Abstract
In this paper, I present a discrete solution for the paradox of Achilles and the tortoise. I argue that Achilles overtakes the tortoise after a finite number of steps of Zeno’s argument if time is represented as discrete. I then answer two objections that could be made against this solution. First, I argue that the discrete solution is not an ad hoc solution. It is embedded in a discrete formulation of classical mechanics. Second, I show that the discrete solution cannot be falsified experimentally.
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Notes
This time series \((t_n)_{n \in \mathbb {N}}\) is for all \(n \in \mathbb {N}^{*}, \ t_n=x_0/v_A + (v_T/v_A)t_{n-1}\) with \(t_0=0\). It is an increasing infinite series which converges to the instant \(t_f=x_0/(v_A-v_T)\). This result may be proved in rewriting the previous recurrence formula as the convergent geometric series : \(t_n=x_0/v_A \sum ^{n-1}_{k=0} (v_T/v_A)^{k}\).
I discuss below in this section the use of the term “overtake” instead of “reach”.
It can be shown that \((t_{k_n})_{n \in \mathbb {N}}\) is \(t_{k_0}=0\) and for all \(n \in \mathbb {N}^{*} \ t_{k_n}=h\left\lceil (x_0/(hv_A))+(v_T/v_A)(t_{k_{n-1}}/h)\right\rceil \), where \(\left\lceil .\right\rceil \) is the ceiling function, which is defined as : \(\forall x \in \mathbb {R}, \left\lceil x\right\rceil =\min \left\{ n \in \mathbb {Z}|n\ge x\right\} \).
One can show that the order of magnitude of \(m\) is \(\ln (\frac{x_0}{hv_A})/\ln (\frac{v_A}{v_T})\).
See also (Salmon 1980, pp. 40–42).
In continuous mechanics, Euler-Lagrange equations and conservation of energy can also be derived from the least action principle—even if it is not the usual derivation. See for example (Marsden and West 2001; p. 467, Chen et al. 2006, p. 227). However, in this case, the latter equation can be deduced from the former. On contrary, this is not possible within DM. This specificity results from the fact that the space coordinate and the time coordinate are treated symmetrically (D’Innocenzo et al. 1987, p. 246). In addition, I would like to emphasize that, as in the continuous case, this framework can be generalized to non-autonomous Lagrangian, i.e time-dependant Lagrangians. In this case, energy is not conserved and the second equation represents the evolution of energy.
A proper introduction of symplecticity requires tools of differential geometry that are beyond of the scope of this introductory presentation. See for example (Marsden and West 2001, p. 477).
Furthermore, there are also different versions of DM depending on the initial conditions for the value of the time step. I go back to this point in Sect. 5.
This results from the fact that, in the case of a free particle, the discrete Euler-Lagrange equations and the equation of energy conservation are degenerate. The discrete solution (5) still holds.
The fact that time step \(h_k\) is allowed to vary within DM guarantees that autonomous systems exactly preserve energy, momenta and symplecticity, as in continuous mechanics. However, as we have seen in the examples above, it does not mean that discrete time step must vary. Even if time step is allowed to vary in the discrete least action principle, the latter can lead to constant discrete time step depending on mechanical systems and versions of DM.
See (Kane et al. 1999) for a comparison between the “simplified” version and the “general” version of DM.
Such cases are currently developed in the context of numerical computation. As it should be clear from Sect. 4.2, many generalizations thus pertain to the “simplified” version of DM. However, this restriction is not mandatory. Generalizations for the general version of DM are also currently developed as it is noticed in (Lee et al. 2009, p. 2017) and started to be achieved in (Pekarek 2010, chapter 3).
If we want to discuss the consequences of quantum limitations on the Achilles paradox, one should investigate a discrete formulation of quantum mechanics. I suggest in the conclusion some perspectives in this direction.
Since the time step is allowed to vary, \(S_d\) tends to \(S\) when the initial time step \(h_0\) and the initial position step \((q_1-q_0)\) tend to zero. For the sake of the argument, I talk about time step \(h\) even if one has to keep in mind that the parameters that go to zero are \(h_0\) and \((q_1-q_0)\).
See (D’Innocenzo et al. 1987, p. 250).
For simplifications, I have used the initial conditions such as the continuous and the discrete solutions are \(q(t)=q_0\cos (\omega _0 t)\) and \(q_k=q_0\cos (\omega _d t_k)\).
According to the model of the harmonic oscillator, \(\omega ^2_0=k/m\), where \(k\) is the constant of the spring and \(m\) the mass of the body. There might be no empirical equivalence if \(k\) tends to infinity and if \(m\) tends to zero. However, such extreme cases might not be compatible with the representation of a physical system by a harmonic oscillator. For example, nonlinearity would have to be taken into account. In any case, as far as \(k\) and \(m\) are non-null and bounded, the empirical equivalence holds. Equally, the empirical equivalence requires that \(T\) is bounded. One can take, for example, the order of magnitude of the age of the universe.
Equally, it is well known that the discretization of non-chaotic differential equations can lead to discrete chaotic behaviour, like the logistic equation. However, the discrete and the continuous solutions share the same behaviour if the time step is sufficiently small, i.e smaller than a critical value.
More precisely, they use the discrete Lagrangian \(L_d(q_k,q_{k+1})=L((q_{k+1}+q_{k})/2,(q_{k+1}-q_{k})/2)\).
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Acknowledgments
I thank the anonymous reviewers for their helpful comments, and their contribution to enhancing the quality of this paper. This work is a part of my PhD dissertation. It has been written while at the IHPST (UMR 8590, CNRS & Université Paris 1 Panthéon-Sorbonne, ENS) and at SND (FRE 3593, CNRS & Université Paris-Sorbonne). I would like to thank my supervisors, Anouk Barberousse and Jacques Dubucs, for their encouragements and their insightful comments. Thanks also to people of the IHPST for their remarks, which helped improve this paper. Finally, I am particularly grateful to my friend Clément Ruef for our discussions on the paper.
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Ardourel, V. A discrete solution for the paradox of Achilles and the tortoise. Synthese 192, 2843–2861 (2015). https://doi.org/10.1007/s11229-015-0688-2
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DOI: https://doi.org/10.1007/s11229-015-0688-2