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317Online philosophy seminar notes, for virtual conference on the Aristotelian philosophy of mathematics, hosted by University of Geneva (organiser Ryan Miller), June 15, 2023
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151"Infinity, Knowledge, and Divinity in the Thought of Cusanus and Cantor" (Manuscript draft of first page of forthcoming book chapter ) (edited book)De Gruyter. forthcoming.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…Read more
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252Absolute Infinity, Knowledge, and Divinity in the Thought of Cusanus and Cantor (ABSTRACT ONLY)In Mirosław Szatkowski (ed.), Ontology of Divinity, De Gruyter. pp. 561-580. 2024.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…Read more
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193This 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.
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245This 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.
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875Knowledge Beyond Reason in Spinoza’s Epistemology: Scientia Intuitiva and Amor Dei Intellectualis in Spinoza’s EpistemologyAustralasian Philosophical Review 4 (Revisiting Spinoza's Rationalism). 2020.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…Read more
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6780Cantor on Infinity in Nature, Number, and the Divine MindAmerican Catholic Philosophical Quarterly 83 (4): 533-553. 2009.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 necessi…Read more
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1170The Epistemology of Geometry I: the Problem of ExactnessProceedings of the Australasian Society for Cognitive Science 2009. 2010.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 th…Read more
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707On the Reality of the Continuum Discussion Note: A Reply to Ormell, ‘Russell's Moment of Candour’, PhilosophyPhilosophy 83 (1): 117-127. 2008.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 pr…Read more
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542Indispensability Without PlatonismIn Alexander Bird, Brian Ellis & Howard Sankey (eds.), Properties, Powers, and Structures: Issues in the Metaphysics of Realism, Routledge. pp. 81-97. 2012.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 …Read more
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3Actual versus Potential Infinity (BPhil manuscript.)Dissertation, University of Oxford. 1997.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 t…Read more
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2369Aristotle and modern mathematical theories of the continuumIn Demetra Sfendoni-Mentzou & James Brown (eds.), Aristotle and Contemporary Philosophy of Science, Peter Lang. 2001.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 inte…Read more
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92Over 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 highe…Read more
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1072Knowledge by Intention? On the Possibility of Agent's KnowledgeIn Stephen Hetherington (ed.), Aspects of Knowing, Elsevier Science. pp. 183. 2006.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).
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925Evans's anti-cartesian argument: A critical evaluationRatio 19 (2): 214-228. 2006.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 misidentificat…Read more
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687Thought, Reference, and Experience: Themes from the Philosophy of Gareth Evans (review)Australasian Journal of Philosophy 84 (1): 5. 2006.This is a very short book review of a recent volume on the philosophy of Gareth Evans with special attention to work on first-person reference.
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548Collaborative Virtual Worlds and Productive FailureIn Proceedings of the CSCL (Computer Supported Cognition and Learning) III, University of Hong Kong. 2011.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…Read more
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568Review of Oppy's Philosophical Perspectives on Infinity (review)Australasian Journal of Philosophy 85 (4): 679-695. 2007.This is a book review of Oppy's "Philosophical Perspectives on Infinity", which is of interest to those in metaphysics, epistemology, philosophy of science, mathematics, and philosophy of religion.
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673Singling Out Objects without SortalsIn Slezak Peter (ed.), International Conference on Cognitive Science (ICCS), . 2003.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.
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2036Interpreting Anscombe’s Intention §32FFJournal of Philosophical Research 34 157-176. 2009.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 ac…Read more
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699This 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…Read more
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671Self-Conscious Self-Reference: An Approach Based on Agent's Knowledge (DPhil manuscript)Dissertation, Oxford University. 2004.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 critic…Read more
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92Evaluating the models and behaviour of 3D intelligent virtual animals in a predator-prey relationship. AAMAS 2012: 79-86Proceedings of the Eleventh International Conference on Agent and Multiagent Systems (AAMAS). 2012.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 con…Read more
Sydney, NSW, Australia
Areas of Specialization
History of Western Philosophy |
Metaphysics and Epistemology |
Philosophy of Mathematics |