In a recent article, Christopher Ormell argues against the traditional mathematical view that the real numbers form an uncountably inﬁnite set. He rejects the conclusion of Cantor’s diagonal argument for the higher, non-denumerable inﬁnity of the real numbers. He does so on the basis that the classical conception of a real number is mys- terious, ineffable, and epistemically suspect. Instead, he urges that mathematics should admit only ‘well-deﬁned’ real numbers as proper objects of study. In practice, this means excluding as (...) inadmissible all those real numbers whose decimal expansions cannot be calculated in as much detail as one would like by some rule. We argue against Ormell that the classical realist account of the continuum has explanatory power in mathematics and should be accepted, much in the same way that "dark matter" is posited by physicists to explain observations in cosmology. In effect, the indefinable real numbers are like the "dark matter" of real analysis. (shrink)
According to Quine’s indispensability argument, we ought to believe in just those mathematical entities that we quantify over in our best scientific theories. Quine’s criterion of ontological commitment is part of the standard indispensability argument. However, we suggest that a new indispensability argument can be run using Armstrong’s criterion of ontological commitment rather than Quine’s. According to Armstrong’s criterion, ‘to be is to be a truthmaker (or part of one)’. We supplement this criterion with our own brand of metaphysics, 'Aristotelian (...) (...) realism', in order to identify the truthmakers of mathematics. We consider in particular as a case study the indispensability to physics of real analysis (the theory of the real numbers). We conclude that it is possible to run an indispensability argument without Quinean baggage. (shrink)
G. E. M. Anscombe’s view that agents know what they are doing “without observation” has been met with skepticism and the charge of confusion and falsehood. Simultaneously, some commentators think that Anscombe has captured an important truth about the first-personal character of an agent’s awareness of her actions. This paper attempts an explanation and vindication of Anscombe’s view. The key to the vindication lies in focusing on the role of practical knowledge in an agent’s knowledge of her actions. Few commentators, (...) with the exception of Moran (2004) and Hursthouse (2000), have gotten the emphasis right. The key to a proper interpretation of Anscombe’s views is to explain her claims within the context of her teleological theory of action. The result is a theory ofintentional action that makes self-knowledge of one’s own actions the norm. (shrink)
In chapter 7 of The Varieties of Reference, Gareth Evans claimed to have an argument that would present "an antidote" to the Cartesian conception of the self as a purely mental entity. On the basis of considerations drawn from philosophy of language and thought, Evans claimed to be able to show that bodily awareness is a form of self-awareness. The apparent basis for this claim is the datum that sometimes judgements about one’s position based on body sense are immune to (...) errors of misidentification relative to the first-person pronoun 'I'. However, Evans’s argument suffers from a crucial ambiguity. 'I' sometimes refers to the subject's mind, sometimes to the person, and sometimes to the subject's body. Once disambiguated, it turns out that Evans’s argument either begs the question against the Cartesian or fails to be plausible at all. Nonetheless, the argument is important for drawing our attention to the idea that bodily modes of awareness should be taken seriously as possible forms of self-awareness. (shrink)
This paper is on Aristotle's conception of the continuum. It is argued that although Aristotle did not have the modern conception of real numbers, his account of the continuum does mirror the topology of the real number continuum in modern mathematics especially as seen in the work of Georg Cantor. Some differences are noted, particularly as regards Aristotle's conception of number and the modern conception of real numbers. The issue of whether Aristotle had the notion of open versus closed intervals (...) is discussed. Finally, it is suggested that one reason there is a common structure between Aristotle's account of the continuum and that found in Cantor's definition of the real number continuum is that our intuitions about the continuum have their source in the experience of the real spatiotemporal world. A plea is made to consider Aristotle's abstractionist philosophy of mathematics anew. (shrink)
The mathematician Georg Cantor strongly believed in the existence of actually infinite numbers and sets. Cantor’s “actualism” went against the Aristotelian tradition in metaphysics and mathematics. Under the pressures to defend his theory, his metaphysics changed from Spinozistic monism to Leibnizian voluntarist dualism. The factor motivating this change was two-fold: the desire to avoid antinomies associated with the notion of a universal collection and the desire to avoid the heresy of necessitarian pantheism. We document the changes in Cantor’s thought with (...) reference to his main philosophical-mathematical treatise, the Grundlagen (1883) as well as with reference to his article, “Über die verschiedenen Standpunkte in bezug auf das aktuelle Unendliche” (“Concerning Various Perspectives on the Actual Infinite”) (1885). (shrink)
We show how an epistemology informed by cognitive science promises to shed light on an ancient problem in the philosophy of mathematics: the problem of exactness. The problem of exactness arises because geometrical knowledge is thought to concern perfect geometrical forms, whereas the embodiment of such forms in the natural world may be imperfect. There thus arises an apparent mismatch between mathematical concepts and physical reality. We propose that the problem can be solved by emphasizing the ways in which the (...) brain can transform and organize its perceptual intake. It is not necessary for a geometrical form to be perfectly instantiated in order for perception of such a form to be the basis of a geometrical concept. (shrink)
G. E. M. Anscombe’s view that agents know what they are doing “without observation” has been met with skepticism and the charge of confusion and falsehood. Simultaneously, some commentators think that Anscombe has captured an important truth about the first-personal character of an agent’s awareness of her actions. This paper attempts an explanation and vindication of Anscombe’s view. The key to the vindication lies in focusing on the role of practical knowledge in an agent’s knowledge of her actions. Few commentators, (...) with the exception of Moran and Hursthouse, have gotten the emphasis right. The key to a proper interpretation of Anscombe’s views is to explain her claims within the context of her teleological theory of action. The result is a theory ofintentional action that makes self-knowledge of one’s own actions the norm. (shrink)
This is a book review of Karen Armstrong's "The Spiral Staircase", the autobiography of a historian of religion. -/- To cite this article: Newstead, Anne. Compassion, Not Belief [Book Review] [online]. Quadrant, Vol. 49, No. 6, June 2005: 88-89. Availability: <http://search.informit.com.au/documentSummary;dn=203690937218529;res=IELLCC> ISSN: 0033-5002. [cited 06 Dec 12].
Do actual infinities exist or are they impossible? Does mathematical practice require the existence of actual infinities, or are potential infinities enough? Contrasting points of view are examined in depth, concentrating on Aristotle’s ancient arguments against actual infinities. In the long 19th century, we consider Cantor’s successful rehabilitation of the actual infinite within his set theory, his views on the continuum, Zeno's paradoxes, and the domain principle, criticisms by Frege, and the axiomatisation of set theory by Zermelo, as well as (...) Zermelo’s assertion of the primacy of potential infinity in mathematics. (shrink)
Genevieve Lloyd’s Spinoza is quite a different thinker from the arch rationalist caricature of some undergraduate philosophy courses devoted to “The Continental Rationalists”. Lloyd’s Spinoza does not see reason as a complete source of knowledge, nor is deductive rational thought productive of the highest grade of knowledge. Instead, that honour goes to a third kind of knowledge—intuitive knowledge (scientia intuitiva), which provides an immediate, non-discursive knowledge of its singular object. To the embarrassment of some hard-nosed philosophers, intellectual intuition has an (...) affective component; it is a form of love, and ultimately given that human beings are finite modes of God/Nature (Deus sive Natura), it is a form of the intellectual love of God (amor Dei intellectualis). Some philosophers do not know what to make of this mysterious aspect of Spinoza’s philosophy, which is nonetheless firmly anchored in a reading of Part V of the Ethics. Nonetheless, this note will insist with Lloyd’s “Reconsidering Spinoza’s Rationalism” that such doctrine is an integral part of Spinoza’s philosophy. Moreover, it will be shown that Spinoza is well aware of the limitations of reason (ratio) in gaining scientific knowledge of the world and requires intuition precisely because of the inability of reason to represent individuals in their full particularity. Imagination too has a role to play in shaping scientific knowledge, although reason performs a vital critical role in disciplining and liberating the human mind from inadequate imaginary ideas. The result is an interpretation of Spinoza’s epistemology as both rationalist and intuitionist. (shrink)
This paper discusses an argument for the reality of the classical mathematical continuum. An inference to the best explanation type of argument is used to defend the idea that real numbers exist even when they cannot be constructively specified as with the "indefinable numbers".
Does Cantorian set theory alter our intuitive conception of number? Yes. In particular, Cantorian set theory revises our intuitive conception of when two sets have the same size (cardinal number). Consider a variant of Galileo’s Paradox, which notes that the members of the set of natural numbers, N, can be put in one-to-one correspondence with the members of the set of even numbers, E.
It is argued that there are ways of individuating the objects of perception without using sortal concepts. The result is an moderate anti-sortalist position on which one can single out objects using demonstrative expressions without knowing exactly what sort of thing those objects are.
This thesis proposes that an account of first-person reference and first-person thinking requires an account of practical knowledge. At a minimum, first-person reference requires at least a capacity for knowledge of the intentional act of reference. More typically, first-person reasoning requires deliberation and the ability to draw inferences while entertaining different 'I' thoughts. Other accounts of first-person reference--such as the perceptual account and the rule-based account--are criticized as inadequate. An account of practical knowledge is provided by an interpretation of GEM (...) Anscombe's account in her landmark monograph "Intention". (shrink)
Coping with everyday life limits the extent of one’s scepticism. It is practically impossible to doubt the existence of the things with which one is immediately engaged and interacting. To doubt that, say, a door exists, is to step back from merely using the door (opening it) and to reflect on it in a detached, theoretical way. It is impossible to simultaneously act and live immersed in situation S while doubting that one is in S. Sceptical doubts—such as ‘Is this (...) really a door?’, ‘Am I really walking?’ — require a reflective withdrawal in thought from the situation at hand. Maintaining sceptical doubt while coping with everyday life requires a split consciousness, a bad faith, with one part of consciousness doubting the existing of things that the other part takes forgranted. For this reason, a sustained lived sceptical doubt is sometimes thought to be impossible. -/- In this article, I examine Wittgenstein's response to scepticism in "On Certainty". I argue that one of his responses is "the response based on action", which is (as other Wittgenstein interpreters have noted) a characteristically pragmatist response. I then evaluate the quality of this pragmatist response to scepticism, noting that actions just as much as representations are susceptible to mis-interpretation. It is argued that despite the insights contained in it, Wittgenstein's contextualism about meaning is inadequate to rescue the Wittgensteinian response to scepticism. (shrink)
This paper reports on an ongoing ARC Discovery Project that is conducting design research into learning in collaborative virtual worlds (CVW).The paper will describe three design components of the project: (a) pedagogical design, (b)technical and graphics design, and (c) learning research design. The perspectives of each design team will be discussed and how the three teams worked together to produce the CVW. The development of productive failure learning activities for the CVW will be discussed and there will be an interactive (...) demonstration of the project's CVW. (shrink)
Over the past decade, teaching and learning in virtual worlds has been at the forefront of many higher education institutions around the world. The DEHub Virtual Worlds Working Group (VWWG) consisting of Australian and New Zealand higher education academics was formed in 2009. These educators are investigating the role that virtual worlds play in the future of education and actively changing the direction of their own teaching practice and curricula. 47 academics reporting on 28 Australian higher education institutions present an (...) overview of how they have changed directions through the effective use of virtual worlds for diverse teaching and learning activities such as business scenarios and virtual excursions, role-play simulations, experimentation and language development. The case studies offer insights into the ways in which institutions are continuing to change directions in their teaching to meet changing demands for innovative teaching, learning and research in virtual worlds. This paper highlights the ways in which the authors are using virtual worlds to create opportunities for rich, immersive and authentic activities that would be difficult or not possible to achieve through more traditional approaches. (shrink)
This paper presents the intelligent virtual animals that inhabit Omosa, a virtual learning environment to help secondary school students learn how to conduct scientific inquiry and gain concepts from biology. Omosa supports multiple agents, including animals, plants, and human hunters, which live in groups of varying sizes and in a predator-prey relationship with other agent types (species). In this paper we present our generic agent architecture and the algorithms that drive all animals. We concentrate on two of our animals to (...) present how different parameter values affect their movements and inter/intra-group interactions. Two evaluations studies are included: one to demonstrate the effect of different components of our architecture; another to provide domain expert validation of the animal behavior. (shrink)
In November, 2009, a prominent group of privacy professionals, business leaders, information technology specialists, and academics gathered in Madrid to discuss how the next set of threats to privacy could best be addressed.The event, Privacy by Design: The Definitive Workshop, was co-hosted by my office and that of the Israeli Law, Information and Technology Authority. It marked the latest step in a journey that I began in the 1990’s, when I first focused on enlisting the support of technologies that could (...) enhance privacy. Back then, privacy protection relied primarily upon legislation and regulatory frameworks—in an effort to offer remedies for data breaches, after they had occurred. As information technology became increasingly interconnected and the volume of personal information collected began to explode, it became clear that a new way of thinking about privacy was needed.Privacy-Enhancing Technologies paved the way for that new direction, highlighting how the universal pr .. (shrink)
Of all my recollections connected with the H of C that of my having had the honour of being the first to make the claim of women to the suffrage a parliamentary question, is the most gratifying as I believe it to have been the most important public service that circumstances made it in my power to render. This is now a thing accomplished.….
As for Avicenna the human soul is a complete substance which does not inhere in the body nor is imprinted in it, asserting its survival after the death of the body seems easy. Yet, he needs the body to explain its individuation. The paper analyzes Avicenna's arguments in the De anima sections, V, 3 & 4, of the Shifā ' in order to explore the exact causal relation there is between the human soul and its body and confronts these arguments (...) with relevant passages in the Metaphysics. It argues that the causal relation between body and soul remains obscure and that, though Avicenna claims that there is a personal immortality and that the disembodied soul remains individuated, he does not provide a satisfactory ontological account for it. (shrink)
Although it may seem from its formalism that game theory must have sprung from the mind of John von Neumann as a corollary of his work on computers or theoretical physics, it should come as no real surprise to philosophers that game theory is the articulation of a historically developing philosophical conception of rationality in thought and action. The history of ideas about rationality is deeply contradictory at many turns. While there are theories of rationality that claim it is fundamentally (...) social and aims at understanding and molding all facets of human psychological life, game theory takes rationality to be essentially located in individuals and to concern only the means to achieve predetermined ends. Thus, there are some thinkers who have made important contributions to this history who do not appear in the story of game theory at all, among them, Plato, Kant, and Hegel. There is, however, a clear trail to follow linking theories of instrumental rationality from Aristotle to the nineteenth-century marginalist economists and ultimately to von Neumann and Morgenstern and contemporary game theorists, that historically grounds game theory as a model of rational interaction. (shrink)
Virtue words, such as justice, fairness, care, and integrity, frequently feature in organizational codes of conduct and theories of ethical leadership. And yet our modern organizations remain blemished by examples lacking virtue. The philosophy of virtue ethics and numerous extant theories of leadership cite virtues as essential to good leadership. But we seem to lack understanding of how to develop or embed these virtues and notions of good leadership in practice. In 2012, virtue ethicist Julia Annas pointed to a training (...) program which she touted as a practical application of virtue ethics. The program Annas identified is called The Virtues Project, and while promising, she warned that in its current state, it lacked theorizing. We address this by aligning its practical strategies to extant theory and evidence to understand what virtues it might develop and how it might facilitate good leadership. Doing so makes two key contributions. First, it lends credence to The Virtues Project’s potential as a leadership development program. Second, it provides a means of applying theories of good leadership in practice. Our overarching objective is to advance The Virtues Project as a means of incorporating virtues into workplace dynamics and embedding virtues in the practice of organizational leadership. (shrink)
Current theories of reasoning such as mental models or mental logic assume a universal cognitive mechanism that underlies human reasoning performance. However, there is evidence that this is not the case, for example, the work of Ford (1995), who found that some people adopted predominantly spatial and some verbal strategies in a syllogistic reasoning task. Using written and think-aloud protocols, the present study confirmed the existence of these individual differences. However, in sharp contrast to Ford, the present study found few (...) differences in reasoning performance between the two groups, in terms of accuracy or type of conclusion generated. Hence, reasoners present an outward appearance of ubiquity, despite underlying differences in reasoning processes. These findings have implications for theoretical accounts of reasoning, and for attempts to model reasoning data. Any comprehensive account needs to account for strategic differences and how these may develop in logically untrained individuals. (shrink)
Many researchers have suggested that premise interpretation errors can account, at least in part, for errors on categorical syllogisms. However, although it is possible to show that people make such errors in simple inference tasks, the evidence for them is far less clear when actual syllogisms are administered. Part of the problem is due to the lack of clear predictions for the solutions that would be expected when using modified quantifiers, assuming that correct inferences are made from them. This paper (...) presents the expected solutions for Gricean, reversible, and reversible Gricean interpretations, and evaluates these using three datasets (two currently available, and one new). The evidence supported the adoption of reversible and reversible Gricean interpretations, but not Gricean interpretations on their own. These results suggest that the categorical syllogism task tends to induce different quantifier interpretations from those identified in simple inference tasks. (shrink)
Four experiments are reported which investigated the types of truth tables that people associate with conditional sentences and the kinds of inferences that they will draw from them. The present studies differed from most previous ones in using different types of content in the conditionals, for example promises and warnings. It was found that the type of content had a strong and consistent effect on both truth tables and inferences. It is suggested that this is because in real life conditionals (...) make probabilistic assertions, and that the strength of the probabilistic link is determined by the situation in which the conditional occurs. The implications of these findings for current theories of reasoning are considered and it is concluded that none of them is entirely satisfactory. It is suggested that more linguistically based theories may prove more successful. (shrink)