17 found
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  1. The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small.Philip Ehrlich - 2012 - Bulletin of Symbolic Logic 18 (1):1-45.
    In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including $-\omega, \,\omega/2, \,1/\omega, \sqrt{\omega}$ and $\omega-\pi$ to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG, it may be said to contain (...)
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  2.  6
    The Rise of Non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of Non-Archimedean Systems of Magnitudes.Philip Ehrlich - 2006 - Archive for History of Exact Sciences 60 (1):1-121.
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  3.  82
    Negative, Infinite, and Hotter Than Infinite Temperatures.Philip Ehrlich - 1982 - Synthese 50 (2):233 - 277.
    We examine the notions of negative, infinite and hotter than infinite temperatures and show how these unusual concepts gain legitimacy in quantum statistical mechanics. We ask if the existence of an infinite temperature implies the existence of an actual infinity and argue that it does not. Since one can sensibly talk about hotter than infinite temperatures, we ask if one could legitimately speak of other physical quantities, such as length and duration, in analogous terms. That is, could there be longer (...)
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  4. From Completeness to Archimedean Completenes.Philip Ehrlich - 1997 - Synthese 110 (1):57-76.
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  5.  7
    Surreal Ordered Exponential Fields.Philip Ehrlich & Elliot Kaplan - 2021 - Journal of Symbolic Logic 86 (3):1066-1115.
    In 2001, the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field ${\mathbf {No}}$ of surreal numbers was brought to the fore by the first author and employed to provide necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of ${\mathbf {No}}$, i.e. a subfield of ${\mathbf {No}}$ that is an initial subtree of ${\mathbf {No}}$. In this sequel, analogous results are established for ordered exponential fields, making use of a slight generalization of (...)
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  6. Real Numbers, Generalizations of the Reals and Theories of Continua.Philip Ehrlich - 1996 - British Journal for the Philosophy of Science 47 (2):320-324.
  7.  49
    Number Systems with Simplicity Hierarchies: A Generalization of Conway's Theory of Surreal Numbers.Philip Ehrlich - 2001 - Journal of Symbolic Logic 66 (3):1231-1258.
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  8.  47
    An Essay in Honor of Adolf Grünbaum’s Ninetieth Birthday: A Reexamination of Zeno’s Paradox of Extension.Philip Ehrlich - 2014 - Philosophy of Science 81 (4):654-675.
    We suggest that, far from establishing an inconsistency in the standard theory of the geometrical linear continuum, Zeno’s Paradox of Extension merely establishes an inconsistency between the standard theory of geometrical magnitude and a misguided system of length measurement. We further suggest that our resolution of Zeno’s paradox is superior to Adolf Grünbaum’s now standard resolution based on Lebesgue measure theory.
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  9.  12
    Number Systems with Simplicity Hierarchies: A Generalization of Conway’s Theory of Surreal Numbers II.Philip Ehrlich & Elliot Kaplan - 2018 - Journal of Symbolic Logic 83 (2):617-633.
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  10. Continuity.Philip Ehrlich - 2005 - In Donald M. Borchert (ed.), The Encyclopedia of Philosophy, 2nd Ed.
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  11.  4
    Are Points (Necessarily) Unextended?Philip Ehrlich - 2022 - Philosophy of Science 89 (4):784-801.
    Since Euclid defined a point as “that which has no part” it has been widely assumed that points are necessarily unextended. It has also been assumed that this is equivalent to saying that points or, more properly speaking, degenerate segments, have length zero. We challenge these assumptions by providing models of Euclidean geometry where the points are extended despite the fact that the degenerate segments have null lengths, and observe that whereas the extended natures of the points are not recognizable (...)
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  12.  52
    Corrigendum to “Number Systems with Simplicity Hierarchies: A Generalization of Conway's Theory of Surreal Numbers”.Philip Ehrlich - 2005 - Journal of Symbolic Logic 70 (3):1022-1022.
  13.  67
    The Palmer House Hilton Hotel, Chicago, Illinois February 18–20, 2010.Kenneth Easwaran, Philip Ehrlich, David Ross, Christopher Hitchcock, Peter Spirtes, Roy T. Cook, Jean-Pierre Marquis, Stewart Shapiro & Royt Cook - 2010 - Bulletin of Symbolic Logic 16 (3).
  14. Investigations Into the Thermodynamic Concept of Temperature.Philip Ehrlich - 1979 - Dissertation, University of Illinois at Chicago
  15.  1
    JL Bell, The Continuous and the Infinitesimal in Mathematics and Philosophy.Philip Ehrlich - 2007 - Bulletin of Symbolic Logic 13 (3):361-362.
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  16.  15
    The Absolute Arithmetic and Geometric Continua.Philip Ehrlich - 1986 - PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1986:237 - 246.
    Novel (categorical) axiomatizations of the classical arithmetic and geometric continua are provided and it is noted that by simply deleting the Archimedean condition one obtains (categorical) axiomatizations of J.H. Conway's ordered field No and its elementary n-dimensional metric Euclidean, hyperbolic and elliptic geometric counterparts. On the basis of this and related considerations it is suggested that whereas the classical arithmetic and geometric continua should merely be regarded as arithmetic and geometric continua modulo the Archimedean condition, No and its geometric counterparts (...)
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  17.  2
    Surreal Ordered Exponential Fields – Erratum.Philip Ehrlich & Elliot Kaplan - 2022 - Journal of Symbolic Logic 87 (2):871-871.
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