300 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
Thermodynamic curvature measures interactions
Thermodynamic fluctuation theory originated with Einstein who inverted the
relation to express the number of states in terms of entropy:
. The theory's Gaussian approximation is discussed in most
statistical mechanics texts. I review work showing how to go beyond the
Gaussian approximation by adding covariance, conservation, and consistency.
This generalization leads to a fundamentally new object: the thermodynamic
Riemannian curvature scalar , a thermodynamic invariant. I argue that
is related to the correlation length and suggest that the sign of
corresponds to whether the interparticle interactions are effectively
attractive or repulsive.Comment: 29 pages, 7 figures (added reference 27
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
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
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
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
Ω-Arithmetization of Ellipses
International audienceMulti-resolution analysis and numerical precision problems are very important subjects in fields like image analysis or geometrical modeling. In the continuation of our previous works, we propose to apply the method of Ω-arithmetization to ellipses. We obtain a discrete multi-resolution representation of arcs of ellipses. The corresponding algorithms are completely constructive and thus, can be exactly translated into functional computer programs. Moreover, we give a global condition for the connectivity of the discrete curves generated by the method at every scale
Thermodynamic curvature and black holes
I give a relatively broad survey of thermodynamic curvature , one spanning
results in fluids and solids, spin systems, and black hole thermodynamics.
results from the thermodynamic information metric giving thermodynamic
fluctuations. has a unique status in thermodynamics as being a geometric
invariant, the same for any given thermodynamic state. In fluid and solid
systems, the sign of indicates the character of microscopic interactions,
repulsive or attractive. gives the average size of organized mesoscopic
fluctuating structures. The broad generality of thermodynamic principles might
lead one to believe the same for black hole thermodynamics. This paper explores
this issue with a systematic tabulation of results in a number of cases.Comment: 27 pages, 10 figures, 7 tables, 78 references. Talk presented at the
conference Breaking of Supersymmetry and Ultraviolet Divergences in extended
Supergravity, in Frascati, Italy, March 27, 2013. v2 corrects some small
problem
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