•  35
    The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least, scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection on examples from elementary…Read more
  •  35
    Reply to James Blachowicz
    The Owl of Minerva 31 (1): 53-54. 1999.
  •  33
    Book reviews (review)
    with Jessica Carter, Jussi Haukioja, and Mariska E. M. P. J. Leunissen
    International Studies in the Philosophy of Science 21 (2). 2007.
    Terence Tao New York, Oxford University Press, 2006xii + 103 pp., ISBN 9780199205615, £37.50 (hardback), ISBN 9780199205608, £12.99 (paperback)This is a book of mathematical problems and their solu...
  •  31
    This is a review of What is a Mathematical Concept? edited by Elizabeth de Freitas, Nathalie Sinclair, and Alf Coles. In this collection of sixteen chapters, philosophers, educationalists, historians of mathematics, a cognitive scientist, and a mathematician consider, problematise, historicise, contextualise, and destabilise the terms ‘mathematical’ and ‘concept’. The contributors come from many disciplines, but the editors are all in mathematics education, which gives the whole volume a discipl…Read more
  •  30
    The formalising tendency in philosophy and experimental psychology
    Phenomenology and the Cognitive Sciences 2 (4): 337-352. 2003.
    This paper is an exercise in the phenomenology of science. It examines the tendency to prefer formal accounts in a familiar body of experimental psychology. It will argue that, because of this tendency, psychologists of this school neglect those forms of human cognition typical of the humanities disciplines. This is not a criticism of psychology, however. Such neglect is compatible with scientific rigour, provided it does not go unnoticed. Indeed, reflection on the case in hand allows us to refi…Read more
  •  28
    Weber and Coyote: Polytheism as a Practical Attitude
    Sophia 59 (2): 211-228. 2020.
    Hyde claims that the trickster spirit is necessary for the renewal of culture, and that he lives only in the ‘complex terrain of polytheism’. Fortunately for those of us in monotheistic cultures, Weber gives reasons for thinking that polytheism is making a return, albeit in a new, disenchanted form. The plan of this paper is to elaborate some basic notions from Weber, to explore Hyde’s thesis in more detail and then to take up the question of the plurality of spirits both around and within us an…Read more
  •  22
    History, methodology and early algebra 1
    International Studies in the Philosophy of Science 8 (2): 113-124. 1994.
    The limits of ‘criterial rationality’ (that is, rationality as rule‐following) have been extensively explored in the philosophy of science by Kuhn and others. In this paper I attempt to extend this line of enquiry into mathematics by means of a pair of case studies in early algebra. The first case is the Ars Magna (Nuremburg 1545) by Jerome Cardan (1501–1576), in which a then recently‐discovered formula for finding the roots of some cubic equations is extended to cover all cubics and proved. The…Read more
  •  19
    From Euclidean geometry to knots and nets
    Synthese 196 (7): 2715-2736. 2019.
    This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or …Read more
  •  18
    Albert Lautman, ou la dialectique dans les mathématiques
    Philosophiques 37 (1): 75-94. 2010.
    Dans cet article, j’explore dans un premier temps la conception que se fait Lautman de la dialectique en examinant ses références à Platon et Heidegger. Je compare ensuite les structures dialectiques identifiées par Lautman dans les mathématiques contemporaines avec celles qui émergent de ses sources philosophiques. Enfin, je soutiens que les structures qu’il a découvertes dans les mathématiques sont plus riches que le suggère son modèle platonicien, et que la distinction « ontologique » de Heid…Read more
  •  17
    Old Maps, Crystal Spheres, and the Cartesian Circle
    Graduate Faculty Philosophy Journal 22 (2): 13-27. 2001.
    It would be a mistake to imagine that the problem of the Cartesian circle lies in Descartes’ suggestion that we cannot know anything unless we know God. It is true that this thought seems fatal to his enterprise; for if we cannot know anything prior to knowing that God exists, then it follows that we cannot know the arguments that prove God’s existence. However the problem of the Cartesian circle does not consist in this logical error. It consists, rather, in the fact that Descartes’ attempts to…Read more
  •  17
    The owl and the pussycat
    Philosophical Quarterly 44 (175): 233-239. 1994.
  •  16
    In, Jeremy Avigad makes a novel and insightful argument, which he presents as part of a defence of the ‘Standard View’ about the relationship between informal mathematical proofs and their corresponding formal derivations. His argument considers the various strategies by means of which mathematicians can write informal proofs that meet mathematical standards of rigour, in spite of the prodigious length, complexity and conceptual difficulty that some proofs exhibit. He takes it that showing that …Read more
  •  12
    Book reviews (review)
    with Peter Lipton, Hans Oberdiek, and Paul Abela
    International Studies in the Philosophy of Science 7 (2): 191-207. 1993.
    The Chances of Explanation: Causal Explanation in the Social, Medical, and Physical Sciences Paul Humphreys, 1989 Princeton University Press x+170 pp., £12.95 (paperback) ISBN 0 691 020286 8; £25.00 (hardback) ISBN 0 69107353 8In Search of a Better World: Lectures and Essays from Thirty Years Karl Popper London, Routledge £25.00 (hardback)Artificial Morality: Virtuous Robots for Virtual Games Peter Danielson, 1992 London, Routledge £35.00 (hardback) ISBN 0 415 034841; £10.99 (paperback) ISBN 0 4…Read more
  •  11
    Three is a magic number
    The Philosophers' Magazine 44 83-88. 2009.
    Logical theory – and philosophical theory generally – is just that, theory. Generations of logic students felt a sort of unease about it without knowing what to do about it. Nowadays, students of mathematical logic feel a similar unease when faced with the fact that in standard predicate calculus, “All unicorns are sneaky” is true precisely because there are no unicorns. Blanché’s analysis reminds us that such feelings of unease may indicate a shortcoming in the theory rather than in the student…Read more
  •  8
    The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least (and plausibly for others), scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection …Read more
  •  8
    The Mathematical Cultures Network Project
    Journal of Humanistic Mathematics 2 (2). 2012.
    The UK Arts and Humanities Research Council has agreed to fund a series of three meetings with associated publications on mathematical cultures. This note describes the project.
  •  8
    Mathematical Cultures: The London Meetings 2012-2014 (edited book)
    Springer International Publishing. 2016.
    This collection presents significant contributions from an international network project on mathematical cultures, including essays from leading scholars in the history and philosophy of mathematics and mathematics education.​ Mathematics has universal standards of validity. Nevertheless, there are local styles in mathematical research and teaching, and great variation in the place of mathematics in the larger cultures that mathematical practitioners belong to. The reflections on mathematical cu…Read more
  •  5
    Critical Friendships Among Beginning Philosophers
    with John Lippitt and Kathryn Weston
    Discourse: Learning and Teaching in Philosophical and Religious Studies 10 (2): 111-146. 2011.
  •  4
    Book reviews (review)
    with Tony O'Connor, William Desmond, Paul K. Moser, Susan Mendus, Gregory McCulloch, George Huxley, Christopher McKnight, John Bussanich, Alison Ainley, Robert Hanna, Attracta Ingram, Dominic Lopes, Vasilis Politis, and Scott A. Shalkowski
    Humana Mente 2 (1): 145-172. 1994.
  •  3
    The Case for Teaching Syllogistic Logic to Philosophy Students
    Discourse: Learning and Teaching in Philosophical and Religious Studies 4 (1): 130-136. 2004.
  •  2
    Manifesto for Higher Education
    Discourse: Learning and Teaching in Philosophical and Religious Studies 6 (1): 225-236. 2006.
  •  2
    Wot u @ uni 4?
    Discourse: Learning and Teaching in Philosophical and Religious Studies 9 (1): 93-109. 2009.
  •  2
    Naturalism
    In Andrew Copson & A. C. Grayling (eds.), The Wiley Blackwell Handbook of Humanism, Wiley-blackwell. 2015.
    Naturalism is sometimes cast as the claim that there is nothing supernatural, nothing ‘spooky’ in the world. One can see that naturalism has two aspects: it makes claims about what there is, and it makes claims about knowledge and explanation. This chapter considers the ontological aspect first, so that one can see what is at stake when comes to the second, epistemological, aspect. The great advantage of methodological naturalism is that it leaves open the question of whether full‐strength, unco…Read more