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338Mathematics, The Computer Revolution and the Real WorldPhilosophica 42 (n/a): 79-92. 1988.The philosophy of mathematics has largely abandoned foundational studies, but is still fixated on theorem proving, logic and number theory, and on whether mathematical knowledge is certain. That is not what mathematics looks like to, say, a knot theorist or an industrial mathematical modeller. The "computer revolution" shows that mathematics is a much more direct study of the world, especially its structural aspects.
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725Aristotelianism in the Philosophy of MathematicsStudia Neoaristotelica 8 (1): 3-15. 2011.Modern philosophy of mathematics has been dominated by Platonism and nominalism, to the neglect of the Aristotelian realist option. Aristotelianism holds that mathematics studies certain real properties of the world – mathematics is neither about a disembodied world of “abstract objects”, as Platonism holds, nor it is merely a language of science, as nominalism holds. Aristotle’s theory that mathematics is the “science of quantity” is a good account of at least elementary mathematics: the ratio …Read more
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633Uninstantiated Properties and Semi-Platonist AristotelianismReview of Metaphysics 69 (1): 25-45. 2015.A problem for Aristotelian realist accounts of universals (neither Platonist nor nominalist) is the status of those universals that happen not to be realised in the physical (or any other) world. They perhaps include uninstantiated shades of blue and huge infinite cardinals. Should they be altogether excluded (as in D.M. Armstrong's theory of universals) or accorded some sort of reality? Surely truths about ratios are true even of ratios that are too big to be instantiated - what is the truthmak…Read more
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7Philosophy, mathematics and structurePhilosopher: revue pour tous 1 (2): 31-38. 1995.An early version of the work on mathematics as the science of structure that appeared later as An Aristotelian Realist Philosophy of Mathematics (2014)
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867Review of N. Wildberger, Divine Proportions: Rational Trigonometry to Universal (review)Mathematical Intelligencer 28 (3): 73-74. 2006.Reviews Wildberger's account of his rational trigonometry project, which argues for a simpler way of doing trigonometry that avoids irrationals.
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671Achievements and fallacies in Hume's account of infinite divisibilityHume Studies 20 (1): 85-101. 1994.Throughout history, almost all mathematicians, physicists and philosophers have been of the opinion that space and time are infinitely divisible. That is, it is usually believed that space and time do not consist of atoms, but that any piece of space and time of non-zero size, however small, can itself be divided into still smaller parts. This assumption is included in geometry, as in Euclid, and also in the Euclidean and non- Euclidean geometries used in modern physics. Of the few who have deni…Read more
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707Diagrammatic Reasoning and Modelling in the Imagination: The Secret Weapons of the Scientific RevolutionIn Guy Freeland & Anthony Corones (eds.), 1543 and All That: Image and Word, Change and Continuity in the Proto-Scientific Revolution, Kluwer Academic Publishers. 2000.Just before the Scientific Revolution, there was a "Mathematical Revolution", heavily based on geometrical and machine diagrams. The "faculty of imagination" (now called scientific visualization) was developed to allow 3D understanding of planetary motion, human anatomy and the workings of machines. 1543 saw the publication of the heavily geometrical work of Copernicus and Vesalius, as well as the first Italian translation of Euclid.
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280Non-deductive logic in mathematicsBritish Journal for the Philosophy of Science 38 (1): 1-18. 1987.Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as Fermat's Last Theorem and the Riemann Hypothesis, have had to be considered in terms of the evidence for and against them. It is argued here that it is not adequate to describe the relation of evidence to hypothesis as `subjective', `heuristic' …Read more
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489Indispensability Without PlatonismIn Alexander Bird, Brian David Ellis & Howard Sankey (eds.), Properties, Powers and Structures: Issues in the Metaphysics of Realism, Routledge. pp. 81-97. 2013.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
Sydney, New South Wales, Australia
Areas of Specialization
Applied Ethics |
Science, Logic, and Mathematics |
Philosophy of Mathematics |
Interpretation of Probability |
Areas of Interest
Philosophy of Mathematics |
General Philosophy of Science |
PhilPapers Editorships
Mathematical Aristotelianism |