Abstract
In a frequently quoted paragraph , Benardete presented the “paradox of the gods” in the following terms.
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Atkinson, D. (2007). Losing energy in classical, relativistic and quantum mechanics. Studies in History and Philosophy of Modern Physics, 38, 170–180.
Benacerraf, P. (1962). Tasks, Supertasks and the Modern Eleatics. The Journal of Philosophy, LIX, 765–784.
Benardete, J. A. (1964). Infinity: An essay in metaphysics. Oxford: Clarendon Press.
Black, M. (1950–1951). Achilles and the tortoise. Analysis, 11(5), 91–101.
Goldstein, H. (1959). Classical mechanics. Reading, Mass: Addison-Wesley Publishing Company.
Grünbaum, A. (1955). Modern science and the refutation of the Paradoxes of Zeno. The Scientific Monthly, LXXXI, 234–239.
Hawthorne, J. (2000). Before effect and zeno causality. Noûs, 34(4), 622–633.
Laraudogoitia, J. P. (2005). Achilles’ Javelin. Erkenntnis, 62(3), 427–438.
Margenau, H. (1950). The nature of physical reality: A philosophy of modern physics. New York: McGraw-Hill.
Mittelstaedt, P., & Weingartner, P. A. (2005). Laws of nature. Berlin: Springer.
Peijnenburg, J., & Atkinson, D. (2010). Lamps, cubes, balls and walls: Zeno problems and solutions. Philosophical Studies, 150(1), 49–59.
Priest, G. (1999). On a version of one of Zeno’s paradoxes. Analysis, 59(1), 1–2.
Shackel, N. (2005). The form of the Benardete dichotomy. The British Journal for the Philosophy of Science, 56(2), 397–417.
Smith, S. R. (2007). Continuous bodies, impenetrability, and contact interactions: The view from the applied mathematics of continuum mechanics. The British Journal for the Philosophy of Science, 58(3), 503–538.
Thomson, J. F. (1954–1955). Tasks and super-tasks. Analysis, 15(1), 1–13.
Uzquiano, G. (2012). Before-effect without zeno causality. Noûs, 46(2), 259–264.
Weyl, H. (1949). Philosophy of mathematics and natural science. Princeton, New Jersey: Princeton UP.
Yablo, S. (2000). A reply to new Zeno. Analysis, 60(2), 148–151.
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Appendix: The Way to Subtasks
Appendix: The Way to Subtasks
A supertask is an infinite sequence of actions (processes) carried out (taking place) in a finite time. A task is a finite sequence of actions (processes). It must necessarily be carried out (take place) in a finite time because, by definition, the carrying out of an action only lasts a finite time. Hereafter, I use the terms “actions” and “processes” indifferently as befits the case. The notions of supertask and task may be formulated in terms of sets of actions instead of in terms of sequences of actions without significant change. Along these lines, we may try to define a subtask as:
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(a)
an infinite set of actions not carried out in a finite time.
The idea is that there exists a certain finite interval of time in which none of the actions of the set has been carried out. But the fact that actions haven’t been brought to completion in a finite interval of time is no more controversial than their non-completion in an infinite interval of time. Considered from another perspective, time is relevant when discussing actions that have indeed been carried out (and these lead to changes in time), but it doesn’t appear to be so when no action is performed at all. This suggests that we should leave time out of the description.
If we defined a subtask as:
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(b)
an infinite set of actions not carried out
subtasks would be uninteresting in a vast majority of cases, except perhaps those in which there are conditions for actions to be carried out. If these conditions are sufficient conditions then we know, that, by definition, they will not be carried out in a subtask. Thus, a subtask would be an infinite set of not carried out actions, with non-carried out sufficient conditions for their completion. In other words, a subtask would be defined as:
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(c)
an infinite set of actions whose carrying out (which happens not to take place) is (materially) conditioned by the realization of certain states of affairs which do not take place either.
Notice that if we do not impose the condition that the states of affairs should be relevant, it follows that, since (as a consequence of the properties of the material conditional) any falsity would be a sufficient condition (as well as a necessary one) for the carrying out of some action in a subtask, (c) would be extensionally equivalent to the definition of a subtask construed as a an infinite set of actions (of processes) whose carrying out does not take place.
But in that case, we would end up with an unnecessarily broad notion (as in the case of (b)). In Sect. 10.7, I defined a subtask as an infinite set of vacuously performed conditioned actions (processes). In more explicit terms, and as a result of the necessary modification to (c), the definition now takes this form:
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(d)
a subtask is an infinite set of actions whose carrying out, which does not take place, is (materially) conditioned to the realization of certain relevant states of affairs that do not take place either.
This suggests that it would be suitable (by analogy) to extend the definition of supertask so that an infinite set of actions, whose completion takes place during a certain interval of finite time, is now taken into account in the definition. In this case, an infinite set of actions whose completion is (materially) conditioned by the realization of certain relevant states of affairs could, in principle, either be a subtask (in case the actions end up not being carried out and the states of affairs end up not being realized), or be a supertask (in case infinite actions are not only brought to completion, but brought to completion in a finite time). A subtask is carried out although none of the infinite actions required by its definition is brought to completion, and none of its relevant states of affairs is realized. By contrast, a supertask is carried out whenever infinite actions are brought to completion (in a finite time), and a task is carried out whenever only a finite number of actions are brought to completion. With respect to the carrying out of actions, a supertask goes beyond a mere task, but a subtask does not even achieve the status of a task, hence its name.
As I suggested above, a subtask will only be non-trivial in case, by way of relevant states of affairs, there are sufficient, or necessary and sufficient conditions, of a certain type. Any infinite set of actions (processes) whose carrying out does not take place would in principle allow us to define a subtask because, in such a case, any infinite set of actions may be (materially) conditioned (at the very least if we take into account sufficient conditions) by the realization of relevant states of affairs that are not realized (and which, furthermore, can be realized in infinitely many ways). The notion of subtask (just like the notion of supertask) therefore includes a multitude of realizations of no interest, as I already noted above. But there are nevertheless interesting cases. An interesting supertask is one that has non-trivial consequences, and the very same thing may be said of interesting subtasks. Although the notion of “trivial” is unmistakably vague, the vagueness merely reflects that of the notion of what is deemed “interesting.”
In any event, (d) provides the concept of subtask with some structure, by allowing it to have non-trivial consequences; this is indeed a progress over the non-structured (b). We’re now facing something which is indeed more interesting than supertasks because it seems a priori impossible for the carrying out of a subtask (as defined as in (d)) to have significant consequences. Just as one must provide an infinite sequence of carried out relevant actions when describing a supertask, one, when describing a subtask, must provide an infinite set of relevant actions (of relevant processes), whose carrying out does not take place, and a set of relevant conditions (relevant states of affairs) for its carrying out, which are not realized either. In many cases, it will be sufficient, given the situation, to specify the relevant conditions in order to deduce that none of the actions of the infinite set of relevant actions will be carried out. This is exactly what we have seen in the paradox of the gods and its generalizations.
I would like to conclude with a note on the terminology. I have talked of “carrying out” a subtask despite the fact that one necessary condition for being a subtask is that none of the infinite actions that play a role in its definition is actually carried out. My intention was to underline the formal parallels with the concept of supertask. It would perhaps be more suitable to talk of carrying out* a subtask, analogously to how some authors use the term “causation*” to describe cases of prevention or omission in causal terms. The expressions “observe” or “comply with” (as when one says that one “observes the rules” or “complies with the law”) might appear to be a suitable alternative to “carry out” in the context of subtasks. In the absence of a clear alternative, I have not followed any of these paths. My purpose was to point out the relevance of the concept of subtask.
Notes
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1.
C (1)n is a universal sentence, namely, C (1)n : for every instant t, with t1 < t < t2, if Q, at t, comes to be at a distance 1/n from L(t), then for every instant t*, with t < t* < t2, godn will ensure that, at t*, Q is at a distance ≤ 1/n from L(t*).
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2.
Since all the material bodies I consider in this paper are assumed to be non-point, Q and godn (in particular) are non-point, which means that they cannot be at the same distance from L(t) and, at the same time, that Q be between L(t) and godn. The small changes required to put this right are trivial and I do not take them into consideration here because they would complicate the argument unnecessarily.
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3.
In other words, everything that is compatible with non-interaction between them at t remains the same.
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4.
There are a number of philosophical subtleties concerning contact and impenetrability that I do not take into account in this paper. My stance on this point is in line with the position defended by Smith, for whom such subtleties have little importance within the framework of the mechanics of continuous media that one finds as an essential chapter of classical physics (see Smith 2007). The principle of interaction by impenetrability (PII) isn’t one of such subtleties. What it does, as the example of particle Q and cylinder C shows, is to facilitate an explanation of why two rigid bodies interact by collision when they do (and do so, furthermore, according to the relevant laws of conservation): PII takes physics seriously.
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5.
I do not explore here questions of differentiability, which would require a more sophisticated treatment, but wouldn’t, in any event, compromise the validity of (V).
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6.
The principle of action and reaction assumes that forces are always given between two bodies (physical systems) such that the total force on a body is always the sum of two-to-two interactions. As Margenau makes clear, if the force between two molecules depends on the distance between them, and molecule 3 introduces a difference in the force of 2 on 1, the relative position of molecules 2 and 1 remaining unchanged, we are faced with a force between three bodies (see Margenau 1950).
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7.
Indeed. For t ≤ tº, the center of mass of the total system of infinite particles Q, P1, P2, P3, P4, … is the point x = 0 (remember that they all have the same mass). Now let Q be at t = tº+ε at point x = δ > 0. If it has not interpenetrated with any particle Pn, this can only be due to all of these particles remaining to its right. But since in tº there were infinite particles Pn between x = 0 and x = δ, they must have moved in the interval of time Δt = ε to the right of the point x = δ. Therefore, the center of masses of the complete system must now be at a point x ≥ δ. This displacement of the center of masses of the isolated system formed by the total set of particles Q, P1, P2, P3, P4, ….. goes against Newton’s 1st law of movement (the law of inertia), which implies that if Q is at t = tº+ ε at point x = δ > 0 (in a more general form, if for instants of time t′ > tº arbitrarily close to t = tº Q is in the region x > 0), then it must have interpenetrated with at least one particle Pn (and, therefore, with a massive part of the set of particles Pn) at instants t′ > tº arbitrarily close to t = tº.
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8.
Alternatively (and rather more elegantly), one might suppose that at no moment shall any of the particles involved move to the left (in the negative direction of the X-axis). Then it is easy to prove that the requirement of the conservation of kinetic energy and momentum leads, at every instant, to one and only one of the particles involved in the model having unit velocity (except, of course, at the instants at which a collision takes place).
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9.
By “conventional process,” I mean a process with interaction by contact and in accordance with Newton’s laws of movement.
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10.
To appreciate the relation between subtasks and supertasks more closely, it is interesting to note that an infinite set of non-vacuously carried out conditioned actions is simply an infinite set of actions simpliciter and, therefore, a supertask. In order to have a subtask, the conditioned actions must be vacuously carried out. Note also that in subtask G2(II), the principle of conservation of energy would appear to be violated (given that the kinetic energy of the system formed by the walls and Q grows constantly), something that is likewise characteristic of many standard examples of dynamic supertasks (see, e.g., Atkinson 2007).
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11.
The idea of nature as actor is not that remote from physical theory proper and has been frequently associated with a teleological formulation of the laws. The paradigmatic case in classical mechanics is Hamilton’s principle of minimum action. All of this is admissible provided it is clear that, as Mittelstaedt and Weingartner note: “Physical nature has no goals in the sense that living organisms have goals” (Mittelstaedt and Weingartner 2005: 144). In the case of subtask G2(II), if we substitute “the gods” by “Nature,” we have a suggestive means of explaining the interaction at a distance between the walls and Q based on the “mechanism” of impenetrability.
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12.
At least in Newtonian worlds which, as I said above, are the only ones I have taken into consideration in this paper.
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13.
The parallels with the history of supertasks are also striking. Authors such as, e.g., Thomson (1954–1955) and Black (1950–1951) have argued against the possibility of supertasks; authors such as Yablo (2000), Shackel (2005) and Peijnenburg and Atkinson (2010) have also refused to accept the possibility of subtasks (using different arguments). Although the critical analysis of such arguments was not the objective of this paper (with the exception of Yablo 2000), I think my generalizations of the Benardete paradox may be taken as a useful starting point for a reply to their objections.
Acknowledgements Research for this work is part of the research project FFI2015-69792-R (MINECO/FEDER)
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Laraudogoitia, J.P. (2016). Tasks, Subtasks and the Modern Eleatics. In: Pataut, F. (eds) Truth, Objects, Infinity. Logic, Epistemology, and the Unity of Science, vol 28. Springer, Cham. https://doi.org/10.1007/978-3-319-45980-6_10
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