311 research outputs found
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
Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond
Many historians of the calculus deny significant continuity between
infinitesimal calculus of the 17th century and 20th century developments such
as Robinson's theory. Robinson's hyperreals, while providing a consistent
theory of infinitesimals, require the resources of modern logic; thus many
commentators are comfortable denying a historical continuity. A notable
exception is Robinson himself, whose identification with the Leibnizian
tradition inspired Lakatos, Laugwitz, and others to consider the history of the
infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies,
Robinson regards Berkeley's criticisms of the infinitesimal calculus as aptly
demonstrating the inconsistency of reasoning with historical infinitesimal
magnitudes. We argue that Robinson, among others, overestimates the force of
Berkeley's criticisms, by underestimating the mathematical and philosophical
resources available to Leibniz. Leibniz's infinitesimals are fictions, not
logical fictions, as Ishiguro proposed, but rather pure fictions, like
imaginaries, which are not eliminable by some syncategorematic paraphrase. We
argue that Leibniz's defense of infinitesimals is more firmly grounded than
Berkeley's criticism thereof. We show, moreover, that Leibniz's system for
differential calculus was free of logical fallacies. Our argument strengthens
the conception of modern infinitesimals as a development of Leibniz's strategy
of relating inassignable to assignable quantities by means of his
transcendental law of homogeneity.Comment: 69 pages, 3 figure
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
Search for CP Violation in the Decay Z -> b (b bar) g
About three million hadronic decays of the Z collected by ALEPH in the years
1991-1994 are used to search for anomalous CP violation beyond the Standard
Model in the decay Z -> b \bar{b} g. The study is performed by analyzing
angular correlations between the two quarks and the gluon in three-jet events
and by measuring the differential two-jet rate. No signal of CP violation is
found. For the combinations of anomalous CP violating couplings, and , limits of \hat{h}_b < 0.59h^{\ast}_{b} < 3.02$ are given at 95\% CL.Comment: 8 pages, 1 postscript figure, uses here.sty, epsfig.st
First measurement of the meson mass
If simplified, every information retrieval problem can be solved when the information need implied by its expression has been identified. We are interested in the criteria used in realising a good information retrieval problem expression. We have listed these criteria through some principles and maxims which first characterized the communication between two persons are applied. We choose to use the gricean maxims because they are the most favoured for this type of situation. Secondly, we have tried to identify some others principles that can be used to realise a good information retrieval problem expression. The principles by Grice can not resolve all forms of error associated with this particular form of communication. In our work, we defined three other principles namely: adhesion principle, reformulation principle, memorization principle. We give some examples of situations where the principles we have formulated are not applicable and the consequences. We present the possible applications of our new model: MIRABEL, which can help in the description of information retrieval problem from. It also compels its user to use essential good expression principle implicitly
Observation of charmless hadronic B decays
Four candidates for charmless hadronic B decay are observed in a data sample of four million hadronic Z decays recorded by the ALEPH detector at LEP. The probability that these events come from background sources is estimated to be less than 10(-6). The average branching of weakly decaying B hadrons (a mixture of B-d(0), B-s(0) and Lambda(b) weighted by their production The average branching ratio of weakly decaying B hadrons (a mixture of B-d(0) cross sections and lifetimes, here denoted B) into two long-lived charged hadrons (pions, kaons or protons) is measured to be Br(B-->h(+)h(-))=(1.7(-0.7)(+1.0)+/-0.2)x10(-5). The relative branching fraction Br(B-d(s)(0)-->pi(+)pi(-)(K-))/Br(B-d(s)(0)-->h(+)h(-)) is measured to be 1.0(-0.3-0.1)(+0.0+0.0). In addition, branching ratio upper limits are obtained for a variety of exclusive charmless hadronic two-body decays of B hadrons
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