•  1
    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.
  •  16
  •  3
    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
  •  10
    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
  •  1
    Wot u @ uni 4?
    Discourse: Learning and Teaching in Philosophical and Religious Studies 9 (1): 93-109. 2009.
  •  1
    Manifesto for Higher Education
    Discourse: Learning and Teaching in Philosophical and Religious Studies 6 (1): 225-236. 2006.
  •  2
    The Case for Teaching Syllogistic Logic to Philosophy Students
    Discourse: Learning and Teaching in Philosophical and Religious Studies 4 (1): 130-136. 2004.
  •  1
    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.
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    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
  •  16
    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
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    The view that a mathematical proof is a sketch of or recipe for a formal derivation requires the proof to function as an argument that there is a suitable derivation. This is a mathematical conclusion, and to avoid a regress we require some other account of how the proof can establish it.
  •  40
    Ineffability and Philosophy, by Andre Kukla
    Mind 118 (472): 1153-1155. 2009.
    (No abstract is available for this citation)
  •  10
    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
  • 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
  •  33
    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
  •  26
    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
  •  34
    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
  •  117
    Why did Kuhn’s S tructure of Scientific Revolutions Cause a Fuss?
    Studies in History and Philosophy of Science Part A 34 (2): 369-390. 2003.
    After the publication of The structure of scientific revolutions, Kuhn attempted to fend off accusations of extremism by explaining that his allegedly “relativist” theory is little more than the mundane analytical apparatus common to most historians. The appearance of radicalism is due to the novelty of applying this machinery to the history of science. This defence fails, but it provides an important clue. The claim of this paper is that Kuhn inadvertently allowed features of his procedure and …Read more
  •  35
    It is difficult to imagine mathematics without its symbolic language. It is especially difficult to imagine doing mathematics without using mathematical notation. Nevertheless, that is how mathematics was done for most of human history. It was only at the end of the sixteenth century that mathematicians began to develop systems of mathematical symbols . It is startling to consider how rapidly mathematical notation evolved. Viète is usually taken to have initiated this development with his Isagog…Read more
  •  196
    Tu quoque, Archbishop
    Think 3 (7): 101-108. 2004.
    Brendan Larvor finds that the Archbishop of Canterbury's recent arguments about religious education are a curate's egg.
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    Reply to James Blachowicz
    The Owl of Minerva 31 (1): 53-54. 1999.
  •  1
    Michael D. Resnik, Mathematics as a Science of Patterns
    International Studies in the Philosophy of Science 12 (3): 287-289. 1998.
  •  119
    How to think about informal proofs
    Synthese 187 (2): 715-730. 2012.
    It is argued in this study that (i) progress in the philosophy of mathematical practice requires a general positive account of informal proof; (ii) the best candidate is to think of informal proofs as arguments that depend on their matter as well as their logical form; (iii) articulating the dependency of informal inferences on their content requires a redefinition of logic as the general study of inferential actions; (iv) it is a decisive advantage of this conception of logic that it accommodat…Read more
  •  17
    The owl and the pussycat
    Philosophical Quarterly 44 (175): 233-239. 1994.