Abstract
Zeno’s paradoxes of motion, allegedly denying motion, have been conceived to reinforce the Parmenidean vision of an immutable world. The aim of this article is to demonstrate that these famous logical paradoxes should be seen instead as paradoxes of immobility. From this new point of view, motion is therefore no longer logically problematic, while immobility is. This is convenient since it is easy to conceive that immobility can actually conceal motion, and thus the proposition “immobility is mere illusion of the senses” is much more credible than the reverse thesis supported by Parmenides. Moreover, this proposition is also supported by modern depiction of material bodies: the existence of a ceaseless random motion of atoms—the ‘thermal agitation’—in the scope of contemporary atomic theory, can offer a rational explanation of this ‘illusion of immobility’. Our new approach to Zeno’s paradoxes therefore leads to presenting the novel concept of ‘impermobility’, which we think is a more adequate description of physical reality.
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Notes
In this article, all translations of Aristotle’s quotations come from Hardie and Gaye (1930).
As an additional statement for ‘The Arrow’ argument (see Sect. 2), the sentence “But it moves neither in the place in which it is, nor in the place in which it is not” (by Diogenes Laertius, Vitae Philos. 9, 72) can also be attributed to Zeno with the help of other primary sources (see Vlastos 1966).
For example, Aristotle’s answer to ‘The Dichotomy’ argument (i.e. distinction between ‘infinite in respect of divisibility’ and ‘infinite in respect of their extremities’; Physics VI:2, 233a25) is considered as “philosophically unsatisfactory” by Kirk and Raven (1957).
These three monographs also treat the paradoxes of plurality (attributed to Zeno as well). The relation between the paradoxes of motion and those of plurality is discussed in detail in a recent article by Hasper (2006). The paradoxes of plurality are also described by Huggett (2010) and by Dowden (2017). The present article focuses only on the paradoxes of motion.
This is true even without taking into account the intense literature about the problems derived from Zeno’s paradoxes (as, for example, the problem of the supertasks see Sect. 4.1).
This conclusion is independent of the existence of any inertial frame of reference or of Galileo’s equivalence.
In the entire paper, the word discrete is a synonym of ‘atomic’ and so, does not refer to any mathematical definition related to ℕ, the set of natural numbers.
Even for quantum physics: quantum discontinuity applies only to energy levels (and to other quantum properties as the spin) of matter components (electrons, neutron, proton, quarks…) but not to space or time (see for example Grünbaum 1967; Grünbaum 1970). One of the cornerstones of the quantum theory is the Schrödinger equation, which is a partial differential equation where time and space coordinates are variables, and such an equation needs a continuous framework to be well defined.
It is possible to reach the same conclusion directly from the hypothesis of discretization of space and time [see, for example, chapter 5 of the monograph by Faris (1996)].
If the latter is chosen, it could seem redundant to refer to ‘indivisible temporal point’ in proposition (d). But according to Vlastos (1966), such odd terms (as indivisible point) are quite usual in Aristotle’s texts.
But curiously, the ‘cinematographic vision of motion’ depicted by Bergson (in a continuous framework) is frequently used by other authors to describe the motion of the flying arrow in the discrete framework (see also footnote 17).
According to Kirk and Raven (1957), the choice of the discrete framework permits us to get a consistent overall picture of the four arguments: for each space–time framework, one of the arguments leads to a paradoxical situation for the motion of a single body (‘The Dichotomy’ in the continuous framework and ‘The Arrow’ in the discrete framework), while another one leads to a paradox when considering the relative motion of two bodies (the ‘Achilles’ in the continuous framework and ‘The Stadium’ in the discrete framework). So, if in each space–time framework the concept of motion is inconsistent, motion should be impossible.
Premise (bd+) may seem obvious but, historically, the hypothesis of a real length contraction of moving body was proposed by George Francis FitzGerald in 1889, and theorized by Hendrik Antoon Lorentz in 1892, for accounting for the null-results of the famous Michelson–Morley experiment (1887). This hypothesis did not survive long; it had been forgotten by the time of Albert Einstein’s view on relative motion (see the history and context of birth of the special theory of relativity for more details).
These two expressions come from Shamsi (1973).
Actually, the term ‘during an indivisible moment of time’ is not adequate as it assumes that some change could be performed during this period of time. But, by definition of the discrete framework, no change can occur ‘during an indivisible moment of time’. A more adequate term should be simply ‘at an indivisible moment of time’, but such a term was not used previously in order to retain the reader in the ‘classical’ fallacious reasoning. Note that describing motion as ‘a succession of rest’ or quoting the famous Bergson’s sentence “movement is made of immobilities” (the ‘cinematographic vision of motion’, see also footnote 13) in the discrete framework is surely, but erroneously, assuming that the term 'being at rest' is defined at an indivisible moment of time.
In such a context, the Diogenes Laertius’s sentence attributed to Zeno “But it moves neither in the place in which it is, nor in the place in which it is not” (see also footnote 2) appears much more understandable.
This also rejects potential future forms of the standard quantum theory incorporating minimal distances (‘hodons’) and times (‘chronons’) (Grünbaum 1967).
See Sect. 5.4 for more details about some principles of classical mechanics.
In the history of mathematics, questions about validity of premise (cc2) led to the ‘Metrical paradox of extension’, which can be briefly summarized by the following question: Does an extended line consist of unextended points? Actually, this questioning is much more related to the ‘paradox of infinite divisibility’ (one of the paradoxes of plurality) than to the paradoxes of motion; the metrical paradox of extension questions only the consistency of Cantor’s continuum in a mathematical framework: there is no notion of motion in such a framework. According A. Grünbaum, the problem is solved: “The set-theoretical analysis of the various issues raised or suggested by Zeno’s paradoxes of plurality has enabled me to give a consistent metrical account of an extended line segment as an aggregate of unextended points” (Grünbaum 1967). Unfortunately, time is not a mathematical concept. So, the solution of this paradox does not really help to define an ontological nature for the concept of time. However, the solution permits, for practical purposes (e.g. in classical kinematics), that periods of time can be described in a consistent way as composed of temporal points. In a similar way, by shooting a picture with a digital camera, reality can be described with the help of colored pixels, but reality is not actually made of pixels.
Bergson was not really interested in explaining why it is absurd. He simply used the paradox to support its pre-established philosophical positions (i.e. claiming that immediate experience and intuition are more significant than rationalism and science for understanding reality) (Tooley 1988).
Such a synonymy is striking in Bergson’s comments: “Yes again, if the arrow, which is moving, ever coincides with a position, which is motionless” or “If it had been there, it would have been stopped there” (italics are added) (Bergson 1907). It is also found in Harrison’s interpretation of ‘The Arrow’ argument: “How can a particle be at a point and also be moving at that point?” (Harrison 1996).
We know from experimental evidence that Achilles will eventually overtake the ‘tortoise’.
However, there also exists a specific physical solution for the ‘Achilles’ argument: a runner will cover a finite distance in a finite number of steps, as noticed already by Bergson (1969), Wisdom (1970) and Kline (1980). By considering the itinerary of the runners in terms of steps (or more precisely in terms of jumps for Achilles and in terms of tiny steps for ‘the tortoise’), it is easy to identify the last step before Achilles overtakes: when the distance separating the two competitors becomes less than the length of Achilles’ jump + length of the tortoise’s step. Thus, the race will finish at Achilles’ next jump. To avoid such an obvious solution of the ‘runners-paradox’, it would be more judicious to exemplify the argument with a sailing race: imagine two sailboats driven by a stable wind on a quiet sea. One of them has a sail five times larger than the other one, which is one mile ahead. The ‘sailboats-paradox’ is then expressed by replacing the word ‘runner’ by the word ‘sailboat’ in Aristotle’s original sentence. With this new illustration of the argument, we are faced with a more continuous motion, which is more in conformity with the continuous framework of space and time chosen to create the paradox.
Obviously, the tricky point is that this region of space is nevertheless defined and limited by the point where the overtaking occurs (in the same manner that the open set [0, 1) is defined by the number 1 but this number 1 is not part of the set). This is why Peijnenburg and Atkinson (2008) wrote that “Any difficulty we have with the Achilles lies in the concept of an open set [a, b)”.
He wrote “but it [the slowest runner] is overtaken nevertheless if it is granted that it traverses the finite distance prescribed” (Physics VI:9, 239b28).
Since, we repeated it, there is no logical connection between the proposition ‘space and time are continuous’ and the propositions ‘Achilles can/cannot catch the tortoise’. They are logically independent.
It is obviously the same formula as the one for the ‘Achilles’ argument presented in the previous section, with d = 1 and r = 2.
However, as one counter-example given by current mathematicians aware of the limit of calculus to solve the paradox, we can quote Joseph Mazur (2007): “I’ll show that while this may seem to be the case on the surface, the math in question—basic algebra—does nothing to address the underlying phenomeno-logical problem that the paradox drives at.”
Note that the only useful information coming from the convergence of the infinite series depicted by Zeno is that the time needed to cross a finite distance is finite, but such information was already available from our usual experience of motion.
Note that there is no agreement in the philosophical community on the question, and the philosophical problem of the ‘supertask’ led to intense discussions in the literature. For example, four of the eleven articles compiled by Salmon (1970) about Zeno’s paradox are dedicated to this problem (“Tasks, Super-Tasks, and the Modern Eleatics” by Benacerraf; “Tasks and Super-Tasks” and “Comments on Professor Benacerraf’s Paper” by Thomson, and “Modern Science and Zeno's Paradoxes” by Grünbaum). Moreover, new articles or books dedicated to the problem are regularly published: some examples are Laraudogoitia (1996), Atkinson (2007), Peijnenburg and Atkinson (2008), Lee (2011) and Romero (2014). A forthcoming short article will be specifically dedicated to this problem.
This proposition (i.e. motion is not a supertask) would be very probably supported by Aristotle and by H. Bergson, since both authors claim (in their own style) that, even if motion can be potentially decomposed in an infinite number of parts, it is not actually composed of parts.
European thinkers focused on such a philosophical aspect of Zeno’s paradoxes during the 18th century, and the regressive version of the argument was much more popular than the progressive version at that time (Blay 2010). But the problem is still considered intriguing nowadays: see, for instance, Medlin (1963), Hamblin (1969), Priest (1985), Mortensen (1985), Jackson and Pargetter (1988) and Smith (1990), or the recent book by Łukowski re-expressing the problem: “In any mental experiment, it is impossible to move imperceptibly from motion to rest or vice versa by some ‘small steps’. Moreover, it is difficult to know how those ‘small steps’—leading us through intermediary states—should be understood. So, in contrast with various ‘hues’ of being a heap, bald or red, there are no ‘hues’ of being [in] motion. Either something is in motion or not” (Łukowski 2011).
For each proposition or conclusion (concerning rest and motion) in this sub-section, the phrase ‘In a given frame of reference…’ should be implicitly added.
See Sect. 4.2.
Of course, we do not consider here possible temporary troubles in perception and/or psychics that may cause hallucinations. Moreover, by specifying “3-dimensional material body”, we exclude here both the cases of some optical illusions, in which a still image seems to move, and digital videos/motion pictures, in which some tricks can be used to induce an illusion of motion. Both cases concern 2-dimensional pictures that are representations of reality rather than the reality itself.
This is the case with Parmenides and Heraclitus.
There are two quotations that support this idea: “All Presocratic thinkers were struck by the dominance of change in the world of our experience. Heraclitus was obviously no exception, indeed he probably expressed the universality of change more clearly than his predecessors; but for him it was the obverse idea of the measure inhering in change, the stability that persists through it, that was of vital importance” (Kirk and Raven 1957, italic is from the quotation), and “It is that some things stay the same only by changing. One kind of long-lasting material reality exists by virtue of constant turnover in its constituent matter. Here constancy and change are not opposed but inextricably connected” (Graham 2015).
This specification of the frame of reference is important since, in the context of modern kinematics, any motion of any object can be cancelled by studying the system in the reference frame attached to this object; so using this arbitrary procedure, any object is by definition at rest (v = 0 and dv/dt = 0) in its own reference frame.
I.e. the instantaneous velocity is, most of the time, not equal to strictly zero, but it can be equal to strictly zero at some durationless instants.
We could argue that the current use of ‘immobility’ is not consistent since it induces problems as soon as we consider immobility as real: see first and second cases in Table 2.
Most of Zeno’s paradoxes of plurality (see footnote 4) are solved by an approach of linguistic philosophy.
The terms ‘impermobility/impermobile’ can also be constructed by inserting the prefix ‘per’—here meaning deviation or destruction—into the words ‘immobility/immobile’.
This sub-section does not contain any bibliographic references since it exposes only encyclopedic knowledge, which is not subject to debate in the scientific community.
As an example of academic reference see Sherwood (1972). Generally, the existence of ‘atomic and/or molecular dynamics’ is part of the current scientific paradigm of most chemists. For them, the proposed ‘philosophy of ontological non-immobility’ will probably appear quite obvious and natural.
Both motions are considered in the same reference frame, but both the reality and the conceptualization of thermal agitation are actually not dependent on the specification of any Galilean reference frame.
According to Bachelard (1934), epistemological obstacles are all representations that are blocking or hindering scientific progress.
This is in agreement with Michael Tooley’s view: “For it seems to me that perceptual knowledge that something is moving involves unconscious use of short-term memory” (Tooley 1988).
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Acknowledgements
The author would like to thank the reviewers of this manuscript for their judicious remarks and constructive comments. Dr. Celine Scornavacca is also greatly thanked for her careful reading and helpful comments.
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Appendices
Appendix 1: Explanation of ‘The Stadium’ Argument
Because this explanation is largely inspired from the one of Faris (1996), the translation of Aristotle ‘s text by this author is given below:
(Part I) The fourth is the one about the two equal rows of bodies that move past each other in opposite directions at equal speeds, the one row from the end of the stadium, the other from the mid-point; in this argument Zeno thinks it follows that the half of a given time is equal to its double.
(Part II) For example, let AA be the stationary bodies, BB those, equal to these in size and number, starting from the mid-point, and CC those starting from the end, equal in size and number to the As and in speed to the Bs.
[Aristotle, Physics VI:239b33-240a1 (for part I); 240a4–240a18 (for part II); Translated by Faris (1996)]
The fourth argument against motion is probably the most difficult to understand directly from the text of Aristotle. Like the ‘Arrow’ argument, a discrete space and time framework have to be used to correctly interpret it… but it is still not obvious to see how Zeno attains the strange conclusion “the half of a given time is equal to its double”. One explanation to get to the same paradoxical conclusion that Zeno propounded is the following:
(a) The first B in the time of its movement has passed 1 A (e.g. B3 was in front of A2 in the initial position and is in front of A3 in the final position, see Fig. 6).
Schematic representation of ‘The Stadium’ argument (when the cardinality of each group is 3) (moments t and t′ are two consecutive indivisible moments of time)
(b) The first C in the time of its movement has passed 2 B’s (e.g. C1 passes B2 and B3).
(c) The time taken by the first C to pass a B = the time taken by the first B to pass an A.
(d) The first C in the time of its movement has also passed 1 A (C1 was in front of A2 in the initial position and is in front of A1 in the final position).
Therefore:
From a,b,c: The first B’s time = half the first C’s time (1).
But, from a,d: The first B’s time = the first C’s time (2).
Therefore, from (1) and (2): Half the first C’s time = the first C’s time (i.e. half of a given time is equal to the whole).
The corner stone of Zeno’s argumentation is obviously proposition (c). Indeed, this one can seem quite dubious at first glance, and it would be more natural to state the following: the time taken by the first C to pass a B = half the time taken by the first B to pass an A. But such a proposition is a true proposition only in a ‘continuous world’: the Stadium argument is trivial in a continuous framework, as the notion of relative speed exists; unfortunately, it is not valid in a discrete space and time framework in which intermediate positions are not allowed (see Fig. 7). Thus, proposition (c) is a direct consequence of discretization of time and space.
Non-existing position in a discrete space and time framework
As noticed already by Adolf Grünbaum (1967, 1970), another way to run into a paradox is to consider the same maneuver but in the reference frame of the B’s instead of the reference frame of the A’s: in this case, the A’s and the C’s are moving in the same direction, and problems occur when considering the speed of these rows. Of course, the A’s are moving at the Bs’ previous rate, but what is the speed of the C’s now? Let’s suppose that we find ourselves looking again at the rows all in the same position (i.e. final position of Fig. 6) but in this case, C1 goes to B1 from B3 without ever passing in front of B2 (remember that there is no time left between two consecutive indivisible moments of time). So, C1 should perform a kind of teleportation! To avoid that, C1 should be in front of B2 after this motion… but in this case, we can remark that the row of C’s is now moving at the same rate as the A’s i.e. it is not moving any faster than the A’s (see Fig. 8).
Schematic representation of ‘The Stadium’ argument in the reference frame of the B’s (Moments t and t’ are two consecutive indivisible moments of time)
Finally, the argument reveals that the hypothesis of an atomic structure of space and time implies automatically that firstly, an absolute reference frame must exist (in the ‘The Stadium’ argument, this absolute reference frame is of course the A row) and secondly, in this absolute reference frame, motion can be carried out at only one speed (change of only 1 element of space between two consecutive indivisible moments of time). The huge problem is that such a situation does not allow the existence of relative motion at any specific speed. So, ‘The Stadium’ argument does not explicitly ban the existence of motion but forbids the existence of relative motion. Of course, such a forbidding remains in discrepancy with our classical physical experience, so the paradoxical situation remains (in the discrete framework).
Appendix 2: Common Situations in Which Movement is not Perceived
Because it is so usual to experience movement in everyday life, we generally do not realize that such a perception of motion requires four conditions, which are depicted in Fig. 9. In a given inertial frame, where an object is physically in motion, the lack of one of these conditions does frequently induce an illusion of immobility (in the same Galilean reference frame).
Schematic representation of the four requirements needed for the perception (by human or animal) or the detection (by any instrument) of motion
One of the simplest examples of such an illusion is the motion of the sun during a very sunny day. When looking at the sun in a perfect blue sky (without any clouds) for a few minutes around noon, one may have the impression that the sun is not moving. But if one stays in the sun, motionless, near a tree and a few minutes later, is in the shadow of the tree, one can easily infer that the sun is moving. In this case, thanks to the tree used as a reference frame, motion can be quickly/easily detected. Without this (local) reference frame, we can still define the position of the sun according to its elevation above the horizon, but without any measurement device to precisely record the azimuth value, we would generally need a few hours to detect actual changes in its position (and if you are less hurried and prepared to extend the duration of observation, waiting for sunset is obviously also evidence of the apparent movement of the sun). So, depending on presence (or absence) of a reference frame, and on relative speed of the motion, the minimum duration of the observation has to be adjusted to correctly perceive the movement and avoid the illusion of stillness.
Moreover, the detection of motion always implies a comparison of two different positions at two different times. This operation of comparison requires having recorded the first position to memory (brain for animals or a specific device for machines).Footnote 53 Such an operation is extremely natural for most people but, unfortunately, not for people affected by akinetopsia (motion blindness). Patients with akinetopsia cannot perceive motion in their visual field, despite being able to see stationary objects. For them, the world is devoid of motion. Even for healthy people, physiological limits of vision can induce illusion of immobility: as the ‘normal’ visual acuity is around 1 arcminute (1′ = 1/60°), any movement of 9 m of spatial largeness cannot be detected if observed from more than 30 km. Moreover, direction of motion in comparison to the axis of observation could also affect the perception of actual motion: let us imagine someone on a beach, bordered by a coast where a lighthouse is located. At a distance of 3000 m there is a 15 m-high sailing boat travelling at 10 knots (~ 5.1 m.s−1) towards the lighthouse (see Fig. 10). For the observer, the boat travels at 5.9′.s−1 and consequently the observer can detect the motion of the boat in less than one second. In this motion the lighthouse acts as local reference frame and its apparent distance from the sailing boat is decreasing quite quickly; but if the boat faces the observer, the apparent distance between the lighthouse and the boat does not change so rapidly anymore. Only changes in the apparent size (or angular diameter) of the boat can give an indication of the actual movement. With the considered configuration (see Fig. 10), a modification of this apparent size of more than 1′ is performed after 32 s. Before this time elapses, the observer has the illusion that the moving boat is actually still.
Schematic representation of an example of geometric effect on the duration of observation to be able to perceive motion
Physiological limitations of human eyes not only restrict the vision of remote objects but also that of close ones. The least distance of distinct vision is the smallest distance at which someone with ‘normal’ vision can comfortably look at something. This distance is typically about 25 cm for a ‘normal’ subject. Combined with the value of ‘normal’ visual acuity, this means that we cannot directly see something smaller than ~ 0.07 mm. However, motion on a microscopic scale exists and anybody can use a microscope to be convinced of that.
Thus, in some common situations, it is easy to be persuaded that stillness is illusory and optico-geometrical effects can mask actual motion.
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Bathfield, M. Why Zeno’s Paradoxes of Motion are Actually About Immobility. Found Sci 23, 649–679 (2018). https://doi.org/10.1007/s10699-017-9544-9
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DOI: https://doi.org/10.1007/s10699-017-9544-9