•  217
    Two Cultures
    Cogito 12 (1): 13-16. 1998.
    The schism between analytic and continental philosophy resists repair because it is not confined to philosophers. It is a local manifestation of a far more profound and pervasive division. In 1959 C.P. Snow lamented the partition of intellectual life in to `two cultures': that of the scientist and that of the literary intellectual. If we follow the practice of most universities and bundle historical and literary studies together in the faculty of humanities on the one hand, and count pure mathem…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.
  •  173
    Williams on Dawkins – response
    Think 9 (26): 21-27. 2010.
    Peter Williams complains that Richard Dawkins wraps his naturalism in ‘a fake finery of counterfeit meaning and purpose’. For his part, Williams has wrapped his complaint in an unoriginal and inapt analogy. The weavers in Hans Christian Andersen's fable announce that the Emperor's clothes are invisible to stupid people; almost the whole population pretends to see them for fear of being thought stupid . Fear of being thought stupid does not seem to trouble Richard Dawkins. Moreover, Williams offe…Read more
  •  144
    Why is there Philosophy of Mathematics at all? Ian Hacking. in Metascience (2015)
  •  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
  •  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
  •  103
    Moral particularism and scientific practice
    Metaphilosophy 39 (4-5): 492-507. 2008.
    Abstract: Particularism is usually understood as a position in moral philosophy. In fact, it is a view about all reasons, not only moral reasons. Here, I show that particularism is a familiar and controversial position in the philosophy of science and mathematics. I then argue for particularism with respect to scientific and mathematical reasoning. This has a bearing on moral particularism, because if particularism about moral reasons is true, then particularism must be true with respect to reas…Read more
  •  94
  •  79
    Proof in C17 Algebra
    Philosophia Scientae 43-59. 2005.
    By the middle of the seventeenth century we that find that algebra is able to offer proofs in its own right. That is, by that time algebraic argument had achieved the status of proof. How did this transformation come about?
  •  79
  •  77
    This article canvasses five senses in which one might introduce an historical element into the philosophy of mathematics: 1. The temporal dimension of logic; 2. Explanatory Appeal to Context rather than to General Principles; 3. Heraclitean Flux; 4. All history is the History of Thought; and 5. History is Non-Judgmental. It concludes by adapting Bernard Williams’ distinction between ‘history of philosophy’ and ‘history of ideas’ to argue that the philosophy of mathematics is unavoidably historic…Read more
  •  73
  •  68
    What is dialectical philosophy of mathematics?
    Philosophia Mathematica 9 (2): 212-229. 2001.
    The late Imre Lakatos once hoped to found a school of dialectical philosophy of mathematics. The aim of this paper is to ask what that might possibly mean. But Lakatos's philosophy has serious shortcomings. The paper elaborates a conception of dialectical philosophy of mathematics that repairs these defects and considers the work of three philosophers who in some measure fit the description: Yehuda Rav, Mary Leng and David Corfield.
  •  65
    Books of essays
    Philosophia Mathematica 10 (1): 93-96. 2002.
  •  64
    Lakatos as historian of mathematics
    Philosophia Mathematica 5 (1): 42-64. 1997.
    This paper discusses the connection between the actual history of mathematics and Lakatos's philosophy of mathematics, in three parts. The first points to studies by Lakatos and others which support his conception of mathematics and its history. In the second I suggest that the apparent poverty of Lakatosian examples may be due to the way in which the history of mathematics is usually written. The third part argues that Lakatos is right to hold philosophy accountable to history, even if Lakatos'…Read more
  •  57
    Three is a magic number
    The 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
  •  50
    Re-reading soviet philosophy: Bakhurst on ilyenkov
    Studies in East European Thought 44 (1): 1-31. 1992.
  •  46
    Lakatos: An Introduction
    Routledge. 1998.
    _Lakatos: An Introduction_ provides a thorough overview of both Lakatos's thought and his place in twentieth century philosophy. It is an essential and insightful read for students and anyone interested in the philosophy of science.
  •  40
    Ineffability and Philosophy, by Andre Kukla
    Mind 118 (472): 1153-1155. 2009.
    (No abstract is available for this citation)
  •  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
  •  35
    Reply to James Blachowicz
    The Owl of Minerva 31 (1): 53-54. 1999.
  •  35
    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.
  •  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
  •  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