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Philosophy of Mathematics

Edited by Øystein Linnebo (Birkbeck College)
Assistant editor: Sam Roberts (Birkbeck College, University of Oslo)
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  1. added 2016-05-22
    Kurt Grelling (1910). Die Axiome der Arithmetik mit besonderer Berücksichtigung der Beziehungen zur Mengenlehre. Dissertation, Georg-Augusts-Universität Göttingen
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  2. added 2016-05-19
    Valérie Lynn Therrien (2012). INVENTING LOGIC: THE LÖWENHEIM-SKOLEM THEOREM AND FIRST- AND SECOND-ORDER LOGIC. Pensées Canadiennes 10.
  3. added 2016-05-13
    Michael J. Shaffer (forthcoming). Lakatos’ Quasi-Empiricism in the Philosophy of Mathematics. Polish Journal of Philosophy.
    Imre Lakatos' views on the philosophy of mathematics are important and they have often been underappreciated. The most obvious lacuna in this respect is the lack of detailed discussion and analysis of his 1976a paper and its implications for the methodology of mathematics, particularly its implications with respect to argumentation and the matter of how truths are established in mathematics. The most important themes that run through his work on the philosophy of mathematics and which culminate in the 1976a paper (...)
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  4. added 2016-05-07
    Stephen Puryear (forthcoming). Finitism, Divisibility, and the Beginning of the Universe: Replies to Loke and Dumsday. Australasian Journal of Philosophy.
    Some philosophers contend that the past must be finite in duration, because otherwise reaching the present would have involved the sequential occurrence of an actual infinity of events, which they regard as impossible. I recently developed a new objection to this finitist argument, to which Andrew Ter Ern Loke and Travis Dumsday have replied. Here I respond to the three main points raised in their replies.
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  5. added 2016-05-05
    Sorin Bangu (2016). On The Unreasonable Effectiveness of Mathematics in the Natural Sciences. In Ippoliti, Sterpetti & Nickles (eds.), Models and Inferences in Science. Springer 11-29.
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  6. added 2016-05-04
    Lior Rabi (2016). Ortega y Gasset on Georg Cantor's Theory of Transfinite Numbers. Kairos. Journal of Philosophy and Science (15):46-70.
    Ortega y Gasset is known for his philosophy of life and his effort to propose an alternative to both realism and idealism. The goal of this article is to focus on an unfamiliar aspect of his thought. The focus will be given to Ortega’s interpretation of the advancements in modern mathematics in general and Cantor’s theory of transfinite numbers in particular. The main argument is that Ortega acknowledged the historical importance of the Cantor’s Set Theory, analyzed it and articulated a (...)
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  7. added 2016-04-27
    Luca Incurvati (forthcoming). Maximality Principles in Set Theory. Philosophia Mathematica.
    In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...)
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  8. added 2016-04-23
    S. Hewitt, A Note on Gabriel Uzquiano's 'Varieties of Indefinite Extensibility'.
  9. added 2016-04-21
    Chris Daly & David Liggins (forthcoming). Dorr on the Language of Ontology. Philosophical Studies:1-15.
    In the ‘ordinary business of life’, everyone makes claims about what there is. For instance, we say things like: ‘There are some beautiful chairs in my favourite furniture shop’. Within the context of philosophical debate, some philosophers also make claims about what there is. For instance, some ontologists claim that there are chairs; other ontologists claim that there are no chairs. What is the relation between ontologists’ philosophical claims about what there is and ordinary claims about what there is? According (...)
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  10. added 2016-04-21
    Araceli Ramirez-Cardenas, Maria Moskaleva & Andreas Nieder (2016). Neuronal Representation of Numerosity Zero in the Primate Parieto-Frontal Number Network. Current Biology 26.
    Neurons in the primate parieto-frontal network represent the number of visual items in a collection, but it is unknown whether this system encodes empty sets as conveying null quantity. We recorded from the ventral intraparietal area (VIP) and the prefrontal cortex (PFC) of monkeys performing a matching task including empty sets and countable numerosities as stimuli. VIP neurons encoded empty sets predominantly as a distinct category from numerosities. In contrast, PFC neurons represented empty sets more similarly to numerosity one than (...)
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  11. added 2016-04-16
    Erhan Demircioglu (2015). Recognitional Identification and the Knowledge Argument. Croatian Journal of Philosophy 15 (3):325-340.
    Frank Jackson’s famous Knowledge Argument moves from the premise that complete physical knowledge about experiences is not complete knowledge about experiences to the falsity of physicalism. Some physicalists (e.g., John Perry) have countered by arguing that what Jackson’s Mary, the perfect scientist who acquires all physical knowledge about experiencing red while being locked in a monochromatic room, lacks before experiencing red is merely a piece of recognitional knowledge of an identity, and that since lacking a piece of recognitional knowledge of (...)
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  12. added 2016-04-14
    Thomas Forster (2016). Mathematical Objects Arising From Equivalence Relations and Their Implementation in Quine's NF. Philosophia Mathematica 24 (1):50-59.
    Many mathematical objects arise from equivalence classes and invite implementation as those classes. Set-existence principles that would enable this are incompatible with ZFC's unrestricted aussonderung but there are set theories which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church's construction provides equivalence classes for “low” sets, and thus, for example, a set of all ordinals. However, that set has an ordinal in turn which is not a member of the set constructed; so no (...)
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  13. added 2016-04-13
    Jaykov Foukzon, Relevant First-Order Logic LP# and Curry’s Paradox Resolution. Pure and Applied Mathematics Journal Volume 4, Issue 1-1, January 2015 DOI: 10.11648/J.Pamj.S.2015040101.12.
    In 1942 Haskell B. Curry presented what is now called Curry's paradox which can be found in a logic independently of its stand on negation. In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Curry’s Paradox and Shaw-Kwei' sparadox without rejection any contraction postulate is proposed. In additional relevant paraconsistent logic C ̌_n^#,1≤n<ω, in fact,provide an effective way of circumventing triviality of da Costa’s paraconsistent Set Theories〖NF〗n^C.
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  14. added 2016-04-04
    Mohammad Husein Saleh, Doaa Mohammad Shokry & Saada A. Rahman Abu Shammala, A Numerical Approach for Solving Classes of Linear and Nonlinear Volterra Integral Equations by Chebyshev Polynomial.
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  15. added 2016-04-02
    Andrew Arana (2016). Imagination in Mathematics. In Amy Kind (ed.), The Routledge Handbook of Philosophy of Imagination. Routledge 463-477.
  16. added 2016-03-26
    Giuseppe Iurato & Giuseppe Ruta (2016). On the Role of Virtual Work in Levi-Civita’s Parallel Transport. Archive for History of Exact Sciences 70:1-13 (provisional).
    The current literature on history of science reports that Levi-Civita’s parallel transport was motivated by his attempt to provide the covariant derivative of the absolute differential calculus with a geometrical interpretation (For instance, see Scholz in ''The intersection of history and mathematics'', Birkhäuser, Basel, pp 203-230, 1994, Sect. 4). Levi-Civita’s memoir on the subject was explicitly aimed at simplifying the geometrical computation of the curvature of a Riemannian manifold. In the present paper, we wish to point out the possible role (...)
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  17. added 2016-03-24
    Burt C. Hopkins (forthcoming). Claire Ortiz Hill and Jairo José da Silva. The Road Not Taken: On Husserl's Philosophy of Logic and Mathematics. Texts in Philosophy; 21. London: College Publications, 2013. ISBN 978-1-84890-099-8 . Pp. Xiv + 436. [REVIEW] Philosophia Mathematica:nkw006.
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  18. added 2016-03-21
    Peter Smith, Category Theory: A Gentle Introduction.
    This Gentle Introduction is very much still work in progress. So far, at least in a rough and ready way, we cover the basic notions of elementary category theory -- explaining the very idea of a category, then treating limits, functors, natural transformations, representables, adjunctions. The long-term plan is to go on to have a chapter on monads, and then move on to say something about categorial logic, explore categories of sets, and edge towards some initial themes in topos theory. (...)
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  19. added 2016-03-21
    Ricardo Crespo & Fernando Tohmé (forthcoming). The Future of Mathematics in Economics: A Philosophically Grounded Proposal. Foundations of Science:1-17.
    The use of mathematics in economics has been widely discussed. The philosophical discussion on what mathematics is remains unsettled on why it can be applied to the study of the real world. We propose to get back to some philosophical conceptions that lead to a language-like role for the mathematical analysis of economic phenomena and present some problems of interest that can be better examined in this light. Category theory provides the appropriate tools for these analytical approach.
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  20. added 2016-03-18
    Burkay Ozturk (2012). On a Perceived Inadequacy of Principia Mathematica. Florida Philosophical Review 12 (1):83-92.
    This paper deploys a Cantor-style diagonal argument which indicates that there is more possible mathematical content than there are propositional functions in Russell and Whitehead's Principia Mathematica and similar formal systems. This technical result raises a historical question: "How did Russell, who was himself an expert in diagonal arguments, not see this coming?" It turns out that answering this question requires an appreciation of Russell's understanding of what logic is, and how he construed the relationship between logic and Principia Mathematica.
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  21. added 2016-03-15
    Andrea Sauchelli (2016). The Definition of Religion, Super-Empirical Realities and Mathematics. Neue Zeitschrift für Systematicsche Theologie Und Religionsphilosophie 58 (1):67-75.
    Providing a precise definition of “religion”—or an analysis in terms of sufficient and necessary conditions of the concept of religion—has proven to be a difficult task, more so in light of the diverse types of practices considered religious by scholars. Here, I discuss Kevin Schilbrack’s recent definition of “religion”, elaborate it and raise several objections, one of which is based on a specific theory in philosophy of mathematics: mathematical realism.
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  22. added 2016-03-14
    John Corcoran (1978-9). CORCORAN's THUMBNAIL REVIEWS OF OPPOSING PHILOSOPHY OF LOGIC BOOKS. MATHEMATICAL REVIEWS 56:98-9.
    PUTNAM has made highly regarded contributions to mathematics, to philosophy of logic and to philosophy of science, and in this book he brings his ideas in these three areas to bear on the traditional philosophic problem of materialism versus (objective) idealism. The book assumes that contemporary science (mathematical and physical) is largely correct as far as it goes, or at least that it is rational to believe in it. The main thesis of the book is that consistent acceptance of contemporary (...)
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  23. added 2016-03-13
    Nigel Vinckier & Jean Paul Van Bendegem (forthcoming). Feng Ye. Strict Finitism and the Logic of Mathematical Applications. Synthese Library; 355. Springer, 2011. ISBN: 978-94-007-1346-8 ; 978-94-007-1347-5 . Pp. Xii + 272. [REVIEW] Philosophia Mathematica:nkw005.
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  24. added 2016-03-09
    Cristian Soto (2014). Mark Colyvan's An Introduction to the Philosophy of Mathematics. [REVIEW] Critica 46 (138).
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  25. added 2016-03-03
    Stefan Buijsman (forthcoming). Accessibility of Reformulated Mathematical Content. Synthese:1-18.
    I challenge a claim that seems to be made when nominalists offer reformulations of the content of mathematical beliefs, namely that these reformulations are accessible to everyone. By doing so, I argue that these theories cannot account for the mathematical knowledge that ordinary people have. In the first part of the paper I look at reformulations that employ the concept of proof, such as those of Mary Leng and Ottavio Bueno. I argue that ordinary people don’t have many beliefs (...)
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  26. added 2016-03-02
    Dan Baras (2016). Our Reliability is in Principle Explainable. Episteme.
    Non-skeptical robust realists about normativity, mathematics, or any other domain of non-causal truths are committed to a correlation between their beliefs and non-causal, mind-independent facts. Hartry Field and others have argued that if realists cannot explain this striking correlation, that is a strong reason to reject their theory. Some consider this argument, known as the Benacerraf–Field argument, as the strongest challenge to robust realism about mathematics (Field 1989, 2001), normativity (Enoch 2011), and even logic (Schechter 2010). In this (...)
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  27. added 2016-02-28
    Raffaele Pisano (2016). The Algebra Between History and Education. [REVIEW] Metascience:1-5.
    ‘‘What Is Algebra?-Why This Book?’’ This is the amazing prelude to Taming the Unknown by Victor J. Katz, emeritus professor of mathematics at the University of the District of Columbia and Karen Hunger Parshall, professor of history of mathematics at the University of Virginia. This is an excellent book; its accurate historical and pedagogical purpose offers an accessible read for historians and mathematicians. [continue...].
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  28. added 2016-02-27
    Nora Berenstain (2016). The Applicability of Mathematics to Physical Modality. Synthese (online):1-17.
    This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The viewpresented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the modal structure of (...)
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  29. added 2016-02-25
    Monica Ugaglia (2016). Aristotle and the Mathematical Tradition on Diastēma and Logos: An Analysis of Physics 3 3, 202a18-21. Greek Roman and Byzantine Studies 56:49-67.
    ARISTOTLE'S PHYSICS 3.3 contains interesting evidence of an open debate in mathematics, concerning the interchangeability of the notions of diastēma and logos in the theory of harmonics. Because of the standard interpretation of the passage, however, this reference to harmonics has gone unnoticed: a slightly different understanding is proposed in this paper, which restores the relevance of the passage and its place in the contemporary debate.
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  30. added 2016-02-25
    Monica Ugaglia (2015). Knowing by Doing: The Role of Geometrical Practice in Aristotle’s Theory of Knowledge. Elenchos 36:45-88.
    Aristotle’s way of conceiving the relationship between mathematics and other branches of scientific knowledge is completely different from the way a contemporary scientist conceives it. This is one of the causes of the fact that we look at the mathematical passage we find in Aristotle’s works with the wrong expectation. We expect to find more or less stringent proofs, while for the most part Aristotle employs mere analogies. Indeed, this is the primary function of mathematics when employed in a philosophical (...)
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  31. added 2016-02-25
    Miro Brada (2000). Personality Model. Problem Paradise:42-43.
    In 1995, as a student of psychology inspired by natural science, I defined a logical model of personality explaining psychosis. I created (for my MA thesis, 1998 and grant research, 1999) new kind of tests assessing intelligence, creativity, prejudices, expectations to show more exact methods in psychology. During my Phd study in economics, I developed 'Maximization of Uniqueness (Originality)' model enhancing the classic utility to explain irrational motivations linking economics and psychology. Later I became computer programmer developing functional programming. According (...)
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