to prove the transitivity of genidentity. Perhaps A really is identical to both B and C, even though B and C are not identical to each other. Or, to be more precise, perhaps A was identical to B and to C (when it was one amoeba) but is not now identical to either of them, and so neither are B and C now identical to each other. Could A have been identical to B in the past but not in the present or future? Is identity the sort of relation that can change, like love or proximity? The transitivity of identity over time depends on the eternity of identity: if A and B are identical at one time, are they identical at all times? If so, identity over time must be transitive, since identity at a time is.
Saul Kripke (1980) gave a very influential argument for the necessity and eternity of identity.
Suppose x = y.
It is a truth of logic that x = x.
So, it is necessarily true that x = x.
So, x has the property of being necessarily identical to x.
Since x and y are identical, y must also have the property of being necessarily identical to x. (By an application of Leibniz's Law to 1 and 4)
So, it is necessarily true that x = y.
We can replace the occurrences of ‘necessarily true’ with ‘permanently true’ in Krikpe's argument in order to demonstrate that temporary identity is impossible. If x = y at any time, then it is permanently true (true at all times) that x = y. Things can't be temporarily identical.
This argument can be extended to demonstrate the eternity of distinctness (see Williamson 1996):
Suppose that b ≠ c.
Assume that at some time in the future b = c, for contradiction.
By the previous argument, we know that b = c entails that it has always been true that b = c. So, b = c at some time in the future entails that at some time in the future it will always have been true that b = c.
If at some time in the future it will always have been true that b = c, then b = c now. What was or will be always the case is now the case.
Contradiction (between 1 and 4). So, it will never be the case that b = c.
By a similar argument, we can show that from 1 it follows that it never was the case that b = c.
Thus, if b ≠ c now, then the two have always been and always will be distinct.
24.2.2.2 Is genidentity transitive?
To sum up: the eternity of identity and distinctness provides an argument for Substratism over Classical Genidentity Theory. In particular, Substratism can provide a simple explanation for the fact that the genidentity relation is reflexive, symmetric, and transitive. Classical Genidentity Theorists must posit this as a brute necessity. However, if genidentity is not transitive, then we have a strong argument for Replacementism of some kind. So, the issue of the transitivity of genidentity is a critical one.
Here are some apparent cases of non-transitive genidentity:
Fission and fusion. If A is split in two and survives twice over, as both B and C, then A = B and A = C, but B ≠ C, since the two are located at different places, with possibly different qualities and properties. Similarly, if B and C are originally distinct (so B ≠ C) but fuse together into one simple thing, A, we might conclude that B = A and C = A, even though B ≠ C.
The Methuselah paradox (based on Thomas Reid's example of the brave officer—Reid 1785). If a person were to live for thousands of years, like Methuselah in the Bible, the younger Methuselah mig
ht be so different from the older Methuselah that the two are different people. However, each year's Methuselah is identical to the following year's Methuselah, so M1 = M2, M2 = M3, and so on until M(n−1) = Mn. However, M1 ≠ Mn, another failure of transitivity. We could also apply Chisholm's paradox to a very long-lived particle, which changes over time from one kind of matter to an entirely different kind.
In the next section, we will consider the problem of the persistence of composite objects, including objects that can gain or lose parts. For the time being, let's suppose that we are only concerned with Hawthorne's quality or dynamically first-class objects. These are the sort of things whose genidentity matters when distinguishing spinning disks and flowing rivers from stationary ones. For such things, we have good reason to suppose that genidentity is necessarily transitive, thanks to the conservation laws (e.g., the conservation of mass-energy, charge, baryon number, and so on). If a dynamically first-class particle A were to split into two, then the two fission products would each have to have less mass than A had (unless one of the two were to have no mass at all). Thus, it is impossible for us to count both B and C as genidentical to A, since at least one, if not both, will differ from A in one or more essential quantities.
Similarly, conservation laws will guarantee that any quality object, no matter how long-lived, will always possess the very same essential quantities. If the particle were at some point to lose mass or charge, then we would have good grounds for saying that the particle had been destroyed at that point.
Thus, there are good grounds for thinking that genidentity, as applied to quality or first-class objects, is necessarily transitive. This provides some advantage to Substratists if they assume that No Temporal Coincidence is necessary. Under these assumptions, the transitivity of identity, which is a law of logic, is sufficient to explain the transitivity of genidentity.
And there is a reason Substratists might have for embracing No Temporal Coincidence. First, we have to assume that Substratists are willing to concede that there are in fact time-slices. That is, we must assume Time-Slice Substratism. According to Endurantists, there can be no question about No Temporal Coincidence because there simply are no time-slices at all. If Substratists do accept the existence of time-slices, they would be wise to consider time-slices to be logical constructions (like ordered pairs of persisting things and instants of time) or dependent entities (like spatial boundaries). If each time-slice is metaphysically dependent on the persisting thing of which it is a part, that is, if time-slices are a kind of internal temporal boundary of persisting things, then No Temporal Coincidence seems to follow, because the identity of the time-slice will be a function of the identity of the thing of which it is a temporal part.
As we have seen, if genidentity is transitive and No Temporal Coincidence is true, Substratists can explain the transitivity of genidentity by means of the logical truth that identity is transitive, while Classical Genidentity Theorists must treat the transitivity of genidentity as a brute metaphysical necessity. We've seen that the conservation laws give us good reason to believe that genidentity is transitive insofar as it connects stages of quality objects.
However, there is an objection to this argument to consider. Couldn't Classical Genidentity Theorists appeal directly to the conservation laws themselves as the ultimate explanation for the transitivity of genidentity? They could certainly appeal to something like the ordinary conservation laws, but not perhaps to the laws themselves. We would normally state a conservation law in something like the following way:
(1) The mass-energy of a particle or other quality object does not change over time, unless it gains or loses a quantity of mass-energy from some other quality objects.
However, Classical Genidentity Theorists can't appeal to such a conservation law, since they don't believe in the fundamental reality of quality objects. They must appeal to something like (2):
(2) The mass-energy associated with one instantaneous object must be equal to the mass-energy associated with any other object genidentical to the first, unless there has been some intervening interaction.
There is something more natural about the form of the conservation law in (1), as compared with (2). The law stated in (1) appeals to an a priori very plausible principle: ifone and the same thing exists at two times, then it will be intrinsically the same at both times, unless there is some causal explanation of the change. The form of the law in (2) can make no such appeal, since the two time-slices are only genidentical to one another. They are not one and the same thing; they just bear some primitive, external relation to each other. So, it seems that Substatists have some advantage over Classical Genidentity Theorists in explaining the basis of the conservation laws, in respect of Ockham's Razor (PMeth 1.4).
24.3 The Metaphysics of Motion
In the last section, we examined the problem of intrinsic change. In this section, we turn to a kind of change that may (or may not) be extrinsic: change of position or locomotion. What is the nature of locomotion? If a thing has a certain velocity, what does this fact consist in? There are two main contenders: the At/At Theory of motion (developed by Bertrand Russell 1917) and Intrinsic Motion.
24.3.1 The At/At Theory vs. Intrinsic Motion
On Russell's At/At Theory, a body's motion consists in its being at different locations at different times, nothing more or less. This means that whether a thing is in motion at a given moment depends on where that thing was and will be at other moments. Its being in motion at a moment isn't intrinsic to its instantaneous condition at that moment. Intrinsic Motion, on the other hand, insists that facts about whether something is in motion at some time are intrinsic to its instantaneous condition at that time.
Intrinsic Motion has its roots in Zeno's paradox of the arrow (discussed by Aristotle in Chapter 9 of Book VI of the Physics). Zeno argued that no arrow is ever in motion, since, at every point of time, the arrow has a single location. Whatever occupies just one position in space is motionless. So, the arrow is motionless at every moment. Whatever is motionless at every moment is never in motion. Hence, the arrow is never in motion. The obvious answer to Zeno is to insist that the arrow is in motion at each instant during its flight. But, what does it mean for the arrow to be in motion at each instant? Is this an intrinsic fact about that arrow at and only at that moment, or does its being in motion at one instant consist in its being located at other places before and after that instant? This is just the dispute between Intrinsic Motion and the At/At Theory.
Def D24.10 Intrinsicality at a Time. A property P is intrinsic to a thing x at time t if and only if x's being P at t is not partially grounded in its state at any other time, or on the state of anything separate from it. [Compare Def D2.3 Intrinsicality]
24.5T Intrinsic Motion. Motion is something intrinsic to the moving thing at each moment of its motion.
24.5A Extrinsic Motion. Motion is not intrinsic to each instantaneous time-slice of the moving thing.
Def D24.11 Instantaneous Location Events. An instantaneous location-event is the event of some things (or some kind of thing's) being at some location at some instant of time.
24.5A.1T The At/At Theory of Motion. The fundamental truths about locomotion are truths about instantaneous location events.
Are the At/At Theory and Intrinsic Motion incompatible because the At/At Theory entails Extrinsic Motion? On the At/At Theory, whether or not a thing is moving, and how it is moving, depends entirely on the set of location events involving the thing. If the At/At Theory is correct, one cannot tell, simply by looking at an instantaneous time-slice of a body, whether or how it is moving.
If we adopt the At/At Theory, it would make sense also to embrace a similar theory of velocity: something's velocity at a moment t is the first derivative (the slope) of the object's trajectory through spacetime at that point. This means that an object could exist at a moment without a single, well-defined velocity if its motion is not continuous or not differentiable at that moment. For example, if so
mething were to abruptly change its velocity from 10 mph to 20 mph, it would have no well-defined velocity at the moment of the change. Similarly, if an object were to jump through space and time in a discontinuous way, moving through space up until moment t, then appearing for an instant at t somewhere in the distance, and then somewhere else in some succeeding instants, the object would have no velocity at all at instant t. Third, suppose an object were created at t and annihilated immediately afterward, having completed no trajectory in spacetime. Such an object could, on the At/At Theory, have no velocity during its sole moment of existence. In contrast, Intrinsic Motion could attribute some definite velocity to the objects in each of these three cases.
Kinetic energy and momentum do seem to be intrinsic properties of basic particles. These quantities are conserved in a variety of transactions, and the conservation of energy and momentum play a role in explaining observed motion. Consider, for example, Newton's cradle, in which three steel balls are suspended from a framework in such a way that the three balls are touching at rest. If one ball is raised and allowed to swing, hitting the second, the kinetic energy flows from the first through the second into the third, causing the third to move, even though the second does not move at all (assuming that the balls are perfectly rigid bodies).
At the same time, kinetic energy and momentum both seem to be functions of velocity (energy equal to mass times velocity squared, momentum to mass times velocity). If so, the middle ball in the cradle must have, for an instant, a positive velocity without actually changing position. At/At Theorists must either deny that kinetic energy and momentum are intrinsic properties of things or else they must deny that a thing's kinetic energy and momentum are determined by (and dependent on) its velocity. Either assumption seems implausible: