-
8Kneebone and Lakatos: at the roots of a dialectical philosophy of mathematicsHopos: The Journal of the International Society for the History of Philosophy of Science. forthcoming.
-
56Book Reviews (review)International Studies in the Philosophy of Science 21 (2): 213-225. 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...
-
20Donald Gillies. Lakatos and the Historical Approach to Philosophy of MathematicsPhilosophia Mathematica 32 (2): 258-262. 2024.
-
6History, Role in the Philosophy of ScienceIn W. Newton-Smith (ed.), A companion to the philosophy of science, Blackwell. 2000.The leading philosophers of science of the first half of the twentieth century had little use for the history of science. There are several possible explanations for this. One is that philosophers of science sometimes (knowingly or not) mimic the methodological habits and values of scientists. Many philosophers of science are motivated by admiration for the perceived rigor and intellectual hygiene of the exact sciences. Historical sense is not normally a cardinal virtue among physicists. Hence, …Read more
-
12NaturalismIn 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
-
65Should philosophy replace religious education? A reply to brenda watson: Larvor Reply to WatsonThink 4 (10): 31-33. 2005.In Issue 7 of Think, Brendan Larvor criticised the Archbishop of Canterbury, Rowan Williams, for suggesting that atheism and humanism ought not to be taught in schools alongside the religious faiths. In Issue 9, Brenda Watson defended the Archbishop's view. Here, Larvor replies to Watson. The numbers below refer to numbered points in Watson's piece.
-
47On the unreasonable reliability of mathematical inferenceSynthese 200 (4): 1-16. 2022.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
-
17The 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.
-
34William Boos. Metamathematics and the Philosophical TraditionPhilosophia Mathematica 29 (2): 292-293. 2021.
-
16Why ‘scaffolding’ is the wrong metaphor: the cognitive usefulness of mathematical representationsSynthese 197 (9): 3743-3756. 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 (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
-
46Weber and Coyote: Polytheism as a Practical AttitudeSophia 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
-
15Wot u @ uni 4?Discourse: Learning and Teaching in Philosophical and Religious Studies 9 (1): 93-109. 2009.
-
13Manifesto for Higher EducationDiscourse: Learning and Teaching in Philosophical and Religious Studies 6 (1): 225-236. 2006.
-
17The Case for Teaching Syllogistic Logic to Philosophy StudentsDiscourse: Learning and Teaching in Philosophical and Religious Studies 4 (1): 130-136. 2004.
-
19Critical Friendships Among Beginning PhilosophersDiscourse: Learning and Teaching in Philosophical and Religious Studies 10 (2): 111-146. 2011.
-
42Book Review: What is a Mathematical Concept? edited by Elizabeth de Freitas, Nathalie Sinclair, and Alf Coles (review)Journal of Humanistic Mathematics 9 (2): 309-322. 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
-
40From Euclidean geometry to knots and netsSynthese 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
-
89Feeling the force of argumentIn Andrea Kenkmann (ed.), Teaching Philosophy, Continuum. pp. 134-152. 2009.
-
66Why the Naïve Derivation Recipe Model Cannot Explain How Mathematicians’ Proofs Secure Mathematical KnowledgePhilosophia Mathematica 24 (3): 401-404. 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.
-
70Ineffability and Philosophy, by Andre KuklaMind 118 (472): 1153-1155. 2009.(No abstract is available for this citation)
-
96Three is a magic numberThe Philosophers' Magazine 44 (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
-
17Mathematical 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
-
55The 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
-
63Weber and Coyote: Polytheism as a Practical AttitudeSophia 1-18. 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
-
147Emily Grosholz and Herbert Breger, editors. The Growth of Mathematical KnowledgePhilosophia Mathematica 10 (1): 93-96. 2002.
-
75From Euclidean geometry to knots and netsSynthese 1-22. 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
-
157Why 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
-
99George Kampis, Ladislav Kvasz, and Michael Stoltzner, eds. Appraising Lakatos: Mathematics, Methodology, and the ManPhilosophia Mathematica 12 (3): 294-300. 2004.
Areas of Specialization
Philosophy of Mathematics |
Areas of Interest
Logic and Philosophy of Logic |
Philosophy of Mathematics |