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  1. J. L. Bell (1994). Introduction. Philosophia Mathematica 2 (1):4-4.
    Continuous as the stars that shine And twinkle on the milky way, They stretched in never-ending line Along the margin of a bay: Ten thousand saw I at a glance, Tossing their heads in sprightly dance.
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  2. John Bell, Chapter.
    Despite the great success of Weierstrass, Dedekind and Cantor in constructing the continuum from arithmetical materials, a number of thinkers of the late 19th and early 20th centuries remained opposed, in varying degrees, to the idea of explicating the continuum concept entirely in discrete terms. These include the mathematicians du Bois-Reymond, Veronese, Poincaré, Brouwer and Weyl, and the philosophers Brentano..
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  3. John L. Bell (2005). Divergent Conceptions of the Continuum in 19th and Early 20th Century Mathematics and Philosophy. Axiomathes 15 (1):63-84.
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  4. John L. Bell (2000). Hermann Weyl on Intuition and the Continuum. Philosophia Mathematica 8 (3):259-273.
    Hermann Weyl, one of the twentieth century's greatest mathematicians, was unusual in possessing acute literary and philosophical sensibilities—sensibilities to which he gave full expression in his writings. In this paper I use quotations from these writings to provide a sketch of Weyl's philosophical orientation, following which I attempt to elucidate his views on the mathematical continuum, bringing out the central role he assigned to intuition.
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  5. John P. Burgess (2000). Critical Studies / Book Reviews. Philosophia Mathematica 8 (1):84-91.
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  6. Piotr Błaszczyk, Mikhail G. Katz & David Sherry (2013). Ten Misconceptions From the History of Analysis and Their Debunking. Foundations of Science 18 (1):43-74.
    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 (...)
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  7. Martin Cooke, To Continue with Continuity.
    The metaphysical concept of continuity is important, not least because physical continua are not known to be impossible. While it is standard to model them with a mathematical continuum based upon set-theoretical intuitions, this essay considers, as a contribution to the debate about the adequacy of those intuitions, the neglected intuition that dividing the length of a line by the length of an individual point should yield the line’s cardinality. The algebraic properties of that cardinal number are derived pre-theoretically from (...)
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  8. S. S. Demidov (1988). On an Early History of the Moscow School of Theory of Functions. Philosophia Mathematica (1):29-35.
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  9. Fausto di Biase (2009). True or False? A Case in the Study of Harmonic Functions. Topoi 28 (2):143-160.
    Recent mathematical results, obtained by the author, in collaboration with Alexander Stokolos, Olof Svensson, and Tomasz Weiss, in the study of harmonic functions, have prompted the following reflections, intertwined with views on some turning points in the history of mathematics and accompanied by an interpretive key that could perhaps shed some light on other aspects of (the development of) mathematics.
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  10. Bradley H. Dowden (1991). A Linear Continuum of Time. Philosophia Mathematica (1):53-64.
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  11. Jens Erik Fenstad (1985). Is Nonstandard Analysis Relevant for the Philosophy of Mathematics? Synthese 62 (2):289 - 301.
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  12. Fernando Ferreira (2008). A Most Artistic Package of a Jumble of Ideas. Dialectica 62 (2: Table of Contents"/> Select):205–222.
    In the course of ten short sections, we comment on Gödel's seminal dialectica paper of fifty years ago and its aftermath. We start by suggesting that Gödel's use of functionals of finite type is yet another instance of the realistic attitude of Gödel towards mathematics, in tune with his defense of the postulation of ever increasing higher types in foundational studies. We also make some observations concerning Gödel's recasting of intuitionistic arithmetic via the dialectica interpretation, discuss the extra principles that (...)
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  13. Han Geurdes, The Construction of Transfinite Equivalence Algorithms.
    Context: Consistency of mathematical constructions in numerical analysis and the application of computerized proofs in the light of the occurrence of numerical chaos in simple systems. Purpose: To show that a computer in general and a numerical analysis in particular can add its own peculiarities to the subject under study. Hence the need of thorough theoretical studies on chaos in numerical simulation. Hence, a questioning of what e.g. a numerical disproof of a theorem in physics or a prediction in numerical (...)
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  14. Jeremy Gwiazda (2013). Throwing Darts, Time, and the Infinite. Erkenntnis 78 (5):971-975.
    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 (...)
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  15. Bob Hale (2002). Real Numbers, Quantities, and Measurement. Philosophia Mathematica 10 (3):304-323.
    Defining the real numbers by abstraction as ratios of quantities gives prominence to then- applications in just the way that Frege thought we should. But if all the reals are to be obtained in this way, it is necessary to presuppose a rich domain of quantities of a land we cannot reasonably assume to be exemplified by any physical or other empirically measurable quantities. In consequence, an explanation of the applications of the reals, defined in this way, must proceed indirectly. (...)
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  16. Bob Hale (2000). Reals by Abstractiont. Philosophia Mathematica 8 (2):100--123.
    On the neo-Fregean approach to the foundations of mathematics, elementary arithmetic is analytic in the sense that the addition of a principle wliich may be held to IMJ explanatory of the concept of cardinal number to a suitable second-order logical basis suffices for the derivation of its basic laws. This principle, now commonly called Hume's principle, is an example of a Fregean abstraction principle. In this paper, I assume the correctness of the neo-Fregean position on elementary aritlunetic and seek to (...)
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  17. Deirdre Haskell (2012). Model Theory of Analytic Functions: Some Historical Comments. Bulletin of Symbolic Logic 18 (3):368-381.
    Model theorists have been studying analytic functions since the late 1970s. Highlights include the seminal work of Denef and van den Dries on the theory of the p-adics with restricted analytic functions, Wilkie's proof of o-minimality of the theory of the reals with the exponential function, and the formulation of Zilber's conjecture for the complex exponential. My goal in this talk is to survey these main developments and to reflect on today's open problems, in particular for theories of valued fields.
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  18. Geoffrey Hellman (1994). Real Analysis Without Classes. Philosophia Mathematica 2 (3):228-250.
    This paper explores strengths and limitations of both predicativism and nominalism, especially in connection with the problem of characterizing the continuum. Although the natural number structure can be recovered predicatively (despite appearances), no predicative system can characterize even the full predicative continuum which the classicist can recognize. It is shown, however, that the classical second-order theory of continua (third-order number theory) can be recovered nominalistically, by synthesizing mereology, plural quantification, and a modal-structured approach with essentially just the assumption that an (...)
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  19. Geoffrey Hellman & Stewart Shapiro (2013). The Classical Continuum Without Points. Review of Symbolic Logic 6 (3):488-512.
    We develop a point-free construction of the classical one-dimensional continuum, with an interval structure based on mereology and either a weak set theory or a logic of plural quantification. In some respects, this realizes ideas going back to Aristotle, although, unlike Aristotle, we make free use of contemporary . Also, in contrast to intuitionistic analysis, smooth infinitesimal analysis, and Eret Bishopgunky lineindecomposabilityCantor structure of ℝ as a complete, separable, ordered field. We also present some simple topological models of our system, (...)
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  20. Friedrich Kaulbach (1967). Philosophisches Und Mathematisches Kontinuum. Philosophia Mathematica (1-2):47-69.
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  21. Michael Kohlhase, A Foundational View on Integration Problems.
    The integration of reasoning and computation services across system and language boundaries has been mostly treated from an engineering perspective. In this paper we take a foundational point of view. We identify the following form of integration problems: an informal (mathematical; i.e, logically underspecified) specification has multiple concrete formal implementations between which queries and results have to be transported. The integration challenge consists in dealing with the implementation-specific details such as additional constants and properties. We pinpoint their role in safe (...)
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  22. Vojtěch Kolman (2010). Continuum, Name and Paradox. Synthese 175 (3):351 - 367.
    The article deals with Cantor's argument for the non-denumerability of reals somewhat in the spirit of Lakatos' logic of mathematical discovery. At the outset Cantor's proof is compared with some other famous proofs such as Dedekind's recursion theorem, showing that rather than usual proofs they are resolutions to do things differently. Based on this I argue that there are "ontologically" safer ways of developing the diagonal argument into a full-fledged theory of continuum, concluding eventually that famous semantic paradoxes based on (...)
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  23. O. B. Lupanov (ed.) (2005). Stochastic Algorithms: Foundations and Applications: Third International Symposium, Saga 2005, Moscow, Russia, October 20-22, 2005: Proceedings. [REVIEW] Springer.
    This book constitutes the refereed proceedings of the Third International Symposium on Stochastic Algorithms: Foundations and Applications, SAGA 2005, held in Moscow, Russia in October 2005. The 14 revised full papers presented together with 5 invited papers were carefully reviewed and selected for inclusion in the book. The contributed papers included in this volume cover both theoretical as well as applied aspects of stochastic computations whith a special focus on new algorithmic ideas involving stochastic decisions and the design and evaluation (...)
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  24. Moshé Machover (1993). The Place of Nonstandard Analysis in Mathematics and in Mathematics Teaching. British Journal for the Philosophy of Science 44 (2):205-212.
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  25. Jean-Pierre Marquis (2006). John L. BELL. The Continuous and the Infinitesimal in Mathematics and Philosophy. Monza: Polimetrica, 2005. Pp. 349. ISBN 88-7699-015-. [REVIEW] Philosophia Mathematica 14 (3):394-400.
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  26. Matthew W. Parker (2003). Three Concepts of Decidability for General Subsets of Uncountable Spaces. Theoretical Computer Science 351 (1):2-13.
    There is no uniquely standard concept of an effectively decidable set of real numbers or real n-tuples. Here we consider three notions: decidability up to measure zero [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382], which we abbreviate d.m.z.; recursive approximability [or r.a.; K.-I. Ko, Complexity Theory of Real Functions, Birkhäuser, Boston, 1991]; and decidability ignoring boundaries [d.i.b.; W.C. Myrvold, The decision problem for entanglement, in: R.S. (...)
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