
162Discussion. Pictures, proofs, and 'mathematical practice': Reply to James Robert brownBritish Journal for the Philosophy of Science 50 (3): 425429. 1999.

138Putnam, realism and truthSynthese 103 (2): 14152. 1995.There are several distinct components of the realist antirealist debate. Since each side in the debate has its disadvantages, it is tempting to try to combine realist theses with antirealist 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

108Poincaré's conception of the objectivity of mathematicsPhilosophia Mathematica 2 (3): 202227. 1994.There is a basic division in the philosophy of mathematics between realist, ‘platonist’ theories and antirealist ‘constructivist’ theories. Platonism explains how mathematical truth is strongly objective, but it does this at the cost of invoking mindindependent mathematical objects. In contrast, constructivism avoids mindindependent 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

58Intuition between the analyticcontinental divide: Hermann Weyl's philosophy of the continuumPhilosophia Mathematica 16 (1): 2555. 2008.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

53Mark Greaves. The Philosophical Status of DiagramsPhilosophia Mathematica 11 (3): 349353. 2003.

47After NonEuclidean Geometry: Intuition, Truth and the Autonomy of MathematicsJournal for the History of Analytical Philosophy 6 (3). 2018.The mathematical developments of the 19th century seemed to undermine Kant’s philosophy. NonEuclidean 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

43Poincare on Mathematics, Intuition and the Foundations of SciencePSA: 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

42Newton and Hamilton: In defense of truth in algebraSouthern Journal of Philosophy 50 (3): 504527. 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

21Book Review: Michael Resnik. Mathematics as a Science of Patterns (review)Notre Dame Journal of Formal Logic 40 (3): 455472. 1999.

20Science, Hypothesis, and HierarchyHopos: The Journal of the International Society for the History of Philosophy of Science 9 (2): 388406. 2019.

18James Robert Brown. Philosophy of mathematics, an introduction to the world of proofs and pictures. Routledge, 1999, vii + 215 pp (review)Bulletin of Symbolic Logic 9 (4): 504506. 2003.

18Ontology, Logic, and Mathematics: Review of M. Schirn (ed.), The Philosophy of Mathematics Today (review)British Journal for the Philosophy of Science 51 (2): 319332. 2000.

17The InfinitePhilosophical 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

7Poincaré's philosophy of mathematicsDissertation, 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 wellfounded 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

4Review article. Ontology, logic, and mathematicsBritish Journal for the Philosophy of Science 51 (2): 319332. 2000.

1Poincaré and the Philosophy of MathematicsRevue Philosophique de la France Et de l'Etranger 183 (3): 631633. 1993.

Towards a Better Understanding of Mathematical UnderstandingIn Gabriele Pulcini & Mario Piazza (eds.), Truth, Existence and Explanation, Springer Verlag. 2018.
Saint Paul, Minnesota, United States of America