A Study of the History and Philosophy of Category Theory Jean-Pierre Marquis. to say that objects are dispensable in geometry. What is claimed is that the specific nature of the objects used is irrelevant. To use the terminology already ...
The aim of this paper is to put into context the historical, foundational and philosophical significance of category theory. We use our historical investigation to inform the various category-theoretic foundational debates and to point to some common elements found among those who advocate adopting a foundational stance. We then use these elements to argue for the philosophical position that category theory provides a framework for an algebraic in re interpretation of mathematical structuralism. In each context, what we aim to show (...) is that, whatever the significance of category theory, it need not rely upon any set-theoretic underpinning. (shrink)
The aim of this paper is to clarify the role of category theory in the foundations of mathematics. There is a good deal of confusion surrounding this issue. A standard philosophical strategy in the face of a situation of this kind is to draw various distinctions and in this way show that the confusion rests on divergent conceptions of what the foundations of mathematics ought to be. This is the strategy adopted in the present paper. It is divided into 5 (...) sections. We first show that already in the set theoretical framework, there are different dimensions to the expression foundations of. We then explore these dimensions more thoroughly. After a very short discussion of the links between these dimensions, we move to some of the arguments presented for and against category theory in the foundational landscape. We end up on a more speculative note by examining the relationships between category theory and set theory. (shrink)
Opponents of human embryonic stem cell (hESC) research claim that such research is incompatible with the moral principle that it is always wrong intentionally to end a human life. In this essay, I discuss how that principle might be revised so that it is subject to as few difﬁculties as possible. I then argue that even the most defensible version of the principle is compatible with the moral permissibility of hESC research.
In this paper, we try to establish that some mathematical theories, like K-theory, homology, cohomology, homotopy theories, spectral sequences, modern Galois theory (in its various applications), representation theory and character theory, etc., should be thought of as (abstract) machines in the same way that there are (concrete) machines in the natural sciences. If this is correct, then many epistemological and ontological issues in the philosophy of mathematics are seen in a different light. We concentrate on one problem which immediately follows (...) the recognition of the particular status of these theories: the demarcation problem between ‘natural kinds’ and ‘artefacts’. (shrink)
Approximations form an essential part of scientific activity and they come in different forms: conceptual approximations (simplifications in models), mathematical approximations of various types (e.g. linear equations instead of non-linear ones, computational approximations), experimental approximations due to limitations of the instruments and so on and so forth. In this paper, we will consider one type of approximation, namely numerical approximations involved in the comparison of two results, be they experimental or theoretical. Our goal is to lay down the conceptual and (...) formal foundations of a local theory of partial truth. This is done by introducing and exploring the concept of truth space. (shrink)
Some concepts that are now part and parcel of mathematics used to be, at least until the beginning of the twentieth century, a central preoccupation of mathematicians and philosophers. The concept of continuity, or the continuous, is one of them. Nowadays, many philosophers of mathematics take it for granted that mathematicians of the last quarter of the nineteenth century found an adequate conceptual analysis of the continuous in terms of limits and that serious philosophical thinking is no longer required, except (...) perhaps when the question of the continuum is transferred to the arena of set theory where it takes the form of the infamous continuum hypothesis. As Philip Ehrlich has recently shown, this conviction goes back to the early writings of Russell who, in 1903 and then again in later writings, forcefully and eloquently pushed the view that mathematicians had given the final answer to immemorial conundrums arising from the continuous and infinitesimals . This proclamation of victory came with what was announced as the necessary defeat of the notion of the infinitesimal, despite the fact that mathematicians like Thomae, Du Bois-Reymond, Stolz, Bettazi, Veronese, Levi-Civita, and Hahn were investigating mathematical structures containing infinitesimals in a mathematically rigorous and logically consistent manner. In this respect Russell was merely walking in the footsteps of Cantor, and many of Russell's contemporaries were only too keen to keep infinitesimals out of Cantor's paradise. However, although Cantor certainly wanted to send the notion of infinitesimal to Hell, the notion kept a low profile in the mathematical purgatory, making its way in the study of non-Archimedean ordered algebraic systems. Of course, nowadays everyone has heard of Robinson's attempt at resurrecting infinitesimals in analysis in the form of non-standard analysis, but Robinson's work, despite the fact that it had, in …. (shrink)
In this paper, the problem of purifying an assumption-based theory KB, i.e., identifying the right extension of KB using knowledge-gathering actions (tests), is addressed. Assumptions are just normal defaults without prerequisite. Each assumption represents all the information conveyed by an agent, and every agent is associated with a (possibly empty) set of tests. Through the execution of tests, the epistemic status of assumptions can change from "plausible" to "certainly true", "certainly false" or "irrelevant", and the KB must be revised so (...) as to incorporate such a change. Because removing all the extensions of an assumption-based theory except one enables both identifying a larger set of plausible pieces of information and renders inference computationally easier, we are specifically interested in finding out sets of tests allowing to purify a KB (whatever their outcomes). We address this problem especially from the point of view of computational complexity. (shrink)
The most comprehensive collection of its kind, Ethics: History, Theory, and Contemporary Issues, Third Edition, is organized into three parts, providing instructors with flexibility in designing and teaching a variety of courses in moral philosophy. The first part, Historical Sources, moves from classical thought (Plato, Aristotle, Epicurus, and Epictetus) through medieval views (Augustine and Aquinas) to modern theories (Hobbes, Butler, Hume, Kant, Bentham, and Mill), culminating with leading nineteenth- and twentieth-century thinkers (Nietzsche, James, Dewey, Camus, and Sartre). The second part, (...) Modern Ethical Theory, includes many of the most important essays of the past century. The discussion of utilitarianism, Kantianism, egoism, and relativism continues in the work of major contemporary philosophers (Foot, Brandt, Williams, Wolf, and Nagel). Landmark selections (Moore, Prichard, Ross, Ayer, Stevenson, Hare, Baier, Anscombe, Gauthier, and Harman) reflect concern with moral language and the justification of morality. The concepts of justice (Rawls) and rights (Feinberg) are explored, as well as recent views on the importance of virtue ethics (Rachels) and an ethic influenced by feminist concerns (Held). In the third part, Contemporary Moral Problems, the readings present the current debates over abortion, euthanasia, famine relief, animal rights, the death penalty, and whether numbers should play a role in making moral decisions. The third edition expands Part II, Modern Ethical Theory, adding essays by Onora O'Neill, Ruth Barcan Marcus, Allan Gibbard, Nicholas L. Sturgeon, and Martha Nussbaum. Part III, Contemporary Moral Problems, features new essays on abortion by Mary Anne Warren, Don Marquis, and Rosalind Hursthouse; an essay on the death penalty by Stephen Nathanson; and a debate between John M. Taurek and Derek Parfit on when and why one should save from harm a greater rather than a lesser number of people. The book concludes with an essay by Judith Jarvis Thomson on the trolley problem. Wherever possible, each reading is printed in its entirety. (shrink)
In this essay, the author joins a conversation started by Martin regarding gender and education seeking to extend the conversation to address sexuality. To do so, the author brings a reading of the Marquis de Sade to challenge the emphasis on reproduction in education as it relates to gendered and sexual norms. The author, following Martin’s approach in Reclaiming the Conversation, reads one particular text of Sade’s—Philosophy in the Bedroom—to argue for queer possibilities that Sade brings to the conversation (...) around gender and sexuality within twenty-first century education. The Marquis de Sade may viscerally seem an inappropriate choice to return to and use to join the conversation, but his novelic imagination incites complications in the conversation. Following in the footsteps of Martin’s re-reading of historic philosophical texts the author offers not only another layer of thinking about “female education,” but also opens up space for engaging gender and sexuality in the history and philosophy of education. As issues around gender and sexed bodies become ever more present and contested in educational discourses, does Sade continue the conversation in ways that corrupt the reproductive aims of education? (shrink)
Modality, morality and belief are among the most controversial topics in philosophy today, and few philosophers have shaped these debates as deeply as Ruth Barcan Marcus. Inspired by her work, a distinguished group of philosophers explore these issues, refine and sharpen arguments and develop new positions on such topics as possible worlds, moral dilemmas, essentialism, and the explanation of actions by beliefs. This 'state of the art' collection honours one of the most rigorous and iconoclastic of philosophical pioneers.
The great contribution Marcus has made to several of intensely discussed topics in philosophy might not have been noticed fully without this collection of some of her most important articles that makes it evident that her achievement is not limited to inventing the famous Barcan formula.
In ‘ Why Abortion is Immoral ’, Don Marquis argues that abortion is wrong for the same reason that murder is wrong, namely, that it deprives a human being of an FLO, a ‘future like ours,’ which is a future full of value and the experience of life. Marquis’ argument rests on the assumption that the human being is somehow deprived by suffering an early death. I argue that Marquis’ argument faces the ‘Epicurean Challenge’. The concept of (...) ‘deprivation’ requires that some discernible individual exists who can be deprived. But if death involves total annihilation, then no discernible individual exists to be so deprived. I argue that the Epicurean Challenge must be addressed before it can be proven that Marquis is correct to claim that abortion and murder are wrong because they deprive someone of an FLO. (shrink)
The Philosophy Now series promises to combine rigorous analysis with authoritative expositions. Ruth Abbey’s book lives up to this demand by being a clear, reliable and more than up-to-date introduction to Charles Taylor ’s philosophy. Although it is an introductory book, the amount of footnotes and references ought to please those who want to study the original texts more closely. Abbey’s book is structured thematically: morality, selfhood, politics and epistemology get 50 pages each. The focus is on the internal (...) coherence of Taylor ’s work, not in its critique of or defence against other positions. The chapters are self-containing, but together they give a good total picture of Taylor ’s position. The concluding chapter is a highly interesting preview of Taylor ’s unpublished work-in-progress on secularity, which according to Abbey is comparable in magnitude to Sources of the Self. (shrink)
In this paper, I defend brain death as a criterion for determining death against objections raised by Don Marquis, Michael Nair-Collins, Doyen Nguyen, and Laura Specker Sullivan. I argue that any definition of death for beings like us relies on some sortal concept by which we are individuated and identified and that the choice of that concept in a practical context is not determined by strictly biological considerations but involves metaphysical, moral, social, and cultural considerations. This view supports acceptance (...) of a more pluralistic legal definition of death as well as acceptance of brain death as death. (shrink)
Sainsbury and Tye (2011) propose that, in the case of names and other simple extensional terms, we should substitute for Frege's second level of content—for his senses—a second level of meaning vehicle—words in the language of thought. I agree. They also offer a theory of atomic concept reference—their ‘originalist’ theory—which implies that people knowing the same word have the ‘same concept’. This I reject, arguing for a symmetrical rather than an originalist theory of concept reference, claiming that individual concepts are (...) possessed only by individual people. Concepts are classified rather than identified across different people. (shrink)
The traditional approach to the abortion debate revolves around numerous issues, such as whether the fetus is a person, whether the fetus has rights, and more. Don Marquis suggests that this traditional approach leads to a standoff and that the abortion debate “requires a different strategy.” Hence his “future of value” strategy, which is summarized as follows: (1) A normal fetus has a future of value. (2) Depriving a normal fetus of a future of value imposes a misfortune on (...) it. (3) Imposing a misfortune on a normal fetus is prima facie wrong. (4) Therefore, depriving a normal fetus of a future of value is prima facie wrong. (5) Killing a normal fetus deprives it of a future of value. (6) Therefore, killing a normal fetus is prima facie wrong. In this paper, I argue that Marquis’s strategy is not different since it involves the concept of person—a concept deeply rooted in the traditional approach. Specifically, I argue that futures are valuable insofar as they are not only dominated by goods of consciousness, but are experienced by psychologically continuous persons. Moreover, I argue that his strategy is not sound since premise (1) is false. Specifically, I argue that a normal fetus, at least during the first trimester, is not a person. Thus, during that stage of development it is not capable of experiencing its future as a psychologically continuous person and, hence, it does not have a future of value. (shrink)
One reason for the persistent appeal of Don Marquis' ‘future like ours’ argument is that it seems to offer a way to approach the debate about the morality of abortion while sidestepping the difficult task of establishing whether the fetus is a person. This essay argues that in order to satisfactorily address both of the chief objections to FLO – the ‘identity objection’ and the ‘contraception objection’ – Marquis must take a controversial stand on what is most essential (...) to being the kind of entity that an adult human being is. Such a stand amounts to a controversial account of personhood. To the extent that FLO's success depends on accepting such a controversial metaphysical view, one apparent attraction of FLO proves illusory. (shrink)
Although championed by the Marquis the Condorcet and many others, majority rule has often been rejected as indeterminate, incoherent, or implausible. Majority rule’s arch competitor is the Borda count, proposed by the Count de Borda, and there has long been a dispute between the two approaches. In several..
Marquis’s account of the ethics of abortion is unsatisfactory but not as Christensen implies baseless. It requires to be amended rather than abandoned. It is true, as Marquis asserts that murder and abortion both might deprive people of something of value to them, in particular, the life of a sort that might have been to them worth living. However, it is mistaken to conclude, as Marquis does, that murder and abortion are thereby morally equivalent. Not all deprivation (...) is wrongful. Not all that is wrongful is wrongful because it deprives someone of something. Contrary to what Christensen asserts, and Marquis seems to accept, it is not solely those discernible people who currently exist who might be deprived by our current actions. It is not only towards and concerning such living discernible people that we can have moral duties. It is not only such living discernible people who can be the beneficiaries of our generosity. Hence, contraception and the emission of greenhouse gases are, like abortion, issues that raise significant moral questions; however, they might each be properly answered. Nonetheless, it does not follow that is morally equivalent to each other far less than they are all morally equivalent to murder. If and when they are morally wrong, they can be different wrongs. (shrink)
When thinking about the intersection of care and Christian bioethics, it is helpful to follow closely the account of Ruth, who turned away from security and walked alongside her grieving mother-in-law to Bethlehem. Remembering Ruth may help one to heed Professor Kaveny?s summoning of Christians to remember ?the Order of Widows? and the church?s historic calling to bring ?the almanahinto its center rather than pushing her to its margins.? Disabled, elderly and terminally ill people often seem, at least (...) implicitly, expendable. By hearing the scriptural account of Jesus? steadfast great-grandmother, readers may recall another way. One may read Ruth?s care for Naomi as a performative, prophetic act of faith. Ruth?s faithful resolve, when set next to Orpah?s prudent way, challenges the notion that a bioethic of care is innately feminine, and may further call women and men corporately to participate in a kind of care that is strenuous work. My thanks to Cathleen Kaveny for allowing me to play off the title of her insightful essay. Thanks also to Willie James Jennings, whose 1998 baccalaureate sermon on Ruth inspired and much informed this essay. I wish also to thank Ellen Davis, who taught me to read Hebrew, and to read Ruth. (shrink)
Ruth Millikan is one of the most interesting and influential philosophers alive. Her work is also hard to penetrate. In this review, I try to present and assess her work on the nature of language, which is collected in this anthology. I also criticize her analysis of “natural convention” as well as her discussion of illocutionary acts.
In an earlier essay in this journal I critiqued Don Marquis's well-known argument against abortion. I distinguished two versions of Marquis's argument, which I refer to as ‘the essence argument’ and ‘the sufficient condition argument’. I presented two counterexamples showing that the essence argument was mistaken, and I argued that the sufficient condition argument should be rejected because Marquis had not adequately responded to an important objection to it. In response to my critique, Marquis put forward (...) in this journal a revised version of his argument. In his modified approach he no longer advocates the essence argument and he offers a new version of the sufficient condition argument. In the current essay, I discuss how Marquis's revised argument deals with my original objections, and I argue that his new sufficient condition argument is unsuccessful. (shrink)
Every body cell of an animal or human being contains the same complete set of genes. In theory any of these cells can be used to start a new embryo. The technique has been employed in the case of frogs. The nucleus is taken out of a body cell of a frog and implanted in an enucleated frog's egg. The resulting egg cell is stimulated to develop into a normal frog, and will be an exact copy of that frog which (...) provided the nucleus with all the genetic information. In normal sexual reproduction, two parents each contribute half their genes, but in the case of cloning, one parent passes on all his or her genes. (shrink)
Recent statistics in South Africa shows that women mostly experience poverty as compared to their male counterparts. In the context of the experience of poverty by women, several Old Testament scholars have convincingly explored the theme of poverty in the Hebrew Bible. In her contextual rereading of the Naomi-Ruth Story, Madipoane Masenya links the issue of poverty to the theme of land. Also, from the historical-critical and partly, the contextual approach to ancient texts, Esias E. Meyer argues that Leviticus (...) 25:8-55 holds liberating possibilities for women who are invisible in such a text. Based on the argument made by the preceding scholars, firstly, this article argues that in the context from which the texts of Ruth 4 and Leviticus 25:8-55 emerged, some women were both landless and poor. Secondly, it is argued in this article that the context of these texts carries a striking resemblance to the situation of women in modern South Africa, as many women do not own productive land and are poor. Thirdly, this article poses the question: What implications do the ideologies of Ernesto 'Che' Guevara and the hermeneutical approach of Fernando F. Segovia to ancient texts bear on the reading of Ruth 4 and Leviticus 25:8-55 in South Africa? (shrink)
This article is a defence of the Fact-Value distinction against considerations brought up by Ruth Anna Putnam in three articles in Philosophy, especially her ‘Perceiving Facts and Values’ January 1998. I defend metaphysical realism about facts and anti-realism about values against Putnam' intermediate position about both and I relate the matter to the logic of imperatives. The motivations of scientists or historians to select fields of investigation are irrelevant to the objectivity of their hypotheses, and so is the goodness (...) or badness of the social consequences of their work though these may affect their motivations. (shrink)