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.

Philosophy of science in the twentieth century tends to emphasize either the logic of science (e.g., Popper and Hempel on explanation, confirmation, etc.) or its history/sociology (e.g., Kuhn on revolutions, holism, etc.). This dichotomy, however, is neither exhaustive nor exclusive. Questions regarding scientific understanding and mathematical explanation do not fit neatly inside either category, and addressing them has drawn from both logic and history. Additionally, interest in scientific and mathematical practice has led to work that falls between the two sides (...) of the dichotomy. We might call these less narrow approaches ‘holistic’. Prior to 1918, work in the philosophy of science was also somewhat more holistic and interdisciplinary in nature. An exemplar of this type of approach is found in the work of Henri Poincaré. In this paper I will consider one particularly flexible tool Poincaré uses throughout his philosophical writings about science and mathematics: the tool of the unifying concept. More specifically, I will focus on one unifying concept: that of structure. I aim to show first, that structure is a unifying scientific concept for Poincaré; second, a focus on structure helps explain his scientific and mathematical success; lastly, this case study may provide a model for how unifying concepts can facilitate progress in mathematics and science. (shrink)

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 also seemed independent of intuition. This paper considers various philosophical responses to (...) these changes, focusing on the idea of modifying Kant’s conception of intuition in order to accommodate the increasing abstractness of mathematics. It is argued that far from clinging to an outdated paradigm, programs based on new conceptions of intuition should be seen as motivated by important philosophical desiderata, such as the truth, apriority, distinctiveness and autonomy of mathematics. (shrink)

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 in general, but his (...) commitment to twodifferent types of intuition, which explains his rather unusual and tormented philosophy of the mathematical continuum. I would like to thank Geoff Gorham, David McCarty, and Rosamond Rodman for reading an earlier draft of this work. I should also thank those who provided helpful comments on several distant ancestors of this paper: Emily Carson, Ulrich Majer, Erhard Scholz, John Schuerman, Stewart Shapiro, and Richard Tieszen. I am indebted to two anonymous referees for pointing out some problems and for pointing me to work on Weyl I did not previously know about. In particular, the recent articles in [Feist, 2004a] turned out to be (somewhat uncomfortably) relevant to the focus of this paper. In this last revision I have tried to show where I agree and disagree with the authors of those papers; I apologize for whatever repetition still exists, but it was tere before I read those papers. This paper has a long history, and comes out of several talks I gave some years ago. Audiences at the Center for Philosophy of Science, University of Pittsburgh (colloquium 1995), St Andrews University Philosophy of Mathematics Workshop (1996), the British Society for the History of Mathematics meeting (1996), the University of Mainz Mathematics Department (colloquium 1996), the Canadian Philosophical Association (1997 and 1999), and the University of British Columbia (colloquium 1998) should be thanked for their helpful comments. I also thank Neil Tennant for encouraging me to resurrect this work. CiteULike Connotea Del.icio.us What's this? (shrink)

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 Hamilton's philosophy of algebra. Namely, (...) Hamilton's defense of algebra, like Newton's defense of geometry, is driven by the claim that a mathematical science must have a proper object and thus a basis in truth. In particular, Hamilton aims to show that algebra is not a mere language, or tool, or a mere “art”; instead, he argues, algebra is a bona fide mathematical science, like geometry, because its methods also provide true and accurate insight into a genuine subject matter, namely, the pure form of temporal intuition. (shrink)

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 that in the philosophical writings of Henri Poincaré there is a coherent (...) argument for an interesting position between the two traditional poles in the philosophy of mathematics. Relying on a semi-Kantian framework, Poincaré combines an epistemological and metaphysical constructivism with a more realist account of the nature of mathematical truth. (shrink)

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 others: hence the hierarchy. (...) This paper attempts to dissolve some potential problems regarding Poincare's hierarchy. One is an apparent epistemological circularity in the hierarchy. A more serious problem regarding the epistemology of analysis is also addressed. (shrink)

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, by outlining (...) Kant's theory of the synthetic a priori, especially as it applies to mathematics. Then, in the main body of the thesis, I explain how the various central aspects of Poincare's philosophy of mathematics - e.g. his theory of induction; his theory of the continuum; his views on impredicativiti his theory of meaning - must, in general, be seen as an adaptation of Kant's position. My conclusion is that not only is there a well-founded philosophical core to Poincare's philosophy, but also that such a core provides a viable alternative in contemporary debates in the philosophy of mathematics. That is, Poincare's theory, which is secured by his doctrine of a priori intuitions, and which describes a position in between the two extremes of an "anti-realist" strict constructivism and a "realist" axiomatic set theory, may indeed be true. (shrink)

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 eternal. Putnam introduces (...) a concept of epistemic idealization — epistemically ideal conditions, or idealized justification — to try to ground the strong objectivity of truth without sliding into metaphysical realism. I argue that the concept of ideal conditions to which Putnam appeals does not cohere with his (anti-realist) commitment to an assertability conditions account of meaning. (shrink)