Analysis

Edited by Nemi Boris Pelgrom (Ludwig Maximilians Universität, München)
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  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 - manuscript
    ¿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. Arguing about Infinity: The meaning (and use) of infinity and zero.Paul Mayer - manuscript
    This work deals with problems involving infinities and infinitesimals. It explores the ideas behind zero, its relationship to ontological nothingness, finititude (such as finite numbers and quantities), and the infinite. The idea of infinity and zero are closely related, despite what many perceive as an intuitive inverse relationship. The symbol 0 generally refers to nothingness, whereas the symbol infinity refers to ``so much'' that it cannot be quantified or captured. The notion of finititude rests somewhere between complete nothingness and something (...)
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  6. Principles and Philosophy of Linear Algebra: A Gentle Introduction.Paul Mayer - manuscript
    Linear Algebra is an extremely important field that extends everyday concepts about geometry and algebra into higher spaces. This text serves as a gentle motivating introduction to the principles (and philosophy) behind linear algebra. This is aimed at undergraduate students taking a linear algebra class - in particular engineering students who are expected to understand and use linear algebra to build and design things, however it may also prove helpful for philosophy majors and anyone else interested in the ideas behind (...)
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  7. 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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  8. Mathematical Analysis and Analytical Science.C. A. Jimenez - forthcoming - Boston Studies in the Philosophy of Science.
  9. A Puzzle about Sums.Andrew Y. Lee - forthcoming - Oxford Studies in Metaphysics.
    A famous mathematical theorem says that the sum of an infinite series of numbers can depend on the order in which those numbers occur. Suppose we interpret the numbers in such a series as representing instances of some physical quantity, such as the weights of a collection of items. The mathematics seems to lead to the result that the weight of a collection of items can depend on the order in which those items are weighed. But that is very hard (...)
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  10. The Banach-Tarski Paradox.Ulrich Meyer - forthcoming - Logique Et Analyse.
    Emile Borel regards the Banach-Tarski Paradox as a reductio ad absurdum of the Axiom of Choice. Peter Forrest instead blames the assumption that physical space has a similar structure as the real numbers. This paper argues that Banach and Tarski's result is not paradoxical and that it merely illustrates a surprising feature of the continuum: dividing a spatial region into disjoint pieces need not preserve volume.
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  11. Luck and Proportions of Infinite Sets.Roger Clarke - 2024 - Erkenntnis 89 (7):2947-2949.
  12. Formal differential variables and an abstract chain rule.Samuel Alexander - 2023 - Proceedings of the ACMS 23.
    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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  13. Functional interpretations.Justus Diller - 2020 - New Jersey: World Scientific.
    This book gives a detailed treatment of functional interpretations of arithmetic, analysis, and set theory. The subject goes back to Gödel's Dialectica interpretation of Heyting arithmetic which replaces nested quantification by higher type operations and thus reduces the consistency problem for arithmetic to the problem of computability of primitive recursive functionals of finite types. Regular functional interpretations, i.e. Dialectica and Diller-Nahm interpretation as well as Kreisel's modified realization, together with their Troelstra-style hybrids, are applied to constructive as well as classical (...)
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  14. 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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  15. The Idea of Continuity as Mathematical-Philosophical Invariant.Eldar Amirov - 2019 - Metafizika 2 (4):87-100.
  16. Do simple infinitesimal parts solve Zeno’s paradox of measure?Lu Chen - 2019 - Synthese 198 (5):4441-4456.
    In this paper, I develop an original view of the structure of space—called infinitesimal atomism—as a reply to Zeno’s paradox of measure. According to this view, space is composed of ultimate parts with infinitesimal size, where infinitesimals are understood within the framework of Robinson’s nonstandard analysis. Notably, this view satisfies a version of additivity: for every region that has a size, its size is the sum of the sizes of its disjoint parts. In particular, the size of a finite region (...)
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  17. Tanabe Hajime no Fukusokansū 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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  18. Transition to analysis with proof.Steven G. Krantz - 2018 - Boca Raton: CRC Press/Taylor & Francis Group.
    Transition to Real Analysis with Proof provides undergraduate students with an introduction to analysis including an introduction to proof. The text combines the topics covered in a transition course to lead into a first course on analysis. This combined approach allows instructors to teach a single course where two were offered. The text opens with an introduction to basic logic and set theory, setting students up to succeed in the study of analysis. Each section is followed by graduated exercises that (...)
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  19. (2 other versions)Zur mathematischen Wissenschaftsphilosophie des Marburger Neukantianismus.Thomas Mormann - 2018 - In Christian Damböck (ed.), Philosophie und Wissenschaft bei Hermann Cohen. Springer. pp. 101 - 133.
  20. 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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  21. 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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  22. 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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  23. An introduction to proof through real analysis.Daniel J. Madden - 2017 - Hoboken, NJ: Wiley. Edited by Jason A. Aubrey.
    An engaging and accessible introduction to mathematical proof incorporating ideas from real analysis A mathematical proof is an inferential argument for a mathematical statement. Since the time of the ancient Greek mathematicians, the proof has been a cornerstone of the science of mathematics. The goal of this book is to help students learn to follow and understand the function and structure of mathematical proof and to produce proofs of their own. An Introduction to Proof through Real Analysis is based on (...)
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  24. Generic properties of subgroups of free groups and finite presentations.Frédérique Bassino, Cyril Nicaud & Pascal Weil - 2016 - In Delaram Kahrobaei, Bren Cavallo & David Garber (eds.), Algebra and computer science. Providence, Rhode Island: American Mathematical Society.
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  25. 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. New York, NY, USA: Oxford University Press UK. 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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  26. 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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  27. 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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  28. Non-standard Analysis.Gert Heinz Müller - 2016 - Princeton University Press.
    Considered by many to be Abraham Robinson's magnum opus, this book offers an explanation of the development and applications of non-standard analysis by the mathematician who founded the subject. Non-standard analysis grew out of Robinson's attempt to resolve the contradictions posed by infinitesimals within calculus. He introduced this new subject in a seminar at Princeton in 1960, and it remains as controversial today as it was then. This paperback reprint of the 1974 revised edition is indispensable reading for anyone interested (...)
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  29. 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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  30. 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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  31. MOD functions: a new approach to function theory.Vasantha Kandasamy & B. W. - 2015 - Bruxelles, Belgium: EuropaNova. Edited by K. Ilanthenral & Florentin Smarandache.
    In this book the notion of MOD functions are defined on MOD planes. This new concept of MOD functions behaves in a very different way. Even very simple functions like y = nx has several zeros in MOD planes where as they are nice single line graphs with only (0, 0) as the only zero. Further polynomials in MOD planes do not in general follows the usual or classical laws of differentiation or integration.
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  32. 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 Vincenzo Fano, Francesco Orilia & Giovanni Macchia (eds.), Space and Time: A Priori and a Posteriori Studies. Boston: De Gruyter. pp. 137-170.
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  33. 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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  34. 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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  35. 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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  36. 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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  37. 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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  38. Towards a Point-free Account of the Continuous.Geoffrey Hellman & Stewart Shapiro - 2012 - Iyyun 61:263.
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  39. The Philosophy of Mathematics - With Special Reference to the Elements of Geometry and the Infinitesimal Method.Albert Taylor Bledsoe - 2010 - Read Books.
    Many of the earliest books, particularly those dating back to the 1900s and before, are now extremely scarce and increasingly expensive. We are republishing these classic works in affordable, high quality, modern editions, using the original text and artwork.
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  40. 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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  41. A constructive theory of continuous domains suitable for implementation.Andrej Bauer & Iztok Kavkler - 2009 - Annals of Pure and Applied Logic 159 (3):251-267.
    We formulate a predicative, constructive theory of continuous domains whose realizability interpretation gives a practical implementation of continuous ω-chain complete posets and continuous maps between them. We apply the theory to implementation of the interval domain and exact real numbers.
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  42. 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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  43. Categoricity in homogeneous complete metric spaces.Åsa Hirvonen & Tapani Hyttinen - 2009 - Archive for Mathematical Logic 48 (3-4):269-322.
    We introduce a new approach to the model theory of metric structures by defining the notion of a metric abstract elementary class (MAEC) closely resembling the notion of an abstract elementary class. Further we define the framework of a homogeneous MAEC were we additionally assume the existence of arbitrarily large models, joint embedding, amalgamation, homogeneity and a property which we call the perturbation property. We also assume that the Löwenheim-Skolem number, which in this setting refers to the density character of (...)
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  44. Ahilej i dvosmislenosti u pojmu beskonačnosti - Meršićev pristup [Achilles and the Ambiguities in the Concept of the Infinite - Meršić's Approach].Srećko Kovač - 2009 - Prilozi Za Istrazivanje Hrvatske Filozofske Baštine 35 (1-2):83-97.
    Mate Meršić (Merchich, 1850-1928) sees the origin of Zeno’s paradox ‘Achilles’ in the ambiguities of the concept of the infinity. According to him (and to the tradition started by Gregory St. Vincent), those ambiguities are resolved by the concept of convergent geometric series. In this connection, Meršić proposes a general ontological theory with the priority of the finite over the infinite, and, proceeding from Newton’s concept of fluxion, he develops a modal interpretation of differential calculus.
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  45. 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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  46. Hilbert’s Epsilon Calculus and its Successors.B. H. Slater - 2009 - In ¸ Itegabbay2009. Elsevier. pp. 385--448.
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  47. 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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  48. Constructing local optima on a compact interval.Douglas S. Bridges - 2007 - Archive for Mathematical Logic 46 (2):149-154.
    The existence of either a maximum or a minimum for a uniformly continuous mapping f of a compact interval into ${\mathbb{R}}$ is established constructively under the hypotheses that f′ is sequentially continuous and f has at most one critical point.
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  49. On the Cauchy completeness of the constructive Cauchy reals.Robert S. Lubarsky - 2007 - Mathematical Logic Quarterly 53 (4‐5):396-414.
    It is consistent with constructive set theory (without Countable Choice, clearly) that the Cauchy reals (equivalence classes of Cauchy sequences of rationals) are not Cauchy complete. Related results are also shown, such as that a Cauchy sequence of rationals may not have a modulus of convergence, and that a Cauchy sequence of Cauchy sequences may not converge to a Cauchy sequence, among others.
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  50. 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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1 — 50 / 113