89 research outputs found
A Cauchy-Dirac delta function
The Dirac delta function has solid roots in 19th century work in Fourier
analysis and singular integrals by Cauchy and others, anticipating Dirac's
discovery by over a century, and illuminating the nature of Cauchy's
infinitesimals and his infinitesimal definition of delta.Comment: 24 pages, 2 figures; Foundations of Science, 201
Stevin numbers and reality
We explore the potential of Simon Stevin's numbers, obscured by shifting
foundational biases and by 19th century developments in the arithmetisation of
analysis.Comment: 22 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1104.0375, arXiv:1108.2885, arXiv:1108.420
A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography
We analyze the developments in mathematical rigor from the viewpoint of a
Burgessian critique of nominalistic reconstructions. We apply such a critique
to the reconstruction of infinitesimal analysis accomplished through the
efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's
foundational work associated with the work of Boyer and Grabiner; and to
Bishop's constructivist reconstruction of classical analysis. We examine the
effects of a nominalist disposition on historiography, teaching, and research.Comment: 57 pages; 3 figures. Corrected misprint
Ten Misconceptions from the History of Analysis and Their Debunking
The widespread idea that infinitesimals were "eliminated" by the "great
triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an
uninterrupted chain of work on infinitesimal-enriched number systems. The
elimination claim is an oversimplification created by triumvirate followers,
who tend to view the history of analysis as a pre-ordained march toward the
radiant future of Weierstrassian epsilontics. In the present text, we document
distortions of the history of analysis stemming from the triumvirate ideology
of ontological minimalism, which identified the continuum with a single number
system. Such anachronistic distortions characterize the received interpretation
of Stevin, Leibniz, d'Alembert, Cauchy, and others.Comment: 46 pages, 4 figures; Foundations of Science (2012). arXiv admin note:
text overlap with arXiv:1108.2885 and arXiv:1110.545
Primary care nurses' experiences of how the mass media influence frontline healthcare in the UK
Background:
Mass media plays an important role in communicating about health research and services to patients, and in shaping public perceptions and decisions about health. Healthcare professionals also play an important role in providing patients with credible, evidence-based and up-to-date information on a wide range of health issues. This study aims to explore primary care nurses' experiences of how mass media influences frontline healthcare.<p></p>
Methods:
In-depth telephone interviews were carried out with 18 primary care nurses (nine health visitors and nine practice nurses) working in the United Kingdom (UK). Interviews were recorded and transcribed. The data was analysed using thematic analysis, with a focus on constant comparative analysis.<p></p>
Results:
Three themes emerged from the data. First, participants reported that their patients were frequently influenced by controversial health stories reported in the media, which affected their perceptions of, and decisions about, care. This, in turn, impinged upon participants' workloads as they had to spend additional time discussing information and reassuring patients. Second, participants also recalled times in their own careers when media reports had contributed to a decline in their confidence in current healthcare practices and treatments. Third, the participants in this study suggested a real need for additional resources to support and expand their own media literacy skills, which could be shared with patients.<p></p>
Conclusion:
In an ever expanding media landscape with greater reporting on health, nurses working in the primary care setting face increasing pressure to effectively manage media stories that dispute current health policies and practices. These primary care nurses were keen to expand their media literacy skills to develop critical autonomy in relation to all media, and to facilitate more meaningful conversations with their patients about their health concerns and choices.<p></p>
Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics
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 his 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 Goedel'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.Comment: 52 pages, 1 figur
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