Euphorion SH 417 deserves to be better known. A curiosity in itself—an apparent poetic reference to number theory—it is also, potentially, one of our earliest references to Euclidean material. On the authority of a late commentator on Aristotle, Euphorion, a mid-third-century b.c. Euboean poet who was also active in Athens and Antioch, is said to have mentioned perfect numbers—i.e. numbers which equal the total of all their factors, including 1. It is a pity that the context in Euphorion (...) does not survive, and that the line is only preserved, and indeed interpreted, by so late a source. But the wording of the fragment—if Westerink's restoration of its various corruptions is plausible—would strongly suggest a reference to the notion of perfect number. The fragment has been known since Westerink published it in 1960, and was included both in Van Groningen's edition of Euphorion in 1977 and in Supplementum Hellenisticum. But its implications have still not been discussed, and when David Fowler came to gather the evidence for references to Euclidean material in and after the third century b.c. in The Mathematics of Plato's Academy, his attention, unsurprisingly, was not drawn to it. Euphorion has had a bad press, as a poet of rebarbatively obscure myth and intractable vocabulary; yet he holds some interest, and we may be missing more insights into the intellectual life of the Hellenistic period which the perverse intelligence and baneful wit of the fragments display. (shrink)

Euphorion SH 417 deserves to be better known. A curiosity in itself—an apparent poetic reference to number theory—it is also, potentially, one of our earliest references to Euclidean material. On the authority of a late commentator on Aristotle, Euphorion, a mid-third-century b.c. Euboean poet who was also active in Athens and Antioch, is said to have mentioned perfect numbers—i.e. numbers which equal the total of all their factors, including 1 . It is a pity that the context in (...) Euphorion does not survive, and that the line is only preserved, and indeed interpreted, by so late a source. But the wording of the fragment—if Westerink's restoration of its various corruptions is plausible—would strongly suggest a reference to the notion of perfect number. The fragment has been known since Westerink published it in 1960, and was included both in Van Groningen's edition of Euphorion in 1977 and in Supplementum Hellenisticum . But its implications have still not been discussed, and when David Fowler came to gather the evidence for references to Euclidean material in and after the third century b.c. in The Mathematics of Plato's Academy, his attention, unsurprisingly, was not drawn to it. Euphorion has had a bad press, as a poet of rebarbatively obscure myth and intractable vocabulary; yet he holds some interest, and we may be missing more insights into the intellectual life of the Hellenistic period which the perverse intelligence and baneful wit of the fragments display. (shrink)

Philosophical theories of counterfactuals have had relatively little to say about counterfactual reasoning in mathematics. Partly this is because most mathematical counterfactuals seem also to be counterpossibles, in that their antecedents deny some necessary truth. In this chapter, I delineate several different categories of mathematical counterfactual (or “countermathematical”) and then examine in detail a case study from mathematical practice that features counterfactual reasoning about “spoof perfect” numbers. I argue that reasoning about spoof perfect numbers presents both a challenge (...) to philosophical analyses of counterfactuals and a resource for thinking more productively about the role of counterfactual reasoning in mathematics. (shrink)

As past president of both the History of Science Society and the American Society of Church History, Ronald L. Numbers is uniquely qualified to assess the historical relations between science and Christianity. In this collection of his most recent essays, he moves beyond the clichés of conflict and harmony to explore the tangled web of historical interactions involving scientific and religious beliefs. In his lead essay he offers an unprecedented overview of the history of science and Christianity from the perspective (...) of the ordinary people who filled the pews of churchesor loitered around outside. Unlike the elite scientists and theologians on whom most historians have focused, these vulgar Christians cared little about the discoveries of Copernicus, Newton, and Einstein. Instead, they worried about the causes of the diseases and disasters that directly affected their lives and about scientists preposterous attempts to trace human ancestry back to apes. Far from dismissing opinion-makers in the pulpit, Numbers closely looks at two the most influential Protestant theologians in nineteenth-century America: Charles Hodge and William Henry Green. Hodge, after decades of struggling to harmonize Gods two revelationsin nature and in the Biblein the end famously described Darwinism as atheism. Green, on the basis of his careful biblical studies, concluded that Ussher's chronology was unreliable, thus opening the door for Christian anthropologists to accommodate the subsequent discovery of human antiquity. In Science without God Numbers traces the millennia-long history of so-called methodological naturalism, the commitment to explaining the natural world without appeals to the supernatural. By the early nineteenth century this practice was becoming the defining characteristic of science; in the late twentieth century it became the central point of attack in the audacious attempt of intelligent designers to redefine science. Numbers ends his reassessment by arguing that although science has markedly changed the world we live in, it has contributed less to secularizing it than many have claimed. Taken together, these accessible and authoritative essays form a perfect introduction to Christian attitudes towards science since the 17th century. (shrink)

This chapter examines the pioneering work of Gottfried Wilhelm Leibniz (1646-1716) on various number systems, in particular binary, which he independently invented in the mid-to-late 1670s, and hexadecimal, which he invented in 1679. The chapter begins with the oft-debated question of who may have influenced Leibniz’s invention of binary, though as none of the proposed candidates is plausible I suggest a different hypothesis, that Leibniz initially developed binary notation as a tool to assist his investigations in mathematical problems that (...) were exercising him at the time, namely those concerning the divisibility of composite numbers, primality, and perfect numbers. The chapter then explores Leibniz’s development of binary, his little-known work on binary fractions and expansions, his use of binary as a symbol for creation in his philosophical-theology, and his response to the suggestion that there was a correlation between binary numeration and the hexagrams of the ancient Chinese divinatory text, the Yijing. The chapter then focuses on Leibniz’s work on other number systems, in particular his invention and exploration of hexadecimal as well as his work on duodecimal. The chapter concludes by revealing a hitherto unknown practical application of binary that Leibniz devised in the last year of his life. (shrink)

In Cavell (1994), the ability to follow and produce Austinian examples of ordinary language use is compared with the faculty of perfect pitch. Exploring this comparison, I clarify a number of central and interrelated aspects of Cavell's philosophy: (1) his way of understanding Wittgenstein's vision of language, and in particular his claim that this vision is "terrifying," (2) the import of Wittgenstein's vision for Cavell's conception of the method of ordinary language philosophy, (3) Cavell's dissatisfaction with Austin, and (...) in particular his claim that Austin is not clear about the nature and possible achievements of his own philosophical procedures, and (4) Cavell's notion that the temptation of skepticism is perennial and incurable. Cavell's reading of Wittgenstein is related to that of John McDowell. Like McDowell, Cavell takes Wittgenstein to be saying that the traditional attempt to justify our practices from an external standpoint is misguided, since such detachment involves losing sight of those conceptual and perceptual capacities in terms of which a practice is understood by its engaged participants. Unlike McDowell, however, Cavell consistently rejects the idea that philosophical clearsightedness can or should free us from that fear of groundlessness which motivates the traditional search for external justification. (shrink)

In a number of passages of the Timaeus Plato states that the world is generated in such a way as to effectively reproduce the intelligible paradigm. The aim of this chapter is to understand in which sense the world is indeed a likeness of the paradigm, especially with regard to three aspects: its unicity, its completeness/perfection (i.e., its being τέλειον), its (a)temporality. My overall claim is that the key feature of the paradigm that the world is meant to reproduce (...) is not its simplicity or stability, but its dynamic and holistic structure. More specifically, I shall argue that: the world is unique because the intelligible structure of the paradigm is holistic in a qualified way; the world is complete/perfect not in the sense that it encompasses the reproduction of all forms, but since it reproduces at the best possible level the rational motion and life of the intelligible world, and this is what Plato actually means by his definition of “time”. This will effectively explain why it is really good for the demiurge to take the intelligible realm as a paradigm, and why Plato can effectively state that the generated world is not a shadowy half-being, but a venerable god. (shrink)

When used next to a predicate at the extreme of a scale such as perfect, the exclusive modifiers just and simply convey a distinctive intensifying effect, presenting a puzzle for theories of exclusivity and alternative-based meanings more broadly. In this article, I develop an analysis of these modifiers as a special kind of alternative-targeting operator, whereby the speaker signals that more specific descriptions than the one they just asserted—modeled here as granularity-based alternatives—are not assertion-worthy in the context—i.e., they need (...) not be asserted in order for a complete representation of the facts to be provided. On this analysis, the intensifying effect of these modifiers are derived from the interaction of exclusivity, granularity, and the distinctive semantic properties of predicates denoting the extreme of a scalar extreme. More broadly, the proposal affords a jumping-off point for further work exploring the interaction between scalarity and exclusive operators, highlighting a number of intriguing lines of future research across semantics and pragmatics. (shrink)

The claim that perfect hallucination is introspectively indistinguishable from perception has been a centrepiece of philosophical theorizing about sense experience. The most common interpretation of the indistinguishability claim is modal: that it is impossible to distinguish perfect hallucination from perception through introspection alone. I run through various models of introspection and show that none of them can accommodate the modal interpretation. Rejecting the modal interpretation opens up two alternative interpretations of the indistinguishability claim. According to the generic interpretation, (...) hallucination is indistinguishable from perception despite the existence of possible exceptions, while according to the actuality interpretation, the indistinguishability of hallucination from perception consists in the actual failure to distinguish hallucination from perception. These alternative understandings of the indistinguishability claim have a number of significant implications for the problem of perception, including the rejection of perfect hallucination and illusion in favour of our ordinary, non-philosophical concepts of these states. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:BOOK REVIEWS The Perfection of the Universe According to Aquinas: A Teleological Cosmology. By OLIVA BLANCHETTE. University Park, Penn.: The Pennsylvania State University Press, 1992. Pp. xvii + 334. $35.00 (cloth). This work represents a significant and most welcome contribution to Thomistic interpretation as well as to the broader study of medieval philosophy. While its tone is unpretentious, its theme, the structure and purpose of the whole created universe, (...) is crucial for Aquinas's philosophy. Those familiar with Thomas's corpus know how often this theme appears-frequently supplying the foundation for the argument-and know equally well how elusive it becomes when one seeks an extended treatment of it. Hence, to have recognized the chief elements of the larger picture, to have identified and collected the pertinent texts, and to have ordered and synthesized them as this book has done represents a major and lasting scholarly achievement. Moreover, while primarily concerned with Aquinas's own philosophical thought, the book also places him in conversation with a number of classical and modern thinkers, from Plato and Aristotle to Hegel and Collingwood. Blanchette begins his study by clarifying the very notion of perfection (Chapter 1). It first refers, in the order of our conceptions, to the completion of a process. A thing is perjectum, "thoroughly made," when its process of generation is finished. Motion, in fact, is precisely a passage from imperfection to perfection. By extension, the term can then be applied apart from motion to anything which lacks nothing it should have, or, as in the case of God, which lacks nothing at all. Analyzing the Greek and Latin terms connected with generation and perfection (poiein/facere; genesthai/fieri; teleion/perfectum; telos/finis; teleiosislperfectio) Blanchette shows why it is that for the Greeks the infinite usually signified imperfection, while Aquinas was able to conceive of an infinity which was perfection. For the present study, a crucial distinction among the kinds of perfection is that between first and second perfection, i.e., between a being's "ontological " perfection, the complete generation of its nature with all its natural powers, and its "operational" perfection, the activities carried out by means of those powers. Only in its operations does a being attain its ultimate perfection, which always lies in a relation to something exterior to itself. This fundamental distinction between first and second perfection underlies the basic 485 486 BOOK REVIEWS structure of the book. Part One describes the universe in its first perfection, i.e., the constituent parts whose presence gives it its integrity. Part Two takes up the activities of the parts which taken together constitute the order within the universe and by which the universe attains to its external good, God himself. At the level of first perfection, the universe is considered to be perfect first of all simply because it is the universe, i.e., the whole of what is (Chapter 2). As a whole, it is perfect (complete) in comparison to any of its parts. Thus it is a priori impossible to have plural universes, since each "universe" would in fact be only a part of the whole which was the universe. Second, simply as a body, the universe enjoys the perfection of having all three dimensions. Third, not having any bodies outside it, the universe does not share the imperfection of being limited by bodies external to it, as does every particular body. Finally, in containing the perfection of each kind of being, the universe possesses a universal perfection not shared by any one kind of being. These perfections belong to the universe simply qua universe. Beyond these, however, we can speak of the perfection of this universe. As Blanchette points out, for Aquinas there is no "best of all possible worlds" in the sense that God could not create a better one. Yet we can speak of the perfection (or completeness) of this world that God, for reasons not accessible to human reason, actually chose to create. The perfection of this universe is such that were any new essential parts, i.e., new species, added, it would no longer be the same universe, but rather a new universe of which the old now constituted a part... (shrink)

An important technique in large cardinal set theory is that of extending an elementary embedding j: M → N between inner models to an elementary embedding j*: M[G] → N[G*] between generic extensions of them. This technique is crucial both in the study of large cardinal preservation and of internal consistency. In easy cases, such as when forcing to make the GCH hold while preserving a measurable cardinal (via a reverse Easton iteration of α-Cohen forcing for successor cardinals α), the (...) generic G* is simply generated by the image of G. But in difficult cases, such as in Woodin's proof that a hypermeasurable is sufficient to obtain a failure of the GCH at a measurable, a preliminary version of G* must be constructed (possibly in a further generic extension of M[G]) and then modified to provide the required G*. In this article we use perfect trees to reduce some difficult cases to easy ones, using fusion as a substitute for distributivity. We apply our technique to provide a new proof of Woodin's theorem as well as the new result that global domination at inaccessibles (the statement that d (κ) is less than 2κ for inaccessible κ, where d (κ) is the dominating number at κ) is internally consistent, given the existence of 0#. (shrink)

: In this paper we give a detailed reconstruction of the first chapter of De Caelo I.1. Aristotle attempts to prove there that bodies are complete and perfect in virtue of being extended in three dimensions. We offer an analysis of this argument and argue that it gives important insight into the role the notion of body plays in physical science. Contrary to other interpretations, we argue that it is an argument about physical, as opposed to mathematical, bodies and (...) that the perfection and completeness of bodies is due to their nature. Moreover, Aristotle heavily relies in his proof on the premise that the number three implies perfection, a view he ascribes to the Pythagoreans. We review the possible sources of this view, as well as its role in Aristotle’s argumentative strategy. (shrink)

A sentence in the Resultative perfect licenses two inferences: (a) the occurrence of an event (b) the state caused by this event obtains at evaluation time. In this paper I show that this use of the perfect is subject to a large number of distributional restrictions that all serve to highlight the result inference at the expense of the event inference. Nevertheless, only the event inference determines the truth conditions of this use of the perfect, the (...) result inference being a unique type of conventional implicature. I argue furthermore that, since the result state is singular, the event that causes it must also be singular, whereas the Experiential perfect is purely quantificational. But in out-of-the-blue contexts the past tense is also normally interpreted as singular. This leads to a certain amount of competition between the Resultative perfect and the past tense, and it is this competition, I suggest, that maintains the conventional (non-truth conditional) result state inference. (shrink)

A number of scholars have claimed that, on the assumption of incompati- bilism, there is a con ict between God's freedom and God's essential moral perfection. Jesse Couenhoven is one such example; Couenhoven, a com- patibilist, thinks that libertarian views of divine freedom are problematic given God's essential moral perfection. He writes, \libertarian accounts of God's freedom quickly run into a conceptual problem: their focus on con- tingent choices undermines their ability to celebrate divine freedom with regard to the (...) essential divine nature. For an Augustinian [i.e., a compat- ibilist], by contrast, God's freedom is not at odds with the necessities of perfect love but ful lled by it."1 Others who argue for similar conclusions include William Rowe and Wes Morriston. Michael Bergmann and Jan Cover have recently argued that divine responsibility and moral perfection are compatible with the absence of divine freedom. In this paper, I argue that the arguments which hold that divine freedom con icts with essen- tial divine moral perfection fail. I develop an account of divine freedom which not only doesn't con ict with God's essential moral goodness but shows that such goodness is a necessary part of perfected freedom. I then show how this understanding of free will takes away a major motivation for Bergmann and Cover's apparent willingness to reject divine freedom. (shrink)

Why is the number seven lucky--even holy--in almost every culture? Why do we speak of the four corners of the earth? Why do cats have nine lives? From literature to folklore to private superstitions, numbers play a conspicuous role in our daily lives. But in this fascinating book, Annemarie Schimmel shows that numbers have been filled with mystery and meaning since the earliest times, and across every society. In The Mystery of Numbers Annemarie Schimmel conducts an illuminating tour of (...) the mysteries attributed to numbers over the centuries. She begins with an informative and often surprising introduction to the origins of number systems: pre-Roman Europeans, for example, may have had one based on twenty, not ten, while the Mayans had a system more sophisticated than our own. Schimmel also reveals how our fascination with numbers has led to a rich cross-fertilization of mathematical knowledge: "Arabic" numerals, for instance, were picked up by Europe from the Arabs, who had earlier adopted them from Indian sources. But the heart of the book is an engrossing guide to the symbolism of numbers. Number symbolism, she shows, has deep roots in Western culture, from the philosophy of the Pythagoreans and Platonists, to the religious mysticism of the Cabala and the Islamic Brethren of Purity, to Kepler's belief that the laws of planetary motion should be mathematically elegant, to the unlucky thirteen. After exploring the sources of number symbolism, Schimmel examines individual numbers ranging from one to ten thousand, discussing the meanings they have had for Judaic, Christian, and Islamic traditions, with examples from Indian, Chinese, and Native American cultures as well. Two, for instance, has widely been seen as a number of contradiction and polarity, a number of discord and antithesis. And six, according to ancient and neo-platonic thinking, is the most perfect number because it is both the sum and the product of its parts. Using examples ranging from the Bible to the Mayans to Shakespeare, she shows how numbers have been considered feminine and masculine, holy and evil, lucky and unlucky. A highly respected scholar of Islamic culture, Annemarie Schimmel draws on her vast knowledge to paint a rich, cross-cultural portrait of the many meanings of numbers. Engaging and accessible, her account uncovers the roots of a phenomenon we all feel every Friday the thirteenth. (shrink)

We investigate uncountable maximal antichains of perfect trees and of splitting trees. We show that in the case of perfect trees they must have size of at least the dominating number, whereas for splitting trees they are of size at least \\), i.e. the covering coefficient of the meager ideal. Finally, we show that uncountable maximal antichains of superperfect trees are at least of size the bounding number; moreover we show that this is best possible.

People v. Strauss-Kahn is an ideal lens through which to examine the operation of a criminal justice system that privileges the presumption of guilt, or, to use the words of the US Supreme Court in the 2012 decisions Lafler v. Cooper and Missouri v. Frye, has become “a system of pleas, not a system of trials.” It is both an excellent example of a transparent and objective invocation of the criminal sanction, and a sharp counterpoint to the vast majority of (...) cases where law enforcement conclusions are privileged and rarely second-guessed. As analyses of wrongful convictions have sadly but inevitably demonstrated, the screening methods of police and prosecutors are fallible. Moreover, the prevalence of “tunnel vision” once the decision to charge has been made means that few cases experience the kind of post-indictment investigation and reevaluation seen in the prosecution of Dominique Strauss-Kahn. Stage by stage, the Strauss-Kahn case illustrates how to counterbalance the presumption of guilt and give expression to the presumption of innocence in the pretrial period through vigilantly-invoked and enforced due process protections. Drawing from this examination, the paper will then explore how to approach this model process in the more standard cases, which typically see a fraction of the judicial, law enforcement, and defense resources afforded Dominique Strauss-Kahn. The Strauss-Kahn prosecution offers several insights, three of which will be sketched at the paper’s conclusion: a requirement that prosecutorial decision-making be subject to a reasonable doubt standard; vigilant and early enforcement of the prosecutor’s obligation to disclose information that is favorable to the accused; and finally, a requirement that a prosecutor explain in writing any decision to dismiss the felony charges in indicted felony cases, so that the factual, legal and policy bases of these decisions (numbering almost one quarter of New York’s superior court felony cases annually) can be analyzed and publicized. (shrink)

We study Lebesgue and Atsuji spaces within subsystems of second order arithmetic. The former spaces are those such that every open covering has a Lebesgue number, while the latter are those such that every continuous function defined on them is uniformly continuous. The main results we obtain are the following: the statement “every compact space is Lebesgue” is equivalent to $\hbox{\sf WKL}_0$ ; the statements “every perfect Lebesgue space is compact” and “every perfect Atsuji space is compact” (...) are equivalent to $\hbox{\sf ACA}_0$ ; the statement “every Lebesgue space is Atsuji” is provable in $\hbox{\sf RCA}_0$ ; the statement “every Atsuji space is Lebesgue” is provable in $\hbox{\sf ACA}_0$ . We also prove that the statement “the distance from a closed set is a continuous function” is equivalent to $\Pi^1_1\hbox{-\sf CA}_0$. (shrink)

Prenatal and preimplantation testing technologies have offered unprecedented access to information about the genetic and congenital makeup of our prospective progeny. Future developments such as preconception testing, non-intrusive prenatal testing and more extensive preimplantation testing promise to increase that access further still. The result may be greater reproductive choice, but it also increases the burden on women and men to avail themselves of these technologies in order to avoid having a child with a disability. The overwhelming question for legislators has (...) been whether and, if so, how to regulate the use of these technologies in the face of compelling but seemingly contradictory claims about the advancement of reproductive choice and the dangers of eugenic or discriminatory effects. This book examines the evolution of this legislative oversight across a number of jurisdictions and explores the tensions and ambiguities that inform these laws. (shrink)

Engaging the thought of the Danish thinker Søren Kierkegaard, I challenge a tendency within the analytic tradition of philosophy on the subject of meaning in life. Taking as a starting point Kierkegaard's insights about meaning in life, the striving needed to attain an imagined ideal self, and his paradoxical conception of the perfection available to human life, I claim that meaning in life is a function of an individual's striving for an ideal self. This continuous effort to achieve myself is (...) marked by suffering, an indispensable part of Kierkegaard's project of identity formation. The imagined grasp of a possible ideal self is essential to this process but insufficient for it because the imagination can only ever glimpse a kind of static perfection, not the lived perfection that only results from willed actualization of an ideal self. The meaning of a human life, then, consists in the suffering that results from a struggle to actualize the ideal I aspire to become in the process of identity formation. I contrast this view with a tendency shared by many contemporary analytic philosophers of meaning in life, for whom meaning in life is constituted by achievement of valued goods, without much attention to one's relation to the process of achieving them. In that respect, I will focus on the position of Iddo Landau. After clearing a number of his misconceptions about Kierkegaard's philosophy, I claim that, for a life to be meaningful, valued goods in life must be complemented by a conscious enactment of the process of the formation of one's identity that includes striving to attain a kind of perfection. I conclude that Kierkegaard's paradoxical account of perfectionism makes him more of an ally to Landau than an opponent. (shrink)

In recent years a number of arguments have been advanced to show that there are conceptual difficulties with a variety of divine attributes. Some have claimed that there is an inherent inconsistency in the notion of omnipotence, others that omnipotence was logically incompatible with omniscience or omnibenevolence, and yet others that omniscience is irreconcilable with immutuability.

The purpose of this work is to offer a critical examination of Immanuel Kant's ethic of suicide. Kant's suicidology marks an influential view regarding the moral stature of suicide, yet one that remains incomplete in important respects. Because Kant's moral views are rationalistic, they restrict moral consideration to rational entities. Many people who commit suicide are not rational at the time of its commission, for they suffer from severe mental illness. Because of this, Kant's suicidology devastatingly excludes certain human demographics (...) from moral consideration, current Canadian statistics indicating that such people mark one of the highest populations at risk of committing suicide in the first place. This work contains a presentation, analysis, and critique of Kant's ethic of suicide, leading to an attempt to state criteria for an adequate suicidology.Dept. of Philosophy. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis2006.T66. Source: Masterss International, Volume: 45-01, page: 0101. Thesis --University of Windsor, 2006. (shrink)

Analyses of biological concepts of disease and social conceptions of health indicate that they are structurally interdependent. This in turn suggests the need for a bridge theory of illness. The main features of such a theory are an emphasis on the logical properties of value terms, close attention to the features of the experience of illness, and an analysis of this experience as action failure, drawing directly on the internal structure of action. The practical applications of this theory are outlined (...) for a number of problems in each of the three main practical areas, clinical work, teaching and research. In each case the resources of the theory suggest new models and generate new results. The full practical significance of the theory, however, is shown to consist in the way in which it ties together biological and social theories into an integrated picture of the conceptual structure of medicine as a whole. It is argued, finally, that practical efficiency of this kind is a test of theory not only in the philosophy of medicine but also in general philosophy. (shrink)

: In Research on Religious Funerary Documents in China, the author studies the religious funerary documents used at funerals for more than two thousand years from the Warring-States period to the present with the truth-seeking and objective attitudes. Besides some criticisms and reasonable doubts, he points out that the previous articles present a somewhat lopsided view with prejudice when the scholars interpret the existing literature. He further explores the underlying implications of the archaeological materials based on studying large numbers of (...) references. In this way, reliable conclusions are obtained with rigorous academic attitude. Resumo: Em Research on Religious Funerary Documents in China, o autor adere às atitudes objetivas e de busca da verdade e questiona as ideias dos principais estudiosos, ao analisar os documentos funerários religiosos usados em funerais por mais de dois mil anos, desde o período dos Estados Combatentes até o presente. O autor critica os pontos de vista de alguns estudiosos famosos e apresenta dúvidas razoáveis a esse respeito. Ele ressalta que seus artigos exibem uma visão um tanto desigual e preconceituosa, na interpretação da literatura existente. Ele ainda seleciona a literatura para um nível mais profundo e explora as implicações subjacentes dos materiais arqueológicos. Dessa forma, são obtidas conclusões, mostrando a abordagem acadêmica dos autores. (shrink)

Why do we need government? A common view is that government is necessary to constrain people's conduct toward one another, because people are not sufficiently virtuous to exercise the requisite degree of control on their own. This view was expressed perspicuously, and artfully, by liberal thinker James Madison, in The Federalist, number 51, where he wrote: “If men were angels, no government would be necessary.” Madison's idea is shared by writers ranging across the political spectrum. It finds clear expression (...) in the Marxist view that the state will gradually wither away after a communist revolution, as unalienated “communist man” emerges. And it is implied by the libertarian view that government's only legitimate function is to control the unfortunate and immoral tendency of some individuals to violate the moral rights of others. (shrink)

A number of results have been obtained concerning Borel structures starting with Silver and Friedman followed by Harrington, Shelah, Marker, and Louveau. Friedman also initiated the model theory of Borel (in fact totally Borel) structures. By this we mean the study of the class of Borel models of a given first-order theory. The subject was further investigated by Steinhorn. The present work is meant to go further in this direction. It is based on the assumption that the study of (...) the class of, say, countable models of a theory reduces to analyzing a single $\omega_1$-saturated model. The question then arises as to when such a model can be totally Borel. We present here a partial answer to this problem when the theory under investigation is superstable. (shrink)

Suppose, we could take a pill that would turn us into morally better people. Would we have a duty to take such a pill? In recent years, a number of philosophers have discussed this issue. Most prominently, Ingmar Persson and Julian Savulescu have argued that we would have a duty to take such a pill. In this article, I wish to investigate the possible limits of a duty to take moral enhancement drugs through investigating the related question of whether (...) it would be desirable to create a world populated entirely with morally perfect people. I argue, drawing on the work of Bernard Williams, Susan Wolf, and Michael Slote, that we have reason to be grateful that we do not live a world in which everyone is morally perfect, as this would prevent people from dedicating their lives to valuable nonmoral projects. I then argue that this thought should serve as a limitation on attempts to morally improve people through the use of technology. Finally, I explore the implications of this discussion for some of the less ambitious forms of moral enhancement currently being explored in the literature. I argue that these forms of enhancement give us no reason to worry about preventing valuable, morally imperfect ways of life. In fact, by acting as a shortcut to moral development, they might serve as an aid to help people fulfill valuable nonmoral goals in a way that is morally permissible. (shrink)

Two models of human perfection proposed by Nietzsche and the Buddha are investigated. Both the overman and the arahant need practice and individual effort as key to their realization, and they share roughly the same conception of the self as a construction. However, there are also a number of salient differences. Though realizing it to be constructed, the overman does proclaim himself through his assertion of the will to power. The realization of the true nature of the self does (...) not lead the overman to seek the way to be released from saṃsara as does the arahant. On the contrary, he rejoices in the eternally recurring situation. The arahant, however, has totally relinquished any attachment to the self, constructed or otherwise. The arahant does not care about the Eternal Recurrence, as he only focuses on the present moment. Finally, they are both beyond good and evil, but in a substantively different way. (shrink)

In modern psychology there are two directions, which from my point of view perfectly complement each other. The founder of the first is KG Jung, and the second - A. Maslow. Jungian psychoanalysis leads to penetration into the hidden soul of the unknown, and the world of archetypal forces and structures. Jung speaks of the process of individuation, of finding the inner center and assimilation of the unconscious by consciousness. The purpose of life is the integration of consciousness, the acquisition (...) of the integrity symbolized by the Self, the Cosmic Man, the guiding center, the archetype whose outer reflection is our "I". A person who has lost touch with the Self lives completely on the surface and her life from the point of view of philosophy is absolutely chaotic and titanic. It is spread over life, is everywhere and nowhere. The stronger the connection with the center, the more we are like our true nature, the more humanity in us, the better we know ourselves and are more holistic. (shrink)

This paper's aim is threefold. First, I wish to show that there is an analogy in section nine that arises out of the interaction of the interlocutors; this analogy is, or has, a certain comic adequatic to the traditional arguments about proofs for the existence of God. Second, Philo's seemingly inconsequential example of the strange necessity of products of 9 in section nine is a perfected analogy of the broken arguments actually given in that section, destroying Philo's earlier arguments. Finally, (...) I raise the question of the designer's intent in creating such a humourous piece. (shrink)

These various paradoxes can be viewed as variants of a ninth to the effect that men ought to do what is absolutely right, although none has sufficient power or knowledge for the purpose. It is to this last paradox that I shall devote the major part of my discourse. I will try to show that the paradox is inescapable, that a number of commonly accepted answers to it are unsatisfactory, and that an adequate answer to it will require a (...) consideration of activities and categories which are outside the province of ethics. Only after these steps are covered will we be in a position to justify our basic contention that ethics and religion are independent and even sometimes incompatible enterprises. (shrink)

Number concepts must support arithmetic inference. Using this principle, it can be argued that the integer concept of exactly ONE is a necessary part of the psychological foundations of number, as is the notion of the exact equality - that is, perfect substitutability. The inability to support reasoning involving exact equality is a shortcoming in current theories about the development of numerical reasoning. A simple innate basis for the natural number concepts can be proposed that embodies (...) the arithmetic principle, supports exact equality and also enables computational compatibility with real- or rational-valued mental magnitudes. (shrink)

One of the problems that Bayesian regularity, the thesis that all contingent propositions should be given probabilities strictly between zero and one, faces is the possibility of random processes that randomly and uniformly choose a number between zero and one. According to classical probability theory, the probability that such a process picks a particular number in the range is zero, but of course any number in the range can indeed be picked. There is a solution to this (...) particular problem on the books: a measure that assigns the same infinitesimal probability to each number between zero and one. I will show that such a measure, while mathematically interesting, is pathological for use in confirmation theory, for the same reason that a measure that assigns an infinitesimal probability to each possible outcome in a countably infinite lottery is pathological. The pathology is that one can force someone to assign a probability within an infinitesimal of one to an unlikely event. (shrink)

The study of the temporal structure of events in natural language is of prime importance in linguistics. Though there has been recent progress on formal theories of events, these theories do not address certain syntactic and semantic properties peculiar to languages such as Hindi. This paper concentrates on properties related to perfectivity. It motivates a small number of semantic features for events and their objects, such as whether an object exists independently of an event, whether it is totally affected (...) by the event, and so on. It then formalizes these features. It also shows how they can be formalized in an algebraic framework and applied in a categorial grammar to derive the properties of verbal and nominal predicates. The result is an integration of descriptive semantics with algebraic theories of objects and events. (shrink)

Given a graph G whose set of vertices is a Polish space X, the weak Borel chromatic number of G is the least size of a family of pairwise disjoint G -independent Borel sets that covers all of X. Here a set of vertices of a graph G is independent if no two vertices in the set are connected by an edge.We show that it is consistent with an arbitrarily large size of the continuum that every closed graph on (...) a Polish space either has a perfect clique or has a weak Borel chromatic number of at most ℵ1. We observe that some weak version of Todorcevic's Open Coloring Axiom for closed colorings follows from MA.Slightly weaker results hold for Fσ-graphs. In particular, it is consistent with an arbitrarily large size of the continuum that every locally countable Fσ-graph has a Borel chromatic number of at most ℵ1.We refute various reasonable generalizations of these results to hypergraphs. (shrink)

The article explores the evolution of the idea of correlating numbers and meanings, from ancient numerological systems to modern models of natural language processing based on vector representations and neural networks. The authors demonstrate that the aspiration to uncover hidden properties of objects by associating them with numbers and performing operations on these numbers has been a common thread across various cultures for millennia. The article traces the stages in the formation of the concept of mathesis universalis (universal mathematics), starting (...) from Aristotle and Proclus’s reflections on general mathematics and culminating in R. Descartes’s ideas about the existence of a universally applicable science of order and measure. Particular attention is devoted to G.W. Leibniz’s ambitious project of creating a universal characteristic – a formalized language in which all knowledge could be precisely expressed and logical inference could be carried out through calculations. It is noted that, despite inherent limitations, some of Leibniz’s intuitions have found embodiment in natural language processing technologies based on vector representations of words. However, the simple correlation of concepts with vectors is insufficient for the full realization of the idea of a calculus of reasoning, as it lacks a mechanism for deriving true statements from others. The development of the transformer architecture, which employs the mechanism of self-attention to construct context-dependent vector representations, has been a significant stride towards the automation of reasoning. Nevertheless, modern language models are based on the principle of maximizing likelihood rather than rigorous logical rules. The authors analyze conceptual solutions proposed for creating a comprehensive calculus of knowledge. In conclusion, it is emphasized that, while the idea of reducing reasoning to mathematical operations is undeniably attractive, one should be cautious about its maximalist interpretations. Cognition is constructive in nature, and even the most perfect model of formalization of reasoning cannot substitute for the practical wisdom and freedom of human thought. (shrink)

We examined how language supports the expression of temporality within sentence boundaries in English, which has a rich inventory of grammatical means to express temporality. Using a computational model that mimics how humans learn from exposure we explored what the use of different tense and aspect combinations reveals about the interaction between our experience of time and the cognitive demands that talking about time puts on the language user. Our model was trained on n-grams extracted from the BNC to select (...) the TA combination that fits the context best. It revealed the existence of two different sub-systems within the set of TA combinations, a “simplex” one that is supported lexically and is easy to learn, and a “complex” one that is supported contextually and is hard to learn. The finding that some TA combinations are essentially lexical in nature necessitates a rethink of tense and aspect as grammatical categories that form the axes of the temporal system. We argue that the system of temporal reference may be more fruitfully thought of as the result of learning a system that is steeped in experience and organised along a number of functional principles. (shrink)

We study the formation of a communication network under perfect foresight. We show the existence of a non-monotonic relationship between the cost of link formation and the total number of links created in stable networks. This result enhances a dilemma between stable and efficient networks.

There is a tension between Dispositionalism––the view that all metaphysical modality is grounded in actual irreducible dispositional properties––and the possibility of time travel. This is due to the fact that Dispositionalism makes it much harder to solve a potentiality-based version of the grandfather paradox. We first present a potentiality-based version of the grandfather paradox, stating that the following theses are inconsistent: 1) time travel is possible, 2) powers fully ground modality, 3) self-defeating actions are impossible, 4) time-travellers retain their intrinsic (...) powers upon time-travelling, and 5) time-travellers are ordinary agents with basic intrinsic potentialities. We then consider a number of potential solutions, and find them wanting. We argue that the metaphysical impossibility of performing a self-defeating action acts as a necessary perfect mask––while time-travel lets us “slip” the potentiality under the mask, thus generating the contradiction. We conclude considering what are the options for the dispositionalist. (shrink)

According to Karl Popper, science cannot verify its theories empirically, but it can falsify them, and that suffices to account for scientific progress. For Popper, a law or theory remains a pure conjecture, probability equal to zero, however massively corroborated empirically it may be. But it does just seem to be the case that science does verify empirically laws and theories. We trust our lives to such verifications when we fly in aeroplanes, cross bridges and take modern medicines. We can (...) do some justice to this apparent capacity of science to verify if we make a number of improvements to Popper’s philosophy of science. The key step is to recognize that physics, in accepting unified theories only, thereby makes a big metaphysical assumption about the nature of the universe. The outcome is a conception of scientific method which facilitates the criticism and improvement of metaphysical assumptions of physics. This view provides, not verification, but a perfect simulacrum of verification indistinguishable from the real thing. (shrink)

Plato believed that behind everything in the universe lie mathematical principles. Plato was inspired by Pythagoras (571 BCE), who developed a school of mathematics at Crotona that studied sacred geometry as a form of religion. The school’s motto was “God is number,” or “All is Number”. Pythagoras believed that numbers represented God in pattern, symmetry, and infinity. When one of its students, Hippasus told the world the secret of the existence of irrational numbers, Greek geometry was born and (...) Pythagoras’ idea of divinity in numbers died because how could God not be perfect and symmetrical? In Plato’s Republic he discusses something called The Divided Line, which is a map, of sorts, for reaching what he calls the highest Good, which is the ultimate truth where one realizes the true state of the universe and can see the world for what it really is. Many mathematicians have attempted to plot Plato’s Divided Line only to come across a litany of problems and conundrums. Some have said that it the Divided Line cannot be plotted and is merely an allegory not meant to be plotted. This paper discusses some of the conundrums preventing the plotting of Plato’s Divided Line (not an exhaustive list), including Whole ‘vs’ Separate, Equality ‘vs’ Ontological Dissimilarity, Linear ‘vs’ Non-linear, and Infinity ‘vs’ Finite. This paper also explores a new understanding of the Allegory of the Cave in light of ‘the problem of the irrational.’ In exploring the link between the Divided Line and the ‘the problem of the irrational,’ I was able to plot it. It was found that the Divided Line is not a line in the linear sense, but a spiral, the Golden Ratio! This paper is an example of a new category of scholarly inquiry I call “Math Theory” based on scholarly mathematical axioms in theory, rather than including actual maths. In my papers I use existing mathematical equations and place them in an encompassing theory, rather than finding new formulae to fit an existing theory. Keywords: Pythagoras, divided line, math theory, highest good, all is number. (shrink)

The aim of this article is both to make a contribution towards a fuller understanding of the use of the perfect in Classical Greek, and to show how this understanding can yield new insights into how a speaker uses language to adapt his presentation of past events to his present rhetorical concerns. First, the semantic value of the perfect and its different basic uses are described. Second, four principles that help accounting for the variation between the perfect (...) and aorist are outlined. Third, a number of typical uses of the perfect in Attic forensic oratory are distinguished. Finally, a textual analysis of De Corona 57-60, 79-121 is given in order to show how Demosthenes uses the perfect to accomplish his rhetorical ends. (shrink)

The scattering number and isolated scattering number of a graph have been introduced in relation to Hamiltonian properties and network vulnerability, and the isolated scattering number plays an important role in characterizing graphs with a fractional 1-factor. Here we investigate the computational complexity of one variant, namely, the weighted isolated scattering number. We give a polynomial time algorithm to compute this parameter of interval graphs, an important subclass of perfect graphs.