Abstract
The objective of this article is to provide a discussion that counters the infinite particle disappearance conclusion argued by Shackel (European Journal for Philosophy of Science, 8(3), 417–433, 2018). In order to do this, clear criteria to disprove the results of the applications of his continuity principles are provided, in addition to the consideration of the fundamental Classical Mechanical principle of mass conservation as an independent and clear basis for this disproof.
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Notes
Please note that my discussion will focus strictly on the second and third sections of Shackel’s article and, thus, will not go into exhaustive considerations about the reasonableness of his particle configuration modelling the Immovable Object meeting the Irresistible Force paradox, which is treated in his fourth section.
Note that this multiple collision is highly indeterministic as there are infinite post-collision states all of which are compatible with the Classical Mechanical conservation laws. Consequently, the position of the particles at t > 1 is by no means knowable in advance. Still, this does not at all imply that there is no point in the space at which the particles can be at t > 1 and, thus, it does not provide any support for the particle disappearance conclusion.
Shackel uses the terminology “Beautiful Supertask” to refer to the outcome of the particle configuration by Pérez Laraudogoitia (1996).
Please note that, given that Shackel’s article provides sufficient evidence to assume that there are two errors in the original formulation of this fragment, my discussion deals with it as if the corrected version included in my text was actually in Shackel’s in which the following is nevertheless stated: “Since for all nearby worlds no particle is at x = 0 at t = 0, for any particle to be at x = 0 at t = 1 in our universe would be a discontinuity and that is ruled out by continuity. Hence for our universe, there is no particle at x = 0 at t = 0.”
My objections to the disappearance of the fn particles are identical to those exposed in the text for the case of the disappearance of the mn because Shackel reasons in favour of the disappearance of the former in an equivalent way he does for the latter.
Still, it is pertinent to note that although this is certainly so, it is because of the specific parameters selected for that configuration. It is not difficult to find another set of parameters for which the limit of the average velocity of the particles over intervals of time bounded by t = 1 is finite, with the configuration still maintaining the fundamental characteristics that Shackel considers sufficient to model an immovable object meeting an irresistible force situation.
For instance, it is the simple and surprising case of the initial velocities of the fn particles being given by vfn = −(n + 1), with n ∈ \(\mathbb {N} - \{1\}\). Note that the dynamic sense of the difference between this new set of parameters and the one selected by Shackel is simply that the velocities assigned to the particles of the fn sequence differ. Still, the difference between the increasing velocities of any two consecutive fn particles is equal and constant in both cases. Accordingly, the particles of this new configuration are also subjected to infinite chains of binary collisions that result in all the mn and fn particles colliding at point x = 0 at instant t = 1. Simple algebraic calculation nevertheless shows that the limit of the average velocity of the particles over intervals of time bounded by t = 1 is now clearly finite.
Note that this implies a general critique to the supposed adequateness of Shackel’s system to actually model the Immovable Object meeting the Irresistible Force paradox because configurations that fulfil the conditions considered sufficient to represent the paradox give drastically different outcomes depending on the selection of the parameters.
For example, it is the simple case of the velocities of the mi and fj particles after the multiple collision being respectively given by: vmi = −i and \(v_{fj}= -\frac {1}{j}\), with i ∈ \(\mathbb {N}\) and j ∈ \(\mathbb {N} - \{1\}\). Note that such velocities are undoubtably possible because no conservation law is violated in this supposed case. Furthermore, they entail that the ordinal disposition of the particles after the scattering at t = 1 is the same they had before the multiple collision; thus, the Classical Mechanical requirement that no particle passes any other is no violated either. Finally, such velocities equally entail that no other collision will occur after the scattering at t = 1; thus, it is clear that the velocity acquired by each particle after the multiple collision will indeed remain constant in this specific case.
Because no justification to rule out the result of a multiple collision at t = 1 has been validated in my discussion, my analysis in these paragraphs depends on the supposition that this is in fact the evolution of the system and that, therefore, particles do continue to exist for any t ≥ 1. Still, it is relevant to emphasise that even if somebody simply assumes that particles do not exist for t > 1 this then implies that the world lines of the particles end at point x = 0 at instant t = 1 and, thus, that they all lack a tangent at this point. Consequently, no instantaneous velocity may be defined for any particle at t = 1 and, thus, the particle disappearance assumption may not justifiably be based on the supposition that their velocity is infinite at this instant.
Note that this indefiniteness of the instantaneous velocity characterises any instantaneous collision of particles independently of whether the collision is deterministic or indeterministic as it is simply a consequence of the fact that at the instant t of the collision the left velocity, which corresponds to the left tangent to the world line of the centre of mass of the particle, is different from the right velocity, which corresponds to the right tangent to the world line of the centre of mass of the particle.
Note that in asserting that the instantaneous velocity of the particles is indeterminate Shackel is now simply abandoning the conclusion previously defended, namely, that this instantaneous velocity is infinite.
The operating of this machine consists in changing the position of a particle infinitely many times from its location at a finite distance to one side of a pivot point to the corresponding location at the same finite distance to the opposite side of the pivot point. Each of these moves is twice as fast as the previous one and right after each of them is carried out the machine stops for an interval of time equal to that employed in the move.
This one-dimensional configuration consist in two sets of countably infinite point particles p±n of the same mass m located at points \(x_{\pm n} = \pm \frac {n}{n+1}\) for any n ∈ \(\mathbb {N}\) ∪ {0} all of which are motionless at time t = 0. Given the possibility of spontaneous self-excitations introduced by Pérez Laraudogoitia (1996) the complete set of particles is supposed to evolve with a very specific pattern of self-excitations. This particular pattern consists in self-exciting at instants ti = \(\frac {1}{2} + \frac {1}{2^{2}}+...+\frac {1}{2^{i}}\) with velocities vi = 2i+ 2 for any i ∈ \(\mathbb {N}\) the direction of which is rightwards for odd values of i and leftwards for even values of it.
References
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Shackel, N. (2018). The infinity from nothing paradox and the immovable object meets the irresistible force. European Journal for Philosophy of Science, 8(3), 417–433.
Acknowledgments
Many thanks to Jon Pérez Laraudogoitia for very helpful discussion and also to the editors of the journal and the reviewers for very useful comments and suggestions.
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Corral-Villate, A. On Shackel’s nothing from infinity paradox. Euro Jnl Phil Sci 10, 26 (2020). https://doi.org/10.1007/s13194-020-0277-1
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DOI: https://doi.org/10.1007/s13194-020-0277-1