Renaissance philosopher, mathematician, and theologian Nicholas of Cusa (1401-1464) said that there is no proportion between the finite mind and the infinite. He is fond of saying reason cannot fully comprehend the infinite. That our best hope for attaining a vision and understanding of infinite things is by mathematics and by the use of contemplating symbols, which help us grasp "the absolute infinite". By the late 19th century, there is a decisive intervention in mathematics and its philosophy: the philosophical mathematician (...) Georg Cantor (1845-1918) says that between the realm of the finite and the absolute infinite, there is an intermediate realm partaking in properties in a certain sense of both the finite and the infinite: the transfinite realm. Like the finite, the transfinite realm is a realm of mathematical objects, numbers, and knowledge. Like the absolute infinite, the transfinite is a form of infinity insofar as transfinite sets and numbers transcend any finite number. Echoing Cusanus and neo-Platonism, Cantor says that the transfinite sequence of all ordinals is a symbol of the absolutely infinite, that is, God. This paper considers how the doctrine of symbolism and the philosophers' different commitments to the laws of logic, especially the Law of Non-Contradiction (LNC), enables these thinkers to articulate a transcendental apophatic approach to divinity. (shrink)

This is a copy of the presentation given at the "Workshop on Agency and Distributed Cognition" at Macquarie University, March 2012. What is noteworthy about this piece of work is that (i) it is a very early foray into the pedagogy, ontology, and epistemology of virtual worlds (it's 2012, way before David Chalmers' book "Reality+" in 2022); and (ii) it was my first foray into "social epistemology" beyond the standard "S knows that p" epistemology, drawing on Vygotskian collaborative approaches to (...) learning and teaching, moving beyond analytic philosophy into educational research. Finally, (iii) there's a hyperlink to the actual virtual world in Unity3D created by a cross-disciplinary team of researchers from Sydney University and Macquarie University for the purposes of teaching scientific inquiry. (shrink)

This volume brings together chapters about aging in many non-Western cultures, from Africa and Asia to South America, from American Indians to Australian and Hawaii Aboriginals. It also includes articles on other issues of aging, such as falling, dementia, and elder abuse. It was thought that in Africa or Asia, elders were revered and taken care of. This certainly used to be the case. But the Western way has moved into these places, and we now find that elders are often (...) left on their own or in institutions, as younger people have migrated to other cities and even countries. Grandparents often find themselves being parents to their grandchildren, a far cry from the kind of life they believed they would have as they aged. This book will explore all these issues and will be of use to students and researchers in this relatively new field. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Poems Ancient and Contemporary HELAINE L. SMITH On the cover of Like: Poems by A. E. Stallings is a double photograph of a double image: two ancient carved heads, in profile and facing each other, of the pole horses of a quadriga, a four-horse chariot, dated about 570 BC, and currently in the collection of The Acropolis Museum. The marble horse in profile on the right side of (...) the cover, looking to the left, has a longer neck and face, and is undamaged except for a hairline crack at the neck. Its counterpart has a broader cheek, with light chipping above the nostril and an ear broken off. But they so mirror each other, they are so “like,” that at first glance one might suppose the photographs to be the right and left sides of the same horse. The bottom half of the cover of Like is an exact replica of the top, but the tinting is different, the heads in the top band cream-colored, in the arion 27.1 spring/summer 2019 Like: Poems by A. E. Stallings was published in 2018 by Farrar, Straus and Giroux, New York. Stallings’s earlier books of poetry include Archaic Smile (1999), Hapax (2006), Olives (2012), and a verse translation of Lucretius’ The Nature of Things (2007). bottom a delicate ochre or sand. There is, then, on the very cover of Like a promise of complex correspondence and subtle variation within the volume itself. Stallings’s title suggests poetry as the art of locating and articulating likeness or as the act of finding one thing in another. And in Stallings’s delicate, robust, witty, and humane use of the past, “like” also and most significantly means that they—the figures of ancient myth—and we are alike in our shared experience and shared humanity. Stallings sometimes presents classical and mythological references glancingly, and that is part of the delight—catching the joke in Hermes’ epithet, “lord of florists,”1 finding in “The Rosehead Nail” that the local blacksmith in Monteagle, Tennessee, “was a god / Before... the tasks / That had been craft were jobbed out to machine.” Likeness is, of course, the mode of Homeric simile, and in “Selvage” Stallings turns the simile of Odyssey 22.468–73, in which Homer compares the hanged maids to thrushes caught in a net, into a different but contextually relevant parallel: “they hang... From the warp like a dozen ancient loom weights,” a likeness that evolves from Stallings’s double entendre in line seven describing the maids as “spinsters of their own doom,” an allusion both to the task, properly theirs, of weaving and to their disclosure to their lovers of Penelope’s secret unravelling. Stallings can expand as well as adapt Homeric similes. At Iliad 2.87–2.90 Homer likens the Achaians’ assembly at Agamemnon’s command to arm and attack Troy to bees leaving the hive: Like the swarm of clustering bees that issue forever in fresh bursts from the hollow in the stone, and hang like bunched grapes as they hover beneath the flowers in springtime fluttering in swarms together this way and that way so the many nations of men from the ships and the shelters along the front of the deep sea beach marched in order by companies to the assembly2 178 poems ancient and contemporary Homer finds likeness in the orderly setting forth—of bees, of men—to do what is required. This, on the other hand, is what Stallings does: Colony Collapse Disorder (Iliad, 2.87–2.90) Just as a swarm pours from a hollow rock In one long beeline for the wild thyme, Alighting in clusters on this purple and that, But is stricken with a mass amnesia That disorients the compass of the sun, And they forget the steps to traditional dances, And each helicopters into a different dimness Taking their saddlebags of sweetness with them, And the hive goes dark, the queen is left to starve, And the drones humbug the whimper of the world, And the palace falls to ruins, broken into By vandals who would loot the golden stores Left in the brittle wax hexameters, Just so... And “so” her... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:“Warn Me If I Approach the Melody” HELAINE L. SMITH In the 1950s on Saturday night TV, Sid Caesar performed comic sketches for a full hour. In one sketch Carl Reiner played Edward R. Murrow interviewing Caesar as the jazz musician Progress Hornsby. At a certain point Murrow asks Hornsby, “To what do you attribute your band’s great success?” and Hornsby answers, “Well, we have special equipment that (...) warns us if we get anywhere near the melody.” And so it must be with Nancy Worman and all those classicists who approach classics and literature as she does—special equipment to warn them whenever they get anywhere near a responsible reading of the text.* Her argument, cast in sentences whose words, like the amphisbaena, turn back on themselves or are unrecognizable as spoken or written English, boils down to (1) Greek men were colonialists and imperialists; (2) Greek tragedy celebrated men of this sort; (3) there arose, quite to the surprise and possibly to the consternation of its authors, women in their plays who were strong and violent rebels against the oppressive social order; (4) Virginia Woolf, whose feminism is not quite as pure as we’d like it, is not so bad after all because, if you look closely, pulling out the odd word here and there, decontextualizing it, inserting its decontextualized meaning elsewhere and mistranslating any Greek phrase that stands in your way, you discover—voila!—that she held views of Greece and Greek tragedy identical to those of radical feminism and shaped her novels, stories, and essays to reflect that view. Worman’s feminist audience, as familiar as it must be with feminist reception theory, would appear, from the unusual number of play summaries, elementary descriptions of the arion 28.1 spring/summer 2020 *Nancy Worman, Virginia Woolf’s Greek Tragedy. London and New York: Bloomsbury Academic, 2019. xii + 152 pages. $102.00. 150 “warn me if i approach the melody” origin and structure of Greek drama, and serious mistranslations in her book, to be unfamiliar with ancient drama, with basic Greek, and perhaps with Virginia Woolf as well. Her book has from such scholars received copious praise. But for those who begin less thematically burdened and who, as a colleague of mine liked to say, “worship at the altar of the text,” Worman’s methods turn scholarly inquiry on its head, not because of what they conclude, but because her conclusions are reached the same way the Queen of Hearts insisted trials be conducted: verdict first, evidence afterwards. Nor does Worman’s text have any discernible structure. Material within chapters bears fleeting connection to chapter titles and subheads. Points appear, vanish, and reappear, like the Cheshire cat, suggesting prior academic papers, here combined. This overall disorganization weakens the book, as does the non-linear quality of individual points. The organization of this review, therefore, is not thematic but chronological : it addresses those works of Woolf’s to which Worman gives attention, and presents them by date of publication. A Room of One’s Own appears nowhere in Worman’s text, and To the Lighthouse receives only minimal attention. Medea is nowhere mentioned in the favored categories of proto-feminist and fierce woman. While The Trojan Women and The Birds would both have served Worman well in arguing that Greek theatre reflected Athenian imperialism, they too make no appearance in her text. Instead, she casts Sophocles’ Electra and Antigone as fierce women and their plays, together with the Agamemnon, as emblematic of Athenian aggression and patriarchy. And so, in the forced nature of Worman’s translations and in the reductiveness of her assessment of Woolf, we see the consequences of a determination to bend texts to serve popular theories. “a dialogue upon mount pentelicus” (1906) woolf’s views of Greece reveal the complexity of great writing. Woolf frequently satirizes classicists who take their Helaine L. Smith 151 learning to be a sign of natural excellence, those for whom knowledge of Greek is simply a marker of position and rank. But for those who love such things for their own sake, or come to love them, as the tourists eventually do in “A Dialogue upon Mount Pentelicus,” she... (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)

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)

First published in 1992. Routledge is an imprint of Taylor & Francis, an informa company.

This book enables readers to understand contemporary Japanese society and culture. Since it is written by experts, it allows readers to start with any chapters they are interested in. It also provides a unique way to introduce Japanese society and culture to those who have never visited or studied Japanese society by reading articles from various authors on topics such as gender, family, economy, natural disasters and politics and laws. It provides scholars, academics, graduate students and the general educated audience (...) all the information required to understand contemporary Japanese society and culture fully and see the diverse perspectives available. (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)

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. DOI: 10.1080/24740500.2021.1962652. (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)

Online philosophy seminar notes, for virtual conference on the Aristotelian philosophy of mathematics, hosted by University of Geneva (organiser Ryan Miller), June 15, 2023.

A fallibilist theory of knowledge is employed to make sense of the idea that agents know what they are doing 'without observation' (as on Anscombe's theory of practical knowledge).

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)

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)

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)

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 is a grant proposal for a research project conceived and written as a Research Associate at UNSW in 2011. I have plans to spin it into an article.

This is a grant proposal written as a post-doctoral application for MQRF at Macquarie University in 2009. As I did so much research for it, I have plans to spin it into an article. I note with pleasure the increased relevance of this kind of research given developments such as Neuralink and ChatGPT.

Stanovich & West's claim that there are two coherent and conceptually distinct types of thinking, System 1 and System 2, is questioned. Some authors equate System 2 with intelligence whereas other do not; and some authors regard the two types of system as distinct while others regard them as lying on a continuum.

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)

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)

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.

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)

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)

AbstractVirtues, anchored in the ancient and robust philosophy of virtue ethics, inform and enable good leadership. However, we are reticent to speak of virtues within the business domain, which hinders virtues-based leadership development. To demonstrate how virtues inform good leadership, albeit usually implicitly, I analyze 25 TED talks promised to make viewers ‘better’ leaders for direct and indirect reference to virtues. My findings illustrate that virtues are implicitly woven throughout popular leadership discourse, but that they are rarely stated explicitly. This (...) is a problem because to develop virtues they need to be explicitly understood and consciously practiced, which necessitates redressing the reticence to speak of virtues and focusing efforts on educating, training, and developing the virtues that enable good leadership. This article advances virtues-based leadership development by proffering a framework of higher-order virtues that we need to make explicit in efforts to develop good leadership. My discussion of the higher-order virtues integrates evidence from the TED talks and extant scholarship and proposes ways to train and develop each virtue. (shrink)

Two studies are reported which demonstrate that analytic responding on everyday reasoning problems can be increased and bias eliminated after training on the law of large numbers. Critical thinking problems involving belief-consistent, neutral, and inconsistent conclusions were presented. Belief bias was eliminated when a written justification of argument strength was elicited. However, belief-based responding was still evident when evaluations of the arguments were elicited using rating scales. This finding demonstrates a dissociation between analytic and belief-based responding as a function of (...) response format. In Experiment 2 an instructional condition designed to foster decontextualised reasoning was included but was ineffective in reducing the degree to which judgements were biased by beliefs. It was concluded that training which makes available the analytic strategies necessary to evaluate a problem has the potential to facilitate performance only if the requested response triggers conscious deliberation of the evidence. (shrink)

(1994). Inductive reasoning, deductive reasoning and mental models. International Studies in the Philosophy of Science: Vol. 8, No. 1, pp. 65-67. doi: 10.1080/02698599408573483.

The aim of the present research was to develop a difficulty model for logical reasoning problems involving complex ordered arrays used in the Graduate Record Examination. The approach used involved breaking down the problems into their basic cognitive elements such as the complexity of the rules used, the number of mental models required to represent the problem, and question type. Weightings for these different elements were derived from two experimental studies and from the reasoning literature. Based on these weights, difficulty (...) models were developed which were then tested against new data. The models had excellent predictive validity and showed the relative influence of rule based factors and factors relating to the number of underlying models. Different difficulty models were needed for different question types, suggesting that people used a variety of approaches and, at a wider level, that both mental models and mental rules may be used in reasoning. (shrink)