Mathematics plays an inordinate role in the work of many of famous Western philosophers, from the time of Plato, through Husserl and Wittgenstein, and even to the present. Why? This paper points to the experience of learning or making mathematics, with an emphasis on proof. It distinguishes two sources of the perennial impact of mathematics on philosophy. They are classified as Ancient and Enlightenment. Plato is emblematic of the former, and Kant of the latter. The Ancient fascination arises from the (...) sense that mathematics explores something ‘out there’. This is illustrated by recent discussions by distinguished contemporary mathematicians. The Enlightenment strand often uses Kant's argot: ‘absolute necessity’, ‘apodictic certainty’ and ‘a priori’ judgement or knowledge. The experience of being compelled by proof, the sense that something must be true, that a result is certain, generates the philosophy. It also creates the illusion that mathematics is certain. Kant's leading question, ‘How is pure mathematics possible?’, is easily misunderstood because the modern distinction between pure and applied is an artefact of the 19th century. As Russell put it, the issue is to explain ‘the apparent power of anticipating facts about things of which we have no experience’. More generally the question is, how is it that pure mathematics is so rich in applications? Some six types of application are distinguished, each of which engenders its own philosophical problems which are descendants of the Enlightenment, and which differ from those descended from the Ancient strand. (shrink)

Throughout history, mathematicians have expressed preference for solutions to problems that avoid introducing concepts that are in one sense or another “foreign” or “alien” to the problem under investigation. This preference for “purity” (which German writers commonly referred to as “methoden Reinheit”) has taken various forms. It has also been persistent. This notwithstanding, it has not been analyzed at even a basic philosophical level. In this paper we give a basic analysis of one conception of purity—what we call topical purity—and (...) discuss its epistemological significance. (shrink)

A provocative critique of Western attitudes about the Orient, this history examines the ways in which the West has discovered, invented, and sought to control the East from the 1700s to the present.

This classic and controversial book examines the roots of the idea of virtue, diagnoses the reasons for its absence in modern life, and proposes a path for its recovery.

In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks, most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good argument (...) for or against platonism, but that we could never have such an argument and, indeed, that there is no fact of the matter as to whether platonism is correct. (shrink)

Platonism is the most pervasive philosophy of mathematics. Indeed, it can be argued that an inarticulate, half-conscious Platonism is nearly universal among mathematicians. The basic idea is that mathematical entities exist outside space and time, outside thought and matter, in an abstract realm. In the more eloquent words of Edward Everett, a distinguished nineteenth-century American scholar, "in pure mathematics we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist (...) there when the last of their radiant host shall have fallen from heaven." In What is Mathematics, Really?, renowned mathematician Rueben Hersh takes these eloquent words and this pervasive philosophy to task, in a subversive attack on traditional philosophies of mathematics, most notably, Platonism and formalism. Virtually all philosophers of mathematics treat it as isolated, timeless, ahistorical, inhuman. Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. Mathematical objects are created by humans, not arbitrarily, but from activity with existing mathematical objects, and from the needs of science and daily life. Hersh pulls the screen back to reveal mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the "humanist" idea of the nature of mathematics more closely resembles how mathematicians actually work. At the heart of the book is a fascinating historical account of the mainstream of philosophy--ranging from Pythagoras, Plato, Descartes, Spinoza, and Kant, to Bertrand Russell, David Hilbert, Rudolph Carnap, and Willard V.O. Quine--followed by the mavericks who saw mathematics as a human artifact, including Aristotle, Locke, Hume, Mill, Peirce, Dewey, and Lakatos. In his epilogue, Hersh reveals that this is no mere armchair debate, of little consequence to the outside world. He contends that Platonism and elitism fit well together, that Platonism in fact is used to justify the claim that "some people just can't learn math." The humanist philosophy, on the other hand, links mathematics with geople, with society, and with history. It fits with liberal anti-elitism and its historical striving for universal literacy, universal higher education, and universal access to knowledge and culture. Thus Hersh's argument has educational and political ramifications. Written by the co-author of The Mathematical Experience, which won the American Book Award in 1983, this volume reflects an insider's view of mathematical life, based on twenty years of doing research on advanced mathematical problems, thirty-five years of teaching graduates and undergraduates, and many long hours of listening, talking to, and reading philosophers. A clearly written and highly iconoclastic book, it is sure to be hotly debated by anyone with a passionate interest in mathematics or the philosophy of science. (shrink)

A summation of the work of one of the most influential and provocative interpreters of science, it aims at saving what is good and valuable in modernity and ...

This book argues against the view that mathematical knowledge is a priori,contending that mathematics is an empirical science and develops historically,just as ...

After discussing the distinction between artifacts and natural entities, and the distinction between artifacts and technology, the conditions of the traditional account of moral agency are identified. While computer system behavior meets four of the five conditions, it does not and cannot meet a key condition. Computer systems do not have mental states, and even if they could be construed as having mental states, they do not have intendings to act, which arise from an agent’s freedom. On the other hand, (...) computer systems have intentionality, and because of this, they should not be dismissed from the realm of morality in the same way that natural objects are dismissed. Natural objects behave from necessity; computer systems and other artifacts behave from necessity after they are created and deployed, but, unlike natural objects, they are intentionally created and deployed. Failure to recognize the intentionality of computer systems and their connection to human intentionality and action hides the moral character of computer systems. Computer systems are components in human moral action. When humans act with artifacts, their actions are constituted by the intentionality and efficacy of the artifact which, in turn, has been constituted by the intentionality and efficacy of the artifact designer. All three components – artifact designer, artifact, and artifact user – are at work when there is an action and all three should be the focus of moral evaluation. (shrink)

Artificial agents (AAs), particularly but not only those in Cyberspace, extend the class of entities that can be involved in moral situations. For they can be conceived of as moral patients (as entities that can be acted upon for good or evil) and also as moral agents (as entities that can perform actions, again for good or evil). In this paper, we clarify the concept of agent and go on to separate the concerns of morality and responsibility of agents (most (...) interestingly for us, of AAs). We conclude that there is substantial and important scope, particularly in Computer Ethics, for the concept of moral agent not necessarily exhibiting free will, mental states or responsibility. This complements the more traditional approach, common at least since Montaigne and Descartes, which considers whether or not (artificial) agents have mental states, feelings, emotions and so on. By focussing directly on mind-less morality we are able to avoid that question and also many of the concerns of Artificial Intelligence. A vital component in our approach is the Method of Abstraction for analysing the level of abstraction (LoA) at which an agent is considered to act. The LoA is determined by the way in which one chooses to describe, analyse and discuss a system and its context. The Method of Abstraction is explained in terms of an interface or set of features or observables at a given LoA. Agenthood, and in particular moral agenthood, depends on a LoA. Our guidelines for agenthood are: interactivity (response to stimulus by change of state), autonomy (ability to change state without stimulus) and adaptability (ability to change the transition rules by which state is changed) at a given LoA. Morality may be thought of as a threshold defined on the observables in the interface determining the LoA under consideration. An agent is morally good if its actions all respect that threshold; and it is morally evil if some action violates it. That view is particularly informative when the agent constitutes a software or digital system, and the observables are numerical. Finally we review the consequences for Computer Ethics of our approach. In conclusion, this approach facilitates the discussion of the morality of agents not only in Cyberspace but also in the biosphere, where animals can be considered moral agents without their having to display free will, emotions or mental states, and in social contexts, where systems like organizations can play the role of moral agents. The primary cost of this facility is the extension of the class of agents and moral agents to embrace AAs. (shrink)

Winner of the Wolf Prize for his contribution to our understanding of the universe, Penrose takes on the question of whether artificial intelligence will ever ...

Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and (...) proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim. (shrink)

I argue here that Alasdair MacIntyre is mistaken when he claims that teaching is not a practice. In particular, I try to throw some doubt on his claim that ‘teaching is never more than a means’ and to challenge his list of things that all students should learn. In the second section, I show how analyses of professionalism endorse MacIntyre’s emphasis on complexity and internal criteria for practices. Finally, building on the doubts observed in the first section and the criteria (...) identified in the second, I describe teaching as a relational practice. (shrink)

I argue here that Alasdair MacIntyre is mistaken when he claims that teaching is not a practice. In particular, I try to throw some doubt on his claim that ‘teaching is never more than a means’ and to challenge his list of things that all students should learn. In the second section, I show how analyses of professionalism endorse MacIntyre's emphasis on complexity and internal criteria for practices. Finally, building on the doubts observed in the first section and the criteria (...) identified in the second, I describe teaching as a relational practice. (shrink)

This truly philosophical book takes us back to fundamentals - the sheer experience of proof, and the enigmatic relation of mathematics to nature. It asks unexpected questions, such as 'what makes mathematics mathematics?', 'where did proof come from and how did it evolve?', and 'how did the distinction between pure and applied mathematics come into being?' In a wide-ranging discussion that is both immersed in the past and unusually attuned to the competing philosophical ideas of contemporary mathematicians, it shows that (...) proof and other forms of mathematical exploration continue to be living, evolving practices - responsive to new technologies, yet embedded in permanent facts about human beings. It distinguishes several distinct types of application of mathematics, and shows how each leads to a different philosophical conundrum. Here is a remarkable body of new philosophical thinking about proofs, applications, and other mathematical activities. (shrink)

A model for the ideal state includes discussions of the nature and application of justice, the role of the philosopher in society, the goals of education, and the effects of art upon character.

According to Alasdair MacIntyre's influential account of practices, ‘teaching itself is not a practice, but a set of skills and habits put to the service of a variety of practices’ (MacIntyre and Dunne, 2002, p. 5). Various philosophers of education have responded to and critiqued MacIntyre's position, most notably in a Special Issue of the Journal of Philosophy of Education (Vol. 37.2, 2003). However, both in that Special Issue and since, this debate remains inconclusive. Much of this earlier discussion seems (...) to accept that teaching is a unique case in being a putative practice that does not fit readily into MacIntyre's account. In fact many supposed practices, including some nominated by MacIntyre himself, do not fit his account. A constructive critique of this account leads to a refurbished, broadly MacIntyrean account of practice. This will clarify the issue of whether teaching and a range of other activities are, indeed, practices. (shrink)

This essay takes issue with Alasdair MacIntyre’s denial that teaching is a practice. It does so less by appeal to MacIntyre’s concept of practice than by criticism of his conception of teaching. It argues that this conception, as reconstructed from adversions to teaching in a range of his writings, does less than justice to what good teachers accomplish; and that, if this inadequacy is rectified—as much else in his writings suggests that it ought to be—there are clearer grounds for acknowledging (...) teaching as a practice. (shrink)

This essay takes issue with Alasdair MacIntyre's denial that teaching is a practice. It does so less by appeal to MacIntyre's concept of practice than by criticism of his conception of teaching. It argues that this conception, as reconstructed from adversions to teaching in a range of his writings, does less than justice to what good teachers accomplish; and that, if this inadequacy is rectified—as much else in his writings suggests that it ought to be—there are clearer grounds for acknowledging (...) teaching as a practice. (shrink)

This work stresses the illogical manner in which mathematics has developed, the question of applied mathematics as against 'pure' mathematics, and the challenges to the consistency of mathematics' logical structure that have occurred in the twentieth century.

This book does three main things. First, it defends mathematical platonism against the main objections to that view (most notably, the epistemological objection and the multiple-reductions objection). Second, it defends anti-platonism (in particular, fictionalism) against the main objections to that view (most notably, the Quine-Putnam indispensability objection and the objection from objectivity). Third, it argues that there is no fact of the matter whether abstract mathematical objects exist and, hence, no fact of the matter whether platonism or anti-platonism is true.