Zeno's Paradox Research Articles

A numerical evaluation of the Finite Monkeys Theorem

The Infinite Monkeys Theorem has long-established the eventual certainty of the complete works of William Shakespeare being reproduced by a monkey randomly pressing keys on a typewriter. This only considers the infinite limit, with either an infinite number of monkeys and/or an infinite time period of monkey labour. Here, we consider the Finite Monkeys Theorem and look at the probability of a given string being typed by one of a finite number of monkeys within a finite time allocation consistent with estimates for the lifespan of our universe. We also calculate the expected number of keystrokes until a target string would first be produced. Given the expected time until the heat death of the universe, we demonstrate that the widely-accepted conclusion from the Infinite Monkeys Theorem is, in fact, misleading in our finite universe. As such, this places the theorem in a class of probabilistic problems or paradoxes, including the St. Petersburg paradox, Zeno's dichotomy paradox and the Ross–Littlewood paradox wherein the infinite-resource conclusions directly contradict those obtained when considering limited resources, however sizeable.

The Meaning of Proofs: Mathematics as Storytelling by Gabriele Lolli

Reviewed by: The Meaning of Proofs: Mathematics as Storytelling by Gabriele Lolli Firdous Ahmad Mala GABRIELE LOLLI The Meaning of Proofs: Mathematics as Storytelling Trans. Bonnie McClellan-Broussard. Cambridge, Massachusetts. MIT Press. 2022. 176 pages. MATHEMATICS HAS BEEN FEARED by many and loved by some for its rigor, symbolism, and brevity. Literature on mathematics abounds with technical, biographical, historical, and recreational titles. This book, The Meaning of Proofs: Mathematics as Storytelling, is not about such things, however. It attempts to present mathematical proofs as forms of narrative. It is a philosophical rumination on why mathematical proofs are expressed the way they are. The purpose of the book, as singled out in the preface, is to convince the reader of the idea that a natural way to conceive and understand a proof is to present and see it as a narrative. However, by presenting mathematical proofs as narratives, one must not mistake that to understand a proof, it is essential to know the biographical details of the one who came up with it. Narrating a piece in mathematics does not involve summarizing what is known about it, nor does it involve an account of the lives of mathematicians or logicians. In essence, it means bringing to the discussion the content and subject matter of that piece in mathematics. The concepts of limits in calculus and the one-to-one correspondence between the whole numbers and natural numbers in set theory are best understood through stories such as Zeno's paradox and Hilbert's grand hotel. These are essentially narratives brought about to bring the notions of limits, bijections, and infinite sets home to readers and students. Narratives have not only been relevant but also instrumental in shaping the pedagogy of mathematics. As a matter of fact, mathematics education, a branch of applied mathematics, aims at a deliberate and extensive use of such narratives to realize its objectives. The book is not aimed at an only-mathematical audience, rather it envisages to address any intellectually curious one. The employment of literature as a proxy for mathematics is a beautiful idea aimed at making it evident that the two subjects are not, in reality, as different as many people think. This idea is, in fact, gaining unprecedented impetus in the form of publications such as this one and Statistical Universals of Language (2021), by Kumiko Tanaka-Ishii, which aims to make the statement that language has a lot more in common with statistics than we realize. Interestingly, the author shows how, contrary to the common belief, metaphors and images pervade the literature of mathematical proofs. In mathematical proofs, visual diagrams and mechanical devices serve a twofold purpose. On one hand, they bring to the fore how narrative tools were viewed in ancient times; and on the other, they are pivotal in the context of contemporary research on automated theorem-proving that attempts to prove theorems in mathematics using computer programs. If mathematics is confined to its symbols and diagrams, it is difficult to listen to it. However, if it is presented as narratives, it is likely to attract more of an audience and have a wider reach. Then, it is also very likely to be embraced with open arms and be held dearer. This book analyzes the narrative of proofs at two levels—the grand and the individual. The grand narratives influence the ramifications of the subject over a considerably long period of time and are brought about, usually, by a large number of authors and papers. A case in point is Klein's Erlangen program, in 1872, that advocated the study of mathematical objects in terms of their symmetries and transformations. This idea soon became the talk of the town in both modern mathematics and theoretical physics. The individual narratives include specific papers highlighting proofs that are written as narratives unique in style and presentation. Concerning the individual narratives, the book analyzes how some individual proofs are modeled on ancient Greek composition forms. As Bruce Springsteen once said, stories save lives. To him, writers tell stories with the aim of saving themselves. It has been critically observed that the ideas expressed in mathematical proofs are common to all [End Page 88] humans...

Some Examples of Materialist Dialectics in the Concept of Higher Mathematics

Some examples of dialectics philosophy in higher mathematics are illustrated in this paper. Firstly, the principle of interconversion between quality and quantity in dialectics philosophy is quantified by the mathematical definition of the limit theory. Secondly, some natural and social phenomena imply the definition of continuous function in incremental form and it is a new explanation for the Zeno paradox. Finally, the dialectics relationship between the local change and the whole change of some variables is discussed in the differential median theorems.

McLaughlin-Millerの運動モデルの位相的側面

McLaughlin and Miller (1992) provided a novel framework for resolving Zeno's paradox by employing Nelson-style nonstandard analysis. The following two principles play the key roles in their model of motion.The first is the ontological principle that every point in space-time is described as a vector of hyperreals. The second one is epistemological: one cannot distinguish two points in space-time that are infinitely close to each other. In this paper, we extract the topological essence from their argument. More precisely, we argue that the above principles correspond to introducing two topologies on the hyperreals, called Q-topology and μ-topology. We also consider Hellman-Shapiro's account of nonstandard analysis in the context of philosophy of continua.

Cube Fest 2022

Cube Fest 2022

Dynamic Quantum Games

Quantum games represent the really twenty-first century branch of game theory, tightly linked to the modern development of quantum computing and quantum technologies. The main accent in these developments so far was made on stationary or repeated games. In this paper, we aim at initiating the truly dynamic theory with strategies chosen by players in real time. Since direct continuous observations are known to destroy quantum evolutions (so-called quantum Zeno paradox), the necessary new ingredient for quantum dynamic games must be the theory of non-direct observations and the corresponding quantum filtering. Apart from the technical problems in organizing feedback quantum control in real time, the difficulty in applying this theory for obtaining mathematically amenable control systems is due partially to the fact that it leads usually to rather non-trivial jump-type Markov processes and/or degenerate diffusions on manifolds, for which the corresponding control is very difficult to handle. The starting point for the present research is the remarkable discovery (quite unexpected, at least to the author) that there exists a very natural class of homodyne detections such that the diffusion processes on projective spaces resulting by filtering under such arrangements coincide exactly with the standard Brownian motions (BM) on these spaces. In some cases, one can even reduce the process to the plain BM on Euclidean spaces or tori. The theory of such motions is well studied making it possible to develop a tractable theory of related control and games, which can be at the same time practically implemented on quantum optical devices.

Crip teleportation: the animal that therefore I am—or I am not

This paper examines how disabled body-minds are discursively dehumanized or superhumanized. It draws on Critical Disability Studies and the Crip Studies scholarship and focuses on invisible mental disabilities, mainly those of the neurodiversity spectrum. The efforts of animalization and super-humanization draw on a mechanism that resonates with Parmenides’s Zeno paradox. As the autistic scholar Melanie Yergeau (2018) discusses, a neuroqueer body-mind is always in constant motion, being relocated from one identity category to another, from humanity to animality and vice-versa, subject to an excessively strict rhetoric model. Concomitantly, animals in slaughtering facilities are humanized to make their deaths seem smoother. To short-circuit the models that evaluate how fit human and animal bodies are for the neoliberal guidelines of productivity, the paper also brings Jasbir Puars’s intersectional approach on capacity and debility into the discussion. Keywords: Animal studies. Neurodiversity. Neurodivergence. Crip studies. Demirhetoric.

The Labours of Zeno – a Supertask indeed?

It is usually supposed that, with his dichotomy paradox, Zeno gave birth to the modern so-called supertask debate – the debate of whether carrying out an infinite sequence of actions or operations in a finite interval of time is physically or even logically possible. I argue that in fact this is not a problem raised by Zeno's dichotomy paradox, and that an account of the dichotomy paradox as a supertask (often implicitly offered also by scholars of ancient philosophy) seriously misconstrues the problems Zeno raises therein. However, comparing Zeno's paradox with a paradigmatic supertask can nevertheless be instructive, since it forces us to make explicit the pre-conditions on which the supertask debate rests and to examine whether these conditions do indeed obtain in the case of a continuous run. I will suggest in the end that the requirements for supertasks and for continuous finite runs are genuinely different.

표준해석학과 비표준해석학의 패러다임 관점에서 본 ‘제논의 역설’

This study has analyzed the phenomenon that Zeno’s paradox is still not completely resolved despite the explanation of a geometric series from the perspectives of the ‘spontaneous’ meaning of ‘everyday language’ as well as ‘standard analysis and non-standard analysis’. As a result of analyzing the philosophers’s arguments, the ‘arriving moment’ on the continuum of time expressed in Zeno’s paradox contained a meaning of non-standard analysis. Kant says that time and space are human transcendental forms of intuition. In Zeno's paradox and its dissolution, time is expressed in terms of non-standard analysis, and space is expressed by the limit of a geometric series in terms of standard analysis. Since the two types of intuition of human beings are expressed in different mathematical theories, Zeno’s paradox is not completely solved by the geometric series. This study suggests that the difficulty of mathematical understanding can be attributed to the use of different mathematical structures or paradigms.

On the Violation of Causality in Quantum Experiments

Four experimental situations are analyzed from the viewpoint of fulfilling the causality principle: instantaneous collapse of the quantum state vector of a system of entangled particles, a quantum eraser, the quantum Zeno paradox, and light transformation by a nonlinear beam splitter. The last is a planar interface of two transparent dielectrics, at least one of which possesses Kerr nonlinearity, that is, its refractive index depends on the intensity of radiation that penetrates it. It has been shown that the causality principle in the first two cases can be violated only in the sense of the instantaneousness of the manifestation of the effect with respect to the cause. For a nonlinear beam splitter, in addition to the directly opposite predictions given by the quantum and classical theories of its description for the behavior of phase fluctuations of the radiation transformed by the splitter, the causality principle is violated in the most general sense: the effect of the following event on the previous one. The quantum Zeno paradox occupies an intermediate position of democratic involvement of the cause and effect in the general cascade of two subsequent events, i.e., the first event can prevent the second event, just as the second event can prevent the first one.

Scheherazade, Achilles, and Borges: "El sur" as an Infinite Narrative

Scheherazade, Achilles, and Borges:"El sur" as an Infinite Narrative Mac J. Wilson ¡jue, la pucha, que trae licionesel tiempo con sus mudanzas! —Martín Fierro Los árabes dicen que nadie puede leerLas mil y una noches hasta el fin. No porrazones de tedio: se siente que el libro esinfinito —Jorge Luis Borges, Siete noches Ser Alonso Quijano y no atreverme a serdon Quijote. —Jorge Luis Borges, "La fama" Like some of his other stories, Jorge Luis Borges's "El sur" has a suggestive yet ambiguous conclusion.1 The last paragraph (and last sentence) reads: "Dahlmann empuña con firmeza el cuchillo, que acaso no sabrá manejar, y sale a la llanura" (919). The narration–with conjectural phrases like "acaso no sabrá manejar"–makes this conclusion suggestive by leading the reader to assume that the protagonist, Juan Dahlmann, dies, without explicitly declaring him dead. Yet because it is left open-ended, it remains ambiguous, allowing the reader to speculate on the next step in the story. Its inconclusiveness is a result of both content and form. First, the narration stops short of spelling out a concrete ending–Dahlmann finds himself coerced into a dagger fight with an apparently more dexterous local tough and the story ends with the anticipation of this fight. Second, the narration switches from the past tense to the present tense, and then speculates by using the future tense, effectively giving the end of the story the uncanny sense of a beginning. The narrator's temporal aside also sets itself apart in a new, though brief, paragraph, which [End Page 47] leaves it isolated from the preceding text. This shift in syntax and structure prolongs the narration in the present and suggests a narrative future, thereby keeping the story, as well as its protagonist, alive. Because many interpretations of "El sur" seemingly disregard this detail, the question of how Dahlmann dies eclipses the question of if he dies at all. And if one assumes that the protagonist expires, one downplays the open-endedness of the story. Taking the peculiar open-ended quality of the conclusion seriously is an important step in qualifying "El sur" as what we will call an infinite narrative. In basic terms, and working off of Borges's idea that The Thousand and One Nights is a "libro infinito," an infinite narrative refers to a story that implicitly and explicitly perpetuates itself transtextually–to use Gerard Genette's terminology.2 As an infinite narrative, "El sur" perpetuates itself intratextually, intertextually, and extratextually. A narrative's intratextual perpetuation operates via the components of its structure: i.e. foreshadowing, narrative arc, plot points, and sequence of events. An open-ended conclusion, for example, would be considered an intratextual perpetuation because of its integral part in the narrative framework. On the other hand, examples of intertextual perpetuation, of course, refer to the story's intertextuality, including, but not limited to, allusions and references to other texts. For the sake of argument, in our study of "El sur," imitative connections with other texts via Genette's notion of hypertextuality–e.g. parody, pastiche, and structural imitation (Palimpsests 7–10)–are included under the terminological umbrella of intertextual perpetuation. Finally, extratextual perpetuation connects with the biographic, geographic, and cultural context of the production and publication of the text, along with–again, to borrow from Genette–paratexts and metatexts, such as interviews, prefaces, post-publication commentary and analysis of the story.3 It goes without saying that many, if not most, texts contain intratextual, intertextual, and extratextual elements, but what qualifies a story as an infinite narrative depends on its intentional elements of narrative perpetuity. By qualifying "El sur" as an infinite narrative, one can fully appreciate its quality of questioning the very nature and scope of storytelling and its relationship with reality. The following article contends that essential clues of narrative perpetuity in the story, such as the role of The Thousand and One Nights, the structural imitation of Zeno's Paradox of the Tortoise and Achilles, and Borges's extratextual influence, confirm "El sur" as an infinite narrative, thus placing it as an exemplar story in his oeuvre. Of all the intratextual and...

Quantum Zeno Dynamics Through Stochastic Protocols

AbstractQuantum measurements play a crucial role in quantum mechanics since they perturb, unavoidably and irreversibly, the state of the measured quantum system. More extremely, the constant observation of a quantum system can even freeze its dynamics to a subspace, effectively truncating the Hilbert space of the system. It represents the quantum version of the famous flying arrow Zeno paradox, and is called quantum Zeno dynamics. In general, it can be obtained by applying frequent consecutive quantum measurements that are equally spaced in time. Here, we introduce time disorder in the measurement sequence, and analytically investigate how this temporal stochasticity may affect the confinement probability of the system in the subspace. As main result, we then exploit how different dissipative and coherent Zeno protocols can be generalized to this stochastic scenario. Finally, our analytical predictions are numerically tested on a paradigmatic spin chain where we find a trade‐off between a probabilistic scheme with high fidelity (compared to perfect subspace dynamics) and a deterministic one with a slightly lower fidelity, moving further steps towards new schemes of Zeno‐based control for future quantum technologies.

Zeno of Elea Shines a New Light on Quantum Weirdness

After a brief reference to the quantum Zeno effect, a quantum Zeno paradox is formulated. Our starting point is the usual version of Time Dependent Perturbation Theory. Although this theory is supposed to account for transitions between stationary states, we are led to conclude that such transitions cannot occur. Paraphrasing Zeno, they are nothing but illusions. Two solutions to the paradox are introduced. The first as a straightforward application of the postulates of Orthodox Quantum Mechanics; the other is derived from a Spontaneous Projection Approach to quantum mechanics previously formulated. Similarities and differences between both solutions are highlighted. A comparison between the two versions of quantum mechanics, supporting their corresponding solutions to the paradox, shines a new light on quantum weirdness. It is shown, in particular, that the solution obtained in the framework of Orthodox Quantum Mechanics is defective.

Zeno's paradox in decision-making.

Classical probability theory has been influential in modelling decision processes, despite empirical findings that have been persistently paradoxical from classical perspectives. For such findings, some researchers have been successfully pursuing decision models based on quantum theory (QT). One unique feature of QT is the collapse postulate, which entails that measurements (or in decision-making, judgements) reset the state to be consistent with the measured outcome. If there is quantum structure in cognition, then there has to be evidence for the collapse postulate. A striking, a prioriprediction, is that opinion change will be slowed down (under idealized conditions frozen) by continuous judgements. In physics, this is the quantum Zeno effect. We demonstrate a quantum Zeno effect in decision-making in humans and so provide evidence that advocates the use of quantum principles in decision theory, at least in some cases.

LIFE AND CONSCIOUSNESS AS QUALITIES

It may be presumed in this work that at the moment of death, everything in the body ( viz. molecules and their motions ) remains identical to what it was at the last moment of life, the difference being that life has something more than the basic underlying physical and chemical elements. It is a totality, which is more like a flow, (i.e. a time derivative, or differential, as in Zeno's paradox of “an arrow in flight”). This is what may be called, in a dialectical sense, a new quality that is ‘life’. A similar picture was presented about consciousness, which is a function of the central nervous system ( not the autonomic ). In both, we assumed the movement of an arrow as being at point at an instant and not there the next instant.The application of science in the hierarchy of life and consciousness is currently one of the most active areas of research in viewing the topic from the bottom‐up; i.e. chemistry >biology>neurology > consciousness. This is as opposed to a top‐down, philosophical and/or phenomenological approach to consciousness.Here we show that both ‘life’ and ‘consciousness’ belong to a category called ‘phenomenon’, both lost instantly and may also represent a new ‘quality’ in the ‘dialectical’ sense. Based on this approach, transformations in nature can be classified into the types: A union of opposites that can generate a new quality, as in the case of an arrow in flight. Quantity to quality; random to organized, viz. molecules to life without consciousness; is reducible. Mechanical to dialectic with transcendental properties; life with consciousness; consciousness from neural inputs; qualitative; is irreducible. Support or Funding Informationnone

Light Velocity and Zeno's Paradox

Light Velocity and Zeno's Paradox

Quantum Zeno effect for a free-moving particle

Although the quantum Zeno effect takes its name from Zeno's arrow paradox, the effect of frequently observing the position of a freely moving particle on its motion has not been analyzed in detail in the frame of standard quantum mechanics. We study the evolution of a moving free particle while monitoring whether it lingers in a given region of space, and explain the dependence of the lingering probability on the frequency of the measurements and the initial momentum of the particle. Stopping the particle entails the emergence of Schr\"odinger cat states during the observed evolution, closely connected to the high-order diffraction modes in Fabry-P\'erot optical resonators.

The Race to Fill the Blanks: On (Animal) Testing in Science Fiction

Abstract In systems of meaning that run on a regular setting, allegory is about filling in the blanks that disclose the “other story.” In the modern setting that Walter Benjamin tracked (back to the seventeenth century), allegory must turn significance out of the blank itself, working the blank as a turning point for drawing the reading onward. The “psy-fi” genre is the hub where bona fide science fictions, documentations of psychosis (memoirs and psychiatric and psychoanalytic studies), and tracts on mass psychology or psychological warfare, meet and cross over toward the evolution of new norms. Is it possible to construe a series of references in works of the “psy-fi” genre to Zeno's paradox of a con-test involving human and animal subjects as allegory of the test situation in which blanks secure the last or new step, which ultimately is taken toward mourning, the final frontier? Yes.

Categorical Semantics for Functional Reactive Programming with Temporal Recursion and Corecursion

Functional reactive programming (FRP) makes it possible to express temporal aspects of computations in a declarative way. Recently we developed two kinds of categorical models of FRP: abstract process categories (APCs) and concrete process categories (CPCs). Furthermore we showed that APCs generalize CPCs. In this paper, we extend APCs with additional structure. This structure models recursion and corecursion operators that are related to time. We show that the resulting categorical models generalize those CPCs that impose an additional constraint on time scales. This constraint boils down to ruling out $\omega$-supertasks, which are closely related to Zeno's paradox of Achilles and the tortoise.

An Attitude of Complexity: Thirteen Essays on the Nature and Construction of Reality Under the Challenge of Zeno's Paradox

This book is about the construction of reality. The central aim of this study is to understand how gravity works and how it may be focused and manipulated. While I do not have an answer to this question, the discoveries along the way have been worth collecting into a single volume for future reference.