•  20
    Science, Hypothesis, and Hierarchy
    Hopos: The Journal of the International Society for the History of Philosophy of Science 9 (2): 388-406. 2019.
  •  7
    Poincaré's philosophy of mathematics
    Dissertation, St. Andrews. 1986.
    The primary concern of this thesis is to investigate the explicit philosophy of mathematics in the work of Henri Poincare. In particular, I argue that there is a well-founded doctrine which grounds both Poincare's negative thesis, which is based on constructivist sentiments, and his positive thesis, via which he retains a classical conception of the mathematical continuum. The doctrine which does so is one which is founded on the Kantian theory of synthetic a priori intuition. I begin, therefore…Read more
  • Towards a Better Understanding of Mathematical Understanding
    In Gabriele Pulcini & Mario Piazza (eds.), Truth, Existence and Explanation, Springer Verlag. 2018.
  •  47
    After Non-Euclidean Geometry: Intuition, Truth and the Autonomy of Mathematics
    Journal for the History of Analytical Philosophy 6 (3). 2018.
    The mathematical developments of the 19th century seemed to undermine Kant’s philosophy. Non-Euclidean geometries challenged Kant’s view that there is a spatial intuition rich enough to yield the truth of Euclidean geometry. Similarly, advancements in algebra challenged the view that temporal intuition provides a foundation for both it and arithmetic. Mathematics seemed increasingly detached from experience as well as its form; moreover, with advances in symbolic logic, mathematical inference al…Read more
  •  138
    Putnam, realism and truth
    Synthese 103 (2): 141--52. 1995.
    There are several distinct components of the realist anti-realist debate. Since each side in the debate has its disadvantages, it is tempting to try to combine realist theses with anti-realist theses in order to obtain a better, more moderate position. Putnam attempts to hold a realist concept of truth, yet he rejects realist metaphysics and realist semantics. He calls this view internal realism. Truth is realist on this picture for it is objective, rather than merely intersubjective, and eterna…Read more
  •  42
    Newton and Hamilton: In defense of truth in algebra
    Southern Journal of Philosophy 50 (3): 504-527. 2012.
    Although it is clear that Sir William Rowan Hamilton supported a Kantian account of algebra, I argue that there is an important sense in which Hamilton's philosophy of mathematics can be situated in the Newtonian tradition. Drawing from both Niccolo Guicciardini's (2009) and Stephen Gaukroger's (2010) readings of the Newton–Leibniz controversy over the calculus, I aim to show that the very epistemic ideals that underpin Newton's argument for the superiority of geometry over algebra also motivate…Read more
  •  17
    The Infinite
    with A. W. Moore
    Philosophical Quarterly 41 (164): 348. 1991.
    Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical
  •  1
    Poincaré and the Philosophy of Mathematics
    Revue Philosophique de la France Et de l'Etranger 183 (3): 631-633. 1993.
  •  58
    Though logical positivism is part of Kant's complex legacy, positivists rejected both Kant's theory of intuition and his classification of mathematical knowledge as synthetic a priori. This paper considers some lingering defenses of intuition in mathematics during the early part of the twentieth century, as logical positivism was born. In particular, it focuses on the difficult and changing views of Hermann Weyl about the proper role of intuition in mathematics. I argue that it was not intuition…Read more
  •  108
    Poincaré's conception of the objectivity of mathematics
    Philosophia Mathematica 2 (3): 202-227. 1994.
    There is a basic division in the philosophy of mathematics between realist, ‘platonist’ theories and anti-realist ‘constructivist’ theories. Platonism explains how mathematical truth is strongly objective, but it does this at the cost of invoking mind-independent mathematical objects. In contrast, constructivism avoids mind-independent mathematical objects, but the cost tends to be a weakened conception of mathematical truth. Neither alternative seems ideal. The purpose of this paper is to show …Read more
  •  53
  • Russell Reread
    Philosophical Quarterly 40 (61): 502. 1990.
  •  43
    Poincare on Mathematics, Intuition and the Foundations of Science
    PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1994. 1994.
    In his first philosophy book, Science and Hypothesis, Poincare provides a picture in which the different sciences are arranged in a hierarchy. Arithmetic is the most general of all the sciences because it is presupposed by all the others. Next comes mathematical magnitude, or the analysis of the continuum, which presupposes arithmetic; and so on. Poincare's basic view was that experiment in science depends on fixing other concepts first. More generally, certain concepts must be fixed before othe…Read more
  •  3
    Review: Russell Reread (review)
    Philosophical Quarterly 40 (161). 1990.
  •  5
    Poincaré and the Philosophy of Mathematics
    Philosophical Quarterly 46 (183): 251-255. 1996.