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A Quantum Argument Against Zeno's Paradox of Dichotomy

Ryan Christie-Muller

Published Apr 1, 2026• 0 citations
OpenAlex

Abstract

During the pre-Socratic period, Parmenides created his radical idea of an eternal, unchanging, and motionless world. This was met with much doubt, contempt, and counterarguments that sought to disprove his statements claiming that the concept of past and future were false, and that both change and movement were impossible. Yet his student, Zeno, came after him and sought to further prove his teacher Parmenides' arguments, creating what are now talked about thousands of years later known as Zeno's paradoxes. This paper will specifically talk about Zeno's dichotomy paradox, claiming that movement cannot exist due to the inherent need to be capable of traversing between a theoretically infinite series of midpoints in order to reach one's destination. The infinite series of points can be overcome with the use of atomist theory to break it down into finite, indivisible points. However, the question then becomes how exactly do we move from one finite point to another? This issue can be addressed using superposition principle, a quantum theory that requires an object to be observed in order for its state to be known, and Planck time, the current smallest measurement of time. Yet even with Planck time being an incredibly small unit of time, our observation is still limited. By understanding movement as a phenomenon that occurs between these lapses of observation, the quantum particles that are resting in superposition are capable of moving from one finite point to another.

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During the pre-Socratic period, Parmenides created his radical idea of an eternal, unchanging, and motionless world. This was met with much doubt, contempt, and counterarguments that sought to disprove his statements claiming that the concept of past and future were false, and that both change and movement were impossible. Yet his student, Zeno, came after him and sought to further prove his teacher Parmenides' arguments, creating what are now talked about thousands of years later known as Zeno's paradoxes. This paper will specifically talk about Zeno's dichotomy paradox, claiming that movement cannot exist due to the inherent need to be capable of traversing between a theoretically infinite series of midpoints in order to reach one's destination. The infinite series of points can be overcome with the use of atomist theory to break it down into finite, indivisible points. However, the question then becomes how exactly do we move from one finite point to another? This issue can be addressed using superposition principle, a quantum theory that requires an object to be observed in order for its state to be known, and Planck time, the current smallest measurement of time. Yet even with Planck time being an incredibly small unit of time, our observation is still limited. By understanding movement as a phenomenon that occurs between these lapses of observation, the quantum particles that are resting in superposition are capable of moving from one finite point to another.
Abstract Parmenides’s insinuation of an unchanging universe is assertively supported by Zeno with various logical paradoxes that question the very nature of plurality, space, time, and the reality of apparent motion. The dichotomy is his most famous paradox. To begin a trip, say, from here to the door, a traveler must travel the first half of it, but before she does that she must travel half of the first half, and in fact half of that, ad infinitum. Since there will always exist a smaller first half to be traveled first, Zeno questions whether a traveler can ever even start a trip. Zeno’s analysis is logical; on the other hand, things everywhere appear to be moving. Hence, either Zeno’s reasoning is wrong or appearances are deceptive. Empowered by the uncertainty principle of quantum mechanics, it will be argued that, at best, the phenomenon of motion is experimentally unverifiable!
When the concept of Zero is analyzed, it is frequently related to another equally deep and complicated concept – Nothing. If in science, Zero is closely associated with mathematics, then Nothing is primarily related to the material world. The question of whether Nothing can be found in the real physical world is often discussed in relation to the vacuum which, at some point in the history of the development of physics, is thought about as an empty space. Nevertheless, theories of modern physics have demonstrated that even in a vacuum, when all the matter is evacuated from a certain region of space, and this region is isolated from ordinary fields – gravitation field, electromagnetic fields etc – there still remain physical fields called vacuum zero fluctuations that can not be removed in principle. These vacuum fields are properties of space. In this chapter I will analyze another physics theory – namely, Quantum Physics – and will consider its less discussed relations to the concept of Nothing. One of the very first Greek philosophers, Thales (624–548 BCE) from Miletus, declared that Something can not appear from Nothing and it can not disappear, turning into Nothing. This idea, in some aspects, was further developed by Zeno from Elea (495–430 BCE) when he created his paradoxes of motion. These paradoxes can be understood as an analysis of the possibility of infinite division of time and space intervals. When we come to Quantum Physics, the very first appearance of the idea of the smallest possible portion of energy, quantum, was stimulated by the analysis of the spectrum of radiation of heated body – black-body radiation. German physicist Max Planck (1858–1947) demonstrated in 1899 that if we assume that there exists a smallest portion of radiation energy that can not be further divided into smaller parts, we arrive at the description of spectrum of the black-body radiation that perfectly matches the observable reality. In contrast, if we assume that radiation energy can be divided an infinite number of times and there does not exist the smallest possible portion, we arrive at the radiation spectrum which is in a sharp contradiction with observable reality. This contradiction is so strong that it has acquired a special term – the ultraviolet catastrophe. Certain parallels in Zeno’s reasoning and approach used in Quantum Physics can clearly be seen and will be discussed. In the conclusion, I will briefly touch on a similar situation in another field of physics when at the other extreme, departure from the idea of the possibility of the infinitely large quantities, this time velocities of motion, gave birth to another theory of modern physics – Einstein’s theory of special relativity.
The article discusses the new approaches to the four Zeno’s Paradoxes: “The Ar­row”, “The Achilles”, “The Dichotomy” and “The Moving Rows”. Paradoxes are analyzed in relation to the current research in the field of elementary particle Physics and promising areas of the quantum theory of gravity development. Pro­vided with the necessary philosophical interpretation Physics concepts are used to clarify Zeno’s Paradoxes and to search for answers to them. The involvement of modern approaches is shown to be ineffective in solving the paradoxes due to their scope losses when analyzed by the microworld Physics on ultra-small scales. The work’s main part is devoted to demonstrating this circumstance – the questions posed by paradoxes are deprived of the possibility of getting the answers (at least, in their classical interpretation). As a possible explanation stands the ground of mixing the properties of the macroworld and the microworld in the paradoxes statements (which is historically justified given the intuitive homogeneity of big and small, and the origin of non-classical Physics – quantum mechanics – only in the 20th century), meaning the transition from observation of the large physical objects to the infinitely small in terms of discreteness and continuity. However, the principles of space organization on ultra-small scales are beginning to be clarified in general terms only now, and, perhaps, these prin­ciples may turn out to be quite far from classical ideas on fundamental physical reality.
In the 5th century B.C.,the philosopher and logician Zeno of Elea posed several paradoxes which remained unresolved for over two thousand five hundred years. The 20th20^{th}20th century saw some resolutions to Zeno's mind boggling problems. This long journey saw many significant milestones in the form of discoveries like the tools of converging series and theories on infinite sets in mathematics. In recent times, the Zeno effect made an intriguing appearance in a rather unlikely place - a situation involving the time evolution of a quantum system, which is subject to "observations" over a period of time. Leonid Khalfin working in the former USSR in the 1960s and ECG Sudarshan and B. Misra at the University of Texas, Austin, first drew attention to this problem. In 1977, ECG Sudarshan and B. Misra published a paper on the quantum Zeno effect, called "The Zeno's paradox in quantum theory". Their fascinating result revealed the bizarre workings of the quantum world. Misra and Sudarshan's 1977 paper activated over two decades of theoretical and experimental explorations into the subject and still continues to evoke a lot of interest. In the following, the quantum Zeno effect is described and a brief outline of some of the work following Misra and Sudarshan's paper is given. The quantum Zeno effect is yet another example of the myriad unimaginable possibilities that lie waiting in the magical world of the quantum.
``No one has ever touched Zeno without refuting him''. We will not refute Zeno in this paper. Instead we review some unexpected encounters of Zeno with modern science. The paper begins with a brief biography of Zeno of Elea followed by his famous paradoxes of motion. Reflections on continuity of space and time lead us to Banach and Tarski and to their celebrated paradox, which is in fact not a paradox at all but a strict mathematical theorem, although very counterintuitive. Quantum mechanics brings another flavour in Zeno paradoxes. Quantum Zeno and anti-Zeno effects are really paradoxical but now experimental facts. Then we discuss supertasks and bifurcated supertasks. The concept of localization leads us to Newton and Wigner and to interesting phenomenon of quantum revivals. At last we note that the paradoxical idea of timeless universe, defended by Zeno and Parmenides at ancient times, is still alive in quantum gravity. The list of references that follows is necessarily incomplete but we hope it will assist interested reader to fill in details.
Albert Einstein's relativity theory has been tested to all sorts of limits and has never yet failed. It shows beyond reasonable doubt that what one observer may see as simultaneous events in his/her 'present' can appear to another observer as separated in time. Similarly, the usual interpretation of the Zeno effect is to suppose that the measurer is stationary and the wave function is flowing past its observational 'window', but maybe what's actually going on is that the measurer is the mover and the wave stationary. This chapter describes a couple of oddities about human relationship to time. The first oddity is highlighted by the moving coloured dot illusion. People's brains do a whole lot of work on the momentary physical happenings before a person gets to experience them consciously. The second oddity goes by 'pre-sponse', 'presentiment', 'feeling the future' and, though undramatic in itself, is totally at odds with the current world view of 'man a machine' aficionados.