Peirce and Dewey were generally more concerned with the process of scientific activity than purely mathematical work. However, their accounts of knowledge production afford some insights into the epistemology of mathematical postulates, especially definition and axioms. Their rejection of rationalist metaphysics and their emphasis on continuity in inquiry provides the pretext for the pragmatic a priori – hypothetical and operational assumptions whose justification relies on their fruitfulness in the long run. This paper focuses on the application of this idea to (...) the epistemology of definitions and an account of progress in mathematics, although it has broader implications for the study of conceptual change and the function and basis of presuppositions in the sciences. (shrink)

In this paper, I present a puzzle involving special relativity and the random selection of real numbers. In a manner to be specified, darts thrown later hit reals further into a fixed well-ordering than darts thrown earlier. Special relativity is then invoked to create a puzzle. I consider four ways of responding to this puzzle which, I suggest, fail. I then propose a resolution to the puzzle, which relies on the distinction between the potential infinite and the actual (...) infinite. I suggest that certain structures, such as a well-ordering of the reals, or the natural numbers, are examples of the potential infinite, whereas infinite integers in a nonstandard model of arithmetic are examples of the actual infinite. (shrink)

This paper evaluates Howe and Leota/Turp's accounts of gamesmanship by examining case studies of gamesmanship from professional darts. While Leota and Turp make some substantial improvements on Howe in reconceptualizing the idea of sporting excellence, I claim that there are points of criticism that must be addressed, notably in their claims that sports do not prescribe necessary skills, and that it is impossible to distinguish between legitimate sporting strategy unaccounted for by the rules on the one hand, and gamesmanship (...) on the other. Leota and Turp criticise Howe’s account of rules for being misconceived: rather than rules prescribing necessary skills as Howe claims, rules actually proscribe skills and set limits on behaviour, rather than prescribing skills. I use darts and other sports to make the case that rules actually do both things. Elsewhere, in the phenomenon of ‘grouping’, I argue that we find a skill completely unaccounted for by the rules of darts and not a necessary skill for playing darts, which nonetheless counts as excellence for darts. This problematizes some claims from Leota and Turp on which their account of gamesmanship hinges. An aspect of Howe, Leota and Turp’s accounts on which they all agree is the importance of psychological resilience in sporting endeavours, which I discuss with reference to darts and rule changes in professional sport. The article ends with a discussion of an example from darts of a potential gamesmanship strategy, from a match between Michael van Gerwen and Darius Labanauskas, that unquestionably remains within the rules of the sport and which could not be eradicated through rule changes because it would violate the spirit of the sport. This is an interesting, controversial example for studies of the different forms of gamesmanship and their categorization, and indicates some limitations on what I say in my argument. (shrink)

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...) Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)

The definition and the assessment of the quality of argumentative texts has become an increasingly crucial issue in education, classroom discourse, and argumentation theory. The different methods developed and used in the literature are all characterized by specific perspectives that fail to capture the complexity of the subject matter, which remains ill-defined and not systematically investigated. This paper addresses this problem by building on the four main dimensions of argument quality resulting from the definition of argument and the literature in (...) classroom discourse: dialogicity, accountability, relevance, and textuality (DART). We use and develop the insights from the literature in education and argumentation by integrating the frameworks that capture both the textual and the argumentative nature of argumentative texts. This theoretical background will be used to propose a method for translating the DART dimensions into specific and clear proxies and evaluation criteria. (shrink)