Zeno of Elea knows well, like every skilled archer, that only steadiness and immobility can be the premises for hitting the target, and indeed he has undoubtedly struck home, and he has done so with his arguments, among them the one on the immobility of the arrow. But which target has he actually hit? What is the meaning of his aporias? What is the aim of his arguments?
A great deal has been written on this, and perhaps the most convincing interpretation is that of Giorgio Colli, who attributes to Zeno the aim of “developing a wholly negative theory of human reason”. “Every thesis in Zeno concludes, Colli writes, with a refutation, and his ultimate purpose would be to develop a wholly negative theory of human reason, to prove that all knowledge is destroyed by rational analysis”. Thus a nihilistic vision of human reason, of a lógos whose argumentative ratio produces incurable aporias.

I will not go into the merits of Colli’s interpretation, nor will I try to defend others; this is not meant to be an article on Zeno of Elea, but a reflection on some possible rereadings of the argument of the arrow, with particular reference to Emanuele Severino’s neo-Parmenidean interpretation of becoming.

The Arrow: Heisenberg and Donald Duck

In Einstein’s Veil, the physicist Anton Zeilinger writes:

“Heisenberg’s principle concerns the precision with which we can simultaneously know the position of a particle and its velocity. The Greek philosopher Zeno of Elea, more than 2000 years ago, had already posed the problem; for him an arrow in flight could not be found, at any moment, at a point that could be precisely determined, because if it were, motion would not be possible. Heisenberg’s uncertainty principle expresses precisely this concept […]. If we know very precisely where it is, then its velocity is highly indeterminate, so much so that the particle does not have a well-defined velocity; conversely, if the particle has a very precise velocity, then, literally, one does not know where it is!”

Before commenting on the connection between Heisenberg and Zeno, it is worth hearing again the problem raised by the Greek philosopher.
“Zeno denies motion by saying: ‘that which moves does not move either in the place where it is, or in the place where it is not’” (Diog. IX 72). Why, according to Zeno, does the moving body move neither in the place where it is nor in the one where it is not? It does not move in the place where it is because it occupies it, and it does not move in the place where it is not simply because it is not there. Therefore the arrow, and this is the conclusion, does not move at all. Another formulation of the aporia of the arrow, this time reported by Aristotle, is the following:

“everything that occupies a space equal to itself is either at rest or in motion; but it is impossible for it to move if it occupies a space equal to itself, and therefore it is at rest. The moving arrow, then, occupying a space equal to itself at every instant of the time during which it moves, will be still; and if it is at rest at all the instants of time, which are infinite, it will be at rest for the whole of time. But it had been assumed that it was in motion, and therefore the arrow in motion will be still”

(in Arist. Phys.)

Here the formulation of the aporia is more elaborate and includes the concept of time, but in substance it is equivalent to the one reported by Diogenes Laertius, which does not contain the concept of the instant.
The connection between the uncertainty principle (the principle according to which one can never be certain of both the position and the velocity of a particle) and the paradox of the arrow is interesting, even though it ends up betraying the sense of Zeno’s argument. Zeilinger’s is in fact a transposition of the paradox that literally shifts the terms of the question; if in Zeno it is the position of the moving body that is incompatible with motion, in particle physics it is instead the exact determination of the position of the moving body that is incompatible with the exact determination of the velocity. The velocity, and therefore the motion, may turn out to be indeterminate, but certainly not impossible. The uncertainty principle does not therefore intend to deny the possibility of motion, but only the possibility of establishing with exactness the velocity and position of the particle. If we know exactly where the particle is, then we cannot precisely determine its velocity, a velocity that is nonetheless considered real. Real the velocity, though indeterminate, real the motion.

The physicist Julian Barbour, too, in the volume The End of Time: The Next Revolution in Physics, restores the paradox of the arrow, but proposes in this regard a far more interesting position, which then finds, in an ontological sense, its place in Emanuele Severino’s neo-Parmenidean interpretation of becoming. Barbour writes that “the arrow never reaches the target” because “the arrow shot from the bow is not (italics mine) the same arrow that strikes the target”. The distinction between the arrow shot and the one that strikes the target is the element of novelty in this reworking of the paradox. What really happens when we observe the “motion” of the arrow? What happens when we see the arrow “go” from A to B? What happens, this is what Barbour seems to maintain, is more or less what occurs when we watch a Disney cartoon, perhaps with Donald Duck running. We interpret the progressive change of configurations ( = alteration) as an expression of motion. The change of configurations is given by the following succession: first Donald Duck¹ appears in position P¹, then Donald Duck¹ disappears together with the position it occupies, and Donald Duck² appears in position P², and so on. This sequence in succession, in which Donald Duck¹ is not identical to Donald Duck², literally manufactures the impression of motion. Conclusion: Donald Duck, like the arrow, does not move at all, yet becoming is preserved, and this thanks to the replacement of the concept of motion with that of alteration. Note that in this case no “discrepancy” would be produced between the logical plane and the physical-real one, because changing reality would know no form of motion; motion would be only the product of an untruthful interpretation of the real and, therefore, the argument of the arrow would no longer constitute an aporia but a mere paradox.

The Arrow and the “Melissian’s Beard”
Severino’s conception of becoming, too, ends up restoring the paradox of the arrow in these terms, except that in this case alteration takes shape, though in substance nothing changes, as the appearing and disappearing of things within the transcendental horizon of consciousness.
As is well known, Severino, in order to overcome the nihilistic interpretation of becoming (an interpretation the philosopher considers untenable because of the numerous contradictions it presents and presupposes), reads the alteration of things as the appearing and disappearing of eternal entities within the transcendental horizon of consciousness (as regards the reasons for this neo-Parmenidean reading, the reader is referred to texts such as The Original Structure and The Essence of Nihilism). Things, all things, according to Severino, are neither generated nor corrupted, nor even transformed, but appear and disappear; this incessant appearing and disappearing would have been mistakenly interpreted by Western thought as an emergence of being from nothingness and a dissolving of being into nothingness. Severino writes:

“This body burns, and this body is replaced by its ashes: appearing attests to nothing but a succession of events: the piece of white paper, the approaching of the flame, the flame that grows, a smaller piece of paper of a different shape, a slighter flame, a still smaller piece of a still different shape, the ashes. Each event is followed by another, in the sense that a second event begins to appear when the first no longer appears. After the last fire, the ashes; that is: when the last fire no longer appears, the ashes appear. But that this, which no longer appears, no longer is either, this appearing does not reveal. This is interpreted on the basis of the way in which something appears and disappears: when something that had never appeared appears, it is said to be born and to have been a nothing before; when something disappears and returns no more, it is said to die and to become nothing […]. But this is the non-veridical interpretation of becoming: only the intervention of δόξα compels us to posit as a nothing (before and after its appearing) that which appears and disappears in a certain way”

This “non-veridical” interpretation of becoming is opposed, according to Severino, to the one according to which all things, “from the humblest and most shadowy to the noblest and greatest”, are eternal. The transposition of becoming in terms of appearing and disappearing eliminates, this is Severino’s position, that contradiction proper to nihilistic thought which ends up positing the identity of the different. And if on the one hand it removes the contradiction, on the other it saves the reality of becoming, incontestably affirmed by experience. Naturally everything has a price, and here the price to be paid consists in renouncing the concept of transformation because it is contradictory. Wood, Severino maintains, not only is not annihilated but does not even undergo any process of transformation. But then, what happens when becoming shows the reduction of wood to ashes? What happens is that the wood disappears, while continuing to be, and the ashes appear; where the wood and the ashes are different entities (a difference nonetheless underlain by an identity) that alternate in the “circle of appearing”. And what happens to the wood when it disappears? Severino answers by using a metaphor:

“When the sun sets, it leaves the vault of the sky and disappears from sight. What becomes of it? What fate befalls it when, sinking into the sea or the earth or behind the mountains, it is no longer visible? These questions leave us today wholly indifferent, also because the Copernican theory assures us that the motion of the sun is apparent and that the sun therefore continues to exist even when it is not visible. But if we wished to answer that question solely on the basis of what appears in the vault of the sky when it has been abandoned by the sun, what could we say of the fate of the sun that has made itself invisible? […]. Nothing! Abandoned by the sun, the vault of the sky is silent about its fate, attests to nothing concerning it”

Similarly, becoming is silent “about the fate of that which does not appear. And if these fates, Severino writes further, are passed over in silence by appearing as such, they are unveiled […] by the truth of being”. A truth that, to return to the analogy of the sun, would perform the same function as the Copernican theory, that is, it would unveil the destiny of what is hidden and, in this case, the continued existence of the wood now vanished and replaced, on the scene of appearing, by the ashes. And now, if we replace the wood with the arrow, here Zeno’s argument re-emerges in the reading proposed by Julian Barbour.

To what extent is such a transposition of becoming in terms of appearing and disappearing intelligible? It has been objected several times to the Severinian conception, above all by Gustavo Bontadini, that such a transposition, contemplating no form of transformation, would end up shattering the unity of the manifold and positing a “proliferation” of entities. Bontadini writes: “It is […], perhaps, worth considering what a strange conception of the worldly real comes of it. Let us refer to what we may by now call the ‘aporia of the Melissian’s beard’. Every Melissian breaks apart into multiple (infinite?) Melissians: for the Melissian who wears, eternally, beard a cannot be the same as the one who, also eternally, wears beard b”. Similarly, Socrates standing cannot be the same Socrates who is seated, even if, as Severino maintains, “The manifold constituted by Socrates seated, Socrates standing, Socrates young, Socrates old, and so on, is nonetheless underlain by an identity: being Socrates”. Now, this “strange conception of the worldly real” is in its result identical to the one arrived at by replacing, in the argument of the arrow, the concept of motion with that of alteration. Let us recall that such a replacement is indispensable if one intends to “save” Zeno’s argument without thereby denying the phenomenological evidence. Just as there is not a single Melissian but many and, perhaps, infinite ones, in the same way there is not a single arrow. So that the arrow positioned in the bow is not properly the same as the one that is halfway between the archer and the target, and this, in turn, is not the same as the one that is in the target. Thus, no arrow would move from A to B, but the multiple arrows would alternate on the scene of appearing in succession. There is therefore a convergence between the argument of the arrow (in its present rereading) and the Severinian interpretation of becoming, which restores that argument in a new sense by means of the figure of the “Melissian’s beard”. In both cases one starts from the observation that there is something wrong, on the conceptual plane, in the interpretation of becoming, and then sets out a different reading of the real in order to reconcile reason and experience. While Severino starts from the contradictions of a becoming understood as becoming-other, Zeno’s arguments would start from the observation of the irremediably aporetic character of becoming understood as motion. The results converge on the same point: neither transformation nor motion; only a succession of events. Obviously Severino would reply that the two paths are certainly not equivalent, because Zeno, as Hegel writes, “never dreamed (the italics are Severino’s) of denying motion”. Zeno too is, for Severino, an “inhabitant of time”. But this concerns the general interpretation of the figure of Zeno, which, as already said, we do not intend to enter into.

The Melissian and the Arrow of Time
We might at this point call our reworking of the argument of the arrow the “paradox of the Melissian’s arrow”. Here the paradoxical character shows itself in all its force the moment we literally put ourselves in the Melissian’s shoes, or, if one prefers, in the arrow’s shoes. Indeed, if the Severinian reading of becoming were correct, the shattering of which Bontadini speaks would strike each of us, and the “proliferation of entities” would become infinite. I would come to occupy, like the arrow, multiple fixed positions that would manifest themselves in succession, giving rise to the arrow of time. I would occupy first one position and then another, without, however, passing from this one to that one. I would inevitably end up resembling Disney’s Donald Duck or the frame of a cinematographic film. Not only that, but to all this would be added a kind of ubiquity, because if it is always me we are speaking of, that is, if this infinite multiplicity is nonetheless underlain by an identity (mine), I would have to be both here and elsewhere, perhaps back there in the past rather than up ahead in the future. If everything is eternal, in fact, my being seated, now that I am standing, has not been annihilated but has only disappeared, letting appear my being standing, which existed well before I stood up. My being seated would therefore be absolutely real, even though past, just like my being standing. Let it be noted, in this regard, that the non-nihilistic transposition of becoming in terms of appearing and disappearing does not intend to remove time, that is, the indispensable distinction between past, present, and future. But precisely because such a distinction is indispensable, if everything were eternal I would have to be both here, in the present, and elsewhere, in the past or in the future. It happens, however, and here is the objection, that of this being of mine also elsewhere I have no consciousness. It would be a matter of a ubiquity not experienced, contradicted by the empirical datum. The incontestability of my experiencing contradicts, or at least seems to contradict, my being also elsewhere. As long, then, as we speak of the appearing and disappearing of eternal arrows, rather than of wood, everything seems fairly intelligible, and the image of the cinematographic film conveys well the idea of the illusoriness of motion and transformation. In the film all the frames coexist, both those that, disappearing, go to constitute the past, and those that, appearing, constitute the present, and those that, not yet appearing, constitute the future. But if, instead of considering objects, we consider ourselves in the first person, then everything becomes complicated, and the neo-Parmenidean transposition of becoming becomes barely intelligible, not to say aporetic. There is then another, still more problematic, aspect: it can be objected to the Severinian conception that disappearing too is a form of annihilation. What becomes of the appearing of the thing when it no longer appears? What would be annihilated, in this case, would be not so much the thing that disappears but the appearing of the thing that disappears. To this criticism Severino responds in The Essence of Nihilism, where he devotes ample space to the analysis of what he calls “the authentic structure of appearing”.
To Severino’s observations on the structure of appearing one may add a few concluding considerations that explain our natural propensity to interpret becoming in a nihilistic sense and clarify the reason why, returning to our arrow, we cannot but regard it as being in motion. These last remarks are addressed not only to Severino, but in general to the “eternalists”, according to whom all events, not only present ones but also past and future ones, are real. Time, to put it with Kant, “has only one dimension: different times are not simultaneous but successive (just as different spaces are not successive but simultaneous)”. While space is a system of coexisting entities, time is not. And I would add: if time were a system of coexisting entities it would not be time. Temporal succession cannot be a mere fact, as Severino several times seems to maintain, who writes: “But succession is a fact, and the fan could appear all together, the past together with the present and the future”. The simultaneity of present, past, and future, which now contingently does not appear, could nonetheless appear, Severino says. Nothing would logically prevent it. But from when could such a simultaneity appear? Since it is not there, it would have to begin at a moment of future time, but such a beginning would in any case end up presupposing a before, a before that therefore could not be simultaneous with such a simultaneity and that would therefore prevent such a simultaneity from constituting itself as simultaneity tout court. In other words, the non-simultaneity of the moments of time would in any case and inevitably end up recurring at the very moment when the co-presence of past, present, and future should, absurdly, begin. The coexistence of times is, as Vladimir Jankélévitch says, “An impossible undertaking! Are the ‘moments’ not, by definition, successive? Is succession not made precisely so that they flow one after another, one before and the other after, states that refuse to coexist but that will consent to appear one at a time?”. Now, precisely because time has only one dimension, it cannot in any way be reduced to space. But both Severino and the eternalists end up, even if implicitly, treating time as if it were a kind of space. In space, this is the point, it makes perfect sense not to assimilate absence (that is, non-presence) to non-existence. In space, in fact, not-being-present does not necessarily imply not-being-existent, non-presence is not necessarily non-existence; the sun, now that it is night, is not present, but this does not mean that it does not exist. In time, on the other hand, the degree of reality cannot disregard presence. When 2010 is present, 2009 is not present, and this absence of it makes of it an unreachable; note well, an unreachable not only in fact but in principle. How could we reach it? By going back? But does not the return always come after the going? And if it is true, as it is, that the return always comes after the going, then the return into the past would make that past subsequent to the present and would deny the principle of non-contradiction. “Irreversibility, Jankélévitch says, is the true objectivity of time”. When 2010 is present, 2009 is by now an absolutely unattainable beyond, a veritable elsewhere that, precisely because of this infinite transcendence of its own, ends up constituting itself, in fact, as a non-existent. In Severino’s language, on the other hand, the absence of 2009 is the absence of that which, not appearing, continues to exist; for the neo-Parmenidean philosopher 2009 is absent and yet existent. This, however, is a spatialization of time! It is no accident that Severino, in order to clarify the sense of disappearing, uses the spatial metaphor of the sun. What sense can there be, instead, in positing as real and existent that which, insofar as it is unreachable and unattainable, presents itself as a non-place, as a true utopia? Is the absence of the non-place not, perhaps, simple non-existence? Is this not the reason why we continue to tread what Severino has called the “path of night”? Is it not for this reason that our nihilistic sense of becoming impels us to reject the spatialization of time? And is it not for this reason that we find unconvincing even the latest reworking, just proposed, of Zeno’s arguments? Yes, because it is precisely this way of understanding time as something not reducible to space, as something irreversible, that renders the argument of the arrow suspect even in its latest version, that of the Melissian. At bottom, the paradox of the Melissian’s arrow is nothing but a spatialization of time. Thus Zeno’s arrow ends up colliding once again with that of time, and once again leaves us unsatisfied, disoriented, and vulnerable.

Cristiano Bianconi was born in Perugia in 1968 and graduated in philosophy in Venice. He currently teaches philosophy and history in secondary schools.