Analysis

Edited by Nemi Boris Pelgrom (Ludwig Maximilians Universität, München)
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73 found
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1 — 50 / 73
  1. Computational reverse mathematics and foundational analysis.Benedict Eastaugh - manuscript
    Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of different foundations for mathematics in a formally precise manner. This paper gives a detailed account of the motivations and methodology of foundational analysis, which have heretofore been largely left implicit in the practice. It then shows how this account can be fruitfully applied in the (...)
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  2. Nota: ¿CUÁL ES EL CARDINAL DEL CONJUNTO DE LOS NÚMEROS REALES?Franklin Galindo -
    ¿Qué ha pasado con el problema del cardinal del continuo después de Gödel (1938) y Cohen (1964)? Intentos de responder esta pregunta pueden encontrarse en los artículos de José Alfredo Amor (1946-2011), "El Problema del continuo después de Cohen (1964-2004)", de Carlos Di Prisco , "Are we closer to a solution of the continuum problem", y de Joan Bagaria, "Natural axioms of set and the continuum problem" , que se pueden encontrar en la biblioteca digital de mi blog de Lógica (...)
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  3. The construction of transfinite equivalence algorithms.Han Geurdes - manuscript
    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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  4. On some historical aspects of the theory of Riemann zeta function.Giuseppe Iurato - manuscript
    This comprehensive historical account concerns that non-void intersection region between Riemann zeta function and entire function theory, with a view towards possible physical applications.
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  5. Continuity of higher order commutators generated by maximal Bochner-Riesz operator on Morrey space.Shihong Zhu - manuscript
    In this papers ,we use the control method of the maximal fractional integral and obtain the boundedness of higher order commutator generated by maximal Bochner-Riesz operator on Morrey space. Moreover , we get it's continuty from Morrey space to Lipschtz space and from Morrey space to BMO space.
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  6. Formal differential variables and an abstract chain rule.Samuel Alexander - forthcoming - Proceedings of the ACMS.
    One shortcoming of the chain rule is that it does not iterate: it gives the derivative of f(g(x)), but not (directly) the second or higher-order derivatives. We present iterated differentials and a version of the multivariable chain rule which iterates to any desired level of derivative. We first present this material informally, and later discuss how to make it rigorous (a discussion which touches on formal foundations of calculus). We also suggest a finite calculus chain rule (contrary to Graham, Knuth (...)
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  7. Mathematical Analysis and Analytical Science.C. A. Jimenez - forthcoming - Boston Studies in the Philosophy of Science.
  8. Hermann Cohen’s Principle of the Infinitesimal Method: A Defense.Scott Edgar - 2020 - Hopos: The Journal of the International Society for the History of Philosophy of Science 10 (2):440-470.
    In Bertrand Russell's 1903 Principles of Mathematics, he offers an apparently devastating criticism of the neo-Kantian Hermann Cohen's Principle of the Infinitesimal Method and its History (PIM). Russell's criticism is motivated by his concern that Cohen's account of the foundations of calculus saddles mathematics with the paradoxes of the infinitesimal and continuum, and thus threatens the very idea of mathematical truth. This paper defends Cohen against that objection of Russell's, and argues that properly understood, Cohen's views of limits and infinitesimals (...)
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  9. The Idea of Continuity as Mathematical-Philosophical Invariant.Eldar Amirov - 2019 - Metafizika 2 (8):p. 87-100.
    The concept of ‘ideas’ plays central role in philosophy. The genesis of the idea of continuity and its essential role in intellectual history have been analyzed in this research. The main question of this research is how the idea of continuity came to the human cognitive system. In this context, we analyzed the epistemological function of this idea. In intellectual history, the idea of continuity was first introduced by Leibniz. After him, this idea, as a paradigm, formed the base of (...)
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  10. Tanabe Hajime no Fukuso-kansu-ron (Tanabe Hajime on complex analysis).Tomomi Asakura - 2018 - RIMS Kokyuroku Bessatsu 71 (B):75-92.
    Tanabe Hajime (1885-1962) in his later years explored the so-called "dialectical" interpretation of complex analysis, an important part of his philosophy of mathematics that has previously been criticized as lacking mathematical accuracy and philosophical importance. I interpret his elaboration on complex analysis as an attempt to develop Leibniz's theory of individual notion and to supplement Hegel's view of higher analysis with the development in mathematics such as the theory of analytic continuation and Riemann surface. This interpretation shows the previously underrated (...)
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  11. Zur mathematischen Wissenschaftsphilosophie des Marburger Neukantianismus.Thomas Mormann - 2018 - In Christian Damböck (ed.), Philosophie und Wissenschaft bei Hermann Cohen, Veröffentlichungen des Instituts Wiener Kreis, Bd. 28. Wien: Springer. pp. 101 - 133.
  12. Controversies in the Foundations of Analysis: Comments on Schubring’s Conflicts.Piotr Błaszczyk, Vladimir Kanovei, Mikhail G. Katz & David Sherry - 2017 - Foundations of Science 22 (1):125-140.
    Foundations of Science recently published a rebuttal to a portion of our essay it published 2 years ago. The author, G. Schubring, argues that our 2013 text treated unfairly his 2005 book, Conflicts between generalization, rigor, and intuition. He further argues that our attempt to show that Cauchy is part of a long infinitesimalist tradition confuses text with context and thereby misunderstands the significance of Cauchy’s use of infinitesimals. Here we defend our original analysis of various misconceptions and misinterpretations concerning (...)
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  13. Emily Rolfe Grosholz. Starry Reckoning: Reference and Analysis in Mathematics and Cosmology.Sébastien Gandon - 2017 - Philosophia Mathematica 25 (3):419-422.
    © The Authors [2017]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected] Grosholz is interested in the growth of knowledge: what happens when reasoning not only orders what we already know, but adds to what we know? In her previous works, especially in her [2007], Grosholz insisted on the fact that working scientists and mathematicians, when they add to what we know, often combine different ‘modes of representation’, taking advantage of the ambiguity that arises when (...)
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  14. A Survey of Geometric Algebra and Geometric Calculus.Alan Macdonald - 2017 - Advances in Applied Clifford Algebras 27:853-891.
    The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics. No knowledge of physics is required. The section Further Study lists many papers available on the web.
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  15. Generality and structures in functional analysis: the influence of Stefan Banach.Frederic Jaëck - 2016 - In Karine Chemla, Renaud Chorlay & David Rabouin (eds.), The Oxford Handbook of Generality in Mathematics and the Sciences. Oxford: Oxford university press. pp. 223-254.
    This article examines Stefan Banach’s contributions to the field of functional analysis based on the concept of structure and the multiply-flavored expression of generality that arises in his work on linear operations. More specifically, it discusses the two stages in the process by which Banach elaborated a new framework for functional analysis where structures were bound to play an essential role. It considers whether Banach spaces, or complete normed vector spaces, were born in Banach’s first paper, the 1922 doctoral dissertation (...)
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  16. Differential Calculus Based on the Double Contradiction.Kazuhiko Kotani - 2016 - Open Journal of Philosophy 6 (4):420-427.
    The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. Next, the (...)
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  17. Laplacian growth without surface tension in filtration combustion: Analytical pole solution.Oleg Kupervasser - 2016 - Complexity 21 (5):31-42.
    Filtration combustion is described by Laplacian growth without surface tension. These equations have elegant analytical solutions that replace the complex integro-differential motion equations by simple differential equations of pole motion in a complex plane. The main problem with such a solution is the existence of finite time singularities. To prevent such singularities, nonzero surface tension is usually used. However, nonzero surface tension does not exist in filtration combustion, and this destroys the analytical solutions. However, a more elegant approach exists for (...)
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  18. Comments on a Paper on Alleged Misconceptions Regarding the History of Analysis: Who Has Misconceptions?Gert Schubring - 2016 - Foundations of Science 21 (3):527-532.
    This comment is analysing the last section of a paper by Piotr Blaszczyk, Mikhail G. Katz, and David Sherry on alleged misconceptions committed by historians of mathematics regarding the history of analysis, published in this journal in the first issue of 2013. Since this section abounds of wrong attributions and denouncing statements regarding my research and a key publication, the comment serves to rectify them and to recall some minimal methodological requirements for historical research.
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  19. Nonconservative Lagrangian Mechanics: Purely Causal Equations of Motion.David W. Dreisigmeyer & Peter M. Young - 2015 - Foundations of Physics 45 (6):661-672.
    This work builds on the Volterra series formalism presented in Dreisigmeyer and Young to model nonconservative systems. Here we treat Lagrangians and actions as ‘time dependent’ Volterra series. We present a new family of kernels to be used in these Volterra series that allow us to derive a single retarded equation of motion using a variational principle.
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  20. Gunkology and pointilism: Two mutually supervening models of the region–based and the point-based theory of the infinite twodimensional continuum.Miloš Adžić & Miloš Arsenijević - 2014 - In Giovanni Macchia, Francesco Orilia & Vincenzo Fano (eds.), Space and Time: A Priori and a Posteriori Studies. De Gruyter. pp. 137-170.
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  21. Ten Misconceptions from the History of Analysis and Their Debunking.Piotr Błaszczyk, Mikhail G. Katz & David Sherry - 2013 - 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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  22. Throwing Darts, Time, and the Infinite.Jeremy Gwiazda - 2013 - 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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  23. The classical continuum without points.Geoffrey Hellman & Stewart Shapiro - 2013 - 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 logic of plural quantification. In some respects this realizes ideas going back to Aristotle,although, unlike Aristotle, we make free use of classical "actual infinity". Also, in contrast to intuitionistic, Bishop, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our "gunky line" into mutually exclusive and exhaustive disjoint parts, thereby demonstrating the independence (...)
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  24. Ambiguities of Fundamental Concepts in Mathematical Analysis During the Mid-nineteenth Century.Kajsa Bråting - 2012 - Foundations of Science 17 (4):301-320.
    In this paper we consider the major development of mathematical analysis during the mid-nineteenth century. On the basis of Jahnke’s (Hist Math 20(3):265–284, 1993 ) distinction between considering mathematics as an empirical science based on time and space and considering mathematics as a purely conceptual science we discuss the Swedish nineteenth century mathematician E.G. Björling’s general view of real- and complexvalued functions. We argue that Björling had a tendency to sometimes consider mathematical objects in a naturalistic way. One example is (...)
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  25. Model theory of analytic functions: some historical comments.Deirdre Haskell - 2012 - 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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  26. Towards a Point-free Account of the Continuous.Geoffrey Hellman & Stewart Shapiro - 2012 - Iyyun 61:263.
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  27. Continuum, name and paradox.Vojtěch Kolman - 2010 - 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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  28. True or false? A case in the study of harmonic functions.Fausto di Biase - 2009 - 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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  29. Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains.Yaroslav Sergeyev - 2009 - Nonlinear Analysis Series A 71 (12):e1688-e1707.
    The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses recently introduced infinite and infinitesimal numbers being in accordance with the principle ‘The part is less than the whole’ observed in the physical world around us. These numbers have a strong practical advantage with respect to traditional approaches: they are representable at a new kind (...)
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  30. A most artistic package of a jumble of ideas.Fernando Ferreira - 2008 - Dialectica 62 (2):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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  31. John L. BELL. The continuous and the infinitesimal in mathematics and philosophy. Monza: Polimetrica, 2005. Pp. 349. ISBN 88-7699-015-. [REVIEW]Jean-Pierre Marquis - 2006 - Philosophia Mathematica 14 (3):394-400.
    Some concepts that are now part and parcel of mathematics used to be, at least until the beginning of the twentieth century, a central preoccupation of mathematicians and philosophers. The concept of continuity, or the continuous, is one of them. Nowadays, many philosophers of mathematics take it for granted that mathematicians of the last quarter of the nineteenth century found an adequate conceptual analysis of the continuous in terms of limits and that serious philosophical thinking is no longer required, except (...)
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  32. Divergent conceptions of the continuum in 19th and early 20th century mathematics and philosophy.John L. Bell - 2005 - Axiomathes 15 (1):63-84.
  33. To Continue With Continuity.Martin Cooke - 2005 - Metaphysica 6 (2):91-109.
    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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  34. Stochastic Algorithms: Foundations and Applications: Third International Symposium, Saga 2005, Moscow, Russia, October 20-22, 2005: Proceedings. [REVIEW]O. B. Lupanov (ed.) - 2005 - 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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  35. An ordinal analysis of parameter free Π1 2-comprehension.Michael Rathjen - 2005 - Archive for Mathematical Logic 44 (3):263-362.
    .This paper is the second in a series of three culminating in an ordinal analysis of Π12-comprehension. Its objective is to present an ordinal analysis for the subsystem of second order arithmetic with Δ12-comprehension, bar induction and Π12-comprehension for formulae without set parameters. Couched in terms of Kripke-Platek set theory, KP, the latter system corresponds to KPi augmented by the assertion that there exists a stable ordinal, where KPi is KP with an additional axiom stating that every set is contained (...)
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  36. Deleuze on Leibniz : Difference, Continuity, and the Calculus.Daniel W. Smith - 2005 - In Current Continental Theory and Modern Philosophy. Northwestern University Press.
  37. Empiricism in arithmetic and analysis.E. B. Davies - 2003 - Philosophia Mathematica 11 (1):53-66.
    We discuss the philosophical status of the statement that (9n – 1) is divisible by 8 for various sizes of the number n. We argue that even this simple problem reveals deep tensions between truth and verification. Using Gillies's empiricist classification of theories into levels, we propose that statements in arithmetic should be classified into three different levels depending on the sizes of the numbers involved. We conclude by discussing the relationship between the real number system and the physical continuum.
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  38. Analysis and Logic.Catherine Finet, Christian Michaux & C. W. Henson - 2003
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  39. Three concepts of decidability for general subsets of uncountable spaces.Matthew W. Parker - 2003 - 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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  40. Real numbers, quantities, and measurement.Bob Hale - 2002 - 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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  41. Black-Scholes PDE: A Finance Application.Quan-Hoang Vuong - 2001 - In International Conference on Differential Equations, Approximations and Applications, DEAA - 2001. pp. 53-53.
    This is a collection of the abstracts of lectures given at the International Conference on Differential Equations, Approximations and Applications, which will be held at the old campus of the Vietnam National University at Hanoi December 10-15, 2001.
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  42. Hermann Weyl on intuition and the continuum.John L. Bell - 2000 - 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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  43. Critical studies / book reviews.John P. Burgess - 2000 - Philosophia Mathematica 8 (1):84-91.
  44. Reals by Abstraction.Bob Hale - 2000 - 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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  45. Analysis and Synthesis in Mathematics: History and Philosophy by Michael Otte; Marco Panza. [REVIEW]Antoni Malet - 2000 - Isis 91:135-136.
  46. Abraham Robinson: The Creation of Nonstandard Analysis, a Personal and Mathematical Odyssey by Joseph Warren Dauben. [REVIEW]Albert Lewis - 1995 - Isis 86:673-674.
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  47. Abraham Robinson: The Creation of Nonstandard Analysis, a Personal and Mathematical Odyssey. Joseph Warren Dauben.Albert C. Lewis - 1995 - Isis 86 (4):673-674.
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  48. Introduction.J. L. Bell - 1994 - 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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  49. Real analysis without classes.Geoffrey Hellman - 1994 - 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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  50. The place of nonstandard analysis in mathematics and in mathematics teaching.Moshé Machover - 1993 - British Journal for the Philosophy of Science 44 (2):205-212.
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