Zeno's paradox as it appears in Aristotle's Physics, Book VI, chapter 9. 239b 15-30.

In ‘Towards a Solution to the Sorites Paradox’, Graham Priest gives us a new account of the sorites based on fuzzy logic. The novelty lies in the suggestion that truth-value assignments should themselves be treated as fuzzy objects, i.e., objects about which we can make fuzzy identity statements. I argue that Priest’s solution does not have the explanatory force that Priest advocates. That is, it does not explain why we find the existence of a cut-off point counter-intuitive. I also (...) argue that this sort of explanation calls for a general theory that goes beyond the special case of linguistic vagueness, for the phenomenon is at bottom not linguistic. (shrink)

"Perhaps not all the stories that follow are true. They could, however, be true, and the Reader is invited to ponder this." So begins _Insurmountable Simplicities_, Roberto Casati and Achille Varzi's colorful incarnation of the many philosophical conundrums that hide in the wrinkles of everyday life. Why do mirrors seem to invert left and right but not up and down? How do we know whether strawberries taste the same for everyone? Where is it written that we must observe the law, (...) and if it is not written, why should we observe it? What if we could swap brains-or the rest of our bodies? _Insurmountable Simplicities_ is filled with stories, dialogues, and epistolary exchanges that cover a range of themes-such as personal identity, causality and responsibility, fortune, the nature of things, the paradoxes of time and space, the interface between logic and language-in captivating and inventive ways. Clear, concise, and intellectually engaging, this internationally acclaimed book brilliantly demonstrates that the beauty of philosophy resides in its thorough engagement with the simplicities of the world, insurmountable as they might initially appear. (shrink)

A dialogue among statements that try to explain to each other the mechanisms and peculiarities of self-referential assertions and, particularly, of their context-dependence.

Analyses, in the simplest form assertions that aim to capture an intimate link between two concepts, are viewed since Russell's theory of definite descriptions as analyzing descriptions. Analysis therefore has to obey the laws governing definitions including some form of a Substitutivity Principle (SP). Once (SP) is accepted the road to the paradox of analysis is open. Popular reactions to the paradox involve the fundamental assumption (SV) that sentences differing only in containing an analysandum resp. an analysans express (...) the same proposition, because analysandum and analysans are the same entity. Following suggestions of Gupta and Belnap it is argued that (SV) should be rejected. (shrink)

Analyses, in the simplest form assertions that aim to capture an intimate link between two concepts, are viewed since Russell's theory of definite descriptions as analyzing descriptions. Analysis therefore has to obey the laws governing definitions including some form of a Substitutivity Principle (SP). Once (SP) is accepted the road to the paradox of analysis is open. Popular reactions to the paradox involve the fundamental assumption (SV) that sentences differing only in containing an analysandum resp. an analysans express (...) the same proposition, because analysandum and analysans are the same entity. Following suggestions of Gupta and Belnap it is argued that (SV) should be rejected. (shrink)

A philosophical dialogue on the functioning, the limits, and the paradoxes of our electoral practices, dealing with such basic questions as: What is a vote? How do we count votes? And do votes really count?

This issue of The Monist is devoted to the metaphysics of lesser kinds, which is to say those kinds of entity that are not generally recognized as occupying a prominent position in the categorial structure of the world. Why bother? We offer two sorts of reason. The first is methodological. In mathematics, it is common practice to study certain functions (for instance) by considering limit cases: What if x = 0? What if x is larger than any assigned value? Physics, (...) too, often studies the (idealized) initial and boundary conditions of a given system: What would happen in the case of a perfect sphere, or a perfectly black body? In the cognitive sciences, research often thrives on the analysis of cognitive errors, perceptual illusions, brain pathologies. Also in logic one can learn a lot by studying special, anomalous scenarios such as those exhibited by the paradoxes: it is unlikely that we actually find ourselves in a soritical context, or in a liar-like situation, but the fact that we might—or simply the fact that we can conceive of such a possibility—is important enough to deserve careful consideration. In short, the odd, the unfamiliar, the extra-ordinary, the limit cases are perfectly at home in scientific and more broadly intellectual discourse at various levels, where they can be fruitfully engaged in a sophisticated way (witness the existence of specific confining and managing strategies for dealing with them); and they are important precisely because they instruct us concerning the normal, the obvious, and the paradigmatic. The same goes for metaphysics, we submit. Although its major concern is, naturally, with such core entities as substances, properties, or hunks of solid matter, a lot may be learned by paying attention to those limit cases where we find ourselves dealing with entities of much lesser kinds, whether real or putative. (shrink)

A brief study of Saul Steinberg’s works on shadows and reflections, and of the seemingly paradoxical world that emerges from such works.

This original and enticing book provides a fresh, unifying perspective on many old and new logico-philosophical conundrums. Its basic thesis is that many concepts central in ordinary and philosophical discourse are inherently circular and thus cannot be fully understood as long as one remains within the confines of a standard theory of definitions. As an alternative, the authors develop a revision theory of definitions, which allows definitions to be circular without this giving rise to contradiction (but, at worst, to “vacuous” (...) uses of definienda). The theory is applied with varying levels of detail to a circular analysis of concepts as diverse as truth, predication, necessity, physical object, etc. The focus is on truth, and hope is expressed that a deeper understanding of the Liar and related paradoxes has been provided: “We have tried to show that once the circularity of truth is recognized, a great deal of its behavior begins to make sense. In particular, from this viewpoint, the existence of the paradoxes seems as natural as the existence of the eclipses” (p. 142). We think that this hope is fully justified, although some problems remain that future research in this field should take into account. (shrink)

In this regular series Michael Clark, editor of the journal Analysis, presents some of the most intriguing philosophical paradoxes. Here we examine the paradox of Achilles and the tortoise.

In this paper, I present a discrete solution for the paradox of Achilles and the tortoise. I argue that Achilles overtakes the tortoise after a finite number of steps of Zeno’s argument if time is represented as discrete. I then answer two objections that could be made against this solution. First, I argue that the discrete solution is not an ad hoc solution. It is embedded in a discrete formulation of classical mechanics. Second, I show that the (...) discrete solution cannot be falsified experimentally. (shrink)

Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal (...) structure with other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll’s paradox) and that of the arrow (for “Achilles and the Turtle”), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure of both, which can be called “ontological”, on which basis “motion” studied by physics and “conclusion” studied by logic can be unified being able to bridge logic and physics philosophically in a Hegelian manner: even more, the bridge can be continued to mathematics in virtue of (2), which forces the equality (for its property of transitivity) of any two quantities to be postponed analogically ad lib and ad infinitum. The paper shows that Hilbert arithmetic underlies naturally Lewis Carroll’s paradox admitting at least three interpretations linked to each other by it: mathematical, physical and logical. Thus, it can be considered as both generalization and solution of his paradox therefore naturally unifying the completeness of quantum mechanics (i.e. the absence of hidden variables) and eventual completeness of mathematics as the same and isomorphic to the completeness of propositional logic in relation to set theory as a first-order logic (in the sense of Gödel (1930)’s completeness theorems). (shrink)

It is widely held that the paradox of Achilles and the Tortoise, introduced by Zeno of Elea around 460 B.C., was solved by mathematical advances in the nineteenth century. The techniques of Weierstrass, Dedekind and Cantor made it clear, according to this view, that Achilles’ difficulty in traversing an infinite number of intervals while trying to catch up with the tortoise does not involve a contradiction, let alone a logical absurdity. Yet ever since the nineteenth century there (...) have been dissidents claiming that the apparatus of Weierstrass et al. has not resolved the paradox, and that serious problems remain. It seems that these claims have received unexpected support from recent developments in mathematical physics. This support has however remained largely unnoticed by historians of philosophy, presumably because the relevant debates are cast in mathematical-technical terms that are only accessible to people with the relevant training. That is unfortunate, since the debates in question might well profit from input by philosophers in general and historians of philosophy in particular. Below we will first recall the Achilles paradox, and describe the way in which nineteenth century mathematics supposedly solved it. Then we discuss recent work that contests this solution, reiterating the dissident dogma that no mathematical approach whatsoever can even come close to solving the original Achilles. We shall argue that this dissatisfaction with a mathematical solution is inadequate as it stands, but that it can perhaps be reformulated in the light of new developments in mathematical physics. (shrink)

Mate Meršić (Merchich, 1850-1928) sees the origin of Zeno’s paradox ‘Achilles’ in the ambiguities of the concept of the infinity. According to him (and to the tradition started by Gregory St. Vincent), those ambiguities are resolved by the concept of convergent geometric series. In this connection, Meršić proposes a general ontological theory with the priority of the finite over the infinite, and, proceeding from Newton’s concept of fluxion, he develops a modal interpretation of differential calculus.

_Paradoxes from A to Z, Third edition_ is the essential guide to paradoxes, and takes the reader on a lively tour of puzzles that have taxed thinkers from Zeno to Galileo, and Lewis Carroll to Bertrand Russell. Michael Clark uncovers an array of conundrums, such as Achilles and the Tortoise, Theseus’ Ship, and the Prisoner’s Dilemma, taking in subjects as diverse as knowledge, science, art and politics. Clark discusses each paradox in non-technical terms, considering its significance and looking (...) at likely solutions. This third edition is revised throughout, and adds nine new paradoxes that have important bearings in areas such as law, logic, ethics and probability. Paradoxes from A to Z, Third edition is an ideal starting point for those interested not just in philosophical puzzles and conundrums, but anyone seeking to hone their thinking skills. (shrink)

The five participants in this dialogue critically discuss Zeno of Elea's paradox of Achilles and the tortoise. They consider a number of solutions to and restatements of the paradox, together with their philosophical implications. Among the issues investigated include the appearance-reality distinction, Aristotle's distinction between actual and potential infinity, the concept of a continuum, Cantor's continuum hypothesis and theory of transfinite ordinals, and, as a solution to Zeno's puzzle, the distinction between infinite and indeterminate or inexhaustible divisibility.

Zeno of Elea's paradoxes of motion are one of the most successful provocations in the history of philosophy. There are exactly four paradoxes, namely, the dichotomy, the arrow, Achilles, and the moving rows. This chapter presents the paradoxes in such a way that their strength, fascination, and profoundness are apparent. After providing some basic information about Zeno, the chapter sketches the research program that is the context of Zeno's paradoxes. It goes back to Parmenides and may be called Parmenideanism. (...) The dichotomy and the Achilles paradox are rather two variants of one thought than two different arguments. The moving rows are complicated and not much can be learned from them. Their presentation in the chapter is restricted to some rough outline after the discussion of reactions to the dichotomy and the Achilles. (shrink)

This essential guide to paradoxes takes the reader on a lively tour of puzzles that have taxed thinkers from Zeno to Galileo and Lewis Carroll to Bertrand Russell. Michael Clark uncovers an array of conundrums, such as Achilles and the Tortoise, Theseus' Ship, Hempel's Raven, and the Prisoners' Dilemma, taking in subjects as diverse as knowledge, ethics, science, art and politics. Clark discusses each paradox in non-technical terms, considering its significance and looking at likely solutions.

Steven Higgins and Vivienne Baumfield have recently attempted to defend the much discussed idea of general thinking skills against attacks from three quarters: what they regard as a priori objections, which they liken to Zeno's paradox that Achilles will not catch the tortoise; domains theories of knowledge, which oppose the idea of thinking skills being general and transcending domains; and the claim that experts use subject specific knowledge, and don't use general thinking skills. We examine these defences and (...) find them flawed and worrying. We conclude that this is a domain in serious need of a priori approaches. (shrink)

Steven Higgins and Vivienne Baumfield have recently attempted to defend the much discussed idea of general thinking skills against attacks from three quarters: what they regard as a priori objections, which they liken to Zeno's paradox that Achilles will not catch the tortoise; domains theories of knowledge, which oppose the idea of thinking skills being general and transcending domains; and the claim that experts use subject specific knowledge, and don't use general thinking skills. We examine these defences and (...) find them flawed and worrying. We conclude that this is a domain in serious need of a priori approaches. (shrink)

The paper presents a new paradox in the sphere of the classical mechanics of infinite systems. The paradox, which implies the existence of an interaction at a distance as well as gravitation, would seem to require an extension of the concept of mechanical explanation. However, it is not clear how such an extension might be carried out.

The paper presents a new paradox in the sphere of the classical mechanics of infinite systems. The paradox, which implies the existence of an interaction at a distance as well as gravitation, would seem to require an extension of the concept of mechanical explanation. However, it is not clear how such an extension might be carried out.

In this paper, we connected Zeno’s paradoxes and motions with the viscous friction force \. For the progressive version of the dichotomy paradox, if the body speed is constant, the sequences of positions and instants are infinite, but the series of distances and time variations converge to finite values. However, when the body moves with force \, the series of time variations becomes infinite. In this case, the body crosses infinite points, approximating to a final position forever, as the (...) progressive version of the dichotomy paradox describes. The same procedures with constant speed and motion with force \ result in different spatial sequences and the same series for the regressive version of the dichotomy paradox. Nonetheless, the regressive version of the dichotomy paradox does not describe a body that approximates a final position forever. Finally, for the Achilles paradox, we find the positional and temporal sequences of the man and the tortoise at constant speeds. Analogously to the progressive version of the dichotomy paradox, the positional and temporal sequences are infinite, but the spatial and temporal series converge to finite values. If Achilles and the tortoise are identical points that move with force \, and the turtle’s position in function of time presents a temporal advantage, the man and the animal will cross successive infinite places forever, and the quick hero will never overtake the slow reptile, as the paradox described. We conclude that the progressive version of the dichotomy and Achilles paradoxes can describe motions when the force is \. (shrink)

Zeno's paradoxes of motion and the semantic paradoxes of the Liar have long been thought to have metaphorical affinities. There are, in fact, isomorphisms between variations of Zeno's paradoxes and variations of the Liar paradox in infinite-valued logic. Representing these paradoxes in dynamical systems theory reveals fractal images and provides other geometric ways of visualizing and conceptualizing the paradoxes.

This double issue of the Carrollian, the journal of the Lewis Carroll Society, is entirely devoted to Lewis Carroll's famous short paper published in the journal Mind in 1895 under the title ‘What...

An example of the second situation is the most famous of the paradoxes of Zeno, the Greek philosopher who lived during the Golden Age of Greece on the island of Elea. Zeno proposed the following "thought experiment". Achilles, a young athlete, runs a race with a tortoise. Achilles can run exactly twice as fast as the tortoise, so to make it fair he gives the tortoise a head start of exactly half the distance from the starting line to (...) the finish line. The starting signal is given and the race begins. Achilles runs to the starting position of the tortoise. In the time it takes to do that, the tortoise has advanced half the distance from his starting position and the finish line. Achilles then advances to the new position of the tortoise. During that time the tortoise again advances half the distance to the finish line. And so on ... Every time Achilles moves ahead by a given distance, the tortoise moves ahead by half that distance. Zeno concluded that Achilles can never catch the tortoise, because in every time interval in which Achilles moves to the tortoise's former position, the tortoise always moves ahead by half that distance. (shrink)

The author argues that, Although zeno's paradoxes on motion cannot be resolved in their own terms, They are nonetheless illegitimate. Examining the paradox of achilles and the tortoise, He finds that the mechanism of zeno's argument consists in an equivocal concept of motion characterized at once by a constant rate and by proportionate segments of movement. He then contends it is illegitimate to treat the concept of motion and its subconcepts like the postulates of a deductive system. However, (...) That the common theory of meaning presupposed by zeno's paradoxes is erroneous would have to be demonstrated metaphysically. (staff). (shrink)

This paper discusses a novel response to two closely related regress arguments from Bolzano’s Theory of Science and Carroll’s What the Tortoise Said to Achilles. Bolzano’s argument aims to refute the thesis that full grounds must include propositions involving notions such as entailment, grounding or lawhood which link the respective grounds to their groundee. This thesis is motivated, Bolzano’s argument is reconstructed, and a response based on self-referential linking propositions is developed and defended against objections concerning self-reference and Curry’s (...) paradox. Finally, the idea is applied to a reading of Carroll’s dialogue and a corresponding solution to the so-called infinite regress problem of inference is proposed. (shrink)

In the article three Zeno's paradoxes are reconstructed. They are: „Achilles and the turtle”, „Arrow” and „Stadium”. Together with the paradox of „Dichotomy” (which was analysed by the author elsewhere) they form the question about the nature of continuum. In the paper the following hypothesis is accepted: „Dichotomy” is principally connected with the mathematical theory of continuum, whereas other paradoxes concern the application of this theory to the description of physical motion.

My aim in this paper is to suggest a new outlook concerning the nature of Zeno’s paradoxes. The attention is directed towards the three famous paradoxes known as “Dichotomy,” “Achilles and the Tortoise,” and “The Arrow.” An analysis of the paradigmatic proposals for a solution shows that an adequate solution has not yet been reached. An answer is provided instead to the question “How Zeno’s paradoxes emerge in their quality of aporiae?,” that is to say in their quality of (...) impasses, of problem situations without an exit, what is the original meaning of the Greek word “aporia.” It is my claim that this is the correct rational approach for solving these conceptual puzzles. In other words, I am not proposing formal solutions by criticizing and/or altering their premises assuming the continuous or discrete nature of space and time, but I try to draw the philosophical attention to the way we possess the phenomenon of motion as a result of the perception of space-time in human experience. (shrink)

This article criticises one of Stuart Rachels' and Larry Temkin's arguments against the transitivity of 'better than'. This argument invokes our intuitions about our preferences of different bundles of pleasurable or painful experiences of varying intensity and duration, which, it is argued, will typically be intransitive. This article defends the transitivity of 'better than' by showing that Rachels and Temkin are mistaken to suppose that preferences satisfying their assumptions must be intransitive. It makes cler where the argument goes wrong by (...) showing that it is a version of Zeno's paradox of Achilles and the Tortoise. (shrink)

In an earlier paper the authors discussed some super-tasks by means of a kinematical interpretation. In the present paper we show a semi-formal way that a more abstract treatment is possible. The core idea of our approach is simple: if a super-task can be considered as a union of (finite) tasks, it is natural to define the effect of the super-task as the union of the effects of the finite tasks it consists of. We show that this approach enables us (...) to handle two of the three super-tasks that we discussed earlier. We also argue that recent objections against our original kinematical interpretation do not hold water. One of our arguments is based on the construction of an elegant correspondence between the first of those three super-tasks and Zeno's Achilles and the Tortois. (shrink)

It has been argued that we find zeno's paradoxes of motion persuasive because physical time is dense and continuous, While time as we experience it is discrete. But we do not experience time as a succession of distinct, Countable, Consecutively ordered mental "nows." nor is it common to attempt the futile mental task of traversing in thought the infinite number of spatial subintervals in zeno's paradoxes, As has also been suggested. Rather, We find the paradoxes persuasive because there are a (...) number of different conceptual and linguistic problems inherent in them. For instance, Understanding how it is possible for achilles to come to the end of an endless sequence of spatial intervals requires disambiguating different senses of the word "end.". (shrink)

This article is an analysis of the logical structure of zeno's arguments without mathematical or metaphysical diversions. Topics of discussion include achilles and the stadium, Achilles and the tortoise, Compositeness of objects, The flying arrow.

Offering clear and reliable guidance to the ideas of philosophers from antiquity to the present day and to the major philosophical systems around the globe, he Oxford Companion to Philosophy is the definitive philosophical reference work for readers at all levels. For ten years the original volume has served as a stimulating introduction for general readers and as an indispensable guide for students and scholars. A distinguished international assembly of 249 philosophers contributed almost 2,000 entries, and many of these have (...) now been considerably revised and updated in this major new edition; to these are added over 300 brand-new pieces on a fascinating range of current topics such as animal consciousness, cloning, corporate responsibility, the family, globalization, terrorism . Here is, indeed, a world of thought, with entries on idealism and empiricism, epicureanism and stoicism, passion and emotion, deism and pantheism. The contributors represent a veritable who's who of modern philosophy, including such eminent figures as Isaiah Berlin, Sissela Bok, Ronald Dworkin, John Searle, Michael Walzer, and W. V. Quine. We meet the great thinkers--from Aristotle and Plato, to Augustine and Aquinas, to Descartes and Kant, to Nietzsche and Schopenhauer, right up to contemporary thinkers such as Richard Rorty, Jacques Derrida, Luce Iragaray, and Noam Chomsky. There are short entries on key concepts such as personal identity and the mind-body problem, major doctrines from utilitarianism to Marxism, schools of thought such as the Heidelberg School or the Vienna Circle, and contentious public issues such as abortion, capital punishment, and welfare. In addition, the book offers short explanations of philosophical terms (qualia, supervenience, iff), puzzles (the Achilles paradox, the prisoner's dilemma), and curiosities (the philosopher's stone, slime). Almost every entry is accompanied by suggestions for further reading, and the book includes both a chronological chart of the history of philosophy and a gallery of portraits of eighty eminent philosophers. An indispensable guide and a constant source of stimulation and enlightenment, The Oxford Companion to Philosophy will appeal to everyone interested in abstract thought, the eternal questions, and the foundations of human understanding. (shrink)

Despite changeability of the world, the human mind also ponders on those quantities that remain constant over time. This was the case in ancient times, in the middle ages, and the same applies in modern physics. This paper discusses i.a. Zenon paradoxes, the principle of inertia, and the Emma Noether theorem, ending with the modern, so-called Zeno’s quantum effect. The foot-notes concern the ancient “Achilles” paradox, spot speed, as well as some of the facts taken out of the (...) life-history of Emma Noether, as well as the example of applying her theorem. (shrink)

Despite changeability of the world, the human mind also ponders on those quantities that remain constant over time. This was the case in ancient times, in the middle ages, and the same applies in modern physics. This paper discusses i.a. Zenon paradoxes, the principle of inertia, and the Emma Noether theorem, ending with the modern, so-called Zeno’s quantum effect. The foot-notes concern the ancient “Achilles” paradox, spot speed, as well as some of the facts taken out of the (...) life-history of Emma Noether, as well as the example of applying her theorem. (shrink)

This paper shows an analysis of some found solutions for the famous aporia of the race between Achilles and the Tortoise. As an introduction, we present the mechanical solution, to establish that it is not in the field of matters of fact where you can resolve a purely rational problem like the one raised by Zeno of Elea. And so, the main part of the article is dedicated to the mathematical solutions, which face the problem under the point of (...) view of the single, mathematical reason. There are two mathematical solutions for Zeno’s paradox. First, we attend to that which we denominate “classical” (because it is the most habitually used by mathematicians), which is based on the calculus of addiction of terms of a geometrical series in progressive decrease. The second considered solution is the one that was proposed by Russell, based upon the theory of the transfinite numbers. The analysis we have made gets us to discover that no one of those solutions can save itself from falling into logical contradictions, therefore it seems that Zeno’s problem is an authentic aporia, which, after so much time, continues challenging human intelligence. As the article’s conclusion, we suggest that the cause of the impossibility of solving Zeno’s problem is the very notion of mathematical continuum, because this notion infringes the logic of facts. (shrink)

This book is an exploration of philosophical questions about infinity. Graham Oppy examines how the infinite lurks everywhere, both in science and in our ordinary thoughts about the world. He also analyses the many puzzles and paradoxes that follow in the train of the infinite. Even simple notions, such as counting, adding and maximising present serious difficulties. Other topics examined include the nature of space and time, infinities in physical science, infinities in theories of probability and decision, the nature of (...) part/whole relations, mathematical theories of the infinite, and infinite regression and principles of sufficient reason. (shrink)