Two of the greatest interpreters of Parmenides, Giovanni Casertano and Franco Ferrari, have given opposite interpretations of the role of the character of Parmenides. For Ferrari, Parmenides would only be a critic of ideas, as he equates them with their sensitive participants (thus, he could not be considered one of the prosopa of Plato). For Casertano, on the other hand, Parmenides expresses the ‘metaphysical’ aspects of ideas in accordance with the young Socrates’ discourse on the “prodigy” in the initial part (...) of the dialogue. Neither of the two interpretations presents a Parmenides that represents the figure of Plato adequate for the “dramatic” development of the dialogue. If, on the other hand, we regard them both as integral parts, the image and role of Parmenides could be more appropriate to the bipolar structure of the dialogue and of Plato himself. To this end, I have assumed that Plato unifies in Parmenides his dialectical-deductive method with that of Zeno: the aporetic-paradoxical one that is more open than his to finding ways of escape from any paradoxical arguments. It would be for this reason that Parmenides encourages the young and inexperienced Socrates to follow Zeno’s tropos when defining virtue, also because it was Socrates himself, in the final part of his discourse on the “prodigy,” who praised Zeno for the double courage shown first in criticizing the common sense and the principles of current physics and then in overcoming the contrast between these principles and the more specific mathematizing logic. I identify in this praise of Socrates an assumption, on the part of Parmenides, that the burden of bringing back into ideas the courageous behaviour that Zeno showed in the world of the sensible. I agree with Vlastos in considering Parmenides to be “the manifesto of Plato’s self-criticism,” and in viewing this Parmenides as the prosopon of a Plato that is willing both to level self-criticism at his own stance and to give voice to those who do not think like him and even to those who reject his position. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Beyond Ratzinger's Republic:Communio's Postliberal TurnSam Zeno Conedera S.J. and Vincent L. Strand S.J.Is the political future of the West a postliberal one? For the past decade, numerous prominent thinkers in America and Europe have been debating this question. Matters that not long ago were merely of historical interest, such as Pope Gelasius I's understanding of the relation between sacral authority and royal power, Thomas Aquinas's thought on monarchy and (...) Robert Bellarmine's theory of the pope's indirect authority in temporal matters are now increasingly studied for their potential answers to contemporary political questions. This development, which would have been nearly unimaginable just a decade earlier, unsurprisingly has been led by Catholics. We say "unsurprisingly" in view of the long pre-modern tradition of Catholic political thought, as well as the Church's long struggle against liberalism during the nineteenth and early twentieth centuries. Some of the current postliberal writers explicitly connect themselves to both of these aspects by using the term "integralist" to describe their position.1 [End Page 889]This "new integralism" represents a radical departure from the mainstream of Catholic political discourse of the last half century, which has largely sought rapprochement with Western liberalism. A group of writers, led by John Courtney Murray, George Weigel, Michael Novak, and Robert George, have held that the American constitutional order is rooted in the natural law tradition, and therefore is basically friendly to Catholic political thought. The Church's role in American life, they say, is upholding the natural law as a public moral philosophy, while embracing the non-establishment clause of the First Amendment as a matter of principle, in accordance with the teachings of the Second Vatican Council. This "Whig Thomism," which for decades represented the predominant outlook of conservative American Catholics, must now face the challenge of a different sort of Thomism represented by the integralists.2While the engagement between Whig Thomists and integralists has received most of the attention from scholars and journalists alike, another postliberal option has emerged in recent years. Its source is a group of Catholic thinkers associated with the Communio theological movement. Their political vision, which has no single name or label, has a complex and subtle relationship with the thought of earlier Communio writers, most of whom shared the Whig Thomist desire for reconciliation with Western liberalism. [End Page 890] The appearance of a Communio postliberalism therefore represents a significant development within this intellectual tradition. The most important representative of the earlier Communio perspective on political questions is Joseph Ratzinger / Pope Benedict XVI, who has recently been lauded as the "most important Catholic thinker of the twentieth century" on matters at the intersection of theology and politics.3In an article titled "Ratzinger's Republic" that appeared in these pages in 2020, we identified a significant tension in his thought about the Church's contribution to political life.4 Exploring Ratzinger's writings on natural law, we demonstrated that two currents are present. In accord with his fundamental theological commitments about nature and grace and about faith and reason, Ratzinger often questioned the effectiveness of arguments from natural law. He claimed that, since reason is historically conditioned, and therefore in the current order marked by sin, reason needs grace and faith to function properly. Yet, on other occasions, Ratzinger strongly endorsed arguments from natural law, highlighting its role as the foundation of a just democratic political order. We traced this ambivalence about natural law back to a dilemma at the heart of Ratzinger's political thought. On the one hand, in accord with his thinking about the relations of nature and grace and faith and reason, Ratzinger argued that the state needs the Church to supply moral values that the state cannot furnish on its own. On the other hand, he believed it belongs to the very nature of the Church to be separate from the state, and he endorsed nonconfessional, religiously pluralistic states.5 In light of these two commitments, Ratzinger argued that the contribution Christians are to make to the state is to supply moral values that were discerned under the influence of grace and divine revelation but... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:NEWMAN'S PSYCHOLOGICAL DISCOVERY: THE ILLATIVE SENSE (V. Continued) 15. The Universals. A long and vehement dispute once raged about the reality of universals. Are they only mental creations, forged by the human brain, without any reality outside them, or have they some independent existence apart from their mental reality? Anyhow, there was an apparent contradiction between die universal character of our ideas and the individual character of concrete things. (...) The Scholastics solved this problem by drawing clear distinctions. An idea is an abstraction; as soon as it becomes applicable to a number of objects or things in the same sense, it is a universal idea. The "direct universal" (universale directum seu metaphysicum) as such has no existence apart from our minds, because an existing thing is always bound up with concrete, determining, individualizing notes, and it is these very circumstances or accidents which are ignored by the mind as it strips the thing of its individual characteristics. When the mind reflects on this "direct universal," i.e., on the abstract essence of the thing, it sees that this is to be found in many more things and that it is identical in all those things, and in this way the "reflex or logical universal" is formed (universale reflexum seu logicum). This same essence or quiddity is not really different from the thing itself, but the mind makes abstraction of all accidents, and this foundation makes it possible to apply this idea in die same sense to all other beings of the same kind, whether they are actually existing or not. In other words the universal as a universal has only reality in the mind but it has a foundation or a ground in the thing itself. Newman was not so well acquainted with Scholasticism that he knew and mastered the solution of the problem. As a matter of fact he approached it from quite another angle. Hence when scholastically trained minds read some of his statements on universals, they may feel shocked. They sound very curious, they seem incon418 DR. ZENO, O.F.M. CAP.419 sistent with each other, they apparenüy imply a negation of the objective reality of universals. This makes it necessary to compare the texts and to explain his meaning. a. Newman Inconsistent. We should never lose sight of the fact that Newman had no intention at all to enter the province of philosophy in the restricted sense of the word. "I am only contemplating the mind as it moves in fact, by whatever hidden mechanism,"1 that is what could be put as a heading to every page of the Grammar of Assent. And when he virtually deals with metaphysics, he wants us to consider his statements as hypotheses, or as propositions stated only by the way, and in this department he always wishes "to speak under correction." This explains how he could refer to the universale metaphy sicum and the universale directum in a very correct way without realizing he was teaching Scholastic doctrine. As far as I know, it is Professor Walgrave who has first drawn attention to this fact,2 when he dealt with Newman's alleged conceptualism. Newman states that "even one act of cruelty, ingratitude, generosity or justice reveals to us at once intensive the immutable distinction between those qualities and their contraries; that is, in that particular instance and pro bac vice."3 The same holds for the opposites: right and wrong, true and false, just and unjust, beautiful and deformed.4 In one experience, therefore, he sees a thing as an immutable, absolute quiddity, which objectively and in the same sense belongs to all possible things of the same kind, although the mind does not yet explicidy confirm this universal applicability. The contemplated quality remains limited to the particular case; but in its absoluteness it is stripped of all individual, changeable and concrete characteristics. This is the metaphysical universal of the Scholastics. But Newman also mentions the logical universal when he says: "As we form our notion of whiteness from the actual sight of snow, a lily or a cloud, so, after experiencing the sentiment of approbation which arises in us on the sight of certain... (shrink)

Zeno’s paradoxes of motion, allegedly denying motion, have been conceived to reinforce the Parmenidean vision of an immutable world. The aim of this article is to demonstrate that these famous logical paradoxes should be seen instead as paradoxes of immobility. From this new point of view, motion is therefore no longer logically problematic, while immobility is. This is convenient since it is easy to conceive that immobility can actually conceal motion, and thus the proposition “immobility is mere illusion of (...) the senses” is much more credible than the reverse thesis supported by Parmenides. Moreover, this proposition is also supported by modern depiction of material bodies: the existence of a ceaseless random motion of atoms—the ‘thermal agitation’—in the scope of contemporary atomic theory, can offer a rational explanation of this ‘illusion of immobility’. Our new approach to Zeno’s paradoxes therefore leads to presenting the novel concept of ‘impermobility’, which we think is a more adequate description of physical reality. (shrink)

The influence of St. Ignatius of Loyola's Spiritual Exercises on Descartes’ work, including the Meditations, has been recognized and discussed by many historians. I just mention a few fairly recent and easily accessible instances. In The Metaphysics of Descartes, J. L. Beck suggests that the literary form of the Meditations is most likely due to the Ignatian meditations to which Descartes had been exposed during his training at the Jesuit college of LaFlèche. Arthur Thomson in ‘Ignace de Loyola et Descartes’ (...) traces some elements in Descartes’ method and psychology to Ignatian sources, mainly focusing on the Discourse. (shrink)

This work presents an operational and geometric approach to logic. It starts from the multilinear elective decomposition of binary logical functions in the original form introduced by George Boole. A justification on historical grounds is presented bridging Boole’s theory and the use of his arithmetical logical functions with the axioms of Boolean algebra using sets and quantum logic. It is shown that this algebraic polynomial formulation can be naturally extended to operators in finite vector spaces. Logical operators will appear as (...) commuting projection operators and the truth values, which take the binary values \, are the respective eigenvalues. In this view the solution of a logical proposition resulting from the operation on a combination of arguments will appear as a selection where the outcome can only be one of the eigenvalues. In this way propositional logic can be formalized in linear algebra by using elective developments which correspond here to combinations of tensored elementary projection operators. The original and principal motivation of this work is for applications in the new field of quantum information, differences are outlined with more traditional quantum logic approaches. (shrink)

Almost everything that we know about Zeno of Elea is to be found in the opening pages of Plato's Parmenides. There we learn that Zeno was nearly 40 years old when Socrates was a young man, say 20. Since Socrates was born in 469 BC we can estimate a birth date for Zeno around 490 BC. Beyond this, really all we know is that he was close to Parmenides (Plato reports the gossip that they were lovers when Zeno was young), (...) and that he wrote a book of paradoxes defending Parmenides' philosophy. Sadly this book has not survived, and what we know of his arguments is second-hand, principally through Aristotle and his commentators (here I have drawn particularly on Simplicius, who, though writing a thousand years after Zeno, apparently possessed at least some of his book). There were apparently 40 ‘paradoxes of plurality’, attempting to show that ontological pluralism — a belief in the existence of many things rather than only one — leads to absurd conclusions; of these paradoxes only two definitely survive, though a third argument can probably be attributed to Zeno. Aristotle speaks of a further four arguments against motion (and by extension change generally), all of which he gives and attempts to refute. In addition Aristotle attributes two other paradoxes to Zeno. Sadly again, almost none of these paradoxes are quoted in Zeno's original words by their various commentators, but in paraphrase. (shrink)

Randomization is a generally accepted principle of sound experimental design and common practice among working scientists. But Bayesian statisticians reject it, most often because of decision theoretic argument against randomization. I trace it back to Abraham Wald's Theory of Inductive Behavior and argue that Bayesians should concur with Ronald Fisher 's criticism of Wald's analysis of randomization. The paper ends with a Bayesian argument in favor of randomization: randomization can lead to an increase in expected utility.

Even in a theory corroboration context, attention to effect size is called for if significance testing is to be of any value. I sketch a Popperian construal of significance tests that better fits into scientific inference as a whole. Because of its many errors Chow's book cannot be recommended to the novice.

After a brief outline of the book, Criticism is offered on the following points. 1) whether or not the verb 'say' is a general performative? 2) can belief-Objects be represented adverbially? 3) does ziff succeed in refuting grice's theory of meaning? 4) ziff's omission of the illocutionary domain.

Contemporary readers of Descartes can hardly fail to notice that the author uses the word pensée, or cogitatio, in a much broader sense than French speakers use pensée nowadays, or we use the closest English word, thought. He consistently maintains that feelings, sensations, as well as the products of one's fancy, are one and all modes of thought.Yet, clearly, according to the normal use of the word, sensations of light, of sound, of hunger, and so forth, are not regarded as (...) part of one's thinking, nor do we so regard the spontaneous flight of the imagination one might experience in daydreams or real dreams, or while thinking about unrelated matters. Some of these sensations, notably aches, pains, pangs of hunger, blinding light, and strong noise, are not only not counted among our thoughts, but they are apt to interfere with our thinking, and, in extreme cases, might stop it altogether. In a similar way, the lascivious play of St. Antony's imagination did not embellish his meditations on the holy mysteries; he had to overcome or ignore it to be able to pursue the train of his thought. (shrink)

If a speaker says something in a language and one of the listeners knows L but another does not, then, normally, A will understand what S said but B will not. What is it, exactly, that A , but not B , succeeds in doing in this case, and how to account for the difference? This is a fundamental problem, which the philosophy of language should be able to solve, yet, to my knowledge, has not done so to date.

Zeno’s Paradoxes In the fifth century B.C.E., Zeno offered arguments that led to conclusions contradicting what we all know from our physical experience—that runners run, that arrows fly, and that there are many different things in the world. The arguments were paradoxes for the ancient Greek philosophers. Because many of the arguments turn crucially on … Continue reading Zeno’s Paradoxes →.

I offer a novel solution to Zeno’s paradox of The Arrow by introducing nilpotent infinitesimal lengths of time. Nilpotents are nonzero numbers that yield zero when multiplied by themselves a certain number of times. Zeno’s Arrow goes like this: during the present, a flying arrow is moving in virtue of its being in flight. However, if the present is a single point in time, then the arrow is frozen in place during that time. Therefore, the arrow is both (...) moving and at rest. In “Zeno’s Arrow, Divisible Infinitesimals, and Chrysippus,” White suggests using an infinitesimal value as the length of the present. Contra Zeno, this allows the arrow to be moving in the present, rather than frozen in place. In this paper, I follow the basic outline of White’s solution but argue that his solution suffers from arbitrariness and a related theoretical artificiality in relation to the system of infinitesimals he invokes, viz. in relation to the hyperreal infinitesimals of nonstandard analysis. After arguing that any solution to the paradox must satisfy certain theoretical requirements, I examine White’s solution alongside two nilpotent solutions. One of these solutions is inspired by F.W. Lawvere’s Smooth Infinitesimal Analysis and the other is inspired by Paolo Giordano’s ring of Fermat Reals. I argue that Giordano’s nilpotents supply the best answer to Zeno’s paradox. (shrink)

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)

ABNER SHIMONY of the Paradox A PHILOSOPHICAL PUPPET PLAY Dramatis personae: Zeno , Pupil, Lion Scene: The school of Zeno at Elea. Pup. Master! ...

A version of nonstandard analysis, Internal Set Theory, has been used to provide a resolution of Zeno's paradoxes of motion. This resolution is inadequate because the application of Internal Set Theory to the paradoxes requires a model of the world that is not in accordance with either experience or intuition. A model of standard mathematics in which the ordinary real numbers are defined in terms of rational intervals does provide a formalism for understanding the paradoxes. This model suggests that in (...) discussing motion, only intervals, rather than instants, of time are meaningful. The approach presented here reconciles resolutions of the paradoxes based on considering a finite number of acts with those based on analysis of the full infinite set Zeno seems to require. The paper concludes with a brief discussion of the classical and quantum mechanics of performing an infinite number of acts in a finite time. (shrink)

Phenomenological psychopathology is a research field that aims to investigate and describe the subjective experience of mental disorders. By suspending the assumptions about etiology and causality as much as possible, and by focusing on the subjective experiences of the patient, it is supposed to offer a profound understanding of the patient’s suffering, and of the disorder in general. Clarity in the description of these experiences is, therefore, a necessity. Traditionally, phenomenological psychopathology was studied mostly by European, and particularly by German (...) and French psychiatrists, such as Karl Jaspers, Wolfgang Blankenburg, and Eugène Minkowski. Today, psychopathologists like Josef Parnas, Giovanni... (shrink)

Zeno’s arguments are generally regarded as ingenious but downright unsound paradoxes, worth of attention mainly to disclose why they go wrong or, alternatively, to recognise them as clever, even if crude, anticipations of modern views on the space, the infinite or the quantum view of matter. In either case, the arguments lose any connection with the scientific and philosophical problems of Zeno’s own time and environment. In the present paper, I argue that it is possible to make sense (...) of Zeno’s arguments if we recognise that Zeno was indeed a close follower of Parmenides, who wanted to show that, if the plurality of beings existed, then various absurd consequences would follow. He intended to highlight the compact and inarticulate nature of the being, and the human character of the system of world partitions producing the entities and the objects on which our knowledge is based. (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)

Perhaps the real paradox of Zeno's Arrow is that, although entirely stationary, it has, against all odds, successfully traversed over two millennia of human thought to trouble successive generations of philosophers. The prospects were not good: few original Zenonian fragments survive, and our access to the paradoxes has been for the most part through unsympathetic commentaries. Moreover, like its sister paradoxes of motion, the Arrow has repeatedly been dismissed as specious and easily dissolved. Even those commentators who have taken it (...) seriously have propounded solutions with which they profess themselves to be perfectly satisfied. So my question is: will Zeno's Arrow survive into the millennium just begun? (shrink)

Professor Grünbaum's much-discussed refutation of Zeno's metrical paradox turns out to be ad hoc upon close examination of the relevant portion of measure theory. Although the modern theory of measure is able to defuse Zeno's reasoning, it is not capable of refuting Zeno in the sense of showing his error. I explain why the paradox is not refutable and argue that it is consequently more than a mere sophism.

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)

It will be shown in this article that an ontological approach for some problems related to the interpretation of Quantum Mechanics (QM) could emerge from a re-evaluation of the main paradox of early Greek thought: the paradox of Being and non-Being, and the solutions presented to it by Plato and Aristotle. More well known are the derivative paradoxes of Zeno: the paradox of motion and the paradox of the One and the Many. They stem from what was perceived by classical (...) philosophy to be the fundamental enigma for thinking about the world: the seemingly contradictory results that followed from the co-incidence of being and non-being in the world of change and motion as we experience it, and the experience of absolute existence here and now. The most clear expression of both stances can be found, again following classical thought, in the thinking of Heraclitus of Ephesus and Parmenides of Elea. The problem put forward by these paradoxes reduces for both Plato and Aristotle to the possibility of the existence of stable objects as a necessary condition for knowledge. Hence the primarily ontological nature of the solutions they proposed: Plato’s Theory of Forms and Aristotle’s metaphysics and logic. Plato’s and Aristotle’s systems are argued here to do on the ontological level essentially the same: to introduce stability in the world by introducing the notion of a separable, stable object, for which a ‘principle of contradiction’ is valid: an object cannot be and not-be at the same place at the same time. So it becomes possible to forbid contradiction on an epistemological level, and thus to guarantee the certainty of knowledge that seemed to be threatened before. After leaving Aristotelian metaphysics, early modern science had to cope with these problems: it did so by introducing “space” as the seat of stability, and “time” as the theater of motion. But the ontological structure present in this solution remained the same. Therefore the fundamental notion ‘separable system’, related to the notions observation and measurement, themselves related to the modern concepts of space and time, appears to be intrinsically problematic, because it is inextricably connected to classical logic on the ontological level. We see therefore the problems dealt with by quantum logic not as merely formal, and the problem of ‘non-locality’ as related to it, indicating the need to re-think the notions system, entity, as well as the implications of the operation ‘measurement’, which is seen here as an application of classical logic (including its ontological consequences) on the material world. (shrink)

After reporting in detail Aristotle’s texts and comments on the well-known motion paradoxes Arrow, Dichotomy, Achilles and Stadium, tracking back to the 5th century BCE and credited by Aristotle to Zeno of Elea, we next explain and dis-cuss traditional continuous solutions of the paradoxes, based on Cauchy’s limit concept. Afterward, the heated philosophical debate on supertasks and infinity machines is reported before the paradoxes are examined within the context of modern quantum theory. Already in 1905, Einstein concluded that matter could (...) not be a continuous thing. Classical continuous spacetime is replaced by ‘granular’ spacetime, and the concept of distance in granular spacetime is discussed. This is followed by a detailed presentation of modern discrete solutions in granular spacetime, several of which are published here for the first time. Finally, a procedure is presented to determine when traditional continuous methods suffice instead of discrete methods, and the handling of discrete versus continuous in physics is briefly reported. (shrink)

A solution of the zeno paradoxes in terms of a discrete space is usually rejected on the basis of an argument formulated by hermann weyl, The so-Called tile argument. This note shows that, Given a set of reasonable assumptions for a discrete geometry, The weyl argument does not apply. The crucial step is to stress the importance of the nonzero width of a line. The pythagorean theorem is shown to hold for arbitrary right triangles.

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)

I SHOW THAT THE COSMOLOGICAL ARGUMENT OF AQUINAS FOR THE EXISTENCE OF GOD COMMITS A RATHER TRIVIAL LINGUISTIC FALLACY, BY SHOWING THAT (1) SOME OF ZENO'S PARADOXES COMMIT A TRIVIAL LINGUISTIC FALLACY, AND THAT (2) THE COSMOLOGICAL ARGUMENT IS SUFFICIENTLY SIMILAR TO THESE PARADOXES THAT IT COMMITS THE SAME FALLACY. COPLESTON'S VIEW THAT "MENTION OF THE MATHEMATICAL INFINITE SERIES IS IRRELEVANT" TO "ANY" OF AQUINAS'S ARGUMENTS FOR GOD'S EXISTENCE IS THUS SHOWN FALSE.

A model of a new version of Zeno's arrow paradox is presented in a plausible extension of Newtonian collision mechanics. In exploring various avenues for resolution of the paradox, it becomes evident that a prerelativistic classical physical topology which is locally deterministic can mechanically generate nonclassical ontological properties such as the appearance of a particle in many places at once. It can also mimic some properties of quantum physics, including unprepared spatially-separated correlations. 1 Zeno's arrow paradox 2 Newtonian collision mechanics (...) and extensions of it 3 Our initial condition (IC) 4 Demonstrating the model paradox 5 Resolving the paradox 6 Unprepared correlations in spatially-separated events 7 Lessons. (shrink)

Recent scholarship on Stoic political thought has sought to explain the relationship between Zeno’s Republic and the concept of a natural law regulating a cosmic city of gods and human beings that is attributed to later Stoics. This paper provides a reassessment of this relationship by exploring the underappreciated influence of Plato’s Laws on Zeno’s Republic and, through Zeno, on the subsequent Stoic tradition. Zeno’s attempt to remove perceived inconsistencies in Plato’s treatment of ‘law’ and ‘nature’ established (...) a philosophical framework that overturned the republicanism of Plato and Aristotle; this same framework established the preconditions for the cosmic city of gods and human beings regulated by natural law. Thus, the early Stoic tradition on the topic of natural law is characterized by continuity rather than by discontinuity. (shrink)

This essay addresses two central issues that continue to trouble interpretation of Zeno’s paradoxes: 1) their solution, and 2) their place in the history of philosophy. I offer an account of Zeno’s work as pointing to an inevitable paradox generated by our ways of thinking and speaking about things, especially about things as existing in the continua of space and time. In so doing, I connect Zeno’s arguments to Parmenides’ critique of “naming” in Fragment 8, an approach (...) that I believe adds considerably to our understanding of both Zeno’s puzzles and this enigmatic aspect of Parmenides’ thought. (shrink)

The present study partly supports, partly corrects, and partly complements recent discussions of Arius Didymus fr. 23 and fr. 25 Diels, Aetius I, 20, 1 and Sextus Empiricus AM X, 3-4 = PH III, 124. It proposes a comprehensive interpretation of the first text (A.I), defends the attribution of its content to Zeno of Citium (A.II), interprets the Stoic definitions of space, place and void to be found in the other sources (B.I) and again vindicates the attribution of the core (...) definitions to Zeno (B.II). The central methodological principle is the presumption of innocence for sources.The main conclusions of (A) are: 1. Arius Didymus' fr. 23 deals only with the coherence and the structure of the cosmos, not with its immobility; 2. The coherence of the cosmos, as that of any object, is determined by its hexis which pushes its parts towards its centre; 3. The structure of the cosmos, its stratification into the four concentric spheres of the elements, is determined by the combined effect of (a) the thrust downwards they all undergo from the cosmos' hexis; (b) their own natural weight or lightness and (c) the relative quantitative values of these weights or lightnesses; 4. The reasons adduced against Zeno's authorship are not based on the evidence but on the now prevalent disparaging sceptical approach towards Stobaeus' way of excerpting from Arius.The main conclusions of (B) are: 1. There is no deep contradiction between the various de finitions of space ascribed by our sources to Zeno, Chrysippus and the Stoics in general: what Chrysippus denied was the form of the de finition attributed to Zeno by Aetius, not the concept de fined, even though this form seems to have prevailed later in the school; 2. There is no good reason therefore to question its ascription to Zeno as some modern researchers have done; 3. Here again the error is due to a predominantly sceptical approach towards the reliability of our sources. (shrink)

In this note i argue against harold n. lee's assertion ("mind," october, 1965) that resolution of zeno's paradoxes is closely connected with the modern mathematical distinction between density and continuity. zeno's paradoxes would arise as much if space or time is dense as they do if it is continuous. in fact the paradoxes only arise if one combines a mathematical analysis of space and time with a non-mathematical conception of motion.

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)

A solution of the zeno paradoxes in terms of a discrete space is usually rejected on the basis of an argument formulated by hermann weyl, The so-Called tile argument. This note shows that, Given a set of reasonable assumptions for a discrete geometry, The weyl argument does not apply. The crucial step is to stress the importance of the nonzero width of a line. The pythagorean theorem is shown to hold for arbitrary right triangles.

Aristotle’s solution to Zeno’s arrow paradox differs markedly from the so called at-at solution championed by Russell, which has become the orthodox view in contemporary philosophy. The latter supposes that motion consists in simply being at different places at different times. It can boast parsimony because it eliminates velocity from the ontology. Aristotle, by contrast, solves the paradox by denying that the flight of the arrow is composed of instants; rather, on my reading, he holds that the flight is (...) a unity which is ontologically prior to the temporal parts that may be abstracted from it. I label this position temporal holism; and an entity that is ontological prior to its temporal parts lasting. Lasting, I argue, is the key to Aristotle’s solution to the arrow paradox, but it has received insufficient attention in the relevant literature. I therefore set out here to investigate lasting and highlight its major role in Aristotle’s metaphysics more broadly, and in his solution to other problems of change. I show, too, how lasting within a contemporary context underwrites a parsimonious and intuitive solution to the arrow paradox that retains instantaneous velocity, and hence solves a problem for those who think velocity is required within the ontology to underwrite certain causal roles. (shrink)

We outline Brentano’s theory of boundaries, for instance between two neighboring subregions within a larger region of space. Does every such pair of regions contain points in common where they meet? Or is the boundary at which they meet somehow pointless? On Brentano’s view, two such subregions do not overlap; rather, along the line where they meet there are two sets of points which are not identical but rather spatially coincident. We outline Brentano’s theory of coincidence, and show how he (...) uses it to resolve a number of Zeno-like paradoxes. (shrink)

The relationship between science and philosophy is contentious. Quine saw philosophy as continuous with science (broadly understood), but many philosophers see a dichotomy between them. The present paper discusses cases where the relevance of certain scientific findings has been denied (related to Zeno’s Dichotomy paradox and to the appeal of skeptical arguments) or overlooked (one argument related to the frame problem of artificial intelligence and Nagel’s “bat” argument). The results caution against overly quick dismissal of the import of science (...) on philosophical questions, whether the latter be of a more theoretical or practical nature. (shrink)

In Physics VI.9 Aristotle addresses Zeno's four paradoxes of motion and amongst them the arrow paradox. In his brief remarks on the paradox, Aristotle suggests what he takes to be a solution to the paradox.In two famous papers, both called 'A note on Zeno's arrow', Gregory Vlastos and Jonathan Lear each suggest an interpretation of Aristotle's proposed solution to the arrow paradox. In this paper, I argue that these two interpretations are unsatisfactory, and suggest an alternative interpretation. In particular, I (...) claim that what seems on the face of it to be Aristotle's solution to the paradox raises two puzzles to which my interpretation, as opposed to Lear's and Vlastos's, provides an adequate response. (shrink)

A model of Zeno's dichotomy paradox is presented in Newtonian collision mechanics. One of several resolutions of the paradox illustrates the point that even in Newtonian ontology there is a spacetime weave. In a Newtonian system in which the base rules permit only spatial contact interactions, we find the mechanical emergence of action-at-a-distance effects.