Zeno of Elea is famous for demonstrating that motion, when analysed through a naïve representational lens, seems impossible. Achilles can never catch the tortoise; an arrow in flight is always at rest at any instant; space is composed of infinitely divisible points. These paradoxes have puzzled philosophers and mathematicians for millennia.
Yet, Zeno’s paradoxes do not reflect a flaw in motion, mathematics, or physics. They reflect a conceptual misalignment: a failure to distinguish structured potential from perspectival actualisation. Viewed through relational ontology, they reveal profound truths about the nature of continuity, change, and the perspectival structuring of experience.
1. The representational trap in Zeno’s reasoning
Zeno’s argument assumes:
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Motion can be represented as a sequence of discrete, independent points in space and time.
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Completion is possible by summing these points or instants.
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The system of points or instants itself can be treated as independent of the act of traversal.
Under relational ontology, all three assumptions are flawed:
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Space and time are not pre-existing objects; they are structured potentials for instantiation.
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Motion is not a sequence of independent snapshots; it is a perspectival actualisation of a system.
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Completion assumes that an infinite potential can be exhausted by sequential enumeration — but potential is never exhaustible by instantiation.
In other words, Zeno attempts to treat continuous motion as if it were a discrete, representational object capable of being fully captured from outside.
2. System, instance, and the perspectival cut
Relational ontology provides three categories:
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System (structured potential): the continuum of possible positions and velocities.
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Instance (actualisation): the trajectory or act of movement as experienced or realised.
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Construal (first-order phenomenon): the perception of motion in context.
When Zeno considers Achilles catching the tortoise, he treats the system of points as if it could be summed independently of the actualisation. But motion is not a collection of points, it is the process of actualising the potential along a perspectival trajectory.
The “impossibility” arises only when the distinction between system and instance is collapsed: points are treated as both potential and actual simultaneously, and the cut that produces motion is mistaken for the motion itself.
3. Infinite divisibility as potential, not obstruction
Zeno’s infinite subdivisions (half the distance, half again, ad infinitum) seem to prevent motion. Relationally:
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Each subdivision is a potential, not an instance.
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No traversal requires actualising all subdivisions individually.
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Motion occurs within a single perspectival cut, actualising the system continuously, not point by point.
The paradox arises because Zeno attempts to measure the actualisation using the same frame as the potential, assuming the potential itself is a sequence of discrete instantiations. In relational terms, he is attempting an impossible self-cut: using a cut designed to instantiate a system in order to evaluate the system’s own potential exhaustively.
4. Time, space, and the illusion of static points
Similarly, the arrow paradox — that at any instant the arrow is at rest — is a mis-cut:
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The arrow’s trajectory is an instance of the system of motion, not a sum of static points.
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A static “instant” is not the same as the event of motion; attempting to represent motion as a collection of instants collapses the system-instance distinction.
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Time itself, when treated as an object, obscures the perspectival act that produces the phenomenon of motion.
The arrow is always in motion because motion is the relational event actualised through a cut, not a property distributed over static, pre-existing points.
5. Lessons from relational ontology
Zeno’s paradoxes teach:
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Continuity is potential, not object.Attempts to treat it as object produce apparent contradictions.
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Actualisation is perspectival.Motion occurs only as a relational cut through potential, not as summable points.
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The paradox arises from collapsing system and instance.Infinite divisibility is no obstacle; it is the landscape through which actualisation occurs.
In short, motion is not “made of points” and cannot be fully captured by summing them. Each act of traversal is a perspectival cut actualising a continuum. The infinite sequence is never exhausted; it need not be. Motion flows precisely because potential is never identical with any given instance.
6. Conclusion: Zeno reinterpreted
Through relational ontology:
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Achilles does catch the tortoise,
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the arrow is always in flight,
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and the continuum of space and time is a field of potential, not a lattice of static instants.
Zeno’s paradoxes are not failures of mathematics or physics; they are demonstrations of a representational error:
They attempt to instantiate the system from a cut that ignores the relational nature of actualisation.
In other words:
No paradox arises when the distinction between system, instance, and construal is preserved.Motion is possible precisely because the cut cannot cut itself.
Zeno, like Gödel and Russell, points toward the profound insight of relational ontology:
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Potential and actualisation are distinct yet inseparable,
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infinite subdivision is a feature, not an obstacle,
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and perspectival cuts are the foundation of experience, meaning, and movement.
Motion, completeness, and continuity are relational phenomena — living, emergent, and always more than the sum of any attempted representation.
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