We provide a new interpretation of Zeno’s Paradox of Measure that begins by giving a substantive account, drawn from Aristotle’s text, of the fact that points lack magnitude. The main elements of this account are (1) the Axiom of Archimedes which states that there are no infinitesimal magnitudes, and (2) the principle that all assignments of magnitude, or lack thereof, must be grounded in the magnitude of line segments, the primary objects to which the notion of linear magnitude applies. Armed (...) with this account, we are ineluctably driven to introduce a highly constructive notion of (outer) measure based exclusively on the total magnitude of potentially infinite collections of line segments. The Paradox of Measure then consists in the proof that every finite or potentially infinite collection of points lacks magnitude with respect to this notion of measure. We observe that the Paradox of Measure, thus understood, troubled analysts into the 1880’s, despite their knowledge that the linear continuum is uncountable. The Paradox was ultimately resolved by Borel in his thesis of 1893, as a corollary to his celebrated result that every countable open cover of a closed line segment has a finite sub-cover, a result he later called the “First Fundamental Theorem of Measure Theory.” This achievement of Borel has not been sufficiently appreciated. We conclude with a metamathematical analysis of the resolution of the paradox made possible by recent results in reverse mathematics. (shrink)

A partire dalle testimonianze epistolari ciceroniane, in Varrone accademico e menippeo sono esaminate le dinamiche filologiche, contingenti e psicologiche, sottese alla seconda edizione degli "Academica" a favore della maschera di Varrone. Il "Varro" di Cicerone è l’unica porzione di dialogo superstite riconducibile alla formazione e all’adesione filosofica di Varrone quanto a logica, etica e fisica. Il reatino sarebbe stato spinto a tentare personalmente l’impresa di una trattazione sistematica nel perduto "De philosophia" (di cui resta traccia in Agostino) solo dopo il (...) dono-pungolo ricevuto dall’amico-rivale. Il famoso elogio ciceroniano rivolto a Varrone è riletto alla luce dell’agonismo e dell’antagonismo non tanto filosofico quanto personale tra i due accademici, risolvendosi in una sostanziale "deminutio" del dedicatario e della sua produzione. Dal punto di vista del conduttore del dialogo, le Menippee, in cui Varrone si era esposto anche filosoficamente (in qualche frammento, perfino canzonando i filosofi) e in cui Menippo era stato imitato e non tradotto, avrebbero funto “ad impellendum satis, ad edocendum parum”. L’imitazione di un’imitazione, dunque. Tale venata allusione prelude anche all’ostile tradizione riguardante la fosca ombra di Menippo, il “cane senza coda”, caldeggiato e difeso da Varrone in una menippea dedicata alla dipartita sui generis del filosofo. Nelle "Eumenides", la menippea meglio conservata, è presente il frammento che riassume anfibolicamente la duplice facies della formazione varroniana: la "cana Veritas" sarebbe la bianca verità platonica, ma anche la mordace verità cinica, portata alle estreme conseguenze dal Diogene pazzo. La riflessione si conclude platonicamente focalizzandosi sull’istantaneamente del suddetto frammento attraverso il calco varroniano "de inproviso", che rimanda anche alla Lettera VII e alle dottrine non scritte di Platone. -/- . (shrink)

This paper divides the sophisms and paradoxes put forth by Chinese thinkers of the pre-Qin period of China into six groups: paradoxes of motion and infinity, paradoxes of class membership, semantic paradoxes, epistemic paradoxes, paradoxes of relativization, other logical contradictions. It focuses on the comparison between the Chinese items and the counterparts of ancient Greek and even of contemporary Western philosophy, and concludes that there turn out to be many similar elements of philosophy and logic at the beginnings of Chinese (...) and Greek civilizations. (shrink)

Sources for Stoicism present conflicting accounts of the Stoic principles. Some suggest that the principles are inseparable from each other. Others suggest that they are separable. To resolve this apparent interpretive dilemma, I distinguish between the functions of the principles and the bodies that realize those functions. Although the principles cannot separate when realizing their roles, the Stoic theory of blending entails that the bodies that realize those roles are physically separable. I present a strategy for further work on the (...) principles in light of this interpretation. (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)

In Naming and Necessity Kripke argues 'intuitively' that names are rigid. Unlike Kripke, Ben-Yami first introduces and justifies the Principle of the Independence of Reference (PIR), according to which the reference of a name is independent of what is said in the rest of the sentence containing it. Ben-Yami then derives rigidity, or something close to it, from the PIR. Additional aspects of the use of names and other expressions in modal contexts, explained by the PIR but not by the (...) rigidity claim, are then discussed. Ben-Yami next examines a difficulty in accepted definitions of rigidity, stemming from the fact that the same name can be used to name different particulars. This difficulty might force us to adopt a revised form of the rigidity claim. (shrink)

Recently we proposed “quantum language”which was characterized as the metaphysical and linguistic turn of quantum mechanics. This turn from physics to language does not only realize the remarkable extension of quantum mechanics but also yield the quantum mechanical world view. And thus, the turn urges us to dream that Western philosophies (i.e., Parmenides, Plato, Descartes, John Locke, Berkeley, Hume, Kant, Wittgenstein, etc.) can be understood in quantum language. In this paper, from the quantum linguistic point of view, we give the (...) answers of the following two problems. (i): Has western philosophy made progress? Or what is the measure of "progress"? (ii):Why has useless western philosophy been prospering? (shrink)

I reconstruct the original versions of Zeno's Racetrack and Achilles paradoxes, along with Aristotle's responses thereto. Along the way I consider some of the consequences for modern analyses of the paradoxes. It turns out that the Racetrack and the Achilles were oral two-party question-and-answer dialectical paradoxes. One consequence is that the arguments needed to be comprehensible to the average person, and did not employ theses or concepts familiar only to philosophical specialists. I rely on this fact in reconstructing the original (...) dialectical versions of the paradoxes. I show that both paradoxes rely for their success on forcing the dialectical answerer to reflect on his own potentially unending experience of imagination, and show that this renders the most popular contemporary critique of each paradox unworkable in an ancient dialectical context. In responding to the Racetrack, Aristotle seeks to replace the answerer's first-person experience with something more objective, a visible diagram. He then argues that the Racetrack involves an equivocation, an equivocation resulting from the fact that one and the same visible diagram can be interpreted in two ways. He frames his charge of equivocation in standard dialectical fashion, but his use of a diagram is his own innovation. While first employing his response to the Racetrack to construct a response to the somewhat different Achilles paradox, Aristotle later proceeds to offer a revised critique of the Racetrack itself. His revised critique is heavily influenced by his reaction to a variant pre-Aristotelian version of the paradox, a version involving counting. This leads him to reflect on the possibility of mentally experiencing an infinite collection. He also reflects on his earlier diagrammatic methodology, and on the conditions that individuate points, especially halfway-points. He concludes that points which are individuated in diagrams need not represent points that are individuated in reality. His revised critique of the Racetrack hinges on the distinction between actual points and potential points, and I show that this critique maintains the dialectical form of its predecessor, while constituting a reflection on the potentially misleading nature of diagrams. (shrink)

I argue that relations between non-collocated spatial entities, between non-identical topological spaces, and between non-identical basic building blocks of space, do not exist. If any spatially located entities are not at the same spatial location, or if any topological spaces or basic building blocks of space are non-identical, I will argue that there are no relations between or among them. The arguments I present are arguments that I have not seen in the literature.

According to Renan, the day of Marcus Aurelius’ death could be considered as the decisive moment in the downfall of the ancient civilization. He, thus, wonders: “If Marcus Aurelius, the unique emperor-philosopher, did not succeed in saving the world, who else, then, could have saved it?” He notes that the emperor’s death was followed by the succession to the throne of his corrupted son, Commodus, and his friends, who all were all ignorant. Renan observes that the emperor’s kindness could not (...) have prevented the unfortunate fate that befell the Roman Empire after his death. What we have here is the perennial problem, already established in Plato, regarding the role of the philosopher-king in establishing a good state and educating good citizens. However, the case of Marcus Aurelius, as demonstrated by Renan in his book, shows the inability of philosophy to serve the real needs, which ultimately leads to disastrous and irreparable consequences. The present paper attempts to reconstruct the reasons for the unsuccessful application of philosophy, especially the philosophy of the Hellenistic era, to the administrative system of the Roman Empire. It is argued that the failure is mainly due to political, religious and cultural problems. (shrink)

We apply a framework developed by C. S. Peirce to analyze the concept of clarity, so as to examine a pair of rival mathematical approaches to a typical result in analysis. Namely, we compare an intuitionist and an infinitesimal approaches to the extreme value theorem. We argue that a given pre-mathematical phenomenon may have several aspects that are not necessarily captured by a single formalisation, pointing to a complementarity rather than a rivalry of the approaches.

This dissertation investigates issues raised by these two questions: what kinds of propositions are ordinarily expressed by uses of sentences that contain proper names; and what kinds of beliefs are ordinarily on the minds of speakers when they use sentences that contain proper names? It develops a new view about the connections between beliefs, linguistic behavior, and propositional content, one that explicitly denies that the kinds of propositions typically expressed by uses of such sentences are the objects of the beliefs (...) typically on the minds of the speakers who use them. ;Chapter I presents both the Millian and the description theories of proper names, and reviews the advantages and disadvantages of each. ;Chapter II critically evaluates Dummett's defense of the description theory against the Modal Objection. ;Chapter III introduces Kripke's puzzle about beliefs and proper names. It shows that Kripke's puzzle is not solved by the theory of proper names recently presented by Devitt. It critically evaluates the "consistency solutions" proposed by Chisholm, Harrison, Noonan, and Over. ;Chapter IV continues the discussion of Kripke's puzzle. It critically evaluates the "inconsistency solution" proposed by Marcus. It examines a commentary on the puzzle by Lewis. Finally, it presents an "inconsistency solution" based on views suggested by the Lewis commentary. ;Chapter V compares my view about the connections between beliefs, linguistic behavior, and propositional content to the "naive view" and the "Russellean view." It applies my view to solve two major problems for the Millian theory of proper names. (shrink)

Michael Dummett claims that the classical model of time as a continuum of instants has to be rejected. In his view, “it allows as possibilities what reason rules out, and leaves it to the contingent laws of physics to rule out what a good model of physical reality would not even be able to describe.” This paper argues otherwise.

The work interprets experimental data for the heat capacity of Zn22 in the region of second-order phase transitions. The proposed understanding of the processes occurring during phase transitions may be helpful to reveal quantum Zeno effects in metal–organic frameworks with evolving structural subsystems and to establish relations between quantum measurements and the entropy of phase transitions.

“Deus fons veritatis”: the Subject and its Freedom. The Ontic Foundation of Mathematical Truth is the title of Gaspare Polizzi’s long biographical-theoretical interview with Imre Toth. The interview is divided into eight parts. The first part describes the historical and cultural context in which Toth was formed. A Jew by birth, during the Second World War Toth became a communist and a partisan, enduring prison, torture, and internment in a concentration camp from 1940 until 6 June 1944. In the second (...) part Toth presents his mathematical training as a “vocation” that led him to rethink the whole tradition of mathematical thought critically, on the basis of non-Euclidean geometry. In the third part Toth describes his research in the history of mathematics, which begin with his studies on Aristotle and mathematical thought and on Plato and the negative ontology of the irrational recognizable in the theory of the infinite dyad and of the One. In the fourth part Toth criticizes the positions of Frege, who came to deny non-Euclidean geometry, viewing it as an expression of irrationality and mysticism. In the fifth part Toth maintains that mathesis and poiesis have similar ontological structures, and he speaks of his collages métaphysiques. In the sixth part Toth recalls how the birth of the idea of freedom made possible the highest political, social and artistic achievements, as well as the entire movement of human emancipation. And this was thanks to philosophy, which is not a science but a knowledge of the subject on the subject’s part. In the seventh and eighth parts Toth speaks of the value and the role of mathematics in the affirmation of the phenomenology of freedom, and remarks on his relations with French and Italian cultures. (shrink)