This Element, written for researchers and students in philosophy and the behavioral sciences, reviews and critically assesses extant work on number concepts in developmental psychology and cognitive science. It has four main aims. First, it characterizes the core commitments of mainstream number cognition research, including the commitment to representationalism, the hypothesis that there exist certain number-specific cognitive systems, and the key milestones in the development of number cognition. Second, it provides a taxonomy of influential views within mainstream number cognition research, (...) along with the central challenges these views face. Third, it identifies and critically assesses a series of core philosophical assumptions often adopted by number cognition researchers. Finally, the Element articulates and defends a novel version of pluralism about number concepts. (shrink)

Hofweber (Ontology and the ambitions of metaphysics, Oxford University Press, 2016) argues for a thesis he calls “internalism” with respect to natural number discourse: no expressions purporting to refer to natural numbers in fact refer, and no apparent quantification over natural numbers actually involves quantification over natural numbers as objects. He argues that while internalism leaves open the question of whether other kinds of abstracta exist, it precludes the existence of natural numbers, thus establishing what he calls “restricted nominalism” about (...) natural numbers. We argue that Hofweber’s internalism fails to establish restricted nominalism. Not only is his primary argument for restricted nominalism invalid, the analysis of quantification proposed threatens to collapse internalism into either a traditional form of error theory or realism. (shrink)

The Renaissance was a period of great intellectual change and innovation as philosophers rediscovered the philosophy of classical antiquity and passed it on to the modern age. Renaissance philosophy is distinct both from the medieval scholasticism, based on revelation and authority, and from philosophers of the seventeenth and eighteenth centuries who transformed it into new philosophical systems. Despite the importance of the Renaissance to the development of philosophy over time, it has remained largely understudied by historians of philosophy and professional (...) philosophers. This anthology aims to correct this by providing scholars and students of philosophy with representative translations of the most important philosophers of the Renaissance. Its purpose is to help readers appreciate philosophy in the Renaissance and its importance in the history of philosophy. The anthology includes translations from philosophers from the thirteenth to the seventeenth centuries, and it ranges from works on moral and political philosophy, to metaphysics, epistemology, and natural philosophy, thereby providing historians and students of philosophy with a sense for the nature, breadth, and complexity of philosophy in the Renaissance. Each translation is accompanied by an introduction by a historian of Renaissance philosophy, as well as select secondary sources, in order to encourage further study. (shrink)

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Nevertheless, some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is more “legitmate” in virtue of being “more basic” or “more fundamental”. This paper addresses two related issues. First, (...) we review some of these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege’s foundationalist epistomology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally “legitimate”. (shrink)

In this paper, we outline and critically evaluate Thomas Hofweber’s solution to a semantic puzzle he calls Frege’s Other Puzzle. After sketching the Puzzle and two traditional responses to it—the Substantival Strategy and the Adjectival Strategy—we outline Hofweber’s proposed version of Adjectivalism. We argue that two key components—the syntactic and semantic components—of Hofweber’s analysis both suffer from serious empirical difficulties. Ultimately, this suggests that an altogether different solution to Frege’s Other Puzzle is required.

Number words seemingly function both as adjectives attributing cardinality properties to collections, as in Frege’s ‘Jupiter has four moons’, and as names referring to numbers, as in Frege’s ‘The number of Jupiter’s moons is four’. This leads to what Thomas Hofweber calls Frege’s Other Puzzle: How can number words function as modifiers and as singular terms if neither adjectives nor names can serve multiple semantic functions? Whereas most philosophers deny that one of these uses is genuine, we instead argue that (...) number words, like many related expressions, are polymorphic, having multiple uses whose meanings are systematically related via type shifting. (shrink)

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of _which number_ is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of _notation_. The purpose of this article is to explore the relationship between (...) the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects. (shrink)

One of the more distinctive features of Bob Hale and Crispin Wright’s neologicism about arithmetic is their invocation of Frege’s Constraint – roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. In particular, they maintain that, if adopted, Frege’s Constraint adjudicates in favor of their preferred foundation – Hume’s Principle – and against alternatives, such as the Dedekind-Peano axioms. In what follows we establish two main claims. First, we show (...) that, if sound, Hale and Wright’s arguments for Frege’s Constraint at most establish a version on which the relevant application of the naturals is transitive counting – roughly, the counting procedure by which numerals are used to answer “how many”-questions. Second, we show that this version of Frege’s Constraint fails to adjudicate in favor of Hume’s Principle. If this is the version of Frege’s Constraint that a foundation for arithmetic must respect, then Hume’s Principle no more – and no less – meets the requirement than the Dedekind-Peano axioms do. (shrink)

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is ‘more basic’ or ‘more fundamental’ than the others. This paper addresses two related issues. First, we review some of (...) these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege's foundationalist epistemology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally ‘legitimate’. (shrink)

Clarke and Beck import certain assumptions about the nature of numbers. Although these are widespread within research on number cognition, they are highly contentious among philosophers of mathematics. In this commentary, we isolate and critically evaluate one core assumption: the identity thesis.

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship between (...) the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects. (shrink)

A core commitment of Bob Hale and Crispin Wright’s neologicism is their invocation of Frege’s Constraint—roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. According to these neologicists, if legitimate, Frege’s Constraint adjudicates in favor of their preferred foundation—Hume’s Principle—and against alternatives, such as the Dedekind–Peano axioms. In this paper, we consider a recent argument for legitimating Frege’s Constraint due to Hale, according to which the primary empirical application of (...) the naturals is transitive counting, or answering ‘how many’-questions using numerals. We make two claims regarding Hale’s argument. First, it fails to legitimate Frege’s Constraint in virtue of resting on unsupported and highly contentious assumptions. Secondly, even if sound, Hale’s argument would vindicate a version of Frege’s Constraint which fails to adjudicate in favor of Hume’s Principle over alternative characterizations of the naturals. (shrink)

The dividing walls of hostility -- The roots of othering -- The forms of othering -- The violent consequences of othering -- Voices of the moral imperative -- And the walls come tumbling down.

Is evil evidence against the existence of God? Even if God and evil are compatible, it remains hotly contested whether evil renders belief in God unreasonable. The Evidential Argument from Evil presents five classic statements on this issue by eminent philosophers and theologians and places them in dialogue with eleven original essays reflecting new thinking by these and other scholars. The volume focuses on two versions of the argument. The first affirms that there is no reason for God to permit (...) either certain specific horrors or the variety and profusion of undeserved suffering. The second asserts that pleasure and pain, given their biological role, are better explained by hypotheses other than theism. -/- Contributors include William P. Alston, Paul Draper, Richard M. Gale, Daniel Howard-Snyder, Alvin Plantinga, William L. Rowe, Bruce Russell, Eleonore Stump, Richard G. Swinburne, Peter van Inwagen, and Stephen John Wykstra. (shrink)

A descriptive polytheist thinks there are at least two gods. John Hick and Richard Swinburne are descriptive polytheists. In this respect, they are like Thomas Aquinas and many other theists. What sets Swinburne and Hick apart from Aquinas, however, is that unlike him they are normative polytheists. That is, Swinburne and Hick think that it is right that we, or at least some of us, worship more than one god. However, the evidence available to me shows that only Swinburne, (...) and not Hick, is a cultic polytheist: he actually worships more than one god. I conclude that only Swinburne is a polytheist par excellence. (shrink)

This 9,000+ word entry briefly assesses five models of the Trinity, those espoused by (i) Richard Swinburne, (ii) William Lane Craig, (iii) Brian Leftow, (iv) Jeff Brower and Michael Rea, and (v) Peter van Inwagen.

By the middle of the nineteenth century, the opinion of science, as well as of philosophy and even religion, was, at least in Britain, firmly in the camp of the plurality of worlds, the view that intelligent life exists on other celestial bodies. William Whewell, considered an expert on science, philosophy and religion, would have been expected to support this position. Yet he surprised everyone in 1853 by publishing a work arguing strongly against the plurality view. This was even stranger (...) given that he had endorsed pluralism twenty years earlier in his contribution to the Bridgewater Treatises. In this paper I show that the shift in Whewell’s view was motivated by three factors: the influence of Richard Owen’s theory of archetypes on Whewell’s view of the argument from design, and Whewell’s perception of the need to strengthen such arguments in light of evolutionary accounts of human origins; important developments in his view of philosophy and his role as a scientific expert; and new findings in astronomy. An examination of the development of Whewell’s position provides a lens through which we can view the interplay of theology, philosophy and science in the plurality of worlds debate.Keywords: William Whewell; Plurality of worlds; Natural theology; Analogies; Archetypes; Conjectures; Evolution. (shrink)

Faith in God conflicts with reason—or so we’re told. We focus on two arguments for this conclusion. After evaluating three criticisms of them, we identify an assumption they share, namely that faith in God requires belief that God exists. Whether the assumption is true depends on what faith is. We sketch a theory of faith that allows for both faith in God without belief that God exists, and faith in God while in belief-cancelling doubt God’s existence. We then argue that (...) our theory, unlike the theory of Thomas Aquinas, makes sense of four central items of faith-data: (i) pístis in the Synoptics, (ii) ʾemunāh in the Hebrew scriptures, (iii) exemplars of faith in God, including Abraham, Jesus, and Mother Teresa, and (iv) the widespread experience of people of faith today. We close by assessing revisions of the two arguments we began with, revisions that align with our theory of faith, and find them dubious, at best. (shrink)

Basic concepts -- Identifying arguments -- Logic and language -- Informal fallacies -- Categorical logic: statements -- Categorical logic: syllogisms -- Statement logic: truth tables -- Statement logic: proofs -- Predicate logic -- Induction -- Probability.

For many people the existence of God is by no means a sufficiently clear feature of reality. This problem, the fact of divine hiddenness, has been a source of existential concern and has sometimes been taken as a rationale for support of atheism or agnosticism. In this collection of essays, a distinguished group of philosophers of religion explore the question of divine hiddenness in considerable detail. The issue is approached from several perspectives including Jewish, Christian, atheist and agnostic. There is (...) coverage of the historical treatment of divine hiddenness as found in the work of Maimonides, St. John of the Cross, Jonathan Edwards, Kierkegaard, and various Biblical writers. A substantial introduction clarifies the main problems of and leading solutions to divine hiddenness. Primarily directed at philosophers of religion, theologians, and scholars of religious studies, this collection could also serve as a textbook for upper-level courses in philosophy of religion. (shrink)

This paper summarizes a version of the argument from evil for atheism and then assesses several theodicies, including those that appeal to punishment, evil as a necessary counterpart for good, free will, natural evil as natural consequence, natural law, higher-order goods, and the conjunctive "Big Reason" including all the above and more beside.

Written across the disciplines of art history, literature, philosophy, sociology, and theology, the ten essays comprising the collection all insist on multidimensional definitions of evil. Taking its title from a moment in Shakespeare's Tempest when Prospero acknowledges his responsibility for Caliban, this collection explores the necessarily ambivalent relationship between humanity and evil. To what extent are a given society's definitions of evil self-serving? Which figures are marginalized in the process of identifying evil? How is humanity itself implicated in the production (...) of evil? Is evil itself something fundamentally human? These questions, indicative of the kinds of issues raised in this collection, seem all the more pressing in light of recent world events. The ten essays were originally presented at the First Global Conference on Perspectives on Evil and Human Wickedness, held in March 2000 in Lady Margaret Hall, Oxford University. (shrink)

The theory of action underlying Immanuel Kant's ethical theory is the subject of this book.

In their recent article ‘Resolving Frege’s other Puzzle’ Eric Snyder, Richard Samuels, and Stewart Shapiro defend a semantic type-shifting solution to Frege’s other Puzzle and criticize my own cognitive type-shifting solution. In this article I respond to their criticism and in turn point to several problems with their preferred solution. In particular, I argue that they conflate semantic function and semantic value, and that their proposal is neither based on general semantic type-shifting principles nor adequate to the data.

This book offers students a philosophical introduction to the ethical foundations of business management. It combines lessons from Kant with virtue ethics and also touches upon additional approaches such as utilitarianism. At the core of the book lies the concept of the nexus of imperfect managerial duty: building and reinforcing the virtuous managerial team, engaging in reasoned discourse among all stakeholders, and diligently pursuing business responsibilities, including the creative efforts necessary for modern organizations. Case illustrations of these applications are presented (...) throughout the book, including chapter appendices. Ancillary videos, test and answer banks and sample syllabi are available online via the author’s website. (shrink)

The concept of minimal risk has been used to regulate and limit participation by adolescents in clinical trials. It can be understood as setting an absolute standard of what risks are considered minimal or it can be interpreted as relative to the actual risks faced by members of the host community for the trial. While commentators have almost universally opposed a relative interpretation of the environmental risks faced by potential adolescent trial participants, we argue that the ethical concerns against the (...) relative standard may not be as convincing as these commentators believe. Our aim is to present the case for a relative standard of environmental risk in order to open a debate on this subject. We conclude by discussing how a relative standard of environmental risk could be defended in the specific case of an HIV vaccine trial among adolescents in South Africa. (shrink)

Within each of the great religions there is a well established doxastic practice (DP) of taking experiential inputs consisting of apparent direct perceptions of God (M experiences) as giving prima facie justification, subject to defeat by overriders supplied by that religion, for belief outputs that God exists and is as he presents himself. (This DP is abbreviated as "MP.") William Alston's primary aim in his excellent book, Perceiving God, is to establish that we have epistemic justification for believing that MPs (...) are reliable in that for the most part their belief outputs are true and moreover true of an objective or experience independent reality, unlike the belief outputs of the DPs based on sensations and feelings, along with the introspective DP whose intentional accusatives, although existing independently of being introspected, fail to be objective because they are themselves conscious states. (shrink)

According to the Christian religion, Jesus was “crucified under Pontius Pilate; he suffered death and was buried. On the third day he rose again”. I take it that this rising again—the Resurrection of Jesus, as it’s sometimes called—is, according to the Christian religion, an historical event, just like his crucifixion, death, and burial. And I would have thought that to investigate whether the Resurrection occurred, we would need to do some historical research: we would need to assess the reliability of (...) the New Testament documents and related manuscripts; we would need to look into the credibility of the witnesses to his post-mortem appearances, the empty tomb, and the like. The task looks rather daunting. But, according to David Hume, we don’t have to do anything of the kind. For no matter how strong the testimony in favor of a miracle is—indeed, even if we “suppose…that the testimony considered apart and in itself, amounts to an entire proof”—we have at our disposal a “full proof…against the existence of any miracle”. So we can avoid all that tedious historical work. We can simply use Hume’s shortcut, a proof against the existence of any miracle, and hence a proof against the Resurrection. So Hume says he has a “a decisive argument,” “an everlasting check to all kinds of superstitious delusion,” superstitious delusions like the Resurrection. There is good news and bad news in Hume’s proclamation. The good news is that he really has two arguments, not just one—or, at any rate, many scholars discern in his writings two arguments. The bad news is that neither succeeds; at any rate, try as I might, I can’t see how they do. (shrink)

Highlighting key writings from Professor Richard Pring's international career in education, the texts in this book provide a historical perspective in relation to current debates about philosophy of education in the UK and internationally, drawing attention to issues of current concern. The text explores key themes such as critical realism, teachers as researchers and a way forward for policy through carefully selected examples from Richard Pring's writings. A short introduction is provided for each chapter to help readers to (...) understand the significance of what is presented and how this relates to other chapters in the book. (shrink)

First things -- Emanuel Bach and the allure of the irrational -- Between enlightenment and romance -- Beethoven : confronting the past -- Fragments -- Death masks.

The notion of a "person" is in deep philosophical trouble. And this has posed a deepening crisis for believers: Christian beliefs are, after all, irreducibly about persons. In response to this situation, Prust proposes a new way to reason about persons, one based on identifying persons as characters of action. Employing a phenomenology of action he calls "character logic," he develops a powerful new tool for thinking through some of the intractable dilemmas that have long befuddled belief: - Can we (...) avoid being arbitrary and parochial in claiming that God is the only source of moral value? - Can we reconcile natural evil in the world with God's absolute power? - Can we continue to honor the historicity of faith-based claims in the face of critical history? - Can our personal life be eternal when neither timeless nor everlasting life is conceivable? - Can we accept our personal mortality and still affirm our destiny as eternal? Wholeness: The Character Logic of Christian Belief argues that character logic shows us a reasonable way to think about persons, one that puts theology on a new footing and gives affirmative answers to all these questions! (shrink)

Oxford Approaches to Classical Literature (Series Editors: Kathleen Coleman and Richard Rutherford) introduces individual works of Greek and Latin literature to readers who are approaching them for the first time. Each volume sets the work in its literary and historical context, and aims to offer a balanced and engaging assessment of its content, artistry, and purpose. A brief survey of the influence of the work upon subsequent generations is included to demonstrate its enduring relevance and power. All quotations from (...) the original are translated into English. Plato's Symposium tells of a dinner party at a crucial point in Athenian history at which the guests decide that they will each in turn deliver a speech in praise of love. The humorous and brilliant work that follows points the way towards all Western thinking about love. The Symposium is also one of Plato's most sophisticated meditations on the practice of philosophy. This book introduces the literary and historical context of Plato's work, surveys and explains the arguments, and considers why Plato has cast this work in a highly unusual narrative form. A final chapter traces the influence of the Symposium from antiquity to the modern day. (shrink)