What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions. Written by some of the world's leading philosophers of mathematics, psychologists, and mathematicians, Mathematical Knowledge gives a lively sense of the current state of debate in this fascinating field. Contents 1. (...) Mary Leng: Introduction 2. Michael Potter: What is the problem of mathematical knowledge? 3. Tim Gowers: Mathematics, memory, and mental arithmetic 4. Alan Baker: Is there a problem of induction for mathematics? 5. Marinella Cappelletti and Valeria Giardino: The cognitive basis of mathematical knowledge 6. Mary Leng: What's there to know? A fictionalist account of mathematical knowledge 7. Mark Colyvan: Mathematical recreation versus mathematical knowledge 8. Alexander Paseau: Scientific platonism 9. Crispin Wright: On quantifying into predicate position: Steps towards a (new)tralist position. (shrink)
Science and mathematics: the scope and limits of mathematical fictionalism Content Type Journal Article Category Book Symposium Pages 1-26 DOI 10.1007/s11016-011-9640-3 Authors Christopher Pincock, University of Missouri, 438 Strickland Hall, Columbia, MO 65211-4160, USA Alan Baker, Department of Philosophy, Swarthmore College, Swarthmore, PA 19081, USA Alexander Paseau, Wadham College, Oxford, OX1 3PN UK Mary Leng, Department of Philosophy, University of York, Heslington, York, YO10 5DD UK Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.
Aristotle and Confucius are pivotal figures in world history; nevertheless, Western and Eastern cultures have in modern times largely abandoned the insights of these masters. Remastering Morals is the first book-length scholarly comparison of the ethics of Aristotle and Confucius. May Sim's comparisons offer fresh interpretations of the central teachings of both men. More than a catalog of similarities and differences, her study brings two great traditions into dialog so that each is able to learn from the other. This is (...) essential reading for anyone interested in virtue-oriented ethics. (shrink)
Mary Leng offers a defense of mathematical fictionalism, according to which we have no reason to believe that there are any mathematical objects. Perhaps the most pressing challenge to mathematical fictionalism is the indispensability argument for the truth of our mathematical theories (and therefore for the existence of the mathematical objects posited by those theories). According to this argument, if we have reason to believe anything, we have reason to believe that the claims of our best empirical theories are (...) (at least approximately) true. But since claims whose truth would require the existence of mathematical objects are indispensable in formulating our best empirical theories, it follows that we have good reason to believe in the mathematical objects posited by those mathematical theories used in empirical science, and therefore to believe that the mathematical theories utilized in empirical science are true. Previous responses to the indispensability argument have focussed on arguing that mathematical assumptions can be dispensed with in formulating our empirical theories. Leng, by contrast, offers an account of the role of mathematics in empirical science according to which the successful use of mathematics in formulating our empirical theories need not rely on the truth of the mathematics utilized. (shrink)
It has often been suggested that people's ordinary understanding of morality involves a belief in objective moral truths and a rejection of moral relativism. The results of six studies call this claim into question. Participants did offer apparently objectivist moral intuitions when considering individuals from their own culture, but they offered increasingly relativist intuitions considering individuals from increasingly different cultures or ways of life. The authors hypothesize that people do not have a fixed commitment to moral objectivism but instead tend (...) to adopt different views depending on the degree to which they consider radically different perspectives on moral questions. (shrink)
This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti-Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions that imply (...) the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the 'no miracles' argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific 'realists' should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism. (shrink)
For many philosophers not automatically inclined to Platonism, the indispensability argument for the existence of mathematical objectshas provided the best (and perhaps only) evidence for mathematicalrealism. Recently, however, this argument has been subject to attack, most notably by Penelope Maddy (1992, 1997),on the grounds that its conclusions do not sit well with mathematical practice. I offer a diagnosis of what has gone wrong with the indispensability argument (I claim that mathematics is indispensable in the wrong way), and, taking my cue (...) from Mark Colyvan''s (1998) attempt to provide a Quinean account of unapplied mathematics as `recreational'', suggest that, if one approaches the problem from a Quinean naturalist starting point, one must conclude that all mathematics is recreational in this way. (shrink)
This paper responds to John Burgess's ‘Mathematics and Bleak House’. While Burgess's rejection of hermeneutic fictionalism is accepted, it is argued that his two main attacks on revolutionary fictionalism fail to meet their target. Firstly, ‘philosophical modesty’ should not prevent philosophers from questioning the truth of claims made within successful practices, provided that the utility of those practices as they stand can be explained. Secondly, Carnapian scepticism concerning the meaningfulness of metaphysical existence claims has no force against a naturalized version (...) of fictionalism, according to which our ordinary standards of scientific evidence may show that we have no reason to believe the mathematical existence claims made within the context of our mathematical and scientific theories. (shrink)
In this article, I argue that Wang Yangming'sNeo-Confucian religious beliefs can bewarranted, and that the rationality of hisreligious beliefs constitutes a significantdefeater for the rationality of Christianbelief on Alvin Plantinga's theory of warrant. I also question whether the notion of warrantas proper function can adequately account fortheories of religious knowledge in which theaffections play an integral role. Idemonstrate how a consideration of Wang'sepistemology reveals a difficulty forPlantinga's defense of the rationality ofChristian belief and highlights a limitation ofPlantinga's current conception of (...) warrant asproper function. (shrink)
Comparing Aristotle's and Confucius' ethics, where each represents an ethics of virtue, I show that they are not susceptible to some of the frequent charges against them when compared to non-virtue ethical theories like utilitarianism and deontology. These charges are that virtue ethics: (1) lack universal laws; they cannot (a) provide content for actions, and (b) they do not consider actions in the evaluation of morality. (2) Virtue ethics cannot provide the resources for dealing with social justice and human rights (...) practices. Contrary to these charges, I show how these thinkers' ethics can account for the first generation civil and political rights, and the second generation economic, social and cultural rights. (shrink)
This discussion note responds to Mark Colyvan’s claim that there is no easy road to nominalism. While Colyvan is right to note that the existence of mathematical explanations presents a more serious challenge to nominalists than is often thought, it is argued that nominalist accounts do have the resources to account for the existence of mathematical explanations whose explanatory role resides elsewhere than in their nominalistic content.
A phenomenological approach to mathematical practice is sketched out, and some problems with this sort of approach are considered. The approach outlined takes mathematical practices as its data, and seeks to provide an empirically adequate philosophy of mathematics based on observation of these practices. Some observations are presented, based on two case studies of some research into the classification of C*-algebras. It is suggested that an anti-realist account of mathematics could be developed on the basis of these and other studies, (...) locating the substance of mathematics in the various informal argument methods used by mathematicians. (shrink)
My purpose is to argue the following theses: (1) Habituation into virtue, social relations, and paradigmatic persons are central for both Aristotle and Confucius. Both therefore need a notion of self to support them. (2) Aristotle’s individualistic metaphysics cannot account for the thick relations that this requires. (3) The Confucian self, if entirely relationistic, cannot function as a locus of choice and agency; if fully ritualistic, it cannot function as a source of moral norms that might help assess existing social (...) proprieties. I shall suggest that each offers some corrective for the other and urge further dialogue between the friends of Confucius and Aristotle. (shrink)
Rather than “selfishness,” a more accurate and revealing interpretation of Wang's use of siyuis “self-centeredness.” One of the main goals in Wang's model of moral cultivation was to attain a state devoid of self-centered desires. Wang relied a great deal on the exercise and cultivation of an emotional identification and feeling of oneness with others. In this paper, I first provide a brief summary of the role of Wang's concept of siyu in his moral psychology. I then examine key passages (...) in Wang's writings that reveal his nuanced understanding of siyu and, along the way, I draw on empirical research in psychology to help illuminate the significance of Wang's view of siyu to his overall model of moral cultivation. (shrink)
Commentators who find in Confucianism the resources for cross-cultural dialogues about human rights frequently tend to be divided in their emphases on liberal or conservative aspects of this tradition. Those who pursue individuality, even autonomy, in Confucianism, I call liberals. Those who stress collectivity or harmony in Confucianism I call conservatives. Despite these rival paths in appropriating Confucianism for human rights, I show that both liberal and conservative characterizations, properly understood, are present in this tradition. Corresponding to each group’s stress (...) on particular dimensions in Confucianism are the respective negative (civil and political) and positive (social, economic and cultural) human rights that each position defends. If my thesis that both the liberal and conservative accounts of human rights holds, then a Confucian account of human rights would entail that both the negative and positive rights will bear practical and political implications for it. (shrink)
In this paper we describe a method for the specification of computationalmodels of argument using dialogue games. The method, which consists ofsupplying a set of semantic definitions for the performatives making upthe game, together with a state transition diagram, is described in full.Its use is illustrated by some examples of varying complexity, includingtwo complete specifications of particular dialogue games, Mackenzie's DC,and the authors' own TDG. The latter is also illustrated by a fully workedexample illustrating all the features of the game.
What does "postmodernism" mean? Why is it so important? Now in its second edition, The Routledge Companion to Postmodernism combines a series of in-depth background chapters with a body of A-Z entries to create an authoritative, yet readable guide to the complex world of postmodernism. Following full-length articles on postmodernism and philosophy, politics, feminism, religion, post-colonialis, lifestyles television, and other postmodern essentials, readers will find a wide range of alphabetically-organized entries on the people, terms and theories connected with postmodernism, including: (...) Peter Ackroyd; Jean Baudrillard; Chaos Theory; Death of the Author; Desire; Fractals; Michel Foucault; Frankfurt School; Generation X; Minimalism; Poststructuralism; Retro; Gayatri Chakravorty Spivak ; and Trans-avant-garde; Students interested in any aspect of postmodernist thought will find this an indispensable resource. (shrink)
Upshot: Leng attacks the indispensability argument for the existence of mathematical objects. She offers an account that treats the role of mathematics in science as an indispensable and useful part of theories, but retains nonetheless a fictionalist position towards mathematics. The result is an account of mathematics that is interesting for constructivists. Her view towards the nominalistic part of science is, however, more in conflict with radical constructivism.
This paper has attempted to present Wonch'uk's Ban-ya pa-ra-mil-da sim gyeong chan (般若波羅蜜多心經贊) or Commentary on the Heart Sūtra which was written in classical Chinese in the 7th century. As an example of the intellectual analysis of a sūtra, Wonch'uk's Commentary is an important text that has exerted asignificant influence on East Asian Buddhist thought. A prominent Korean Yogācāra scholar, Wonch'uk authored twenty-three works during his lifetime; unfortunately, all but three have been lost. The Commentary on the Heart Sūtra is (...) the shortest among his extant writings, yet it clearly reflects his incomparable erudition. To date, there has been very limited research on Wonch'uk and his thought in both the East and West. Utilizing Wonch'uk's original Chinese text,this paper will examine the distinctive features of Wonch'uk's Commentary which may offer the contemporary readers an opportunity to remind the importance of sūtra study and the engagement in sūtra exegesis. (shrink)
Nick Bostrom argues that if we accept some plausible assumptions about how the future will unfold, we should believe we are probably not humans. The argument appeals crucially to an indifference principle whose precise content is a little unclear. I set out four possible interpretations of the principle, none of which can be used to support Bostrom’s argument. On the first two interpretations the principle is false, on the third it does not entail the conclusion, and on the fourth it (...) only entails the conclusion given an auxiliary hypothesis that we have no reason to believe. (shrink)
Jonathan Haidt ( 2001 ) advances the 'Social Intuitionist' account of moral judgment , which he presents as an alternative to rationalist accounts of moral judgment , hitherto dominant in psychology. Here I consider Haidt's anti-rationalism and the debate that it has provoked in moral psychology , as well as some anti-rationalist philosophical claims that Haidt and others have grounded in the empirical work of Haidt and his collaborators. I will argue that although the case for anti-rationalism in moral psychology (...) based on the work of Haidt and his collaborators is plausible , a decisive case has yet to be made. It will require further experimental evidence before a decisive case could be made. My assessment of anti-rationalist philosophical arguments that are grounded in the empirical work of Haidt and his collaborators is much more negative than this. I will argue that this body of empirical work is a very unpromising basis for such arguments. (shrink)
En este trabajo se suscribe la tesis de que la filosofía de la ciencia—al igual que las demás ramas de la filosofía—consiste en discusiones interminables sobre problemas que no se pueden resolver, pero se sostiene también que, a pesar (o a causa) de eso, tiene derecho a existir debido a que cumple funciones importantes, entre ellas precisamente la de dar lugar a discusiones interminables sobre problemas que no se pueden resolver, actividad que a las personas con genuina vocación filosófica les (...) produce una satisfacción intelectual difícil de entender para quienes no comparten esa vocación. (shrink)
‘‘Theoretical biology’’ is a surprisingly heter- ogeneous field, partly because it encompasses ‘‘doing the- ory’’ across disciplines as diverse as molecular biology, systematics, ecology, and evolutionary biology. Moreover, it is done in a stunning variety of different ways, using anything from formal analytical models to computer sim- ulations, from graphic representations to verbal arguments. In this essay I survey a number of aspects of what it means to do theoretical biology, and how they compare with the allegedly much more restricted (...) sense of theory in the physical sciences. I also tackle a recent trend toward the presentation of all-encompassing theories in the biological sciences, from general theories of ecology to a recent attempt to provide a conceptual framework for the entire set of biological disciplines. Finally, I discuss the roles played by philosophers of science in criticizing and shap- ing biological theorizing. (shrink)
Does mathematics ever play an explanatory role in science? If so then this opens the way for scientific realists to argue for the existence of mathematical entities using inference to the best explanation. Elsewhere I have argued, using a case study involving the prime-numbered life cycles of periodical cicadas, that there are examples of indispensable mathematical explanations of purely physical phenomena. In this paper I respond to objections to this claim that have been made by various philosophers, and I discuss (...) potential future directions of research for each side in the debate over the existence of abstract mathematical objects. Introduction: Mathematical Explanation Indispensability and Explanation Is the Mathematics Indispensable to the Explanation? 3.1 Object-level arbitrariness 3.2 Concept-level arbitrariness 3.3 Theory-level arbitrariness Is the Explanandum ‘Purely Physical’? Is the Mathematics Explanatory in Its Own Right? Does Inference to the Best Explanation Apply to Mathematics? 6.1 Leng's first argument 6.2 Leng's second argument 6.3 Leng's third argument Conclusions CiteULike Connotea Del.icio.us What's this? (shrink)
According to Haidt's (2001) social intuitionist model (SIM), an individual's moral judgment normally arises from automatic 'moral intuitions'. Private moral reasoning - when it occurs - is biased and post hoc, serving to justify the moral judgment determined by the individual's intuitions. It is argued here, however, that moral reasoning is not inevitably subserviant to moral intuitions in the formation of moral judgments. Social cognitive research shows that moral reasoning may sometimes disrupt the automatic process of judgment formation described by (...) the SIM. Furthermore, it seems that automatic judgments may reflect the 'automatization' of judgment goals based on prior moral reasoning. In line with this role for private moral reasoning in judgment formation, it is argued that moral reasoning can, under the right circumstances, be sufficiently unbiased to effectively challenge an individual's moral beliefs. Thus the social cognitive literature indicates a greater and more direct role for private moral reasoning than the SIM allows. (shrink)
In this manuscript we initiate a systematic examination of the physical basis for the time concept in cosmology. We discuss and defend the idea that the physical basis of the time concept is necessarily related to physical processes which could conceivably take place among the material constituents available in the universe. As a consequence we motivate the idea that one cannot, in a well-defined manner, speak about time ‘before’ such physical processes were possible, and in particular, the idea that one (...) cannot speak about a time scale ‘before’ scale-setting physical processes were possible. It is common practice to link the concept of cosmic time with a space-time metric set up to describe the universe at large scales, and then define a cosmic time t as what is measured by a comoving standard clock. We want to examine, however, the physical basis for setting up a comoving reference frame and, in particular, what could be meant by a standard clock. For this purpose we introduce the concept of a `core' of a clock (which, for a standard clock in cosmology, is a scale-setting physical process) and we ask if such a core can - in principle - be found in the available physics contemplated in the various `stages' of the early universe. We find that a first problem arises above the quark-gluon phase transition (which roughly occurs when the cosmological model is extrapolated back to $\sim 10^{-5}$ seconds) where there might be no bound systems left, and the concept of a physical length scale to a certain extent disappears. A more serious problem appears above the electroweak phase transition believed to occur at $\sim 10^{-11}$ seconds. At this point the property of mass (almost) disappears and it becomes difficult to identify a physical basis for concepts like length scale, energy scale and temperature - which are all intimately linked to the concept of time in modern cosmology. This situation suggests that the concept of a time scale in `very early' universe cosmology lacks a physical basis or, at least, that the time scale will have to be based on speculative new physics. (shrink)
It is often assumed that empiricism in the philosophy of mathematics was laid to rest by Frege’s stinging attack on Mill. I will argue that empiricism is alive and well and able to deal with almost everything that’s thrown at it. In particular, I will show how the brand of empiricism I subscribe to is able to give a satisfying account of mathematical knowledge. This brand of mathematical empiricism has a rather curious feature though: some parts of mathematics (e.g., analysis, (...) modern algebra, ZFC set theory) are taken to be theories about which we have genuine mathematical knowledge, while others (e.g., set theory with large cardinal axioms) are (following Quine) treated as “mathematical recreation”. I will defend this demarcation against some recent criticisms from Mary Leng. (shrink)
