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)

This is a survey of the concept of continuity. Efforts to explicate continuity have produced a plurality of philosophical conceptions of continuity that have provably distinct expressions within contemporary mathematics. I claim that there is a divide between the conceptions that treat the whole continuum as prior to its parts, and those conceptions that treat the parts of the continuum as prior to the whole. Along this divide, a tension emerges between those conceptions that favor philosophical idealizations of continuity and (...) those that are more readily available for mathematico-scientific use. (shrink)

The primary purpose of this paper is to analyze the relationship between the familiar non-Archimedean field of hyperreals from Abraham Robinson’s nonstandard analysis and Paolo Giordano’s ring extension of the real numbers containing nilpotents. There is an interesting nontrivial homomorphism from the limited hyperreals into the Giordano ring, whereas the only nontrivial homomorphism from the Giordano ring to the hyperreals is the standard part function, namely, the function that maps a value to its real part. We interpret this asymmetry to (...) mean that the nilpotent infinitesimal values of Giordano’s ring are “smaller” than the hyperreal infinitesimals. By viewing things from the “point of view” of the hyperreals, all nilpotents are zero, whereas by viewing things from the “point of view” of Giordano’s ring, nonnilpotent, nonzero infinitesimals register as nonzero infinitesimals. This suggests that Giordano’s infinitesimals are more fine-grained. (shrink)

I argue for the possibility of hypergunk: that is, it is possible that there exists an x such that every part of x has a proper part and, for any set S of parts of x, there is a set S′ of parts of...

We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like (...) Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves. (shrink)

For almost twenty years, Penelope Maddy has been one of the most consistent expositors and advocates of naturalism in philosophy, with a special focus on the philosophy of mathematics, set theory in particular. Over that period, however, the term ‘naturalism’ has come to mean many things. Although some take it to be a rejection of the possibility of a priori knowledge, there are philosophers calling themselves ‘naturalists’ who willingly embrace and practice an a priori methodology, not a whole lot different (...) from traditional conceptual analysis. Along a different line, some take naturalism to involve the rejection of abstract objects—a sort of physicalism—while other naturalists not only allow the existence of abstracta; they take this existence to be all but obvious.For present purposes, we can begin with W.V.O. Quine, who once characterized naturalism as ‘the abandonment of the goal of first philosophy’ and ‘the recognition that it is within science itself …that reality is to be identified and described’. For Quine, the ‘naturalistic philosopher begins his reasoning within the inherited world theory as a going concern …[The] inherited world theory is primarily a scientific one, the current product of the scientific enterprise’ [Quine, 1981, p. 72]. This characterizes at least the bulk of contemporary philosophers who call themselves ‘naturalists’, and it characterizes the targets of many who oppose what they call ‘naturalism’. But as Maddy notes at the start, ‘the term has come to mark little more than a vague science-friendliness’.An attempt to define naturalism more fully would surely require a characterization of science—at least so that the reader can see when someone sins against that naturalism by being unscientific. But, as Maddy notes, one ‘lesson won through decades of study in the philosophy of science [is that] there is no …. (shrink)

The second edition of The Theory and Practice of Husserl's Phenomenology is a clear and concise introduction to the theoretical background and the rigorous method of Edmund Husserl , perhaps the most influential philosopher of the twentieth century and the founder of the phenomenological movement. According to Husserl phenomenology is not a body of knowledge but a scientific practice based in a rigorous and difficult method, a method that takes long effort and practice to enter into and in which to (...) develop some significant results. Since many scholars and students do not read many of Husserl's published works, this text is meant to show the core unity of focus and method throughout decades of his phenomenological practice and refinement. The present work covers the historical origin of phenomenology and some main thematic results that appear in Husserl's works, and provides an introduction to the "how to" of producing phenomenological description. (shrink)

Various philosophers have wanted to move from a (P.F.) “Strawsonian” understanding of the “practices of moral responsibility” to a non-skeptical result. I focus on a strategy moving from a “response-dependent” theory of responsibility. I aim to show that a key analogy associated with this strategy fails to support a compatibilist result. It seems clear that nothing could show that nothing we have been laughing at has really been funny. If “the funny” is similar to “the blameworthy”, then perhaps it would (...) follow that nothing could show that no one we have ever blamed has really been blameworthy. The comparison is interesting, but inconclusive: even if we grant that these properties are normatively similar, the latter has substantive empirical presuppositions, whereas the former does not. One important upshot: even if the standards operative in our practices cannot be mistaken, it could nevertheless be that no one has ever met these standards Ruling out this possibility seemingly requires re-engaging in all of the so-called "metaphysical" debates Strawsonians are keen to avoid. (shrink)

This modern, advanced textbook reviews modal logic, a field which caught the attention of computer scientists in the late 1970's.

Moral latitude is only ever a matter of coincidence on the most popular decision procedure in the literature on moral uncertainty. In all possible choice situations other than those in which two or more options happen to be tied for maximal expected choiceworthiness, Maximize Expected Choiceworthiness implies that only one possible option is uniquely appropriate. A better theory of appropriateness would be more sensitive to the decision maker’s credence in theories that endorse agent-centred prerogatives. In this paper, we will develop (...) and defend a modified version of this view that allows for agent-centred prerogatives and supererogation as part of its verdicts about appropriateness. (shrink)

"'There is only one thing that is certain, namely that we can have nothing certain; and therefore it is not certain that we can have nothing certain,' Samuel Butler once said, expressing in that mindbloggler all the elements required to form a classical paradox. Throughout the ages wise men and jesters alike have been intrigued by such mental twists and riddles which defy common sense and yet appear to be true." -- Dust jacket.

Inductively defined structures are ubiquitous in mathematics; their specification is unambiguous and their properties are powerful. All fields of mathematical logic feature these structures prominently: the formula of a language, the set of theorems, the natural numbers, the primitive recursive functions, the constructive number classes and segments of the cumulative hierarchy of sets. -/- This dissertation gives a mathematical characterization of a species of inductively defined structures, called accessible domains, which include all of the above examples except the set of (...) theorems. The concept of an accessible domain comes from Wilfried Sieg's analysis of proof-theoretic practices, starting with his dissertation. In particular, he noticed the special epistemological character of elements of an accessible domain: they can always be uniquely identified with their build-up. Generally, the unique build-up of elements justifies the principles of induction and recursion. -/- I use category theory to give an abstract characterization of accessible domains. I claim that accessible domains are all instances of initial algebras for endofunctors. Grounded in the historical roots of Sieg's discussions, this dissertation shows how the properties of initial algebras for endofunctors and accessible domains coincide in a satisfying and natural way. -/- Filling out this characterization, I show how important examples of accessible domains fit into this broad characterization. I first characterize some accessible domains by relatively simple functors (e.g. finite, polynomial, those that preserve certain colimits) where we can see how iterating the functor can produce the accessible domain. Then I describe accessible domains that result from more involved specifications (e.g. ordinals and segments of the cumulative hierarchies associated with CZF, IZF, and ZF) by relying heavily on algebraic set theory. I end with a discussion of some of the methodological features of category theory in particular that helped characterize accessible domains. (shrink)

What kind of persons could engage in political torture? Not only the morally impaired who lack empathy or compassion, or even the merely obedient, but also the righteous who struggle with conscience, and the realists who set morality aside.

This volume centers on debates about how far moral judgments bind across traditions and epochs. Nowadays such debates appear especially volatile, both in popular culture and intellectual discourse: although there is increasing agreement that the moral and political criteria invoked in human rights documents possess cross-cultural force, many modern and postmodern developments erode confidence in moral appeals that go beyond a local consensus or apply outside a particular community. Often the point of departure for discussion is the Enlightenment paradigm of (...) a common morality, in which it is assumed that certain unchanging beliefs inhere in the structure of human reason. Whereas some thinkers continue to defend this paradigm, others modify it in diverse ways without abandoning entirely the attempt to address a universal audience, and still others jettison virtually all of its distinguishing features. Exhibiting a range of positions Western participants take in these debates, this volume seeks to advance the substance of the debates themselves without prejudging the outcome. Rival assessments of the Enlightenment paradigm are offered from various philosophical and theological points of view. In addition to the editors, the contributors include Robert Merrihew Adams, Annette C. Baier, Alan Donagan, Margaret A. Farley, Alan Gewirth, David Little, Richard Rorty, Jeffrey Stout, and Lee H. Yearley. (shrink)

We present an argument in support of the legal prohibition of infant male circumcision (IMC) in developed Western countries. We submit that all IMC, irrespective of whether the motivation behind it be secular or religious, violates children’s rights to self-determination (autonomy) and bodily integrity and is therefore morally illegitimate. And while IMC’s being morally wrong does not entail that it ought to be criminalised, we contend that it should be legally proscribed so as to protect children against harm and to (...) uphold their rights. We consider and reject an argument that an exemption from the legal ban should be granted to religious parents who perceive themselves to be under a moral obligation to subject their male infants to IMC. We argue that since IMC sets back children’s basic interests and violates their rights, there is no room, and no case, for the conferring of a religious exemption. (shrink)

Thomas Aquinas has argued that the passion of hope is the movement of the sensitive appetite and the first of the irascible passion. The first part of the article aims to explore the cause and the mechanism of the passion of hope, and tries to clarify the relationship between the passion of hope and the perception. In human beings, it is possible that the passion of hope is caused by false judgement of the perception, which will lead to the result (...) of false hope. In Aquinas’s argument, the problem of false hope could be solved through the moral virtue and the prudence. In the second part of this article, the aim is to analyse how the virtue of magnanimity, humility and the prudence work impact on the passion of hope and perfect it.Contribution: This research concentrates on the topic of the passion of hope of Thomas Aquinas, and provides a new perspective on the understanding of the relationship between the passion and the virtue. (shrink)

At least since Aristotle's famous discussion of the sea-battle tomorrow in On Interpretation 9, philosophers have been fascinated by a rich set of interconnecte.

_"One of the most important political books of 2018."—Rod Dreher, ___American Conservative__ Of the three dominant ideologies of the twentieth century—fascism, communism, and liberalism—only the last remains. This has created a peculiar situation in which liberalism’s proponents tend to forget that it _is _an ideology and not the natural end-state of human political evolution. As Patrick Deneen argues in this provocative book, liberalism is built on a foundation of contradictions: it trumpets equal rights while fostering incomparable material inequality; its (...) legitimacy rests on consent, yet it discourages civic commitments in favor of privatism; and in its pursuit of individual autonomy, it has given rise to the most far-reaching, comprehensive state system in human history. Here, Deneen offers an astringent warning that the centripetal forces now at work on our political culture are not superficial flaws but inherent features of a system whose success is generating its own failure. (shrink)

This volume centers on debates about how far moral judgments bind across traditions and epochs. Nowadays such debates appear especially volatile, both in popular culture and intellectual discourse: although there is increasing agreement that the moral and political criteria invoked in human rights documents possess cross-cultural force, many modern and postmodern developments erode confidence in moral appeals that go beyond a local consensus or apply outside a particular community. Often the point of departure for discussion is the Enlightenment paradigm of (...) a common morality, in which it is assumed that certain unchanging beliefs inhere in the structure of human reason. Whereas some thinkers continue to defend this paradigm, others modify it in diverse ways without abandoning entirely the attempt to address a universal audience, and still others jettison virtually all of its distinguishing features. Exhibiting a range of positions Western participants take in these debates, this volume seeks to advance the substance of the debates themselves without prejudging the outcome. Rival assessments of the Enlightenment paradigm are offered from various philosophical and theological points of view. In addition to the editors, the contributors include Robert Merrihew Adams, Annette C. Baier, Alan Donagan, Margaret A. Farley, Alan Gewirth, David Little, Richard Rorty, Jeffrey Stout, and Lee H. Yearley. (shrink)

"A thematic exposition focused on the "whole man," especially in his practical, aesthetic, ethical, and religious dimensions, moving from consideration of the stream of consciousness and consciousness as selective according to interests, through the ethical and religious aspects of man's aspiration and experience, to the humanistic bases of James' pragmatism and radical empiricism ... Dooley's account is remarkably clear and streamlined, stressing the consistency rather than the tensions in James' thought. Thus, while James' own texts provide at once the most (...) authoritative and the most attractive approach to his thought for readers at all levels, Dooley's book seems most likely to be helpful as supplementary reading for teachers and students in introductory courses.". (shrink)

Averill (1990) argues that not every property is a disposition. I claim here that his reasoning is faulty, suffering at one point from a logical error and at other points from an inadequate account of counterfactuals.

The Handbook of Modal Logic contains 20 articles, which collectively introduce contemporary modal logic, survey current research, and indicate the way in which the field is developing. The articles survey the field from a wide variety of perspectives: the underling theory is explored in depth, modern computational approaches are treated, and six major applications areas of modal logic (in Mathematics, Computer Science, Artificial Intelligence, Linguistics, Game Theory, and Philosophy) are surveyed. The book contains both well-written expository articles, suitable for beginners (...) approaching the subject for the first time, and advanced articles, which will help those already familiar with the field to deepen their expertise. Please visit: http://people.uleth.ca/~woods/RedSeriesPromo_WP/PubSLPR.html - Compact modal logic reference - Computational approaches fully discussed - Contemporary applications of modal logic covered in depth. (shrink)

Focusing on a 1628 Lenten sermon, this essay explores Donne’s handling of the complex problem of how we should conceive of time antecedent to the creation. Read in the light of Donne’s remarks elsewhere, and of those by St. Augustine, this sermon shows him considering pre-time time as an argument for the importance of vocation. Contemplation of the mystery of God’s activity before the creation yields a powerful idea of one’s life and calling as the surpassingly meaningful culmination of a (...) vast process. Additionally, such contemplation helps Donne cope with the vagaries of his own life, especially his ties to Roman Catholicism, by contextualizing his own past-ness within the larger story of humanity’s sin and redemption. (shrink)

Focusing on a 1628 Lenten sermon, this essay explores Donne’s handling of the complex problem of how we should conceive of time antecedent to the creation. Read in the light of Donne’s remarks elsewhere, and of those by St. Augustine, this sermon shows him considering pre-time time as an argument for the importance of vocation. Contemplation of the mystery of God’s activity before the creation yields a powerful idea of one’s life and calling as the surpassingly meaningful culmination of a (...) vast process. Additionally, such contemplation helps Donne cope with the vagaries of his own life, especially his ties to Roman Catholicism, by contextualizing his own past-ness within the larger story of humanity’s sin and redemption. (shrink)

This book launches a sustained defense of a radical interpretation of the doctrine of the open future. Patrick Todd argues that all claims about undetermined aspects of the future are simply false.

One approach to the problem of differentiating a religious from a non- religious ethic would be to formulate a definition of religion that would clearly distinguish between religious and nonreligious traditions; however, a broad definition of religion would include some moral traditions, such as Marxism, commonly thought to be forms of secular humanism. A second approach would argue that some moral beliefs are independent, both in content and justification, of religious convictions; such a set of moral beliefs could be described (...) as a secular version of natural law. This approach would be rejected by those who argue that religious convictions go "all the way down." While the first can be debated by appealing to the heuristic value of various definitions of religion, the second involves familiar issues of moral epistemology. The author explores the consequences of holding a broad definition of religion and an "all the way down" epistemology. (shrink)

El sentido como cabe o entre en el orden de lo público, en laCondición humana de Hannah Arendt, es de suyo esclarecedorde las posibilidades del lenguaje; y, de retorno, el lenguaje comofundamento para la comprensión de lo político (público); en estosdos lados se presenta como una dialéctica del fluir de la acción–que quizá por igual puede llamarse: la experiencia humana delmundo. Interesa, igualmente, ver la distinción entre el fluir –en elmodo del río heraclíteo– del sentido y la emergencia –en el (...) mododel nacimiento del agua en una fuente. Nuestra hipótesis es queen el pensamiento de Arendt tiene más el segundo que el primersentido. ¿Cómo se relacionan en este caso tiempo y sentido? Es,como telón de fondo, la pregunta que atraviesa nuestro “ensayode comprensión”1. La ponencia se desarrollo en cinco apartados;en el apartado, I La preeminencia del derecho, se mira el enlaceentre la teoría política de la autora y juicio de Eichmann; en elapartado II, La identidad narrativa, se estudia la primacía de lajustificación en mediante la cual el sujeto se construye en el relatoy, así mismo, el ir hacia o la apertura de sentido que ofrece elmismo; en el apartado III, Lo público, se estudia cómo la intersubjetividades terreno común para llegar a construir lo propio, la esfera de propiedad; en el apartado IV, El lenguaje, se mira sucarácter de entre, en medio de o cabe tanto en las relacionesintersubjetivas como en la comprensión subjetiva de mundo; finalmente,el apartado V, La persona, y a manera de conclusión sediscute la preeminencia de las tesis agustinianas sobre la persona–su carácter de hipóstasis al ser, al tiempo, entre y suidad– encuanto relación, así mismo se discuten los alcances y limitacionesde la tesis de la autora siempre que se evalúen a la luz de la perspectivafenomenológico-genética para el estudio de la políticacomo cosa misma. (shrink)

The author uses social epistemology to develop the cognitive authority theory. The fundamental concept of cognitive authority is that people construct knowledge in two different ways: based on their first-hand experience or on what they have learned second-hand from others. What people learn first-hand depends on the stock of ideas they bring to the interpretation and understanding of their encounters with the world. People primarily depend on others for ideas as well as for information outside the range of direct experience. (...) Much of what they think of the world is what they have gained second-hand. All that people know of the world beyond the narrow range of their own lives is what others have told them. However, people do not count all hearsay as equally reliable; only those who are deemed to “know what they are talking about” become cognitive authorities. --. (shrink)

As well as disagreeing about how much one should donate to charity, moral theories also disagree about where one should donate. In light of this disagreement, how should the morally uncertain philanthropist allocate her donations? In many cases, one intuitively attractive option is for the philanthropist to split her donations across all of the charities that are recommended by moral views in which she has positive credence, with each charity’s share being proportional to her credence in the moral theories that (...) recommend that particular charity. Despite the fact that something like this approach is already being used by real-world philanthropists to distribute billions of dollars of donations, it is not supported by any of the approaches to moral uncertainty that have been proposed thus-far in the philosophical literature. In this paper, we will develop a bargaining-based approach to moral uncertainty that honors the proportionality intuition in favor of splitting one’s donations. We also show how this bargaining-based approach has several further advantages over the best alternative proposals. (shrink)

In a sample of 615 Belgians a model for fair trade buying behaviour was developed. The impact of fair trade knowledge, general attitudes towards fair trade, attitudes towards fair trade products, and the perception of the quality and quantity of fair trade information on the reported amount of money spent on fair trade products were assessed. Fair trade knowledge, overall concern and scepticism towards fair trade, and the perception of the perceived quantity and quality of fair trade information, influence buying (...) behaviour directly and indirectly through product attitudes. Interest in fair trade products, price acceptability and product liking have a significant impact on fair trade buying behaviour. Product interest is the most important variable influencing buying behaviour. Implications for the campaigns of governments and for the marketing strategy of fair trade organisations are offered. (shrink)