Abstract
The arrow paradox is an argument purported to show that objects do not really move. The two main metaphysics of motion, the At–At theory of motion and velocity primitivism, solve the paradox differently. It is argued that neither solution is completely satisfactory. In particular it is contended that there are no decisive arguments in favor of the claim that velocity as it is constructed in the At–At theory is a truly instantaneous property, which is a crucial assumption to solve the paradox. If so the At–At theory faces the threat that most of our physical theories turn out to be non-Markovian. Finally it is considered whether all those threats and paradoxes are dispelled if only a new metaphysics of persistence is taken into account, namely four-dimensionalism.
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Notes
“Modified” because the original At–At-theory does not require continuity, whereas the Russellian version—that arguably deserves the title of “received view”—does.
See Footnote 12.
To jump a little bit ahead we will argue that this metaphysics of motion includes the introduction of a kinematical quantity, four-dimensional velocity, which is not an instantaneous property (thus having something in common with MAA) but is also not reducible to the derivative of the position function (thus having something in common with VP).
We cannot enter into the subtleties of different formulations of the argument, nor we can give a hint of different proposed solutions.
This will be the focus of Sect. 2.
But see the (sound) criticisms in Meyer (2003: 94–98).
White (1992: 177) points out that classical mechanics neither presuppose, nor deny instantaneous velocities. In that context instantaneous velocities are simply a metaphysical addition.
Note that Tooley (1988: 251) mentions the problem, even though in a slightly different form. However, as it stands, it is not a challenge to his own account because he is willing to supplement relativity theory with an unobservable privileged frame.
Naturally we do not mean that they are numerically the same.
Note that this non-Markovianism is not due to the complexity of the system, as is often the case in statistical mechanics. A referee of this journal has pointed out to us that this way of phrasing the argument is somewhat ambiguous. Indeed it can lead to some confusion. Arntzenius writes: “What is crucial for our purposes is that we do not think Markovianism must fail in all cases in which the probabilities of future developments depend on instantaneous velocities. And surely the At-At theory is wrong if it entails that” (Arntzenius 2000: 191). We took it to mean the following. In theories such as classical mechanics the future development of states is fixed iff the state of a physical system at an instant contains a specification of an instantaneous velocity. This is the sense in which future development of states depend on instantaneous velocities in classical mechanics. Since in the At-At theory according to Arntzenius there is no such a thing as an instantaneous velocity it turns out that the At-At theory entails that classical mechanics is not Markovian.
For an introduction see Sanford (2011).
For example it could be position and momentum.
Arntzenius contends that it is unable to explain why the ball moves towards \(r_{3}\) at \(t_{2}\), whereas it moves towards \(r_{1}\) at \(t_{4}\). Thanks to an anonymous referee for this journal for having drawn our attention to this point.
Though he also insists on the third as well, in Smith (2003b).
The position function.
Smith recognizes that there are indeed properties that could be attached non arbitrarily to a temporal point but that we will in fact not consider instantaneous. He considers associating at every point the definite integral over the position function with limits of integration \((t - k,t + k)\). However he notes that the value will indeed depend on \(k\), so that for different values of k we will have different values of the property in question. He contends that in such cases having a particular value is a property of the time interval \((t - k,t + k)\) rather than an instantaneous property of \(t\) itself. This is not the case with velocity though.
In this reconstruction of the argument, claim (9) is, strictly speaking, redundant.
This claim is somewhat hasty. Let us explain a little bit better. Smith claims, contrary to Russell himself, that Russellian velocity is instantaneous in that it is part and parcel of the instantaneous state of an object. If that’s the case we could resist the arrow paradox by simply claiming that occupying a single spatial region at an instant is neither necessary nor sufficient to be at rest at that instant. Rather being at rest at instant \(t\) amounts to “having Russellian velocity = 0” at \(t\), which resembles the solution of Velocity Primitivism. Now, this might seem a dissolution of the paradox, rather than a solution of it, for Russell himself grants Zeno that there are no truly instantaneous states of motion. This is indeed true but Smith is actually explicit in recognizing that he does not think, as Arntzenius does, that Russellian velocity is an addition to the At–At theory of motion. He explicitly writes: “Thus, rather than being a supplementation of the At-At view with calculus, this is a rejection of the view” (Smith 2002: 267). The assessment of such claims go beyond the scope of this paper. Thanks to an anonymous referee for having drawn our attention to this point.
The following discussion is highly indebted to some remarks of an anonymous referee for this journal.
See Hughes and Creswell (1985).
Four-dimensionalism is surely a controversial metaphysical thesis, yet one that has a significant philosophical pedigree which includes—but it’s not limited to—Russell (1914), Whitehead (1920), Quine (1976), Armstrong (1980), Lewis (1986), Heller (1990) and Sider (2001). Probably the most influential consideration in favor of four-dimensionalism is its unmatched ability to solve different puzzles of coincidence and material constitution. Lewis (1986) and Sider (2001) present a direct argument from vagueness to four-dimensionalism that has occupied a central stage in contemporary metaphysics. Recently it has been forcefully argued, most notably in Balashov (1999), Gilmore (2007) and Balashov (2010), that the special theory of relativity favors some variant of a four-dimensionalist ontology. We will return to this last point briefly in the conclusion.
Neither does Sider (2001). But, he claims that since three-dimensionalists usually use the time indexed notion, they would not be able to accuse him of begging the question against them. We believe that the use of time-indexed notions faces almost insurmountable difficulties in relativistic spacetimes. For different formulations of metaphysics of persistence in relativistic settings see Calosi and Fano (forthcoming). These formulations can be easily generalized to non-relativistic spacetimes.
The following axiom is assumed: \(P(x,y,t) \to TL(x,t) \wedge TL(y,t)\).
We believe that this definition of temporal part is unsatisfactory in relativistic spacetimes and should be abandoned, though we cannot argue for it here.
From now on when we refer to four-dimensional objects we mean continuous four-dimensional objects, unless otherwise specified. We will then omit this last clause.
The locus classicus is Lewis (1986: 202–04).
Note that as it stands (20) does not, by itself, force the continuity of motion. Its modification is however entirely straightforward. It just needs the introduction of a topological predicate of connection for spatial regions. If we had that we could say that for every instant of its existence there is a temporal part that is located at each region in between \(r_{1} ,r_{n}\) where this betweeness should be cashed out in topological terms. Note that the existence of a temporal part for each instant in the interval throughout which the object is allegedly in motion is guaranteed by (19).
Actually this is how the argument was originally formulated.
We could try to define Markovianism by referring to the instantaneous states of temporal parts of four-dimensional objects. However Markovianism will turn out to be the claim that the instantaneous state of a particular temporal part should constrain the state of another distinct temporal part. And it is far from clear whether this is even a useful ideal to follow in our scientific practice. It could be that it is possible to define a notion of quasi-Markovianism in a four-dimensionalist setting where the relevant properties are ascribed to temporally extended temporal parts, where the temporal extension in question can be small as you like.
This might seem uncontroversial at first. It depends on whether location is additive. There is however some controversy about the additivity of location, especially in a setting that allows for multilocation. We cannot enter into the details of this debate here.
This follows from the fact that proper parthood is irreflexive.
If Smith’s argument is not correct. Thanks to an anonymous referee.
A starting point can be Butterfield (2006).
This was suggested to us by the physicist Nicola Semprini.
Maybe it could according to a strong operationalistic viewpoint. But it seems that all attempts to gather metaphysical consequences from scientific theories are committed to at least a mild variant of scientific realism.
Arntzenius (2003: 282) makes the same point.
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We are especially grateful to two anonymous referees of this journal for their insightful and detailed suggestions.
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Calosi, C., Fano, V. Arrows, Balls and the Metaphysics of Motion. Axiomathes 24, 499–515 (2014). https://doi.org/10.1007/s10516-014-9240-0
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DOI: https://doi.org/10.1007/s10516-014-9240-0