
1The Mathematical Cultures Network ProjectJournal 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.

16William Boos. Metamathematics and the Philosophical TraditionPhilosophia Mathematica 29 (2): 292293. 2021.

3Why ‘scaffolding’ is the wrong metaphor: the cognitive usefulness of mathematical representationsSynthese 197 (9): 37433756. 2020.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

10Weber and Coyote: Polytheism as a Practical AttitudeSophia 59 (2): 211228. 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

1Wot u @ uni 4?Discourse: Learning and Teaching in Philosophical and Religious Studies 9 (1): 93109. 2009.

1Manifesto for Higher EducationDiscourse: Learning and Teaching in Philosophical and Religious Studies 6 (1): 225236. 2006.

2The Case for Teaching Syllogistic Logic to Philosophy StudentsDiscourse: Learning and Teaching in Philosophical and Religious Studies 4 (1): 130136. 2004.

1Critical Friendships Among Beginning PhilosophersDiscourse: Learning and Teaching in Philosophical and Religious Studies 10 (2): 111146. 2011.

25Book Review: What is a Mathematical Concept? edited by Elizabeth de Freitas, Nathalie Sinclair, and Alf ColesJournal of Humanistic Mathematics 9 (2): 309322. 2019.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

16From Euclidean geometry to knots and netsSynthese 196 (7): 27152736. 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

79Feeling the force of argumentIn Andrea Kenkmann (ed.), Teaching Philosophy, Continuum. pp. 134152. 2009.

35Why the Naïve Derivation Recipe Model Cannot Explain How Mathematicians’ Proofs Secure Mathematical KnowledgePhilosophia Mathematica 24 (3): 401404. 2016.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.

40Ineffability and Philosophy, by Andre KuklaMind 118 (472): 11531155. 2009.(No abstract is available for this citation)

10Three is a magic numberThe Philosophers' Magazine 44 8388. 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 20122014 (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

33The 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

26Weber and Coyote: Polytheism as a Practical AttitudeSophia 118. 2018.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

46Emily Grosholz and Herbert Breger, editors. The Growth of Mathematical KnowledgePhilosophia Mathematica 10 (1): 9396. 2002.

34From Euclidean geometry to knots and netsSynthese 122. 2017.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

117Why did Kuhn’s S tructure of Scientific Revolutions Cause a Fuss?Studies in History and Philosophy of Science Part A 34 (2): 369390. 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

76George Kampis, Ladislav Kvasz, and Michael Stoltzner, eds. Appraising Lakatos: Mathematics, Methodology, and the ManPhilosophia Mathematica 12 (3): 294300. 2004.

28EMILY R. GROSHOLZ. Representation and Productive Ambiguity in Mathematics and the Sciences. Oxford: Oxford University Press, 2007. ISBN 9780199299737. Pp. viii + 313 (review)Philosophia Mathematica 20 (2): 245252. 2012.

35Michel Serfati. La Révolution Symbolique: La Constitution de l'Ecriture Symbolique Mathématique. Preface by Jacques Bouverasse. Paris: Éditions Petra, 2005. Pp. ix + 427. ISBN 2847430067 (review)Philosophia Mathematica 15 (1): 122126. 2007.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

196Tu quoque, ArchbishopThink 3 (7): 101108. 2004.Brendan Larvor finds that the Archbishop of Canterbury's recent arguments about religious education are a curate's egg.

1Michael D. Resnik, Mathematics as a Science of PatternsInternational Studies in the Philosophy of Science 12 (3): 287289. 1998.

119How to think about informal proofsSynthese 187 (2): 715730. 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

26What Philosophy of Mathematical Practice Can Teach Argumentation Theory About Diagrams and PicturesIn Andrew Aberdein & Ian J. Dove (eds.), The Argument of Mathematics, Springer. pp. 239253. 2013.

21After Popper, Kuhn and Feyerabend: Recent Issues in Theories of Scientific MethodMetascience 100104. 2002.
Areas of Specialization
Philosophy of Mathematics 
Areas of Interest
Logic and Philosophy of Logic 
Philosophy of Mathematics 